[Contemporary Mathematics] Fluids and Plasmas: Geometry and Dynamics Volume 28 ||

Fluids and Plasmas: Geometry and Dynamics

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Titles in this Series

Volume

CONTEMPORARY MATHEMATICS

1 Markov random fields and their applications, Ross Kindermann and J. Laurie Snell

2 Proceedings of the conference on integration, topology, and geometry in linear spaces, William H. Graves. Editor

3 The closed graph and P-closed graph properties in general topology, T. R. Hamlett and L. L. Herrington

4 Problems of elastic stability and vibrations, Vadim Komkov. Editor

5 Rational constructions of modules for simple Lie algebras, George B. Seligman

6 Umbral calculus and Hopf algebras, Robert Morris. Editor 7 Complex contour integral representation of cardinal spline

functions, Walter Schempp 8 Ordered fields and real algebraic geometry, D. W. Dubois and

T. Recio. Editors 9 Papers in algebra, analysis and statistics, R. Lidl. Editor

10 Operator algebras and K-theory, Ronald G. Douglas and Claude Schochet. Editors

11 Plane ellipticity and related problems, Robert P. Gilbert. Editor 12 Symposium on algebraic topology in honor of Jose Adem,

Samuel Gitler. Editor 1l Algebraists' homage: Papers in ring theory and related topics,

S. A. Amitsur. D. J. Saltman and G. B. Seligman. Editors 14 Lectures on Nielsen fixed point theory, Boju Jiang 15 Advanced analytic number theory. Part 1: Ramification

theoretic methods, Carlos J. Moreno 16 Complex representations of GL(2, K) for finite fields K,

llya Piatetski-Shapiro 17 Nonlinear partial differential equations, Joel A. Smoller. Editor 18 Fix~t' points and nonexpansive mappings, Robert C. Sine. Editor 19 Proceedings of the Northwestern homotopy theory conference,

Haynes R. Miller and Stewart B. Priddy. Editors 20 Low dimensional topology, Samuel J. Lomonaco. Jr .. Editor

http://dx.doi.org/10.1090/conm/028

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Titles in this series

Volume

21 Topological methods in nonlinear functional analysis, S. P. Singh. S. Thomeier. and B. Watson. Editors

22 Factorizations of b" ± 1, b = 2, 3, 5, 6, 7,10,11,12 up to high powers, John Brillhart. D. H. Lehmer. J L. Selfridge. Bryant Tuckerman. and S. S. Wagstaff. Jr.

23 Chapter 9 of Ramanujan's second notebook-Infinite series identities, transformations, and evaluations, Bruce C. Berndt and Padmini T. Joshi

24 Central extensions, Galois groups, and ideal class groups of number fields, A. Frohlich

25 Value distribution theory and its applications, Chung-Chun Yang. Editor

26 Conference in modern analysis and probability, Richard Beals. Anatole Beck. Alexandra Bellow and Arshag Hajian. Editors

27 Microlocal analysis, M. Salah Baouendi. Richard Beals and Linda Preiss Rothschild. Editors

28 Fluids and plasmas: geometry and dynamics, Jerrold E. Marsden. Editor



I Volume28

Fluids and Plasmas: Geometry and Dynamics

Jerrold E. Marsden1 Editor

AMERICAn MATHEMATICAL SOCIETY Providence • RhOde Island



EDITORIAL BOARD

R. 0. Wells, Jr., managing editor

Jeff Cheeger Adriano M. Garsia

Kenneth Kunen James I. Lepowsky Johannes C. C. Nitsche Irving Reiner

PROCEEDINGS OF THE AMS-IMS-SIAM JOINT SUMMER RESEARCH CONFERENCE IN THE MATHEMATICAL SCIENCES

ON FLUIDS AND PLASMAS: GEOMETRY AND DYNAMICS

HELD AT THE UNIVERSITY OF COLORADO, BOULDER JULY 17-23, 1983

These proceedings were prepared by the American Mathematical Society with partial support from the National Science Foundation Grant MCS 8218075.

1980 Mathematics Subject Classification. Primary 58Fxx, 76Exx.

library of Congress Cataloging in Publication Data Main entry under title: Fluids and plasmas: Geometry and dynamics.

(Contemporary mathematics, ISSN 0271·4132; v. 28) "AMS-SIAM-IMS Summer Research Conference, Boulder, Colorado, July 17-23, 1983"-lncludes bibliographies. 1. Fluid dynamics-Congresses. 2. Plasma dynamics-Congresses. 3. Differentiable

dynamical systems-Congresses. 4. Geometry, Differential-Congresses. 1. Marsden, Jerrold E. II. AMS-SIAM-IMS Summer Research Conference ( 1983: Boulder, Colo.) II I. American Mathe-matical Society. IV. Society for Industrial and Applied Mathematics. V. Institute of Mathemati-cal Statistics. VI. Series: Contemporary mathematics (American Mathematical Society); v. 28. OA911.F57 1984 532'.05 84-3011 ISBN 0-8218-5028-8

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this pub-lication in reviews provided the customary acknowledgement of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathe-matical Society. Requests for such permission should be addressed to the Executive Director, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940.

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Copyright © 1984 by the American Mathematical Society Reprinted 1988

Printed in the United States of America All rights reserved except those granted to the United States Government

This volume was printed directly from author prepared copy, The paper used in this journal is acid-free and falls within the guidelines

established to ensure permanence and durability"§



CONTENTS

Introduction . . . . . . vii Conference Participants xiv

Part I. Geometric-Analytic Methods

A.l~einstein, Stability of Poisson-Hamilton equilibria . D.Holm, J.Marsden, T.Ratiu and A.Weinstein, Stability of rigid

body motion using the energy-Casimir method D.Holm, Stability of planar multifluid plasma equilibria by

Arno 1 d's method . . . . . . . . . . . . . . . . . . . A.Kaufman* and R.Dewar, Canonical derivation of the Vlasov-Coulomb

noncanonical Poisson structure ........... . J.Marsden, T.Ratiu and A.Weinstein, Reduction and Hamiltonian

3

15

25

51

structures on duals of semidirect product Lie algebras 55 R.Montgomery, J.Marsden and T.Ratiu, Gauged Lie-Poisson structures . 101 J.Marsden,* P.Morrison and A.Weinstein, The Hamiltonian structure of

the BBGKY hierarchy equations . . . . . . . . . . . . . . 115 M.Grmela, Particle and bracket formulations of kinetic equations 125 J.Marsden and P.Morrison,* Noncanonical Hamiltonian field theory

and reduced MHO . . . . . . . . . . . . . . . . . . . . . 133 R.Littlejohn, Geometry and guiding center motion . . . . . . . . 151 A.Kaufman* and B.Boghosian, Lie-transform derivation of the gyro-

kinetic Hamiltonian system . . . . . . . . . . . . . . . . 169 M.Mayer, Poisson structures for relativistic systems . . . . . . . 177 G.Goldin, Diffeomorphism groups, semidirect products and quantum theory. 189

Part II. Analytic and Numerical Methods

N.Zabusky, Contour dynamics for two dimensional flows .. Y.Wan, On the nonlinear stability of circular vortex patches . T.Beale* and A.Majda, Vortex methods for fluid flow in two or

three dimensions .................. .

*An asterisk indicates the author who delivered a lecture, in case of multiple authors.

v

211 215

221



vi CONTENTS

P.Olver, Hamiltonian perturbation theory and water waves ... S.Wollman, Results on existence and uniqueness of solutions to the

Vlasov equation ................. . R.Glassey and W.Strauss, Remarks on collisionless plasmas H.Segur, Toward a new kinetic theory for resonant triads . P.Spalart, A spectral method for external viscous flows . R.Miller, Forecasting the ocean's weather: numerical models for

application to oceanographic data ..... .

Part III. Bifurcation and Dynamical Systems

231

251 269 281 315

337

H.Swinney, Geometry and dynamics in experiments on chaotic systems 349 J.Guckenheimer, Dimension estimates for attractors . . . . . . . . 357 D.Mclaughlin, J.Moloney and A.Newell,* Solitary waves as fixed

points of infinite-dimensional maps in an optical bistable ring cavity . . . . . . . . . . . . . . . . . . . . 369

J.Crawford, Hopf bifurcation and the beam-plasma instability . . . 377 P.Holmes, Some remarks on chaotic particle paths in time-periodic,

three-dimensional swirling flows 393 E.Siggia, A universal transition from quasi-periodicity to choas . 405 J.Curry* and E.Wayne, On the nonpathological behavior of Newton's

method . . . . . . . . . . . . . . . . . . . . . . . 407 J.Scheurle, Successive bifurcations in the interaction of steady

state and Hopf bifurcation J.Swift, Convection in a rotating fluid layer ........ .

419 435



INTRODUCTION

The intention of this conference was to foster interaction among

people working on mathematical, numerical and physical aspects of fluid

and plasma dynamics. To this end, the organizing committee consisting

of Jerry Marsden (Chairman), Philip Holmes and Andy r~.ajda, with Alex

Chorin and Alan Weinstein as advisors, chose 27 speakers from the three

sub-areas whom we felt would foster good interaction. We worried, though,

that the conference would degenerate into three subconferences with spe-

cialists talking to only themselves and leaving the audience bewildered.

As it turned out, the opposite happened. The conference developed a

sense of camaraderie and the speakers made every effort to bridge communi-

cation gaps, despite the inevitable differences in taste and background

needs that could not all be met. Another worry, prompted by our common

experiences at numerous conferences, was that speakers, in their enthu-

siasm, would run overtime. He offered a non-NSF sponsored prize of $100

for the best lecture .with the imposed necessary condition of not running

overtime. This worked marvelously-- it was worth every penny. The

conference participants voted in the last session to award Harry Swinney

the prize for his lecture "Observations of instabilities and chaos in

hydrodynamic and chemical systems". Runners-up were Alan Weinstein, Allan

Kaufman, Norman Zabusky, and Alan Newell, who all presented exceptional

lectures.

It is impossible for me to give a fair and adequate survey of the

highlights of the conference, but I shall try to convey the flavor of a

few points that I knew about or caught my attention.

vii Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.


viii INTRODUCTION

The organizing committee envisioned bringing together three groups

of people working on the following topics in fluid and plasma dynamics:

1. Geometric aspects; Hamiltonian structures, perturbation theory

and nonlinear stability by variational methods,

2. Analytical and numerical methods; contour dynamics, spectral

methods, and functional analytic techniques,

3. Dynamical systems aspects; experimental and numerical methods,

bifurcation theory, and chaos.

Of course, we could have easily spent our entire budget on any one of

these areas. But our purpose was to emphasize interaction rather than

comprehensiveness.

Let me comment a little on some of the background for these three

items, why they are all exciting developing areas, and how they inter-

re 1 ate.

The geometric methods center on outgrowths of Arnold's article "Sur

la geometriedifferentielledes groupes de Lie de dimension infinie et ses

applications a l'hydrodynamic des fluids parfaits", Ann. Inst. Fourier,

Grenoble,_!_§. (1966) 319-361. Arnold discovered the relationship between

the Lagrangian and Eulerian description of an incompressible fluid in

group theoretic terms. In the Lagrangian description, the phase space

is the tangent or cotangent bundle of the group of volume preserving

diffeomorphisms with its usual canonical symplectic structure. Each such

diffeomorphism represents a possible fluid configuration relative to a

fixed reference configuration. In the Eulerian description, the phase

space is its Lie algebra (or its dual), the space of divergence free

vector fields (or the space of vorticities). The passage from the canonical

Lagrangian description to the noncanonical Eulerian description is an

example of what we now call reduction, a general procedure for elimina-

tion symmetries in a system (see the books on classical mechanics by

Arnold, "~'lathematical Methods of Classical Nechanics", Springer (1978),



INTRODUCTION

and Abraham and Marsden, "Foundations of Mechanics", Addison Wesley,

(lg78), for accounts). Arnold worked with the Lie algebra but it is

now generally preferred to use its dual, which carries a natural bracket

structure on real valued functions on the dual; this is the lie-Poisson

bracket discovered by Lie in 1890. For incompressible fluids, this dual

is identified with the space of vorticities and in two dimensions (for

example) the Lie-Poisson bracket on functions of scalar vorticity w is

given by the vorticity bracket

{F,G} (w) = J0 w{~~, ~~}xy dx dy

where 0 c IR 2, is the domainfor the fluid, 6F/6w is the functional

derivative and { , } is the standard Poisson bracket in the plane with xy x and y as conjugate variables. The vorticity equations of motion

become F {F,H} , where H is the kinetic energy, expressed in

terms of the vorticity.

Arnold used Hamiltonian methods (constrained second variations and

ix

convexity estimates) to study the stability of two dimensional incompres-

si5le flows, obtaining a nonlinear version of the classical Rayleigh

inflection point criterion for linearized stability. This was a brilliant

achievement that received only a fraction of the attention it deserved.

Arnold's work is found in several references around 1966 that are cited

in Appendix 2 of his mechanics book. In one especially important work,

Arnold supplies rigorous convexity estimates; in English translation, it is

"On an a priori estimate in the theory of hydrodynamic stability",

Trans. Am. Math. Soc. !J.. (1969) 267-269.

Formal stability results based on second variation methods occurred

in the plasma literature, independently of Arnold's ideas. Results of

Newcomb, Rosenbluth, Kruskal, Bernstein, Gardner and others, were pub-

lished between 1958 and 1965. An account of this development to 1969

can be found in the book of Clemow and Dougherty "Electrodynamics of



X INTRODUCTION

Particles and Plasmas", Chapter 10, Addison-Wesley (1969). Similar methods

are in common use in other areas of applied mathematics. For example,

the proof of the nonlinear stability of the single KdV soliton due to

Benjamin and Bona (see Proc. Roy. Soc. Lon. 328A [1972] 153-183 and

344A [1975] 363-374) has many features in common with Newcomb and Arnold's

constrained variational methods.

Poisson brackets for MHD and the Maxwell-Vlasov equations governing

plasma motion were found in 1980 by Morrison and Greene. Allan Kaufman

played an important role in bridging the mathematics-physics gap by

explaining this work to Alan Weinstein and me. We subsequently showed

(Physica 4D (1982) 394-406) how to obtain the Maxweli-Vlasov bracket by

Arnold's methods (again a reduction from a Lagrangian to an Eulerian

description) and by utilizing this method, corrected one of the terms in

Morrison's bracket. The method of Clebsch variables was developed shortly

afterwards by Morrison, Holm and Kupershmi dt. These various approaches

quickly became united and were applied to a variety of systems as the

understanding of Hamiltonian structures deepened and the stability results

were extended. In fact, they are currently being applied to rather

exciting problems such as tokamaks, three dimensional multifluid plasmas,

internal waves in the ocean, and to externally stabilized plasmas.

The above setting provides a backdrop for the subjects treated in

the lectures or contributions of Darryl Holm, Robert Littlejohn, Richard

Montgomery, Phil Morrison, Meinhard Mayer, Allan Kaufman, Alan l~einstein,

and Tudor Ratiu. Peter Olver talked about a way one might bridge the

gap between these bracket structures and those for water waves, by

asymptotic expansion methods. Gerald Goldin explained how one might

use these classical structures in quantum field theoretic situations via

represenations of semi-direct products involving the diffeomorphism

group. Chuck Leith described how enstrophy and its generalizations are

used in geostrophic turbulence. Generalized enstrophy is, not coin-

cidentally, a key ingredient in Arnold's stability method. It is a Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.


INTRODUCTION

Casimir in the sense that, using the vorticity bracket, it commutes with

every function of vorticity. The papers of Miroslav Grmela, Harvey

xi

Segur, and the joint paper of myself, Phil Morrison and Alan Weinstein deal

with various aspects of kinetic theory, emphasizing Hamiltonian structures.

The analytic and numerical areas stressed were Hose that had some

relationship with the basic mathematical structures for fluids and plasmas.

Techniques available for specific numerical implementation naturally

came up, but were not emphasized. The interatction between theory and

practice is nicely illustrated by the work of Glimm and Chorin. The

random choice methods they use for both compressible and incompressible

flow are based on a deep understanding of the basic theory and are very

successful numerically. (See the books of Chorin and ~1arsden, "A Mathe-

matical Introduction to Fluid Mechanics," Springer (1979) and Smaller

"Shock Waves and Reaction-Diffusion Equations", Springer (1983) for

further details and references). Not only do existence and uniqueness

theorems tie into these methods, but so do the geometric aspects of

the equations. For example, asking that a code be consistent with the

Hamiltonian structure as far as possible could be a useful way to improve

or debug it, or even to design new numerical algorithms.

Existence and Uniqueness theorems for the Poisson-Vlasov equations

were discussed by Steve Wollman, Robert Glassey and Walter Strauss.

Tom Beale and Andy Majda discussed the obstruction to continuability

of three dimensional solutions to the Euler equations in terms of sharp

bounds on the vorticity. This is related to numerical studies of the

problem that were reported by Dan Meiron. Numerical aspects of Chorin's

vorticity algorithm were presented by James Sethian. Zabusky described

his program for numerical implementation of contour dynamics. His

methods have had a very useful influence on the theory, as was demon-

strated in Steve Wan's lecture in which he used inspiration from both

Zabusky's work and that of Arnold to show the dynamical stability of

circular vortex patches. Phillip Spallart and Steve Orszag presented Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.


xii INTRODUCTION

state of the art methods for numerical fluid problems using spectral

methods, while Jerry Brackbill and John Dawson concentrated on particle

methods in plasma problems and Bob Miller dealt with numerical methods

i n oceanography.

Most of the remaining talks fell into the third category of dynamical

systems aspects. Thirteen years ago, when Ruelle and Takens first intro-

duced ideas of chaos into fluid mechanics, many people thought it was a

crazy idea. However, even by then, dynamical systems methods were already

making large strides. Lorenz in his famous 1963 paper (J. Atmos Sci. 20

(1963), 130-141) had already very convincingly shown the presence of chaos

in a deterministic system. In the late 1960's and early 1970's, Judovich,

Sattinger, Joseph, Iooss, and Marsden had shown how the Hopf bifurcation

and ideas of infinite dimensional dynamical systems can be rigorously

applied to yield an understanding of fluid oscillations. Presently "chaos

is in"; skeptics have been largely converted to the useful ness of the

ideas.

In John David Crawford's talk, dynamical systems ideas and the Hopf

bifurcation especially, were applied to the beam-plasma instability. In

Harry Swinney's lecture we saw the Hopf bifurcation used as a basic building

block toward understanding the more complex chaotic dynamical behavior he

was onserving in his experiments. His data on Couette flow indicates the

presence of a strange attractor of small fractal dimension, this dimension

ranging continuously from about 4 to 7 in the experiments reported. Alan

Newell gave a marvelous illustration of the mixture of ideas from solitons

and chaos in laser optics. Philip Holmes' contribution deals with chaotic

particle paths near a solitary wave that occurs in helical flow in a cylindri-

cal tube. In John Guckenheimer's lecture we saw a program emerging for

how one might distinguish noise from deterministic chaos. In this regard,

one should keep in mind that in many systems (such as the Henan attractor)

it is not a priori clear if the chaos is due to a genuine strange attrac-

tor or to a slightly noisy but complicated tangle of horseshoes and sinks. Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.


INTRODUCTION xiii

It is thus important to develop tests and basic theory which try to make

these distinctions. As the lectures anG contributions of James Curry,

Ed Ott, Jurgen Scheurle, Eric Siggia and Jim Swift demonstrated, while great

strides in the basic theory of bifurcations and chaos have been made, the

full story is by no means complete. For example, how strange attractors come

and go and are related to the more analytically tractible transverse homo-

clinic bifurcations is still a subject of research interest.

I wish to thank all the conference participants for their energetic

and thoughtful lectures, contributions, questions and interaction. Special

thanks go to the Ar~s for administering the conference, especially Carole

Kohanski who did most of the detailed work and saw that things ran smoothly.

Ruth Edmonds was a great help with organizing the conference and this volume.

Connie Calica did a beautiful job typing many of the papers. Finally, the

NSF is gratefully acknowledged for their wisely spent financial support.

Jerrold E. Marsden Berkeley, January 1984



CONFERENCE PARTICIPANTS

BEALE, Tom Department of t1athematics Duke University Durham, North Carolina 27706

BRACKBILL, Jeremiah Group x-1, MSE531 Los Alamos National Lab Los Alamos, New Mexico 87545

BRENIER, Yann liMAS- Universidad Nacional Autonoma de Mexico Mexico D.F., Mexico

CHANG, Ina Sacramento Peak Observatory Oxford University Sunspot, New Mexico 88349

COURANT, Ted Department of Mathematics University of California Berkeley, California 94720

CRAWFORD, John David Lawrence Berkeley Laboratory University of California Berkeley, California 94720

CURRY, James Department of Mathematics University of Colorado Boulder, Colorado 80309

DAWSON, John M. Physics Department UCLA Los Angeles, CA 90024

EHRLICH, ·Paul E. Department of Mathematics University of Missouri Columbia, Missouri 65201

GLASSEY, Robert Department of Mathematics Indiana University Bloomington, Indiana 47401

xiv

GOLDIN, .Gerald A. Department of Math Science Northern Illinois University DeKalb, Illinois 60115

GREENGARD, Claude Department of Mathematics University of California Berkeley, California 94720

GRMELA, Miroslav Ecole Polytechnique of Montreal Montreal, Canada

GUCKENHEIMER, John M. Department of Mathematics University of California Santa Cruz, California 95064

HOLM, Darryl Center for Nonlinear Studies MS B 258 Los Alamos National Lab Los Alamos, New Mexico 87545

KAUFMAN, Allan N. Physics Department University of California Berkeley, California 94720

KOSTELICH, Eric Department of Mathematics University of Maryland College Park, Maryland 20742

LANGFORD, William F. Department of Math & Statistics University of Guelph Ontario, Canada NlG 2Wl

LATHAM, Peter Center for Studies of

Nonlinear Dyanmics La Jolla, CA 92037

LEITH, Cecil E., Jr. NCAR PO Box 3000 Boulder, Colorado 80307



CONFERENCE PARTICIPANTS

LEWIS, H. Ralph Los Alamos National Lab MS-F642 Los Alamos, New Mexico 87545

~1J'.IlHEir,IER, Ha 11 ace NRL Code 4790 Washington, D.C. 20374

MARSDEN, Jerrold E. Department of Mathematics University of California Berkeley, California 94720

MAYER, Meinhard E. Department of Physics University of California Irvine, California 92717

MEIRON, Daniel Department of 1'1athematics University of Arizona Tucson, Arizona 85721

MILLER, Robert Department of Mathematics Tulane University New Orleans, Louisiana 70118

MONTGOMERY, Wallace Richard Department of Mathematics University of California Berkeley, California 94720

MORRISON, Philip Institute for Fusion Studies University of Texas Austin, Texas 78756

NAGATA, Wayne Department of Mathematics Colorado State University Ft. Collins, Colorado 80523

NEWELL, Alan C. Department of Mathematics University of Arizona, Bldg 89 Tucson, Arizona 85721

OLVER, Peter School of Mathematics University of Minnesota Minneapolis, Minnesota 55455

OMOHUNDRO, Stephen M. University of California Berkeley, California 94720

OTT, Edward Department of Physics University of Maryland College Park, Maryland 20742

PALAIS, Robert Department of Mathematics University of California Berkeley, California 94720

PALMORE, Julian Department of Mathematics University of Illinois Urbana, Illinois

PRICE-JONES, Neil Department of Math & Statistics University of Guelph Ontario, Canada NlG 2Wl

RATIU, Tudor Department of Mathematics University of California Berkeley, California 94720

RIEDEL, Kurt Department of Mathematics New York University New York, NY 10012

ROBINSON, Clark Department of Mathematics Northwestern University Evanston, Illinois 60201

ROBERTS, Stephen Department of Mathematics University of California Berkeley, California 94720

SANDERS, Jan Free University Wiskundig Seminarium, VU Postbus 7161, 1007 MC Amsterdam

SCHEURLE, JUrgen Division of Applied Mathematics Brown University Providence, Rhode Island 02906

SCHMID, Rudolf Yale University and MSRI 2223 Fulton Street Berkeley, California 94720

SEGUR, Harvey ARAP PO Box 2228 Princeton, New Jersey 08540

XV



xvi CONFERENCE PARTICIPANTS

SETHIAN, James A. Lawrence Berkeley Laboratory University of California Berkeley, California 94720

SIGGIA, Eric Department of Physics Cornell University Ithaca, New York 14853

SPALART, Philippe Roland NASA Ames Laboratory Mountain View, California 94043

STONE, Alexander P. Department of Mathematics New Mexico University Albuquerque, New Mexico 87131

SWIFT, James Department of Physics University of California Berkeley, California 94720

SWINNEY, Harry Department of Physics University of Texas Austin, Texas 78704

THOMPSON, Russell Department of Mathematics Utah State University Logan, Utah 84322

THOMAS, James Department of Mathematics Colorado State University Ft. Collins, Colorado 80523

TURKINGTON, Bruce E. Department of Mathematics Northwestern University Evanston, Illinois 60201

WAN, Yieh-Hei Department of Mathematics State University of New York Buffalo, New York

WAYNE, Clarence {Gene) Institute for Math & Its Application 504 VH, 207 Church Street SE University of Minnesota Minneapolis, MN 55455

WEINSTEIN, Alan Department of Mathematics University of California Berkeley, California 94720

WOHL, Randy Department of Mathematics University of California Berkeley, California 94720

WOLLMAN, Stephen Department of Mathematics City University of New York New York, New York 10010

ZABUSKY, Norman, Dept. of Math, University of Pittsburgh, PA 15260



Part I. Geometric-Analytic Methods



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Contemporary Mathematics Volume 28, 1984

STABILITY OF POISSON-HAMILTON EQUILIBRIA!

Alan Weinstein2

ABSTRACT. The only effective way to establish stability of an equilibrium point of a hamiltonian system in many degrees of freedom is to find a conserved quantity with a local maximum or minimum point at the equilibrium. On a Poisson manifold, the hamiltonian function may not even be stationary at the equilibrium, so it is necessary to add other functions in the search for the desired conserved quantity. Examples are given in finite dimensions to show how stability may depend upon the Poisson structure as well as the hamiltonian. The paper concludes with a discussion of stability results for fluids and plasmas.

The aim of this talk is to present some of the geometric background behind the results on stability of fluids and plasmas which are described more fully, and more analytically, in the lectures of this conference by Holm, Ratiu, and Wan. The basic ideas here are due to Arnold, but recent work has provided refinements of his pictures which have suggested new applications.

1. THE CANONICAL CASE Pit equilibrium point x0 for a flow (or any group action) on a topological

space is called (Liapunov) stabLe if, for each neighborhood u of x0 there is a smaller neighborhood V such that every orbit which intersects V is contained in U. This kind of stability applies both forward and backward in time, as is appropriate for the conservative systems we will be studying.

We recall the classical stability criterion of Lagrange. In IR2n with coordinates (q 1 , ••• ,qn, p1 , ••• ,pn) and hamiltonian H: IR2n ~ IR, Hamilton's equations

(1) aH

Pi = - aqi

1980 Mathematics Subject Classificatlons: Primar,y 58F05, 58F10, Secondary 70K20, 67E30.

1This paper is dedicated to the memory of the Governor of New York State, 1848-1852; U.S. Secretary of State, 1869-1877.

2Research partially supported by National Science Foundation grant MCS 80-23356 and Department of Energy contract AT03-82ER12097.

3

@ 1984 American Mathematical Society 0271-4132/84 $1.00 + $.25 per page

http://dx.doi.org/10.1090/conm/028/751971



4 ALAN WEINSTEIN

have an equilibrium point at x0 E IR2n if and only if the differential H'(x0 )

is zero. Lagrange's sufficient condition for stability of such an equilibrium point is that the second derivative matrix, or hessian H"(x ) be positive

0 definite or negative definite. In fact, Lagrange's condition implies that each level surface of H which intersects a neighborhood of x0 has a compact component entirely contained near x0 • Since H is a constant of motion for (1), stability follows immediately.

In function spaces, it is necessary that H"(x0 ) be definite in a strong sense determined by the topology of the space, otherwise H might not have a local maximum or minimum at x0 • A simple example is the function defined on the space of square integrable sequences (y1 ,y2 , •.• ) by

~ (y~ y~) H(y1 .Yz , •.. ) = t'1 2k- -4-

0ther examples of this phenomenon more closely related to physically realistic problems are given by Ball and Marsden [2].

Lagrange's sufficient condition implies that the linearized Hamilton equations at x0 have 0 as a stable equilibrium point- this property is referred to as linea.P stability. On the other hand, linear stability can occur in the absence of Lagrange's condition, such as for the hamiltonians ~(q: + p:) - ~ (q! + P!), k > 0. Such stability is sometimes called gyroscopic.

In fact, linear stability for hamiltonian systems is always neutral rather than asymptotic (i.e. eigenvalues lie on the imaginary axis rather than to the left of it) and does not necessarily imply stability. The example

H = ~(q2 + p2) _ (q2 + p2) + q p q + ~(q2 _ p2)p I I 2 2 112 1 I 2

was given by C. L. Siegel [15], p.88. If a = q~ + p~ and S = q~ + p~, then Ci = 4cxS + S2, so (i;;;>-Q and o.>O unless S=O; instability follows immediately.

The main results on the stability of linearly stable equilibria where the hessian of the hamiltonian is indefinite are due to Arnold. He showed that, generically, there is a set of large measure of invariant tori near the equilib-rium point. If n =2, this is enough to insure stability, while if n;;;>-3, orbits are expected to "diffuse" around the tori and hence escape from the equilibruim. (See [1], Appendix 8, and references therein.) In infinite dimensions, even the existence of the invariant tori is unproven.

The examples and discussion above show that the only effective way to prove the stability of an equilibrium point of a hamiltonian system is by showing that the hamiltonian (or some other conserved quantity) has a local minimum or maximum at that point.



STABILITY OF POISSON-HAMILTON EQUILIBRIA

2. POISSON STRUCTURES A Poisson structure on a differentiable manifold P is an antisymmetric

bilinear operation { , } on the space c"'(P) of c"" functions from P to IR

5

such that the operation ~H: C00 (P) + C00 (P) defined by ~H(F) = {F,H} satisfies the following two "derivation" identities for each HE C00 (P):

(Leibniz)

(Jacobi)

(~HF)G + F~HG

{~HF,G} + {F,~HG}

The Leibniz identity means that the operation I;H is that of differentia-tion by a vector field, which we also denote by ~H. The Jacobi identity means that the operation { , } is a Lie algebra structure on C00 (P) and that this structure is preserved by the flow of ~H; in other words, for all t,

where exp ti;H is the time t map (possibly defined only locally) for the flow generated by ~W This flow is called a hamiltonian flow, and H is called its generating function or hamiltonian. ?ne has immediately d~ ( F o exp ti;H) = {F,H} oexp t~H or, more concisely, F = {F,H}.

Another way to write the Jacobi identity is

which means that the map H ~ ~H is a homomorphism from the Lie algebra C00(P) to the Lie algebra of vector fields with the negative commutator bracket opera-tion-[,].

This is essentially the form in which Jacobi originally discovered his identity; when combined with the antisymmetry of { , } it yields immediately the theorem

for which Poisson had given an extremely complicated proof about 20 years earlier. The work of Poisson and Jacobi concerned the particular Poisson structure on IR2n defined by

(2) { F ,G} f: ( aF aG aF aG ) . 1 aq. ap. - ap. aq.-1= 1 1 1 1

Any Poisson structure which is locally isomorphic to this structure is called symplectic;* coordinates in which a symplectic structure takes the form (2) are

*The term symplectic is a word of Greek origin adopted by Hermann Weyl as a substitute for the Latin complex. The earliest use in English of the word seems to be in reference to the symplectic bone of a fish (Oxford Engl. Diet.).



6 ALAN WEINSTEIN

called canonical. In canonical coordinates, the general rule F = {F,H} gives in particular

and

so the discussion in §1 is directly applicable to hamiltonian systems on symplectic manifolds.

If (x 1 , ••• ,xm) are local coordinates on any Poisson manifold, the Leibniz identity and antisymmetry imply the following general formula:

m {F,G} = L {x.,x.}.l£_ 1.§... .. 1 1 J ax. ax.

1,J = 1 J

The Poisson structure is thus determined by the functions nij = {xi,xj}. The nij are the entries of an antisymmetric-matrix-valued function or, more invariantly, the components of an antisymmetric contravariant tensor 1r of rank two, called the Poisson tensor. (See the lectures of Morrison and Olver in these proceedings for an approach to Poisson structures where the tensor n rather than the operation { , } is taken as the fundamental object.) The rank of this matrix, or tensor, is a function on the Poisson manifold with values in the even numbers 0,2,4,6, ....

3. STABILITY AT REGULAR POINTS A point near which the rank of a Poisson structure is constant is called

reguLar; the other points are called singular. It was shown by Lie [101 that near a regular point one can always find coordinates (q 1 , ••• ,qn,P1 , ••• ,pn, c1 , ... ,c~) for which the matrix 1rij has the form

( -~ 0

0

In other words, Poisson brackets are still given by the formula (2), derivatives with respect to the ci's being ignored. Given a hamiltonian H(q,p,c), the equations of motion for the hamiltonian flow are

• aH P· =-- ' 1 aqi

What we have, then, is a canonical hamiltonian system depending on R. parameters. If R. =0, the Poisson structure is symplectic; this result is sometimes called Darboux's theorem.

As observed by Arnold [1], Lagrange's test for stability extends easily to



STABILITY OF POISSON-HAMILTON EQUILIBRIA 7

Poisson manifolds near regular points. The condition for a point x0 to be an

equilibrium is that aapH (x ) and aaH (x 0 ) be zero for all i. ... aH (x ) need i 0 qi oCi 0

not vanish; however, we can manufacture a function with a critical point at x0

R. aH by taking H- ~ ac· (x0 ) • ci; the hamiltonian system generated by this

1=1 1 function is exactly the same as the one generated by H. If the (q ,p) part of H "( x0 ) is definite (say, positive definite), then

" ~ <lH ~ 2 H = H- .LJ ac. (x 0 )ci + .LJ [ci -ci(x0 )] 1=1 1 1=1

is a conserved quantity having a strict local minimum at x0 , so the stability of the equilibrium at x0 follows.

The manifolds defined by setting the ci's equal to constants are determined by the Poisson structure, independent of coordinates, and are called symplectic

leaves, since they are the leaves of a foliation on the set of regular points. The functions which are constant on symplectic leaves (i.e. functions of (c1 , ••• ,cR.)} may also be invariantly defined as those C for which {F ,C} =0 for all FE C00(P). They are called Casimir functions.

Consider, for instance, the Poisson structure on lR3 given by

The Euler equations for a free rigid body pre a hamiltonian system with respect to this structure and the hamiltonian ~ .r M~/Ii . The Poisson matrix

1=1

has rank 0 at the origin and rank 2 elsewhere, so lR\{0} consists entirely of regular points and the origin is singular. The function M~ + M: + M: is a Casimir function, so it may be taken as c1 • The symplectic leaves are the spheres centered at the origin. The canonical coordinates q1 and p1 can only be defined locally; for instance, on the complement of the set where M1 ~ 0 and M2 =0 we may take q1 = tan- 1 (M/M 1 ), ranging from -1r to 1r, and p1 = M3 •

We may include M1 < 0 by allowing q1 to be a "cyclic" variable defined modulo 21r. The stability of equilibria for hamiltonians H(Ml'M 2 ,M 3 ) may now be determined very easily; this is done by Arnold [1] for the free rigid body. (See also the article by Holm, Marsden, Ratiu, and Weinstein in these proceedings. )



8 ALAN WEINSTEIN

4. SINGULAR POINTS A point x0 of rank 0 in a Poisson manifold is an equilibrium· point for

every hamiltonian system. The stability of x0 may then be dependent on both the hamiltonian and the Poisson structure itself. For example, the origin in ffi 3 with the structure of §3 is always a stable equilibrium point because the Casimir function M~ +M~ +M~ has a strict local minimum there: each motion beginning near 0 is confined to a small sphere which is entirely contained near 0.

An "unstable" structure in JR 3 is given by the rules {M ,M } = -M , ' 1 2 3

{M 2 ,M 3 } = M1 , {M 3 ,M 1 } = M2 • The origin is once again the only singular point, the basic Casimir function is now C = M~ + M~- M!, and the symplectic leaves are the components of the cones and hyperboloids which make up the level surfaces of C. The hamiltonian system determined by H = M1 is M1 = 0, M2 = M 3 ,

M3 =M 2 ; the solution M1 =0, M2 =et, M3 =et shows the instability of the origin. On the other hand, if the hamiltonian is H=M 3 , we can combine it with the Casimir to form the constant of motion C +2H 2 = M~ +M~ +M~, so the origin is stable for this particular hamiltonian.

Another example, more physically motivated, is

which yields the motion of a planar pendulum if H = ~ M!- M2 • Here the Casimir function C = M: + M~ represents the distance from the center of mass to the center of suspension, M3 is the angular momentum, and 8 = tan- 1 (M 2 /M 1 )

is the angle from the vertical. All the points on the M3 axis are singular, and the right circular cylinders having this axis are the symplectic leaves. Each singular point (O,O,a) is stable for this hamiltonian, since (H -~a 2 ) 2 + C has a strict local minimum there. (\~hen a=O, the minimum is weaker than quadratic.) With the same Poisson structure but the hamiltonian H = -M 2 ,

the equations of motion become M1 = 0, M2 =0, M3 = M1 , and the equilibrium points are now all unstable.

The general Poisson manifold P near a singular point x0 is a direct product of the rank zero case discussed above and the symplectic case of §1. It is shown in [17] that near any point x0 one can introduce coordinate functions (q 1 , ••• ,qn ,p1 , ••• ,pn ,y1 , ••• ,yk) for which the structure matrix '~~"ij

has the form

0

0




The "transverse" Poisson structure {yi ,yj} = aij(y) is uniquely determined up to isomorphism by the structure on P, but the particular submanifold q =p =0 depends on the choice of coordinates. On the other hand, the submanifold y = 0 is independent of the choice of coordinates. In [17] it is called the symplea-

tia leaf through x0 • In this paper we will use the term "leaf" only when x0

is a regular point; when x0 is a singular point we will use the term sympleatia

bone, in deference to the previously cited etymology. We thus have the following picture of an arbitrary Poisson manifold.

The regular points form an open dense subset foliated by symplectic leaves. These form the meat of the manifold, and the remainder of the Poisson (the skeleton?) is a disjoint union of symplectic bones.

The stability of an equilibrium which is a general singular point can depend on a combination of the transverse Poisson structure and the behavior of the hamiltonian along a symplectic leaf. Consider, for example, the five-dimensional Poisson manifolds with coordinates (q 1 ,p1 ,M1 ,M2 ,M 3 ) and brackets {q 1 ,p1 } = 1, {MpM2 } = E:M 3 , {M2 ,M 3 } = M1 , {M 3 ,M 1 } = M2 • We will consider the cases E: = 1, E: = -1, and E: =0; in each of them the manifold M1 = M2 = M3 = 0 is a two-dimensional symplectic bone, and the symplectic leaves have dimension 4. If the hamiltonian is q~ + p~ - M2 , which has positive definite second derivative along the symplectic bone, the origin is an unstable equilibrium point when E: = -1 or 0 and is stable when E: = 1.

A Poisson structure like the case E: =0 occurs for the three dimensional rigid body in a gravitational field, except that there is an extra dimension corresponding to one more Casimir function (see [17]). A stability analysis of the Lagrange top based on the techniques of this paper is carried out in the paper by Holm, Marsden, Ratiu, and Weinstein in this volume.

5. FLUIDS AND PLASMAS The preceding examples have indicated that care must be taken in analyzing

the stability of equilibria near singular points of a Poisson manifold, even in the finite dimensional case. As yet, there is no general theory for this kind of analysis. For the infinite-dimensional problems arising in fluid and plasma dynamics, we are even further from having a comprehensive theory, but the basic ideas can be combined with analytical tools to yield interesting results.

We will consider the Poisson manifold V' of distributions on IR2 , dual to the space V of smooth functions. This example is applicable to one-dimensional plasmas and two-dimensional incompressible fluids. The space V has the struc-ture of a Lie a 1 gebra under the canoni ca 1 Poisson bracket {F ,G} = ~: ~; - ~~ ~~ . The Poisson bracket of functi ona 1 s F and G on V' wi 11 be denoted by { {F ,G}}. To define it, we first note that if F is a functional of fEV', then the



10 ALAN WEINSTEIN

derivative dF at f can be identified with an element ~~ of v by the rule

<dF, g > def lim .!_ [ F( f + s g) - F( f)] = J §£ • g dqdp s-+ 0 s IR2 Of

Now we define

{{F,G}}(f) = J f { ~~ , ~~} dqdp IR2

(See [11] for a discussion of the Lie-algebraic background of this definition.) It turns out that the symplectic leaf or bone through a given f 0 E V'

consists of all those f E V' which have the form f 0 o 1/J, where 1/J: IR2 -+ IR2

is a smooth canonical (i.e. area and orientation preserving) transformation. 2

For instance, if f 0 {q,p) = e-P, then the leaf or bone through f will contain only f's which attain the maximum value of 1 along a curve stretching to infinity at both ends. If f 0 (q,p) = e-(q 2 +p 2

), then its leaf or bone should consist of these f's having a unique nondegenerate maximum with value 1 andforwhich Area{(q,p)jf{q,p)~b} = Area{(q,p)jf 0 (q,p)~b} forall b between 0 and 1. {See [6 J . )

How do we distinguish between regular and singular points? If we are dealing with a finite dimensional Lie algebra instead of V, the following test would apply:

Test (see [17]). The point f 0 is regular if and only if either of the following conditions holds, where Vf consists of those smooth functions 6 for which 0

()fo ()6 ()fo ()6 0 = {fo,6} = aq a-p- Clp aq

i) Vf is an abelian subalgebra of V. 0

i i) V f depends "smoothly on f in a neighborhood of f 0 •

In the finite dimensional case, the analogues of {i) and (ii) are equivalent when Vf has an ad-invariant complement (P.Molino has pointed out the need for

0 this complement, omitted in [17]), but in the infinite dimensional case, {ii) is stronger and should probably be taken as the definition of regularity. It must be properly interpreted, though, especially as regards the topology to be used in defining the "neighborhood of f 0 ." If f 0 is smooth, it is reasonable to restrict attention only to smooth f near f 0 , in which case condition {ii) for regularity says that f 0 should be "structurally stable" in the sense that smooth f near f 0 have the same local topological type. In this sense, e{-q 2 +p 2

), or any other function with only nondegenerate critical points, is structurally

2 stable, but e-p is not because nearby functions will have isolated maxima and minima instead of a whole line of maxima. {In the theory of one-dimensional plasmas, these nearby functions could be "BGK" modes [5] near a homogeneous equilibrium.)




Although a general theory is not yet available, it has been found that the method of §3 can be applied in many cases, independent of whether the Poisson structure is regular or singular at an equilibrium point of interest.

The following idea can actually be traced to the plasma physics literature of the 1950's, as well as to Arnold's work on fluids beginning in 1965. We saw in §3 that if x0 is a regular equilibrium point for the hamiltonian H, then there can always be found a Casimir function C with c'(x0 ) = H'(x0 ), so that H-C is a conserved quantity having a critical point at x0 • Even if x0 is not regular, though, there is nothing to stop us from looking for a Casimir function C such that C'(x 0 ) = H'(x 0 ). It turns out that, in many interesting cases, such a Casimir can be found, and it may even be unique within a prescribed class of Casimirs.

For the Poisson manifold V', a large family of Casimirs is given by the functionals

$(f) = J ~(f(x,y)} dxdy IRz

where q,: IR + IR is a smooth function. (Strictly speaking, these functionals are defined only for continuous f, not arbitrary distributions.)

For f to be an equilibrium for the hamiltonian X, the bracket

{f0 , ~~(f 0 )} must vanish, i.e. ~~(f 0 ) and f0 should be locally functionally

dependent. Suppose that ~~ (f0 ) and f0 are globally functionally dependent,

i.e. that [~~(f 0 )J(x,y) = Q(f0 (x,y)} for some function Q. If~ is to

satisfy (X-~)'(f 0 ) = 0, then we must have

<li 1 ( f O ( X ,y) } = [ ~~ ( f 0 ) J (X ,y) = Q {f 0 (X ,y ) }

so q,' = Q on the range of f , which determines ~ up to an inessential 0

constant on this range. Once ~ has been found, the next step in the analysis is to test the

convexity of X-~ near f. One may begin by testing for definiteness of (x-$)". 0

As we already saw in §1, this definiteness may not be sufficient for stability. For true stability, some kind of convexity argument must be added. This was done for incompressible fluids by Arnold [1]. New applications to compressible fluids, magnetohydrodynamics, and plasmas are presented in [7] and [8]. A related stability theorem for vortex patches is proven in [16]; the methods here are inspired by the Poisson structure, but a deeper kind of analysis is necessary due to the discontinuity of the vorticity function.

To end this discussion, we shall look a little more closely at the case of plasmas. We use (q,p) instead of (x,y) as variables on IR2 • The hamiltonian



12 ALAN WEINSTEIN

in suitable units is

X(f) JJ~ P2 f(q,p) dpdq + JJJJ [V(q 1 ,q 2 )(f(q ,p )-e)(f(q ,p )-e) do dp ] dq dq 1 1 2 2 '1 2 1 2

where e is a neutralizing background charge density and V(q 1 ,q 2 ) is the interaction potential. An important class of equilibria consists of the spatially homogeneous densities in phase space f (q,p) = y(p) with

0

Jy(p)dp =e. For these equilibria ~~ (f0 ) = ~p 2 , so the function Q above is determined by ~p 2 = Q{y(p)). * Now ~'(8) = Q(8), and the convexity condition for JC- ~ turns out to be ~"(8) < 0, or Q'(8) <0. Differentiating the equation ~P 2 = Q(y(p)) with respect top gives p = Q'(y(p)) ·y'(p), or Q'{y(p)) = p/y'(p); for this to be negative, y'(p) and p must have opposite signs. Thus our sufficient condition for stability will be satisfied if the ... graph of y has a maximum at p=O and no other critical points.'

The argument above was introduced by Newcomb in the Appendix to [4] for the case y(p) = e-P 2

, in which case $ turns out to be the entropy. It was observed by Rosenbluth [131 that the same proof applies whenever y is a "one-humped" distribution. A paper published by Rowlands [14] claimed to extend the reasoning to arbitrary y, but this result cannot possibly be correct. First of all, the function If> would have to be multiple-valued, which invalidates several steps in the reasoning. Second, it is well known [9) [121 that an equilibrium distribution with two local maxima is already linearly unstable if the humps are far enough apart.

There remains an interesting question about the two-humped equilibria. It is known that these can be linearly stable if the humps are close enough together, even though the Lagrange-type test definitely fails. This indicates that the linear stability is of gyroscopic type, with a hamiltonian which is indefinite along symplectic bones, so that nonlinear stability may fail because of Arnold diffusion. In fact, recent numerical studies [3] suggest that nonlinear instability actually occurs. We also refer to Crawford's article in this volume for a further discussion of these instabilities.

*Notice that we must assume Y(p) to be an even function of p for the following argument to work.

tRemember that our condition for stability is sufficient but not necessary. If y has a unique maxi mum at some p r 0, we can add to $ a conserved quantity of linear momentum type to prove stability. This procedure, which corresponds to changing to a moving reference frame, works because both the equation and the equilibrium solution are translation invariant.




BIBLIOGRAPHY

1. V.I.Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York (1978}.

2. J.M.Ball and J.E.Marsden, "Quasiconvexity, second variations and nonlinear stability in elasticity," Arch. Rat. Mech. (to appear).

3. R.H.Berman, D.J.Tetreault, T.H.Dupree and T.ooutros-Ghali, "Computer simulation of nonlinear ion-electron instability," Phys. Rev. Letters 48 (1982), 1249-1252.

4. I.B.Bemstein, "\~aves in a plasma in a magnetic field," Phys. Rev. 109 {1958)' 10-21.

5. I.B.Bernstein, J.M.Greene and M.D.Kruskal, "Exact nonlinear plasma oscillations," Phys. Rev. 108 (1957), 546-550.

6. V.W.Guillemin, "Band asymptotics in two dimensions," Adv. in Math. 42 {1981), 248-282.

7. D.D.Holm, J.E.Marsden, T.Ratiu, and A.Weinstein, "Nonlinear stability conditions and a priori estimates for barotropic hydrodynamics," Phys. Letters A (to appear).

8. D.D.Holm, J.E.Marsden, T.Ratiu and A.Weinstein, "A priori estimates for nonlinear stability of fluids and plasmas," in preparation.

9. J.D.Jackson, "Longitudinal plasma oscillations," J. Nuclear Energy C 1 {1960), 171-189.

10. S.Lie, Theorie der Transformationsgruppen, Zweiter Abschnitt, unter mitwirkung von Prof. Dr. Friedrich Engel, B. G. Tuebner, Leipzig {1890).

11. J .Marsden and A .Weinstein, "Coadjoi nt orbits, vortices, and Clebsch variables for incompressible fluids," Physica 7D (1983), 305-323.

12. O.Penrose, "Electrostatic instabilities of a uniform non-maxwellian p 1 asma," Phys. Fluids 3 ( 1960) , 258-265.

13. M.Rosenbluth, "Topics in microinstabilities," in Advanced Plasma Theory, M.Rosenbluth, ed., Academic Press, New York {1964), 137-158.

14. G.Rowlands, "Extension of the Newcomb entropy method of stability analysis," Phys. Fluids 9 ( 1966) , 2 528-2529.

15. C.L.Siegel, Vorlesungen Uber Himmelsmechanik, Springer-Verlag, Berlin ( 1956).

16. Y.H.Wan, J.E.Marsden, T.Ratiu and A.Weinstein, "Nonlinear stability of circular vortex patches," Preprint (1983).

17. A.Weinstein, "The local structure of Poisson manifolds," J. Diff. Geom. (to appear) .

DEPARMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720







STABILITY OF RIGID BODY MOTION USING THE ENERGY-CASIMIR METHOD

Darryl Holm, Jerrold Marsden, Tudor Ratiu, and Alan Weinstein

ABSTRACT. The Energy-Casimir method, due to Newcomb, Arnold and others is illustrated by application to the motion of a free rigid body and the heavy top.

§1. INTRODUCTION

In the preceding paper of Weinstein, a general framework for cal-culating stability criteria is reviewed. In this note we illustrate the method in the concrete cases of a rigid body and heavy top. The classical stability results are obtained. Tt:e purpose of this note is to illustrate

the basic ideas of the method with simple "hands-on" examples that should aid in the understanding of fluid and plasma examples in Holm's lecture that follows .

Let us recall ~he basic procedures used in the "Energy-Casimir method".

Step A. Equations of Motion and Conserved Quantities Write the equations as evolution equations

dx dt = X(x)

where x E P, the phase space and X is a vector field on P. Find a conserved energy H: P-+ IR; i.e.

d d t H (X ( t)) = 0

for any solution x(t) of (EM), and a family of conserved quantities

(EM)

(H)

F: P-+ IR. (These conserved quantities are typically Casimirs or are gen-erated by symmetry groups -- See Weinstein's lecture for the definitions of these and the definition of Liapunov stability).

1980 Ma thema tics Subject Class ifi cation 58F05, 58Fl 0

15

© 1984 American Mathematical Society 0271-4132/84 $1.00 + $.25 per page




16 D. HOLM, J.MARSDEN, T. RATIU AND A. WEINSTEIN

Step B. First Variation Let x be an equilibrium e point; i.e. X(x ) = 0, whose (Liapunov) e

stability we wish to ascertain. Find all F in step A with the property that HF = H + F has a critical point at xe

Step C. Second Variation Compute the second derivative d2HF(xe) and see if it is definite,

either positive or negative for some F satisfying step B. If P is finite dimensional then x is Liapunov stable -- this follows from con-

* e servation of HF. (If P is infinite dimensional, as for fluids and plasmas, then this second variation test is not sufficient for nonlinear staoility; this deficiency can be remedied by convexity estimates.)

In the next two sections we shall go through these three steps for our two examples.

§2. RIGID FREE BODY

A. Equations of Motion and Conserved Quantities The free rigid body equations are

~ = d~_/dt = m x w

(VP)

( 2 .1 )

where m, wE IR3 , w is the angular velocity and m the angular momentum both viewed in the body; the relation between m and w is given by

- -mi = Iiwi, i = 1 ,2,3, where I = (1 1 ,I 2,I 3) is the diagonalized moment of inertia tensor, I1 , I 2, I 3 > 0. This system is Hamiltonian in the Lie-Poisson structure of IR 3 considered as the dual of the Lie algebra of the rotation group S0(3). Explicitly, for F,G:IR 3 ->- IR,

{F ,G}(m) = -m·(VF(m) X vG(m)) (2.2)

and with respect to this bracket, (2.1) is easily verified to be Hamiltonian in the sense that (2.1) is equivalent to F = {F,H} where the H is equal to the kinetic energy:

1 3 2 H(m) = -2 m·w = I m./l. i = 1 l l

(2.3)

For any smooth function ¢:IR ->- IR, the function

2 c ¢ (~) = ¢ (I ~I I 2) ( 2.4)

~See, e.g., Siegel and Moser [1971], p. 208.



STABILITY OF RIGID BODY MOTION USING THE ENERGY~CASIMIR METHOO 17

is a Casimir function for (2.2), i.e. its bracket with any other function G is identically zero, as an easy computation shows. Thus, for any ¢, C<P is a conserved function.

B. First Variation We shall find a Casimir function C¢ such that HC := H + C¢ has a

critical point at a given equilibrium point of (2.1). Such points occur when ~ is parallel to ~· We shall assume without loss of generality, that ~ and ~ point in the Ox-direction. Then, after normalizing if necessary, we may even assume that the equilibrium solution is m = (1,0,0). -e The derivative of

is DHC (~)-IS~=(~+~¢'( lml 2 t2))·o~

<P .

This equals zero at m = (1,0,0), provided that

¢'(1/2) = -1/Il

C. Second Variation Using (2.5) and (2.6), the second derivative at the equilibrium

~ (1,0,0) is

D2Hc (m ) •(om) 2 = ow·om + ¢' lm 12!2) lom1 2 +(m ·am) 2if>"( 1m 1212) 4>-e - -- e - --e- e

This quadratic form is positive definite if and only if

¢"(1/2) > 0 and

(2.5)

(2.6)

( 2. 7)

( 2.8)

(2.9)



18 D. HOLM, J. MARSDEN, T. RATIU AND A. WEINSTEIN

2 Consequently, ¢(x) = (-2/1 1 )x + (x - ~) makes the second derivative of He at (1 ,0,0) positive definite, so stationary rotation around¢ the longest axis lS (Liapunov) stable.

The quadratic form ( 2. 7) is i ndefi ni te if

(2.10)

or the other way around. Consequently, we cannot show by this method that rotation around the middle axis is stable. {In fact, it is unstable.)

Finally, the quadratic form is negative definite, provided

¢"(~) < 0 (2.11) and

(2.12)

It is obvious that we may find a function ¢ satisfying the requirements 2 (2.6) and (2.11 ); e.g. ¢(x) = (-2/1 1 )x- (x- ~) . This proves that

rotation around the short axis is (Liapunov) stable. ~/e summarize the results in the following well-known theorem.

Rigid Body Stability Theorem. In the motion of a free rigid body,

ro-tation around the long and short axes is (Liapunov) stable.

Remarks. 1) It is important to keep the Casimirs as general as possible, because otherwise (2.8) and (2.11) could be contradictory. Had we simply chos2n ¢{x) = -(2/1 1 )x, (2.8) would be verified, but (2.11) not. It is only the choice of two different Casimirs that enables us to prove the two stability results, even though the level surfaces of these Casimirs are the same.

2) In this case,rotations about the intermediate axis are unstable. This is true even for the linearized equations as an eigenvalue analysis shows.

3) The same stability theorem can also be proved by working with the second derivative along a coadjoint orbit in IR\ i.e. a two-sphere; see Arnold I1966 ]. This coadjoint orbit method also suggests instability of rota-tion around the intermediate axis, but it has the deficiency of being inappli-cable where the rank of the Poisson structure jumps. (See Weinstein's lecture in this val ume.)



STABILITY OF RIGID BODY MOTION USING THE ENERGY-CASIMIR METHOD 19

§3. LAGRANGE TOP

A. Equations of Motion and Conserved Quanti ties The heavy top equations are

dm/dt = m x ~ + Mgt_r x X

dr/dt = y x w

3 where !E.• r_, ~· _x E IR . Here m and ~ are the angular momentum and angular velocity in the body, m. = I.w1., I. > 0, i = 1, 2, 3, with

1 1 1 ] = (I 1,I 2,I 3) the moment of inertia tensor. The vector I represents the motion of the unit vector along the Oz-axis as seen from the body, and the constant vector ?5_ is the unit vector along the 1 i ne segment of 1 ength t connecting the fixed point to the center of mass of the body;

(3.la)

(3.lb)

M is the total mass of the body, and g is the strength of the gravitational acceleration, wtiich is along Oz pointing down.

This system is Hamiltonian in the Lie-Poisson structure of IR 3 x IR3

regarded as the dual of the Lie algebra of the Euclidean group E(3) = S0(3) D< IR 3 (IX denotes semidirect produc~). The Poisson bracket is given by (see Holmes and Marsden [1983] and references therein):

{F,G}(~.~) = -m·(.:?mF x .YmG)

-Y·(v F x 11 G + 11 F x _'IZ..,G). - ~ -x -J.. --. .. (3.2)

The Hamiltonian of this system is the total energy

H(!}!,y) = ~~·~ + Mgtr•f. (3.3)

This can be easily verified directly. For further information, see Ratiu's lecture in this volume. The functions !!!"I and 1Yi 2 are Casimir functions

- 3 3 for (3.2), i.e. their brackets with any function G:IR x IR + IR vanish. Hence the same is true for

C(_!!!,'J..) =<!>(_!!!·I· lyi 2l (3.4)

2 where <!> is any function from IR to IR. We shall be concerned here only with the Lagrange top. This is a heavy

top for which r1 =I 2, i.e. it is symmetric, and the center of mass lies on the axis of symmetry in the body, i.e. X = (0,0,1). This assumption simplifies the equations of motion (3.la) to



20 D. HOLM, J. MARSDEN~ T. RATIU AND A. WEINSTEIN

m 1 (I 2 - I 3)m2m3/I2 I 3 - MgR-y 2

m2 = (I 3 - I1)m1m3;I1I 3 + MgR-Y1

m3 = (I 1 - I 2)m1m2;r1r2 .

Since I1 = 12, we have ~ 3 = 0; thus m3 and hence any function ~(m 3 ) of m3 is conserved.

B. First Variation We shall study the equilibrium solution ~ = (O,o,m3), Ye = (0,0,1),

which represents the spinning ofa symmetric top in its upright position. To begin, we look for conserved quantities of the form H$,~ = H+$(.!!!_-r_, lyi 2) + ~(m 3 ) which have a critical point at the equilibrium.

The first derivative of H$,~ is given by

(3.5)

where $ = a$;a(.!!!_·.!'_), $' = a$;a(i:ri 2). At the equilibirum solution .!!!.e• Y.e• the first derivative of H$,~ vanishes, provided that

-w3 + ~(m 3 ,1) + ~·(m 3 ) = o; -w3 = m3n 3

MgR- + ¢(m3,l)m3 + 2$'(m3,ll = o;

(The remaining equations, involving indices 1 and 2 are trivially verified.) Solving for ¢(m3,1) and $'(m3,1) we get the conditions:

C. Second Variation

~(m 3 ,1) =- (} + ~(m 3 l)m 3 3

$'(m3,1) = ~(f + ~·(m 3 l)m~- ~MgR-3

( 3.6)

We shall check for definiteness of the second variative of H$,~ at the equilibrium point ~ = (O,O,m3), ..Ye = (0,0,1). To simplify notation we shall set

a = ~"(iii ) 3




b = 4~"(iii3,1)

c = ~(iii 3 , 1 )

d = 2~'(iii3,1).

Wtth this notation, (3.5), and {3.6), we find that the matrix of the

second derivative at .!!!e• 1:e is

1/11 0 0 ~{iii3' l) 0 0

0 1/12 0 0 ~{iii3, 1 ) 0

0 0 (1/1 3)+a+c 0 0 ~(iii 3 , 1 ) +2m3c+d

«:m3, 1) 0 0 2~'{iii3,1) 0 0

0 ~(iii3, 1 ) 0 0 24>' (iii2 '1 ) 0

0 0 i(m3, 1) +2m3c+d 0 0 2~' {iii3 , 1 ) +b+iii~c+2iii 3 d

( 3. 7)

lf this form is definite, it must be positive definite since the (1,1)-entry i's postttve. The six principal determinants have the following values, (recall that 11 = r2):

1 Ill

1 /I 2 1

(1/1 3 +a + c)/Ii

f(f- +a+ c)(/ 4>'(m3,l)- t(m3,1) 2) 1 3 1

( 1 ~ 4>'(m3,1)- i(iii 3 ,1) 2 ) 2 ( 1 ~ +a+ c)

Consequently, the quadratic form given by (3.7) is positive definite, if and only if



22 D. HOLM, J. MARSDEN, T. RATIU AND A. WEINSTEIN

_!_+a+c>O (3.8). I3

2 . 2 1 ' (m3, 1 ) - <t>(m3, 1 ) > a

1 (3.9)

(3.10)

Conditions (3.8) and (3.10) can always be satisfied if we choose the numbers a, b, c, and d appropriately; e.g. a = c = d = 0 and b sufficiently large and positive. Thus, the determining condition for stability is (3.9). By ( 3. 6 ), this becomes

* KI~ + cp'(m3))m;- Mg~J- (I~+ <P'(m 3 ))2m~ > o.

We can choose cp' (m3) so that f + cp' (m3) = e has any value we wish. 3

The left side of (3.ll) is a quadratic polynomial in e, whose leading coefficient is negative. In order for this to be positive for some e, it is necessary and sufficient for the discriminant

to be po s i t i ve ; that is ,

(3.ll)

which is the well-known stability condition for a fast top. We have proved the following. Heavy Top Stability Theorem. An upright spinning Lagrange top is stable

provided that the angular velocity is strictly larger than 14Mg~/I 1 •

Remarks. 1) The method suggests but does not prove that one has instability when m~ < 4Mg~Il. In fact, an eigenvalue analysis shows that the equilibrium is linearly unstable and hence nonlinearly unstable in this case.

2) When I 2 = r1 + £ for small £, the conserved quantity cp{m3) is no longer available. In this case, a sufficiently fast top is still linearly stable, but nonlinear stability can only be established by KAM theory.

Other regions of phase space are known to possess chaotic dynamics in this case (Holmes and Marsden [1983]).




REFERENCES 1. V. Arnold, Sur la geometrie differentielle des groupes de Lie

de dimension infinite et ses applications a l'hydrodynamique des fluids parfaits, Ann. Inst. Fourier, Grenoble.!.§_, (1966), 319-361.

2. P. Holmes and J. Marsden, Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups, Ind. Univ. Math. J., E· (1983), 273-310.

3. C.L. Siegel and J.K. Moser [1971]. Lectures on Celestial Mechanics, Springer-Verlag.

Center for Nonlinear Studies, MS B-258 Los Alamos National Laboratory Los Alamos, N.M. 87545

Department of Mathematics University of California Berkeley, CA 94720

Department of Mathematics University of Arizona Tucson, AZ 85721

Department of Mathematics University of California Berkeley, CA 94720







STABILITY OF PLANAR MULTIFLUID PLASMA EQUILIBRIA BY ARNOLD'S METHOD

1 Darryl D. Holm

ABSTRACT. A method developed by Arnold to prove nonlinear stability of certain steady states for ideal incompressible flow in two dimensions is extended to the case of barotropic, compressible, multifluid plasmas.· This extension is accomplished by constructing conserved functionals derived from degeneracy of Poisson brackets. The results are applied to planar shear flows of the plasma.

I. INTRODUCTION. Arnold [ 1965a, 1969] formulates a method for establishing sufficient conditions for stability of stationary (i.e., steady) motions of an

ideal fluid against disturbances of small but finite amplitude. Stability is established by finding a priori estimates (expressed in a certain norm

depending on the problem being considered) that place bounds on the subsequent size of the disturbances, as they develop in time. These estimates· apply for

as long as the solutions of the disturbed flow continue to exist. When such estimates have been established, the stationary motions are said to be "stable

by Arnold's method." Arnold's method is based on the construction of a conserved functional· (a

constant of the motion) that has a given stationary flow as its extremum (critical point). If this extremum is a true minimum or maximum relative to nearby flows within a neighborhood whose topology must be determined for each problem, then the corresponding stationary ·now is stable in that topology. Such stability can be understood geometrically by a heuristic argument. Imagine the level surfaces of the conserved functional in function space, in a neighborhood of the point representing a given stationary flow. For a maximum or minimum, these level surfaces will be nested and closed, surrounding the equilibrium point. If the steady state flow is disturbed at some instant, the corresponding phase point in the function space will shift onto a nearby level

1980 Mathematics Subject Classifications: 58F05, 58F10

1 Work performed under the auspices of the U.S. Department of Energy under contract W-7405-ENG-36 and the Office of Basic Energy Sciences, Department of Applied Mathematics.

25





26 Darryl D. Holm

surface and will remain on it throughout the subsequent time of motion, by

conservation of the functional. If a priori estimates can establish that the

distance in an appropriate norm from the equilibrium point to the nearby level

surface upon which the disturbed motion takes place subsequently remains

bounded, then the equilibrium point is stable by Arnold's method.

Bounded in a certain norm, motions stable by Arnold's method are also

stable in the sense of Lyapunov: for each g > 0 there exists a o > 0, such

that if the initial values are disturbed by less than o (in the norm

determined by the a priori estimates) then the solution deviates from a

specified solution (e.g., the stationary one) by less than c during the entire

subsequent motion. Having found by Arnold's method a norm I ·I in which the

perturbations ox0 at time zero, and ox at timet satisfy lox! < Klox0 1, with

K > 1 and for all time, one may choose lox0 1 < o; then lox! < c = Ko. This is

the type of stability result derived by Arnold's method.

Arnold [ 1965a, 1969] studies incompressible planar fluid motion, where

stability is established, among other examples, in the case of stationary

flows satisfying Rayleigh's inflection point criterion. Dikii [ 1965] shows

this type of stability for incompressible zonal circulation on a spherical

surface, provided the stationary flows there satisfy a spherical analog of

Rayleigh's criterion. Holm et al. [1983] establish conditions for stability

by Arnold's method for compressible (barotropic) planar flows. Abarbanel

et al. [1984] prove stability criteria by this method for two and three

dimensional, stratified, incompressible flows, with buoyancy effects included. Holm et al. [ 1984] deal with additional examples of stability of stationary

flows by this method: three dimensional adiabatic compressible hydrodynamics,

magnetohydrodynamics, and multifluid plasma dynamics; two dimensional

magnetohydrodynamics, both compressible and incompressible; Poisson-Vlasov, and Maxwell-Vlasov plasma equations; and multilayer quasigeostrophic systems.

Wan et al. [1984] prove stability conditions for incompressible circular

vortex patches in the plane by a method similar to Arnold's, but requiring

more delicate analysis.

Arnold's stability method is assembled from several well known elements:

extremal principles for conserved functionals, definiteness in sign of their

second variations, and convexity arguments that establish a priori estimates.

However, the success of this method in fluid dynamics derives from a less

familiar element: degeneracy of Poisson brackets. Degeneracy of Poisson

brackets for a given dynamical system means that certain quantities - the so-called "Casimirs" - are constants of the motion for ang Hamiltonian. Thus,

the Poisson bracket vanishes when taken between a Casimir and any other quantity depending upon the given dynamical variables. Casimirs are discribed



PLASMA STABILITY BY ARNOLD'S METHOD 27

from a geometrical viewpoint with finite-dimensional examples, in Weinstein [1984], in these proceedings. Construction of degenerate Poisson brackets for various fluid theories and their association to certain Lie algebras is treated in Marsden, Ratiu, and Weinstein [1983], Holm and Kupershmidt [1983], and Holm, Kupershmidt, and Levermore [1983]. Explicit derivation of Casimirs for Poisson brackets in fluid theories is discussed by Ratiu in these proceedings, see Marsden, Ratiu, and Weinstein [1984].

Arnold's stability method uses the Casimirs to construct conserved functionals. It imposes the Casimirs (as well as other constants of motion) essentially as Lagrange multiplier constraints for a variational principle that seeks conditional critical points of the energy. Denote by H this constrained energy, so that H is the sum of the energy and certain constants of motion. For stationary states, the first variation of H vanishes, i.e., H has a critical point, for appropriately chosen Lagrange multipliers. This critical point is locally a minimum, a maximum, or a saddle point, depending on whether the second variation of H at the critical point is, respectively, positive definite, negative definite, or indefinite.

Under certain conditions on the stationary states, the second variation at the critical point may be definite in sign. Under these conditions, the second variation defines a norm, which induces a weak type of stability, called "formal stability." Formal stability implies linearized stability against infinitesimal disturbances at the critical point, since the norm of the second variation is preserved by the linearized equations. This is only neutral stability, though, since the spectrum of the linearized ideal fluid equations lies on the imaginary axis. Formal stability in fluids and plasmas had been considered by a number of authors, even before Arnold [1965a]. For plasma theory, see, e.g., Kruskal and Oberman [1958], Newcomb (in Appendix I of Berstein, et al. [1964]), and Rosenbluth [1969]. For incompressible planar shear flows, formal stability is discussed in a geophysical context by Blumen [1971], and, more recently, for multilayer quasigeostropic flows, by Benzi et al. [1982].

Fortunately, the conditions on the stationary states that give formal stability via definiteness in sign of the second variation, can often be strengthened sufficiently to provide the desired a priori estimates; thereby expressing Lyapunov stability against disturbances of small but finite amplitude. These estimates are obtained via convexity arguments involving the constrained energy, H.

The present work establishes sufficient conditions for stability by Arnold's method for planar stationary plasma equilibria, as described by the ideal, compressible, multi fluid plasma equations. This problem exemplifies



28 Darryl D. Holm

the kind of results available for stability of fluids that are coupled self

consistently with other fields, and displays the role in Arnold's method

played by degenerate Poisson brackets possessing Casimirs. In the next

section, after a brief introduction of energy principles in the context of

potential flows, this role is reviewed for vortical incompressible flows in

three dimensions (Beltrami flows) and in two dimensions (Arnold's case). In

section 3, we study the multifluid plasma problem.

II . HOMOGENEOUS INCOMPRESSIBLE FLOWS

II.A. Potential Flows. The problem of establishing sufficient conditions for

stability in ideal incompressible hydrodynamics can be introduced conveniently

by recalling a result due to Lord Kelvin. Kelvin [1849] shows that ideal

incompressible potential flows (~ = ~cp, div ~ = 0, ~ the velocity, cp its

potential) satisfy a minimum energy principle among divergenceless flows in a

simply connected domain D.C IR3 with prescribed normal flux at the surface.

Euler's equations for an ideal incompressible fluid are

a v = -(~ • ~)~ - ~ t-

div ~ = 0 (1)

where p is pressure, and the constant fluid density has been set equal to

unity. These equations conserve the kinetic energy

In Kelvin's minimum energy principle for potential flows, the kinetic energy

is minimized subject to the two conditions that div v = 0 in domain n and

~ • ~ = Q(~) on the .boundary an, where n is the unit vector normal to the boundary and Q is the prescribed normal flux consistent with conservation of

energy. These two conditions will be imposed by choosing Lagrange multiplier

functions, cp,x, respectively .. Thus, one considers the functional

H.._ (v) .... x - = f [~1~1 2 + cp(~) div ~]d 3 x + f X(~)(~ • ~- Q(~))d 2 x n an



PLASMA STABILITY BY ARNOLD'S METHOD

The first·variation of H~ is, for arbitrary variations 6v_, 6~, 6x, •• x '

6H~ := DH~ (v) • (6v, 61fl, 6x) •• x .... x - -

= f [(~ - ~~) • 6v + 6~ div ~] d3x D

+ J an

[(• + x)<'l~ • n + <'lx(~ 2 • n - Q)]d x

29

(2)

The first variation 6Hifl,x vanishes for an equilibrium velocity, ~· which is a

stationary potential flow,

under the conditions imposed by the Lagrange multipliers,

div v = 0 in D -e

on an

provided also ci>+X = 0. Note that if Q - 0, e.g., for a fixed, impermeable

boundary, then the equations l!.cj) = 0 in D and n • '!ell = 0 on an imply that ell will be constant, so ~e will vanish. Plainly, this static solution ~ - 0

would be a trivial minimum of H~ . We seek nontrivial minima • .... x Taking the second variation of H~ leads to .... x

2 D H~ (v ) ... ,x -e

D an

which is positive definite in the clas8 of divergenceless velocity v•riations

(div 6~ = 0 in D) for the prescribed flux (6~ • ~ = 0 on an). So the kinetic

energy has a conditional minimum for potential flows. This is Kelvin's

minimum energy principle.

Kelvin's minimum energy principle indicates how to establish stability of

these stationary potential flows.

following quantity is also conserved

Noting that H~ (v) is conserved, the ....x -

= H•,x(_ve + <'lv_) - H. (v ) - DBa (v ) • (~v, ~., &x) 'I' ... ,x -e . ...,x -e -

(3)



30 Darryl D. Holm

Letting a!o denote the initial value of a velocity perturbation that takes a value a! at a certain time t later, one has

(4)

Thus, Euler's equations conserve an energy norm (3), which is an 12 norm in

av. In this norm, ideal, stationary, potential flows are stable, according to the a priori estimate (4).

II .B Beltrami Flows: Introduction of Casimirs·.

velocity is an eigenfunction of the curl operator:

curl v = av , a = const.

Thus, expressing Euler's equations (1) as

For Beltrami flows, the

(5)

(1')

where w := curl v is vorticity, one sees that Beltrami flows are stationary

states of Euler's equations, when V(lv 1212 + p ) vanishes. We shall show - -e e that Beltrami flows extremalize the kinetic energy, constrained by the

"helicity", Fh, defined as

v := -A- 1 curl ~ ., (6)

in a finite domain DC IR3 , with vanishing normal flux at the fixed boundary, an. However, we shall see that this helicity constraint will not be enough to establish the norm required to prove stability of Beltrami flows in three dimensions by Arnold's method. Nevertheless, stable Beltrami flows may still exist. We wish to use this apparently negative example to emphasize that even when successful, in most cases Arnold's method provides conditions that are only sufficient, not necessary and sufficient, for stability. This example also introduces the use of Casimirs, for a Poisson bracket in terms of the

vorticity. Euler's equations are Hamiltonian in terms of the vorticity. Namely,

upon taking the curl of (1') and identifying v = -A-1 curl winD, one finds

(7)



with Hamiltonian

E(~) = f ~ D

PLASMA STABILITY BY ARNOLD'S METHOD

-1 3 (-A ~)d x

and Poisson bracket {·,·} defined by

{F,G} = f ~ · (curl 6F x curl 6G)d3x D &u &u

31

(8)

for functionals F(~) and G(~) of ~. where 6F/~ and M/~ are variational derivatives, defined by

[ d F( ) ] _ f 6F d3 de ~ + e9. - cSw • 9. x e=O D -

for an arbitrary function, 9.· The time development of a functional F(~) thus obeys

The helicity Fh in (6) is a Casimir for the Poisson bracket (8), i.e.,

the Poisson bracket {Fh,G} vanishes for every Hamiltonian G(~),

In particular, the bracket {Fh,E(~)} vanishes, so the helicity is a constant of motion for Euler's equations (7) in D, with the boundary conditions of zero normal flux on an. In addition, the Hamiltonian formed by the sum

HA = E + AFh , A = const,

that is,

(9)

also generates the Euler equations {7) via Poisson bracket (8), with E

replaced by HA for any value of A, which we now regard as a Lagrange

multiplier for the conserved constraint, Fh.



32 Darryl D. Holm

Taking the first and second variations of the constrained kinetic energy HA in (9) yields the formulas

c5H = f A D

-1 3 (-6 ~ + 2A~) • c5w d x = f (~ + 2A curl ~) 3

0~ d x, (10)

= f [~ D

with definitions

oRA = DHA(~) .

2c52H 2 = D HA(~) A

c5v -1 curl = -a

c5w

(~)2

c5w

D

3 o~ld x (11)

and with surface terms having been set equal to zero whenever they appear due

to integration by parts, according to the boundary conditions. From the first variation, which vanishes for equilibrium velocity ~e such that v + 2A curl v = -e -e 0, one sees that Beltrami flows do

given Beltrami flow (5) with eigenvalue extremalize HA, and for a

-1 a, one has A= (2a) for the Lagrange multiplier, A.

The second variation c5 2HA is indefinite unless A = 0, in which case the equilibrium flow is static. Indeed, o2HA is equal to the following conserved quantity

where w is -e perturbation.

for any ~.

the equilibrium ~orticity distribution and ~ can now be a finite

The quantity HA is

is merely a

vanishes. With the quantity HA

conserved, since HA(~e + ~) is conserved

constant real number, and DHA (~) c5w

indefinite, no norm is established and

constancy of HA does not restrict the growth of perturbations.

Besides introducing Casimirs into the construction of the conserved

quantity HA, this example illustrates the following point: when HA is indefinite, no conclusion is indicated by Arnold's method about either

stability, or instability. In particular, one cannot draw the conclusion now that all Beltrami flows with Cl :f. 0 are unstable, cf. Arnold [1965b]. Such indefiniteness, though, does suggest employing a complementary technique. For

example, one could seek sufficient conditions for linear instability of Beltrami flows, using, say, normal mode analysis.




II.C. Arnold's Theorem. Arnold's theorem uses an extremal energy principle to obtain stability criteria for stationary, planar, vortical flow of an ideal, incompressible fluid. Arnold [1965a,1965b,1969] considers incompress-ible fluid motion in a fixed domain DC: IR2 , in the (x,y) plane, with velocity tangent to the boundary, aD. In this case, vorticity is defined by a scalar function w, as

curl v = z w(x,y,t)

where z is the unit vector normal to the (x,y) plane. The Poisson bracket (8)

becomes

6F 6G {F,G} = f w[6w'6w]dx dy D

(12)

for functionals F ,G of w, with [ ·, •] being the jacobian (or the canonical Poisson bracket), defined by

(f,g] - a£ £& - £& a£ (13) - ax ay ax ay

for functions f(x,y), g(x,y). Also, the energy E in (7) becomes

E(w) = \ f w(-a- 1w)dx dy D

whereby the equation of motion results,

. -1 atw = {w,E} = [-A w,w] (14)

using (12). Defining the stream function ljl such that w = -&II leads to the standard formula,

Consequently, a certain functional dependence exists for stationary flows ljle,we, expressible as

ljl = ~(w ) e e

since the jacobian [ljl ,w ] vanishes for stationary flows. e e

(15)



34 Darryl D. Holm

A Casimir for the Poisson bracket (12) is, with an arbitrary function

<l>(w)'

= I <l>(w)dxdy D

(16)

By direct computation, one shows that F<l> satisfies {F,GJ = 0 for every

Hamiltonian, G,

=I 6G [w,<l>'(w)]dx dy D&J

= 0 V G(w), <l>(w)

upon using the properties of the jacobian (13) and integrating by parts. In

particular, the bracket {F,EJ vanishes so that F<l> in (16) is a family of

constants of motion for the two dimensional Euler equations.

Following Arnold [ 1969 J , one defines the sum

(17)

which is a conserved functional. Taking the first and second variations of

Hcp(w) yields

6Hm := DH (w) • 6w =I [-~- 1 w + <l>'(w)J&J dxdy "' D

:= D2H(w) • (&.1) 2 = I D

-1 2 [&.1(-~ &.1) +'' (·u)(&J) ]dxdy

(18)

(19)

The first variation 6H vanishes, provided w takes equilibrium values, we,

satisfying

That is, for stationary flows 6H vanishes, and <l>(we) is determined for a

given stationary flow satisfying (15), by

<l>'(w) = -~(w) (21) e e




As mentioned in the introduction, either negative, or positive

definiteness of the second variation o 2 H~ suggests that Lyapunov stability can

be established. Both cases are shown to be possible in Arnold [1969]. In

each case, a convexity argument for the function ~(w) is used in combination

with the conserved quantity H~,

to establish Lyapunov stability in a certain norm. Here, &w is considered to be a vorticity disturbance at a certain time, t, which has the value &w0 at

time zero. The quantity H~ is conserved, since H~(we + &w) is conserved for

any &w, H~(we) is merely a constant real number, and DH~(we) • &w vanishes, by (20).

Case 1. According to (19), the second variation o 2 H~ will be positive

definite, provided

~· '(w ) > 0 e

-1 since (-~ ) is a positive operator. By using (15) and (21), this condition

can be expressed as

'' (w ) e = -~· (w ) = e (23)

For example, flows parallel to the x-axis in the strip {Y1 ~ y ~ Y2 } and periodic in x have

Consequently, for such flows (23) becomes

<l>''(w(y))=~ >O e v'' (y) (24)

provided an inertial frame can be chosen so that the sign of v is everywhere

the same as the sign of v''. o2H positive definite.

~ Positive definiteness

Thus, all flows having no inflection points have

of 2 o H~, by itself, does not imply Lyapunov

stability. Arnold [1969] supplies a convexity argument which does prove



36 Darryl D. Holm

Lyapunov stability criteria in this case. Strengthening the condition (23) to

0 <a~ ~''(C)< A<~ (25)

and extending the definition of ~(C) over the entire C axis subject to inequality (25), implies that, for any h,

(26)

Hence, according to the definition (22)

-1 2 [&w(-a &w) + a(&w) ]dxdy > 0

(27)

and H~(t) = H~(O), so that the growth of a disturbance &w is bounded in terms of its initial value &w0 . The estimate (27) implies Lyapunov stability of

stationary flows with V~ /V~ > a > 0. - e- e-

Case 2. Consider stationary flows with ~e~~~e < 0. Let a stationary flow be such that

(28)

and extend the definition of .,(t) over the entire t axis, subject to (28).

Then one bounds -2H.,, to find

-2;~(t) ~I [-&w(-a-1&w) + a(&w) 2]dxdy ~I D D

-2 2 (-k . + a)(&w) dxdy m1n

-2H.,(O) ~I [-&w0(-a-1&w0) + A(&w0)2 Jdxdy ~I A(&w0) 2dxdy D D

(29)

where k2 . is the minimum eigenvalue of minus the Laplacian (-d) in domain D. m1n Consequently, if

a= min I.,''(C)I > k-~ m1n

then perturbation growth is bounded, since again H.,(t) = H.,(o). The estimate (29) establishes Lyapunov

-2 -~e~~~e ~ a > kmin

stability of stationary flows with




III. COMPRESSIBLE MULTIFLUID PLASMA STABILITY IN TWO DIMENSIONS The multifluid plasma (MFP) equations describe motion of a system of

ideal, charged fluids interacting together via selfconsistent electromagnetic forces. The fluid species are labeled by superscript s (Note: no summation convention is imposed on the superscript s in this section.); each species is composed of particles of mass ms and charge qs, with charge to mass ratio a8 = qs/ms. Dynamical fluid variables are: fluid velocity vs; mass density ps (with barotropic partial pressure ps = p8 (ps) and specific internal energy es = es(ps), each depending only on ps); electric field!; and magnetic field

B. The MFP equations consist of dynamical Maxwell equations for the

electromagnetic fields; a continuity equation for each species; and the MFP

motion equations:

l:\.!! = -curl E

s s s a E = curl B - I a p ~ t- s

The static Maxwell equations

div B = 0

(30)

(31)

although nondynamical, are compatible with the flow, i.e., if true initially (31) will remain true under MFP dynamics.

The MFP equations are shown to be Hamiltonian in Spencer [ 1982] with { } s s s s } Poisson bracket F,G defined in terms of {p .~ :=p ~ ,!,_!! by

(32)

+ f d3x (~i · curl ~~ - ~ • curl gi)



38 Darryl D. Holm

and Hamiltonian energy function

(33)

The time development of any functional, F, of the MFP dynamical variables obeys

Moreover, one readily shows that the static Maxwell equations (31) correspond to the following Casimirs,

GE = f .(!)(-div ~ + l as ps)d3x s

Each of the quantities GE' GB' for arbitrary functions .,$, Poisson commutes using (32) with every Hamiltonian H[ps, ~s, ~. !!_). Thus, not only the equations of motion, but the Poisson bracket (32) itself preserves the static Maxwell equations.

III.A. Planar MFP Flows. We consider now planar MFP motion in some domain DC IR2 in the (x,y) plane. In order that such motion remain planar, each of

s s } s the dependent variables {p .~ .~.~ must be functions only of (x,y,t); ~ and E must lie in the (x,y) plane; and ws and B must be directed normally to the plane, along z,

(34)

B = z B(x,y,t)

The planar MFP equations are

curl ~ = E1 2 - E2 1 ' '

s (35) s s s <\P = -div p ~




where hs(ps) is specific enthalpy,

internal energy es by

s related to pressure p and specific

hs = es + ps/ps

dhs = (ps)-ldps (36)

For a single fluid species and when 1~1 and B are absent, these equations reduce to the equations for planar motion of a barotropic fluid, whose stability criteria are proven by Arnold's method in Holm et al. [ 1983]. Taking the curl of the planar motion equation and using the continuity equation leads to the advected quantitites ns, the so-called "modified vorticities",

(36)

with species material derivative

v (37)

along the flow lines of each species. In view of (36) and the continuity equation for each s, for every real valued function of a real variable ~s(t),

each functional

(38)

is conserved by the planar MFP equations (provided the integral exists and the solutions are smooth; O.s would be created at a discontinuity). Another conserved quantity is the energy (33) expressed in two dimensions,

E := ff {l [\p8 l!s1 2 + p8 e8 (p8 )] + \1~1 2 + \B2}dxdy D s

Either by direct computation from the Poisson bracket (32) specialized to planar motion, or by showing invariance under the coadjoint action of the semidirect of (32),

product group whose Lie-Poisson bracket is a one may readily show that each functional F (O.s)

~s

key ingredient in (38) is a



40 Darryl D. Holm

Casimir,

{F ,G} = 0 cps

s s V G[p , ~ , ~' B]

Likewise, Gauss's Law in (31) corresponds to the following Casimir,

GE = f $(!)(-div ~+I asps) dxdy - D s

(39)

The Casimir GB mentioned earlier is identically zero in two dimensions with B normal to the -plane.

Equilibrium States. The equilibrium states ps vs E , B , of the system (35) - e' -e' -e e in the (x,y) plane are the stationary, two-dimensional, barotropic MFP flows. For such stationary flows, one has the relations

~ = - ycpe

VB xz - e

s v - e

s

s v -e v rl = o - e

(40)

According to the last two equations in (40), the gradient vectors '\70s and - e YC~I~I 2 + hs(p:) + ascpe) are orthogonal to the equilibrium species velocity vs. Consequently, these two gradient vectors are collinear, provided they or -e the velocity do not vanish. A sufficient condition for such collinearity in

the plane is the functional relationship

(41)

for certain functions ks(C) , C € IR; these are called the Bernoulli functions

and (41) operator

represents Bernoulli's Law for each species. Either applyinf! the (Os)- 1 ; x V to (41), or simply vector multiplying by z the e -

stationary motion equation,

( s s ) A s ( I s 12 + hs( s) + as~ ) we + a Be z x ~ = -y \ ~ Pe '~'e (42)




gives the relation

ks'(Os) A

psvs = ---"'e_ z x VOs = _1_ z e-e 0 s - e 0 s (43)

e e

where prime ' denotes derivative of a function with respect to its stated argument. Substitution of (43) into the second equation in (40) (i.e., Ampere's Law) leads to another relation for stationary flows,

VB - e (44)

Relations (43) and (44) will be useful in establishing the following proposition.

Proposition. For smooth solutions with velocitg fields parallel to the

boundarg and fixed circulation on the boundarg, a stationa:r::T solution (vs, pse' -e B , B ) of the ideal planar MFP equations is a conditional extremum of the -e e total energy B for fixed Casimirs F and G8, and an absolute extremum of

iPS H = B + F + G8, where cp = cpe and

F t'Ps

t s = CCJ k (t) dt + const) t2

k8 being the Bernoulli function of species s.

The functional HF in the Proposition is, explicitly,

HF(~s. ps, ~.B)= JJ {I[\psl~s 1 2 + pses(ps) + pst'Ps(Os)] D s

(45)

+ \1~1 2 + \B2 + cp(!)(-div ~+I asps)} dxdy . (46) s

After integration by parts, the variational derivative lilly in the direction



42 Darryl D. Holm

s s (o~ , op , o~, oB) becomes

+ l[ps~s _ z x ~ ~s'(Qs)] s

ovs + [B + l as~s'(Qs)] oB s

+ (~ + ~cp) oE + (-div ~ + l asps)ocp} dxdy s

+ f l ~s'(Qs) ovs aD s

d.e + J cpo~ an

(z x d.£)

(47)

where d.£ is the line element along the boundary aD. For a stationary

solution, the connected components of the boundary an are both streamlines and

equipotential lines. Thus, Qs and cp are constants on an and the boundary e e integrals become

(48)

Let the variations o~s and 0~ satisfy faD o~s • d.£ = 0 and Ian 0~ • z X

d.£ = 0, respectively. Then the boundary integrals in (48) each vanish. In

equation (47), the op8 coefficient vanishes for a stationary flow obeying (41), provided that ~s is related to the Bernoulli function ks by

from which equation (45) in the Proposition follows. Differentiating with respect to {; implies C1 ks I({;) - ~Sit({;) = 0. Then the ovs and oB

coefficients in (47) each vanish, by (43) and (44), respectively, since v~s'(Q) = (Q8 )-l V ks(Qs). If cp = cpe, the o~ coefficient vanishes. Finally, - e e - e the ocp coefficient in (47) vanishes, by Gauss's Law in (31). D

The quadratic form defined by the second derivative of ~ at the

stationary solution is

( ~ s ~ s ~E ~B)2 __ f {'[psl~vs + vs~ s1 s 12 uv_ , up , u_, u L u up p e - -e e D s

+ (oB) 2 + 16~1 2 + (-div oE + I asops)ocp}dxdy s

(49) Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.



The last term vanishes for variations that satisfy Gauss's Law. Sufficient

conditions for this quadratic form to be positive definite are:

(i)

where cs is the sound e by shs, ( s) = (cs)2 pe pe e '

(ii)

(SO)

speed of species s for the stationary solution, defined

i.e. , the stationary flow is everywhere subsonic; and

(51)

i e the two collinear gradient vectors V_(~lv_esiZ + hs(ps) T as$ ) and V(Qs)Z · ., e e - e point in the same direction throughout the flow. For a single, incompressible

fluid without charge (s = 1, ps = 1, 6ps = 0, 6B = 0, {j~ = O), formula (49) e reduces to Arnold's formula, equation (19) in Section III.C, discussed

earlier.

A priori estimates expressing stability. Lyapunov stability criteria for

planar stationary flows of MFP in the smooth regime can be proved by

establishing sufficient conditions that imply certain a priori estimates

bounding perturbation growth in terms of the Bernoulli functions ks. These

estimates can be obtained readily by following the same convexity argument as

in Theorem 1 of Holm et al. [ 1983] for planar barotropic flows. Thus, one

obtains the following result.

THEOREM. Assume that each Bernoulli function ks in (41) and internal energg

densitg function es := pses(ps) satisfies

(52)

where qs and Qs are positive constants and similarly,

(53)

with constants rs,Rs, and for all values of the arguments. Let (&y_s, &ps, 6~,

68) be a small, but finite, smooth perturbation of a stationary solution (vs, -e

ps, E , B ) and denote its value at t = 0 bg (6v0s, 6p0s, &E0 , &80 ). Let the e -e e "' -

circulation faD o~ · d£ and integral faD &~ 0 · z x d! each vanish. Then the

perturbation (ovs, 6ps, 6~, &B) of the stationary solution (vs, ps, E , B ) at - -e e-e e



44 Darryl D. Holm

s s ang time t is estimated in terms of (5!0, 5p0, 5~, 580 ) bg

+ (5B) 2 + 15~( 2 }dxdy

15(psvs) 12 f {I[ - O + (Rs -D s p: + 5p~

< (54)

Just as in Holm et al. [1983], the proof of the Theorem proceeds by

showing that a conserved functional

s - IL(v --y e' (55)

is bounded from below (above) by the left (right) hand side of (54). The a

priori estimate (54) then implies Lyapunov stability for smooth solutions, provided ps + 5ps remains finite and bounded away from zero. Under such an e additional hypothesis on the density, one has the following result.

Corollary 1. Let a stationary solution satisfg (41) for smooth functions

ks(t). Assume that

(56)



and

for all

(Cs. )2 m~n '


(57)

C € IR and ts such that 0 < p:in < ts < p:ax < oo where qs, Qs,

(cs )2 s s ·t· t t Also assume that P P are pos~ ~ve cons an s. max ' min' max'

(58)

for some other positive constants hs,As. Then with the same definitions as in

the Theorem, the following estimates obtain,

<

(59)

s s s for solutions with densities satisfging pmin < P < Pmax·

Corollary 1 follows immediately from the Theorem by replacing (53) by

(57), imposing (58), and bounding ps.

Remark. The a priori estimate (59) in Corollary 1 implies stability for

smooth, planar, MFP solutions, in the sense of a norm estimate of small, but

finite, circulation-preserving perturbations obeying Gauss's Law, that develop

from a perturbed, initially steady flow. Because of the method of proof for

the Theorem, the right hand side of the inequality (59) in Corollary 1 can be

minimized by replacing it with ~(6~~, op~, 6~ 0 , oB0). Thus, we have shown

another corollary.



46 Darryl D. Holm

Corollary 2. Under the assumptions of Corollary 1 and the Theorem, the

following a priori estimates obtain,

(60)

When there is only a single fluid species and electromagnetic fields are

absent, the result of the Theorem reduces to the estimate in Holm et al.

[1983] for planar barotropic flow. These estimates can break down when smooth

solutions cease to exist; for example, upon occurence of cavitation, and/or

the formation of shocks from an initially-smooth, steady flow. When these

phenomena occur, however, it is questionable whether the barotropic

approximation should still be used. One could exclude cavitation by replacing

(54) by an estimate as in Holm et al. [1983], modeling an elastic fluid. None

of the estimates in this section apply to three-dimensional phenomena. That

topic is discussed in Holm et al. [1984].

A stationary solution of the MFP equations Example. Subsonic Shear Flows.

(35) in the strip {(x,y) C IR2 Y1 2 y 2 Y2} is a plane parallel flow along

x, admitting arbitrary velocity profile vs(x,y) = (~?(y) ,0), electrostatic -e - s -s potential $e(x,y) = $(y), and density pe(x,y) = p (y). The density profile is

subject only to the subsonic condition (SO), expressible as

d-s -s 2 ~(y) - (v (y)) > 0 (61) dp

and depending on the barotropic relation ps = p s (ps). In this domain, the

independent variable x can be either unrestricted on the entire real line, or

periodic. The former case requires that initial perturbations be sufficiently ~ s s

integrable for ~(6~ 0 , op0 , 6~ 0 , oB0 ) to be finite and, thus, give a

meaningful upper bound in (60).

To determine the limits of stability for subsonic stationary planar MFP -s - -s

flows, we proceed as follows. (i) Choose profiles v (y), <j)(y), and p (y),

satisfying the subsonic condition (61). Relations (43) and (44) then imply

y-dependence only, for magnetic field and modified vorticity: Be(x,y) = B(y),

Q 5 (x,y) = Qs(y). (ii) Use Ampere's Law in the form (44) to determine B(y) e




-s -s -s from p (y) and v (y), then compute 0 (y) from its definition (36) in terms of -s -s - -s -1 s -s p , v , B. (iii) Solve for an expression for the quantity (O ) k 1 (0 )

appearing in condition (52) of the stability theorem and consider its sign, thereby determining the limits of stability in terms of the profiles ps (y),

~?(y), B(y). Given the profiles ~s(y), ps(y), and ~(y), one finds ws(y) and Os(y) from

their definitions

ws = z • curl vs e -e -s

: - V I (y) -s =: w (y) (62)

and

-s -1 -s s-(p (y)) (-v 1 (y) +a B(y)) =: ns(y) (63)

Equations (43) and (44) give the relations

(64)

and

Bl(y) =I asps(y)~s(y) (65) s

(-s)2 ~s = ----------~~~~------------- (66)

where, e.g., ~s, = d 2 ~s(y)/dy 2 , B' = dB(y)/dy, etc. Thus, control of positivity of (05 )-lks'(05 ) in (56) and, hence, of stability for MFP involves

an interplay among velocity, density and magnetic field profiles, through the positivity condition,

> 0 (67)



48 Darryl D. Holm

Given that an inertial frame can be chosen so that condition (67) holds,

planar MFP flows will be stable, provided

OS' (y) :;:. 0 (68)

We consider several cases.

Case A. In the case of neutral fluids (as = 0) and stationary flows with

constant -s -s -1 s -s density (p '(y) = 0), positivity of (Q ) k '(Q ) (67) reduces to

-s -s v (y)/v "(y) > 0 (69)

Provided an inertial frame can be chosen so that (69) holds throughout domain

D, one recovers Rayleigh's criterion (24) for stability of shear flows: all

flows in this case with no inflection points in their velocity profile are

stable.

Case B. (ps'(y)

For the case of charged fluids (as :;:. 0) at constant density

0), positivity in (67) reduces to

-s 2-s Cp ) v > 0 (70)

Provided an inertial frame can be chosen in which (70) holds throughout D, one

obtains the following criterion for stability in this case,

-s s-v "(y):;:. a B'(y) (71)

Case C. In the general MFP case, with charged, compressible fluids, (as :;:. 0,

ps'(y):;:. 0), when an inertial frame exists in which (67) holds, the stability

condition (68) becomes

-s -s -s s- s-:;:. (p '/p )(v ' -a B)+ a B' (72)

which involves all three stationary profiles.

Note that the conditions obtained here by Arnold's method are sufficient

for stability. Thus, violation of these conditions would be necessary for the

onset of instability but no~ necessary and sufficien~, except in the fortunate

event where they coincide with instability conditions found by linear

analysis.




ACKNOWLEDGMENTS It is a pleasure to thank my collaborators in this work, Jerry Marsden,

Tudor Ratiu, and Alan Weinstein, for their help and comraderie. I would also like to thank Henry Abarbanel, Allan Kaufman, and Philip Morrison for their

interest, correspondence, and helpful comments.

REFERENCES H. D. I. Abarbanel, D. D. Holm, J. E. Marsden, and T. Ratiu [1984). Stability analysis of stratified incompressible ideal fluid flow. (In preparation).

V. I. Arnold [1965a). Conditions for nonlinear stability of stationary plane curvilinear flows in an ideal fluid. Sov. Math. Dokl. _!_62 (5), 773-7.

V. I. Arnold [1965b). Variational principle steady-state flows of an ideal fluid. Prikl. Translated in J. Appl. Math. Mech. 29, 1002-8.

for Mat.

three Mekh.

dimensional 29' 846-51.

V. I. Arnold [1969). On an a priori estimate in the theory of hydrodynamical stability. Am. Math. Soc. Transl. 79, 267-9.

R. Benzi, S. Pierini, A. Vulpiani, and E. Salusti [1982). On nonlinear hydrodynamic stability of planetary vortices. Geophys. Astrophys. Fluid Dyn. 20, 293-306.

1. A. Dikii [1965). On the nonlinear theory of the stability of zonal flows. Fiz. Atmos. Oceana _!_, 1117-22. Translated in Izv. Atmos. Oceanic Phys. _!_, 653-5.

D. D. Holm and B. A. Kupershmidt [1983). Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid phasmas, and elasticity. Physica ~D, 347-63.

D. D. Holm, B. A. Kupershmidt, and C. D. Levermore [1983). Canonical maps between Poisson brackets in Eulerian and Lagrangian descriptions of continuum mechanics. Phys. Lett. A, 98A, 389-395.

D. D. Holm, J. E. Marsden, T. Ratiu, and A. Weinstein [1983]. Nonlinear stability conditions and a priori estimates for barotropic hydrodynamics. Phys. Lett. 98A, 15-21.

D. D. Holm, J. E. Marsden, T. Ratiu, and A. Weinstein estimates for nonlinear stability of fluids and plasmas.

[ 1984] A priori (In preparation.)

Lord Kelvin (W. Thompson) [1849]. On the vis-viva of a liquid in motion. Camb. and Dub. Math. Journ. [Papers, i. 107]. Referenced by H. Lamb [1932] Hydrodynamics (Dover, New York), sixth ed. p. 47.

M.D. Kruskal and C. R. Oberman [1958]. On the stability of plasma in static equilibrium. Phys. Fluids_!_, 275-80.

J. E. Marsden, T. Ratiu, and A. Weinstein [1983]. Semidirect products and reduction in mechanics. Transactions of AMS (to appear).

J. E. Marsden, T. Ratiu, and A. Weinstein [1984]. Reduction and Hamiltonian structures on duals of semidirect-product Lie algebras. (these Proceedings).



50 Darryl D. Holm

M. N. Rosenbluth [1969]. Advanced Plasma Theory (Academic Press, New York), p. 137.

R. G. Spencer [1982]. The Hamiltonian structure of multi-species fluid electrodynamics. In "Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems," ed. by M. Tabor and Y. M. Treve. (AIP Conference Proceedings, No. 88). pp. 121-6.

Y. H. Wan, J. E. Marsden, T .. Ratiu, and A. Weinstein [1983]. stability of circular vortex patches (preprint).

A. Weinstein [1984]. Proceedings).

Stability of Poisson-Hamilton equilibria.

THEORETICAL DIVISION, MS-B284 LOS ALAMOS NATIONAL LABORATORY LOS ALAMOS, NM 87545

Nonlinear

(these




CANONICAL DERIVATION OF THE VLASOV-COULOMB NONCANONICAL POISSON STRUCTURE

Allan N. Kaufman* and Robert L. Dewar**

ABSTRACT. Starting from a Lagrangian formulation of the Vlasov-Coulomb system, canonical methods are used to define a Poisson structure for this system. Successive changes of representation then lead systematically to the noncanonical Lie-Poisson structure for functionals of the Vlasov distribution.

The Vlasov system with Coulomb interaction is not only the standard model for plasma theory, but it is also a paradigm for the understanding of noncanonical Poisson structures. The Vlasov distribution f(~,~) is its dynamical field variable and has no canonical conjugate. Yet it possesses a noncanonical Lie-Poisson structure for observables A(f), functionals off:

{Al'A2} = /d3rd3p f(~.~) [a1 (~.~), a 2 (~.~)], (1)

where a(~.~) = oA/of(~.~) is the functional derivative, and

is the canonical bracket on functions of(~.~).

Since a knowledge of the Poisson structure is essential for a Hamiltonian field theory, it is highly desirable to understand the derivation of this structure. In this paper, we present such a derivation, based on old-fashioned canonical field theory. This derivation is rigorous, and is accessible to readers ignorant of modern mathematics.

Before beginning, we survey the earlier derivations of the Poisson structure (1). In 1980, this structure was discovered by Morrison and Kaufman by manipulating Morrison's results for the Vlasov-Maxwell sytem [9]; the latter had been obtained by inspired guesswork. (Tne guesswork method was sytematically reviewed by Morrison in [10]. It has been extended to wave systems by McDonald and Kaufman [8].)

Independently, the structure (1) was ootained by Gibbons [3], using

1980 Mathematics Subject Classification. 76X05, 70H05, 58F05. *Work supported by U.S. DOE under Contract No. DE-AC03-76SF00098.

51





52 ALLAN N. KAUFMAN

Lie-group concepts. This approach was extensively investigated by Marsden and Weinstein [7], who have applied these concepts to other systems as well.

A third approach to deriving the structure (1) was introduced by Bialynicki-Birula and Hubbard [1]. They began with the canonical structure for a system of particles, derived from it the noncanonical structure for the Klimontovich distribution, and then postulated the identical structure for the Vlasov distribution. The success of this approach led Kaufman to apply it to multi-fluid electrodynamics [4] and to waves [5].

The method adopted here is to begin with a field Lagrangian for the Vlasov-Coulomb system. our form is a modification of the Low Lagrangian due to Dewar [2]. With an eye on the application to gyrokinetics [6J, we consider one species (for simplicity) of particle, moving non-relativistically in a specified {possibly nonuniform) static magnetic field ~{~). with only Coulomb interaction. At some reference time (t = 0), we specify the smooth Vlasov distribution f0 of initial conditions (~,~)

in six-dimensional phase space. The dynamical field in a Lagrangian description is ~(t;~.~), the corresponding position at any time t.

The Coulomb interaction energy C is thus [z0 = (~,~)]

C = (e 2 ;2)jd 6 z 0 jd 6 z~ f 0 (z 0 )f 0 {z~) l~(t;z 0 )-R(t;z~ll- 1 , (2)

while the noninteractive field Lagrangian L0 is

L0 = j iz0 f0 (z0 ) {mA212 + (e/c)~·~(~(t;z 0 ))), (3)

where ~(t;z 0 ) = a~(t;z 0 )/at, and ~(~) = \JX ~(~). We consider the total Lagrangian L = L0 - C, and apply the standard canonical procedure.

The first step is to define the momentum field ~(z 0 ) canonically conjugate to the position field ~(z 0 ):

~(z 0 ) = ol/6~(z 0 ) = f0 (z0 ) ~(z 0 ), (4)

. e where~= m~ + c ~(~) is the usual canonical particle momentum.

The next step is to express the canonical Poisson Bracket relation for the position field and its conjugate (at equal times):

{~{z 0 ), ~(z~l} = .!_ o6(z0 - z~),

with other brackets vanishing. Inserting (4), we obtain the interesting relation:

(5)



CANONICAL DERIVATION OF POISSON STRUCTURE

which requires that f 0 (z0 ) > 0 everywhere. Tne final conventional step is to form the field Hamiltonian:

H = jd6z0 ~(z 0 ) .!3_(z0 ) - L

53

jd6z0 f0 (z0 ) (~(z 0 )-(e/c)~(.!3_(z 0 ))) 2 /2m + c. (6)

We now wish to transform our representation to the Vlasov field f(~,£;t):

f(.!:_,£;t) = jd6z0 f 0 (z0 ) 6 3 (~- .!3_(t;z0 ) )63(£- ~(t;z 0 )) • (7)

Thus f is a function on (!:_,£)-space, and simultaneously is a functional of the fields .!3_(z0 ), ~(z 0 ). Setting aside the former property, let us consider general functionals G (.!3_(z 0 ),~(z 0 )). The Poisson Bracket on such functionals is obtained by using its property of differentiation, and the chain rule:

{G1,G2} = jd 6 z 0 jd 6 z~ (6G 1 /6.!3_(z 0 ))(6G 2 /6~(z~)):{.!3_(z 0 ),~(z~)}

+ ( oG1/ o~(z 0 ) )( oG2/ .s.!3_(z~)) :{ ~(z 0 ) ,.!3_(z~ l} = jd6z0 ((.sG 1 /.s.!3_(z 0 ))·(.sG 2 /.s~(z 0 ))

- (6G 1 /6~(z 0 )) (.sG2/6.!3_(z0 )))/f0 (z0 ),

where we have used the fundamental Poisson Bracket (5). We now may immediately apply (8) to the special functional (7),

choosing G1 = f(z1), G2 = f(z2), where zj = (.!:.j•£j) is a parameter for purposes of the functional differentiation in (8). After a few straightfoward steps, we obtain

{f(z1),f(z2l} =/d6z0 f0 (z0 ) (a 2 /a.!:_(a~- a 2 ;a~·a£ 1 )

x o3(.!:_1-.!3_(zo) )63(£1-~(zo) ).s3(~-.!3_(zo) ).s3(~-~(zo))

(8)

Finally, we consider functionals A(f), and their Poisson Bracket. We have

{A1,A2} = /d6z1jct6z2 oAtf6f(z1) oA2/.sf(z2) {f(z1), f(z2)} ,

substitute (9), and integrate by parts, obtaining (1), the desired result.



54 ALLAN N. KAUFMAN

The Hamiltonian (6), (2) reads, in terms of f,

H(f) = /d6z f(z) (!- (e/c)~(.!:_)) 2 /2m + C,

C(f) = (i/2)/d6z/d6z• f(z)f(z') l.!:.-.!:.'r1•

This completes the canonical derivation of the Hamiltonian structure of the

Vlasov-Coulomb system.

BIBLIOGRAPHY

1. I. Bialynicki-Birula and J. c. Hubbard, "Gauge-independent canonical Formulation of Relativistic Plasma Theory," Phys. Rev. A (in press).

2. R. L. oewar, "A Lagrangian Theory for Nonlinear Wave Packets in a Collisionless Plasma", J. Plasma Physics 7 (1972) 267-284.

3. J. Gibbons, "Collisionless Boltzmann Equations and Integrable Moment Equations," Physica 3D (1981) 503-511.

4. A. N. Kaufman, "Elementary Derivation of Poisson Structures for Fluid Dynamics and Electrodynamics," Phys. Fluids 25 (1982) 1993-1994.

5. A. N. Kaufman, "Natural Poisson Structures of Nonlinear Plasma Dynamics,"Physica Scripta T2/2 (1982) 517-521.

6. A. N. Kaufman and B. M. Boghosian, "Lie-Transform Derivation of the ~rokinetic Hamiltonian System," this volume.

7. J. Marsden and A. Weinstein, "The Hamiltonian Structure of the Maxwell-Vlasov Equations," Physica 4D (1982) 394-406.

8. S. W. McDonald and A. N. Kaufman, "Hamiltonian Kinetic Theory of Plasma Ponderomotive Processes," in Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems, M. Tabor andY. M. Treve, eds. (AlP Conf. Proc. 88, 1982), 117-120.

9. P. J. Morrison, "The Maxwell-Vlasov Equations as a Continuous Hamil toni an System", Phys. Lett. 80A (1980) 383-386.

10. P. J. Morrison, "Poisson Brackets for Fluids and Plasmas," in Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems, M. Tabor andY. M. Treve, eds. (AlP Conf. Proc. 88, 1982), 13-46.

* PHYSICS DEPARTMENT AND LAWRENCE BERKELEY LABORATORY UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720

** DEPARTMENT OF THEORETICAL PHYSICS RESEARCH SCHOOL OF PHYSICAL SCIENCES AUSTRALIAN NATIONAL UNIVERSITY CANBERRA ACT 2600, AUSTRALIA




REDUCTION AN 0 HAMIL TON IAN STRUCTURES ON DUALS 0 F SEMI DIRECT PRODUCT LIE ALGEBRAS

Jerrold E. Marsden, 1) Tudor Ratiu, 2) and Alan Weinsteinl)

ABSTRACT. With the heavy top and compressible flow as guiding examples, this paper discusses the Hamil toni an structure of systems on duals of semidirect product Lie algebras by reduction from Lagrangian to Eulerian coordinates. Special emphasis is placed on the left-right duality which brings out the dual role of the spatial and body (i.e. Eulerian and convective) descriptions. For example, the heavy top in spatial coordinates has a Lie-Poisson structure on the dual of a semidirect product Lie algebra in which the moment of inertia is a dynamic variable. For compressible fluids in the convective picture, the metric tensor similarly becomes a dynamic variable. Relationships to the existing liter-ature are given.

l. INTRODUCTION. There are natural brackets {f,g} defined for f,g:~*->- IR where r>{ is the dual of a Lie algebra (finite or infinite dimensional); these were discovered by Lie in 1887 and are now called Lie-Poisson brackets. These brackets arise by reduction of canonical Poisson brackets un T*G, the cotangent bundle of the corresponding group, by left or right invariance (giving structures differing in sign) and are compatible with the Kirillov-Kostant symplectic structures on coadjoint orbits (Marsden and Weinstein [1974]). We review some features of this theory in §4.

Lie-Poisson structures in mechanics have a complex history due, in part, to lack of communication and ignorance of Lie's original discovery. We are concerned here with the line of investigation initiated by Arnold [1966], [1969] in which he gave a clear presentation of the reduction from material (i.e. Lagrangian) coordinates to spatial (i.e. Eulerian) and body (i.e. convective) coordinates for i ncompress i b 1 e fluids and the rigid body. Arnold used symplectic structures on coadjoint orbits but did not use the Lie-Poisson bracket. In spite of this, Kuznetsov and Mikhailov [1980], for example, attribute it to him, we think quite appropriately.

1980 Mathematics Subject Classification 58F05, 58Fl0, 70K20. l) Research partially supported by DOE contract DE-AT03-82ER12097. 2) Research partially supported by an NSF postdoctoral fellowship.

55





56 Jerrold E. Marsden, Tudor Ratiu, and Alan Weinstein

Lie-Poisson structures for semi-direct products have the following his-tory. They were noted for the heavy top in Vinogradov and Kupershmidt 11977]. They appear, using a quantum mechanical motivation, in Dzyaloshinskii and Volovick [1980] (see also Dashen and Sharp [1968], Goldin [1971] and Goldin, Menikoff and Sharp [1980]). For our development, the papers of Guillemin and Sternberg [1980] and Ratiu [1980, 1981, 1982] are crucial. They started developing the abstract setting in which Lie-Poisson structures asso-ciated with semi-direct products appear. Simultaneously, Morrison and Greene [1980] and Morrison [1980] published brackets for MHO and the Maxwell-Vlasov equation. It was well-known to workers in the area that the bracket for ideal compressible flow was the Lie-Poisson bracket for the semi-direct product of the diffeomorphism group with functions. Marsden and Weinstein [1982] were the first to put the bracket structures for the Maxwell-Vlasov equation back in the spirit of Arnold and in doing so, corrected one term in the bracket--a correction necessary to ensure Jacobi's identity. The Morrison-Greene bracket for MHO was derived using Clebsch variables and was observed to be a Lie-Poisson bracket for a semidirect product by Holm and Kupershmidt [1983a].

In Ratiu [1980,1981 ,1982] and Guillemin and Sternberg [1980] a general scheme b.egan emerging in which semi-direct products arose by reduction from T*G by a subgroup. For example, a special case of their result shows that when T*so(3) is reduced by an s1 subgroup, corresponding to invariance under rotationsabout the direction of gravity for the heavy top, one automatically gets the Lie-Poisson structure on the dual of the Euclidean group. Some improvements in this theory were given by Ratiu and Van Moerbeke [1982] and Holmes and Marsden [1983]. The sharpest results however, were given by Marsden, Ratiu and Weinstein [1983], who also incorporated the aforementioned fluid and plasma examples into the same scheme.

In the present paper we take the point of view of Poisson manifolds and shall be as concrete as possible, using the heavy top and compressible flow as detailed motivating examples for the general theory. In Marsden, Ratiu and Weinstein [1983] we studied the role of symplectic reduction and deter-mined the symplectic leaves of the reduced spaces for T*G divided by an isotropy subgroup of a representation of G on a vector space V. These were shown to be symplectically diffeomorphic to the coadjoint orbits in the dual of the semi-direct product OJ- 1>< V, This provided a satisfactory ex-planation of why semi-direct products occur in so many examples. Indeed, T*G represents the basic Lagrangian phase space and reduction by the subgroup of symmetries represents the passage from Lagrangian to Eulerian or convective coordinates. In addition to the Poisson point of view, the new results in the present paper are:



REDUCTION AND HAMILTONIAN STRUCTURES 57

a) A demonstration is given (in §4.4) that a generalization of the Poisson map of Holm, Kupershmidt and Levermore [1983] can be directly constructed from the setting of ~larsden Ratiu and Weinstein [1983]. The basic idea is that by re-ducing T*(G x V) by V one can pass from a Poisson map of T*(G x V) +(OJ~ V)* to one from T*G x v* + (~ t>< v)*. In addition, we give both the right and left reductions--they are not trivially related. The generalization to allow V to be a non abelian group is not hard; for example, it is covered by Montgomery, Marsden and Ratiu's contribution to these proceedings, dealing with semi-direct product bundles.

b) A derivation is given (in §5.2) of the Hamiltonian structure for the heavy top in spatial coordinates (it is usually given in body coordinates). Here the moment of inertia tensor is a dynamic variable; cf. Guillemin and Sternberg [1980].

c) A derivation is given (in §5.4) of the Hamiltonian structure for the equations of compressible flow in convective ("body") coordinates (it is usually given in Eulerian ("spatial") coordinates.) Here the metric tensor is a dynamic variable. In a future paper this formulation will be connected with the results of Sima and Marsden [1983] on the Doyle-Eri cksen formula for the stress tensor (o = 2pCle/dg, where e is the internal energy and g is the metric tensor), which is closely related to the co-variant Hamiltonian formulation of elasticity (see also Marsden and Hughes [1983]).

The left-right duality which is emerging as a basic, yet usually over-looked, ingredient in the Lagrangian to Eulerian map is summarized as follows:

( Lagrangian ) re pres entation

Left tcoosl~ ~tcooslotioos

( body or convective) (spatial or. Eulerian) representation represenat1on

If the basic Lagrangian space one starts with is T*G, as is appropriate for

i) the free rigid body (G = S0{3)) ii) incompressible flow (G =volume preserving diffeomorphism)

or iii) the Poisson-Vlasov equation (G = canonical transformation), the picture specializes to




T*G left translat~on / ""right translation

implemented by ~ ~by JL

* * ~- bJ.+

implemented

where -?: is bj-* with + or - Lie-Poisson structures (this is reviewed briefly in §4.2 below) and JR and JL are the momentum maps associated to the right and left actions of G respectively.

If the basic Lagrangian space one starts with is the cotangent bundle * T (G x V) i.e. the basic configuration variables are G x V, then we get

a more detailed picture:

T*G x v* « PL T*(G X V)

PR > T*G x v*

'· l / JL J,

L

(f1xV)~ (19) x v):

(convective) (spatial)

Here, JR' JR, JL and JL are momentum maps for the left and right actions of G 1>< V on T*(G x V) and T*G x V*. These maps include, as special cases, Poisson maps found by Guillemin and Sternberg, Ratiu, Kupershmidt, and Holm, Kupershmidt and Levermore. The maps PL and PR are Poisson maps imple-menting the reduction by V; while PL just projects out V, PR involves a fiber translation by a differential (such maps play an important role in Guillemin and Sternberg [1980] and in Marsden and Weinstein [1982]). This asymmetry between left and right occurs because we chose G to act on V on the left.

The plan of the paper is as follows. In sections 2 and 3, concrete and detailed expositions of the Hamiltonian structure for the heavy top and compressible flow are given. Here things are done more or less by bare hands both to motivate and show the power of the abstract theory, presented in §4. In §5 we return to the examples to show how the theory works. There are other examples as well. See Marsden, Ratiu and Weinstein [1983] for MHD, multifluid plasmas and the Maxwell-Vlasov equations, Montgomery, Marsden and Ratiu [1984, and this volume] for the Yang-Mills-Vlasov equations, Marsden and Weinstein [1983] for incompressible flow, and Abarbanel, Holm, Marsden



REDUCTION AND HAMILTONIAN STRUCTURES

and Ratiu [1984] for stratified flow.

Acknowledgements. We thank Darryl Holm and Richard Montgomery for useful comments.

2. THE HEAVY TOP. 2.1 Configuration Space. A top is, by definition a rigid body moving about a fixed point in three dimensional space. A reference configuration B of

59

the body is the closure of an open set in IR 3 with piecewise smooth boundary. Points in B denoted ~ = (X1, x2, x3) E B are called material points and xi, i = 1,2,3 material coordinates. A configuration of B is a mapping n:B ~ IR 3 that has certain smoothness properties, is orientation preserving, and is invertible on its image. The points of the target space of n are called spatial points and are denoted by lower case letters .11. = (x1 ,x2,x3) IR 3; xi, i = 1 ,2,3, are called spatial coordinates. A motion of B is a time dependent family of configurations, written ~ = n(~,t) = nt(.!) or simply ~(~,t) or ~t(~). Spatial quantities are maps whose doman is IR3, i.e. they are functions of .1\.. They are lcwer case letters such as z (if scalar valued) or ~ (if vector valued). By composition with nt' spatial quantities become functions of the material points ~·

Dually, one can consider material quantities such as scalar maps Z:B~IR or vector maps ~:B~IR 3 . Then we can form spatial quantities by

. . z -1 z -1 compos1t1on: zt = tont and ~t = ~ont . In addition to the material and spatial coordinates, there is a third

set, the convected or body coordinates. These are the coordinates associated with a moving basis. Although these are defined in general (Marsden and Hughes [1983] p. 41) we shall first consider them in the context of a rigid body.

In §4,5 we shall see the following picture emerge of which the present discussion is a special case:

(Lagrangian or material coordinates)

T*G

left ation / ~ ;:;~; transl~ ~lation

~* ~* (body or convected (Eulerian or spatial

coordinates) coordinates)




Rigidity of the top means that the distances between points of the body are fixed as the body moves. This says that if the configuration ~(!:_,t)

represents the position of a particle that was at X when t = 0, we have

~(~.t) = A(t)~ i.e. Xi = A i_ ( t ) xj , 1· J. 1 2 3 J. , = , , , sum on J

( 2 .1)

where A(t) = (Aij(t)) is an orthogonal matrix. Since the motion is assumed to be at least continuous and A(O) is the identity matrix, it follows that det (A(t)) = l and thus A(t) E S0(3), the proper orthogonal group. Thus, the configuration space of the heavy top may be identified with S0(3).

* Consequently the phase space of the top is the cotangent bundle T (S0(3)), which will be described in 2.4.

Now we are ready to define convected, or body coordinates. Let ~l, ~ 2 •

~ 3 be an orthonormal basis relative to which material coordinates 1 = (X1 ,x2 ,X 3) are defined and ~l, ~ 2 • ~ 3 be an orthonormal basis asso-ciated to spatial coordinates. Let the time dependent basis ~l, ~ 2 , ~ 3 be defined by

~- = A( t)E. -1 -1

so ~- move attached to the body. The body coordinates of a vector in IR 3 -1

E IR 3 are its components relative to ~-. For '!. its spatial coo rdi nates -1 ' i related to its body coordinates ) by v are

vi = A i . vj J

where Aij is the matrix of A relative to ~i and ~i· Of course the components of a vector 'j_ relative to ~i are the same as the components . of Ay relative to ~i. In particular, the body coordinates of x are X1 •

2.2 Euler Angles are the traditional way to express the relationship between space and body coordinates, i.e. to parametrize S0(3). In what follows we shall adopt the conventions of Arnold [1978] and Goldstein [1980] which are different from those of Whittaker [1917].

One can pass from the spatial basis (~ 1 ,~ 2 .~ 3 ) to the body basis (~ 1 ,~ 2 ,b), by means of three consecutive counterclockwise rotati.ons per-formed in a specific order: first rotate by the angle ¢ around ~ 3 and denote the new position of ~l by ON (line of nodes), then rotate by the angle e around ON, and finally rotate by the angle 1)! around ~ 3 (see Fig. 1). Consequently 0 ~ ¢,1)! < 2n and 0 ~ 8 < n. Note that there is

a bijective map between the (¢,1)!,8) variables and S0(3). However, this




~3 ~3

Figure l

bijective map does not define a chart, since its differential vanishes, for example, at cp = 1)! = e = 0. The differential is non-zero for 0 < ¢ < 2n,

61

0 < ~ < 2n, 0 < e < n and on this domain, the Euler angles do form a chart. Explicitly this is given by (¢,~,e)~ A, where A is uniquely determined by ~ = A:!_ and has the rna tri x relative to ~i and ~i given by

A

[ cos• coso - cose sin¢ sin~ cos~ sin¢+ cose cos¢ sin~

= -s1n1jJ cos¢ - cose sin¢ cos~ -sin~ sin¢ + cose cos¢ coslji

sine sin¢ -sine cos¢

sine s i nlji l sine coslji

cose

(2.2)

With the aid of the chart given by Euler angles we induce a natural chart (cp,~,e, ¢,~,6) on the tangent bundle T(S0(3)) of the proper rotation group 50(3}. Then using a Legendre transformation given by a certain metric on 50(3} uniquely determined by the mass distribution of the top, we will define a mapping to the natural chart (cp,~,e, Pep• plji, Pel on the cotangent bundle T*(50(3)) which is the canonical phase space. This will be done in §2.4.

2.3 The Lie Algebra so(3) and Its Dual. In order to simplify the computations and identify the geometrical structure of the Hamiltonian of the heavy top, a summary of the Lie algebra .6o(3) and its dual are needed.

The proper rotation group 50(3) has as Lie algebra the 3 x 3 infini-tesimal rotation matrices, i.e. the space so(3) of 3 x 3 skew-symmetric



62 Jerrold E. t-'arsden, Tudor Ratiu, and Alan Weinstein

matrices; the bracket operation is the commutator of matrices. The Lie algebra Jo(3) is identified with IR3 by associating to the vector v = (vi,l,v 3) E IR3 , the matrix v E M(3) given by

Then we have the following identities:

A

u•v = u x v

1 AA

~·..': = - 2 Tr(~).

( 2. 3)

(2.4)

( 2. 5)

(2.6)

( 2. 7)

3 Moreover if A E 50(3) and '!._ E IR , then the adjoint action (conjugation) is given by

A A A -1 (Av) = Ad v := AvA - A- - (2.8)

Consequently, since the adjoint action is a Lie algebra homomorphism, for all

A E 50(3), ~·~ E IR3 we recover the vector algebra identity

In what follows we shall identify the dual inner product, i.e. mE Jo(3)* corresponds to for all v E IR 3• Th;n the coadjoint action of sented by-the usual action of 50(3) on IR 3,

Ad* m = Am -1- -A

Jo(3)* with IR 3

.!!1_ E IR 3 by ~('!._) 50(3) on Jo(3)*

i.e.

(2.9)

by the = .!!!_•'!._, is repre-

( 2.1 0)

2.4 The Hamiltonian. If .X. E B is a point of the body, then the trajectory followed by .X in space is ~(t) = A(t)!, where A(t) E 50(3). The material or Lagrangian velocity !(!,t) is defined by

(2.11)




the spatial or Eulerian velocity !(!,t) by

!(!,tl = a~(~,t)/at = 1(~,t) = A(t)A(tf 1 ~. and the body or convective velocity ~(~t) by

(2.12)

~(~,t) = -a~(_~,t);at = A(tf1A(t)A(tf\

= A(tf 1 A(t)~ = A(t)- 1 1(~,t) = A(tf\:.(~,t). (2.13)

Let p0{20 denote the density of the body in the reference configura-tion. Then the kinetic energy at time t is, by (2.11), (2.12), (2.13), and the invariance of the Euclidean norm under S0(3),

1 I -1 2 3 = 2 P0(A(t) ~~l~(!,t)ll d! (spatial) A( t)B

(2.14)

(2.14)

(2.16)

Differentiating A(t)TA(t) =Identity and A(t) A(t)T = Identity,it follows that both A(tf1A(t) and A(t)A(t)-l are skew-symmetric. Moreover, by (2.12), (2.13), (2.5) and the classical definition of angular velocity, it follows that the vectors ~S(t) and ~ 8 (t) in IR 3 defined by

~s(t) = A(t)A(t)-1

~ 8 (t) = A(t)- 1A(t)

(2.17)

(2.18)

are the spatial and body angular velocities of the top. Note that ~S(t) = A(t)~(t), or as matrices, ~S =AdA~ = A~ 8 A- 1 . Using the Euler angle parametrization (2.2) of S0(3), (2.17), and (2.18), !!l.s and !!l.B have the following expressions

[8 cos ¢+ 1jJ sin ¢sin 6]

.\!ls= esin¢:~~oscpsin6, ¢ + 1jJ cos 6

[ e cos 1jJ +¢sin 1jJ sin 66] ~B = -~ sin 1jJ + ~ cos 1jJ sin

¢ cos e + w (2.19)

Due to the fact that in (2.14) and (2.16), Po is independent of time, the kinetic energy can be expressed in a simple manner in the material and



64 Jerrold E. Marsden, Tudor Ratiu and Alan Weinstein

reference configurations. We have by (2.16) and (2.5),

( 2. 20)

Using (2.19), the kinetic energy of the body is a function of (¢,l/J,8,~,~.e)

or of ~· To give it a more familiar expression, introduce the following 3 inner product on IR ,

( 2. 21 )

completely determined by the density p 0 (_~) of the body. Then (2.20) becomes

(2.22)

Now define the linear isomorphism I:IR 3 -+ IR 3 by I~·£ =~~.E_}> for 3 all ~.E_ E IR ; this is possible and uniquely determines I, since both

the dot product and~,;,:. are nondegenerate bilinear forms.t It is clear that I is symmetric with respect to the dot product and is positive. To gain a physical interpretation of I we compute its matrix. Let (~ 1 .~ 2 .~ 3 )

be an orthonormal basis for material coordinates. Thus,

(2.23)

j

which are the expressions of the matrix of the inertia tensor from classical mechanics. Thus I represents the inertia tensor. Since it is symmetric, it can be diagonalized; the basis in which it is diagonal is a principal axis body frame and the diagonal elements r1 , 12, I 3 are the principal moments_ of inertia of the rigid body. In what follows we work in a principal axis body frame.

To get from (2.22) a function defined on 60(3)* ; IR 3 we must take into account that 6o(3)* and IR 3 are identified by the dot product and not by ~,;,:.. Consequently, the linear functional ~!%• •}> on 6o(3) ; IR 3 is identified with I~ 8 := _'!! E60(3)*; IR 3 since _'!!·~ =~~s·~> for all a E IR 3. Hence (2.22) becomes, for I = diag(Il'r 2,I 3),

1 -1 1 [m~ m~ m~] K(ffi) = 2 .!!!.·I .'!! = 2 I +I +I l 2 3

(2.24)

tAssuming the rigid body is not concentrated on a line.




which represents the expression of K on -6o(3)*; note that m = I!% is the angular momentum in the body frame.

By the second formula in (2.19) and the definition of m for I = diag(I 1,I 2,I 3),it follows that

e sin

65

[I 1 I 0 s'" .!!! = I 2 (~ sin e cos

• d coq)] 1jJ - e sin 1J! ( 2. 25)

I3(¢ cos e + ~)

This expresses ..m. in terms of coordinates on T(S0(3)). Since T(S0(3)) and T*(S0(3}) are to be identified by the metric defined as the left translate at every point of ~, ~, the canonically conjugate variables ( p¢, pljJ' p8) to (¢,1)!,8) are given by the Legendre transformation p¢ = ClK/Cl~, p1jl = ClK/Cl~, p8 = ClK/ClB of the kinetic energy on T(S0(3)) which is obtained by plugging (2.25) into (2.24). We get the standard formulas

p ¢ = I 1 ( ~ sin e sin 1J! + 8 cos 1J!) sin e sin 1)!

+ I 2(¢ sine cos 1)!- 8 sin 1)!)sin e cos 1)!+ I 3 (~ sin 8+ ~)cos e

P 1J! = I 3 ( ¢ cos e + ~)

Pe = I1(¢ sin 8 sin 1jJ + e cos ljJ)cos 1)!- I 2 (~ sin 8 cos 1)!- e sin ljJ) sin 1jJ

whence by (2.25),

[[(p¢ - pljJ cos

.!!! = [(p¢- pljJ cos

p1jl

e)sin 1)!+ p8 sin e cos 1)!]/sin e ] e) cos 1)!- p8 sin e sin ljJ]/si n e

(2.26)

( 2. 27)

and so by (2.24) we get the coordinate expression of the kinetic energy in the material picture to be

_ 1 { [(p<P- pl/J cos e)sin tj; + p8 sine cos w] 2 K(¢,tj;,e,p~,P.,,'Pel - 2 2

'~' "' I 1 sin e

( 2. 28)

The potential energy V for a heavy top is determined by the height of the center of mass over a horizontal plane in the spatial coordinate system.



66 Jerrold E. Marsden, Tudor Ratiu and Alan Weinstein

Let Q._x denote the vector determining the center of mass in the reference configuration (i.e. the body frame at t = 0), where ~ is a unit vector along the straight line segment of length !1, connecting the fixed point with

the center of mass. Thus, if M = J 3 d~(x) is the total mass of the body, IR

g is the gravitational acceleration, and ~ denotes the unit vector along the spatial Oz axis, the potential energy at time t is

-1 V(t) = Mg~·A(t)Q._x = MgM ~·_x = MgQ.r_-x = MgQ.~·~

-1 where }_=A ~ and ~=AX. Consequently,

V = MgQ.~ _ _.A_x = MgQ.~·:>..

= MgQ.r_x

(Lagrangian or material) (Eulerian or spatial) (convective or body)

Thus, by (2.24) the Hamiltonian has the following expressions

2 1 3 m.

H(f!l_.y) =- L -I~ + Mg!l,y 3 2 j=l J

(body)

(2.29) ( 2. 30) ( 2. 31 )

(2.32)

H 1 {·[(p<P-p\jJ cose)sinlj!+pesinecoslj!] 2 +

2 r1 sin2e

[(p<P- p\jJ cose)coslj!- Pe sine sinl)Ji

12 sin2e

p2 } + ~ + Mg£ case (material) (2.33)

The table at the end of this subsection (w.hich appears in Holmes and Marsden [1983]) summarizes and completes the relations between m, y, <P, lji, 8, <P, iJ.I,

e, Pf piJ.I, Pe· We close with a study of the invariance properties on H on T*(s0(3)).

By (2.7) (2.18), (2.24), and (2.29), the Hamiltonian in the material configura-tion equals

1 -1·-l· H =- 4 Tr(IA AA A) + MgQ.k·A_x. (2.34)

Consequently, if B is a constant matrix in S0(3) and we replace A by BA (left translation), it is easily seen that the kinetic energy does not depend on B, i.e. it is invariant under the maps A I+ BA. The potential energy however is only invariant if Bk = k, i.e. under rotations about the spatial Ox 3 axis. The corresponding ~ons~rved quantity is, by Hamil ton's




equation written in terms of (~.~.e. p~, p~, Pel• p~ = ~·r· since p~ = -aH/3~ = 0 by (2.33). There is one more conserved quantity in body coordin-ates namely llrll 2 = 1. The importance of ~·r. and llr_ll 2 will become apparent in 2.6 and in §4.

Finally, let us note that H depends on the parameter MgR.!. What happens if this parameter is changed will be explained in Sections 4 and 5.

67

m1 = [(p<P-p~ cos e) sin~+ p8 sine cos ~]/sine= I 1 (~ sine sin~ +e cos~)

m2 = [{p~-p~ cos e) cos ~- Pe sin e sin ~]/sin e = I2(¢ sin e cos ~- 6 sin ~)

m3 = p~ = I 3 (~ cose + ~)

yl = sin e sin ~

y2 =sin e cos ~

y3 = cos e

p~=~·.:r= I 1 (~ sin esin ~+ 8cos ~)sin 8sin ~+ I 2 (~sin ecos ~-

- e sin ~) sin e cos ~+ I 3 (~ sin 8+ ~)cos e

P~ = m3 = I 3 (~cos e+ ~)

'2 .. Pe = (y2m1-y1m2);Jl-Y3 = I 1 (~sln e sin ~ + e cos ~)cos ~

- I 2 ( ¢ s i n 8 cos ~- 8 sin ~) sin ~

2.5 Equations of M:Jtion. Hamilton's canonical equations

· aH . aH · aH ~ = ap' ~ = ap' e =-

<P ~ ape (2.35)

· aH aH · aH P ~ = - acp' P ~ = - a~· Pe =as

in a chart of T*(s0(3)) with H given by (2.33) become after a 1 engthy com-putation in which ¢,~,8, p<P' pl/J, p8 are replaced by (~.r), the Euler-Poisson equations



68 Jerrold E. Marsden, Tudor Ratiu, and Alan Weinstein l r;

( 2. 36)

m3y2 m2y3 yl = r;-- ~

mly3 m3yl y2 = 11-- r;-

m2yl mly2 Y=-----

3 I 2 I l

Note that these equations are on M( 3) * x IR 3 ;;; IR 3 x IR 3 whereas the canonical equations were on T*so(3)). This is an instance of a general fact that will be explained in section 4.

2.6 Poisson Bracket in Body Coordinates. For F, G: T*(S0(3))->- IR, i.e., F, G are functions of (¢,\jl,e, pep' p\jl, Pel' the canonical Poisson bracket is given by

{F,G}("','I',e, P"'sP,J,•Pel = ~ ~- _E <lG + ~ <lG - -~f. <lG + ~ ~ - _££__ ~ "' "' "' "' acp ap cp ap cp d¢ <l\jl a p\jl ap\jl <l\jl ae ape ape ae

This bracket becomes after the change of variables

{F,G}(m,y) = -m·(\7 F x 1.7 G) - y•(\7 F x 1.7 G + 1.7 F x 1.7 G), -- - ~ --.!!! - --.!!! -~ ~ --.!!!

where 1.7 and 1.7 denote the gradients with respect to m and y. -m -y - -(2.39) defines a oilinear, skew-symmetric operation on functions of A computation shows that it also satisfies the Jacobi identity, i.e. is a Poisson bracket on IR 3 x IR 3 . Moreover

F = {F,H}

(2.37)

( 2. 38)

(2.39)

Clearly (~.~). ( 2. 39)

with H given by (2.32) yields, for F equal to m., ..r., i = l ,2,3, the 1 'i

equations of motion (2.36). Note that whereas (2.37) is non-degenerate, i.e., { F,G }= 0 for all G implies F =constant, the bracket in (2.39) is degen-erate. It is in fact easy to see that




{F,G}(~,j:) = 0

if G(~,.:y) = ¢( llyll 2) or G(~,_r) = '¥(~ ·_r) for a rbi tra ry rea 1 va 1 ued functions ¢, '¥ of a real variable. Note that unlike the case of the free rigid body where the bracket consists only of the first term of (2.39), an arbitrary function of llm11 2 does not commute with every function of m and y. Recall also that ~·r-and llyll 2 are the only conserved quantities-for the-heavy top, if no otfler symmetries are present. The geometric reason of (2.39.) and tile existence of the above two functions will be given in Section 4.

In Section 5 we shall discuss the equations in the spatial picture.

3. IDEAL COMPRESSIBLE ADIABATIC FLUIDS 3.1 Configuration Space. Let rl be a compact region in IR 3 with smooth boundary an, filled with a moving fluid free of external forces. A con-figuration of the fluid is chosen and called the reference or Lagrangian configuration; its points, called material or Lagrangian points, are denoted by ~=(X ,x2,x3); Xi are referred to as material or Lagrangian coordinates. A configuration of the fluid is an orientation preserving diffeomorphism n of rl with certain smoothness properties.* We shall not be specific here about the correct choices of function spaces and refer the reader to Ebin and Marsden [1970] and Marsden [1976] where this is discussed in great detail for i ncompressi bl e fluids; obvious changes have to be made for the compressible case. The manifold rl, thought of as the target space of a configuration n, i.e. a configuration of the fluid at a different time, is called the ~patial or Eulerian configuration, whose points, called spatial or Eulerian points, are denoted by lower case letters x. A motion of the fluid is a time dependent family of diffeomorphisms, written

or simply !(!,t). Given the mass density p0(X.) and entropy o 0 (_~) of the fluid in

the reference configuration, both functions of !• denoting by Jn (D the t

* In principle, one can develop the theory of fluids as we did for rigid bodies in Section 3.1, considering the fluid and the containing space as two different manifolds. The configuration space is then a space of mappings from the fluid manifold to the container manifold, and it becomes a group only when a reference configuration is chosen. Although this viewpoint is actually necessary in elasticity theory, we have used the more conventional approach here, in which the fluid particles are identified with their positions in space at t = 0.




Jacobian determinant d~/d~ of the motion nt at !. we shall see in 3.3 that the mass and entropy density satisfy

Consequently, the Eulerian mass density and entropy p and o are completely determined by the motion and Po and o 0 respectively. Hence, the con-figuration space of compressible fluid flow with a given mass and entropy density in the reference configuration is the group of diffeomorphisms Di ff(S1) of n. Consequently the phase space is the cotangent bundle T*(oiff(n)).

There are two problems with this approach. First, the configuration space requires a choice of Po and o0 . But Po and o0 have to be changed in accordance with the choices of initial conditions. How this is done will be explained abstractly in Section 4. The change of p0 and o0 is akin to the change of the parameter Mg£.!5. in the heavy top problem. We shall think of Po and o0 (exactly as we did of Mg£l in the previous section) as a parameter. The second prob 1 em is much more serious. We think of the fluid as moving nicely in n, at any time filling n. However, under certain conditions, shocks and cavitation can occur. The present approach cannot deal with such problems and represents a serious limitation.

For a motion ~ = nt (~) one defines three ve 1 oci ties:

a) the material or Lagrangian velocity

~(!,t) = _'{_t(!) = ;Jn(!_,t);Jt;

b) the spatial or Eulerian velocity

c) the convective or body velocity

( 3. 2)

( 3. 3)

(3.4)

-1 ) Taking the derivative of (ntont C?Sl = ~ and denoting by Txnt the Jacobian matrix dx/dX of nt at ~. we get

( 3. 5)

i.e.

Note that both _'{_t and ~t are tangent to n at x = nt(~). This




means that ~t is a time dependent vector field on ~. On the other hand, tangency of ;!_/~) and nt(_~) says that ;{_t is a vector field over nt on ~. i.e. ;{_t is a map from ~ to the tangent bundle m such that ;{_t(X) is tangent to 11 not at ~. but at ntC~J. Finally, notice that ~t is a tangent vector at ~. i.e. ~t is also a time dependent vector field on 11.

We summarize the relations between ;{_, ~· and ~ in the following commutative diagram, in which vertical arrows mean vector fields.

Figure 2

Let Z(~, t) be a material quantity, i.e. a given function of (~, t)

71

and let z(~,t) = Z(~,t) be the same quantity expressed in spatial coordinates. Then by the chain rule

az = az + (_v·_")z, i e az _ az + az j at at v • • at - at -;;;r v ( 3.6)

In particular, if Z represents different components of a vector !:_, we have

( 3. 7)

The right hand side of (3.6) or of (3.7) is called the material derivative of z or !. and is usually denoted by z = Dz/Dt or i = 0!_/Dt; it represents the time-derivative of z holding the material point ~ fixed. As opposed to that, the usual partial derivative az;at represents the time-derivative of z holding the spatial point ~ fixed. One can develop analogous formulas for the convective velocity. We will return to this point in §5.

We shall determine the phase space T*(oiff(l1)) and elementary Lie group operations on Diff(~). on its Lie algebra, and its dual.



72 Jerrold E. Marsden, Tudor Ratiu, anc Alan Weinstein

3.2 The Lie Group Diff(st). There are two ways in which Diff(st) can be made into a Lie group. The most obvious one is to consider only c"' diffeomor-phisms. It turns out that in this way Diff(st) becomes a Frechet manifold, i.e. its model space is a locally convex,Hausdorff,complete vector space. Composition of diffeomorphisms and taking the inverse are smooth operations, so Diff(st) becomes a Frechet Lie group (see e.g. Leslie [1967], and Omori [1975]). The main drawback of this approach is that in Frechet spaces special hypo theses are needed for in verse function theorems to hold; the same is true of existence and uniqueness theorems for integral curves of differ-entia 1 equations. Use of the Nash-Moser theory is not necessary.

The second approach is to use diffeomorphisms of Sobolev or Holder class. It turns out that if the Sobolev class Ws,p or Holder class Ck+a is high enough so that such diffeomorphisms are at least c1, then they form a c"" Banach manifold and one has the usual existence and uniqueness theorems for solutions of differential equations. Unfortunately only right translation is smooth whereas left translation and taking inverses are only continuous. Thus Ws,p_ Diff(st) (or Ck+a_Diff(st)) is now a topological group which is a Banach mani-fold on which right translation is smooth. One may now make Diff(st) into a "Lie" group by taking the inverse limit as the differentiability class goes to oo (Ebin and Marsden [1970], Omori [1975]).

We next determine the tangent space Til( Di ff(st)) of Di ff(st) at ll. Let t 1-+ llt be a smooth curve with llo = 11. Then (d11t/dt) lt=O is, by defi-nition, a tangent vector at 11 to Diff(Si). If ~En, then t 1-+ llt(~)

is a smooth curve in n through 11(D and thus

dl1t (~) I _d_t_ _ E Til( X)st'

t-0 -

where Tll(X)n is the tangent space to st at ll(~). Consequently we have a map ~ E-Q »- (dl1t(D/dt) lt=O E T11 (X)n' i.e. (dll/dt) lt=O is a vector field over ll· Thus -

T (Diff(st)) = {V :n + TstJV (X) E T (X)Q}. ll -11 -11 - 11 -( 3 .8)

In coordinates, if x = 11(X), V (,~) = Vi(X)(a;axi). - - -11 -In particular, if e denotes the identity map of n, Te(Diff(n)) =*(n),

the Lie algebra of vector fields on n. It turns out that the Lie algebra bracket of J(n) is minus the usual Lie bracket of vector fields, i.e. [U,V]i = Vj(aui;axj)- Uj(avi;axj). Thus the Lie algebra of Diff(Q) may be iden-tified with ~(n), with the negative of the usual Lie algebra structure.

To determine the dual of X(n) and the cotangent bundle of Di ff(Q), we



REDUCTION AND HAMIL TONI AN STRUCTURES 73

take a geometric point of view. Instead of considering the functional analytic dual of all linear continuous functionals on¥(~). we will be content to find anotner vector space')((~)* and a weakly non-degenerate pairing

) ::J((~)* X X(~) -+ IR;

this means that ( , > is a bilinear mapping such that if (_!:!,_'{_ > = 0 for all _'{_ E J(~), tnen !:! = 0. Clearly l(~)* is a subspace of the functional analytic dual. With this definitior., it is easy to see that X(~)* consists of all one-form densities on ~. i.e.

(3.9)

The notation in (3.9) is the standard one: Ai (~) denotes the set of all exterior i-forms on n and IA3(n) I denotes the densities on ~. Thus a one-form density is of the form ad3x with a a one-form on n, so locally it is (a;t_~) dXi) d3x_. The pai~ng-( , > -between )!:(~)* and ~(~) is <~d 3 ~,_'{_ > = J~ ~(_'{_)(Dd 3 ! or in local coordinates, J~ a;(_!)Vi (Dd3_!.

Finally, in view of (3.9), T*(Diff(~)) consists of all one-form densities over n. i.e.,

This means that ~ = t. d3_!, where ~ is a one-form over n on n, i.e. * n 41 ni 3 i

.S./~) E \(.X.)~' i Locally, ~n = (t;:i (_!)dx )d f, where (x ) = ~ = n(_!) and t;:n(_!) = t;:i (Ddx . We shall denote the action of one-forms _s_ over n on vector fields 1 over n by t;; (V ); the result is a function of X which n . ~~ -locally equals s;V1 • The pairing ( , } between T~(Diff(~)) and T (Diff(rl)) is given by (a, V >=Jr. t;; (V )(X)d3X if a = t;; d3X; locally this\as -n-n "4l_n_- -n ~-

' ( i 3 the expression J~ r;;i D~. (Dd ~·

Left and right translations are defined by

L :Diff(~)-+ Diff(Q), L (¢)=no¢ n n R :Diff(n) -+ Diff(n), R (¢) = </J 0 n n n

for n.<P E Diff(n). Both are diffeomorphisms of the Lie group Diff(n). It is easy to see that their derivatives have the following expressions:

{ 3 .ll) and

( 3 .12)




for _:{<P E Tq/Diff(rl)). The physical interpretation of these formulas is the following. Think of <P as a relabelling or rearrangement of the particles i·n r2 and of n as a motion. Then ( 3.ll) says that the material derivative of tne motion n followed by the relabelling cp equals coordinates, if ¢(D = '!_ and n('!_) = 1..• then !cp(9

~no!cp· I~ local v1 (D( a;av1 ) and

(Tri o_:{<P) \~) = ~ (Y._)Vj(_~) -;. ()y ()y

(3.13)

On the other hand, (3.12) says that the material derivative of the relabelling cp followed by the motion n equals_ ~on. ~n local coordinates, if n(~) =~. ¢(~) =x_, then !cp(~) = V 1 (~){d/Cly 1 ) and

C~cp on) i (~l = (vi onl (~l ( a;ayi l. ( 3.14)

Simply put, left translation by n transforms Vcp(.~). a vector at cp(~) to a vector at n(¢(!)) whereas right translation merely changes the argument from !_ to n(K).

By (3.12), the derivative of right translation is again right transla-tion, so Rn is C00

• However, if n and cp are diffeomorphisms of a given finite Sobolev class, Tn looses one derivative. (This is basically the reason why left translation is only continuous in Ws,p_Diff(rl).) In C00-0iff(n) however (with differentiability suitably interpreted), left trans-lation is C00

•

As an application, note that by (3.3) and (3.5), the material velocity ~t is the right translate of the spatial velocity ..!'.t and the left translate of the convective velocity ~t·

If _:{ E,;1<(Q), a diffeomorphism n E Diff(rl) acts on ~ by the adjoint action, the analogue of conjugation for matrices. The definition combined with (3.ll) and (3.12) gives

AdV:=T(LoR 1)V=T 1Ln(TeR 1(V)) n- e n n- - n- n- -

i.e. the adjoint action of n on V is the push-forward of vector fields:

Ad V = n*V . (3.T5) n- -For example, by (3.3) and (3.5), -""-t = Adn..!:t• which is similar to the

t formula which relates ~B to .<!ls in the previous section. Finally, we compute the coadjoint Ad~-la action of n on _s: E~rl)*. By the change of variables formula we have




here a•V in the integrand signifies the pairing between one-form densities and vector fields so that 9:_•:!_ is a density on n. Thus

75

* Ad_ 1 ~=n*~; (3.16) n

n*~ is the push-forward of the one-form density 9:_; the push-forward operates separately on the one-form and the density.

3.3 Equations of Motion. We review the derivation of the equations of motion in Eulerian coordinates from four principles: conservation of mass, entropy, and momentum. Conservation of energy will follow by imposing the adiabatic equation of state.

a) The principle of conservation of mass stipulates that mass can be neither created or destroyed, i.e.

for all compact W with non-empty interior having smooth boundary. Changing variables, this becomes

(3.17)

where J( nt) = jd~/ d~! is the Jacob ian determinant of nt and n; is pull-back of forms or functions as the case may be. Using the relation between Lie deri va ti ves and flows, ( 3.1 7) is equi va 1 ent to the continuity equation

-¥t + d i v ( Pt:!) = 0 . (3.18)

b) By the principle of conservation of entropy, the heat content of the fluid cannot be altered, i.e.

for all compact W with non-empty interior having smooth boundary. By a change of variables this becomes

and by (3.17) we get




or <la + v•'Va = O· <lt -- t ' (3.19)

the second relation follows by taking the time derivative of the first. The last relation says that no heat is exchanged across flow-lines.

c) Balance of momentum is described by Newton's second law: the rate of change of momentum of a portion of the fluid equals the total force applied to it. Since we assume that no external forces are present, the only forces acting on the fluid are forces of stress. The assumption of an ideal fluid means that the force of stress per unit area exerted across a surface element at ~· with outward unit normal .!! at time t, is -p(~,tl.!! for some function p(~,t) called the pressure. With this hypothesis, the balance of momentum becomes Euler's equations of motion:

1 d~ + (~·yl~ = -. p ~·p av

(3.2

with the boundary condition ~ parallel to an (no friction exists between fluid and boundary) and the initial condition ~(~,0) = ~(~) on n.

The proof of conservation of energy is standard. The kinetic energy of the fluid is ~In p(~)ll~(~)ll 2 d3.!S.· The assumption of an adiabatic fluid

means that the internal energy of the fluid is Jn p(~)w(p(~), a(~))d 3 ~ with tlie equation of state p(~) = p(~) 2 (aw/<lp)(~) satisfying <lw/<lp > 0. In the next computati~n the following two vector identities will be needed:

(11 v11 2) (~·.'::_)~ = ~\T + ~ x :!_, where ~ = curl ~· the vorticity

2(w + pClw/aal = 2P/p + (aw/aa)yo.

We have by (3.18), (3.19), and (3.20) 2

~tc:!_~ + pw(p,a)) = -div(p~)G 11~11 2 + w(p,o) + p ~;]- P:!_·[(v·'V)~ + ~ _l?_p +~~:?_a]

= -div(p~H 11v11 2 + w + p ~;]) Consequently, the total energy

H(~,p,a) = ~ Jn p(~)ll~ll 2 d 3 ~ +In p(~)w(p(~), a(~)) d 3 ~, which represents the Hamiltonian of the system, is conserved.

( 3. 21)




The physical problem to be solved now consists of the continuity equa-tion (3.18), entropy convection (3.19),and Euler's equations (3.20) with p = p2'?!-H/'ap, where the internal energy density w(p,o) is a known function; the boundary condition says that:!... is tangent to an and the initial condi-tion is :!_(x,O) = :!...oC~), :!...o a given vector field on n.

Recall that 'ap/'ap is the square of the sound speed, so that ap;ap > 0 represents a very reasonable physical condition. We also mention that 'ap/Clp > 0 is exactly the condition needed to prove local existence and uni'queness of so 1 utions.

3.4 Hamiltonian in Lagrangian Coordinates. Theequations of motion just described are not on T*(Diff(n)) which is the phase space of the problem. To describe the dynamics in T*(Diff(n)) using Hamilton's equations, the Hamil toni an ( 3. 21 ) must be expressed on T* ( Diff( n)), i .e. in materia 1 coordinates.

We start with the potential energy. Perform the change of variables ~ = nt(!) in the potential energy and use (3.17) to get

77

(3.22)

The right hand side is a function of nt and hence definedon Diff(n) so that by lifting we get a function on T*(Diff(n)).

To express the kinetic energy as a function on the cotangent bundle, we need first its expression in terms of the material velocity. This is accomplished by performing the same change of variables ~ = nt(!). We have by (3.3) and (3.17)

~ Jn p(~)ll:!...t(~)H2d3~ = ~ Jn Po(!)llyt(~)ll2d3~

But 1t E Tnt(Diff(n)) so that (3.23) represents the expression of the

kinetic energy on the tangent bundle. Note that the mapping

for V , W E T (Diff(n)) and the dot in the integrand signifying the 11 11 n

(3.23)

(3.24)

metric on n (in our case the usual dot product), defines a weak Riemannian metric on Diff(n) and (3.23) is its kinetic energy.

In finite dimensions, a metric on a manifold induces a bundle metric on the cotangent bundle as we have seen in Section 2. In infinite dimensions,




as in the present case, this bundle metric does not exist in general and in examples it must be constructed explicitly. Let a , B E T*(Diff(Q)), -n -n n i.e. a = ~ d3X, Bn = r;: d3X with ~ and r;: one-forms over n. Con--n -n - - 3 n - -n 3 -n sequently, a !(P0 d X)=~ !Po B /(Pod X)=s !Po are one-forms over n, -n - -n n -n so evaluated at ~ they are elements of T~(X)~"L But n is a finite dimensional Riemannian manifold (with the EucTidean metric in our case) so to every one-. form at n(~) there exists a unique vee tor at n(.X) associ a ted by the metric. Explicitly, if u ET n, the one-form u'b ET*n is defined

'b -x ~ -~ ~ by .!!..!(\!~) = ~x-~x for all ~x ET~n. In this way, the index lowering action

b:Tn + T*n is a bundle isomorphism. The inverse of 'b is denoted by #:T*n + TQ and is called the index raising action. In coordinates, if g = (g .. ) is the metric and (gij) is the inverse matrix of (g .. ), we

1 J . . . 1 J ltave for u = u1 (a;ax1 ), a= a.dx\

- - 1

'b j i # . . . .!!. giju dx, Q = g1 Jaj(a;ax1 ).

Now define the bundle metric on T*(Diff(Q)) by

(_~n·~n) = Jn Po(~)in(!)·!'!_n(~)d3K ( 3. 25)

for .Y_n = (~n;p 0 d 3 D#, !'!_n = (~n;p 0 d 3 X)# E Tn(Diff(n)). Denote by 11·11 the bundle norm defined by the metric (3.25) and let

be the materialmomentum density of the fluid. With this notation, (3.23) becomes liM 11 2;2 and so by (3.22) the expression of the Hamiltonian on * 11 T (Diff(Q)) becomes

(3.26)

( 3. 27)

We want to investigate the symmetry properties of H. We shall prove that H is right invariant under the subgroup

( 3. 28)

For the potential energy this is easily seen, for if one replaces n by noljl with ¢ as in (3.28), both arguments of w do not change. To right translate M by ¢ means to compute the dual map of (3.12). Let ljJ be an arbitrary -n diffeomorphism and M = ~ d3x. By a change of variables, we have for any ljJ -n -n -




Consequently, if

M = !; d3x, then -n -n -

Thus, if Mn = p~~d 3 ! and cp satisfies (3.28), then T* 1RcpC!:!n)

79

(3.29)

-1 'b 3 n"cp- ( ) p 0 (_~nocp ) d _l, so that by (3.25), a change of variables, and 3.29 we have

liT* -1Rcp(Mn)ll2 =I Po{_~)ll:!_n(cp-l(X)II2 d3X. nocp n

=In Po(tP(!)II:!_n(Y)112JcpC0d3! = Jn Po(!)ll:!_n(!)D2 d3!

= 11~11 2 •

i.e. the kinetic energy is invariant by right translations with diffeomorphisms of the form (3.28).

3.5 Poisson Bracket in Eulerian (Spatial) Coordinates. The dynamics of the Hamiltonian (3.27) on T*(Diff(n)) is equivalent to

F = {F,H} ( 3. 29)

* for F an arbitrary function on T (Diff(n)) and { , } the canonical Poisson bracket of the cotangent bundle. If ~ is a function space on n modelling the manifold Diff(n), then I x_g_* models T*(Diff(n)); the dual E* has to be taken in the same geometric manner as we discussed in 3.3. If .!! E E,

* y E~ , the Poisson bracket (3.29) is given by

{F,G}(n,v) = J (oF oG _ _§£ oG) d3X - - on 0\! 0\) on -r.. - - - -( 3. 30)

* where the functional derivatives oF/o_!!_ E ~ and oF/ov E E are defined by




where D F, D F denote the Frechet deri va ti ves of F with respect to n n v -and v. -By (~.26}, (3.3), (3.17}, and (3.29}, the expression of M in - -n Eulerian coordinates ! = n(~) is

~n(~) = Po(X}.Y_n(~)'bd3~ = pO(~)~(!)'b J -1 (_~_)d3X. n

where the quantity

p(!)v(!)bd 3! = T;R _ 1 (~l(D, n

M(x) = p(x)v(x)bd 3x = T*R (M )(X) (3.31) --- - - - e n n -is called the Eulerian momentum density of the fluid. Consider the map

M (X)~ (M(x),p(x)d 3x, o(x)) -n- -- - - - (3.32)

from Lagrangian to Eulerian coordinates, where R is given by (3.31) and

( -1 -1 P = J _1 p0an ); o = o0an . (3.33) n

Note that (3.33) is simply a rewriting of (3.17). Then a computation shows that the bracket (3.30) via the change of variables (3.33) becomes

{F,G}(M,p) = J M· ~(o~·~) o~- (o~. ~) o~J d3~ ~ C oM oM oM oM

+ Jr P[o~·(J.. oF)_ o~.(~ oG)] d3~ ~ oM 8p oM op

(3,34)

+ where M(~) = p(!) ~(!) is identified with ~ in (3.31). The computation that transforms the bracket (3.30) via (3.31), (3.32), (3.33) to (3.34) is tedious; see Kaufmann's lecture in these proceedings for a different example where such a computation is carried out. An even longer computation shows that (3.34) which is bilinear and skew-symmetric, also satisfies the Jacobi identity. In §4 we shall give an abstract theorem which includes these



REDUCTION AND HAMILTON IAN STRUCTURES

results and allows one to efficiently bypass such computations yet obtain the

correct answers. +

The equations of motion (3.18), (3.19), (3.20)in terms of M, Panda can be obtained from (3.34) and the dynamics F = {F,H} by taking for

F the functions Jn p(~) d 3 ~, Jn P(~) a(~) d 3 ~, Jn Mi (~) d 3 ~. i = 1 ,2,3. Also, note that the map (3.32) is defined on T*(Diff(n)) with values

in '}t*(n) x F*(n) x F(n), where l*(n) denotes the one-form densities on n in accordance with (3.9), F(n) denotes the space of smooth functions on n, and F*(n) denotes the geometric dual of F(n), the densities on n. The pairing between F(n) and F*(n) is integration of the product.

If ~=curly denotes the Eulerian vorticity, the Eulerian potential vorticity is defined by

81

n: = ~·Y.._a/ p. ( 3.35)

From the equations of motion, it is easy to see that a and n are conserved. In fact, a computation shows that any functional on F*(n) x F*(n) x F(n) commutes (using the bracket (3.34)) with

F(a,n) = Jn p(~)(a(~) ,n(~)d 3 ~, (3.36)

where <1> is an arbitrary real valued smooth function of two real variables. Consequently, (3.34) is a degenerate bracket, unlike the canonical bracket (3.30). The significance of the functionals F will be explained in the next section.

4. MECHANICAL SYSTEM ON DUALS OF SEMIDIRECT PRODUCT LIE ALGEBRAS 4.1 Poisson tlanifolds and Momentum Maps. Throughout this section we employ the following standard notations and conventions. For a smooth manifold P, P(P) and ~(P) denote the ring of functions and the Lie algebra of vector fields on P respectively. The Lie algebra bracket of ~(P) is minus the usual Lie bracket for vector fields i.e. minus the bracket given by

( 4.1)

A Lie group G is a smooth manifold which is a group in which multiplication and taking inverses are smooth maps. The tangent space TeG at the identity e E G has a bracket operation obtained in the following way. For ~.nE TeG, one defines vector fields Xr-(g) = T L (~).X (g)= T L (n), where

-s e g -n e g L :G + G, Lg(h) = gh is left translation and T L :TeG + T G is the g e g g derivative of Lg, a linear map from T G to the tangent space T G to e g G at g. Then [~.nJ = [~,~](e). With this bracket, TeG becomes a Lie




algebra, called the left Lie algebra of G, or simply Lie algebra of G, and is denoted by bJ.. Of course the same construction can be performed with right translations Rg(h) = hg, and again a Lie algebra structure on TeG would result. The latter structure is anti-isomorphic to~· i.e. its bracket has the opposite sign of that of'}·

lt(P) is the right Lie algebra of Diff(P); (see Ebin and Marsden [1970], or Abraham and Marsden [1978], ex. 4.l.G, page 274.) Since Lie algebras are usually thought of as left Lie algebras of Lie groups, we must introduce certain minus signs in the definitions that follow, in order to obtain the standard formulas in the literature;in particular, the left Lie algebra bracket of x(P) is minus the usual Lie bracket of vector fields.

Let P be a smooth manifold. A Poisson bracket on P is a multiplica-tion { , } on F(P) making ( F(P), { , } ) into a Lie algebra and a map f1-+ Xf EJ:(P) such that Xf(g) = {g,f} that is a Lie algebra homomor-

phism of F(P) into )((P), i.e. X{f,g} = -[Xf,Xg]. A manifold P endowed with a Poisson bracket is called a Poisson manifold. A map a:(P1,{ , }1)-+ (P 2,{ ,}2) between Poisson manifolds is called canonical, if

* * * a {f,g}2 = {a f,a g}1 (4.2)

for any f,g E F(P 2), where the upper star denotes the pull-back operation. Functions C E F(P) such that {C,f} = 0 for all f E F(P) are called Casimir functions. Note that a canonical map a:P1-+ P2 takes trajectories of hE F(P1) into trajectories of a*h E F(p2).

A Lie group action on a manifold P is a group homomorphism <P:G-+ Diff(P), where Diff(P) denotes the group of diffeomorphisms of P, such that the map (g,p) 1-+ <P (p) is smooth. If P is a Poisson manifold, :OJ-+~(P) such that the map (Cp) ~ <t>UJ(p) is smooth. If bf happens to be the (left) Lie algebra of a Lie group G

acting on P, then <P = <P', where the upper prime denotes the Lie algebra homomorphism induced by <P i.e. <P' = Te<P. If P is a Poisson manifold, the Lie algebra action <P is said to be canonical if for any E; E OJ. and fl, f 2 E F(P),

If the Lie group G with Lie algebra &'). acts canonically on the Poisson manifold P, a momentum mapping J:P-+ ?-* is a map satisfying

<f>( E;) = XJ( E;) (4.4)




for all ~ E'), where J(~) E F(P) is defined by J(O{p) = (J{p), ~ >, where< , > denotes the pairing between ~ * and ?· equi variant, if

Joq> =Ad* 1oJ g g-

J is said to be

( 4. 5)

for all g E G; here Adg: 61-+ &(--denotes the adjoint action of G on 6-J-and Ad~: h}* -+'}* is its dual map. If we deal with a canonical Lie algebra action <P of ~ on P, the definition of the momentum mapping is unchanged, but equivariance is replaced by

T J(<P(~)(p)) =-(ad ~)*(J(p)) p ( 4.6)

for all !; E b'J.• pEP; here TPJ:TPP-+ b(* denotes the tangent map (dif-ferential) of J at pEP. Lie group (algebra) actions on a Poisson mani-fold admitting equivariant momentm maps are called Hamiltonian actions.

In duals of the Lie algebras, a Casimir function is characterized by the property of being invariant under the coadjoint action. This means that C E F(bJ*) is a Casimir function if and only if

* C oAd _ l = C, or g

* C(Ad _1ll) g

C(ll)

for all g E G and l1 E 0')-*. We now give examples of the concepts above. Any symplectic manifold,

in particular any cotangent bundle, is a Poisson manifold, the Poisson bracket being defined by the symplectic 2-form. The Casimir functions are constants. As we shall see in the next subsection, duals of Lie algebras are Poisson manifolds. If G = 50(3) and P = rn 3, an example of an action of G on P is q,A(l} = A_!, where A E 50(3) and _! E m3. If G =

Diff(n) and P = ~(n), F(n) or F*(n) (the densities on n), an action of P on G is given by push-forward. The adjoint action of S0(3) on .60(3) is conjugation, and the adjoint action of Diff(n) an *(n) is push-forward.

The momentum maps used in this section are all defined by actions which are cotangent lifts. This means that G acts on the manifold Q and one considers the induced action on T*Q. Thus, if :G-+ Diff(Q) is a (left or right) action, then

* ( g, aq) 1-+ T g ( q) g _ 1 ( aq ) (4.7)

is also a (left or right) action; here g E G, aq E T;Q. where T;Q denotes the cotangent space at q to Q and T; (q) _1 is the dual of the

g g




tangent map (derivative} of$ _1• In particular, if Q = G, then the g

left and right translations Lg and Rg can be lifted to left and right actions also denoted by L and R , of T*G by (4.7}, namely g g

(4.8}

(4.9}

Noether 's theorem gives a formula for the momentum map of the actions (4.7}. If ~ denotes the infinitesimal generator of the action of G on Q, ~ E ~· then the equivariant momentum map

J:T*Q + 'J*

is ( 4.1 0}

for a.q ET;Q, ~ E ~· and<,> the pairing between T*Q and TQ. In particular, the two commuting actions (4.8} and (4.9} have the equivariant momentum maps

* ( T R } (a. } for L e g g ( 4.11)

* (T/g) (ag) for R. (4.12)

4.2 Duals of Lie Algebras. The dual 11)* of a Lie algebra ?-is a Poisson manifold with respect to the ± Lie-Poisson bracket given by

(4.13)

for f.! E b}* and f,g functions on "}*; here ( , > denotes the pairing between b'J and bj-*· The "functional derivative" Of/of.IE "J- is the derivative Df(f.l) regarded as an element of 6} rather than ~**, i.e.

Of Of( f.!)· v = <v, of.! > (4.14)

for f.l,V E 1>(1*· (For infinite dimensional~· the pairing is with respect to a weakly non-degenerate form and the existence of Of/of.! is a bona fide hypothesis on f.) The space or* endowed with the ± Lie-Poisson bracket is denoted by~=· The Hamiltonian vector field defined by the function h on ~* is given by




(oh )* Xh(ll) =+ad 611 (\.!), (4.15)

where ad(~)·n = [~.n] is the adjoint action of~ on"). and (ad(~))*:b).* +&)* its dual map.

An important property of equivariant momentum maps is that they are canonical. More precisely, if J:P +OJ* is an equivariant momentum map of a left Lie group or algebra action then 'J:P + ~: is canonical, i.e.

* * *{ } {J f,J g} = J f,g + (4.16)

for all f,q E F( 'J*l and { ,} the Poisson bracket on P. An equivalent formulation is

J([~.n]) = {J(~) ,J(n)}

for all ~.nEb]-· If left actilil;ns are replaced by right actions, all the signs in (4.16) and (4.17) have to be changed; J:P +?: is canonical. For example

and

(4.17)

given by (4.ll) and (4.12) are canonical maps. By (4.11) and (4.12) JL is right invariant and JR is left invariant. Another important example is provided by the following.

Let ~· ~ be Lie algebras and a:~+ -5. a linear map. The dual map a*:-}+~* is canonical if and only if a is a Lie algebra homomorphism.

For a study of the local structure of Poisson manifolds the reader is referred to Weinstein [1984] and his lecture in this volume.

4.3 Semidirect Products. Let V be a topological vector space and assume that is a left Lie group action on V such that each g is linear, i.e. :G + Aut{V) is a group homomorphims, where Aut{V) is the Lie group of all linear continuous isomorphism of V. Such an action is called a (left) representation of G on V. The Lie algebra of Aut(V) is the space End(V) of all linear continuous maps of V into itself, with bracket the commutator of 1 inear maps [A,B] = AB-BA, A,B E End(V). The group repre-sentation induces a Lie algebra representation ': OJ+ End(V), so ' is a Lie algebra homomorphism.

Given G,V, and , we define the semidirect product S as the Lie group with underlying manifold G x V and multiplication




(4.18)

where g1 , g2 E G, u1 , u2 E V. S is usually denoted by G rx V, the action of G on V being known. Let .6 = 1rx V be the Lie algebra of S; its bracket equals

(4.19)

for ~l, ~ 2 E 1Jf, v1, v2 E V. The adjoint and coadjoi nt actions of S on .6 and .6* are given by

Ad( )(~,v) (Ad ~.4l(g)v- ¢'(Ad ~)u) g,u g g (4.20)

and

where g E G, u,v E V, v E &{, and a E V*; ¢~:? + V is given by¢~(~) =

4l'{~)u and (g,u)-l = (g-l, -4l(g-1)u). Recall that a Casimir function is characterized by being invariant under the coadjoint action. Formula (4.21) will be used in examples to determine whether a given function is a Casimuir.

The ± Lie-Poisson bracket of F,H:-6* + IR is, by (4.13) and (4.19), equal to

{ _ +( [OF oHJ •(oF) oH •(oH) OF F,H}±{J.l,a) -- J.l, o].l' Of!)± (a,¢ ()jj •()a) =F (a,¢ Oil "oa) (4.22)

where f.l Eo;.*, a E v*, and as in (4.15), ~~ EOJ.and ~= E V. From formula (4.15), we compute the Hamiltonian vector field of H:-6* + IR to be

XH(f.l,a) = =F(ad(~~)\- 4'' oH* a, Oa

*

(4.23)

where ¢' oH: b)-+ V is Oa

given by ¢' oH( ~) = ¢' ( ~) ·~~· 6a

and 4l' 0H is its adjoint. Oa

The left and right translations on T*G X v X v* are

(ah,v,a) E T(h,v)(G x V) = S, for

h '

L ( ( g, u) , ( ah, v, a ) ) = ( (T gh L _ 1 ) *ah , u + ¢ ( g) v, ¢( g- 1 ) *a) , g

R((g,u),(ah,v,a)) = (c\ R _/"ah- dfa _1 (hg),v +¢(h)u,a) gg 4l(g )u

(4.24)

( 4. 25)



REDUCT! ON AND HM1I L TONI AN STRUCTURES 87

where f~(g) is the "matrix element" (a,<P{g)u) and df~(g) its differen-

tial. The corresponding momentum mappings are by (4.11) and (4.12)

JL:T*s-+ '-'+*' JL(a ,v,a) = (T( O)R( ))*(a ,v,a) = ((T R )*a + (q,')*a,a) g e, g, v g e g g v (4.26)

and

(T( O)L( ))*(a ,v,a) ((T L )*a, q,(g)*a). e, g,v g e g g

Recall that JR is left invariant and JL is right invariant; both are canonical.

(4.28)

4.4 The Theorems. In many physical examples a Hamiltonian system on T*G is given whose Hamiltonian function Ha depends smoothly on a parameter a E v*. In addition, Ha is left invariant under the stabilizer Ga {g E GI<P(g- 1)*a =a} whose Lie algebra is &)a={~ E ~I<P'(~)*a = Q}. We can think of this Hamiltonian also as a function H:T*G x v*-+ IR, H (ag) = H(a ,a), where T*G x v* has the direct sum Poisson structure: a g the bracket of two functions on T*G x v* is their bracket on T*G. (If V were a Lie algebra, a case not discussed in this lecture, v* would be endowed with its own Lie-Poisson structure.) We wish to study the motion determined by H on a "flat" space whithout losing any information about the original motion on T*G x v*. The key to this approach is the momentum maps (4.26) and {4.27).

We start with the left action of the semi-direct product S =GO< V on T*s. The momentum map JR is invariant under the left action and the sub-group V c S acting on the left on T*s has a momentum map given by the second component of JL (see (4.26)), i.e. (ag,u,a) 1-+ a. This is a canonical map if v* is thought of as having the +Lie-Poisson structure, which is trivial since V is an abelian Lie algebra. ~oreover, the canonical projec-tion T*s -+ T*G is clearly canonical, so that the map

(4.28)

is canonical. Now it is easily seen that JR factors through PL:

(4.29)

i'.e. the following diagram commutes




r*G x v* ---.,.,._--~ JR

Consequently, JR is a canonical map.

* .6

A similar situation occurs when one considers the right action of S on T*s. The momentum map JL is right invariant, and the subgroup V c S

* acts on the right in a Hamiltonian manner on T S with momentum map given by the second component of (4.27), i.e. (a ,u,a) t+ ~(g)*a. This map is there-g fore canonical. Moreover, the map

(a ,u,a) >+a + dfa 1 (g)= a + T*R 1 (~')*a g g ~( g- ) ( u) g g g- u

being a projection followed by a translation with an exact differential on tile fit5ers, is a canonical map T*s-+ T*G. Consequently

is canonical.

* T* x v* PR: T S -+ G

PR(ag,u,a) =(a + T*R 1 (~•}*a, ~(g)*a) g g - u g It is easily shown that JL factors through PR:

(T*R (a), ~(g- 1 )\) e g g

i.e. the following diagram commutes

T*s

/~ T*G x v*

( 4. 30)

( 4. 31)

Consequently, JL is the reduction of JL by V, so is a canonical map. A few comments are in order regarding the difference between right and

left in the previous construction. The space T*G X v* is diffeomorphic to tile orbit space of T*S by the left or right V-action. The explicit diffeomorphisms are *[ag,u,a]t-r (ag,al for the left V-action and [ag,u,a] 1+

(a + dfa _1 (g), ~(g) a) for the right V-action, where [a ,u,a] denotes the g ~(g ) g

left or right V-orbit through (ag,u,a). Via these diffeomorphisms




the canonical projections become PL and PR respectively. (As we remarked

in the introduction, the asymmetry between left and right is because we have chosen a left action of G on V.). We refer the reader to Marsden, Ratiu, Weinstein [1983] for an analysis of the symplectic leaves of T*G x v*. We summarize the res-ults in th.e following.

Theorem 1. The maps

JL,JR:T*G X v* ~ ~; ' JL(ag,a) * ( -1 * ( T Rg a ) , <P( g ) a) e g

- * * JR(ag,a) = (T L (a ), <P(g) a) e g g

are canonical; in fact, these maps are reductions of the momentum maps

by the action of V and are themselves momentum maps for the action (left

or right) of G IX V on the Poisson manifold T*G x v*.

89

See Holm, Kupershmidt,and Levermore [1983] for a direct verification of the canonical nature of JL in some examples.

After this kinematic theorem we turn our attention to dynamics. Let H:T*G x v* ~ IR be a Hamiltonian and assume that the function H :T*G ~ IR, a H (a) = H(a ,a), a E v*, is invariant under the lift to T*G of the left a g g

action of the stabilizer G on G. Then it is easily seen that H induces a -

a Hamiltonian function HL:~~ ~ IR defined by HLoJR = H, i.e. HL(T*L (a), <P(g)*a) = H(a ,a). For right invariant Hamiltonians interchange e g g g - -"left" by "right", and "-" by "+". However, si nee the maps JR and JL

- * -1 * are different, we have HRoJL = H, i.e. HR(TeRg(ag)' <P(g ) a)= H(afa). It is of interest to investigate the evolution of a E v* in ~_;we

work now with a left action. Let c (t) E T*G denote an integral curve of a H and let g (t) be its projection on G. Then t t+ {c {t),a) is an a a * a_

integral curve of H on T G x v* so that the curve t f+ JR(ca {t) ,a) is an integral curve of HL on ~*. Thus tt~ <l>(ga{t))*a is the evolution of the initial condition a in ~*. For right actions, if c (t) and

- - -1 * a g {t) are as above, the curve t t+ JL(c {t), <P(g (t) ) a) is an integral a a a curve of H on T*G x v* so that t t+ JL{c {t), <P{g (tf 1)*a) is an integral a a curve of HR on ~*. Hence t t+ <l>{g (t)-l)*a is again the evolution of + a the variable a in ~:. The difference between the integral curves of H for left and right actions is due to the different formulas for PL and PR. We have proved the following.

Theorem 2. Let H: T*G X v* ~ IR be left invariant under the action on T*G of the stabilizer G for every a E V • Then H induces a Hamiltonian a HL E F{~~) defined by H{T;Lg{ag)' <P{g)*a) = H(a9,a), thus yielding Lie-




Poisson equations on f.>:. 'l"ne eza>Ve ca(t) E T*G is a solution of Hamilton's

equations defined by H on T*G if and only if JR(c (t),a) is a solution a a of the Hamiltonian system defined by HL on f.>:. In particular, the evolu-

ti'cm of a E V* is given by <P( g ( t)) *a where g ( t) is the projection a a of ca{t) on G. For right invariant systems, interchange everywhere "left"

by "right,""-" by"+", set HR E F{<), HR(T:Rg(a.g)' <P(g-l)*a) = H(a.g,a), and the evolution of a is given by <P{ga{t)-l)*a.

We conclude this section with some general remarks. In many examples one is given the phase space T*G, but it is not obvious a priori what V and <P should be. The phase space T*G is often interpreted as 'material' or 'Lagrangian' coordinates, while the equations of motion may be partially or wholly derived in 'spatial' ('Eulerian') or 'convective' ('body') coordinates. This means that the Hamiltonian might be given directly on a space of the form o;* x v*, where the evolution of the v* variable is by 'dragging along' or 'Lie transport' i.e. it is of the form t I+ cp{g(t))*a for left invariant systems (or t ~+ <P{g{t)- 1)*a for right invariant ones), where a E v* and g( t) is the sol uti on curve in the configuration space G. This then determines the representation <P and shows whether one should work with left or right actions. The relation between HLpr HR) and Ha in Theorem 2 uniquely determines H , which is automatically G -invariant, a a and (4.20), (4.21) give the corresponding Lie-Poisson bracket and equations of motion. The parameter a E v* often appears in the form of an initial condition on some physical variable of the given problem.

5. APPLICATIONS. In this section we shall consider the heavy top and the adiabatic fluid equations in both space and body coordinates. The convective picture for the heavy top and the Eulerian picture for fluids are classical. The other two pictures are less common but are also interesting; see Guillemin and Sternberg [1980] for some indications in this direction.

5.1. Heavy Top in Body Coordinates. We shall apply the theorems of the previous section first directly and then backwards.

The direct approach starts with the Lagrangian picture. The Hamiltonian (2.34) (or (2.33) in terms of Euler angles) is invariant under rotations about the spatial Oz-axis. This means that we deal with the standard left repre-sentation of S0(3) on IR 3,

<P(A)~ = A!, (5 .1)




A E S0(3), ~ E IR 3• By Theorem 2, H defines a Hamiltonian HL on e(3)*, wfiere e(3) = .so(3) e< IR3 is the Euclidean Lie algebra. The Lagrangian to body (convective) map JR:T*(s0(3)) x IR 3 + e(3)* is given in this case by

- -1 JR(~A'!) = (A~A' A _!). (5.2)

To gain a physical interpretation of this map, we must determine 5!:_A' if t 1+ A(t) is a solution of the problem. If A(t) is the tangent vector

A -1 o * to S0(3) at A(t), then ~B(t) = A(t) A(t) E.so(3) and .!!! =I~ E.so(3). Now recall that by definition when working in body coordinates, 5!:.A = T;L _1(,!!!), i.e. 5!:.A is the momentum in the material picture. Thus, if !!:A is ~s above,

In coordinates this is the map (2.38). Thus, the Hamiltonian HL has the familiar expression (2.32). By Theorem 2, the evolution of ~ is given by A(t)- 1k, where A(t) is the solution of the problem in the configuration space S0(3). Note that 1 = A(tr 1 ~ is the dynamic variable in e(3)* in accordance to the general theory. It is clear that Mgt~ is a parameter i'n the problem. It represents the direction of gravity and the momentum of the body around the fixed point.

91

By the general theory, HL given by (2.32) defines Lie-Poisson equations on e(3):. The bracket is given by (4.20) and the equations of motion by (4.21). To write the bracket and the Lie-Poisson equations explicitly, we note first that ~':.so(3) + End(IR 3) is given by ~'(s)x = sx, s E.so(3), A E IR3. For F,G:e(3)* + IR ~F = (yJ(, ~F = V F(V ,; d~note the usual • u,!!! --... uY -v m y 3 - J... -

gradients with respect to .!!!• y E IR ) and hence

With these formulas, the bracket (4.20) becomes (2.30). The Lie-Poisson equations (4.23) become for this case

l ~ = -~H y = -V H -m

xm-vHxv - -:r ...!..

x:r




m1 m2 m3 or, explicitly taking into accountthat ~H = I' I' I tn is system i s ( 2 . 36) . l 2 3

and 'V H = Mg£X, -r -Theorem 2 of the previous section can also be applied backwards. Then

one starts with the Hamiltonian (2.32) on IR3 x IR 3 and the equations of motion (2.36). The last three equations say that y is dragged along 5y the group action. So 1 = A(tf 1 ~ where A(t) -is the sol uti on of the problem in tne configuration space S0(3). This implies by Theorem 2 that we are dealing with a left invariant sys~m and the standard representation of S0(3) on IR 3. Consequently, one easily computes JR and H which, of course, turns out to be (2.34). Thus, onceagain, (2.36) are Hamiltonian with respect to the bracket (2.39).

To determine the Casimir functions, note that conserved by (2.36). By (4.21) applied to e(3)*, given by

Ad* -1 (~ • .r) = (A_!!! + !! X A.r. A}). (A ,.!J.)

2 11111 and ~ . .":X. are the coadjoint action is

With this formula, it is easy to see that 2 c1(_1!!,:x_) = <P(II_yll ), c2(!!1.,1) = 1/J(!!I.•y)

are invariant under the coadjoint action, for arbitrary real-valued functions of a real variable <P and II'. In other words, c1 and c2 are Casimir functions.

5.2 Heavy Top in Space Coordinates. To study the motion of the heavy top in space coordinates, we again apply the theorems of the previous section. As remarked at the end of §4, we first have to investigate the invariance properties of H under right translations A~ AB, BE S0(3), a constant matrix. By ( 2. 34) ,

H(AB) - { Tr(BIB-lA- 1AA- 1A) + Mg£~·AB~

so that H(AB) = H(A) if and only if

-1 BIB = I, B_x = _x.

Thus, the parameter a in the general theory is, in this case, the pair (I,_x). So far, we have thought of I as being a diagonal matrix, which was consistent with the body coordinate approach: an observer sitting on the body perceives I as constant, so he can choose once and for all a body coordinate system in which I is diagonal. However, an observer who is spatially fixed sees I moving. Thus, even if I is initially diagonal,




ft wtll not stay so; i.e. I must be a general symmetric covariant (indices down) two tensor, the vector space of all them being denoted by s2(IR 3 ).

( 3) 3 * . ( 3 We see that (I.~) belongs to s2 IR x IR = V • The dual of s2 IR ) is s2(IR 3), the space of contravariant symmetric two-tensors on IR3 , with the pairing given oy contraction on both indices, i.e. the trace of tlie product. Consequently V = s2(IR3) x IR 3 , and the action of S0(3) on V i·s conjugation in the first factor and the standard action on the second. The Lagrangian to Eulerian map JL :T*{S0(3)) x s2(IR 3) x IR 3 -+

(.60(3) ~ (s 2(IR3) x IR 3)): is given by

- * -1 JL (~A, L ,~) = ( T RA (~A), ALA , A~)

To gain a physical interpretation of this map, recall that I was computed in a body frame. The spatial frame is obtained by a rotation by A(tf1 , where t 1-+ A(t) is the trajectory of the motion. Consequently, the moment

-1 -of inertia I5 in space coordinates is Is =AlA . Thus, the map JL oecomes

where ,\ = AX and !!1-s = Is~s = A!Jl. The Hamil toni an becomes

and thus

H (m ,I ,,\) = m ·I-~ + MgR._k•_,\ s -s s - -s s-s

-1 I ms s - Mg.Q.k ,

where a ® b represents the symmetric rna tri x whose entries are a. b .• - - 1 J

The + Lie-Poisson bracket on (.60(3) x (s 2(rR3 ) x IR 3 ))* is by (4.21)

equal to

{F,G}(~s,Is,2J = ~ 5 ·(17~/ x ~ 5G) + Tr(rs([(~;f, 0 ~~]- [(~~): ~])) + ~·(.Y1TI F X VAG+~ G X ~,\F).

-s - -s -The equations of motion are by (4.22)

[ oH rs]~ m =V H m- -r.-1ss, -s -m s x -s o1 -s

1s = [rs • G~)A] ,\ = ~ H5 x .2,..

-s

93




where ":.6o(3) .... IR 3 is the inverse of ~. Using i·Y.. = ..!!..xy_ and the "back-cab" identity fl x (j! x f) = .!!_(~·0 - £(~·!2..), one sees that for our Hamiltonian the first two terms of the Jils equation cancel. Thus, we have

where m = I w • -s s--s

= Mgt~ X~

= [Is'~]

= w X A -s

This is the spatial form of the heavy top equations and they are thus in Euler-Poisson form; i.e. Lie-Poisson for a semi-direct product.

The coadjoint action is given by

Let nl' n2, n3 be the three invariants of the matrix Is. Since they are invariant under conjugation, they are invariant under the above coadjoint action. Consequently, these give Casimirs. There are in fact six in all:

Cl(~s'Is.~) <Pl (nl)

C2(.!!!s ,Is.~) <P2(n2)

C/~s ,Is.~) <P3(n3)

C4(!!ls ,Is.~) < IR3 ))* is six dimensional (the coadjoint action has, at each point, a six dimensional isotropy subgroup with A =Identity and u =I !., - s-J = >-® !., (one dimension); Q = Q, [J,I 3] = 0 (three dimensions) and ~1 A· J = 0 (two dimensions)) which is consistent with the existence of six ~asimirs. There is, in addition, the constant of motion _!!!s·~ for our special Hamiltonian corresponding to invariance under rotations about the z-axis. Thus, we can reduce again getting back to the four dimensional reduced phase space (T*s 2) of the heavy top. (For the Lagrange top there is, of course, an additional conserved quantity).




The above shows concretely the duality between the spatial and body descriptions of a heavy top. In fact they form a dual pair (Weinstein [1983]). We shall see a similar situation for fluids in Sections 5.3 and 5.4 following.

5.3 Ideal Compressible Adiabatic Fluids in Eulerian Coordinates. The Hamiltonian (3.27) was shown in §3 to be right invariant under the subgroup of Diff(O) given by (3.28). This means that we deal with the representation of Diff(O) on F(O) x F*(o) by push-forward, i.e.

for f E F(O), f.! E F*(O). The induced Lie algebra representation is by mi·nus the Lie derivative. The Lagrangian to Eulerian map

is given in this case by

- -1 3 * * JL(a. ,f.!, f) = ((snon )J _1d ~· n ll, n f), -n n where .<.:/~) = ~n(~)d 3 K, ~ = n(~). Thus, if -'b, is the material velocity, (3.26) gives the material momentum density, ~n = p~ 0 d 3 ~ and the above formula becomes

JL (~n' Pod3~·ool = (~. pd3~,o)

where M(~) = p(x)vb(x)d 3_x. This is exactly the map (3.32). By Theorem 2 th l . -f -d3X d . . *( d3 ) * e evo ut1on o Po _ an o0 1s glVen by t 1+ 1'\ P0 ~, t ~-+ nto0, where nt is the solution curve in Diff(O). The Lie-Poisson bracket given by (4.21) is easily seen to equal (3.34) and the equations of motion (4.22) are (3.18), (3.19) and (3.20). The change of the parameters p and

0 o0 corresponds to choosing different initial conditions.

The coadjoint action of Diff(o) ex (F(O) x F*(o)) on ~(0) ~ (F*(o) x

F(O))* is given by (4.20), (3.16)

* 3 3 3 Ad 1 (!1_, pd x,o) = (n .t1. + fn (pd x) + lln*o' n (pd x), n*o). (n,f,f.l)- - * * - * -

Let now w =curl v be the vorticity and denote by n = ~·YO/P It is then easy to see that the functional

C(!:l_,p,o) = JJfn p(~)~(o(~). n(~))d3~·

div(o~)/p.

for an arbitrary real-valued function ~ of two real variables, is invariant under the Ad*-action. Consequently, the functional C is a Casimir function.




The theorems of §4 can also be applied backwards in order to interpret (3.18), (3.19), (3.20) as Lie-Poisson equations. Start with the configura-tion space Diff{rl), the physical energy function H(!i,p,o) given by (3.21), !1{x) = p(x)vb(x)d3x, conservation of mass and entropy (3.18), (3.19) and balance o~ mom;ntu; (3.20) with equation of state p = p2aw(p,o)jClp. Then remark that (3.18), (3.19) are equivalent to L (p{x)d 3x) = 0, L {a(x)) = 0, v - - v -i.e. n;(a(~)d 3 _!0 = a 0 {~)d 3 !, n~(a(~)) = a 0 (~). -for p0 , a0 the-initial mass and entropy density. Hence the dual of the representation space is F*(rl) x F(rl) so that V = F(rl) x F*(rl). Moreover, the prior push-forward formulas show that the left representation of Diff(n) on v* is push-forward so that by Theorem 2, the representation of Diff{rl) on V is also push-forward. Then, again by Theorem 2, since HP is invariant under

0 ' 0 0 Diff(rl) , equations (3.18), {3.19), (3.20) are+ Lie-Poisson equations

Po· 0 o * on (;((n) IX ( F(n) x F (n)):.

5.4 Ideal Compressible Adiabatic Fluids in Convective Coordinates. To study the motion in convective coordinates we have to investigate the invariance properties of H under left translations n 1+ \j;on, \j! a time independent (orientation preserving) diffeomorphism. Since

T* 1 L,,,(~) =T*\j!(F, )d 3X -'f'T) --n-nolj;

for H = F, d3X, formula (3.27) yields -n -n -

H(T* 1L (~))=~liT* 1L U1 )II+ Jr~ p 0 (.~_)w(p 0 (.~)J~.~n(~). a0(!))il. nof \j! n nol/!- lj;-lj; " 'V

If g is the metric on n and t denotes adjoints with respect to g, then if ~ = p 0 ~d 3 ~. using the definitions in §3.4 we see that

so that

( * 3 # T -l L\j;(~n)/pod X) nol/!

Thus the Hamiltonian becomes

= ~ J p0(Dli(Tlj;)t V/X.lll 2d3.X + J p 0 (!)w(p 0 {_~)J~\~)J~ 1 (n(Dl. a 0 (~l)ct 3 ~ ·




Thisexpression coincides with the one for H(_!i11) if and only if (Tl)!)t = Tl)i and hence Tl)! is an isometry,and p(f) J~\~l = p(x), where p(~) = P 0 (_~)J- 1 (~). If 1)! is an isometry, then J\)! = l; consequently, H is left invariant under tfie group

Diff(l1) ={l)J E Diff(l1) 11)! is an isometry for g}. g

Thus, the parameter a in the general theory is in this case g. The space of all the g's does not form a vector space, but it is an open cone in the vector space of all symmetric covariant 2-tensors s2(11). The dual of s2(11) is s2(11) 0 IA3{11) I the vector space of contravariant symmetric two-tensor densities. The convected p and a have trivial equations of motion corresponding to the dependence of a only on g. However, we can include them for comparison with the spatial case (this is analogous to adding the trivial equations I= 0 to the heavy top equations in body coordinates.) Thus, we take v* = s2(11) x F*{l1) x F(l1) and hence V = s2(11) ® IA3(11) I x

* F(l1) x F(l1) . The representation of Diff{l1) on V is by push-forward in every factor.

- * * The Lagrangian to convective map JR:T (Diff(l1)) x s2(11) x F (n) x F{11) .... (Diff(l1) ~ (s 2(11) 01A3(11)1 x F(11) x F(11)*))_* is given by

JR(M ,g,p,a) = (Tel (M ), n*g, n*p n*a) := (M,G,R,S) -n n """rl

where M. is the convected momentum density, related to the spatial momentum density ~ by

.M = n*11 Using the identity Lvy'b = t.z.Y.y)"b + ~ dlly_ll 2, we find that the equations of motion for (M_,G,R,S)- are

Cl.M = .l_ Rd( IIMII 2) - dP at 2 - G

aG = L G at g

aR _ as at - 0 ' at = 0

where II,MII~ is the length of .M. in the metric G, P = 11*p = R2aw;aR is the convected pressure and JL is the convected velocity: .M = Rf. By the general theory, these equations are Lie-Poisson on the space of tuples (tf,G,R,S); of courseR and S are 'cyclic' variables. As in the heavy top, what were Casimirs in the spatial picture now become special constants of the motion, and new Casimirs appear (integrals of functions of G, R and




S). Note finally that G, analogous to .r_ for the heavy top, is advected in the convective (body) picture, but is static in the spatial (Eulerian) picture. Likewise R and S, analogous to picture and dynamic in the spatial picture.

are static in the convective

As mentioned in the introduction, the duality between the spatial and convective pictures and its relationship to the stress formulas of Doyle-Ericksen-Simo-Marsden (see Sima and Marsden [1983]) as well as to covariance of energy balance under body (right) and spatial (left) diffeomorphisms will be the subject of a future publication.

REFERENCES l. H. Abarbanel, D. Holm, J. Marsden and T. Ratiu, Nonlinear stability

of stratified flow (to appear), (1984).

2. R. Abraham and J. Marsden, Foundations of Mechanics, Second Edition Addison-Wesley, (1978).

3. V. Arnold, Mathematical methods of classical mechanics. Graduate Texts in Math. No. 60, Springer, {1978).

4. V. Arnold, The Hamiltonian nature of the Euler equations in the dynamics of a rigid body of an ideal fluid, Usp. Mat. Nauk, 24, (1969), 225-226.

5. V. Arnold, Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a 1 'hydrodynamique des fluids parfaits, Ann. Inst. Fourier, Grenoble~. (1966), 319-361.

6. R.F. Dashen and D.H. Sharp, Currents as coordinates for hadrons, Phys. Rev. 165, (1963), 1857-1866.

7. I.E. Dzyaloshinskii and G.E. Volvick, Poisson brackets in condensed matter physics, Ann. of Phys. ~. (1980), 67-97.

8. D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid. Ann Nath. ~. (1970), 102-63.

9. G.A. Goldin, Nonrelativistic current algebras as unitary representa-tions of groups, J. Math Phys. l.?_, (1971), 462-487.

10. G.A. Goldin. R. Menikoff and D.H. Sharp, Particle statistics from in-duced representations of a local current group, J.Math. Phys. rL· (1980), 650-664

11. H. Goldstein, Classical Mechanics, 2nd Ed. Addison-Wesley, (1980).

12. V. Guillemin and S. Sternberg, The moment map and collective motion, Ann. of Phys. 127, (1930), 220-253.

13. E.A. Kuznetsov and A.V. Mikhailov, On the topological meaning of canonical clebsch variables, Physics Letters, 77a, (1980), 37-38.

14. D.O. Holm and B.A. Kupershmidt, Poisson brackets and Clebsch repre-sentations for magnetohydrogynamics, multifluid plasmas, and elasticity, Physica 6D, (1983), 347-363.




15. D.O. Holm, B.A. Kupershmidt and C.D. Levermore, Physics Letters, 98A{l983) 389-395.

16. P.J. Holmes, J.E. Marsden, Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups. Indiana Univ. Math., 32, (1983a), 27 3-310.

17. J. Leslie, On a differential structure for the group of di ffeomorphisms. Topology_£ :263-271 (1967).

18. J. Marsden Well-posedness of the equations of a non-homogeneous perfect fluid, Comm. P.D.E., l• (1976), 215-230.

19. J. ~1arsden and T. Hughes, Mathematical Foundations of Elasticity, Prentice-Hall (1983).

20. J.E. Marsden, T. Ratiu and A. Weinstein, Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc. (to appear), (1983).

21. J.E. Marsden and A. Weinstein, The Hamiltonian structure of the f•laxwell-Vlasov equations, Physics 40, (1982), 394-406.

22. J.E. Marsden and A. Weinstein, Coadjoint orbits, vortices and Clebsch variables for incompressible fluids, Physica l.Q_, 305-323, (1983).

23. J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. i· 121-130, (1974).

24. J.E. Marsden, A. Weinstein, T. Ratiu, R. Schmid and R.G. Spencer, Hamiltonian systems with symmetry, coadjoint orbits and plasma physics, Proc. IUTA~1- IS IMM Sympos i urn on Modern Deve 1 opments in Ana lyti ca 1 Mechanics, Torino, June 7-11, 1982.

25. P.J. Morrison, Poisson brackets for fluids and plasmas, in Mathematical Methods in Hydrodynamics and Integrability in Related Dynamical Systems, AlP Conf. Proc., #88 La Jolla, M. Tabor (ed)., (1982).

26. P.J. Morrison, The Maxwell-Vlasov equations as a continuous hamiltonian system, Phys. Lett. 80A, (1980), 383-386.

27. P.J. Morrison and J.M. Greene, Noncanonical hamiltonian density formulaion of hydrodynamics and ideal magnetohydrodynamics, Phys. Rev. Letters, 45, (1980), 790-794.

28. H. Omori, Infinite dimensional Lie transformation groups, Springer Lect. Notes Math., vol. 427, (1975).

29. T. Ratiu, Euler-Poisson equations on Lie algebras, Thesis, Berkeley, (1980).

30. T. Ratiu, Euler-Poisson equations on Lie algebras and theN-dimensional heavy rigid body, Am. J. Math., 104, (1982), 409-447, 1337.

31. T. Ratiu, Euler-Poisson equations on Lie algebras and theN-dimensional heavy rigid body, Am. J. Nath, 104, (1981), 409-448.

32. T. Ratiu and P. van Moerbeke, The Lagrange rigid body motion, Ann. lnst. Fourier, Grenoble, _R, 211-234.

33. J.C. Sima and J. E. Marsden, The rotated stress tensor and a material version of the Doyle Ericksen formula, Arch. Rat. Mech. An. (to appear), (1983).

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100 Jerrold E. t4arsden, Tudor Ratiu, and Alan Weinstein

34. A.~1. Vinogradov and B. Kupershmidt, The structure of Hamiltonian mechanics, Russ. Math. Surveys. 32, 177-243.

35. A. Weinstein, The local structure of Poisson manifolds J. Diff. Geom. (to appear)~ (1983).

UNIVERSITY OF CALIFORNIA DEPARTMENT OF MATHEMATICS BERKELEY, CA 94720

UNIVERSITY OF ARIZONA DEPARTMENT OF MATHEMATICS TUCSON, AZ 85721

article




GAUGED LIE-POISSON STRUCTURES

Richard Montgomery, 1 Jerrold Marsden1 and Tudor Ratiu 2

ABSTRACT. A global formula for Poisson brackets on reduced cotangent bundles of principal bundles is derived. The result bears on the basic constructions for interacting systems due to Sternberg and Weinstein and on Poisson brackets involving semi-direct products for fluid and plasma systems. The formula involves Lie-Poisson structures, canonical brackets, and curvature terms.

l. INTRODUCTION. Let G be a Lie group and~ its Lie algebra. The right (resp. left) reduction of T*G by G produces the+ (resp. -) Lie-Poisson structure on Of*:

{F,G}{~) = ±(~, [~~· ~~]). This construction is now well-known and has been reviewed in the lectures of Weinstein, Ratiu and Morrison in these proceedings. This paper concerns the Poisson structure on the reduction of T*B, where n:B --*X is a principal bundle. The Poisson structure on the (right) reduced space G\T*B is a mixture of Lie-Poisson and canonical structures and will be computed explicitly.

There are several motivations for considering the constructions presented here. First of all, these reduced spaces occur in the construction of phase spaces for i nterac ti ng sys terns : see Sternberg [1977] and Weinstein [l 978] for a particle in a Yang-Mills field and Marsden and Weinsten [1982] for the Maxsell-Vlasov equation. The link between the approaches of Sternberg and Weinstein and the physicist's equations (Wong's equations) was gi_ven in Montgomery [1983] and provides a basis for this paper.

The second mo ti va ti on was to better understand the role of Lie-Poisson structures associated with semi-direct products of groups G ~ H. The way these arise in examples was first systematically explored by Guillemin and Sternberg [1980] and Ratiu [1980,1981 ,1982]. The symmetry breaking mechanism

1980 Mathematics Subject Classifications: 58F05, 58Fl0. 1Research partially supported by DOE contract DE-AT03-82ER12097.

Research supported by an NSF postdoctoral fellowship.

101





102 Richard Montgomery, Jerrold Marsden and Tudor Ratiu

behind their occurrence is now well-understood for examples whose underlying configuration space is a Lie group such as the heavy top, compressible fluids and MHD (see Marsden, Ratiu and Weinstein [1983] and Ratiu's lecture in these proceedings). However, semi-direct products occur in somewhat more mysterious ways as well; for example in the last section of Marsden, Ratiu and Weinstein [1983], it is observed that in momentum representation the brackets for the Maxwell-Vlasov equations and for multifluid plasmas, involve semi-direct products. This paper in fact began on the road to Boulder as an attempt to Better this understanding.

The third motivation is to provide a setting for understanding limits of Poisson structures and for averaging. For example,the limit c-+ co in the Maxwell-Vlasov to Poisson-Vlasov transition can be understood as rescaling the bracket so the motion on the base X freezes (electrodynamics becomes electrostatics) leaving only Lie-Poisson motion in the fiber. The dis-cussions and examples in Weinstein [1983] seem to be consistent with this scheme. Also, if one averages the Hamiltonian H over the fiber by the G action, then the average H drops to T*x by reduction. Hopefully, systems where fast time sea 1 es can be smeared out can be unders toad in this context. As is well-known (see Kummer [1981]), this reduction may involve a modifica-tion of the Poisson structure by rna gneti c (or curvature) terms, a phenomenon we shall see explicitly. In particular, we think one can understand the guiding center equations of Littlejohn [1979] in this way, as well as other situations involving averaging, such as MHD and guiding center plasmas.

In the scheme for interacting systems proposed by Sternberg [1977], Weinstein [1978] and used in Marsden and Weinstein [1982], one starts with a phase space of the form

T*s x ~*

where~ is the Lie algebra of a Lie group H and n:B-+ X is a principal G-bundle, with G acting by a canonical action on ~*. In the cases of multifluid plasmas and the Maxwell-Vlasov equations, elements of "?-* repre-sent matter fields, while T*B represents the pure fields (Maxwell or Yang-Mills fields). After reduction by G, the couplingmanifests itself in the Poisson structure on the reduced space

T*B xGf·

An important idea in this paper is to think of r*s X~* as T*(B X H) reduced by H. As in Guillemin and Sternberg [1980], GO< H acts on B x H making it a principal G 0< H bundle, so



GAUGED LIE-POISSON STRUCTURES 103

T*B XG i* = G IX H\T*(B X H)

which reduces the study of T*B xG }* to the case G\T*B. (Warning. As explained in Ratiu's lecture in these proceedings, the right reduction of T*(B x H) to T*B x ~* by H is not simply by projection if G acts on }* on the left, but it is if G acts on~* on the right).

In this paper we shall describe the reduced brackets on G\T*B in both the Weinstein and Sternberg representations. (See Marsden [1981] for a synopsi"s of the two viewpoints.) On the Weinstein side we deal directly with G\T*B where G acts by the cotangent lift. On the Sternberg side one selects a connection A to split T*B into hori zonta 1 and verti ca 1 covec tors be fore reduction. The main new results of this paper are formulas for the Poisson bracket on the Sternberg side (see §4).

In a more comprehensive paper in preparation we shall a. Give an intrinsic proof of the global formula in §4; b. show how the semi-direct bracket formulas in §5 apply to fluids and

plasmas, and c. obtain a formula for the brackets for free boundary problems and for

Yang-Mills fluids and plasmas in reduced variables (the analogs of E and B).

In future publications, we hope to apply the ideas herein to study limits of Poisson structures and averaging, continuing the program begun by Weinstein [1983].

2. BRACKETS IN THE WEINSTEIN REPRESENTATION. Let n:B ~X be a principal (right) G bundle. We are interested in the bracket structure on the reduced space

W = G\T*B

in a local trivialization. This is essential for understanding the Sternberg side. To begin then, assume B =X x G, so T*B = T*x x T*G and we can identify

G\ T*B = T*x x G\ T*G

= T*x x OJ.:· The second equality occurs because T*G right trivialized is canonically isomorphic to ~* with its +Lie-Poisson structure. Thus, in this choice of trivialization, the Poisson structure on the Weinstein side is canonical on T*X and Lie-Poisson on ~::




{F,G}(x,p,Jl) = cSF.cSG _ cSG.~ + (Jl [cSF, ~]) ox cSp ox c5p ' O]J O]J

The first two terms denote, of course, the canonical bracket on T*x. For computational purposes later we will need to make this a bit more precise. Assume in the trivialization that X is also a coordinate neighborhood, so without loss of generality, X is Banach space. Then T*x = X X x*, with (x,p} EX X x*. So ~ means the first partial derivative dlF(x,p,]J): X-->IR, anelementof X*. Likewise ~FEX** andweassumeitliesin X, just as one assumes ~~ E ~- P

3. BRACKETS IN THE STERNBERG REPRESENTATION, LOCAL VERSION. Our version of the Sternberg space is

s

where B is the pullback bundle of B to T*X:

B ----' 1~-----'> B

n I I n v v

T*x --1'----:> x

The bundle B has a concrete realization as the subbundle of T*B which annihilates vertical vectors in TB. Here, t is the cotangent projection, and :;: is the restriction of the cotangent projection T*B ->B. The map n is defined by

where

and

-Note that B is a principal bundle over T*X where the G action is the restriction to B of the (lifted) G action on r*s and G acts on OJ-* by the coadjoint action. S is then a vector bundle over r*x. It is an



GAUGED LIE-POISSON STRUCTURES 105

associated Bundle, also known as the coadjoint bundle to B. It consists of G orBits in B x oj* where the action is

To define a Poisson structure on S we need a connection A on B. Such a connection can be viewed as an equivariant splitting of TB into hori-zontal and vertical vectors, or dually, as an equivariant splitting of T*B:

* BxOJ*---->TB

where Ab:TbB + ~ is the connection one-form. We use this isomorphism to pull Ei.ack the canonical symplectic structure on T*B in order to get an

- * A-dependent symplectic structure on B x OJ . If we now mod out by G, we get a Poisson isomorphisim S + W.

As before, we are interested in the Poisson brackets in a local trivialization. So we will assume B =X x G with X a Banach space. Then

- * * * * B = X X X X GoC..+ T X X T G = T B

where G is embedded as the zero section in T*G. And

s =X X x* X G\(G x"f)".:) X X x* x~*·

In the previous section we showed that the same trivialization induces * anidentificationof W with xxx xo;* also. Itwasshownin

Montgomery [1983] that the isomorphism S ::;. W is then given by

(X, p, ll) ......- (X, p + ~(X)* ll)

where A is theOJ-valued one-form on X induced by the trivialization. Since this is a Poisson isomorphism we can now calculate the

Local formula for the Sternberg bracket:

{F G}( ~ = *oG __ oG.oF + , ' x,p, ll1 x op 6x Tp

[ OF oG] [ oG oF]> (ll,- ~(x)·<Sp· Oll + ~(x)·oP' oll (.0 F o G)

+ < ll• fl_(x) loP' oP >




reF oG] + ( ~. [6~. 6~ )

Here n is the local expression for n, the curvature of A.

Proof. Let

F{x,p,v) = F{x,p- ~{x)*v,v)

denote the pushforward of the function F on S to F on W. Then

[oF OG] + < ~. o~· o~ >

From the definition of F we read off

= (6F- d Fod {A*~~·x + oF.P + <6F- A{x) ~. ~ > ox 2 x - 1 op o~ - op

Plugging these results into the previous equations we get

{F,G}S = reF- d Fod {A*~~· _o.§- [oG- d God (A*ll~· _6£ [ox 2 x - ~ op ox 2 x - 1 op

_ oF .oG oG oF - ox oP - 8i<" oP

< -[A{x) • .§£ oGJ + [A{x)· oG oF]> + < [OF oGl > + ~. - op• o~ - op• o~ ~. all· o~J

( * oF { * ) oG + d2Godx ~ l.l)• op- d2Fod2 ~ ll •oP

+ ( l-1. [~(x)• ~:. ~(x) ~~] )




Comparing this with the alleged local formula, we see that it suffices to prove that the last two terms equal the curvature term in the local formula. Since E = d ~ + [~.~] we need only show:

d2God (A*i-1) .~F - d2Fod (A*JJ) ~G X - up X- up

The left hand side is

( 6F [A·6GJ _ 6G [A·6F] _ A·[6F 6G ]> JJ, op - op op - op - op' op

Now

6F [ 6G] _ 6F [ 6G J ( Jl - A•- ) - - ( JJ A•- > • 6p 6p 6p • - 6p

= _§£ [<A* 6G >] 6p - jJ, 6p

= ( d (A* JJ) • ~F' ~G > + (A* Jl' d 6G ~ ) X - up up - X 6p 6p

= d God (A *JJ)•.§£_ + ( A (d (6G). 6F)J ) 2 X - 6p jJ, - X 6p 6p

Subtracting the similar expression with F and G switched and recalling the local expression for the Lie bracket of the vector fields [~~· ~~] yields the result. •

4. BRACKETS IN THE STERNBERG REPRESENTATION, GLOBAL VERSION. The global formula for these brackets requires some more terminology. In this section

n :s -+ T*x

denotes the vector bundle projection. Using the trivialization of the previous section, n is given by

107




(x,p, f.l) 1+ (x,p)

(See Montgomery [1983] for this calculation). Let

A

denote the pullback connection on B. (It is trivial in the momentum direc-tions of T*x.) Since S is an associated vector bundle to B, we have a

* horizontal lift h = hA of vectors on T X to vectors on S. Using this, we define the covariant differential of a function F:S + IR at s E S to be that covector dAF(s) at p = n(s) E T*x given by

This may be thought of as the horizontal part of dF(s). The vertical part may be thought of as an element in the dual bundle to S which is the adjoint bundle

It is given by

_§£(s)•s'=ddt/ F(s+ts'). ov t=O

The curvature of A is

where S1 is the curvature of A. We may consider S1 to be a two-form on T*x with values in s* by the mapping

( v , wp) 1+ [ ( b, n( b) ( h v , hw ) ] G · p p p

Here bE B, iT(b) p, h denotes horizontal lift to B, and the brackets denote an equivalence class in B x OJ- under the G- action. From the transformation law for curvatures, this equivalence class is independent of which bE B is picked. [In the case where B is the frame bundle p this formulation of the curvature is the usual Riemann tensor.]

* # Finally, if S is a covector at p on T X, then S denotes the symplectically dual vector at p given by




# S = w( P) ( •, S ) ,

where w is the canonical two-form on T*x.

Global bracket formula on S

+ <s, [oF (s) oG (s)J) ov ' ov

where n(s) = p.

In a future paper a global proof will be presented. Here we will prove the formula by showing that it agrees locally with the formula given above:

In the local trivialization of the previous section,

(oF [oF ]> oF) di\F(x,p,v) = (ox- <v, ov' ~(x). , op

This is seen by considering F as a G-invariant function on B)(. cg.*, which we will denote F. Then

dAF(x,p,v)•(x,p) = dF(x,p,e,v)·(h(x,p),O)

where h(x,p) (i,~,- ~(i)·x) is the horizontal lift of (i,~) to (x,p,e) EB. By the G invariance of F,

dF(x,p, -A(x)·x,O) = dF(x,p,O, adA(x)·i*v)

= ~~ x + <v, [~(x) ·x, ~~] > + ~~·p

* The covector bracket on T X is

* * * where a= (a1 ,a2) E TP(T X) = X x X, and likewise with 8. If we set a = dAF' 8 = dAG and use the formula for dA we find the first term of the global bracket formula equals the first two terms of the local bracket.

To check that the curvature terms of the two formulas match, note that

109



110

and that

Richard Montgomery, Jerro 1 d Marsden and Tudor Ra ti u

# oF TT·d-F(s) = -A op

Finally, it is clear that the last, pure Lie-Poisson terms of the two formulas are equal.

5. COUPLING AND SEMI-DIRECT PRODUCTS. Suppose p* is a canonical right action of G on the "matter-fields"{:. Then we can reduce the total phase space T*B xj: by the canonical G action:

We want an expression for the brackets on this reduced phase space and also one for the Sternberg side, namely on B xG (b(* x'*)+.

-* Assume p is induced by an action

p:G +Lie algebra automorphisms of }

which in turn is induced by a right action

p:G +Aut H

of G on H by a utomorphi sms. That is

and

-* - * ..f.* 1. * p (g) = p(g) :'(-+ 'i

is its dual. If one thinks of ): as a reduced Poisson space; i.e.

~: = H\ T*H,

then p* is the action induced on }* by the lift of p acting on T*H.




Using p, one forms the semi-direct product group G ~ H with multiplication

and the semi-direct principal bundle B k H whose underlying manifold is

B x 1-1; it has the right GosH action given by

(b,k)·(g,h) =. (bg, p(g-1) (kh)).

~

The connection A on B induces the semidirect product connection A on

111

B ~><H. A is uniquely determined by the fact that the embedding B = B x {e} ~

c B x H maps horizontal subspaces of A onto hori zonta 1 subs paces of A (see Kobayashi-Nomizu [1963], p. 79). Somewhat lengthy calculations prove the formula

where

p': OJ-+ Lie algebra of Aut H ~'!(H)

is given by

p'(~)(h) = d~~ p(exp t ~)·hE ThH. t=O

We now apply the results of the previous section with B IX H in place of B. Note that the pullback bundle is

~ - * B X H = B IX H ~ T (B IX H)

where as a manifold B IX H = B x H c r*s x TH with H c T*H as the zero section. Now we reduce by G P< H in two steps, first by the normal subgroup H = {e} x H, then by G. Calculations show that this results in the com-mutativity of:



112

reduce by

G to< H

"""-'

Richard Montgomery, Jerrold Marsden and Tudor Ratiu given b A ~

B X H X (OJ X~):-,....-....._;/---;> T*B X

llced"oe by H

T*H = T*(B 0< H)

llced"oe by H

8 X Dt: X 1: > T*B x}:

F"'"" by

G ll"'"" by G

reduce by

G 0< H

(CJ X~):= B (f X$*)+ * xG ~: Bx H x X T B G to< H\T*(B rxH) GrxH G

where the central horizontal map is the isomorphism ~ x 9* + T*B given by A on these factors, and the identity on the ~* factor.

The Lie bracket on OJ. x ~ is

where p 1 :0J-+der-$ isthederivativeof p:G +Aut~. Inthelocal Weinstein formula, we replace ~ by (~.v) and replacement of the bracket there by this bracket leads to the following

Local form of the bracket on T*B xG $*::X x X* x "J* x~*.

{ G}( ) = if:. oG _ _Q_§_.Qf + < [if: oG] > F, x,p.~.v oX op oX op ~. o~· 0~

+ <v [oF oG] + I troF~).oG _ I (oG).oG > , ov' ov p o~, ov p lo~ ov

To calculate the brackets on the Sternberg side note that

~ = (~,0), a 1-form on X with values in ~x-}

where A is as before and A is the pullback of A on B 0< H X x G x H by the identity section x '+ (x,e,e). And that

A

rl = (_fl,O).




Plugging these results into the local formula with hats (") on !2 and _B, fl replaced by (f.I,V), and 19;1. brackets replaced by OJ IX~ brackets,we obtain the following

Local form of the coupled brackets on the Sternberg side B xG (~ * x~*r:: * * -1. * XxX x~ x.t:

{ }( ) = oF. oG _ oG.~ + ( [A( ) oF oGJ [A( ) oG o FJ ) F,G x,p,f.!,\1 ox op ox op f.l,-- x • op' of.! + - x • op' Of.!

+ <v p'(A(x)• 0G ~- p' (A(x)·~F~._§ ) ' - op ov ~ op J ov

f oF oG1 +(f.!, ~t(x) ~· T - \.up uP,

+ ( f.l, I oF, oGJ > LOf.l Of.!

+ <v, [oF oGJ + p' (oFl.oG _ p' troGj'·~ ) ov' ov (of.!_ ov Of.! ov

As we have mentioned in the introduction, these formulas give, in particular, the semi direct product formulas appearing in the last section of Marsden, Ratiu and Weinstein [1983]. Details concerning this and other applications to Yang-Mi 11 s fluids and plasmas and to free boundary problems will be the subject of another publication.

References

l. V. Guillemin and S. Sternberg [1980]. The moment map and collective motion, Ann. of Phys . .!22., 220-253.

2. S. Kobayashi, and K. Nomizu [1963]. Foundations of differential geometry, Wiley.

3. M. Kummer [1981 ] . On the cons true ti on of the reduced phase space of a Hamiltonian system with symmetry. Indiana Univ. Math. J. 30, 281-292.

4. R.G. Littlejohn [1979]. A guid ing center Hamiltonian: A new approach. J. Math. Phys. 20, 2445-2458.

5. ··J.E. Marsden [1981]. Lectures on geometric methods in mathematical physics, SIAM, CBMS Conf. Series #37.

6. J. Marsden, T. Ratiu and A. Weinstein [1983]. Semidirect products and reduction in mechanics. Trans. Am. Math. Soc. (to appear).

113



114 GAUGED LIE-POISSON STRUCTURES

7. J. Marsden and A. Weinstein [1982]. The Hamiltonian structure of the Maxwell-Vlasov equations, Physica 0, i• 394-406.

8. R. Montgomery [1983]. Canonical formulations of a classical particle in a Yang-Mills field and Wong's equations, Letters in Math. Physics (to appear).

9. T. Ratiu [1980]. Euler-Poisson equations on Lie algebras, Thesis, U.C. Berkeley.

10. T. Ratiu Il98l]. Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body, Proc. Natl. Acad. Sci. U.S.A. 78, 1327-1328.

ll. T. Ratiu [1982]. Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body, Am. J. Math. lQi, 409-448.

12. S. Sternberg [1977]. On minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field, Proc. Nat. Acad. Sci . .zi, 5253-5254. --

13. A. Weinstein [1978]. A universal phase space for particles in Yang-Mills fields, Lett. Math. Phys. ~. 417-420.

14. A. Weinstein [1983]. The local structure of Poisson manifolds. J. Diff. Geom. (to appear).




THE HAMILTONIAN STRUCTURE OF THE BBGKY HIERARCHY EQUATIONS

Jerrold E. Marsden, 1 Philip J. Norrison, 1 •2 and Alan Weinstein1

ABSTRACT. The BBGKY hierarchy equations for the evolution of the i-point functions of a plasma with electrostatic interactions are shown to be Hamiltonian. The Poisson brackets are Lie-Poisson brackets on the dual of a Lie algebra. This algebra is constructed from the algebra of n-point functions under Poisson bracket and the filtration obtained by considering subspaces of i-point func-tions , l < i < n .

§1. Introduction The purpose of this paper is to show that the BBGKY (Bopliubov-Born-

Green-Kirkwood-Yvon) hierarchy equations are Hamiltonian with ct Poisson bracket associated to a certain Lie algebra. For background and the original references on the hierarchy, the reader may consult one of the standard texts, such as Clemmow and Dougherty [1969], Ichimaru [1973] or Van Kampen and Felderhof [1967]. For background on Lie-Poisson structures on duals of Lie algebras, see Marsden and Weinstein [1982], Marsden et. al. [1983] and the lectures of Morrison, Ratiu and Weinstein in these proceedings.

In the present paper, we simply exhibit the Hamiltonian structure of the hierarchy equations making use of the theory of momentum mappings. Eventually, we hope to show how this structure is inherited by trun-cated systems, providing a statistical basis for recently discovered bracket structures for plasma systems (Morrison and Greene [1980], Morrison [1980], Marsden and Weinstein [1982], Morrison [1982] and Marsden, et. al. [1983]).

2. The Hierarchy Equations Let P be a finite dimensional symplectic manifold; for example, the

position-momentum space rn6 for a single particle. Let Pn = P x P x ••• x P (n times) be thought of as the phase space for n particles. Points in Pn will be denoted (z1 , ... , zn). Consider a Hamil toni an on Pn of the form

l Research partially supported by DOE Contract DE-AT03-82ER-l2097. 2Research partially supported by DOE Contract DE-FG05-80ET-53088.

115





116 J. MARSDEN, P. MORRISON, AND A. WEINSTEIN

H ( z1 , .•• , z ) n n

where H1 :P-+ IR and H2:P x P (minus the diagonal) -+ IR are given and H2 is symmetric in its arguments. For example, on m6 with z = (q,p) and z' = (q',p'), the functions

2 H(z)=lE.t. and 1 2m lq - q I I

describe the dynamics of identical particles of mass m and charge e under electrostatic interaction. (The simple generalization to an arbitrary number of different species is omitted here.)

Hamilton's equations on Pn give the Liouville equation for the evolu-tion of a smooth symmetric function,

namely

()f a~ + {fn,Hn}n = 0, (L}

where { , }n denotes the Poisson bracket on pn, i.e. the n-particle Poisson bracket. The moments of fn are defined by the following equations

one-point function: f1(z;t) = n ffn(z,z 2 , ... , zn;t)dz 2 ... dzn

two-point function: f 2(z,z';t) = n(n-l)Jfn(z,z',z3, ... , zn;t)dz 3 dz n

where dz denotes Liouville measure. The hierarchy equations can be obtained by differentiating these equations in t using the evolution equation for fn. For example, the first equation is

a~ J ~ (z;t) + {f1,J{(f1 )}(z;t) = {f1(z;t)f1(z';t)- f 2(z,z';t), H2(z,z')} dz'

( H 1 )

where :ffj (f1 )(z) = H1 (z) + Jf(z' )H 2(z,z' )dz' and the braces denote the

Poisson bracket on P (see the appendix).



THE HAMILTONIAN STRUCTURE OF THE BBGKY HIERARCHY EQUATIONS

§3. Lie-Poisson Equations A Lie-Poisson 5racket is the natural bracket on functions defined on

the dual of a Lie algebra. These brackets play a fundamental role in the Hamiltonian description of rigid bodies; fluids. and plasmas, (see the references cited earlier). If G is a Lie group with Lie algebra 0/ and dual 0(*. then for F,G:~* + IR, their Lie-Poisson bracket at ll Eo;* is defined 5y

( LP)

where oF EM is defined by o]J -<f

OF( \.1) is the Frechet deri va ti ve, < , > is the pairing between 1J* and 8f, and [,] is the Lie bracket on~·

The Lie-Poisson bracket for the group Sym(P) of canonical transforma-tions of a symplectic manifold P may now be described as follows. Except for constants, the Lie algebra sym (P) may be identified with (generating) functions K:P + ffi and its dual sym(P)* with densities fd!J, where f:P + rn and d\.1 is Liouville measure on P. Then we set

f {oF oG} {F,G}(f) = f Of' of d!J. Jp

(PV)

117

This is the bracket for the Poisson-Vlasov eq~ation; it is also a fundamental ingredient in the Maxwell-Vlasov bracket (Morrison [1980], Marsden and Weinstein [1982]). With P replaced by Pn, it also describes the Liouville equation (L). In fact one can check either by a direct calculation or from considerations of reduction of dynamics on Sym(Pn) that (L) is equivalent to

F = {F,J( }(f ) n n

where F is a functional of f , } (f ) is given by the bracket (PV) with n n fn in place of f, pn in place of P and

(

Jfn(fn) = jpn Hn(z1 , ... , zn)fn(z1, ... , zn)dz1 ... dzn

Here Sym(Pn) may be replaced by Syms(Pn), those elements of Sym(Pn) that commute with perm uta ti ons and sym( Pn) by sym5 ( Pn), the symmetric functions on Pn.




4. The Hierarchy Algebra Suppose that An is a real Lie algebra and

linear subspaces. Below we shall choose A to n

A1 c A2 c ... cAn are be the algebra sym (Pn) s .

of symmetric functions under Poisson bracket and Ai to be the space syms(P1 )

embedded as a subspace by the map

where the sum is over distinct subsets of {l,

! ji. For example

... ,

.. . . '

z. ) J· 1

n}; i.e .

One checks that in this example we have a filtration; i.e.

[A. ,AJ.J cA .. l 1 1 + J-

(Note that only A1 is a Lie subalgebra.) In general given such a filtra-tion there is a Lie algebra structure on

such that the map

a :A ..,. A n n n defined by

is a Lie algebra homomorphism. Indeed, set

(F)

(HLA)

where [K.,L.] is to 1 J

i + j- 1.;;; n. If i be put i n the k!b. s 1 o t i f k = i + j - 1 and i f + j - 1 > n, the term is to be put in the last (nth)

slot; one has some options here that will be the subject of our ~ork on truncations.

One can check directly that (HLA) defines a Lie algebra structure and that an is a Lie algebra homomorphism.



THE HAMILTONIAN STRUCTURE OF THE BBGKY HIERARCHY EQUATIONS 119

§5. The Moments Comprise a Momentum Map The dual a~ of an in our example is determined as follows. We have

a :A1 (!) ••. (!)A -+ A and so n n n

is given by

From the definitions it follows that

* * * * a :A -+ A1 (!) •.. (!) A n n n

dz n

where f1, .•• , fn are the moments of fn and the embeddings Ei are sup pressed.

Thus, the process of taking moments is given by the dual of a Lie algebra homomorphism and is therefore a momentum map (this is a standard fact; cf. Guillemin and Sternberg [1980] or Marsden et. al. [1983]).

§6. The Hierarchy Equations are Lie-Poisson Since a~ is a momentum map, it is a Poisson map; i.e. it preserves

brackets. We have the following maps

sym*( Pn) s

taking moments I a~ "'

j( n IR

n * . j( A = I sym ( P 1 ) ---~ IR n i =1 s

dz n

+]n I H2(z.,z.)f (z1 , ... , z )dz1 ... dz p i <j 1 J n n n




so that Jfn to a map J(

From general

"collectivizes" in the sense of Guillemin and Sternberg [1980] on A; <±) ••• <±) A~ that depends only on the first two arguments. properties of momentum maps and reduction, it follows that the

equations of motion for J( are Lie-Poisson. the equations for fn written out in terms the hierarchy equations. We summarize:

But these equations are just of the moments; they are thus

Theorem. The BBGKY hierarchy equations for the moments f1, ••. , fn of an n particle distribution function fn(z1, ... , zn) are equivalent to the Hamil toni an equations

where F is a functional of (fl' ... , fn) (regarded as independent variables), Jf is given by

. ) = J H1 (z)f1 (z)dz + ~ J 2 H2(z,z' )f2(z,z' )dz dz' p p

and { }A* is the Lie-Poisson bracket on the dual of the hierarchy Lie n

algebra with Lie bracket given by (HLA).

(LP)

Remark. The present formalism i5 appropriate for electrostatic interactions and has brackets compatible with those for the Poisson-Vlasov equation. Electromagnetic interactions require a Poisson structure compatible with that for the Maxwell-Vlasov equations, with fully incorporated electromagnetic field variables.

Appendix. Direct Verification of the Main Theorem for the First Two Hierarchy Equations

From (L) we have

+ I H2(z.,z.)} o, (L) . . 1 J 1 <J z1 , ... , zn

where we have replaced the brace subscript n by the explicit variable dependence z1 , ... , zn. Thus




= nJr-{f (z1 , .•• , z ;t), L H1(z.) + L H2(z.,z.)} ctz2 .. ctz n n. 1 .. 1J n 1 1 <J z1 , ... , z n

Using the identity J {f ,g}zk dzk = 0, we obtain

a = -nf{fn(z1, z ;t),L H1(zi) + .L. H2(zi,z.)} atf1(z;t) ... , n i 1 <J J zl

=-nJ{fn(z1, ... , z ;t), H1 ( z1 )} zl ctz 2 ... dz n n

-n J {fn(z1, ... ' zn;t), 2 H2(zl,z.)} d dz 1 <j J z z2 · · · 1

This is equivalent to the first hierarchy equation (Hl). For f 2 we similarly compute (assuming n ~ 3)

+ L H2(z.,z.)} ctz 3 ... dz i<j 1 J z1, ... ,zn n

+ 2 H2(z 2,zk)} z dz 3 ... dz k>2 zl , 2 n

ctz2 ... dz

n

n

dz n




Let us TJOW verify that the Lie-Poisson structure also gives (Hl) and (H2). Indeed, let F(f1,f2) be a functional of f1 and f 2. Then

Also,

{F,J{} (f1 , A* n

... ,

o:JC _ o:JC 1 Now 611 _ H1 and Of2 = 2 H2, so

. . . , fn) = <(f1 , ... , fn).[[~~ 1 • ~~ 2 • 0,0, ... o) .

h , ~ H 2, 0, ... , 0 ]}

The bracket is obtained by embeddirg as functions of z1 , ... , zn' taking the Poisson bracket there and then identifying the answer as an embedded function. The embedding introduces various combinatorial factors. For example, we find

Remark.

second slot to give another Lie algebra for which the theorem remains valid. Thus




+ 2Jf 2 ({{~ (z1), ~ H2(zl'z2)}z l l

+ {~~2 (zl,z2)' Hl(zl)}zJdzldz2

Comparing coefficients of ~~l and ~~ 2 gives (Hl) and (H2).

REFERENCES 1. P.C. Clemmow and J.P. Dougherty, Electrodynamics of Particles and

Plasmas, Addison-Wesley, (1969).

2. V. Guillemin and S. Sternberg, The moment map and collective motion, Ann. of Phys . .!1I· (1980), 220-253.

3. S. Ichimaru, Basic Principles of Plasma Physics, Benjamin, (1973).

4. J.E. Marsden, T. Ratiu and A. Weinstein, Semidirect products and reduction in mechanics, Trans. Amer. Math Soc. (to appear), (1983).

5. J. E. Marsden and A. We.instein, The Hamiltonian structure of the Maxwell-Vlasov equations, Physica D, i· (1982), 394-406.

6. J.E. Marsden, A. Weinstein, T. Ratiu, R. Schmid and R.G. Spencer, Hamiltonian systems with symmetry, coadjoint orbits and plasma physics,




Proc. IUTAM-ISIMM Symposium on "Modern Developments in Analytical Mechanics," Torino, June 7-ll, (1982).

7. P.J. Morrison, Poisson brackets for fluids and plasmas, in Mathematical Methods in Hydrodynamics and Inte rability in Related Dynamical ys ems, onf. Proc., 8 La Jolla, M. Tabor e 1982).

8. P.J. Morrison, The Maxwell-Vlasov equations as a continuous namiltonian system, Phys. Lett. BOA, (1980}, 383-386.

9. P. J. Morrison and J.M. Greene, Noncanical hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics, Phys. Rev., Letters, 45, (1970}, 790-794.

10. N.G. Van Kampen and B.U. Felderhof, Theoretical Methods in Plasma Physics, Wiley-Interscience, (1967}.

DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720 (J.E.M. and A.W.)

DEPARTMENT OF PHYSICS AND INSTITUTE FOR FUSION STUDIES UNIVERSITY OF TEXAS AUSTIN, TEXAS 78712 (P.J.M.)




PARTICLE AND BRACKET FORMULATIONS OF KINETIC EQUATIONS

Miroslav Grmela

ABSTRACT. The particle and the bracket formulations of the Boltzmann-Vlasov and the Enskog-Vlasov kinetic equations are studied.

I. INTRODUCTION

The Knudsen kinetic equation

af(x,t) _ af(x,t) at - -va. ar

a. ( 1 )

describes the time evolution of a gas of non-interacting particles (Knudsen gas). By f we denote one particle distribution function, x = (r.~), r is the position vector,~ is the velocity, a= 1,2,3, the summation convention is used throughout this paper. We shall consider only the boundary conditions that make all the integrals over the boundaries, arising in by parts integra-tions, equal zero. Two different, but equivalent, formulations of Eq. (1) bring an insight into the structure of its solutions.

The first formulation is called a particle formulation. We note that Eq. (1) is a continuity (Liouville) equation corresponding to the equations

f'a. = va. v = 0, a.

(2)

a = 1 ,2,3, that govern the time evolution of one particle. The dot denotes the time derivative. We note also that Eqs. (2) arise also as equations deter-mining characteristics corresponding to Eq. (1).

The second formulation is called a bracket formulation. By script capital letters we denote sufficiently regular functionals of f. The time evolution of a functional F is governed by 1

where

125

(3)





126 MIROSLAV GRMELA

~ = Ekin = Jdx i v2f(x,t),

is the kinetic energy and

{B,cl = Jdxf(x)[a;- c~~rx)) a~ (6~Zx)) (l (l

a ( 6B ) a ( 6C )] def - 3V MlxT ar MlxT = {B,C}LP (l (l

By 6F/6f(x) we denote the Volterra functional derivative (i.e. (OfF)¢=

{4)

{5)

!dx oF/6f(x) ¢(x), Of is the Frechet derivative). It is an easy matter to verify that Eq. {5) implies Eq. (1). An important advantage of the bracket formulation is that it reveals the geometrical structure (determined by the bracket (5)) that is naturally associated with the kinetic equation (1). 1 The bracket (5) makes the space of functionals into a Lie algebra. 1

In order to describe the physical systems that are more complex than the Knudsen gas, the Knudsen kinetic equation (1) has to be modified. We shall consider in this paper three modifications leading to the Vlasov kinetic equa-tion, the Boltzmann-Vlasov kinetic equation, and the Enskog-Vlasov kinetic equa-tion. The aim of this paper is an attempt to find both particle and bracket formulations of the above three kinetic equations. By viewing the modifications in the particle and the bracket formulation, their physical and mathematical nature is clarified.

Both formulations are well known for the Vlasov kinetic equation. 1 Section II, devoted to the Vlasov kinetic equation, is thus only a review. In Section III both particle and bracket formulations are found for the Boltzmann-Vlasov kinetic equation. Only partial results concerning the bracket formulation are obtained in Section IV for the Enskog-Vlasov kinetic equation.

II. VLASOV KINETIC EQUATION

The Vlasov kinetic equation2

af(x,t) = -v af(x,t) + af(x,t) aE(f;x) at n arn avn arn

describes a gas of particles interacting via the mean-field (i.e., depending on f) potential function E(f;x). The particle formulation of Eq. (6) is:

r = v (l (l

v = _ aE(f;x) (l ar (l

The bracket formulation1 of Eq. (6) is Eq. (3) with the bracket (5) and

(6)

(7)



PARTICLE AND BRACKET FORMULATIONS OF KINETIC EQUATIONS 127

def ~ = Ekin + Epot = Etot• (B)

where the potential energy · Epot is related to E(f;x) introduced in Eq. (6) by

oEpot (f) _ . of(x) - E(f,x),

Etot is the total energy.

III. BOLTZMANN-VLASOV KINETIC EQUATION

In addition to the mean-field-type Vlasov interaction, the Boltzmann-Vlasov kinetic equation3

af(x,t) = _ af(x,t) + af(x,t) aE(f;x) at va ar av ar a a a

+ J dx' I dy' I dy[W(x' ,y' ;x,y)f(x' ,t)f(y' ,t)

- W(x,y;x' ,y')f(x,t)f(y,t)]

(9)

( 1 0)

takes into account also collision-like interactions. We have used in Eq. (10) the following notation: y = (g_,y), x' = C!:.' ,y_'), y' = (.8_' ,yo). The quantity W(x,y;w',y') is the probability of the transition of two particles from the state (x,y) to the state (x' ,y'). The transition occurs due to the colli-sion-like interactions. The quantity W is required to satisfy the following seven properties: 4

(1) 8 : W>O forall x,y,x',y',

(2) 8: W(x,y;x' ,y') = W(y,x;x' ,y') = W(x,y;y' ,x') = W(y,x;y' ,x')

(3) 8 : W = 0 unless ~· :: ~· .8_' :: R

(4)B:

(5)B:

(6)B:

(7)B:

W = 0 unless v2+v2 = v' 2+V' 2 and v + V = v' + V' - - - -W(~,y_,g_,y_; ~· ,y_' ,R' ,yo) = W(~,-y_,g_,-y_; ~· ,-y_' ,.8_' ,-Y.') W(x,y;x' ,y') = W(x' ,y' ;x,y)

W = 0 unless r :: R.

It has been observed in Ref. 5 that

r = v a a v = _ aE(f;x) + F(B)(f·x) a ar a ' ' a

( 11)



128

where

MIROSLAV GRMELA

F~B)(f;x) = f(),t) I dy1 I dx 1 I dy I: dn v~ W(x(n),y(n);

x 1 ( n) ,y I ( n)) [ f ( x ( n) , t) f (y ( n) , t)] , ( 12)

dn) = r_, .B_(n) = R = r_, ~(n) = ~- n~ 1 , ~~ (n) = ~ + (l-n)~ 1 , is the particle formulation of the Boltzmann-Vlasov kinetic equation (10). It is easy to verify that a~ (f(x,t)F(B)(f;x)) equals the third term on the right hand

a a side of Eq. (10). The "friction force" F(B) introduced in (12) is distin-guished from other admissible friction f~rces F(B) (i.e., from those forces

for which - a~ (fF~B)) equals the third term on the right hand side of (10))

of the followin~ property: [(B)(fde;x) = 0, where fde is a solution of

I dx 1 I dy 1 I dy W(x,y;x 1 ,y 1 )[f(x 1 ,t)f(y 1 ,t)- f(x,t)f(y,t)] = 0.

We note that because of the appearance of the friction force [(B), the system (11) is non-Hamiltonian. The energy Etot introduced in (8) remains, however, a constant of motion.

Now we turn our attention to the bracket formulation of the Boltzmann-Vlasov kinetic equation (10). It is an easy matter to verify that Eq. (3) with

def ~ = Ek . + E t - To® = A 1n po

and the bracket

{B,C} = {B,C}LP + {B,C}8 ,

( 13)

( 14)

is the bracket formulation of the Boltzmann-Vlasov kinetic equation (10). The functionals Ekin and Epot have been introduced in (4) and (9),

I def ® = - dx f(x,t) log f(x,t) = e 8

T0 is a positive constant, { }LP is the Lie-Poisson bracket (5) and

{B,C}8 =~I dx I dx' I dy I dy 1 W(x,y;x 1 ,y 1 )

[( oB oB ) ( oB oB )] X rrrxT + MTYT - MTX'T + 6f'CY'T ( 15)

[ 1 ( oc oc ) 1 ( oc oc )] x exp To - Of(x') - 6f'CY'T- exp To - MfXT- MTYT '




The functional A has the physical meaning of the non-equilibrium free energy. If A is evaluated at an equilibrium state feq (feq is a time independent solution of Eq. (10) that is invariant with respect to the transformation f(r_,y) ~ f(r_,-y) - see Refs. 5,6) then A becomes the free energy intro-duced in equilibrium thermodynamics. In order to verify that Eq. (3) with ~

and { , } introduced in (13) and (14) implies Eq. (10), we use the following observations:

( ~eB ~eB ) ( ~eB ~eB ) f(x)f(y) - f(x 1 )f(y') = exp - MTXT - mYT - exp - MTX'T- mY'1 ,

{B,®}LP :: 0 for al 1 B,

{B,Ekin + Epot}B :: 0 for all B.

We shall end this section by noting some properties of the bracket (14).

REMARK III.l. The following inequality

( 16)

( 1 7)

( 1 8)

( 19)

holds for all functionals B; the equality in (19) holds only for solutions of

~B + ~B _ ~B + ~B MTXT mYT - MTX'T mY'T (20)

This observation implies that

{A,A} ~ 0 (21)

and consequently dA df ~ 0. (22)

The equality in (21) and (22) holds only for solutions of Eq. (20) in which the functional B is replaced by the functional e 8 (see (14)). The inequal-ity (22) (the famous Boltzmann H-theorem3) together with some other results (see Refs. 7,6) then implies that the Boltzmann-Vlasov kinetic equation is compatible with equilibrium thermodynamics. One of the advantages of the bracket formulation is that it implies immediately the result (22) containing a significant information about solutions of the Boltzmann-Vlasov kinetic equa-tion. In fact, the result (22) becomes a property of the bracket and thus a property of the geometrical structure that is naturally associated with the Boltzmann-Vlasov kinetic equation.

REMARK III.2. If we linearize the Boltzmann-Vlasov kinetic equation (10) about an equilibrium state (i.e., the time independent solution of (10) that is



130 MIROSLAV GRMELA

moreover invariant with respect to the transformation f(.!:_,~, t) ~ f(.!:_,-~, t)) we obtain the linearized form

{B,c}ilin) = 4io I dx I dx' I dy I dy' W(x,y;x' ,y')

[( 6B 68 ) ( 6B 6B )] mxT + fflYT - fflX'T + 81\y'f (23)

X [( 6~ 6~ ) ( 6~ 6~ )] - 8f(XT - fflYT - - 8f(XT} - 81\y'f

of the bracket (15). Since

{B,C}~lin) {C,B}~lin) (24)

the bracket ( 1 · ) def ( 1 · )

{ ' } 1 n = { ' } LP + { ' } 8 1 n (25)

makes the space of functionals into a Lie-admissible algebra. 8•9

REMARK III.3. In the context of the Knudsen or the Vlasov kinetic equa-tion the functional ~ in Eq. (3) can be either the total energy (see (4) and (8)) or the sum of the total energy and a functional of the type Jdx F(f), where F is an arbitrary, sufficiently regular, function.

Thus in particular, the free energy functional A (see (13)) can serve as the functional ~ in Eq. (3). On the other hand, in the context of the 8oltzmann-Vlasov equation (10), only the free energy functional A can serve as the functional ~ in Eq. (3).

IV. ENSKOG-VLASOV KINETIC EQUATION

The Enskog-Vlasov kinetic equation has the same form as Eq. (10). The transition probability W satisfies, however, another list of seven properties (we shall denote them (l)E- (?)E). The properties (l)E- (4)E remain the same as the properties (1) 8 - (4) 8. The remaining three properties are the following:

(5)E: W(.!:_,~,~.y_; .!:.' ,~' ,~' ,y_') = W(.!:_' ,-~' .~' ,-_IL'; .!:_,-~.~.-.'L)

(6)E: Ws(x,y;x' ,y') satisfies the properties (1) 8 - (6) 8 (7)E: f dx' f dy' Wa(x,y;x',y') = (va- Va)(ra-Ra)w(.!:_,~).

The quantities Ws and Wa are introduced as follows:




W(x,y;x' ,y') = Ws(x,y;x' ,y') + Wa(x,y;x' ,y')

Ws(x,y;x',y') = ~ (W(x,y;x',y') + W(x',y';x,y)) (26)

Wa(x,y;x',y') = ~ (W(x,y;x',y')- W(x',y';x,y))

w is an arbitrary function of ~ and R satisfying w(~.Rl = w(R.~).

By comparing the property (l)B- (7)B and (l)E- (6)E we see that the Boltzmann-Vlasov kinetic equation is in fact a special case of the Enskog-Vlasov kinetic equation. For the Boltzmann-Vlasov equation, Wa = 0 and ~ = R· From the physical point of view, the Erskog-Vlasov kinetic equation takes into account finiteness of the size of the colliding particles (i.e., ~ ~ Rl. It is also the simplest kinetic equation that implies the thermo-dynamic equation of state of the van der Waals-type (see more in Ref. 6). Thus the Enskog-Vlasov kinetic equation is the simplest kinetic equation describing a gas that undergoes a gas-liquid plase transition (provided the quantities W and E are appropriately chosen).

Now we shall try to find the bracket formulation of the Enskog-Vlasov kinetic equation. We look for a bracket { , } and a functional Q such that Eq. (3) is the bracket formulation of the Enskog-Vlasov kinetic equation with the functional ~ given in Eq. (13) and the entropy functional

0 = 0 6 - Q, (27)

where 0 8 has been introduced in Eq. (14). We shall look for the bracket in the form

{ ' } = { ' }LP + { ' }B + { ' }E' (28)

{ , }LP and { , }B have been introduced in (5) and (15) respectively. The transition probability W appearing in (15) is now however replaced by Ws introduced in (26). The functional Q is assumed to depend on f only through its dependence on

n(!:_,t) =I d~ f(!:_,~,t). (29)

It is easy to see that Eq. (3) is the bracket formulation of the Enskog-Vlasov kinetic equation provided

{F,A}E = {F,T0Q}LP - I dx I dy I dx' I dy'

w ( , , ) oF a x,y;x ,y of(x,t) [f{x' ,t)f(y' ,t) + f(x,t)f(y,t)]

holds for all sufficiently regular functionals F. We note that if W = 0 a

(30)

and Q = 0 then {F,A}E = 0 and thus we indeed return to the bracket formula-



132 MIROSLAV GRMELA

tion of the Boltzmann-Vlasov kinetic equation that was introduced in the pre-vious section. If Wa t 0 then we are free to choose Q so that the bracket { , }E (and consequently also the bracket { , } introduced in (28)) possesses some chosen properties and the equation (30) is satisfied. We shall choose the property (21) (and consequently also (22)) so that the observations made in Remark III.l will be valid also in the case of the Enskog-Vlasov kinetic equation. By using the property (7)E and the results obtained in Ref. 7,6 we find that the inequality (21) holds provided

( 31 )

We note that Eq. (31) relates w and Q since vi enters the bracket in Eq. (3) and Q the functional ~ in Eq. (3), the equation (31) represents a relation between the bracket and the functional ~.

REFERENCES

1. J. E. Marsden and A. Weinstein, Physica D, ~. 394 (1982). 2. A. A. Vlasov, Many Particle Theory and its Application to Plasma, Gordon

and Breach Sci. Publ., New York (1961). 3. L. Boltzmann, Wissenschaftlichen Abhandlungen von Ludwig Boltzmann, Chelsea,

New York, Vol. 2 (1968). 4. L. Waldmann, in Handbuck der Physik, vol XII, ed. S. Fluge, Springer-

Verlag (1958). 5. M. Grmela, Ann. de la Fondation Luis de Broglie z, 293 (1982). 6. M. Grmela and L. S. Garcia-Colin, Phys. Rev. A, 22, 1295 (1980).

M. Grmela and W. G. Laidlaw, J. Chern. Phys. 78, 5151 (1983). 7. M. Grmela, Can. J. Phys. ~. 698 (1981). 8. M. C. Tomber, Hadronic Journal~. 360 (1982). 9. R. H. Oehmke, Hadronic Journal~. 518 (1982).

ECOLE POLYTECHNIQUE OF MONTREAL MONTREAL, QUEBEC, H3C3A7 CANADA




NONCANONICAL HAMIL TONI AN FIELD THEORY AND REDUCED NHD

Jerrold E. Marsden1 and Philip J. Morrison1•2

ABSTRACT. Aspects of noncanonical Hamiltonian field theory are reviewed. t·1a ny sys terns are Hamiltonian in the sense of possessing Poisson brae ket structures, yet the equations are not in canonical form. A particular system of this type is considered, namely reduced magnetohydrodynamics (RtiHD) which was derived for tokamak modelling. The notion of a Lie-Poisson bracket is reviewed; these are special Poisson orackets asso-ciated to Lie groups. The RMHD equations are shown to oe Hamiltonian for brackets closely related to the Poisson bracket of a semi-direct product group. The process by which this bracket may oe derived from a canonical Lagrangian description by reduction is descrioed.

l. INTRODUCTION. The basic idea underlying noncanonical Hamiltonian field theory is that systems which are not Hamiltonian in the traditional sense can be made so by generalizing the Poisson bracket. In fact, Poisson brackets for most of the major non-dissipative plasma systems have now been obtained. Four of the most basic systems are as follows, in chronological order:

l. Ideal ~1HD - Morrison and Greene [1980]. 2. Maxwell-Vlasov equations - Morrison [1980] and ~~arsden and ~ieinstei n

[1982]. 3. Multifluid Plasmas - Spencer and Kaufman [1982]. 4. BBGKY hierarchy- t~arsden, Morrison and Weinstein (in these proceed-

; ngs ) . For additional historical information and other systems, see Sudarshan

and ~1ukunda [1983] and the reviews of Morrison [1982], Marsden et al., [1933] and the lectures of Holm, Ratiu and Weinstein in these proceedings. The purpose of this paper is to discuss some of the basic ideas and apply them to reduced magnetohydrodynamics (RHMD).

1980 Mathematics Subject Classification 76W05, 58F05.

1Research partially supported by DOE contract DE-AT03-82ER-12097. 2Research partially supported by DOE contract DE-FG05-80ET53088.

133





134 Jerrold E. Marsden and Philip J. Morrison

We now describe some of the uses for Poisson structures that are now surfacing.

l. Categorizing fields. To specify a Hamiltonian field theory, a Hamiltonian and a Poisson bracket are chosen. The structure of the bracket can shed light on the theory, so a categorization by the bracket form is natural.

2. Casimirs. To each bracket there are functions that Poisson commute with every function; these are called Casimirs (see Sudarshan and Mukunda [1983], Littlejohn [1982] and Weinstein's lecture). Casimirs are invariants for any Hamiltonian system when a given bracket is used.

3. Stability. Casimirs are useful in testing for linear and nonlinear stability by a method going back to Arnold in the mid 1960's. See the lectures of Holm and Weinstein, Holm et. al. [1983], [1984] Abarbanel et. al [1984] and Hazeltine, Holm and Morrison [1984] for further information.

4. Quantization. Dashen and Sharp [1968] use noncanonical brackets for quantum observables in the context of current algebras. Goldin's lecture in these proceedings indicate how Poisson structures may be useful in quantization. The quantum approach also can be used to derive classical brackets, as in Dzyaloshinskii and Volovick [1980].

5. Chaos. As in Holmes and Marsden [1983], noncanonical Poisson struc-tures can be used to prove the existence of chaos in perturbations of integrable systems.

6. Limits, Averaging and Perturbations. As in Littlejohn [1979] and Kaufman's lecture in these proceedings, Poisson structures can play a role in understanding the processes by which one passes to averaged systems or limiting systems and to what degree these more idealized models are good approximations to a more encompasing model. A general framework in which these processes are hoped to be under-stood is given in Montgomery, Marsden and Ratiu's paper in these proceedings.

7. Numerical Schemes. It is hoped that a deeper understanding of Hamiltonian structures will enable one to design algorithms with superior accuracy. For example it is known that algorithms which are energy preserving have better stability properties (see Lewis [1970], Chorin et. al. [1978] and references therein). Also, the successful vorticity algorithms of Chorin-Hald-Beale-Majda are known to be Hamiltonian (see ~~rsden and Weinstein [1983]). See Holm, Kuperschmidt and Levermore [1984] for some related results.



NONCANONICAL HAMILTONIAN FIELD THEORY AND REDUCED MHO 135

A tokamak uses a toroidal magnetic field configuration to confine hot plasma (see, for example, Chen [1974]). The physics of a tokamak is compli-cated and encompasses a wide range of scales. Kinetic and fluid models are typically used. In particular, RMHD is a simple fluid model that is obtained by approximating three dimensional incomressible t"HD with the goal of high-lighting the dominant physics (Strauss [1976, 1977]). RMHD is a member of a family of such fluid models that strive to explain major tokamak features and yet remain tractable (see Rosenbluth et. al. [1976], Hasegawa and Mirna [1977], Hazeltine et. al. [1983] and Hazeltine et. al. [1984]). RMHD has achieved notable success (see Carreras et. al. [1979]). The reader will notice that R~1HD is a generalization of the two dimensional Euler equations; perhaps the techniques discussed in the lectures of Zabusky and Beale can be adapted to RMHD.

The paper is organized as follows. In §2 we review some features of canonical and noncanonical Hamil toni an field theory. R~1HD and its non-canonical brackets are presented in §3. In §4 the theory of Lie-Poisson brackets is reviewed and the brackets for RMHD are shown to consist of two pieces, one of which is a Lie Poisson bracket for a semi-direct product group. This group is related to the helical lagrangian paths followed by fluid particles in an idealized limit. The methods by which these brackets are obtai ned from the Lagrangian description by reduction and from ide a 1 ~1HD

by a limiting procedure are outlined in §5.

2. HAMILTONIAN DESCRIPTION OF CLASSICAL FIELDS. As in classical texts such as Wentzel [1949] and Goldstein [1980], a system of evolution equations (partial differential or integral equations for example) is said to be in canonical Hamiltonian form if they can be written in the form

ank oH ank - oH Tt = onk , -at = -k, k = 1 , 2,

on ••• , N ( 2.1 )

where nk(x,t) are the basic field variables and nk(x,t) are their conju-gate momenta, x belonging to a region V of three space. Here H is a func tiona 1 of the fields n and n, the dependence being denoted H [n, n]. We recall that the functional derivatives are defined by

J oH - d3x = onk nk v

limit H(n,n + £n) - H(n,n) £-+0 £

( 2. 2)

(sum on k), with a similar definition for oH/onk. The reader should consult one of the aforementioned texts for basic examples of this formalism



136 Jerrold E. Marsden and Phi 1 i p J. ~~orri son

such as the Klein-Gordon field. This theory from the point of view of symplectic geometry, along with additional examples, is found in Chernoff and Marsden [1974] and Abraham and Marsden [1978, Section 5.5].

Poisson brackets are defined for functionals F and G of the fields ll,7T by

{F,G} = f (§£___ _§§_ - _j§_ _iE_) d\ V onk 6\ onk oTik

(sum on k); note that {F,G} is a real valued function of (n,TI). readily verified that the evolution equations (2.1) are equivalent to

F = {F,H}

It is

The bracket (2.4) assigns the new functional {F,G} to two given ones F and G, and has the following basic properties:

(i) {F,G} is linear in F and G (bilinearity) ( i i) {F,G} = - {G, F} (anti symmetry)

(iii) {E, {F,G}} + {F, {G,E}} + {G, {E,F}} = 0 (Jacobi's identity) (iv) {EF,G} = E{F,G} + {E,G}F (derivation).

(2.4)

( 2.5)

(i), (ii) and (iii) define a Lie algebra. A bracket on functionals defined on a phase space P (the space of (n,n) above being an example) satisfying (i)-(iv) is called a Poisson structure. (See ~~einstein's lecture in these proceedings).

The four basic plasma physics examples listed in the introduction have equations that can be written in Hamiltonian form (2.5) for a suitable Poisson structure {F,G}; however, this Poisson structure does not have the canonical form (2.4) and correspondingly, the evolution equations do not have the canonical form (2.1). These examples clearly demonstrate the need for taking the wider view of non-canonical Hamiltonian field theory -- one demands only a Poisson structure and a Hamiltonian functional such that the equations of motion have the form (2.5). If the basic fields of the theory are denoted 1/Ji(x,t), i- 1, ... , n, then the Poisson structure is often of the form

{F,G} = J ~ oij ~ d3x v 61)!1 61/!J

where Oij isamatrixoperator of 1j!=(1j!i). Properties(i)and(iv)

(2.6)

are automatic from the form (2.6), and (ii) holds if Oij = -Oji. On the other hand, Jacobi's identity is a relatively complicated condition on Oij that requires ingenuity or a deeper insight into how bracket structures arise. Of course (2.6) includes (2.4) as a special case. A common class of Poisson



NONCANONICAL HAt~ILTONIAN FIELD THEORY AND REDUCED ~1HD

structures have the form (2.6) where

0 i j ,J,k i j = '+' ck

137

where c~j are structure operators for a Lie algebra. For these, Jacobi's i j identity follows from Jacobi's identity for ck . These Lie-Poisson structures

are examples of Poisson structures and will be considered in §4. There are three ways to obtain Poisson structures for a given system.

First of all, one can proceed by inspection and analogy with known brackets. The verification of Jacobi's identity can be done directly or with the assis-tance of Lie-Poisson structures. Second, one can introduce potentials (i.e. Clebsch variables) and induce a bracket on functionals of the physical fields by means of canonical brackets on functionals of the potentials and their conjugate momenta. See, for example, Morrison [1982], Holm and Kupershmidt [1983] and Marsden and Weinstein [1983] for accounts of this method. Thirdly, and perhaps most fundamentally, one can first write the theory in terms of a Lagrangian (or material) representation for the matter fields with the basic fields be1ng the particle displacement field nk and its conjugate momentum nk. The canonical bracket (2.4) then induces a non-canonical bracket on the Eulerian (or spatial) fields by means of the map taking the Lagrangian to the Eulerian description. This procedure is a special case of reduction and was the method Marsden and Weinstein [1982] used to obtain the l~axwell-Vlasov

bracket and which Spencer [1982] used to obtain the multifluid plasma bracket. Marsden, Ratiu and Weinstein [1983] used this method for several other basic systems as well and its basic features are described in Ratiu's lecture in this volume. See the article of Kaufman and Dewar in these proceedings for a related approach.

3. REDUCED MHD AND ITS BRACKET. As noted in the introduction, the RMHD equations are obtained by approximating the ideal incompressible MHO equations with the goal of describing the dominant tokamak physics. The approximation is tailored to the tokamak toroidal geometry and is discussed in the original papers of Strauss [1976,7]; see also 1·1orrison and Hazeltine [1983].

The tokamak geometry is sometimes described by toroidal coordinates: (r,e) represent polar coordinates in a plane perpendicular to the major toroidal axis; this plane is called the poloidal plane. The ilngular coordinate along the major axis of the torus is denoted 1: and is called the toroidal ~ngle. Thus, 8 and c; are 2n-periodic while 0 .::=_ r .::=_a, where r =a represents the torus boundary.

The RMHD fields are obtained by considering the components of the three dimensional velocity field v and magnetic field S in the poloidal plane.




The divergence free assumption on v (to lowest order) and the equation V·S = 0 for B leads one to consider corresponding potentials for their poloidal projections, namely

(i) a scalar vorticity U(r,8,~;;,t) (so v x (U2) is the poloidal velocity, where ~ is a unit vectur in the ~;; direction)

and (ii) a poloidal flux function (or magnetic potential) 1)!(r,8,i;;,t) (so 'V x 1J!2 is the poloidal magnetic field).

The toroidal components in the RMHD approximation to leading order are regarded as constant.

The RNHD equations in what is called the low B limit (i.e. neglecting pressure effects) are

au _ aJ at- [U,¢] + [1J!,J] - a1;

C!1)! - C!¢ at - [1J!,¢] - a~;;

where [f,g] = l(C!f .£.9.- af ~)is the canonical Poisson bracket in ·the r ar 38 a8 ar poloidal plane and where

')

(3.la)

(3.lb)

and V.i¢ = U, so ¢ is the velocity stream function

2 1/11)! = J, the toroidal current

ACJ lAa A A Here v1 = r- +- 8- is the poloidal gradient operator and r and 8 ' ar r (!8 are unit vectors along the r and e coordinate axes. We recall that the l·lHD current is J = 1/ x B so for B in the poloidal plane, J points in the toroidal direction.

The equations (3.1) are to be supplemented with appropriate boundary conditions on ¢ and 1)! at the boundary r =a.

We now describe the sense in which equations (3.1) are Hamiltonian. There is a conserved Hamiltonian, which is just the kinetic energy of the fluid plus the magnetic field energy:

H =; J (I Ill <PI2 + IV'11J!I2)d3x v

(3.2)

where V is the torus, 0 _:: r _::a, 0 _:: G _:: 2n, 0 _:: s _:: 2n. There are addi-tional constants of the motion that are important (for the stability analysis for example) which won't be discussed here; see Morrison and Hazeltine [1933].

Poisson brackets for the RMHD equations (3.1) are as follows; let F and G be functionals of U and l/! and set



NONCANONICAL HAMILTONIAN FIELD THEORY AND REDUCED t-lHD 139

The bracket is due to Morrison and Hazeltine [1983]. Using the fact that oH oH 8TI = -cp and o1)i = -J, it is easy to show that the equations (3.1) are

equivalent to the lfamil toni an form

F = {F,H} ( 3.4)

The only property of the bracket (3.3) which is not obvious is Jacobi's identity. It is verified directly in Morrison and Hazeltine [1983]. In the next section we shall verify that the first two terms of (3.3) are a Lie-Poisson bracket for a semi-direct product; this will give another proof of the Jacobi identity. In the final section we shall discuss the derivation of (3.3) by reduction and approximation (the method of Clebsch potentials is discussed in Morrison and Hazeltine [1983]).

4. LIE-POISSON BRACKETS AND SEMI-DIRECT PRODUCTS. A key feature of the first two terms of (3.3) is the linear dependence on U and 1)1. Brackets of this type are called Lie-Poisson and the associated phase space is the dual of a Lie algebra. We shall describe this construction in this section and shall show that the first two terms of (3.3) are Lie-Poisson brackets on the dual of a semi-direct product Lie algebra. The last term of (3.3) will be discussed in the final subsection.

A. Lie Poisson Brackets. Let G be a Lie group and b} its Lie algebra. ~Je recall (see Abraham and Marsden [1978, Sect. 4.1] for background) that ~ is the tangent space to G at the identity and that for ~.n E 0}• Lie bracket of ~and n is given by the formula

the

d d -1 I [~.n] = dS dr g(s) h(r) g(s) r=s=O ( 4.1)

where g(s) and h(r) are arbitrary smooth curves in G such that

g(O) = e, h(O) = e, g'(O) = ~ and h' (0) = n

Let b;}-* be the dual space of linear functionals on OJ- with the pairing between elements )J E IJ)* and ~ E Of- being denoted ()J,~). In the infinite dimensional case we choose ~* together with a pairing satisfying: (]J,E; > = 0 for all )J implies ~ = 0 (a non-degeneracy condition) in a



140 Jerrold E. tt.arsden and Philip J. t·1orrison

way appropriate for the problem at hand.

For F:~*-+IR, wedefine ~~E1 by

d -1

<- oF -d F()J + E)J) -Q = \l, T} E E- ulJ ( 4. 2)

? which is consistent with (2.2) if ( , >

Lie-Poisson bracket is defined by is taken to be the L'--pairing. The

- 4 ( I i':: oG J ) {F, G}± - - \l, L O\l' O)J . (4.3)

There are two choices, + or -. For this paper, we shall use the + bracket, but the - bracket is also used. To understand the ± distinction we need to recall how (4.3) is derived.

If F and G are real valued functions on ti(*• we can extend them by left translation to functions FL and GL on· T*G, so FL restricted to T;G = 6J--* is F. But T*G carries canonical Poisson brackets { ·, • }T*G and we have

{FL' GL} restricted to T*G is {F ,G} T*G e

Similarly, extending by right invariance,

{FR,GR} restricted to T*G is {F ,G }+ T*G e

(see Marsden, Wei ns te in e t. a l . [1983] for details of the proof) . Thus, the ±Lie-Poisson brackets are naturally obtained from canonical brackets on T*G. The process just described of getting brackets on &(* from those on T*G is a special case of a more general procedure called reduction (~1arsden and Weinstein [1974]). Thus, whether one uses the ± bracket depends on whether the system under investigation corresponds to a left (-) or right(+) invariant system on T*G. In fact the space T*G often corresponds to material, or Lagrangian coordinates. The above picture relating T*G and d}-* has its origins in the fundamental work of Arnold [1966]; see Ratiu's lecture in these proceedings for further information.

The Lie-Poisson brackets (4.3) make~* into a Poisson manifold. The properties (i)-(iv) of §2 can all be verified directly. For example, Jacobi's identity follows from symmetry of the second variations and from Jacobi's identity for the Lie bracket ] on~- Alternatively one cansimply observe that T*G, being a canonical manifold (cotangent space), is a Poisson manifold and that the Poisson bracket properties are inherited on OJ-* from T*G by the reduction procedure described above.



NONCANONICAL HAMILTONIAN FIELD THEORY AND REDUCED t~HD

B. The Lie-Poisson bracket for the group of canonical transformations. The first term of (3.3) conforms to the Lie-Poisson format (4.3). A bracket of this type occurs for the Vlasov Poisson equation (see Morrison [1980] and t.Jarsden and vJeinstein [1982]) and for the two dimensional incompressible

141

Euler equations (see Morrison [1982] and Harsden and Weinstein [1983]. vie note that Arnold [1966] discussed the Hamiltonian formuation of Euler's equations, but did not explicitly give this bracket.)

If G is the group of canonical transformations of IR 2 then the Lie algebra 6-J of G consists of the Hamiltonian vector fields. (See Ebin and r~larsden [1970] for the function space topologies used to make these asser-tions precise). If we identify Hamiltonian vector fields with their gener-ating functions (a constant is dropped in making this identification) then the Lie algebra Of is identified with functions and the Lie bracket is the Poisson bracket (see Marsden and Weinstein [1982]; here we use the standard left Lie bracket, while they used the right Lie bracket). The dual of OJ:* is identified with functions on IR 2 (or more properly densities on IR2 ) and the pairing of 6)-* with ? is the usual L2 pairing. Thus, we conclude that the first term of (3.3) is the(+) Lie-Poisson bracket for the group of canonical transforamtions on IR2 . How this term arises from a Lagrangian description is discussed in §5.

C. Semi-dire::t products. We now want to show that the first two terms of (3.3) taken together still define a Lie-Poisson bracket. This involves the notion of a semi-direct product, so we review the abstract construction first.

Let G be a group and V a vector space. Let p be a representation of G on V, so p is a homomorphism from G to the group of invertible linear transformations of V. We write p (v) for p{g){v) for notational g convenience. The semi-direct product G IX V is, as a set, G x V, and has group multiplication given by

{g,ul) ·(g2,u2) = {glg2' ul + Pg 1 (u2)) · (4.4)

One easily checks that G P< V is a group. Using formula (4.1) and (g,u)-1 = (g- 1 ,-p _1 ), one can readily prove that the Lie bracket for G ~ V is given

g by

(4.5)

where pi:V + V is defined by

Pk(v) = d~ Pg{o:)(v)jo:=O



142 Jerrold E. Marsden and Philip J. t'lorrison

where g( E) is any curve in G satisfying g(O) = e and g' (0) = r;,. This Lie algebra is denoted 6). 0< V.

For example, the Euclidean group E(3) of rigid motions of m3 is the semi-direct product of the rotation group 0(3) and the translation group IR 3 ; E ( 3) = 0 ( 3) IX IR 3 where 0 ( 3) acts on IR 3 by rna t r i x multi p l i cation . This of course is well-known (see, for example, Talman [1968] or Sudarshan and Mukunda {1983, p. 25lff]).

If we identify("}- x V)* with &)-* x V*, combining (4.3) and (4.5), we see that the Lie Poisson brackets on (OJ- 0< V)* are given by

where ( 11,a) E OJ-* x V* and , > denotes the pairing on the appropriate space.

( 4.6)

Now let G = Sym(IR 2) be the group of canonical transformation n of IR2 or of a region in IR2 and let V = F(IR 2) be the space of functions

2 -1 k on IR and let G act on V by Lie transport: pn(k) = kon . (The inverse is to make it a (left) representation). The induced action of (1( on V is by Lie differentiation of vector fields or in terms of

functions, by Poisson brackets:

( 4. 7)

where { , } is the standard Poisson bracket in IR 2 (the same as [ , ] used in 3.1). Substituting (4.7) in (4.6) with a= \jl, v = U and using the + Lie-Poisson structure, (4.6) reduces to the first two terms in (3.3). In summary, we have proved that the first two terms of (3.3) are the Lie-Poisson bracket associated with the semi-direct product of canonical trans-fa rma ti ons and functions, Sym(IR2 ) e< F(IR 2 ) •

D. Helical Symmetry. Finally, we consider the last term of (3.3). First of all, this term is in almost canonical form and Jacobi's identity for it is readily checked. Combined with the result of part C, this verifies that indeed (3.3) defines a Poisson structure.

If we confine ourselves to solutions with helical symmetry, then the last term of (3.3) can be transformed away and the entire bracket then becomes Lie-Poisson. This proceeds as follows: fix a number q0 and consider the additive group IR acting on (r,e,c:) space by

qo -1 H5 (r,e,;;;) =(r,e + sq 0 , ;;; + s). (4.8)



NON CANON I CAL HAMIL TOtliAN FIELD THEORY AND REDUCED ~'1HO

This is the group of helical transformations with pitch q0• If l)Jh is invariant under H , it has the form s

143

( A( -1 1)1 r,e.~.t) = 1)1 r,e- q0 ~.t) (4.9)

as is easily checked, and similarly for U. One can check that for helically symmetric functions, the transformation 1)1 + l)ih' u + uh given by

2 l)Jh(r,e,t) = ~(r,e,t) + ~q

0

Uh(r,8,t) = U(r,8,t)

transforms the bracket (3.3) to

J, [cSF cSGJ ([ cSF cSG] [cSG M ]) 2 {F,G} = u uh wh, wh + 1/!h Lc51/.h' wh - 61/!h' wh d x

( 4.10)

( 4.11)

Thus, in the single helicity case, the bracket (3.3) transforms via (4.9) to the Lie-Poisson bracket (4.11). (See Morrison and Hazeltine [1983]).

One can also transform away the third term in (3.3) by using Lie trans-forms. One attemps to solve the equation

( 4 .12)

for l)Jh given 1/J. In general this is impossible because ~ must be a periodic variable. However, if it were possible, one sees that formally, this transforms away the third term of (3.3) (see the Appendix for this calculation). Following the dictates of Lie transform theory, we get a good approximation to (4.12) by averaging (see Guckenheimer and Holmes [1983, Chapter 4]). Since the helicity condition (4.9) gives the solution (4.10) to (4.12), it is natural to average 1j; first with respect to H~:

J2n q l)Jav(r,e,~) = 0 l)JoHso(r,e,~)ds

where t is suppressed. Then l)Jav is helically invariant. :-.low let 1/Jh 2 lj!av + r /2q0, Uh = Uav. The map

transforms the bracket (3.3) into (4.11), which is shown just as in the appendix.

( 4.13)

In fact, one can verify that (4.14) is a momentum map for the action of




the semi-direct product Sym(IR 2 ) x F(IR 2) on the space of U(r,B,t;) and l)J(r,e,~;;) with the bracket (3.3) given as follows. (See Ratiu's lecture, Abraham and Marsden [1978] and ~1arsden, Weinstein, et. al. [1983] for the basic definitions and properties of momentum maps). If n ESym(IR2) and f E F(IR 2), let them act on (\)!, U) by

where

and

(l)J,U) 1---7 (1/Jav, Ua)

-1 ) 1/J (r,e,~;;) = l)J(n(r,e + f(r,e)q 0 ,~;; + f(r,B)) av

U (r,B,~;;) = U(n(r,e),~;;) av

This remark is consistent with the fact that momentum maps are always Poisson maps and the fact that Lie transforms, averaging and reduction are closely related.

S. REI'·'i.~RKS AND CONJECTURES ON LAGRANGIAN COORDINATES, REDUCTIONS AND APPROXH1ATIONS. The preceeding discussions still leave open the question of how to derive brackets like (3.3) or (4.11). For the single helicity case, the derivation of (4.11) from canonical Lagrangian coordinates proceeds as follows. LJe assume that individual fluid particles follow trajectories that commute with the helical group and that the magnetic field is Lie tra sported. Thus, the particles move by means of volume preserving diffeomorphisn.~

3 3 qo qo ¢:IR ->- IR such that H o¢ = ¢ 0 H for each s. Call the group of such s s ¢'s, :K. Now we add a constraint that is consistent with the RMHD approxi-mation namely that the toroidal speed of the particles is fixed; thus the configuration space for the fluid is Jt;s1 where s1 is the group of H~ 0 But Jf/Sl is the group of transformations of the helices (orbits of the action (4.8)), which is isomorphic to Sym(IR 2 ). JC;s 1 then is the basic configuration space for a single helicity fluid.

Thus, the phase space is T*Sym(IR 2). Nowthemagnetic potential is Lie transported by the helical action of Sym(IR2 ) as in §40. Thus, one can reduce T*Sym(IR 2 ) as described in Ratiu's lecture to obtain a Lie-Poisson structure for the semi-direct product of sym(IR2 ) and the space on which the magnetic potential lives. This produces exactly the structure (4.11).

To obtain the bracket ( 3. 3) from a canonical Lagrangian picture we proceed as follows. As above, we build the Rfi,HD approximation we have in mind into the Lagrangian configuration space. Choose q0 = oo so H; = Hs is just translation in the ~;;-direction; these Hs form an s1 group. Now to allow ;:; dependence we choose the basic configuration space to be the



NONCANONICAL HAMILTONIAN FIELD THEORY AND REDUCED ~HD 145

group c of volume preserving transformations that map s1 orbits to s1

orbits. Again the magnetic potential is Lie transported. However, our magnetic field will be assumed to have a dynamic component only in the poloidal plane so it is consistent to choose the subgroup S of C consisting of di ffeomorphisms that are the identity in the r, 8 variables ("streaming" diffeomorphisms: (r,8,s) ~ (r,e,s + g(r,8))). Our basic configuration space is then c;s, which is roughly, speaking, the s-dependent diffeomor-phisms of the (r,e) plane, and so the phase space is T*( CIS). However, the magnetic potential is Lie transported, so we need to reduce T*( C/S) by the further symmetry group corresponding to the magnetic potential and the s1

invariance. We note that c;s is a bundle over the s axis with fiber Sym(IR 2), the diffeomorphism group of the s =constant poloidal planes. We now perform the reduction procedure described in Ratiu's lecture fiberwise. By the formulas in the paper of t4ontgomery, Marsden and Ratiu in these pro-ceedings, the bracket on the quotient space is the semi-direct Lie-Poisson bracket plus a canonical bracket for the s dependence. Finally, the s1

symmetry quotient inserts a 3/3s in this canonical bracket, to produce the bracket (3.3).

The last step in this construction can be illustrated by the wave equa-tion on the s axis. The canonical bracket on the phase space F(S1) x F(S1)* is

{FG} = J(_Q£_Q§__~cSG)d ' 6¢ on on 6¢ s

sl However the bracket on the reduced space with ~;;-translations divided out is

{F G} = Jr (cSF _E_ cSG - _§£ _E_ _Q£) di;; ' cS¢ as on cS¢ as on

sl i.e. we change the cosymplectic operator as follows:

( 0 I) ( 0 i_) -I o >+ - ~r; ~s

This is proved the same way as the corresponding assertion for Maxwell's equation (see Marsden and Weinstein [1982]).

We just mention that there is another way of getting (3.3) directly from the (incompressible homogeneous version of) the Morrison-Greene bracket for MHD. Namely, one inserts the decomposition }!__=vue+ 2 X Vl¢ and B = B112 + 2 x v1 lj! into that bracket. With div B = 0, s 11 = 1 and v 11 = 1, the expression (3.3) results. One can also use this procedure to derive




more complex brackets with a v11 dependence. (These are related to the hamiltonian structure of the Hazeltine equations [1983] which will be the subject of anotherpublication (Hazeltine, Holm and Morrison [1984]).) The rough idea is that if a factor s11 is inserted in the last term of (3.3), linearity of the brackets in the field variables is restored. This is con-sistent with the fact that the Morrison-Greene bracket is Lie-Poisson for a semi-direct product (Holm and Kupershmidt [1983]) and can be derived from canonical brackets in a Lagrangian representation (Marsden, Ratiu and Wein-stein [1983]). The procedure of neglecting o/ov 11 and o/oB 11 terms in this bracket can be viewed as an approximation procedure analogous to the 1 imit c + oo which converts the ~axwell-Vlasov to the Poisson-Vlasov bracket.

We hope that the bundle point of view sketched in the paper of ~1ontgomery,

Marsden and Ratiu in these proceedings will shed light on how these processes of averaging, reduction and limits all tie together into a coherent picture.

APPENDIX

If ~ is a given function then the formal solution to (4.12) can be obtai ned by integrating the characteristic equations where ~ acts as a Hamiltonian and ~ plays the role of time. Cne obtains ~h(r 0 (r,e,s),

eo(r,e.~). 0), where we have suppressed the parametric time dependence. Shortly we will implicitly differentiate (4.12) in order to formally transform

variational derivatives into derivatives with respect to ~ . n Let us suppose that P is some functional of 1jJ that has the first

variation

( A.l)

In the second equality we assume P is a functional of ~h through (4.12). If we define the operator £ by, £f = Clf/a~ + [f,~], then linearization of (4.12) yields

-1 -1 where we have used £ to mean the inverse of £. In practice £ is obtained by integrating over characteristics. For our purposes it will be sufficient to pretend that the appropriate analysis has been done and that we can freely invert £ when needed. Inserting (A.2) into (A.l) yields

(A.2)



NONCANONICAL HAtHL TON IAN FIELD THEORY AND REDUCED MHD 147

(A. 3a)

(A. 3b)

where (.C 1)+ is the formal adjoint of £ 1 . Equation (A.3b) can be further transfomred by using the identity J/[g,h] d3x = fv g[h,f] d3x; we obtain

(A.4)

If Eq. (A.3b) is to hold for all variations 6t/! then evidently

oP [ -1 + oP ] ow = ( £ ) owh, l)!h . (A. 5)

Operating on both sides of (A.5) with £ and using £[f,g] [£f,g] + [f,£g], which is not difficult to establish, yields

(A.6a)

(A.6b)

Equation (A.6b) follows from the fact that an anti-self-adjoint linear oper-ator will ~ave an anti-self-adjoint inverse. From Eq. (A.6b) it follows immediately that the RMHD bracket becomes

= J { [oF oGJ + ([1 .§§.]+[oF oGJ)} 3 {F,G} V U OU' oU ljlh oljlh' oU oU' oljlh d x (A. 7)

Hence we have transformed away the non Lie-Poisson term and the bracket possesses the algebraic interpretation of Section 4C. Moreover, it appears that we have replaced a three-dimensional problem with a two-dimensional problem!

In spite of the rosey picture painted above, there is a catch, which is associated with a periodicity constraint on (4.12). Recall ~· was required to be periodic in e and 1;; if ljlh is to be single-valued then it too must be periodic. Flows with periodic Hamiltonian's typically are not periodic-- indeed such would be an exception. So our problem lies in the fact that appropriate ljlh do not in general exist. There are, however, the special single helicity solutions discussed in Section 40.




REFERENCES

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2. R. Abraham and J. t~arsden, 1978. Foundations of t~echanics, Second Edition, Addison-Wesley.

3. V. Arnold, 1966. Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique des fluids parfaits, Ann. Inst. Fourier, Grenoble, 16, 319-361.

4. B. Carreras, B. Waddell and H. Hicks, 1979. Poloidal Magnetic Field Flucutations. Nuclear Fusion ..!2_, 1423-1430.

5. F. F. Chen, 1974. Introduction to Plasma Physics, Plenum.

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7. R.F. Dashen and D.H. Sharp, 1968. Currents as coordinates for hadrons, Phys. Rev.~. 1857-1866.

8. I.E. Dzyaloshinskii and G.E. Volovick, 1980. Poisson brackets in condensed matter physics, ~nn. Phys. ~. 67-97.

9. E. Ebin and J. t1arsden, 1970. Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math. 92, pp. 102-163.

10. H. Goldstein, 1930. Classical Mechanics, Second Edition, Addition-Wesley.

ll. J. Guckenheimer and P. Holmes, 1983. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer Appl. Math. Sciences Vol. 42.

'12. A. Hasegawa and K. ~lima, 1977. in magnetized nonuniform plasma.

Stationary spectrum of strong turbulence Phys. Rev. Lett. 39, 205-208.

13. R.D. Hazeltine, 1983. Reduced magnetohydrodynamics and the Hasegawa-Mirna equation, Physics of Fluids,~. 3242-3245.

14. R.D. Hazeltine, M. Kotschenreuther and P.J. Morrison, 1983. Reduced Fluid 1·1odels and Tearing-interchange Instability. Manuscript under preparation. Bull Am. Phys. Soc. 28, 1090.

15. R.D. Hazeltine, D. Holm and P.J. Morrison, 1984. work in progress.

16. D.O. Holm and B.A. Kupershmidt, 1983a. Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity, Physica .§_Q, 347-363.

17. D. Holm, B. Kupershmidt and D. Levermore, 1984. (To appear in Advances in Applied Math.}

18. D.O. Holm, J.E. ~:arsden, T. Ratiu, and A. Weinstein, 1983. Nonlinear stability conditions and a priori estimates for barotropic hydrodynamics, Physics Letters ~SA, 15-21.



NONCANONICAL HAMILTONIAN FIELD THEORY AND REDUCED ~~HD

19. D. Holm, J. Marsden, T. Ratiu, and A. Weinstein, 1984. Nonlinear stability of equilibria in fluid and plasma systems (in preparation).

20. P.J. Holmes, and J.E. l-1arsden, 1983. Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups, Indiana Univ. 11ath. J. B.· 273-310.

21. H.R. Lewis, 1970. Energy-conserving numerical approximations for vlasov plasmas, Journal of Computational Physics,~. 136-141.

22. R. Littlejohn, 1979. A guiding center Hamiltonian: a new approach. J. Math. Phys. 20, 2445-2458.

23. R.G. Litlejohn, 1982. Singular poisson tensors, from Mathematical Methods in Hydrodynamics and Integrability in Dynamical Sys terns, AIP Conf. Proc. 88, 47-66.

24. J. Marsden, T. Ratiu and A. Weinstein, 1983. Semi-direct products and reduction in mechanics, Trans. Am. ttath. Soc. (to appear).

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26. J.E. Marsden and A. Weinstein, 1982. The Hamiltonian structure of the t·1axwell-Vlasov equations, Physica D, ±• 394-406.

27. J.E. Marsden and A. Weinstein, 1983. Coadjoint orbits, vortices and Clebsch variables for incompressible fluids, Physica 7D, 305-323.

28. J.E. Marsden, A. Weinstein, T. Ratiu, R. Schmid and R.G. Spencer, 1983. Hamiltonian systems with symmetry, coadjoint orbits and plasma physics, Proc. IUTAM-ISI~1t1 Symposium on Modern Developments in Analytical Mechanics, Torino,June 7-ll, 1982.

29. P.J. Morrison, 1980. The Maxwell-Vlasov equations as a continuous hamiltonian system, Phys. Lett. BOA, 383-386.

30. P.J. Morrison, 1982, Poisson brackets for fluids and plasmas, in ~1athematical 1·1ethods in Hydrodynamics and Integrability in Related Dynamical Systems, AIP Conf. Proc., #88, La Jolla, M. Tabor (ed).

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149

34. R.G. Spencer, 1982. The Hamiltonian· structure of multi-species fluid electrodynamics in r.',athematical Methods in Hydrodynamics and Integrability in Related Dynamical Systems. (M. Tabor and Y.M. Treve, eds.) AIP Conf. Proc., La Jolla Institute 1981,88,121-126.

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150 Jerrold E. ~1arsden and Philip J. t'.orrison

36. H. Strauss, 1976. Nonlinear, three-dimensional Magnetohydrodynamics of Noncircular Tokamaks, Physics of Fluids, .}1, 134-

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40. G. Wentzel, 1949. Introduction to Quantum Field Theory, Wiley.

DEPARH1ENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720

INSTITUTE FOR FUSION STUDIES, and DEPARTMENT OF PHYSICS UNIVERSITY OF TEXAS AUSTIN, TEXAS 78712




GEOMETRY AND GUIDING CENTER MOTION

Robert G. Littlejohn

1. INTRODUCTION. Over the years, plasma physicists have developed a number of computational techniques for solving various problems of physical interest. Often these involve perturbation methods of one sort or another, which are seldom systematized or clearly articulated. As one examines these mathemati-cal methods of plasma physics, however, one finds that there is a surprising amount of interesting mathematics in them, especially in the area of differen-tial geometry. A good example of this concerns guiding center motion, which is one of the most important approximation schemes in plasma physics, and which forms the subject of this paper.

To preview the main points of this paper, it turns out that guiding center motion not only involves some intersting features of sympletic geometry, as one might expect for a mechanical system, but there is also some interesting metrical geometry. It even happens that guiding center motion is a gauge theory.

Th~ physical principles of guiding center motion have been well understood for many years, and are reviewed in the excellent book by Northrop. 1 So here I will approach the subject from a mathematical standpoint, where there is much that is new. I will begin by giving a brief overview of the subject for the non-plasma physicist.

2. WHAT IS GUIDING CENTER MOTION? Guiding center motion concerns the motion of a single charged particle in a given electromagnetic field. It is, therefore, represented by a Hamiltonian system of three degrees of freedom. The essence of the approximation involved in guiding center motion can be seen by examining the Lagrangian for a particle in a magnetic field,

i L(x, v) =r+ v. A(x).

1980 Mathematics Subject Classification 76X05, 70H05, 58F05. This work was supported by the Director, Office of Energy Research, of the U. S. Department of Energy under Contract Number DE-AC03-76SF00098.

(1)


151




152 ROBERT G. LITTLEJOHN

For simplicity I have neglected any electric field, and I have treated the par-ticle nonrelativistically. The position of the particle is x, its velocity is v, and the magnetic field B is represented by the vector potential A, so that B = \1 x A.

I have also suppressed the physical constants, e, m, and c from the Lagran-gian in order to make the mathematics more clear. However, when the physical constants are restored and the two terms of the Lagrangian are numerically evaluated for many plasmas of physical interest, it turns out that the second term v ·A, representing the coupling of the particle to the magnetic field, greatly dominates the first term, v2/2, which is the kinetic energy of the particle. Therefore for these physical situations one can achieve a good ap-proximation to the particle motion simply by neglecting the kinetic energy al-together. Doing this and carrying out the variational principle, one finds the approximate equation of motion,

v X B = 0. (2)

The physical meaning of this is that the particle moves parallel to the direc-tion of the magnetic field, i.e. that it is constrained to lie on the integral curves of B, the magnetic field lines. This, in a nutshell, is the essence of plasma confinement by magnetic fields.

A more exact analysis of the particle motion shows that the component of the particle velocity perpendicular to the magnetic field does not really vanish, as indicated by (2), but rather undergoes high frequency oscillations. However, the average of this velocity component does, in fact, vanish to a good approximation, so that (2) is still correct in an averaged sense. The actual particle moves in tight circles around a magnetic field line, producing overall a helical trajectory as it moves along the field line, as shown in fig. 1.

Practically speaking, one must have E3 a more refined picture of the particle

motion than that indicated by (2), and so various schemes have been invented to analyze the motion perturbatively. These schemes effectively treat the kinetic energy as a perturbation, although they seldom explicitly acknowledge this. The physical effect of the perturbation is to cause the particle to "drift," i.e. to move slowly in a direction perpendicular

XBL 8312-2513 to the magnetic field lines, at the same time that it moves rapidly along the




field lines according to (2). The motion is pictured physically in an averaged sense, with the time average of the particle position being called the "guiding center."

3. THE SYMPLECTIC FORM IN GUIDING CENTER THEORY

153

Now, in what sense can there be interesting symplectic geometry in guiding center motion, as I claim? After all, if you've seen one symplectic structure, you've seen them all. Darboux's theorem guarantees that local canonical co-ordinates always exist, so all symplectic structures look alike locally, i.e. they are all simply given by dp A dq.

The answer is that the canonical coordinates whose existence is guaranteed by Darboux's theorem have different physical meanings in different contexts, so that it is the physical interpretation of the symplectic structure which provides interest and variety. To say this another way, the quantities of most immediate physical interest do not always form canonical coordinates, so that the symplectic structure expressed in terms of physical variables is not simply dp A dq. For example, in the case of guiding center motion, the sym-plectic 2-form has four contributions of different physical origin: one is the (kinetic) mechanical action, the second is the magnetic flux, the third is as-sociated with a symmetry group producing rotations in the plane perpendicular to the magnetic field, and the fourth is a curvature form associated with the transport of triads of unit vectors in Euclidean space.

The decomposition of the symplectic form into physically interesting pieces can be seen in simpler form in the case of the motion of a charged particle in a magnetic field (completely apart from any guiding center approximation). For this system, the canonical momentum p is often not considered a physical quan-tity, because it is not invariant under a change of gauge of the magnetic field. The velocity v, however, is physical. In terms of the canonical momen-tum, the symplectic structure in phase space is just w = l: dpi A dqi. But in terms of the physically meaningful velocity, the symplectic structure is

(3)

The first term may be called the kinetic action; it is the only term present for mechanical problems with Lagrangians of the form L = T- V, where T is the kinetic energy and V is the potential energy. Indeed, this is the only class of problems considered in older treatises on mechanics. The second term is the magnetic flux (or rather the lift of it to phase space); it is responsible for the magnetic forces on the particle, and by our previous discussion of the Lagrangian (1), it is the dominant term when the guiding center approximation




is valid. The tensor Bij is expressed in terms of the usual magnetic field vector B by Bij = EijkBk.

It is useful to pursue a little further the notion of using only physically significant quantities to represent a dynamical system. Consider, for example, the equations of motion. In canonical coordinates these are just Hamilton's equations, but what do we do if the canonical coordinates are physically un-desirable? The best answer, I believe, is to use the Poincare-Cartan 1-form ~. which is defined in terms of canonical coordinates by

{4)

where H is the Hamiltonian. This form lives on the odd-dimensional space which is phase space augmented by time, for which I will write P x R. As explained by Arnold, 2 the equations of motion are implicitly contained in the 1-form ~ through the construction of its "vortex lines." That is, if the 2-form rl = d~ is of maximal rank, it defines a 1-dimensional distribution on P x R. A vector field X lying in this distribution satisfies ix rl = 0, and any such X speci-fies the equations of motion through the relation

(951 .92. dt ) X = \d s' ds' CiS '

for some parameter s (which has no physical significance). also deals nicely with time-dependent Hamiltonians.

( 5)

This formulation

It is significant that the equations of motion depend only on rl = d9, so that 9 can be subjected to a transformation 9 ~ ~ + dS, for any scalar S, with-out changing any physical results. I call this a "gauge transformation on phase space," to distinguish it from a gauge transformation on the magnetic vector potential A in physical space.

A completely equivalent formulation, and one that is easier to use in practice, is the variational principle,

0 J ~ = 0, (6)

where the variations of the path in P x R are required to vanish at the end-points. In canonical coordinates this variational principle is trivially equivalent to Hamilton's equations, and it is discussed in many mechanics books as a curiosity. However, one can easily use physically interesting variables in this variational principle, and as a result one can clearly see the physical ingredients in the symplectic structure as well as in the Hamiltonian.

Consider again, for example, the motion of charged particle in a magnetic field. Setting p = v +A in (4) and {6), we have

6 J [(A + v) · dx - { dt] = 0. ( 7) Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.



It is easy to show that this is equivalent to the usual Newton-Lorentz equa-tions of motion. But notice that the symplectic structure neatly breaks up into the kinetic action, represented by v·dx, and the magnetic flux, represented by A •dx. Furthermore, the Hamiltonian H = v212 is just the kinetic energy.

Now let us consider the same system in the guiding center approximation. Just as with the Lagrangian (1), the Poincare-Cartan forme has a dominant term and a perturbation, for which we write e = e0 + e1• The dominant term is e0 = A·dx, and the perturbation is e1 = v·dx- (v2/2)dt. The physical picture we drew above for the guiding center approximation is easily verified in the formulation (7).

4. GEOMETRY AND PERTURBATION THEORY

155

Now we are ready to get serious about a perturbation calculation. It turns out that this is not just an unhappy exercise in algebra, but that there is some interesting geometry involved. This is mainly because we require a perturbation theory which is applicable to 1-forms like e = e0 + e1, and this problem in turn causes us to think about the structure of perturbation theory in general. A more complete accounting of this analysis may be found in Refs. 3-4.

A great deal is known about standard forms of Hamiltonian perturbation theory, which are applicable to problems for which the Hamiltonian H consists of a dominant term H0 and a perturbation H1• In terms of the Poincare-Cartan for e, we could write e0 = Zpidqi - H0dt, and e1 = -H1dt. That is, for these standard problems in Hamiltonian perturbation theory, the symplectic structure is given exactly in terms of some physically significant canonical coordinate system, and all approximations are focused on the Hamiltonian. Therefore for these problems, one typically uses a sequence of canonical transformations to transform the perturbation into something easier to solve, the virtue of canonical transformations being that they exactly preserve the canonical form of the symplectic structure.

For the guiding center problem of (7), however, the symplectic structure itself is perturbed, with the perturbation being given by v • dx, and the entire Hamiltonian is also treated as a perturbation. Therefore we require a generalization of canonical perturbation theory which allows us to transform not only the time component of e, i.e. the scalar Hamiltonian, but the whole 1-form in all of its components.

It is easiest to see how to do this by examining the geometric foundations of Lie transforms, which are often used in canonical perturbation theory. The




goal of canonical Lie transforms is to perform a change of coordinates in order to simplify the scalar Hamiltonian. The Lie transforms themselves are a speci a 1 kind of coordinate tranformati on. They are given by the advance map of some vector field G. which is called the generator of the transformation. That is, what is often viewed as a change of coordinates can also be seen as the application of the pullback of the advance map associated with G:

H T*H, ( 8)

where r = exp(G). (9)

For time-dependent Lie transforms, the vector field G is seen as a vector field on the space P x R. In the canonical theory G must be a Hamiltonian vector field, since one requires the map exp(G) to be a canonical transformation. That is, we must have Gf = {g, f} for some g and any f. (The bracket shown is the Poisson bracket.) Typically, the scalar g, and hence the vector field G and the transfonnation T = exp(G), is chosen so as to make H easier to solve than the original H. Usually this involves finding coordinates in which an approximate symmetry of the original problem becomes exact.

These notions are easily generalized and applied to the Poincare-Cartan form of (7). Again we define a Lie transform as a mapping T exp(G) for some vector field G, and we apply the obvious generalization of (8),

a= T*e = exp(LG)e. (10)

i~ow, hmvever, G should not be a Hamiltonian vector field, because it is precisely the form of the symplectic structure which we wish to change (so as to deal with the perturbation in it).

Thus, the practical perturbation program for guiding center motion is the following. We use (10) to obtain an explicit relation connecting e and G in terms of the given e = e0 + e1. Then we choose G to make a easier to understand or solve than e. This comes down to a kind of averaging of a over

the rapid oscillations; an analogous averaging procedure is often applied in canonical perturbation theory and gives the "averaged Hamiltonian" H. Here we produce an averaged symplectic structure as well as an averaged Hamiltonian, both contained in e. To go to higher order in the perturbation series, the process has to be repeated with a new G. The result is the Poincare-Cartan form e for guiding center motion.

The use of Lie transforms in this process has some unexpected benefits. If on simply takes the 1-form e of (7) and tries to transform it by some arbitrary change of coordinates (not necessarily a Lie tranfonn), one quickly finds a proliferation of magnetic gauge-dependent terms, coming from the




transformation of the term A • dx. These can always be eliminated by peform-ing a gauge transformation in phase space, 9 ~ 9 + dS, but it is not always easy to find the appropriate S. Using Lie transforms, however, we are apply-ing the Lie derivatives Lg to forms such as e, as shown by expanding the exponential in (10). This allows us to use a nice formula from differential geometry,

157

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The second term represents a gauge transformation in phase space, and can be dropped without any effect on the dynamics. But by doing so, one finds that all the magnetic gauge dependencies are dropped also. Thus, by using Lie transforms, one easily obtains a perturbation expansion which is magnetic gauge invariant to all orders (except for the original magnetic flux term, A· dx, which stays around.) The ability to guarantee gauge invariance to arbitrary order is an important benefit of the geometric approach to perturbation theory.

Now I shall discuss the actual results of the perturbation calculation have just outlined. These results consist of a sequence of generating vector fields G. which I shall not display, and an averaged Poincare-Cartan form 9. The generators specify the coordinate transformation (or map of P x R onto itself) which connects the original set of phase space coordinates (x,v) with a new set of averaged coordinates. For the latter it is convenient to take the set (X,U~ ~.~). in which X is the guiding center position, U is the component of the guiding center velocity parallel to B, ~ is the magnetic moment, and ~ is the "gyrophase." The magnetic moment is approximately given by ~ = vi /2B; its dynamical significance is that it is the generator of the U(l) symmetry group whose action consists of rotations in the plane perpendicular to the magnetic field. The gyrophase ~ is some conventional angle in this plane, so that the Hamiltonian vector field associated with ~ is a/a~. The conjugate variable pair (~ .~) describe the symmetry, and indicates that the perturbation calculation has achieved a "reduction." Thus, ~is an ignorable coordinate, ~ is a constant of motion, and the four remaining variables (X,U), for fixed ~. can be taken as coordinates on the reduced phase space.

The averaged Poincare-Cartan form is given by

(12) where

H = ~B + ~ 2 + E( ••• ). (13)




In deriving this result I have introduced a formal expansion parameter e into the original 1-form e, i.e. I have set e = e0 + E e1 so that the final 1-form e appears as a series in E • The vector R will be discussed presently; additional notation appearing in (12) is defined by B = IBI, b = B/B. The ellipsis in (12) represents higher order terms which I have not displayed.

When this 1-form is applied to the variational principle of (6), the result is the set of drift equations familiar in plasma physics. A practical benefit of this formulation of the drift equations is that several important conservation laws (those for energy, angular momentum, and phase volume) emerge easily and naturally. The status of these conservation laws has been obscure in traditional guiding center theory.

The symplectic structure appearing in (12) consists of several contri-butions with different physical interpretations. The dominant term is A·dX, which is still the magnetic flux, although it can now be interpreted as living on the reduced phase space. The next term is Ub • dX, which is in a sense the average of the kinetic action (since the perpendicular velocity components average to zero). At next order, the term ~ds can be interpreted in terms of the symmetry group generated by ~. the term -~R·dX will be the same order; it will be discussed in the next section. finally, the Hamiltonian (the coefficient of dt) is still the kinetic energy, now expressed in terms of the averaged variables.

5. GYROGAU~E INVARIANCE: THE GEMOEfRICAL PICTURE I turn now to the issue of defining the gyrophase, which involves some

interesting metrical geometry. Because this section contains material of occasional use in plasma physics, I will express the results in tenns of three-dimensional vector calculus.

Physically, the gyrophase is an angle which represents the rapid circular motion of the particle in the perpendicular plane. Figure 2 shows the

geometrical situation; the unit vector

8

XBL 8312~2511

b is parallel to the magnetic field, and the unit vectors e1, e2 span the plane perpendicular to b. The set (e1, e2, b) form an orthonormal triad. The use of such a triad of unit vectors is necessary to coordinatize the motion of the particle, but only the vector b has an immediate physical and geometrical significance. The other two vectors, e1 and e2, are



GEOMETRY AND GUIDING CENTER MOTION 159

constrained to form an orthonormal triad with b, but otherwise their particular orientation in the perpendicular plane is immaterial. Nevertheless, in a practical problem some specific choice for the vectors e1 and e2 must be made, in order to define the gyrophase r;. For Example, r; may be taken as the angle between e1 and the perpendicular component of the particle velocity.

It is intuitively clear that no physical results should depend on the orientation of e1 and e2 in the perpendicular plane. It turns out that this is not a problem when the guiding center theory is carried only to lowest order because e1 and e2 do not appear in the drift equations anyway. But when the equations are carried to next order, these vectors do appear, and a number of plasma physicists over the years have wondered what to do about them. Some people have suggested using some privileged choice for e1 and e2, such as the principal normal and binormal vectors of the field line. But such choices have certain esthetic drawbacks, and they do not simplify any of the calculations or results. A better answer is to let the arbitrariness in e1 and e2 be a free parameter of the problem, and then to study the invariance principle which results. It turns out that this invari-ance principle involves a kind of gauge transformation.

The arbitrariness inherent in e1 and e2 is that they can be rotated about by an arbitrary angle 1/J with no effect on the physics. Furthermore, the amount of rotation can vary from one point of space to another, i.e. 1jJ is allowed to depend on x. Explicitly, the transformation is

e' 1 + el cos 1jJ + e2 sin 1jJ '

e' 1 - el sin 1jJ + e2 cos 1jJ ' (14)

I ca 11 this a II gyrogauge transformation. 11 It is easy to see the effect of a gyrogauge transformation on the gyrophase r;; one is simply redefining the origin of gyrophase, i.e. the reference direction in the perpendicular plane which corresponds to r; = 0. Thus, we have

r;' = 1'; + 1jJ (X) (15)

under a gyrogauge transformation. We see that r; is not gyrogauge invariant. Naturally, we expect that any quantity which is gyrogauge invariant can be expressed purely in terms of the vector b and other physical quantities.

Now, when one carries out the perturbation analysis of guiding center motion, the first quantity to appear which is not gyrogauge invariant and which requires further interpretation is a certain vector which I call R:

(16)



160 ROBERT G. LiffLEJuHN

This vector is not gyrogauge invariant, because, as one easily verifies from ( 14)'

R' = R + VljJ • (17)

One can see already that R looks like a 1-form, and that (17) is a kind of gauge transformation. But to proceed from a physical point of view, let us consider the equation of motion for the gyrophase ~ . This equation can be obtained from (12) or by other means; in any case, the result is

• B • ~ =- + X • R + other terms. (18)

E

The first term on the right hand side, B/E, shows the rapid evolution of the gyrophase due to the rapid orbiting of the particle around the magnetic field line. The frequency of this motion is B, and the term appears perturbatively at order £-1.

The second term is more subtle. As the guiding center moves along with velocity X, the local e1 and e2 vectors, to which the definition of ~ is tied, change from point to point. In a certain sense, this change is due to two causes. One is the fact that b itself changes from point to point, and e1 and e2 are constant to be perpendicular to b. A second is that the orientation of e1 and ex in the perpendicular plane can also change, i.e. these vectors may rotate in their plane of definition as one moves about. will show momentarily that it is the latter effect which is represented by the . term X· R. Accepting this for a moment, we can now interpret Rasa 1-form by writing

p = R·dX, (19)

and we see that the integral of p along some path is the net angle of rotation which e1 and e2 undergo in their plane of definition along the given path.

But what sense does it make to talk about an angle of rotation, when the plane in which e1 and e2 lie is changing from point to point? Clearly we need some concept of transport, so that e1 and e2 lying in the perpendicu-lar plane at one point can be compared to their neighbors a short distance away, lying in another plane. Nor can this be the usual parallel transport of Riemannian geometry, because physical space is flat.

Instead, we find the following geometrical picture. Consider two neighboring points x and x + ~ x, and the corresponding vectors e1(x), e2(x) and e1(x + ~x), e2(x + ~x). Take e1(x) and e2(x) and move them parallel to themselves over to the point x + ~x. Then project these vectors onto the perpendicular plane at the new point, to create new vectors,




f1, f2. The angle ba between f1, f2 and e1(x + t,x), e2(x + t,x)

is interpreted as the rotation of e1 and e2 on passing between the two points. Indeed, a simple calculation shows that

t:,. a = R • t:,. x,

which confirms the interpretation of R given above.

161

(20)

At this point one might be tempted to say that e1 and e2 rotate as one moves about in space only because they were poorly defined. If we were somehow able to define a set of unit vector fields e1 and e2 which were "rotation-less," then the troublesome terms involving the vector R would vanish. Never-theless, it turns out that it is impossible, in general, to define such rota-tionaless vector fields. One way to see this is to take the curl of (17):

'i/ X R I = 'i/ X R.

Thus, although R itself is gyrogauge dependent, its curl is gyrogauge invariant. If this curl is nonzero, as it sometimes is, then clearly no choice of e1 and e2 can makeR vanish.

( 21)

Another way to see the same thing is to attempt a geometrical construction of a roationaless set e1, e2 and see what happens. We begin by choosing e1 and e2 at a single point x0. Next, we take some curve passing through x0 and extend the definition of e1 and e2 to points on this curve, in such a way that the vectors are rotationless along the curve. This is done geometrically by first moving e1 and e2 a small distance along the curve parallel to themselves, next projecting them down onto the perpendicular plane at the new point, and then by repeating the first two steps for the next and successive small increments of distance. This process can be described in terms .of a transport differential equation. If we let V be a vector we wish to transport along the curve (it might be e1 or e2 or something else, but it should satisfy b·V = 0 at x0) and we let s be the arc length, then the desired transport equation is

dV = -b (~ • V) ds as · (22)

This equation follows from a simple analysis of the gemoetrical picture of the transport process which I have just given.

This transport equation has several notable features. First, if b • V = 0 at x0, as we require, then b•V = 0 everywhere on the curve. Thus, e1 and e2 transported by (22) are actually in the perpendicular plane at each point. Next, if v1 and v2 are two perpendicular vectors created by the transport process, then the scalar product vl • v2 is constant along the curve. Thus, e1 and e2 created by (22) remain orthogonal to each other along the curve. Finally, by taking v1 = v2 = e1 or e2, we see that




the length of e1 and e2 is preserved by the transport, so these vectors remain unit vectors. Thus we obtain an orthonormal triad at each point on the curve, which is rotationless along the curve.

Equation (22) has the form of a parallel transport equation in Riemannian geometry, if we define Christoffel symbols by

i r .k = b.bk . J 1 .J (23)

Unlike the usual Christoffel symbols, however, these are not symmetric, i.e. i i rjk t rkj• unless vx b = 0. But in that case it happens that

there exist surface which are everywhere perpendicular to b (actually b·V x b = 0 is sufficient), and rjk can be given its usual interpreta-tion in terms of parallel transport along curves which lie in those surfaces. Altogether, we see that the transport and connection given by (22)-(23) is a kind of generalization of parallel transport on two-dimensional surfaces in Euclidean R3.

The transport process yields a rotationless triad along a given curve, but not a field of triads. One could fill up a finite volume of space with triads by drawing many curves radiating from x0, but these triads would be rota-tionless only along the given curves. Along some other curve, such as the path of the guiding center, they might not be rotationless. Thus, the way to see if a rotationless field of triads can be set up is to consider the transport of a triad along a closed curve (and not to worry that actual guiding centers might never follow a closed curve), in order to see if the property of being rotationless is path dependent.

Let us transport a perpendicular vector V around a small parallelogram defined by two small di sp 1 acements, !l x1 and !l x2. The area of the parallelogram is represented by the vector llS = llx1 x llx2. In taking V

around this small parallelogram, one must carry (22) out to second order in the displacements, because the first order terms cancel. We know that when we bring V back to its starting point it must have the same length as when it started, and it must still be perpendicular to b. Therefore at worst it has rotated in the perpendicular plane by a cartain angle lla. This angle is given by ll a = N • ll S, where

N = _k ~b. . b. . ) - ( 'V • b2 ~ + ( 'V • b) b • 'V b - b • 'V b • Vb iC ~ 1 ,J J, 1 J

Thus, if N ~ 0, it will be impossible to set up a field of triads which is rotationless along every path.

(24)

A complementary point of view is to imagine we are given an arbitrary field of triads (e1, e2, b), which no one has tried to make rotationless, and find what angle of rotation the triad suffers around a closed curve. By



GEOMETRY AND GUIDING CENfER MOTION

the argument surrounding (19)-(20), this is the line integral of R around the closed curve, and Stokes' theorem can be applied:

163

~ R • dx = J (17 x R) • dS. (25)

From this it is clear that N = 17 x R (26)

and indeed we see that 17 x R, which is gyrogauge invariant by (21), can be expressed purely in terms of b. One can also verify {26) directly, by taking the curl of {16) and using some arcane vector identities.

The vector N is related to a 2-form v by 1 }: (

v = "l ijk e:ij kNkdxiA dxj. 27)

v represents the angle of rotation a triad suffers on being transported around the boundary of a 2-dimensional region. If surfaces exist perpendicular to b and the regions considered lie in these surfaces, then vis the curvature form {there is only one) of the surface. We note in (12) that v = dp forms the fourth and final contribution to the symplectic structure of the guiding center motion.

Finally, I would like to summarize some algebraic properties of the vectors Rand N, which cannot be found anywhere else. By {26), we must have

17 ·N = 0. However, when we work this out explicitly from {24), we find 3 1 3 17• N = -2 (17 • b)( b . . b . . ) - -2 { 17 • b ) - { b . . b k . b . k ) . { 28) 1,J J,1 J,1 ,J 1,

To see that this actually does vanish {it is not obvious), we call on the following formula. Let M be a 3 x 3 matrix with components Mij" Then

det M = ~(TrM} 3 - 3(TrM}(TrM2) + (TrM3}]. (29}

Therefore by setting Mij = bj,i' we have

1 17 • N = - "'5" det( b. . ) . .:> 1 ,J

And this in turn vanishes because the matrix b .. has an eigenvector with 1 ,J

eigenvalue zero. This is none other than b itself:

vb • b = }17{b2) = o,

since b is a unit vector.

{30)

(31)

On the other hand, N is closely related to the characteristic polynomial of bi,j" Let the eigenvalues of bi,j be A0, A1, A2, and suppose AO = 0 (one of them must be zero). Then it turns out that {24) can be written in the form




(32)

Finally, we obtain another useful identity by multiplying Vb on the left by this. The result must vanish, because any matrix satisfies its own secular equation. Thus,

N • V b = 0,

and we see that N is a left eigenvector of vb, just as b is a right eigenvector.

6. GYR OGAUGE I NVAR I ANCE : D YNJ!M I CAL CONS ID ERA TIO NS

(33)

We have succeeded in dealing nicely with the interpretation of the gyrogauge dependent quantity R, and in showing that it cannot be transformed or defined away. But if we were to carry the guiding center theory out to higher order, would we keep running into other gyrogauge dependent quantities which would have to be analyzed similarly, or can we settle the issue once and for all? In a similar vein, is it possible to find drift equations which are gyrogauge invariant to all orders? To begin, the Poincare-Cartan form of the particle, shown in (7), is certainly gyrogauge invariant, as it must be. Similarly, the Poincare-Cartan form of the guiding center, shown in (12) is also gyrogauge invariant, although here some discussion is called for. The components of e are not individually gyrogauge invariant, because the vector R appears. But the coordinate differential ds is not gyrogauge invariant either, for by (15) we have

ds' = ds + R • dX. (34)

However, when we examine the behavior of e under a gyrogauge transformation, we find that the transformation of R and that of ds exactly cancel one another, showing overall gyrogauge invariance for e.

The result is that the drift equations coming from a are gyrogauge invariant. For example, the equations of motion for X and U involve the vector R only through its curl, which is given purely in terms of b by (24). Similarly, the equation for s, shown in (18), is form invariant under a gyrogauge transformation, even though the vector R appears in it; the transformation properties of the two sides of the equation cancel one another.

But how did this gyrogauge invariance come about in the averaged Poincare-Cartan form of (12), and would it persist to higher order This gyrogauge invariance was not automatic; rather, it came about by using gyrogauge invariant generators G in the Lie transforms. As long as such generators are used in (10) for the perturbation transformations, the result will be gyrogauge invariant guiding center dynamics to arbitrary order.




A generating vector field G appearing in (10) can be written

G=G ·~+G k+G !_+G!_, X oA U o u II oil I;; o(;

165

in the form

(35)

All of the partial derivative operators appearing here are gyrogauge invariant with the exception of a/aX, which transforms according to

(~x )' = ~x - vw:~;; . (36)

Therefore the overall vector field G will be gyrogauge invariant if the component G <;;compensates for (36) by transforming according to

G'=G + VI/J•Gx. (37) I;; <;;

And this will be the case if all the components of G consist of gyrogauge invariant quantities, except for G<;; which must contain a term equal to R·Gx. By this definition of a gyrogauge invariant generator, one can construct a gyroguage invariant guiding center theory to arbitrary order.

It is interesting that the gyrogauge transformation (17) is mathematically identical to an ordinary magnetic gauge transformation. Thus, the vector R is analogous to the vector potential A, and the divergence free and completely physical vector N = 'J x R is analogous to the magnetic field B. Furthermore, these two types of gauge fields are coupled to each other in the guiding center Poincare-Cartan form of (12), and the coupling constant is the magnetic moment.

This analogy is even more striking when time-dependent fields are considered. Then the vecotrs e1 and e2 depend on time as well as space, and one finds a gyrogauge dependent scalar,

ae2 a =at . el.

Under a time-dependent gyrogauge transformation, one has

which taken with (17) shows that a is like an electric potential. Indeed, just as N = 'J x R is gyrogauge invariant, so now is the vector F, given by

aR F = - 'Ja -at'

Evidently, F is like an electric field, just as N is like a magnetic field. Since F is gyrogauge invariant, it can also (likeN) be expressed purely in terms of b; the result is

(38)

(39)

( 40)




Finally, there is another gyrogauge equivalent of a Maxwell equation, the complement of V • N = 0:

aN VXF=-at

(41)

( 42)

Thus, by using the quantities a and F, it is straightforward to extend gyrogauge invariance to time-dependent systems. One finds, for example, the term ~a in the Hamiltonian.

It turns out that it is impossible to find a set of variables to describe the guiding center motion which are both canonical and gyrogauge invariant, if one of these variables is the magnetic moment and another is some gyrophase canonically conjugate to it. This was the principal esthetic difficulty with my earlier work in guiding center motion, which used Darboux's theorem to construct averaged variables. In the present formalism, the Poisson brackets {X, c;;} and {U, d, which can be derived from the 1-form of (12) in a manner described in Ref. 3, are nonzero. If one redefines these variables so that these brackets vanish, then the variables (X,U) cannot be made gyrogauge invariant. This conclusion may have interesting consequences for the applicability of canonical coordinates and canonical transformation theory to other dynamical systems with a symmetry and for the reduction process in general.

7. CONCLUSIONS The principal practical goal of guiding center theory has been to address

specific problems in plasma physics, and therefore I have not considered possible mathematical generalizations. Let me now suggest a few of these, and raise some questions. Some of these are vague and not clearly thought out, but perhaps they will be suggestive.

In guiding center motion, the reduction of the phase space by means of the symmetry associated with the ignorable coordinate z:; has produced a reduced phase space, coordinates (X,U), which is the guiding center phase space. Each point of this space can be considered to have a fiber attached to it, for which the ignorable coordinate 1; serves as a coordinate. That is, we have a circle bundle on the reduced phase space, each circle being a copy of the symmetry group U(l). For an arbitrary dynamical system with a symmetry, would we in the same manner obtain a group bundle on the reduced phase space?

If so, then the identity element in each copy of the group could be redefined from point to point on the reduced phase space with no physical effect, just as we used the field ~ to redefine the origin of the gyrophase.



GEOMETRY AND GUIDING CENTER MOTION 167

~would now be a field of group elements. Similarly, the 1-form p = R·dX would generalize to a Lie algebra-valued 1-form.

Would there then be a connection? The metrical structure of Euclidean R3 seemed to play an essential role in the connection we have discovered here; can this be generalized? What role does the curvature form of the group bundle generally play in the symplectic structure of the dynamical system? It seems to have played a role in guiding center theory, through the form v = dp, but then here v is closed. I will leave these and further issues to my mathematical colleagues.

ACKNOWLEDGMENTS would like to thank Jerry Marsden for giving me the opportunity to com-

municate this material to a mathematical audience. I would also like to thank fed Frankel and Alan Weinstein for several useful discussions.

This work was supported by the Office of Basic Energy Sciences of the U.S. Department of Energy under contract No. DE-AC03-76SF00098.

BIBLIOGRAPHY

1. T. G. Northrop, The Adiabatic Motion of Charged Particles (Inter-science, New York, 1963).

2. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1978 .

3. Robert G. Littlejohn, J. Plasma Physics 29, 111 (1983).

4. John R. Cary and Robert G. Littlejohn, Ann. Phys., to be published, 1983.

*PHYSICS DEPARTMENT AND LAWRENCE BERKELEY LABORATORY UNIVERSITY OF CALIFORNIA, BERKELEY, BERKELEY, CA 94720







LIE-TRANSFORM DERIVATION OF THE GYROKINETIC HAMILTONIAN SYSTEM

Allan N. Kaufman* and Bruce M. Boghosian**

ABSTRACT. The Hamiltonian structures of the self-consistent gyrokinetic equations, and of the cold guiding-center plasma, are derived from first principles. The Vlasov system is first formulated as an action principle, and is then subjected to a single-particle Lie transform. The resulting expression then yields the Poisson structure, on functionals of the guiding-center distribution, and the system Hamiltonian. The self-consis-tent field is eliminated by a subsidiary condition.

The Lie transform [1,3,4,11] has proved to be a very successful tool for systematically reducing the Hamiltonian description of single-particle motion. In the area of plasma physics, the chief applications have been to the concepts of oscillation-center motion [2,6,7] and guiding-center motion [10,12]. This method transforms to higher order the effects of oscillatory motion and gyration, respectively. These higher order terms then contribute, for example, to ponderomotive effects [2,6] and to wave coupling [7].

In order to obtain a (self-consistent) system Hamiltonian corresponding to this reduced description, it is necessary to imbed the single-particle Lie transform in a many-particle or continuum framework which includes the self-consistent fields. In the companion paper [9], a Lagrangian principle is presented for the Vlasov equation with Coulomb interaction, which leads to the accepted Hamiltonian structure of the Vlasov equation. However, that principle utilizes particle orbits in configuration space, and thus is of no help for the Lie transform, which operates in particle phase space.

In this paper, the action principle for the Vlasov system is first formulated in particle phase space. The invariance of the particle action integral under canonical transformations then enables us to effect the Lie

1980 Mathematics Subject Classification 76X05, 70H05, 58F05. This work was supported by the Director, Office of Energy Research, of the U. S. Department of Energy under Contract Number DE-AC03-76SF00098.

169

© 1984 American Mathematical Society 0271-4132/84$1.00 + $.25 per page




170 ALLAN N. KAUFMAN

transform directly. The treatment of the self-consistent field requires special care, of course. In the new variables, a system Hamiltonian is then obtained in terms of the reduced description. The standard Poisson structure [8] then yields the desired nonlinear Vlasov equation.

To illustrate this methodology, we treat the gyrokinetic model studied by Dubin, Krommes, Oberman, and Lee [5]. These authors consider a Vlasov plasma in a uniform magnetic field, with only Coulomb interaction. The self-consistent Coulomb potential is assumed to vary slowly on the gyrofrequency time-scale, and in addition to be weak relative to the particle kinetic energy. Further, its spatial variation along the magnetic field is weak.

These authors present a careful treatment of the Lie transform to the guiding-center description, wherein the gyrophase is systematically eliminated from the particle Hamiltonian. The Poisson equation for the self-consistent potential is then expressed in terms of the Vlasov guiding-center distribution. Finally, an energy invariant is found for the coupled Vlasov and Poisson equations. As we shall see below, the energy is actually the system Hamiltonian which generates the nonlinear evolution.

Our starting point is an action principle for the self-consistent Vlasov system:

S(E_,_g_,¢} =fdtfd 3 x!v¢(~,t}! 2 /81T + fdtfd6z/0 (z0 }(E_(z0 ,t}j_(z0 ,t}-h(E_,_g_,¢}} (1}

Here .2. and .9. are two conjugate vector-valued fields on (z0 ,t} space, where z0 represents the six components of particle-initial-condition. The initial Vlasov distribution is f 0 (z0 }. The particle Hamiltonian function ish, expressed in terms of .2.·.9. and the scalar potential field ¢(~,t}.

(Appropriate summation over species is implicit}. We consider independent variations of the three fields ¢• E.• .9.• and demand

that S be stationary. Varying ¢(~,t), we obtain

0 = 6S/6¢(~,t} =- v 2 ¢(~,t}/41T- Jd6z/0 (z0 ) 6(/hdt)/6¢(~,t). (2}

We write the particle Hamiltonian as

h(E_,_g_,¢} hkin(E_,_g_) + fd3x ¢(~, t} p(~;E_,_g)' where 3

p(~;E_,_g_) = e 6 (~ - _c(E_,_g_}}

is the single-particle charge density, while _c(E_,g) is particle position in terms of the chosen representation .2.·.9.· Inserting (3) in (2), we obtain

( 3}

(4)

- v 2 ¢(~,t)/41T = Jd6z0 f 0 (z0 ) p(~;E_(z 0 ,t),g(z 0 ,t)). (5)



LIE TRANSFORM DERIVATION OF THE GYROKINETIC HAMILTONIAN SYSTEM 171

We define the Vlasov distribution in ~·S space by

f(~ps 1 ;t) = Jd6z0 f 0 (z0 ) o 3 (~ 1 - ~(z 0 ,t)) o3<s1 - s<z0 ,t)) (6)

Thus (5) reads

- ·lq,(~.t)/411 = p(~;f) (7)

where

p(~;f) = Jd6z f(~.s;t) p(~;~.s> is the system charge density. The Poisson equation (7) enables ¢to be expressed as a functional of f.

(8)

Returning to (1), we vary S with respect to the fields~ and S· obtaining

E_(z0 , t) = - ah(~·S•¢)/as.

the particle Hamiltonian equations. As a result, the Vlasov distribution (6) satisfies the Vlasov equation

af(~.s;t)/at =- [f,h],

where [ ] aa ab aa ab a , b = aS . a~ - a2_ . aS

is the canonical bracket in particle phase space.

( 9)

( 10)

( 11)

Considering the action S as the time integral of a Lagrangian, we see from (1) that the field canonically conjugate to s<z0 ,t) is f 0 (z 0 )~(z 0 ,t),

in agreement with Eq. (4) of the companion paper. The methods of that paper then lead again to the Poisson bracket on system functionals:

{A1 ,A2} = Jd6z f(z) [aA1/af(z), aA2/af(z)]. (12)

The system Hamiltonian H is read off from the action (1), interpreted as

S = Jdt{fd6z0 .9. · ~f 0 ) - Hdt. ( 13) We thus obtain

H(f) = Jd6z f(z) h(z,¢) - Jd3x IV¢! 2/811, ( 14)

where we have used (6), and ¢is a functional of f from (7). Considering the right side of (14) as H(f,¢(f)), we note that the particle Hamiltonian

h(z) = oH/of(z) ( 15) can be separated into explicit and implicit parts:

h(z) = oH(f,¢)/of(z) + Jd3x oH(f,¢)/o<P(~) o<P(~)/of(z) (16)




But the factor oH/a vanishes, as a result of Eq. (2). Hence, the first term of (16) yields the identity h =h. Although the form (14) is more convenient for calculation, we note that, from (3) and (7), it can be expressed as

H(f) = Jd6z f(z) hkin(z) + Jd3x lvcp(f) 12/811, (17)

in agreement with the final equation of the companion paper. Having assured ourselves that the action principle (1) yields correct

results, we now perform a canonical transformation on particle phase space: E_, .9.• h(E_, .9.•</J) "' f., .Q., K(f_, .Q.,¢). (18)

We shall use the Lie transform to effect this transformation, but for now we may treat the transformation as arbitrary. We use the covariance property of the particle action integral: for given z0 ,

Then, defining the Vlasov distribution in f_,.Q. space analogously to (6): 6 3 3 F(f.l'.Q.1;t) = Jd z0 f 0 (z0 ) 6 (f_1- f_(z 0 ,t)) 6 (g_1- .Q.(z0 ,t)), (20)

we obtain the corresponding Vlasov equation,

aF(f_,.Q.;t)/at = - [F ,K] (21)

on varying S with respect to the fields f_(z0 ,t), .Q.(z0 ,t). In (21), [,]is the canonical bracket in g_, f. space.

Before varying S with respect to </J, we expand the new particle Hamiltonian K in powers of¢· If the Lie generating function is linear in </J, the Hamiltonian K contains terms bilinear in <1> and higher order. We express this expansion in <1> formally as

K(f_,.Q.,<t>l = hkin(f.,.Q.l + Jd3x <t>(~l j;(sf. • .Q.l

The action principle thus reads

S(f_,.Q_,¢) = JdtJd6z0 f 0 (z0 ) (f_(z0 ,t) . .Q_(z0 ,t)- hkin(f_,.Q_))

(23) where

(24)



LIE-TRANSFORM DERIVATION OF THE GYROKI~ETIC HAMILTONIAN SYSTEM 173

Variation of (23) with respect to¢, at constant ~.Q, now tie1ds

Jd3x• e:(~·~';F) <P(~') = ~(~;F) (25)

Tnis e.xpresses tile potential in tenns of the guiding-::enter charge density p,

and the dielectric kernel e. By (24), we see that E includes the polarization kernel ·J of the guiding-center Hamiltonian (22). As before, we suppose that (25) is solved formally for q, as a functional of F.

The derivation of the system Poisson bracket proceeds as before, and we obtain, for functionals A(F):

{A1(F),A2(F)} = Jd6Z F(Z) [oA1/oF(Z), oA2/or(Z)]. (26)

From (23), we read off the system Hamiltonian

( 27)

The guiding-center Hamiltonian is thus

K(Z;F) = nH(F)/oF(Z) = hkin(Z) + Jd3x <!>(~) 6~(~)/oF(Z)

- }Id 3 xJd 3 x'cp(~)<f>(~')od~.~·)/oF{Z) + h.o.t. (28)

The last term of (28) expresses the previouslt discovered relation between linear susceptibility and ponderomotive Hamiltonian. Again, we may use (25) to re-express the system Hamiltonian as

H(F) = Jd6Z F(Z) hkin(Z) +} Jd3xJd3x' <f>(~;F) <P(~;F)d~.~·;F) + h.o.t. (29)

Since the Hamiltonian (27) or {29) contains no explicit time dependence, it is invariant in time.

It is particularly instructive to treat the cold plasma limit, wherein the gyro radius r9 vanishes, and the motion along the magnetic field is ignored. We choose (~,9,>~) as the four original phase space coordiOdtes of a parti:le; ! represents the guiding center position:

X=r-r, - - --g ( 30)

9 i s the gy rop ha se:

vx = - v sin 9, vy = - v cos 9, (31)




~: is the :JY romomentum: 1 2

11 = t"v In,

and the gyrofrequency is !J = eB/mc. The kinetic energy is thus

hkin = 1-1rl '

while the potential energy is

e ¢(.!:_,t) = e d! + rg.t) = e ¢(!,tl- ~- f(!,t),

Hhere 1l =

is the electric dipole moment of the gyration relative to X.

(32)

(33)

(34)

(35)

The dipole term in (34) is transformed to higher order by the Lie generat-ing function~~. satisfying

31 ~ l- (11 rl) = rr(v.,e) · E(X,t) ae a11 - --

One then obtains, by standard Lie transform tee hni ques, the new Hamil toni an

1 mc 2 2 K(ll.!.tl = 11n- e<H!,tl - 2 - 2- E (!,tl + h.o.t. B

(36)

( 37)

The ter.n quadratic in the electric field represents the net polarization energy

1 1 A 2 - 2 <~ . f = - <~> . f + 2 m ( f x b c /8 ) , ( 38)

which is the sum of tile potential energy of the mean dipole <~>and the kinetic energy of drift. To deter:nine the mean dipole, we apply the Lie transform to (35) and average over gyrophase, obtaining

<rr> = [w, e .!:_g] = mc 2 I(!,t)/32. (39)

This mean dipole represents the displacement of the center of gyration from the guiding center, due to the polarization drift. In terms of the new variables, the mean particle position is

<.!:_> = ! + <~le . (40)

The polarization drift is now obtained by differentiating (40): 2

d<r>/dt = X + me 2 df./dt; (41) eB

thus the guiding-center drift X excludes the polarization drift. On substitut-ing (39) into (38), we obtain the polarization energy of (37).



LIE-TRANSFORM DERIVATION OF THE G~ROKINETIC HAMILTONIAN SYSTEM 175

We now take the zero temperature limit~ ~ 0 in (37), and obtain the

guiding-center Hamiltonian

1 JOC 2 2 K(!, t) = e¢(!, t) - 2" if E (!, t) •

By the arguments 1~hich led to (27), we obtain the system Hamiltonian

H(N) = Jd2X ¢(!) ~(~;N) - Jd2X d!;N) E2(!l/81T,

(42)

(43)

where N(X) is the guiding-center density, the guiding-center charge density is

.J(!;N) = ; esNs(!l (44)

{we now include species label); and the perpendi:ular dielectric function is

2 411Ns(!)msc d!;N) = 1 + l: 2 (45)

s B

The Poisson equation relating ¢to N is obtained, as before, by

0 = oH(N,¢)/o¢(!) = P"(!;N)- V·(d!;N) ~_{!l)/411. (46) This expresses the field~ in terms of the guiding-center charge density p and

the dielectric shielding E·

The Poisson structure is now

{A1(N), A2(N)} = l: JiX Ns(!) [oA1/oNs(!l. oA2/oNs(!)Js' s

where

[a 1(!l,a2(!)Js =- {c/e5B) b· va1x va2•

The Vlasov equation reads

{47)

(48)

aNs(!,tl/at =- [Ns,Ks], (49)

with Ks given by (42). The Hamiltonian can be expressed concisely as

H(N) = Jd2X d!;N) E2(!;N)/811. (50)

In conclusion, we have seen how the single-particle Lie transform can be used in the system action principle, to obtain a Hamiltonian structure for the guiding-center distribution. This structure consists of a Poisson bracket (47) or (26) on functionals, a Hamiltonian (43) or (27), and a subsidiary

condition (46) or (25) for the self-consistent field. from these ingredients, one obtains a self-consistent Vlasov equation (49) or (21) for the guiding-center distribution.



17G ALLAN N. J<AUF~1AN

The extension of this Lie transform method to 1vave and oscillation-center distributions [8] is currently under investigation. Further reductions and applications are planned for future publication.

BIBLIOGRAPHY

1. J. R. Cary, "Lie Transfonn Perturbation Theory for Hamiltonian Systems", Physics Reports 79, No.2 (1981) 129-159.

2. J. R. Cary and A. N. Kaufman, "Ponderomotive effects in collisionless plasma: A Lie transform approach", Phys. Fluids 24 (1981) 1238-1250.

3. R. L. Dewar, "Kenormalized canonical perturbation theory for stochastic propagators," J. Phys. A9 (1976) 2043-2057.

4. A. J. Dragt and J. rvt. Finn, "Lie series and invariant functions for analytic symplectic maps," .J. l~ath. Phys. 17 (1976) 2215-2227; "Normal forms for mirror machine Hamiltonians," J. Math Phys. 20 (1979) 2649-2660.

5. D. H. Dubin, J. A. Krornmes, C. Oberman, and W. W. Lee, "Nonlinear gyrokinetic equations," Pilys. Fluids, in press.

6. C. Grebogi, A. N. Kaufman, and R. G. Littlejohn, "Hamiltonian Theory of Ponderomotive Effects of an Electromagnetic Wave in a Nonuniform Magnetic Field," Phys. Rev. Lett. 43 (1979) 1668-1671.

7. S. Johnston and A. N. Kaufman, "Lie-Operator Approach to Mode Coupling in Nonuniform Plasma," Phys. Rev. Lett. 40 (1973) 1266-1269.

8. A. N. Kaufman, "Natural Poisson Structures of Nonlinear Plasma Dynamics," Physic a Scripta T2/2 ( 1982) 517-521..

9. A. N. Kaufman and R. L. Dewar, "Canoni:al Derivation of the Vlasov-Coulomb Noncanonical Poisson Structure," this volume.

10. R. G. Littlejohn, "Hamiltonian formulation of guiding center motion," Phys. fluids 24 (1981) 1730-1749.

11. B. McNamara, "Superconvergent adiabatic invariants with resonant denominators by Lie transforms,'' .J. Math. PhJS. L9 (1978) 2154-2164.

12. H. E. Mynick, "Guiding-center Hamiltonian for large gyroexcursion particles in mirror configurations," Phys. Fluids 23 (1980) 1888-1896.

*PHYSICS DEPARTMENT AND LAWRENCE BERKELEY LABORATORY UNIVERSITY Of CALifORNIA, BERKELEY BERKELEY, CA 94720

**LAWRENCE LIVERt40RE NATIONAL LABORATORY UNIVERSITY OF CALifORNIA li VER140RE, CA 94550




1.

POISSON STRUCTURES FOR RELATIVISTIC SYSTEMS <Relativistic Superfluids)

Meinhard E. Mayer

ABSTRACT. In this lecture some features of Poisson structures for relativistic systems are discussed on the eaample of a model for a relativistic superfluid. Such a model may be useful for the description of the phenomenology of neutron stars, which may be the only eaample of a relativistic superfluid. The two-fluid hydrodynamics of superfluids is described. The dynamical variables are a set of Clebsch potentials together with the phase of the wave function of the condensate and its conjugate variable -- the superfluid density. These variables facilitate the rewriting of the Landau-Khalatnikov equations and their relativistic generalizations in a Poisson bracket form, which is adequate both for special and general relativity. The Poisson structure is derived both heuristically, and by means of symmetL·y and momentum map. General features of relativistic systems are abstracted from this eaample. The role of dissipation is briefly mentioned.

INTRODUCTION. Poisson brackets have been around for almost two centuries, and

have been heavily used by physicists, there are periodic revivals of interest in them,

such as the upsurge in activity in the 1920-s in connection with the invention of

quantum mechanics <Maa Born and Pascual Jordan wrote a book on this subject which has

become a classic), Dirac's profound studies on constrained dynamical systems DIRAC

[1948]. Jest's "unpedagogical lecture" (JOST [19641, cf. also MACKEY [1963], which is

closely related), and the appearance of Poisson brackets in condensed-matter physics

<LEBEDEV and KHALATNIKOV [1978, 19801, DZYALOSHINSKII and VOLOVICK [19601, HOLM

and KUPERSHI1IDT [19821). However, a really important step forward was made in the

recognition by mathematicians that there is a close relation between Poisson systems,

symplectic manifolds, and the coadioint orbits and representations of Lie groups. This

development, started by Lie in the last century, was due essentially to BEREZIN (19671,

ARNOL'D (19681, KIRILLOV [19701, KOSTANT (19701, and SOURIAU [19701, and the latter

was responsible for the introduction of the "momentum map", an important generalization

of Noether's theorem to Poisson structures. The real power of the Poisson approach

<compared to the somewhat less intuitive symplectic manifold approach> to dynamical

systems and continua is of more recent vintage <to quote a few representative papers:

STERNBERG [19771, GUILLEMIN and STERNBERG [19801, MARSDEN and \./EINSTEIN [1974,

1982a, bl, MARSDEN, RATIU, and WEINSTEIN [19621, NOVIKOV [19821, etc.) and was

applied to a number of special cases.

1980 Mathematics Subject Classification 76. 02, 76 Y05, 83C55, 70H99

177





178 Meinhard E. Mayer

The power of the momentum map and symplectic reduction technique manifests

itself in constrained systems with gauge symmetries, such as classical non abelian

gauge theories, plasma kinetic theory and magnetohydrodyanmics, both relativistic

<BIAL 'iNICKI-BIRULA et al. C1982l, MAYER C1983l>, and nonrelativistic <MARSDEN and

WEINSTEIN [19821, MORRISON and WEINSTEIN [1982]), in fluid dynamics <MARSDEN and

WEINSTEIN C1982b]) and other fields (cf. the many other articles in the list of

references).

With the proliferation of Poisson structures <PSl in areas ranging from classical

particle and continuum mechanics, plasma physics, and gauge theories to condensed

matter physics, including such diverse areas as liquid crystals, supefluids, and

neutron stars, one might legitimately aslc what reason there is what reason there is

for a lecture on PS in relativistic systems. One reason is the laclc of a

systematic treatment of general-relativistic systems (particularly of neutron

stars, where matter is both highly relativistic and which are assumed to have a

superfluid core, the example on which I will concentrate). A second reason is that

one might learn something useful in relativistic quantum field theory, which in

turn might lead to progress in quantum gravity. General relativity itself may only

play a very reduced role in any but astrophysical applications; but, writing

equations in a generally covariant form <or better still in a completely

coordinate-independent form), may have applications such as dynamics in curvilinear

coordinates.

Relativistic constrained systems (both special-relativistic and general-

relativistic) play an important role both in elementary particle physics and in

astrophysics, and it is important to learn to deal with such systems in a frame-

independent way. In this lecture I will try to illustrate the construction of a

Poisson algebra. for a relativistic superfluid -- a system that might seem contrived,

were it not for its possible application to the explanation of pulsar glitches. But

even if it had no direct application, the system is a. nice pedagogical exercise,

illustrating the power of the methods, as well as the fact that by judicious choice of

variables, the transition from special relativity to curved, general-relativistic,

spacetimes is achieved simply by an occasional introduction of a 4-g factor in

expressions involving densities <this "miracle", i. e. , the disappearance of

Christoffel symbols from Ma.xwell's curved-space equations) is well known in

electrodynamics, and no surprise if the equations are written as exterior

differential equations, rather than pa.rtial differential equations for tensor

components. I will not attempt to discuss general relativity itself as a Poisson

system -- this is a formidaLle problem, which is well worth tackling, and I will

limit myself to some remarks in the last section.

Before entering the subject proper, let me briefly discuss the nature of the phase

spaces describing the physical systems, beginning with the most fundamental one, the

spacetime in which physics operates. It is usually assumed that, at least at scales



Relativistic Superfluids

-14 exceeding, say, 10 em, spacetime is a four-dimensional pseudoriemannian ma.nlfold,

with the local structure determined by the Einstein equations which couple the loca.l

geometry to the energy-momentum tensor of matter Un distinction from some other

authors, I mean by matter both fermionic matter and massless bosonic "radiation",

i. e. , leptons, quarks, photons, gluons, W-s, Z-s, and whatever will be discovered;

avoid mentioning the energy-momentum tensor of gravity itself). At much smaller

scales, which according to present-day theoretical prejudices a.re expressed as

179

15 reciprocals of "symmetry-breaking masses", i. e. , energies of the order of 10 CeV or

larger, popular models involve "hidden dimensions". At such small scales the spacetime

manifold is conceivably a fiber bundle with the usual four-dimensional manifold 114 as

base and a homogeneous space CIH as fiber, where C is some compact Lie group, and H

is a closed subgroup - the stabilizer of the symmetry reduction. This concept of

hidden dimension and dimensional reduction may not play a. role In the macroscopic

physics discussed here, but in the future it may be important for the understanding of

small length scales or luge energy scales.

Section 2 describes the phenomenology of neutron stars. In Section 3 the two-

fluid approach to the hydrodynamics of superfluids is briefly reviewed and the

equations are rewritten in relativistically invariant form. The section contains a

discussion of some aspects of relativistic thermodynamics. Ciebsch variables are

introduced, allowing us to recast the Landau-Khalatnikov <and Lebedev-Khala.tnikov)

equations in Poisson bra.cket form. In Section 3 the group of phase-volume preserving

and gauge diffeomorphisms Is introduced, and the momentum map is briefly described; the

reader is referred for details to the papers by MARSDEN et al. , and by HOLM and

KUPERSHMIDT [1983]. Section 4 discusses the lessons one can learn from this model for

other relativistic systems, and the problems posed by relativity itself. The list of

references at the end is representative, but by no means eKhaustive. Further

references can be found in the review papers quoted.

2. PULSAR GLITCHES AND SUPERFLUID NEUTRON STARS. My pretext for discussing an

outlandish topic such as a relativistic superfluid is not only the mathematical

interest of the subject, involving pseudoriemannian geometry, nontrivial topology <in

the presence of vortex lines, vortex rings, or even more complicated structures, such

as linked vortex rings>, a gauge in variance, and the possible existence of solitary

waves, but also has potential astrophysical uses in explaining the "glitches" that

have been observed in the rate of pulsars. I am no expert on astrophysics, but let me

try to summarize what seems to be known about pulsars.

It is the consensus of the astrophysicists that pulsars are neutron stars, i. e. ,

objects of approximately 1. 4 solar masses and a radius of the order of 15 km, and thus 1011 -3 of high density <larger than g. em ) . They are believed to have a solid crust

approximately 5 km thick, containing either neutron-rich nuclei in a periodic array

and free electrons, or, with increasing density, free neutrons. The core is liquid,




with a den5ity ju!it below that of nuclear matter, and c.onsi5t largely of superfluid

neutrons. Since neutron!i are fermion!i, they do not eKhibit Bose-Ein!itein conden5a.tion

and must "Cooper-pair" near the Fermi surface to conden!ie. Neutrons can pair in two

ways: with antiparallel spins -- responsible for the S-paired iiuperfluid, and with

parallel spins forming the so-called P-paired super fluid. The fir!it forms a superfluid

permeating the cru5t <at densities 4. 3x1011 < p < Zx1014 g. cm-3 they coeKi!it

with neutron-rich nuclei, the P-paired superfluid is found dominantly in the core.

Neutron stars are rotiiting, thus accounting for the pulses of synchrotron radiation

observed on Earth. This rotation is also !iupposed to be responsible to a compleK of

vortices aligned with the rotation uis and exhibiting quantized vorticity <in units of

h/Zm), with vortices "pinned" in the crust (for a complete discussion of these compleK

phenomena, including the "spinup" or "!ipindown" thought to describe the "glitches" -

period changes, cf. PINES et al. [19801, where references to earlier work can be found) .

The "lambda-point", i. e., temperature below which a super fluid component appears in

the liquid (i. e. some pairs "condense" into a coherent state, with a single wave

function, the phase of which will be one of our dynamical variables> lies at about

MeV. This shows that the energy density is so large that curvature effects are not

negligible. The opinion of the astrophysics community on the validity of this model is

divided. But even if it is far removed from reality, it poses nice mathematical

problems, to which I now turn my attention.

3.THE NONRELATIVISTIC AND RELATIVISTIC SUPERFLUID.A. The two-fluid model. A

nonrelativistic superfluid may be considered as a miKture of a normal classical fluid

<characterized by its normal density, P1 , normal velocity four-vector u1 ,

a.nd energy-momentum tensor T 1 , as well as the entropy density s, and reciprocal \IV

temperature p ·= 1/lcT) and a superfluid component -- the Bose-Einstein condensate --

a coherent qua.ntum state, characterized by its state vector 'i! and thus having zero

entropy. Ouasiclassically, the state of the superfluid is described by its mass density

p5 and by its three-velocity vector v s which is the gradient of the phase a of the

wave-function of the Bose-Einstein condensate:

v = va. <3.1> 5

In the presence of vortices the superfluid velocity will have vorticity concentrated on

the "cores" of the vortex filament, or put differently, the phase of the wave function

ceases to be single-valued -- or the space on which it is defined simply connected --

much in the same way the electron wave function acquires a multi-valuedness described

by a period integral of the magnetic vector potential (or the magnetic flux, the two-

form related to it by Stokes' theorem) in the presence of a magnetized whislcer in the

Bohm-Aharonov effect, or in the presence of a monopole. I will return to the gauge-

theory aspects of the superfluid velocity. Classically a velocity field with vortices

can be described by means of two additional Clebsch potentials <originally introduced

by CLEBSCH [18591 and used in the description of superfluids by THELLUNG [19531 and




ZIMAN C1953l; for a modern point of view, cf. MARSDEN and WEINSTEIN (19821> which

allows one to write the fluid equations in terms of Poisson braclcets, or as canonical

equations. In his original paper, Clebsch showed that the most general fluid velocity

field has the form:

V = -Va- ~Vll

or, for the associated one-form }

vb = -da - ~dll

( 3 . 2)

( 3 . 3)

181

which allows one to derive the hydrodynamic equations from a variational principle with

the Lagrange density

( 3 . 4 )

where W<?>, the internal energy, is the following function <cf. ,

e. g., LONDON p. 114-133, ABRAHAM, MARSDEN and RATIU C1983l -- abbreviated as

AMR p. 81-497, particularly p. 491):

p -1 \./(p) = -PJ(p - pO)dp

Po ( 3 . 5)

The Eulerian equations following from the variational principle yield the vorticity

equations <here t. denotes one of the quantities ~, 11

att, + v•vt. = o the continuity equation,

atp + v•VP

and the Euler equation

0

atv + <v•Vlv + (lfp)VP = 0

( 3 . 6 )

( 3. 7>

( 3 . 8)

, with the vorticity two-form (using notations close to those of AMR>

J(>.,\1) = d~Ad\1. ( 3 . 9)

This last equation shows that the vorteK cores are the lines along which the surfaces

>. = canst and 11 = canst intersect. The equations <3. S> show that in a

comoving reference frame these surfaces are time-independent (this is a version

of the Helmholtz-Friedmann theorem on vortea line conservation>.

The canonical momenta associated to the fields a and \1 are the

density p5 <this corresponds to the well-lcnown quantum-mechanical conjugacy

between phase and particle-number), and pi, respectively

leading

1T 3Ltaa1 p ' a s

1T aLta11 1 p >.' ll 5 to a. Hamiltonian density

2 T 00 = H = 11 a [ ( V a - ( 11 11 I 11 a ) V 11 )

and three-momentum density

T 01. = g 1. = P v . = -11 a. 11 - 11 a . a s1 11 1 a 1

and the following set of Poisson bra.clcets:

(3.10)

+ W( 11 ) I (3.11> a

( 3 . 1 2 )



182 Heinhard £. Mayer

{a(K, t ) • 11a<y,t)} .s ( ll - y)

{ ll ( K I t ) 1 1111 ( y. t)} & (X - y) ( 3 . 1 3)

<a< x 1 t ) 1 ll ( y. t)) = 0 1 {a(x, t ) • 11]1 ( y • t)} = 0

{ ll ( ll • t ) • lla<y~ t ) } = 0 {lla<K, t ) 1 11\l(J, t ) } 0

Alternatively, by using the representation <3. ll1 Poisson brackets involving the

superfluid velocity and density can be written, but these will involve gradients of the

delta-function 1 cf. 1 e. g. 1 KHALATNIKOV and LEBEDEV [1980] (cf. also HOLM and

KUPERSHMIDT [1982]; I would like to thank Darryl Holm for calling my attention to this

work during the meeting 1 work which I had previously overlooked>. For the

normal component one may consider the entropy density and the Boltzmann

temperature P = 1/kT as a. canonically conjugate pair. In the relativistic case this will

be replaced by an appropriate one-form <four-vector) w 1 see below.

The el!istence of vorticity, and hence nontt"ivial second de Rham cohomology

("nonsimple-connectedness") of the space in which the flow occurs I requires a more

detailed topological consideration. Vortex lines must begin i.nd end on boundaries <or

in case of unbounded flow extend to infinity) I or be closed, i. e. , the vortices are

concentrated on vorteK rings. In the case of linked vortex rings the Clebsch

representi.tion becomes inadequate, and I will not consider this c.ase here.

B. The Re Ia. t i vi 5 tic. Ca. 5 e . The normal fluid will be described in the familiar way

(cf. , e. g. , LANDAU-liFSHITZ [1959], ch. XV or any of the standard treatises on

general relativity, such as MTW or Weinberg) by its mass-density four-vector

< P, j), its four-velocity vector u 11 <subject to the constraint u 11 u = -1. c 1), 11

its energy-momentum tensor

T = wu u \IV 11 V ( 3 . 1 4)

where w e + p is the enthalpy per unit volume in a comoving frame, p is the

pressure <in the same frame) and guv is the metric tensor, and by the

entropy density 3-form *s and the (reciprocal) temperature 1-form P <see below).

The superfluid component is described by its mass-density three-form *P

and the superfluid velocity, defined as the four-dimensional gradient of the phase of

the wave function of the Bose-Einstein condensate. lt was pointed out by KHALATNIKOV

&nd r.EBEDEV [1983] that this three-vector should not be interpreted as the spatial part

of a four-velocity <which satisfies the constraint ullu = -1, c = 1). Therefore l1

v may eKceed the speed of light . s b

dimensional one-form v s

I prefer to interpret v5 as the four-

d/;; assuming that the phase is locally a smooth function.

It should however be clear, that the mass transport by the superfluid cannot occur with

a speed exceeding 1 <this is equivalent to the requirement that pl - jz > 0,

where p is the mass density, and / is the mass current density, and in

the rest frame this Lorentz-invariant requirement is obviously satisfied).

It is convenient to rewrite the mass-density and mass current, both for the

normal and the superfluid components as a three-form *J; in terms of local coordinates



Relativistic Superfluids 183

I Z 3 .I Z 3 D *J = pdx 11dx 11dx + J dx 11dx 11dx + cycl perm <0,1,2,3), (3.15)

p being the mass density and ji the mass flux density>, &nd the entropy-

density of the norma.l component is written u a. three-form I l 3 ll 3 0 *s = sdx11dK11dK + sv dxlldK!IdK + cycl perm <0,1,2,3) (3.16)

D i v 0 being the velocity of the normal component of the fluid. The relativistic entropy

density is considered a. three-form, <the dual of the usu&l entropy flux density

four-vectoJ:' collinear with the reciprocal temperature four-vector au (for a.

detailed discussion of the thermodynamics, cf. LEBEDEV and KHALATNIKOV [19831

and ROTHEN U968ll .

In the relativistic case the prassure plays the role of the Lagrange density and the

Clebsch potentials a,~, and 1.1 describe not the normal velocity but rather a

thermodynamic one-form wb <in the sequel I shall omit the "flat" denoting the one-form

or the 4-covector by w> defined by LEBEDEV and KHALATNIKOV [19831 so that the

fundamental thermodynamic identity can be written in covuia.nt form

dP = *J11dv + *S11dW <3.17)

(recall that the pressure is a. three-form, and so is the entropy density Sl. This leads

to the following splitting of the one-forms vb and wb into components b i 0

v <1.1 + v .v ldx b 5 11 0 0

w "' (T + v w. ldK

- V dHj 5 i .

- w.dx 1

(3.18)

(3.19) D I I

allowing dP to be rewritten in terms of the interior products

(3.20)

The covectors s and P are proportional to the four-velocity u, hence they a.re

collinear and satisfy the orthogonality constraint p wl.l = 1. 1.1

This allows us to Legendre transform (3. 20> to

dP j ..ldvb -Tsdp .Jw, <3.21)

<where Ts = wl.ls >, showing tha.t the energy density E qualifies as l.l

a. Hamiltonian density

(3.22)

The Clebsch representation for w is introduced simil&rly to Eq. <3. 3> by

w = -dt; - ltdY (3.23)

It is now easy to verify that the "potentials" are paired with the density three-form

*P, the entropy-density three-form *s, and the three-form *P to yield

the following set of Poisson brackets

{*plla} = &

{lt~pAp} : &

<*sAt;} = 8

(3.24)

(3.25>

(3.26)

(3.27)

with all other brackets vanishing. The right-h&nd side of equ&tions <3. 24> - <3. 27> is

to be interpreted as "de Rha.m delta-current" with support at coincident points <i. e.,

a. three-dimensional delta-function with a. four-dimensional volume element



184 Heinhard E. Hayer

attached> 1 and the wedge is to be interpreted much in the same manner as the bn.clcet

of Lie-algebra. valued differential forms.

Alternatively 1 one can write the Hamiltonian equations of motion which show that

the chancteristlc quantities are advected with the normal four-velocity u. This topic

is treated in detail in the paper by LEBEDEV and KHALATNIKOV t1983ll to which I refer

the reader for details. It can be shown that the energy-momentum tensor has the same

form in flat and curved space.

4. SYMMETRY CROUP AND MOMENTUM MAP. The symmetry group with respect to

which our dynamical variables are invariant is the product of the hydrodynamic group of

locally Lorentz diffeomorphisms of M4 and the gauge group U<l) of phase

transformations. More precisely 1 the group acting on the hydrothermodynamic variables

consists of all the transformations of the normal and superfluid velocity fields

leaving the conserved quantities inva.ri;;.nt. Since the normal fluid is described by the

velocity four-vector u 1 the <reciprocal> temperature one-form n 1

and the entropy flux three-form "'s 1 the admissible diffeomorphism& of the base space M4 must be transformations which preserve volume <otherwise the forms will acquire

Jacobia.ns> and at the same time preserve in the hngent or cot&ngent space the four-

velocity normalization <ub 1 u> = -1 1 i. e. 1 must le&ve hyperboloids invariant. Such

transformations are "gauged Lorenh transformations" 1 i. e. 1 in each local frame the

transformation reduces to a Lorent:r; transformation 1 but the parameters may depend on

the point in spacetime (and if the latter is not flat 1 curvature and holonomy effects

will malce themselves felt. The superfluid velocity is not a four-velocity 1 i. e. 1 is

not constrained to lie on a hyperboloid 1 but rather is the gradient one-form of the

phase a of the wave function <it is convenient to divide the phase by h/2ml.

Thus the physical observables a.re invariant with respect to the group of gauge

tra.nsforma.tions g<x> = expia<x> 1 where a<x> is a smooth function.

In the case of vortex lines the function a<x> is defined in the multiply-connected

region obtained by eltminating the vortex cores 1 a situation tha.t requires a. more

careful discussion.

One ca.n then repea.t the arguments given by MARSDEN and WEINSTEIN C1982bl for

ordinary fluids 1 or in a.n infinitesimal setting by KHALATNIKOV a.nd LEBEDEV [1978 1

1980]1 DZYALOSHINSKII and VOLOVICK [1980] 1 and recently by HOLM and

KUPERSHMIDT [1982] 1 for superfluids I to derive the Poisson a.lgebra. for the

Clebsch va.riables from the coa.djoint representation of the symmetry group.

Since the method is straightforward 1 but the calculations sometimes tedious 1

will not carry this out in detail here.



Relativistic Superfluids 185

5. CONCLUDING REMARKS. If one wants to apply the Poisson str.,cture derived here to

a "realistic" situation, such as a neutron star, one is forced to take into account

additional physical phenomena, such as the electromagnetic radiation produced, which in

turn leads to dissipation, i. e., we are taken out of the realm of conservative

dynamics. It is certainly worthwhile considering the role of dissipa.tion in the

framework discussed here.

Another aspect which requires more detailed analysis is the role the gauge group

plays in the theory. In the case considered here, like in elt!ctrodynamics, the gauge

group is abelian. In the case of nonabelian gauge theories, such as pure Yang-Mills

theories or Yang-Mills fields coupled to spinor sources <quantum chromodynamicsl, the

situation is more complicated because the "field strengths" are not observables: their

values depend on the choice of gauge. Only invariants are gauge-independent, and

thus a. discussion of nonabelian hydrodynamics (as initiated by GIBBONS, HOLM,

and KUPERSHMIDT [19831> requires particular care. The problem is closely related to

the problem of quantized gauge theories, which I will be discussing elsewhere.

Finally, a remark about general relativity itself. The standard Hamiltonian

approaches to general relativity are beset by even greater problems than nonabelian

gauge theories. The Arnowitt-Deser-Misner approach, is the best-known. It would be

worthwhile to investigate this problem further, and I expect substantial progress to be

made in the near future.

ACKNOWLEDGEMENTS. would like to thank Jerry Marsden for the opportunity

to participate in this meeting, and to numerous pa-rticipants for stimulating

discussions.

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Department of Physics University of California Irvine, CA, 92717, USA




DIFFEOMORPHISM GROUPS, SEMIDIRECT PRODUCTS, AND QUANTUM THEORY

Gerald A. Goldin

ABSTRACT. Unitary representations of diffeomorphism groups and their semidirect products describe a wide variety of quantum systems. Several ways of obtaining such groups are discussed, with particular attention to demidirect products of vector spaces of tensor fields T~(M) with Diff(M) . In an overview of the represen-tation theory, the roles of measures on configuration spaces, gauge groups, and inducing constructions are summarized. Physical interpretations are given for a number of specific representations, and relationships are drawn with the method of coadjoint orbits.

1. INTRODUCTION. In several talks at this conference, we have seen how diffeomorphism groups and their semidirect products occur in the study of classical fluids. For example, the config-urations of a compressible fluid confined to a region n can be labeled by the elements of Diff(n) , and the phase space for such a fluid is the ~otangent bundle T*(Diff(n)) . After re-ducing the phase space by means of a "momentum map," one is able to obtain a Poisson structure on coadjoint orbits of Diff(Q) This permits the analysis of problems in hydrodynamics using physical variables directly, rather than in terms of auxiliary

. 1-4 variables such as Clebsch coord1nates. In this talk I shall be concerned with continuous unitary

representations of diffeomorphism groups and their semidirect products, which describe quantum-mechanical systems. As in the case of the classical fluid, this permits quantum theory to be discussed using gauge-independent, observable quantities such as local current densities, rather than in terms of auxiliary vari-ables such as canonical fields. 5- 7

1980 Mathematics Subject Classification 58D05, 81C40.

189

© 1984 American Mathematical Society 0271-4132/84 $1.00 + S.25 per page




190 GERALD A. GOLDIN

One way to obtain such a semidirect product group in quantum + + E 3 theory is the following. Let W(x) for x ~ be a second-

quantized non-relativistic scalar field at a fixed time t ' and + let W*(x) be its adjoint. Let

time canonical commutation tions,

(-) w and w* satisfy the equal-

or anticommutation {+) rela-

+ + + + [ljJ(x), wf•<y> J± = Hx - y) (1)

+ w<y>J±

+ 1/J'~(y) ]± [ljJ (x)' = [ljJ>'•(x)' = 0

Now define the mass density + P {x) and the momentum density j(;{) by:

+ + + p{x) = mw*Cx)ljJ(x) (2)

= + + (VljJ*(x) )tjJ (x))

As usual in quantum field theory, we interpret p(~) and j(~) as operator-valued distributions; that is, they become bona fide self-adjoint operators when averaged with C00 test functions. Therefore, let

I + -+ 3 p(f) = p(x)f(x)d x (3)

J(g) = J j(;'{) • g(~)d 3 x

where f space S

k -+ and the components g of g are elements of Schwartz' From Eqs. (l) formal calculations now yield the Lie

algebra of current commutation relations,

= 0

= ili.p(g • Vf) (4)

+ + . where g • Vf is the directional derivative of f in the g-dl.-rection, and where [g1 , g2] denotes the Lie bracket g2 • Vg1 -+ + g1 • Vg 2 of the vector fields.



DIFFEOMORPHISM GROUPS AND QUANTUM THEORY 191

It should be remarked that this formal proced~re for obtain-ing Eqs. (4) from Eqs. (1) is open to question. Analo~ous pro-cedures for obtaining relativistic current commutation relations from underlying fields at equal times result in the omission of "Schwinger terms," which are generally of infinite magnitude and cannnot be disregardect. 8 •9 Thus the relationship between Eqs. (l) and (4) must be verified in a representation of the field oper-

+ 00 + + ators. Let the Fock Hilbert space H- = Ef) H- '-'he H- l" n=O n ' " re n ~ the Hilbert space of symmetric (-) or antisymmetric (+) L2

+ functions of n vector variables. An element ' E H- may be written ':l' = (':l' 1 , ':l' 2 , ':l' 3 , ... ) with (':l' , ':l') = I~= 0 (':l' n, '!' n) < co For the canonical commutation relations, the Fock represen-tation is given in H by:

( 5) .... .... .... .... .... ....

(lj!(x)':l')n(xl, ... , X ) = In + l ':l'n+l(xl, ... ' X n' x) n

.... l n .... .... .... I" ~.) (lj!1•(x)':l' )n (xl, ... , X ) = rn L o(x X n j=l J

A

':l' .... .... .... ) n-l(xl, ... , X. ' ... , X

J n

defining operator-valued distribtuions lj! and \j!''' Likewise for the canonical anticommutation relations, the Fock represen-tation is given in H+ by:

( 6) + + ....

li1"+l 'I' n+l (~l' .... ~) (lj!(x)':l')n(xl, ... , X ) = ... ' X n' n

(-l)n+l n <-l)j+ 1 o<~ .... .... + ) I ~.) (1/J'''(x)'¥ )n (xl, ... ' X = X n ,tn j=l J

A

+ + .... ) "'n-l(xl, ... ' x.' ... ' X

J n

From either Eqs. ( 5) or Eqs. ( 6 ) ' direct calculation gives

n .... (p(f)':l')n = m I f( X. )':l'

j=l J n ( 7)

.... h n + .... + .... (J(g)':l')n = 2i I [g(x.) • II . + 1/ • •g(x.)]'¥

j=l J J J J n

from which Eqs. (4) can be verified.




+ + The subspaces H- of n H- are invariant for the represen-

+ tations given by Eqs. (7). Restricted to H- we obtain irreduc-n ible representations of the local current algebra, called the n-parti~le Bose (-) or Fermi (+) representations.

Too write down the local current group associated with Eqs. -+

(4), set

on F 3 m = h = l (for simplicity), and let ¢ be the flow

-+ s generated by the vector field g , for s E F That is,

;s 1s the one-parameter group of C00 diffeomorphisms satisfying the differential equation

-+ + (d ¢ )(x)

s s -+ + -+ = g(¢ (x)) s

+ +

( 8)

with the boundary condition ¢s=O (~) = x If g has compon--+ +

ents 1n S , then for any s , ¢ (x) s tends rapidly to x as

1~1- 00 The desired group is now a semidirect product S ~ DiffCF 3 ) , where S is Schwartz' space under addition, and DiffCF 3 ) is the group of diffeomorphisms of F 3 having the cor-rect asymptotic behavior, under composition. Like S , DiffCF 3 ) is endowed with a topology of uniform convergence in all deriva-tives. The group law for S ~ DiffCF 3 ) is given by:

vJhere + H

-+ f o ¢ and

-+

=

Let U(f)V(¢) be a continuous unitary representation of

( 9)

s 1\ DiffCJR 3 ) . Then U(sf) and V(;s) are continuous one-para-meter unitary groups. Stone's theorem asserts a unique cor-respondence between self-adjoint operators and continuous one-pa-

+ rameter unitary groups; thus, U(sf) = exp[isp(f)] , V(¢ ) = s exp[isJ(g)] , and conversely,

p(f) lim U( sO - I = is s+O +

V(¢ ) - I -+ s J(g) = lim is s+O

(lO)

The n-particle Bose and Fermi representations of S ~ DiffCF 3 ) + in H- are: n



DIFFEOMORPHISM GROUPS AND QUANTUM THEORY

U(f)'l' n = n

exp[i L j=l

f(~.)]'l' J n

193

(ll) ....

V(cjl)'i'n =

where ;;ex.) is the Jacobian matrix defined by [JtCx)]k = k _,.'+' J 'I'

caR.cp )(x) •

One noteworthy feature of the preceding development is that the same Lie algebra (4) and semidirect product group (9) occur whether we use canonical commutation or anticommutation relations. Thus, information about the particle statistics which was embod-ied in the algebru of fields, is no longer embodied in the algeb-ra of currents. Instead, this information is contained in the choice of unitary representation of the local current group. Furthermore, the local currents preserve particle number in the Fock representation but no information about the number of par-ticles is embodied in the algebra (as it would be, for example, in a Heisenberg algebra of q's and p's ) . This information, too, is contained in the choice of unitary representation of the group.

These observations are suggestive of the viewpoint taken in this paper -- that the diffeomorphism group can be taken as a universaL group for quantum theory, in the sense that the pos-sible quantum systems in ~ 3 can all be obtained as continuous unitary representations of Diff~ 3 ) or its semidirect products.

A related context in which unitary representations of dif-feomorphism groups occur in quantum physics is in the "super-space" formulation of quantum gravity. Here the manifold is an asymptotically flat spacelike surface M. Two 3-metrics g ..

l] on M are defined to be equivalent if they are related by means of a diffeomorphism of M , and the resulting collection of equivalence classes of 3-metrics is called superspace. One ex-pects that in a canonical description of quantum gravity, the field operators would act on state functionals defined on super-space. The appropriate diffeomorphism group Diff(M) would then serve as a gauge group for the theory, and the state func-tionals would carry a unitary representation of Diff(M) . Uni-tarily inequivalent representations would correspond to physical-1 d . . l . 10-14 y lStlnct so utlons.




In the next section I shall survey some of the unitary rep-resentations of S /\ DiffOR 3) which have been studied thus far, together with their physical interpretations. The roles of measures on configuration spaces, gauge groups, and inducing con-structions are summarized, and relationships are mentioned with the method of coadjoint orbits. Much of this work has been per-formed in collaboration with R. Menikoff and D.H. Sharp. In the last section, I shall step back from the study of particular rep-resentations to discuss various semidirect products of the dif-feomorphism group, especially semidirect products of vector spaces of tensor fields T~(M) with Diff(M) .

2. REPRESENTATIONS OF S ,.. DiffOR3) AND THEIR INTERPRETATION (see References 15-17). To represent the group S ,.. DiffOR3 ) ,

one can consider the dual space S' of tempered distributions (generalized functions} on ~ 3 • This space is equipped with a a-algebra of measurable sets, generated from cylinder sets by countable processes, which turns out to coincide with the o-al-8ebra of Borel sets in the weak dual topology. DiffOR3 ) acts on S' by:

.. .... ( F, f) <F, fo<j>) (12)

for FE S' and f E S . Thus S' acquires an orbit structure under DiffOR3 ) .

The configuration space for a representation of DiffOR 3) is a manifold IJ. in S' invariant under IJ. may be finite- or infinite-dimensional. One also needs a probability measure ~ on IJ. , quasi-invariant for the action

is, the family of measure zero sets must be diffeomorphism. For the representation to be

of DiffOR3 ) that preserved by any irreducible, ~ must be ergodic; that is, the measure of any set invariant under DiffOR3 ) must be either 0 or l . This leaves open two possibilities: either IJ. is a single orbit, or it is the union of uncount~bly many orbits. Both situations oc-cur among the repsentations which have been studied. Now the Hilbert space for the representation can be realized as an L2-space over IJ. with respect to ~ , and the operators U(f) and V{;) are given (for ~ E L2(S') ) as follows:

il




(U(f)ljl)(f) = exp[i( F, f)]'¥(f) (13)

... (V(¢)\ji)(F) =

where d~;/d~ is the Radon-Nikodym derivative of the transformed measure ~; with respect to ~ .

Let us consider some examples. EXAMPLE 1. The n-particle Bose representation discussed

above is realized on the orbit

l'l(n) = {F = n L

j=l o-> x.

J

... x. J

all distinct} (14)

where o-> denotes the evaluation functional (Dirac 8-function) X

(o->, f)= f(~) . This orbit may also be regarded as the space of X

unoPd@Ped configurations £~ 1 , ... , ~n} . The measure ~ can 3 3 be any measure equivalent to Lebesgue measure d x1 ... d xn

More generally, one can describe N distinct species of Bose particles, characterized by their masses (or charges, if p is interpreted as the charge density and J as the electric cur-rent density), with nk particles havinB mass (or charge) qk . This is accomplished merely by introducing real coefficients in front of the evaluation functionals in Eq. (14):

= {F =

...

~. all distinct} Jk

(15)

The action of a diffeomorphism ¢ on these orbits is simply ... ...,

to move the points x. at which the particles are located: x. = J J ... ...

¢ (X.) • J

EXAMPLE in JR 3 is a

2. A locally finite configuration {~.} of points J

subset such that only finitely many points are con-tained in any bounded region. The configuration has average den-sity p if limy_."" (N/V) = p , where V is the volume of an appropriate bounded region in JR 3 , and N is the number of points in the region. Let rP be the set of all locally finite configurations in JR 3 with average density p •




The infinite free Bose 8as is described by means of a measure on the infinite-dimensional manifold

( 00) 6-

p = { F = 00

I L

j=l o-+ x.

J (16)

This manifold is the union of uncountably many orbits, because ->

the action of ~ preserves the asymptotic behavior of a configu-ration. The measure ~ is a Poisson measure, which can be char-acterized as follows: let NR(F) denote the number of points in the infinite sum for F which are located in region R , and let V be the volume of R . Then

(17)

that is, the probability of finding N particles ln region R - 18,19 lS given by a Poisson distribution with parameter pV .

EXAMPLE 3. A composite quantum particle having an internal degree of freedom, the dipole moment, is described by the follow-ing representation of S /\ DiffCJR 3 )

(U(f)'l')(~, !) = exp[i! • (V'f)(~)]'l'(~, !)

·+ 7 -+ (V(¢)'l')(x, A) =

Here we think of -> X as the position coordinate and

(18)

as the dipole moment coordinate, and 'l' is a square-integrable function

+ -> with respect to Lebesgue measure on (x, A)-srace. The transform-• -> (->)-> • • ('')k c~o"k)( .... x)'Q, ed dlpole moment A1 = J¢ x A lS glven by A = oN'P A

summation over repeated indices being assumed. This representation is realized on the orbit

/':,(dipole) = { r = -t · Vo-+ t ~ o} (19) X

-> c-X·vo-+) c-t' ln S' where -> ) In analogy ' X <j>(x) with Eq. (14)' n identical Bose dipole particles are described by means of the orbit

(20) n /':,(n dipoles) = {F = I -> x.

J all distinct}

j=l




Representations of SA DiffOR3) can also describe particles with fixed net charge as well as variable dipole moment. The orbit for a single-particle representation of this type is:

197

(21) ~(charged dipole) = { F = qo~ - r . vo~ 1 r .,_ o}

X X

EXAMPLE 4. Quadrupole and higher multipole particles are likewise described by means of representations of S A Diff0R3) There are nine distinct orbits for a quadrupole particle, corres-ponding to distinct group representations. In each case the measure is concentrated on a single orbit in S 1 of the follow-ing form, where Q is a 3x3 symmetric real matrix:

~(quadrupole) = (22)

The action of a diffeomorphism ; on ~(quadrupole) is given ~ ~ ~ ~ ~I by $ : ( x, A , Q) ~ ( x 1 , A , Q 1 ) , where :

~I ~ ~

= <jl(x)

(A 1 )k = (() R,<flk) (~)A R, + l_qmn(<l () <jlk) (~) 2 m n (23)

(Q 1 )mn = <a 1<Pm> <~> (<lk<fln) Cx>QR.k

An orbit under Eq. (23) is determined by specifying the signs of the three eigenvalues of Q ; that is, two generalized functions of the form shown in Eq. (22) are in the same orbit if and only if their quadrupole matrices have the same number of positive, negative, and zero eigenvalues.

As before, one can write representations for n quadrupole particles, and one can consider particles with fixed net charge in addition to variable dipole and quadrupole moments. 20

EXAMPLE 5. When the manifold ~ is finite-dimensional, there is no difficulty in specifying a measure. We have also seen one example of an infinite-dimensional manifold, where Poisson measure satisfies the requirement of quasi-invariance with respect to diffeomorphisms. A plausible generalization is to consider spaces of non-pointlike configurations such as strings, loops, surfaces, or extended regions. Specifically,

~ let M be a manifold or manifold with boundary, and let x(p)




for p E M be a continuous mapping from M . 3 lnto :R . Let v be a measure on M, and define the generalized function FE S' by

( F, f) = r + J f(x(p) )dV(p)

M

+ A diffeomorphism ~ acts on such a configuration by taking

+ + + to x'(p) = ~(x(p))

For example, string configurations could be obtained by letting M be the unit interval and V 18 Lebesgue measure; loop configurations are obtained by requiring ~(0) = ~(1) .

(24)

+ x(p)

+ Note that x(p) need not be differentiable in order for the gen-eralized function to be defined. The spaces 6 of all string configurations, all loop configurations, configurations with various knottedness properties, etc. all define formal represen-tations of S ~ Diff<:R3) . It remains, however, to obtain suit-able measures on such spaces which have the property of quasi-invariance for diffeomorphisms.

Having examined various examples of configuration spaces, we can now observe how additional representations of S A Diffe:R3> are obtained by means of an inducing construction. In this way, inequivalent representations are associated with the same configuration space, and particles having different statis-tics or internal degrees of freedom such as spin can be de-scribed.21•22

The general procedure is to consider a particular element F E 6 The little gPoup GF is the subgroup of Diff(:R 3) whose elements leave F fixed. A representation of GF should now induce a representation of Diff<:R 3) , but in the general case measure-theoretic difficulties stand in the way. The prob-lem is that because neither Diff<:R 3> nor GF is locally com-pact, there is no Haar measure on these groups. Thus, even when we have a measure on 6 , we are not guaranteed as in the Mackey theory that the inducing construction will work for an arbitrary representation of GF

If, however, there is a continuous homomorphism from GF to a locally compact group K , then those unitary representa-tions of GF which factor through representations of K can be used to induce representations of Diffe:R3) with fewer measure-




theoretic difficulties. The group K is the gauge group for the

theory. Let us see how this works to yield representations describ-

ing particles with different statistics, as well as particles with spin.

EXAMPLE 6. Referring to the orbit ~(n) in Eq. (14), we

observe that it is not simply connected. In fact, its fundamen-tal group is S , the group of all permutations of n objects. n There is a natural homomorphism from GF , the group of all dif-feomorphisms leaving ~~ 6~ fixed, onto S and S LJ=l x. n n

J serves as a gauge representation of

group for the theory. Let T be a unitary S in a Hilbert space M n then T also

defines a unitary representation of GF . The Hilbert space for -the induced representation is the space of functions on ~ (the universal covering space of ~ ) taking values in M , which

The ac-transform according to the representation T of S ~ n -

tion of a diffeomorphism ¢ on ~ lifts to ~ ; indeed, ~ ~

is just the coordinate space of ordered n-tuples (x1 , When T is the trivial one-dimensional representation of by unity, we recover the n-particle Bose representation of

~

' X ) • n s n

S A DiffOR3 ) When T is the one-dimensional alternating rep-resentation of S , we recover the n-particle Fermi represen-n tation of S A DiffOR3 ) And when T is a higher-dimensional

we obtain representations of S A representation of S , n DiffOR3 ) corresponding to "parastatistics."

EXAMPLE 7. A special situation occurs when, instead of S A DiffOR3 ) , we consider S 1\ DiffOR 2 ) . Now the fundamental group of an n-particle orbit is larger than Sn ; it is the "braid group" B whose elements can be thought of as braided n paths connecting n points in a two-dimensional configuration with each other. There is a natural homomorphism from B onto n Sn , but of course not every representation of Bn factors through a representation of S The one-dimensional unitary n

can thus induce representations of S A representations of B n DiffOR2 ) describing not only Bose and Fermi statistics, but al-so the unusual statistics which even distinguishable particles

. 23-26 moving in two-dimensional space can theoretlcally obey. EXAMPLE 8. Another special situation comes about when,

instead of S 1\ DiffOR3 ) , we consider SA Diff(M) , where M is a non-simply connected manifold. One way to arrive at this




situation is through the generalization of Eq. (2) to the case + of an external, non-quantized electromagnetic field A , where

B = V x A . If in (2) we substitute:

(25)

then the third commutation relation in (4) becomes (26)

=

Eq. (4) is recovered in the field-free region. But in the situ-+ ation of the Aharonov-Bohm effect, where the B-field is confined

to a tightly wound, infinite solenoid behind a high potential barrier, the particles move in a non-simply connected field-free region M . Now the fundamental group of even the one-particle orbit is nontrivial. Its representations induce representations of S A Diff(M) in which the orbital angular momentum spectrum is shifted from integer multiples of h by an amount proportion-

1 h fl h h h 1 . d 2 5, 2 5 a to t e · ux t roug t e so enol . EXAMPLE 9. To appreciate the final example, consider first

another generalization of Eqs. 0)-(4). Let 1/J(}t) be a "spinor" field for a representation of a Lie group G for example, G might be SU(2) . Let L a= 1, ... , N denote N inde-a pendent generators of G . Set

"i': -+ -+-= 1/1 (x)L ljJ(x) a ( 27)

along with Eqs. (2), and

T(h) = N I

a=l + + 3 T (x)h (x)d x a a (28)

along with Eqs. (3), for test functions h E S' . Then we ob-a tain, in addition to Eqs. (4), the following additional commuta-tion relations:

[p(f), T(h)] = 0

= ihT(g • Vh) (29)

= Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.


where


= N I eabchlah2b

a=l b=l

in which eabc are the structure constants of G .

201

(30)

The local current group associated with the third commuta-tion relation of Eq. (29) is the group of C00 mappings from R 3

into G under pointwise multiplication. Calling this group 3 MapOR , G) , the full local current group for Eqs. (4) and (29)

is CS ® MapOR 3 , G)) f\ DiffOR3 ) For the case G = SU(2) , the operators T(h) are spin

density operators. But the question arises as to whether par-ticle spin can be obtained directly from the study of representa-tions of S A Diff0R3 ) , without adjoining spin density opera-tors "by hand." The following example shows that this is pos-

'bl 28 s1 e. Let ~ be the !-particle configuration space {6~

_,. 3 x E R } .

_,. The little group for a point x is G-+ =

X {$ I $ <~) =

~} . Define the map h-+ DiffOR 3 ) - SL( 3, R) X

by

Restricted to G-+ h-+ defines a homomorphism from X ' X

SL(3,R) More generally, consider a path r from oo

G-+ X to

to

( 3l)

_,. X _,. _,.

Then h-+(¢) for y E r is a path from the identity _,. y • "TD3 ln .l" .

element to h-+(¢) in SL(3, R) X

and thus defines an element of the universal covering group SL(3, R) which we denote - _,. h-+(¢) . Now SL(3, R) plays the role of the gauge group, and a

X representation of SL(3, R) induces a representation of DiffOR 3 )

But representations of SL(3, R) can be decomposed with re-spect to SU(2) . This decomposition permits the local current operator to be written

where: -+ J (g) 0

-+ l _,. _,. l 2 -+-+ = J (g) + -( v X g) • l: + 2 I G (g) (X) T 0 2 )1=-2 -)l )l

is the orbital term, l -+ _,. 2cv x g) • l: is the spin

contribution, and the last term is a spin-changing term with

( 3 2)




_,. _,. functional coefficients G (g)(x) . Such terms may have appli-

-11 cation to excited states of nuclei, to supermultiplets of hadrons lying on Regge trajectories, or to particles in a strong non-

'f . . l f' l 29-33 unl orm gravltatlona le d. _,.

Alternatively, the class of test functions g can be re-stricted so that on appropriate domains, the spin-changing terms vanish. On the resulting SU(2)-invariant subspaces, the surviv-ing terms generate representations of Eqs. (4) and (29); that is, representations of orbital momentum density and spin density op-erators. Particles with both integral and half-integral spin are described this way.

It appears that the analogous construction for the group SA Diff0R2) would lead to representations describing particles

. h f . l . 34 Wlt ractlona spln.

The various representations of S A DiffOR3) described here can also be obtained by the method of coadjoint orbits. First, it should be noted that in this method of group representation, one starts with a classical theory, and proceeds by means of "geometric quantization." Thus, the successful description of classical fluids by means of Poisson structures on coadjoint orbits of diffeomorphism groups and their semidirect products strongly suggests the use of coadjoint orbits to study quantum hydrodynamics.

In broad outline, the procedure is to consider orbits in the coadjoint representation of Diff0R 3 ) or one of its subgroups, such as the group sDiffOR 3 ) of volume-preserving diffeomor-_,. phisms. A point F in a coadjoint orbit is an element of the dual of the Lie algebra of the group; i.e., a "generalized vector field." Associated with the generalized vector field f is a "little group" of diffeomorphisms which leave f invariant. To proceed with geometric quantization, one next seeks a "polari-zation" -- that is, a suitable group (and corresponding Lie alge-bra) intermediate between the little group and the full group of diffeomorphisms. When the orbit has the property called "integ-rality," it is possible to obtain a character of the identity component of the "polarization" group from the action of f on Lie brackets of vector fields. Combined with a representation of the fundamental group of the orbit, a representation of DiffOR 3 ) . . 35-37 lS lnduced.




Kirillov has described representations associated with fi-nite-dimensional coadjoint orbits of the group of diffeomor-phisms of a manifold. He points out that unlike the case of finite-dimensional solvable and semisimple Lie groups, the fun-damental group of a coadjoint orbit of the diffeomorphism group can be non-commutative. He also notes the role played by the symmetric group.

When the coadjoint orbit formalism is used to represent DiffOR3> , the semidirect product structure with S is no lon-ger of essential importance. It seems that particle statistics will now be described by means of the choice of representation of the fundamental group of an orbit, and spin will be described by the choice of the orbit itself. This is still true when the coadjoint orbit formalism is used to represent S A DiffOR3> directly, but the choice of polarization for the coadjoint or-bits appears to be unique in this case, singling out the indi-cated semidirect product structure.

Representations of sDiffOR3) in the coadjoint orbit pic-ture may be especially useful in describing quantum vortices. 38 • 39 Here one is interested in generalized vector fields F , where V x F = 0 except on a closed oriented loop in ~ 3

Again, the problem arises of finding suitable quasi-invariant measures on spaces of such generalized vector fields. It is al-so interesting to observe that in some situations no polariza-tion is possible -- for example, in the case of a point vortex in a two-dimensional, incompressible fluid.

3. SEMIDIRECT PRODUCTS. Let us now look for a moment at the kinds of groups we have encountered, and why the use of the dif-feomorphism group as a "universal group" for quantum theory is such an attractive idea.

Suppose we start with a manifold M (such as ~ 3 at a fixed time, or more generally a space-like surface), and the as-sociation of self-adjoint operators in a Hilbert space H with observations in local regions of M . Then Stone's theorem, which asserts a correspondence between self-adjoint operators and continuous one-parameter unitary groups, leads us to ask what kinds of groups have subgroups associated naturally with regions of M . An immediate candidate, of course, is Diff(l1)




under composition, in that we can define for a bounded

the subgroup of diffeomorphisms having support ln n . the attractiveness of Diff(M) is that it comes to us

M , without additional imposed structure.

region n Thus,

along with

Given a Lie group G , we have also considered the group

Map(M, G) under pointwise multiplication. Its elements are 00

C mappings from M to G which tend suitably rapidly to the

identity at infinity. For a region n , of maps which equal the identity outside

there is the subgroup n . We also have the

semidirect product Map(M, G) A Diff(M) , with the group law

for R1 , R2 E Map(M, G)

we recover S A Diff(M) ,

=

and ¢1 , ¢ 2 E Diff(M)

and when G = SU(2)

local current group for particles with spin.

( 3 3)

When G = ~ we recover the

The introduction of the Lie group G feels rather arbitrary

if one wishes to start only with the manifold M It is, on the

other hand, quite natural to consider the vector space of C00

scalar functions on M , together with the action of Diff(M)

on this space, as being "given" along with the geometry of M This is another way to think of S A Diff(M) . Thus elements of

S' are naturally considered as "spatial configurations" in the quantum theory obtained by representing S A Diff(M) .

There is, however, no reason to restrict our attention to 00

scalar functions. Given the c manifold M we have just as naturally the vector space Tr of tensor fields of contra-s variant rank r and covariant rank s ' and we can consider tensor fields tending rapidly toward zero at infinity. We also have the usual "lifting" of Diff(M) to act on Tr . Thus, it s is natural to consider unitary representations of the semidirect

product Tr A Diff(M) , with S A Diff(M) being the special s case r = s = 0 . Even more generally, the semidirect product ( @ Tr) A Diff(M) could be regarded as a natural object to r,s=O s represent.

Now one must generalize the notion of a spatial configura-tion from an element of S' (a generalized function) to an ob-ject with components ized tensor field).

ln (Tr)' for arbitrary r, s (a general-s This appears to be a useful framework for

understanding configurations describing particles with "internal"




degrees of freedom. Additionally, one could consider actions of Diff(M) on

tensor fields which "contract" tensors of higher rank and "mix" them with tensors of lower rank, as for example in Eqs. (23). One can thus obtain alternate semidirect products of vector spa-ces of tensor fields with Diff(M)

Finally, let us look again at Map(M, G) . Instead of thinking of this group as mappings from M into a fixed Lie group G , let us consider a copy G of G to be associated

X with each point x E M so that we have a fiber space F which is locally M x G and a system of isomorphisms I : G --4 G . xy x y Then for ~ E Diff(M) , we obtain a lifting ~ F ~ F given by (~R)(x) = I , R(x') where x' = ~Cx) and R is a Coo

X X cross-section. Thus all of the semidirect products discussed in this paper can be built up from liftings of Diff(M) to fiber spaces over M , and the notion of a physical configuration in the quantum theory becomes that of a "generalized" cross-section.

ACKNOWLEDGMENTS. The author is indebted to A.S. Wightman for his encouragement and for many stimulating discussions, to B. Kostant and G. Mackey for interesting conversations, and to R. Menikoff and D.H. Sharp for their collaboration in obtaining many of the results summarized in Section 2. He thanks the Physics Depart-ment at Princeton University for hospitality and support during his 1982-83 sabbatical year as a Visiting Fellow, and Los Alamos National Laboratory for its frequent hospitality. Financial assistance was also provided by the Graduate School at Northern Illinois University.

REFERENCES

1. V.I. Arnold, Ann. Inst. Fourier Grenoble 16 (1966), 319.

2. J. Marsden and A. Weinstein, "Coadjoint Orbits, Vor-tices, and Clebsch Variables for Incompressible Fluids," in Pro-ceedings of the Los Alamos Conference "Order in Chaos", ed. by D.K. Campbell, H.A. Rose and A.C. Scott, Physica 7D (1983), 305-323.

3. J. Marsden, A. Weinstein, T. Ratiu, R. Schmid and R.G. Spencer, "Hamiltonian Systems with Symmetry, Coadjoint Orbits, and Plasma Physics," in Proceedings of the IUTAM Symposium on Modern Developments in Analytical Mechanics (to be published).




4. See also the talks in these proceedings by Holm, Kaufman, Marsden, Meyer, Morrison, Ratiu, Weinstein, and Wollman.

5. R.F. Dashen and D.H. Sharp, Phys. Rev. 165 (1968), 1857.

6. G.A. Goldin, "Current Algebras as Unitary Representa-tions of Groups," Ph.D. thesis, Princeton University (1968); J. Math. Phys. 12 (1971), 462.

7. G.A. Goldin and D.H. Sharp, "Lie Algebras of Local Cur-rents and their Representations, " in Group Representations in Mathematics and Physics: Battelle Seattle 1969 Rencontres, ed. by V. Bargmann, Springer-Verlag, New York (1970), 300.

8. A.A. Dicke and G.A. Goldin, Phys. Rev. D 5 (1972), 845. 9. G.A. Goldin, "What We Have Learned About Local Rela-

tivistic Current Algebras,'' in Local Currents and their Applica-tions, ed. by D.H. Sharp and A.S. Wightman, North Holland, Amsterdam (1974), 101.

10. J.A. Wheeler, "Geometrodynamics and the Issue of the Final State," in Relativity, Groups and Topology, ed. by B.S. DeWitt and C. DeWitt, Gordon and Breach, New York (1964).

11. J.L. Friedman and R.D. Sorkin, Phys. Rev. Lett. 44 (1980), 1100.

12. J.L. Friedman and R.D. Sorkin, Gen. Rel. Gravity 14 (1982), 615.

13. C.J. Isham, ''Quantum Geometry," (1982 preprint, to ap-pear in the Festschrift in honor of Bryce DeWitt's 60th birthday, published by A. Hilger).

14. J. Marsden, D.G. Ebin and A.E. Fischer, "Diffeomorphism Groups, Hydrodynamics and Relativity,'' in Proceedings of the Thirteenth Biennial Seminar of the Canadian Mathematical Congres~ ed. by J.R. Vanstone, Montreal (1972).

15. I.M. Gel'fand and N. Ya Vilenkin, Generalized Functions, Vol. 4, Academic Press, New York (1964).

16. G.A. Goldin, R. Menikoff and D.H. Sharp, "Induced Rep-resentations of Diffeomorphism Groups Described by Cylindrical Measures," in Measure Theory and its Applications, ed. by G.A. Goldin and R.F. Wheeler, Northern Illinois University Dept. of Mathematical Sciences, DeKalb, Illinois (1981), 207.

17. G.A. Goldin, R. Menikoff and D.H. Sharp, "Diffeomor-phism Groups, Gauge Groups and Quantum Theory," Phys. Rev. Lett. 51 (1983) 2245.

18. G.A. Goldin, J. Grodnik, R. Powers and D.H. Sharp, J. Math. Phys. 15 (1974), 88.

19. A.M. Vershik, I.M. Gel'fand, and M.I. Graev, Usp. Mat. Nauk 30 (1975), 3. English translation: Russ. Math. Surveys 30 (1975), l.

20. G.A. Goldin and R. Menikoff, "Quantum-Mechanical Rep-resentations of the Group of Diffeomorphisms and Local Current Algebra Describing Tightly Bound Composite Particles," Los Alamos Preprint LA-UR-83-2250 (1983).

21. G.A. Goldin, R. Menikoff and D.H. Sharp, J. Math. Phys. 21 (1980), 650.




22. G.A. Goldin, R. Menikoff and D.H. Sharp, J. Phys. A: Math. Gen. 16 (1983), 1827.

23. F. Wilczek, Phys. Rev. Lett. 48 (1982), 1144. 24. R. Jackiw and A.N. Redlich, Phys. Rev. Lett. 50 (1983),

. 555. 25. G.A. Goldin, R. Menikoff and D.H. Sharp, J. Math. Phys.

22 (1981), 1664. 26. G.A. Goldin and D.H. Sharp, Phys. Rev. D 28 (1983),

830. 27. R. Menikoff and D.H. Sharp, J. Math. Phys. 18 (1977)'

471. 28. G.A. Goldin and D.H. Sharp, Commun. Math. Phys. (1983,

in press). 29. L.C. Biedenharn, R.Y. Cusson, M.Y. Han and O.L. Weaver,

Phys. Lett. 42B (1972), 257. 30. O.L. Weaver, L.C. Biedenharn and R.Y. Cusson, Annals of

Physics 77 (1973), 250. 31. O.L. Weaver, R.Y. Cusson and L.C. Biedenharn, Annals

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4 (1971), 457. 37. A.A. Kirillov, Ser. Math. Sov. l (1981), 351. 38. M. Rasetti and T. Regge, Physics BOA (1975), 217. 39. M. Rasetti and T. Regge, "Quantum Vortices" (1983 pre-

print).

DEPARTMENT OF MATHEMATICAL SCIENCES NORTHERN ILLINOIS UNIVERSITY DeKALB, ILLINOIS 60115






Part II. Analytic and Numerical Methods







CONTOUR DYNAMICS FOR TWO-DIMENSIONAL FLOWS

Norman J. Zabusky

ABSTRACT. Contour dynamics is a boundary integral evolutionary method ideally suited to 2-D inviscid and nearly inviscid flows. Applications in many areas are described. Specific emphasis is given to the Euler equations. A large class of new steady-state solutions with piecewise constant vorticity is described, includ-ing singly-connected rotating and doubly connected translating "V-states". Examples of time dependent calculations are shown which exhibit merger and scattering.

In the contour dynamics (CD) approach, the evolution of plane curves describes the nonlocal and nonlinear dynamics of two-dimensional (20) fluid and plasma systems. Contour dynamics is a free boundary--integral evolu-tionary method that is ideally suitable for incompressible inviscid or nearly-inviscid 20 flows. The contours are the boundaries of constant den-sity or vorticity regions that are the sources of the flow. The velocities of the contours are obtained from integrals or integral equations on the closed contours. It is a natural technique for flows in unbounded media because the Green's function has a simple form. Contour dynamics is a generalization of the "waterbag"method and is being practiced by a variety of groups.

Longuet-Higgins and Cokelet [1] have investigated shallow and deep water waves on boundaries that separate piecewise-constant density regions. Liggett, Liu and collaborators have investigated the flow in 20 and 30 porous media [2,3] and recently the shallow water wave problem with forcing at one end and sloping beaches at the other end [4,5]. Baker, Orszag and Meiron have also recently investigated the water wave problem [6]. The inverse situation, namely the Rayleigh-Taylor instability has been investigated by the last group [7] and by Menikoff and Zemach [8].

1980 Mathematics Subject Classification. This work was supported by the Office of Naval Research, the Naval Research Laboratory and the Army Research Office.

211

© 1984 American Mathematical Society 0271-4132/84$1.00 + $.25 per page




212 Norman J. Zabusky

Zabusky and colleagues [9,10,11,12], Pierrehumbert [13], and Saffman and colleagues [14,15,16] have investigated stationary configurations of the Euler equations with piecewise-constant finite area vortex regions (FAVR's). Deem and Zabusky [9] have called this generic class of solutions of the Euler equations as "V-states". Zabusky, Overman and Wu [17] have developed accurate and faster evolutionary algorithms. These CD algorithms have been applied to study the merger of two like-signed FAVR's and coro-tating V-states [18] and the coaxial scattering of translating "dipolar" V-states [19].

Finally, Overman, Zabusky and Ossakow [20,21] have been studying the evolution of a piecewise-constant, weakly-ionized and strongly-magnetized ionospheric plasma (or a deformable dielectric) in an electric field, a problem mathematically analogous to equations of flow in porous media. There are no existence theorems for the plasma flows and we have no assur-ance that solutions exist for all times. Computational evidence shows that the two fluid Rayleigh-Taylor problem and the magnetized plasma problem are ill-posed and develop interfacial corners after a finite time. To overcome this inadequacy, Zabusky and Overman have introduced a physically moti-vated, dissipative, tangential regularization procedure [22].

REFERENCES

1. Longuet-Higgins, M.S. and E. D. Cokelet, "The deformation of steep surf ace waves on water I. A numer i ca 1 method of computation," Proc. R. , Soc. London Ser. A 350, (1976) 1-26. Also, "The deformation of steep sur-face waves on wateru. Growth of normal mode instabi 1 ities ," Proc. R., Soc. London Ser. A. 364, (1978) 1-28. 2. Liggett, J. A. and P.L-F. Liu, The Boundary Integral Equation Method for Porous Media Flow, George Allen & Unwin, England, 1982 3. Lennon, G. P., P.L-F. Liu and J. A. Liggett, Boundary integral solu-tions to three-dimensional unconfined D'Arcy's flow. J. Water Resources Res. ~. (1980) 651-658. 4. Liu, P.L-F and J. A. Liggett, "Applications of boundary element methods to problems of water waves," Chapter 3, pp. 37-67, in Developments in Boundary Element Methods - 2 ed. by P. K. Banerjee and R. P. Shaw, Applied Science Publishers, England, 1982. 5. Kim, S. K., P.L-F. Liu and J. A. Liggett, "Boundary integral equation solution for solitary wave generation, propagation and run-up." {Preprint, 1983). 6. Baker, G. R. and Meiron, D. I. and Orszag, S. A., "Generalized vortex methods for free surface flow problems." Preprint, Massachusetts Institute of Technology (1982).



CONTOUR DYNAMICS FOR TWO-DIMENSIONAL FLOWS

7. Baker, G. R. and Meiron, D. I. and Orszag, S. A., "Vortex simulations of the Rayleigh-Taylor instability". Phys. Fluids Q (1980}, 1485-1490. 8. Menikoff, R., and C. Zemach, "Rayleigh-Taylor Instability and the use of conformal maps for ideal fluid flow," J. Comput. Physics (1983, in press). 9. Deem, G. S. and N. J. Zabusky, "Stationary V-states: Interactions, recurrence and breaking," Phys. Rev. Lett. 40 {1978), 859. See also G. S. Deem and N. J. Zabusky, "Stationary "V-states", Interactions, recurrence and breaking," in Solitons in Actions, 277-293 (eds. K. Lonngren and A. Scott), Academic Press, Inc., 1978. 10. Zabusky, N. J., M. H. Hughes and K. V. Roberts, "Contour dynamics for the Euler equations in two-dimensions," J. Comput. Phys. 30 (1979} 96-106. 11. Zabusky, N. J., "Recent developments in contour dynamics for the Euler equations," Ann. of the N.Y. Acad. Sci. 373 {1981) 160-170. 12. Wu, H. M., and Overman II, E. A., and Zabusky, N. J., "Steady-state solutions of the Euler equations in two dimensions: Rotating and transla-ting V-states with limiting cases. I. Numerical Results." J. Comput. Phys. {1984) .?1_, xxx, (January).

13. Pierrehumbert, R. T., "A family of steady translating vortex pairs with distributed vorticity," J. Fluid Mech. 99 {1980) 129-144. 14. Saffman, P. G. and R. Szeto, "Equilibrium shapes of a pair of equal uniform vortices," Phys. Flds. 23 {1980) 2339-2342. 15. Saffman, P. G. and R. Szeto, "Structure of a 1 inear array of uniform vortices," Stud. in Appl. Math. LXV, (1981) 223-249. 16. Saffman, P. G. and J. C. Schatzman, "Properties of a vortex street of finite vortices," SIAM J. Sci. Stat. Comput .. Also, "Stability of a vortex street of finite vortices," J. Fluid Mech., 117 {1982) 171-185. 17. Wu, H.-M., E. A. Overman and N. J. Zabusky, "Fast contour dynamical algorithms," in preparation. 18. Overman II, E. A. and Zabusky, N.J., "Evolution and merger of isola-ted vortex structures." Phys. Fluids, 25 (1982) 1297-1305. 19. Overman II, E. A. and Zabusky, N. J., "Coaxial scattering of Euler-equation translating V-states via contour dynamics." J. Fluid Mech., 125 (1982) 187-202. -20. Overman, E. A., II and N. J. Zabusky. "Stability and nonlinear evolu-tion of plasma clouds via regularized contour dynamics," Phys. Rev. Lett. 45 (1980) 1693-1696. 21. Overman, II., E. A. and Zabusky, N.J., and Ossakow, S. L., "Iono-spheric plasma cloud dynamics via regularized contour dynamics. I. Sta-bility and nonlinear evolutin of one contour models." Phys. Fluids, 26 (1983)' 1139-1153. -22. Zabusky, N. J., and Overman II, E. A., "Regularization of contour dynamical algorithms, I. Tangential regularization." J. Comput. Phys., g (1983) xxx, (December). DEPARTMENT OF MATHEMATICS University of Pittsburgh Pittsburgh, PA 15260 Current Address: National Center for Atmospheric Research P. 0. Box 3000 Boulder, CO 80307

213







ON THE NONLINEAR STABILITY OF CIRCULAR VORTEX PATCHES

* Yieh-Hei Wan

What I would like to present here is a joint work with J. Marsden, T. Ratiu, and A. Weinstein. Details of this work will be given elsewhere [6].

Let D c R2 be a disk of radius R, centered at the origin. Consider the motion of an incompressible inviscid flow in D, in the absence of external forces.

The vorticity w evolves according to the vorticity equation

w = uw + vw = 0 t X y >

where, ~ = (u,v) = ($y'-$x) the velocity field,

w = -$ - $ = -~$ the vorticity, XX YY and $ a stream function. Denote by Gw the stream function such that Gw/oD = 0, for a given w.

A vorticity w in the form AXB• with A constant B a subset of D, is called a vortex patch with strength A and size area B.

Let ~t{w) be the vorticity at time t with initial vorticity w. The semiflow ~t(.) preserves vortex patches together with their strength and sizes.

Denote by w0 , a circular vortex patch of radius y {see Figure 1); i.e.,

wo(~) = {

01 i f I ~I < Y,

if 1~1 > y.

w0 is a stationary solution; i.e. ~t{w 0 ) = w0 for all t ~ 0. In 1880 [4], Kelvin showed that a small perturbation to the circle solution

proportional to cos me {in R2) rotated uniformly with angular velocity

* Research supported partially by DOE contract DE-AT03-82ER12097.

215





216

1 m-1 zm·

'IIEH-HEI IvAN

Figure

Thus, the stationary solution w0 is linearized stable (in R2).

PROBLEM. Can one prove in a suitable sense the nonlinear stability for the circular vortex patch?

The numerical work of Zabusky and others [3] shows that, as times evolv-ing, a thin arm may develop. (See Fig. 2.) The theoretical work [5] of Marsden and Weinstein shows that, nearly circular waves do not have strong enough dis-persion to prevent breaking or singularities.

Figure 2

For Area(A~) u (B~) = lxA- xBILl' it suggests that one should put a

L1-norm on the space of vorticities.

STABILITY THEOREM. For any n > 0, there exists o > 0, such that if w is any vortex patch satisfying jw-tc0 1Ll < o, then jt(w)-w01Ll < n, for all t > 0.

Using Green's Theorem, we have the total energy

1 I 2 1 I 1 E = 2 1.!!..1 dxdy = 2 0 wGwdxdy = 2 <w,Gw>.

Let us sketch the geometric idea ([1], [2], [5J) in proving our stability theorem. The coadjoint orbit 0 through w0 consists of those vortex

wo patches w1 with the same strength and size as that of w0. Clearly, ow0 is

an "invariant manifold" through w0. Regarding E has a Liapunov function on Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.


ON THE NONLINEAR STABILITY OF CIRCULAR VORTEX PATCHES 217

0 , the stability of <!>t on 0 near w0 follows provided E has a "non-wo wo degenerate" local maximum at w0 on 0 . In analytic terms, it means that wo there exists c3 > 0 such that for all w1 = x8,

(E)

holds if !w1 = !w0, and !w1-w0! 1 small. L

The same geometric idea has been presented in Holm's (Monday) lecture, where stationary solutions are minima of a convex function on a convex set. Thus, first order conditions for minima become global conditions. In our situa-tion, stationary solutions can be maxima of a convex function on a convex set (i.e., 0 ~ w1 ~ 1, !w1 = !w0). Therefore, we have to verify second order con-ditions.

Now, let us sketch the proof of our inequality (E). Given lw1-w0! 1 L

small, !w1 = !w0, Gw1 is c1-close to Gw0 (from LP-estimates of ~u = f, with p > 2). There exists a unique vortex patch w1 ={~ED I Gw 1 (~) ~ ~ 0 •

for some ~ 0 } such that !w1 = !w1. Notice, w1 may have a thin arm, but w1 becomes nearly circular {cf. Figure 3).

Figure 3

It suffices to establish:

(A) E(w1) - E(w1) ~ c1iw1 - w1!21, for some constant c1 > 0, L

(B) E(w0)- E(w1) ~ c2iw0-w1!21, for some constant c2 > 0. L

Indeed, inequalities (A) and (B) imply

E(w0) - E(w1) ~ c3!w1-w0!21 with c3 =} min(c1,c2). L

(a) E(w1)- E(w1) = <w1-w1,Gw1> + t <w1-w1,G(w1-w1)> ~ <w-w1 ,Gw1>.

Using a~/ar < 0, we can obtain,



218 YIEH-HEI WAN

1 1 1 with 4A-+ lG1r as w1 ... w0 in L norm. Thus, the inequality (A) is valid.

(B) Recall, w1 is a radial perturbation of w0.

Denote the boundary of w1 by r = f(e) = y + h(e) in polar coordinates. Through lengthy computations, with a careful estimate of the remainder term, we obtain,

E(w1) - E(w0) = ~ <h,Lh> + o(ihi~) (1)

The self-adjoint linear transformation L on L2 has eigenfunctions eine with eigenvalues - y2/2 + an,

Ia = l 2(1 - (y/R)2Inl) n 2 Y lnl

2 a0 = y ln(R/y).

1 2 2 ~ 2 y (1 - (y/R) ), n ~ 0

Since, i !(y+h) 2de = ~ !y2de, for lhi 2 small,

one can easily establish,

<h,lh> ~ -( y;- E)ih!~ 2R

I wl-wo I 1 ~ l21r ( y+ ~l I h 12 L

Equations (1), (2) and (3) imply the inequality (B). Thus, we complete the sketch of the proof of our stability theorem.

(2)

(3)

So far, our circular vortex patch is nonlinearly stable within the class of vortex patches. A somewhat stronger stability result is the following.

and

STABILITY THEOREM'. For any n > 0, there exists o > 0 such that if

(i) 0 ~ w(x,y) ~A, with I A-ll < o.

( i i ) I w-w0 I 1 < o , L

then l~t{w)-w 0 i 1 < n for all t > 0. L

As before, it suffices to show: there exists c3 > 0 such that for all w, 0 ~w ~ 1, one has the inequality



ON THE NONLINEAR STABILITY OF CIRCULAR VORTEX PATCHES 219

The proof of this inequality is basically the same, but more tricky.

REFERENCES

[1] Arnold, V. I. (1965). Conditions for nonlinear stability of stationary plane curvilinear flows on an ideal fluid. Soviet Math. Dokl. ~. 773-777.

[2] Arnold, V. I. (1969). On an apriori estimate in the theory of hydrodynami-cal stability, Am. Math. Soc. Transl. 79, 267-269.

[3] Deem, G. S. and N. J. Zabusky (1978). Vortex waves: stationary •v-states•, interactions, recurrence, and breaking. Phy. Rev. Lett. 40, 859-862.

[4] Lamb, H. (1945). Hydrodynamics, Dover Publications, New York.

[5] Marsden, J. and A. Weinstein, (1983). Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids. Physica D (to appear).

[6] Wan, Y. H., J. E. Marsden, T. Ratiu, and A. Weinstein, Nonlinear stability of circular vortex patches (to appear).

DEPARTMENT OF MATHEMATICS STATE UNIVERSITY OF NEW YORK AT BUFFALO BUFFALO, NEW YORK 14214







VORTEX METHODS FOR FLUID FLOW IN TWO OR THREE DIMENSIONS

J. Thomas Beale1 and Andrew Majda2

Vortex methods are a means of simulating incompressible fluid flow which are appropriate for inviscid flows or flows at high Reynolds number. Their use grew out of early attempts to represent fluid phenomena with systems of point vortices. A general method developed by Chorin and others [4,13] uses elements of vorticity of two kinds which are advected according to a velocity computed from their current configuration. The interior flow is represented by a collection of vortices of finite core, or vortex "blobs", w.hile vortex sheets are generated near the boundary to satisfy the no-slip condition. A random walk of the computational elements simulates the effect of viscosity. J. Sethian, in another talk at this conference, has described a typical appli-cation of vortex methods to the combustion of swirling flow. For an exposi-tion of the full method, see [5].

Our emphasis here will be on the design and accuracy of vortex methods for inviscid, incompressible flow in free space, especially in three dimen-sions. Methods for 3-D c~n be motivated by analogy with the more familiar two-dimensional case. After reviewing the formulation, behavior, and analysis of the two-dimensional method, we describe two rather different 3-D methods, both of which are rigorously justified in the sense that they converge to the actual solution as long as the underlying flow is smooth.

To introduce the method in two dimensions, we begin with the Euler equa-tions governing the flow,

ut + (u • V)u + Vp = 0,

v . u = 0 .

Here u = (u1,u2) and we will write coordinates as z = (x,y). The vorticity is the curl of the velocity field; in 2-D this is the scalar

w = u2,x- ul,y .

1980 Mathematics Subject Classification. 65Ml5, 76C05. lSupported by O.N.R. Contract No. N00014-76-C-0316 and N.S.F. Grant No.

MCS-81-01639. 2Supported by A.R.O. Grant No. 483964-25530 and N,S.F. Grant No. MCS-81-

02360.

221

© 1984 American Mathematical Society 0271·4132/84 $1.00 + $.25 per page




222 BEALE AND MAJDA

Two important facts form the basis of the method: the vorticity determines the velocity, and the vorticity is conserved along particle paths. To express the velocity in terms of the vorticity, we first use the divergence condition to introduce a stream function ~{z,t) so that

u = (~y,-41x) .

Then w = -6~, and this relation can be inverted to give

l/J(z,t) = f G(z- z')w(z',t)dz'

with G(z) the Green's function -(2n)-1loglzl. Differentiating, we have

( l ) u(z,t) = f K(z- z')w(z',t)dz'

where

K(z) = (Cl ,-Cl )G = {-y,x) . y x 2nlzl2

It will be convenient to refer to the Lagrangian coordinates of a fluid particle, which we denote by a; i.e., z(t;a) will be the position at time t of a particle which started at a at time 0. The particles follow the flow, so that

( 2) dz _ dt- u{z,t), z(O;a) = a,

and a coordinate mapping ¢t: a ~ z is induced. The conservation of vorticity in 2-D is expressed in the differential form

(3) dw dt + (u • V)w = 0

or equivalently as

( 4)

We will assume that the initial vorticity is nonzero only in a bounded set and is sufficiently smooth.

The vortex method will be a discretized version of (1), (2), (4). Suppose we cover the support of the initial vorticity w0 with a square grid of size h and introduce a particle at the center a; of the ith square. (We think of i as a pair of indices and a; = ih.) He will form a system of ordinary differential equations to approximate the paths zi(t) of this collection of particles. If the ith particle is located at zi(t), then according to (4) the vorticity at that location is wi = LLl0(ai). From these values we interpo-late an approximate vorticity function which is a sum of vortices of finite cores or "b 1 obs" of prescribed shape. These b 1 obs are then advanced according to ( 1), ( 2).

Let ¢ be a smooth approximation to the delta function, of total weight 1, and let ¢0(z) = o-2¢(z/o). The choice of ¢ and 8 will be explained



VORTEX METHODS FOR FLUID FLOW IN TWO OR THREE DIMENSIONS 223

later. Our vortex blob approximation is b 1 ob ( 2 (5) w (z,t) = ~ ~ 0 z- zj(t))wj h .

J

It can be shown that wblob approximates w well when proper choices are made; it is essential that h/o + 0 as both h,o + 0. The corresponding velocity approximation is, from (1),

blob 2 (6) u(z) ~ K * w = L (K * ~ 0 }(z- z.)w· h , j J J

a discretized version of (1) with K replaced by the smooth kernel K0 = K * ~ 0 . In fact, (6) can be viewed as a Riemann sum for (1), with K0 in place of K, over the partition formed by the current images of the initial squares. Since ~t is area-preserving, each of the current cells has area h2• We have arrived at a second interpretation of the approximation which is equivalent to the vortex-blob formulation but rather different conceptually--we smooth out the kernel K and then discretize the velocity integral. The blob function of (5} is analogous to the "shape factor" for the density distribution of plasma simulations [9,12].

In summary, the positions zi(t) of the particles evolve according to the ODE's

(7) zi(O) = ih .

Knowing the current particle positions, we can reconstruct the velocity field according to

(8) - - 2 u(z,t) ~ u(z,t) = ~ K0(z - zj(t))wj h . J

(We have introduced tildes to distinguish computed quantities from the corre-sponding quantities of the actual flow.) The main convergence result, stated more carefully below, is

MAIN RESULT. With proper choice of ~ and o = o(h), zi(t) and u(z,t) converge to z(t;ai) and u(z,t) with high order accuracy.

This method has several advantages, not the least of which is its simpli-city. The computational elements are needed only in regions with vorticity, and they follow the concentration of vorticity. It is not necessary to compute the pressure, and the divergence condition is taken care of automatically. Errors like the numerical diffusion of difference schemes are avoided. In spite of the smoothing effect, the system (7) conserves the energy form

1 - - 4 2 .L. (G * ~ 0 )(zi - zj)wiwj h . 1 ,J



224 BEALE AND MAJDA

In fact, J. Marsden has pointed out that (7) is a Ha~iltonian system; this can be seen from the classical derivation for point vortices, replacing G by G * ¢6. Leonard [13] has noted a number of other conserved quantities.

There is an important disadvantage: Since each particle influences every other, there are O(N2) operations per time step if N is the number of particles. Thus the amount of computation grows rapidly under refinement un-less some modification is made. The "vortex-in-cell'' or "cloud-in-cell" methods allow a reduction to O(N log N) operations by the use of fast Poisson solvers to compute the velocity, but there can be accuracy problems with close-by interactions. See [13] for further discussion. One possible compromise is to use a pure particle method, as above, for nearby interactions and a mesh for the far-field, in an attempt to gain the advantages of both. The P3M method of Hackney and Eastwood [12] for the Vlasov equations of plasma physics has this structure. C. Anderson is presently experimenting with methods of this type for the solution of (7).

The first convergence proof for (7) was given by Hald and Del Prete [11]. It was followed by a more definitive treatment of Hald [10], who showed O(h2) convergence with 6 = h112. The authors [1,2], building on Hald's work, found that methods could be designed which converge with high order accuracy and introduced a 3-0 method with similar convergence results. This method is des-cribed below along with an alternate 3-D method. Cottet and Raviart [7,8,15] have shown the convergence of a particle method for the one-dimensional Vlasov equation and have also made improvements in the earlier arguments, especially a simplified treatment of the consistency part.

In stating the two-dimensional convergence result more precisely, our as-sumptions on ¢ are

(9i) ¢ is smooth and rapidly decreasing;

(9ii) f ¢ dx dy = 1 ;

(9iii) f xa yS ¢{x,y)dx dy = 0 for 0 < a + S ~ m - 1 ,

for some m. It is natural to take ¢ = ¢(r), in which case m is even, since the odd moments vanish by symmetry. If no moment conditions are imposed, then m = 2. The following theorem describes the convergence of the method.

CONVERGENCE THEOREM. Assume u0(z) is smooth and w0 has bounded sup-port. Let ¢ be as above, and take 6 = c0 hq with 0 < q < 1. Then for p < oo and 0 ~ t ~ T,

{r lz-(t) - z.(t)IP h2J1/P < Chmq = C'om, j J J -




where C depends on T, the other parameters, and bounds for the initial data. Simple formulas can be given for modified kernels K6 satisfying the

assumptions above. If ¢ = ¢(r), then K6 has the form

K (z) = K(z)f(r/6) = L-y,x) f(r/6) , 6 2nr2

with f determined by ¢. For example, the simple choice ¢(r) satisfying (9) with m = 2, leads to

2;.r:2 K = K • (1 - e-r u ) • 6

Greater accuracy can be achieved either by using Gaussians of different scales or by allowing polynomial factors. For example,

r2 2162 K = K • {1 + (--- l)e-r } 0 62

is fourth order. We have found that kernels of sixth or eighth order can lead to improved accuracy in simple test problems; see [3] for further formulas and test results.

The essential part of the convergence argument is to estimate the errors in evaluating the velocity field, and these can be written in three terms of rather different character. The error introduced by the smoothing,

f [K6(z- z')- K(z- z')]w(z')dz'

is easily seen to be 0(6m), with m determined by the moment conditions on ¢; this term determines the order of accuracy in the theorem. The discretiza-tion error

2 ~ K6{z- zj)wj h - J K6{z- z')w{z')dz' J

can be treated as a quadrature error and bounded by C hM+l/oM for M large (see [8,15]). Finally, the basic stability estimate is that

- - 2 L: [ K ... ( z . - z . ) - K ... ( z . - z . ) ]w . h , j \J 1 J u 1 J J

as a function of i, is bounded in a discrete LP-norm by II zi - zi II in Lp, provided 6 ~ c0h. The essential reason is that convolution with vK6, which occurs in the linearized stability term, is a bounded operator on LP, accord-ing to the Calderon-Zygmund inequality. One implication of these estimates is that the stability and discretization error are generally improved by increas-ing the radius of smoothing 6, but of course the moment error grows if this is done. Good accuracy is achieved by balancing these effects.

A simple class of test problems which illustrate the behavior of the methoc is obtained by choosing a radial distribution of vorticity, w = w(r). Such a



226 BEALE AND MAJDA

vorticity determines a steady flow whose streamlines are concentric circles. If, e.g., w(r) = (1 - r2)3 for r ~ 1, w(r) = 0 otherwise, the inside rotates four times as fast as the outside, and substantial shearing takes place. Using a fourth order kernel and 208 particles with 6 = 2h we find that the mean-square error in the particle velocities reaches 1.4% by one in-side rotation time but rises rapidly at the end of the second rotation to 5.2%. If we increase 6 to 2.5h, the error in the first period increases to 2.6%, but the maximum error over four rotations is 4.7%. When the number of parti-cles is increased we observe the predicted order of accuracy with second or fourth order kernels for about one inside rotation time, and improvement, but in a less predictable way, for longer times. For example, if we use a fourth order kernel as above with 316 particles, and choose 6 = c0h314 , correspond-ing to 6 = 2h in the earlier case, the error is .8% for the first rotation and reaches 2.3% at the end of the second rotation. Errors at locations other than the particle trajectories are comparable. These results can be explained in terms of the error estimates discussed above. At first only the smoothing error is significant, but at later times the distortion of the particle con-figuration causes the discretization error to dominate. A more detailed des-cription of test results is given in [3]. Closely related experiments are reported in [11,14] and work of M. Perlman to appear.

We will now discuss vortex methods for three-dimensional flows which are close analogues of the above methods in 2-D. In three dimensions the vorticity w = V x u is a vector quantity, and the velocity can be expressed in the form (1) by the Biot-Savart law,

( 1 0) 1 z - z' u(z,t) =-- f x uJ(z',t)dz'; 4n lz - z' 13

the kernel K is now a matrix. We will approximate the particle paths as in (7), but (3) no longer holds, and the vorticity must be updated as well as the positions. There are two usual ways to express the change in vorticity; the more familiar one, which replaces (3), is

(11) ~~ + (u • V)w = (w • v)u.

However, there is also an integral form of this expression. Let a be the Lagrangian position as before and ¢t: a ~ z the flow. Then

( 12) t w(z, t) = V¢ (a) • w0(a),

Thus the vorticity is carried along particle paths but distorted by the Jacobian matrix of the flow.

Either expression (11) or (12) can be used as the basis for a convergent 3-0 method. In any case we use ODE's for the computed particle paths, zi(t),




with a modified kernel

(13)

with initial condition

Ko=K*<Po'

zi = ~ K0(zi J

zi(O) = (li = ih '

just as in (7), except that wj must be computed as well. To find wj from (12), we need to approximate V~t(cxj). Since zj(t) = ~t(cxj) = ~t(jh), we can discretize (12) as

h w(zj) -v V zj • w0j ,

where vh is a difference operator with respect to cxj = jh approximating the gradient and w0j = w0(jh). Then (13) is completed by setting

( 14) wj = if' Z j • wOj .

It was shown in [1,2] that the method (13), (14) converges in a fashion similar to the 2-D case as long as the actual solution remains smooth. Of course, the order of accuracy of the difference operator vh must be at least as great as the intended order of accuracy. Actually, the vorticity was computed in an apparently more indirect way in [1], using differential equations for wj obtained by differentiating (12) in t. C. Greengard has pointed out that the formulation above is equivalent to that of [1], and it is more natural con-ceptually. This method is quite similar to one used by Chorin [6] in modelin~

boundary layer flow, although there are differences. The other method uses ODE's based on (11) to compute wi(t) = w(zi(t)),

which are coupled with (13). To update wi' we need a value for Vu(zi). The approximate velocity field is

(15) u(z) -v ~ K0(z - zj)wj h3 . J

Once an explicit formula for K0 is chosen, we can differentiate analytically to obtain

Vu(z) -v ~ VK0(z - zj)wj h3 . J

Inserting this in (11), we arrive at the ODE for wi' ) d~ ~ ~ ~~ 3

(16 dt wi = wi • E VK0(zi - zj)wj h

This method was suggested by C. Anderson. The authors have recently shown that convergence results similar to those already mentioned hold for this method also, provided that the kernel K0 is at least fourth order. The proof uses techniques as in [1] for the earlier 3-D method, but there are more terms to be estimated.



228 BEALE AND MAJDA

The second method has the advantage that it does not make reference to the initial configuration of the particles, whereas the first depends or the initial positions explicitly through the difference operator.

It is therefore reasonable to expect that the second might perform better when the geometry becomes highly distorted. The second method appears to re-quire more computation, but this impression may be misleading, since in the first method extra neighbors will have to be included to perform the differ-ences. Greengard has carried out sample calculations using both methods, modeling phenomena such as the leapfrogging of a pair of vortex rings. Better understanding of how the two methods compare will have to await further tests. A practical 3-0 method will almost certainly have to use a more efficient means of computing far-field interactions, as described earlier, because of the in-crease in the number of particles.

~Je have seen that in the vortex methods for two-dimensional flows the approximation (5) of the vorticity by a sum of smooth delta functions was equivalent to representing the velocity field by smoothing and discretizing the kernel as in (6). In the 3-0 methods, however, we began with the smooth velo-city kernel, and the most natural approximation to the vorticity is somewhat different. In fact, since the actual vorticity has divergence zero, we might exoect our approximations to have the same property. From the expression (15) for the velocity we can write, as in (16),

2 w(z) ~ V x z K0(z - zj)wj h

On the other hand, (15) can be written as

so that the vorticity expression becomes

(17) w(z) ~ -V x V x 6-1£z ¢0(z- z.)w. h3} . J J J

This is simply the orthogonal projection onto the subspace of divergence-free vector fields of the sum of "blobs" analogous to (5). If we began with (17) as our basic ansatz for three-dimensional vortex methods, in place of (5), we would then be led to (15) and (13), just as (5) led to (6) and (7).

BIBLIOGRAPHY

1. J. T. Beale and A. Majda, "Vortex methods. I: Convergence in three dimensions", Math. Comp., 39 (1982), l-27.

2. J. T. Beale and A. Majda, "Vortex methods. II: Higher order accuracy in two and three dimensions", Math. Comp., 39 (1982), 29-52.




3. J. T. Beale and A. Majda, "High order accurate vortex methods with explicit velocity kernels", to appear.

4. A. J. Chorin, "Numerical study of slightly viscous flow", J. Fluid Mech., 57 (1973), 785-96.

5. A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer-Verlag, New York, 1979.

6. A. J. Chorin, "Vortex models and boundary layer instability", SIAM J. Sci. Statist. Comput., 1 (1980), 1-21.

7. G. H. Cottet, "Methodes particulaires pour 1 'equation d'Euler dans le plan", These de 3e cycle, Universite P. et M. Curie, Paris, 1982.

8. G. H. Cottet and P.-A. Raviart, "Particle methods for the one-dimen-sional Vlasov-Poisson equations", preprint.

9. J. M. Dawson, "Particle simulation of plasmas", Review of Modern Physics, 55 (1983), 403-45.

10. 0. Hald, "The convergence of vortex methods, II", SIAM J. Numer. Anal., 16 (1979}, 726-55.

11. 0. Hald and V. M. Del Prete, "Convergence of vortex methods for Euler's equations", Math. Comp., 32 (1978), 791-809.

12. R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, New York, 1981.

13. A. Leonard, "Vortex methods for flow simulations", J. Comput. Phys., 37 (1980), 289-335.

14. Y. Nakamura, A. Leonard, and P. Spalart, "Vortex simulation of an inviscid shear layer", AIAA/ASME Third Joint Thermophysics, Fluids, Plasma and Heat Transfer Conference, 1982.

15. P. A. Ravi art, "An analysis of particle methods", C. I.M. E., Como, 1983.

DEPARTMENT OF MATHEMATICS DUKE UNIVERSITY DURHAM, N.C. 27706 DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA. 94720







HAMILTONIAN PERTURBATION THEORY AND WATER WAVES

Peter Olver1

ABSTRACT. A general theory of noncanonical perturbations of Hamiltonian systems, both finite dimensional and continuous, is proposed. The results determine a general formula for the deform-ation of a Poisson structure on a manifold. The theory is applied to the Boussinesq expansion for the free boundary problem for water waves, which leads to the Korteweg-de Vries equation. New Hamilton-ian model equations for both uni- and bi-directional propagation of long waves in shallow water are found. An explanation of the complete integrability (soliton property) of the KdV equation,as a consequence of the expansion,is determined.

1. INTRODUCTION. In 1895 Korteweg and deVries first derived their celebrated

equation as a model for the unidirectional propagation of long waves in

shallow water. Their method proceeded by first applying the perturbation expansion introduced by Boussinesq, and then restricting the resulting bi-

directional Boussinesq system to a "submanifold" of approximately unidirect-

ional waves. Hamiltonian methods entered the subject when Zakharov found the Hamiltonian form of the water wave problem. Subsequently, the Korteweg-

de Vries equation was shown to be Hamiltonian, in fact in two distinct ways.

In earlier work with Benjamin, [2], [12], symmetry group techniques used in conjunction with Zakharov's Hamiltonian structure proved that the two-dimensional water wave problem without surface tension has precisely eight nontrivial conservation laws. The present work arose in an ongoing invest-igation as to how these laws behave under the perturbation expansion leading to the KdV equation. This project came to a temporary halt, however, with the surprising discovery that the Hamiltonian structures of these two equations do not match up in any natural way. Indeed, this is first evidenced by the fact

that almost all versions of the Boussinesq system, which is the essential half-way point in the derivation, are not Hamiltonian, in particular do not conserve

energy. Even more striking is the elementary, but apparently unnoticed observation that the perturbation expansion of the energy for the water wave

1980 Mathematics Subject Classification 35Q20, 58F05, 76Bl5· 1 Supported in part by NSF Grant MCS 81-00786.

231

© 1984 American Mathematical Society 0271-4132/84 $ LOO + S .25 per page




232 PETER OLVER

problem does no~ agree to the requisite order with either of the Hamiltonians for the KdV equation. Alternative models such as the BBM or Regularised

Long Wave equation, [1], suffer from the same problem.

In order to better understand this state of affairs, a general theory of noncanonical perturbation expansions of Hamiltonian systems must be developed.

In outline, the theory proceeds as follows. Consider a Hamiltonian system

x = J(x,e)'VH(x,e) , (1.1)

in which e is a small parameter, H(x,e) is the Hamiltonian function and

J(x,e) the skew-adjoint Hamiltonian (or cosymplectic) operator. Since the operator J appears in the cosymplectic two-vector 8J = -21 bT 1\ Jl'l , defining

X X

a Poisson structure, we call ( 1.1) the cosymplectic form of Hamilton's equations, to be distinguished from the symplectic form

K(x,e)i = 'VH(x,e) , (1. 2) 1 T -1 corresponding to the symplectic two-form 0 =- - dx 1\ Kdx , K = J (At 2

first sight, this distinction appears trivial, but the two forms lead to very different types of perturbation equations.)

Consider a perturbation expansion

x = y+ e w(y) + . . . . (1.3)

In standard perturbation theory, one substitutes (1.3) into (1.1) or (1.2), expands in powers of e and truncates to some required order. The resulting system, as simple examples easily show, is not in general Hamiltonian. In order to preserve the Hamiltonian structure we must expand both the Hamiltonian

2 H(x,e) = H (y) +eH1(y) + e H0 (y) + ... 0 ~

and the ~osymplectic operator . 2

J(x,e) ~, J 0 (y) + e J 1 (y) + e J 2(y) + · · ·

and truncate at the required order. (We ignore for the moment the additional complkation that the truncated series for J is not in general a true cosymplectic operator- see section 2B.) To first order,

called the cosymplectic perturbation of (1.1). perturbation expansion

(1.4)

It agrees with the ordinary

y = J 'VH + e (J117H + J 17H1) (1.5) o ·o o o to first order but includes some additional terms in e 2 so as to maintain the Hamiltonian structure. Note that (1.4) is not the second order ordinary



HAMILTONIAN PERTURBATION THEORY AND WATER WAVES 233 2

perturbation of (1.1) - this would include the terms e (J 0 ~H 2 + J 2 ~H 0 ) '

which would again destroy the Hamiltonian form of the system. The symplectic perturbation proceeds along the same lines, leading to

(1.6)

which is always Hamiltonian. For evolution equations, as the examples in section 4 bear out, the cosymplectic form is usually the more desirable because in (1.6) the symplectic operator, which may very well be nonlinear, is applied to temporal derivatives of y .

This Hamiltonian perturbation theory falls between the two main schools of perturbation theory- on the one hand standard perturbation methods, [6], pay no regard to any Ha~iltonian structure in the systems under investigation, whereas in classical and celestial mechanics, [15], all perturbations are canonical and the problems discussed here never arise. Nevertheless, the present theory should prove to be of importance in a wide range of physical applications in which the perturbations are more or less prescribed, but one still wishes to maintain some form of Hamiltonian structure.

In the water wave problem, there are two small parameters a and ~ but the expansions take the same form. If (1.1) represents the original free boundary problew, then the non-Hamiltonian Boussinesq systems are of the form (1.5). To make these Hamiltonian, we must add certain quadratic terms in a 2 ,~,~ 2 , as in (1.4); see (4.15) for the resulting system. Similar remarks apply to the subsequent derivative of the KdV equation (coming from the cosymplectic form of the expansion) or the BBM equation (coming from the symplectic form). In terms of the surface elevation ~(x,t) , the non-Hamiltonian perturbation equation (1.5) is the familiar KdV equation

(1. 7)

To retain the correct Hamiltonian structure according to the general theory, one must include quadratic terms as in (1.4), leading to the "Hamiltonian version" of the KdV equation

~t+~x+ ~ex ~~x+ i t3 ~xxx+ ~6 ~(~ 2 )xxx+ ~ a~~x = 0

This model has Hamiltonian functional .. H[~ J = f ( ~ ~2 + ~ a ~3) dx ' _ .. (1.9)

(1.8)

which is the correct first order expansion of the energy (Hamiltonian) of the water wave problem, and cosymplectic operator

1 1 3 J = - [D + -4 a(~ D + D ~) + 7 /3 D ] • X X X 0 X (1.10)

Note that (1.9) does not agree with either of the usual Hamiltonians for the KdVequation. (Segur, [14], gives a completely different derivation of the



234 PETER OLVER

KdV cquat ion using two time scales. His expansion of the energy leads to a linear combination of the two KdV Hamiltonians. It remains to be seen how the two methods can be reconciled.)

There remains the question of why, in spite of the general theory, the KdV equation is Hamiltonian. Note that the operator (1.10) appearing in the Hamiltonian perturbation resembles a linear combination of the two cosymplectic operators for the KdV equation. Under special circumstances, the non-Hamiltonian perturbation (1.5) can inherit two compatible Hamiltonian structures (corresponding to J 0 and J1 ), and hence, by a theorem of Magri, [9), is automatically completely integrable. This may offer an explanation for the remarkable fact that completely integrable Hamiltonian systems (soliton equations) such as the KdV, sine-Gordon, and nonlinear Schrodinger equations appear so often as model equations in the perturbation expansions to a wide variety of physical systems.

I wish to thank T. Brooke Benjamin and Jerry Bona for valuable comments on the results, and Jerry Marsden for organizing a superb conference.

2. FINITE DIMENSIONAL HAMILTONIAN PERTURBATIOO THEORY. The aim is to set up a Hamiltonian perturbation theory for evolution equations, but to keep things simple we begin with the finite dimensional case. One lesson gleaned from the evolutionary case is that one should not rely on the existence of Darboux coordinates in general, so we take a Hamiltonian structure to be defined by either a symplectic two-form, or, more generally, a cosymplectic two-vector field a la Lichnerowicz. To perturb the Hamiltonian structure, it then suffices to perturb either the symplectic form (which is straight forward or the cosymplectic two-vector (which is less so); in fact, the correct form of the perturbation of the cosymplectic two-vector requires the full theory of Poisson manifolds, which we develop in a form amenable to be immediately generalized to the infinite-dimensional case of evolution equations. A. POISSON STRUCTURES. In the usual theory, Hamiltonian mechanics takes place on a manifold M equipped with a symplectic two-form 0 One immediate complication is that in local (non-Darboux) coordinates, if

0 = - l: dxT A K(x) dx = - l I: K dx A dx 2 2 ij i j '

then both Hamilton's equations

:ic = J 17H(x) , (2.1)

and the Poisson bracket

require the inverse J -1 K of the matrix appearing in 0 . In the infinite-dimensional version, J is a differential operator, so trying to use the



HAMILTONIAN PERTURBATION THEORY AN'.0 1/JATER WAVES 235

symplectic form usually introduces unnecessary complications. These can be avoided by introducing a Poisson structure, as detailed in the

paper by Weinstein in these proceedings. For our purposes, however, it is expedient to adopt the viewpoint of Lichnerowicz, [8], and regard the cosymplectic two-vector field

® = .! l? II J(x)b = ~ I: J .. 't/J II bb (2.2) 2 x x 1J xi xj as the fundamental object determing a Poisson structure, rather than the Poisson bracket, which is easily recovered from ® :

{F,G} = (dF II dG,®) . (2.3)

The requirement that the Poisson bracket satisfy the Jacobi identity translates into a system of nonlinear differential equations for the coefficients J .. (x) lJ of ® . These are most easily expressed using the Schouten-Nijenhuis bracket.

We begin by describing a new invariant definition of this important bracket between multi-vector fields which will readily generalize to the case of

evolution equations. A k-vector field is a section of "kTM , the bundle of contravariant alternating k-tensors. Note that if a is a k-vector fEld and ro a differential (k-1) -form, then the interior product v = ro ..J a is an ordinary vector f~ld. Thus v(e) = (ro~ a)e, will denote the Lie derivative of another differential form e with respect to this vector field. DEFINITION 2.1 Let a be a k-vector field and ~ an t-vector field. The Schouten-Nijenhuis bracket [a,~] is the following uniquely determined (k+t -1)- vector field: For every k+t -1 closed differential one-forms

~·· .. '~+t -1 '

< [a,t3] ·~"· · · "rok+ t - ..)

kt+t ( -1' =-+ I: sign I(a,(~.J t3)roi,)

I (2.4) (-l)k . + -k- I: s1gn J(t3,(roJ.Ja)roJ,) .

J In this formula, the first sum is over all multi-indices I= (il' ..• ,it_1) ,

l_:si1 < ... <it-l.:Sk+t-1, with complement I' =(ii, ... ,ik) such that l.:Sii < .•. <ik_.:Sk+t-1 and (il' ... 'it-l'ii,····ik_) = n(l, ..• ,k+t-1) for some permutation n , and sign I = sign n • Similarly, the second sum is over all J = (j1 , · · · ,jk-l) , l~jl < ... <jk_1 _:sk+ t- 1 with J' , sign J defined similarly.

In the special case k = 1 , so a= v is an ordinary vector field, ( 2.4) still holds with the understanding that in the second summation there is one term, corresponding to I= yj , ~ = 1 (constant). It is easily seen that in this case the Schouten-Nijenhuis bracket [v,t3] is just the Lie derivative of t3 with respect to v . Checking that definition 2.1 agrees with both that of Nijenhuis, [lO],and the invariant definition favored by Lichnerowicz, [8], is a useful exercise. We have chosen this definition because it appears to be



236 PETER OLVER

the only one that readily generalizes to the infinite dimensional formulation needed to treat evolution equations.

Let a,a be k-vector frelds, ~ ant-vector field and y an m-vector field. The basic properties of the bracket follow from (2.4):

a) Bilinearity

(2.5)

b) Super-symmetry

[a,~] = (-l)k-L[~,a] , (2.6)

c) Jacobi identity

( -l)km[ [a,~], Y] + ( -l)tm[ [Y ,a],~]+ ( -l)kt[ [~, y) ,a] 0' (2.7)

d) Pseudo-derivation oLm+ m

(a,~ A Y) = [a,~) A Y+ (-1) ~ A [a,y] (2.8)

These properties, especially (2.8) which does not appear to be as well known, are vital for determining the local coordinate formulae for this bracket. DEFINITION 2.2 A two-vector field S is cosymplectic if

[e ,e] = o . (2.9)

A cosymplectic ·two-vector 9 determines a Poisson structure on M in the sense of Weinstein, [16], via (2.3) and conversely. For a Hamiltonian function H: M .... JR , the associated Hamiltonian vector field is

VH = Fl!ll ( dH) :: dH.J ® ,

with flow given by (2.1) in local coordinates. THEOREM 2.3 Let 9 have constant rank 2m<n

(2.10)

Then there is a foliation of M with 2m-dimensional leaves so that on each leaf L ,el EA2TLI and is of

X X maximal rank for each x£1 . Thus 0 defines a symplectic structure on L Each leaf is invariant under the flow of any Hamiltonian vector field on N, in fact

for any x E LC M. See LichnerCTwicz, [8], for a proof and Weinstein, [16], for a discussion

of the non-constant rank case. The cosy.mplectic two-vector S sets up a complex

Be = 5 : '\'I'M_, '\+lTM '

with 5(a) = [l!ll,a] . The condition (2.9) implies, using the Jacobi identity (2.7), that the complex is closed: 5o 5 = 0 . However, unless S is of maximal rank, this complex is not locally exact . THEOREM 2.4 Let l!ll be cosymplectic, of constant rank. Let a be a k-vector field on M . Then [0,a] = 0 if and only if in any coordinate cube



HAMILTONIAN PERTURBATION THEORY AND WATER WAVES 237 a = [G,S] + a0 for S a (k-1)-vector field and a0 a k-vector field which, in

the given coordinates, is constant on the leaves of the symplectic foliation induced

by 8 . (a0 will in general depend on the choice of local coordinates.)

The proof of this result, as well as a discussion of the global cohomology, can be found in Lichnerowicz, [8]. B. PERTURBATION THEORY.

We now consider perturbation theory for a system of ordinary differential equations in Hamiltonian form. Throughout this section e will be a small parameter, and we allow the possibility of both the Hamiltonian and the cosymplectic form depending on e • The basic system is

x = J(x,e)~H(x,e) = F(x,e) (2.11)

Given a perturbation expansion 2 x = y+ ecp(y) + e t(y) + ... , (2.12)

following standard perturbation methods, we substitute (2.12) into (2.11) and expand the series in e to first order:

(2.13)

Here F0 ,F1 can easily be evaluated from (2.ll) using the chain rule:

F (y) = F(y,O) = J (y)~H (y) , F1 (y) = Fjy,O) +~F(y,O)cp(y) • 0 0 0 ~

We can also invert l+e'Vcp in (2.13) to obtain the alternative system

(2.14)

where F1 = F1 - ~cp • F0 • Unless the expansion (2.12) happens to be canonical, neither (2.13) nor (2.14) will be in general Hamiltonian. If we expand the Hamiltonian

(2.15)

we find that the first order truncation H0 +e H1 is not in general a constant of the motion.

In order to maintain same form of Hamiltonian structure under perturbation, we must investigate how the symplectic or cosymplectic forms themselves are being perturbed. First we look at the easier case when the system is in symplectic form

K(x,e}x = 'VH(x,e)

The symplectic two-form has the perturbation expansion

O(x,e) = 0 0 (y) + e n1 (y) + e2a2(y) + ••• ,

or, in coordinates,

(2.16)



238 PETER OLVER

using (2.12). Since the closure condition dO =0 for a symplectic two form

is linear, we can truncate the expansion (2.16) at any order and (provided e is sufficiently small to ensure nondegeneracy) be assured the truncated form, 0 0 + e 01 say, remains symplectic. This, together with (2.15), yields the first order symplectic perturbation

(2.17)

which is a Hamiltonian system. Note that (2.17) is not the same as (2.13) or (2.14), but does agree with them up to terms of first order in e This is because to lowest order y = F (y)+ O(e) , so whenever we see a term

0

like ey we can replace it by eF (y) and still maintain first order 0

agreement. Note also that it is not permissible to invert K + e K1 in (2.17) -- 0

and truncate and expect to have a Hamiltonian system. As for the cosymplectic form (2.ll), we can similarly expand the two-vector

field

(2.18)

or

.!_ z?AJ(x,e)~ = .!_ 1)TA(J(y)+eJ1(y)+e2J2(y)+ •.• )~ 2X X 2Y 0 y

However, owing to the basic nonlinearity of the cosymplectic condition (2.9) one cannot expect in general to be able to truncate the series (2.18) and have the resulting two-vector field be cosymplectic. Thus the first order perturbation

Y = (J 0 (y) + e Jl (y) )(VH0 (y) + e VHl (y))

= JOVHo+ e(JlVHo+ JoVHl) + e2Jl\7Hl (2.19)

will not in general be Hamiltonian. However, since J 0 + e J1 is still skew-symmetric, the perturbed Hamiltonian H0 + e H1 will always be a constant of the motion of (2.19). LEMMA 2.5 The perturbed two-vector 9 0 + e 91 is cosymplectic if and only if 91 itself is:

0 . ( 2. 20)

PROOF.

The full series (2.18) is certainly cosymplectic. (Indeed, the perturbation expansion (2.12) is in essence just a change of coordinates.) Expanding (2.9) in powers of e , and using (2.5,6), we find the infinite series of relations

[9 0 ,9 0 ]=0, 2[Gl 0 ,Wl1 ]=0, 2[9 0 ,92 ]+ [81 ,81 ]=0, ... , (2.21)

resulting from the fact that (2.18) is cosymplectic for all e On the other hand, the conditions that Gl 0 + e Gl1 be cosymplectic are the first two of (2.21), which are automatically fulfilled, plus (2.20). This proves the



HAMILTONIAN PERTURBATION THEORY AND WATER WAVES 239

lemma.. (Note, by (2.21) we can replace (2.20) by [® 0 ,92] = 0 . ) More generally, if (2.20) fails to hold, yet we still wish to retain the

Hamiltonian property of the perturbation, we are required to include certain higher order terms in e in the cosymplectic two- vector agreeing with (2.15) to first order~ i.·e. of the form

2"" ®0 +e e1 +e 9 2 + ....

To accomplish this, we simplify matters by working locally to avoid global integrability conditions. THEOREM 2.6 Let 9 0 ,®1 be two-vector fields satisfying (2.21) for same ®2 . Then there exists a vector field v1 and a two-vector field Y1 constant on the leaves of the foliation induced by ®0 such that

Moreover, the two-vector field

* 9 = exp(ev1)*(® 0 + e Y1)

is cosymplectic, with expansion

e* = Gl 0 +e e1 +0(e 2) .

PROOF

(2. 22)

(2.23)

( 2. 24)

The existence of v1 ,Y1 follows directly from theorem 2.4. In (2.23) the * refers to the action of the one-parameter (local) group of diffeamorphisms exp(e v1) on the space of two-vector fields. Since the Schouten- Nijenhuis bracket is invariant under diffeamorphisms it suffices to check that ®0 +e Y1 is cosymplectic. Clearly [S 0 ,Y1] =0, so we need only check that [Y1,Y1 ] = 0 Using the Jacobi identity (2.7), and the third equation in (2.21),

Therefore

[91'91]

[[vl,®o)' [vl,®o]]+ 2[[vl,l!llo]'Yl]+ ['l!l,Yl]

[& 0 ,- [vl'[v1 ,& 0 ]] -2[v1,Y1]]+ [Yl'\] •

for same well defined r . But since '1!1 is constant on the leaves induced by 9 , this latter identity is impossible unless both sides vanish. Finally, to

0 establish (2.24) we need only notice that

for any k-vector field a , using the identification of the bracket with the Lie derivative in this case.



240 PETER OLVER

C. SOME QUALITATIVE COMPARISONS. What are some of the advantages of the Hamiltonian theory over standard perturbation methods? The most important is certainly that the Hamiltonian perturbation equations conserves energy, whereas the standard perturbation equation does not in general. (This is also true

when one truncates the cosymplectic form without worrying about the bracket condition; however in this case there is no Poisson bracket.) It is easy to find two-dimensional examples in which the orbits of the unperturbed system

are closed curves surrounding a fixed point. The Hamiltonian perturbation has the same orbit structure, its orbits just being perturbations of the closed curves, whereas the solutions of the standard perturbation equations slowly spiral into or away from the fixed point. In higher dimensions, KAM theory shows that "most" solutions of a small Hamiltonian perturbation of a completely integrable system remain quasi-periodic, whereas the standard perturbation can again result in spiralling behavior. At the other extreme, only Hamiltonian perturbations of an ergodic system stand a chan.ce of being ergodic in the right way as the standard perturbation wi.ll mix up the different energy levels. Of course, both the Hamiltonian and non-Hamiltonian expansions are valid to the same order, and hence give equally valid approximations to the short-time behavior of the system. Based on the above observations, the Hamiltonian perturbation appears to do a better job modelling long-time and qualitative behavior of the system. It remains to see whether any rigorous theorem to this effect can be proved.

3· EVOLUTION EQUATIONS. The Hamiltonian theory of evolution equations is most easily developed using the formal variational calculus introduced in [5], [11].

Here we present a brief outline of the theory, including an extended discussion of multi-vectors and the Schouten-Nijenhuis bracket, the latter being new. For simplicity, we work in Euclidean space, with x = (x1 , ... ,xp) EX~ mP and u = (u1 , ... ,uq) E U ~ mq denoting independent and dependent variables. The infinite jet space J = X X U is the inverse limit of the spaces Jn =X X Un with coo;dinat~s 00(x,u(n)) = ~x, ... ,u~, ... ) , where u~ represents

1 1 I the partial derivative oJu =b ...• b. u , m<n, b. =o ox .. Let J1 Jm ( rl) J J u denote the space of smooth functions P(x,u ) , n arbitrary, and

Ak = ~T*J 00 the space of v:rtical k-forms, i.e. finite sums of the form (n) 1 1 ik

w = I: Pjx,u )duJ A ... A duJ • 1 k

Vector fields are formal infinite sums ?'I i 0 v= E Q. -,..- + !: QJ -. , J uX. ,.. 1

J uUJ

with The standard formulae relating Lie derivatives, exterior derivatives and interior products extend readily to this set-up. In particular



HAMILTONIAN PERTURBATION THEORY AND WATER WAVF.S k

the total derivatives D. can be viewed as vector fields, hence act on A J

by Lie derivatives.

241

The space of functionals J is the quotient space of a by the image of the total divergence, Div Q = n1Q1 + •.. + Dp~ , Qj E a . The projection a~ J is denoted by an integral sign: IPdx E J for P E a . Similarly, the space of functional k-forms is A~ = Ak /Div(Ak) , with projection J (I) dx ,

k k k+l . (I) E A . The deRham complex d:A ~A proJects to a locally exact complex d:A! ~ A!+l The dual space to ~ is the space T0 of evolutionary vector fields

v=Q·bu=I:DJQi bi Q=(Ql, ... ,Qq)' bUJ

uniquely characterized (except for the trivial translational fields 1'> / bx) by the fact that they commute with all total derivatives. Hence they act by Lie derivatives on A! , and again the standard differential-geometric formulae can be readily established. The exponential exp(ev) of an evolutionary vector field can be found by solving the system of evolution equations

bu = Q be u(x,O) = u (x) ,

0

with flow u(x,e) exp(ev)[u0 ] , in same appropriate space of functions. The spaces of multi-vectors, dual to functional forms, are more interesting;

they are not images of the spaces ~T J~ under any projection! Part of the problem is that there is no well-defined exterior product on ~ : J (I) dx A I eax I J((l) M)dx . In particular, '\(A!) I A~ • We are interested in multi-linear, alternating maps on ~ . First, recall that every functional one form is uniquely equivalent to 8ne of the form

(just integrate by parts). Moreover, by the exactness of the d-complex on A! , a function one-form ~ is closed: OO'p = 0 , if and only if ~ = d(IQdx) for same functional, which means that P = E(Q) where E is the Euler operator, or variational derivative, [ll]. EXAMPLE 3.1. A functional one-vector will be determined by q-tuple of differential operators ~ = (~ 1 , ... '~q) , ~i = I: Q~ DJ (finite sums, DJ = D. D. ) with QJ~ E a . Given ~ , consider the linear map

Jr · · Jm

~ 'b = I: ~ . b . Al ~ J u ~ ~ * u

given by ~-b 11 [J(P·du)dx] =I ~P dx= I [I: ~iPi]dx parts shows that

where - J' i Q. = I:(-D) QJ '

~ J

A simple integration by



242 PETER OLVER

* so the space I]_ of functional one-vectors can be identified with T , the 0

space of evolutionary vector fields. (Note that in the above notation we are

regarding [ 1'\ ) u

as the basis of A~ dual to the "basis" [ dui} of 1 "* . ) DEFINITION 3.2 A functional k-vector is a finite, constant coefficient linear combination of the basic k-vectors, defined as follows. Given differential operators ~ 1 , ... ,~k,

~ b a= ~ 1 -m A ••• /1 f)k --m , l<m. <q, bu 1 1'\u k - J-

(3.1)

is defined so that for any

j=l, ... ,k'

we have

a(Cl.]_A ••• A~) =Jdet[t)ip~ ]d.x, i

the determinant being of a kXk matrix with the (i,j)- entry indicated.

EXAMPLE 3. 3 Suppose q = 1 A functional two vector is of the form

a = ~l~u A ~2°u '

with

a(~ 1\wQ) fC~ 1 P ~ 2 Q- t) 2P il1Q]d.x = fCP~Q)d.x * * * where ~ = [) 1[) 2 - i9 2 ~ 1 is skew adjoint (i; = -f;)

two vector is uniquely equivalent to one of the form Thus every functional 1:2 1'J J\{; o for f)

u u skew-adjoint. This integration by parts argument easily generalizes to functional k-vectors.

Once the basic definition of a functional multi-vector has been properly presented, the definition and properties of a Poisson structure readily adapt to this infinite dimensional situation. In particular, the definition 2.1

of the Schouten-Nijenhuis bracket carries over with no change, as it does not rely on the exterior derivative d . (This is the definition used by Gel'fand and Dorfman, [5], in the special case k =.C.= 2 , although they appear to omit the vital assumption that the one-forms (ll. be closed.) Thus a skew-adjoint

J differential operator f) is cosymplectic if and only if the two-vector e = l'lu 1\~. l'lu satisfies [19 ,e] = 0 . In particular, if l.fJ does not depend on u , it is automatically cos~nplectic.

EXM-1PLE 3·4 Consider the KdV equation in the form

This is Hamiltonian in two ways:

in which 9 denotes the variational derivative with respect to u ,



I and

J = D 0 X

HAMILTONIAN PERTURBATION THEORY AND WA'l'ER WAVES

1 2 udx, 2

I 1 3 1 2 H1 = ( b u - 2 u)dx ,

3 2 1 J =D +-uD +-::-u 1 X 3 X 3 X

243

The first operator is cosymplectic since it does not depend on u ; the proof that J 1 is cosymplectic is not difficult and can be found in [5), [9], [ll).

The only part of the theory that has not so far been adapted to this context is the exactness result of the a-complex in theorem 2.4. We still have 5o 5 0 , and I strongly suspect that some version of this theorem is true, but do not have a proof. Thus in the perturbation theorem 2.6, one cannot at present be guaranteed the existence of a vector field v1 and two-vector Y1 , but in all the simple examples I have looked at, v1 is easy to find and Y1 is invariably zero.

Finally, we need to discuss change of variables. For simplicity, assume p = q = 1 , but the result readily generalizes. u =F(v,v , ... ) (e.g. the Miura transformation

X

the differential operator

D* = bF _ D ()F + D2 bF _ • • • , F ()v x ()vx x ()vxx

Given a change of variables 2 u = v + v for the KdV) define

XX

is the adjoint of the Frechet derivative of F . Then the functional multi-vectors transform according to the basic rule

_b_ = D* _b_ bv F ()u

applied to (3.1). For example,

* * * b II'Db =D.,() II'DDFb =b !IDlQDF() · V V rU U U F U

To see this, a one-form clearly transforms by

w = J(P(u,u , ... )du]dx = J[P(F,D F, ... )dF]dx p X X

bF bF * = JP(-=-dv + - dv + ... )dx = JfD (P)dv]dx ()v ()v x F

From this, (3.2) follows by duality. (Often, as

(3.2)

depends on v , (3.2) is not directly useful except in conjuction with some perturbation expansion!)

4. WATER WAVES. The water wave problem means the free boundary problem of irrotational, inviscid, incompressible, ideal fluid flow with gravity. We also omit surface tension effects, although this is not essential- see [13]· The model equations are for long, small amplitude, two-dimensional waves over a shallow horizontal bottom. The basic equations, and subsequent derivation of the KdV equation, are given in Whitham, [17, pp. 464-6], whose notation we use here. After rescaling, the problem takes the form

O<y<l+a'f1 , (4 .1)



244 PETER OLVER

cp - 0 y- ' y =0 ' (4.2)

I"'~ _, 0 I xl ... ., ' (4.3)

1 2 1 -1 2

'1y=l+a11 (4.4) cp + - a cpx + 2 013 cp +'11=0 t 2 y

-1 (4.5) 'llt = t3 cpy - dl]xcpx

Here x is the horizontal and y the vertical coordinate, cp(x,y,t) the velocity potential, l+dl](x,t) the surface elevation. The two small parameters are a= a/h , the ratio of wave amplitude to undisturbed water depth, and t3 = h2 / ~ 2 , the square of the ratio between depth and wave length. A. NON-HAMILTONIAN PERTURBATIONS. In Boussinesq's method, the first step is to solve the elliptic boundary value problem (4.1-3) in terms of the potential W =t8(x,t) = cp(x,8,t) at depth 0<8<1, giving the series solution

cp = w+% 13(82 -/)wxx+ 2t t?>2(584 -682/+y4)wxxxx+ •... (4.6)

(We will not worry about problems concerning the precise domains of definition of the functions - see Lebovitz, [7].) Substituting the series (4.6) into (4.4,5), differentiating the former with respect to x and truncating to first order leads to the following version of the Boussinesq system:

1 2 O=ut+'llx+a uux+ 2 13(8 -l)uxxt,

0 = 'llt+u +a('!lu) + ~·t3(3e 2 -l)u , X X 0 XXX

(4.7)

e in which u=u (x,t) =cp (x,e,t) X

is the horizontal velocity at depth e . The basic system (4.7) can be modified by resubstituting, expanding and truncating again; for instance since to leading order ut = -'llx , the term uxxt in the first equation can be replaced by -'llxxx to yield a purely evolutionary system. See Bona and Smith, [3), for a complete discussion of the possibilities, and the companion paper [13] for the second order terms in the expansion.

To specialize to unidirectional waves, one looks for an expansion of the form 'll = u+aA+13B+ ... such that the two equations in (4.7) become the same up to the requisite order. To first order,

1 2 1 2 'll = u+ tau + El·t3(3B -2)uxx, (4. 8)

leading to the KdV equation

3 1 ut+u+ -auu+-6 r3u =0, X 2 X XXX

(4. 9)

independent of depth e Alternatively, one can express u in terms of 'll, leading to the same equation for 'll, (1.7). Again one can play the same games as with the Boussinesq system, so, for instance, since ut = -ux to



HAMILTONIAN PERTURBATION THEORY AND WATER WAVES 245

leading order, we can replace u by -u , yielding the BBM equation, xxx xxt [1], whose dispersion relation offers some advantages over the KdV model.

B. HAMILTONIAN MODELS. In Zakharov's Hamiltonian formulation of the water wave problem, the basic variables are the surface elevation ~ and the potential on the surface cpS(x,t) = cp(x,l+dl](x,t) ,t) , the values of cp within the fluid being determined from cps by solving the auxilliary boundary value problem (4.1-3), cf. [2]. The Hamiltonian is the energy

H = J [! cp(l3-lcp -a11 qi ) + _! ~ 2 }dx . (4.10) S 2 Y XX 2

(The S on the integral means all terms are evaluated on the free surface y = 1 + dTJ • ) The water wave probe 1m (4 .1-5) is now in canonical form

(4.11)

First consider bidirectional Boussinesq systems. Substituting (4.6) into (4.10), and truncating, we get the first order expansion

( 1) Joo[ 1 2 1 2 1 2 1 2 2} H = 2 u+ 2 '11+ 2 anu+ 6 13(2-38)uxdx (4.12) -oo

for the energy. For the symplectic version of the Boussinesq system, we expand the two form 0 = d~ A ~S appropriate to (4.1D, leading to

O(l) = d~ A (dw+ .!2 13(82-l)dW ) = d~ A (D-l+ 12 13(82-l)D )du . (4.17) XX X X

(We omit the integral sign from O(l) for simplicity.) This yields

0 = ut+~ +a uu + 113(82-l)u t , (4.13) X X 2 XX

o = 'llt+ u +a(u~) + 1 f3(8 2-l)'Tl t+ 13(82- _g )u X X2 XX 3 XXX

(We have differentiated both equations with respect to x here.). Note that

the "symplectic Boussinesq" system (4.13) agrees to first order with (4.7)

after manipulations similar to those discussed earlier. Alternatively, we can perturb the cosymplectic two-vector @) =l'l !\(')

'11 cps Using (4.6) again, from (3.2) we find

@J(l) = O'll A [D + _! .f3(l-82)D3}(') , (4.14) 'I X 2 X U

which is cosymplectic since the underlying operator is constant coefficient. This yields the "cosymplectic Boussinesq" system

O=ut+'Tlx+auux+ %13(1-82)'Tlxxx+ ~aj3(l-B 2 )(u 2 )xxx, (4·15)

O=~t+ u +a(~u) + ~ 13(382-l)u + .!ai3(1-82)(Tiu) - 113(384-582 + 2)u X X b XXX 2 XXX 3 XXXXX'

differing from (4.7) by the inclusion of quadratic terrns. The special case 8 = 1 is of special note, as remarked by Broer [4], since to first order the expansion (4.6) is equivalent to a canonical expansion in the variables 'Tl,CJls ~



246 PETER OLVER

the Hamiltonian systems (4.13,14) reduce to versions of the u~ual system (4.7). The more general (e fl) Hamiltonian Boussinesq systems are new.

As for unidirectional models, since we are still matching the two equations to first order in the Boussinesq system, the definition (!+.8) of the submanifold of unidirectional solutions remains the same. Thus we need only substitute (4.8) into the energy and the (co-) symplectic form and expand to first order. The appropriate Hamiltonian is

co

H(l) = J [u2 + t cx u3 + ( ..§_ -e2 )t3 u~}dx, -CO 3

(4.16)

where we have integrated one term by parts. For the cosymplectic model, note first that from (4.8)

__Q_ = ( 1 + _! CtU + (_! 82 l)AD2) b bu 2 2 - 3 f-' x bTl '

cf. (3.2), hence to first order

l = (1- .! cx u+ ( .! - .! e2)t3D2)l bT) 2 3 2 x bu Therefore, substituting into (4.14), we find

~(1) 1 h !ill = b A (D - -4 cx( uD + D u) + ( 26 U X X X

which can be proved to be cosymplectic, [5), [9]. Combining (l.+-16,17) we find the following "cosymplectic version" of the Korteweg-de Vries equation.

ut + [Dx- t cx(uDx + Dxu) + ( i -8 2 )t3D~] [u+ ~ cxu2 + (e 2 - } )t3ux) = o ,

or, in detail,

ut + u + l o:uu + ~ pu +-J, 132( -18e4 + 2782 - lO)u + X 2 X 0 XXX lo XXXXX (4 ·18)

+ ( 53 - ll e2)Ctt3uu + ( 139 - 782)0,f3u u - 45 .-:/u2u = 0 21+ lt XXX 24 X XX 32 X

(In deriving (4.18) we have multiplied by f- this is rigorously justified since we are restricting the system to a submanifold.)

The symplectic form, which resembles more closely the BBM equation, is more complicated. We find

()(l) = du" rn:1+ fcx(un:1 +n:1 u)+ (82 - t )t3D~]du, hence, formally,

-41 -c:t(uDx-l + D-l u) + (8 2 - 2. )D ](ut) + u+ 2 au2 + (e2 - ..§. )t3u = o X 6X 8 3 XX

To convert this into a bona-fide differential equation, recall differentiate:

u = b W , and X

1 1 2" . 9 2 2 w + -<tV w + - c:tW lit + ( 8 - 5.- ) t3W t + 111 + -4 a w Iii + ( 8 - - ) t3ilt =0. xt 2 X xt 4 XX t 0 XXX XX X XX 3 XXXX

This example illustrates well the previous remarks that the symplectic perturbation is easier to handle theoretically, but the resulting equations are much more unpleasant.



HAMILTONIAN PERTURBATION THEORY AND v!.tl:I'ER ',-lAVES

There are a lot of open questions concerning these models, most of which

are probably only amenable to numerical investigation. What are their

247

solitary-wave solutions like, and h~1 do they interact? (Only the ~-equation

(l.8) can be solved "explicitly" in terms of a hyperelliptic integral.) How do the solutions compare with those of the KdV or BBM equation? In particular, do they give any truer indication of the qualitative or long time behavior of water waves? Does the dependence of (4.18) on the depth e have any relevance to the breaking of water waves, in that solitary waves of the same amplitude may move a different speeds at different depths, thereby setting up some kind of shearing instability? (See also [13].) All those questions must await further research.

5· COMPLETE INTEGRABILITY. We now turn to the question of why the KdV equation happens to be Hamiltonian. Returning to the geneTal set-up, as summarized in (1.4,5), we see that one possibility for (1.5) to be Hamiltonian is if the first order terms are multiples of each other:

(5 .1)

This of course cannot be expected in general, but if it does happen, the situation can be handled by the theorem of Magri on complete integrability of hi-Hamiltonian systems, [9), [5). THEOREM 5.1 Suppose a system x = K1 (x) can be written in Hamiltonian form in two distinct ways: K1 = J 0VH1 = J 1VH0 • Assume also that the two Hamiltonian structures are compatible, meaning that J 0 +1J. J 1 is cosymplectic for all constant infinite

1-1 • Then the recursion relation Kn = J 0 VHn = J 1VHn-l defines an sequence of commuting hi-Hamiltonian flows x = K (x) , with mutually n

conserved Hruniltonians H (x) n in involution (with respect to either the J 0

or J 1 Poisson bracket). (One also needs to assume that J 0 in the recursion relation always invertible, but this usually holds.)

Thus, in this special case, both the noncanonical perturbation equation (1.5) and the cosymplectic version (1.6) are linear combinations of the completely integrable flows K0 , K1 , K2 , and hence, provided "enough" of the Hamiltonians Hn are independent, are both completely integrable Hamiltonian systems.

For the water wave expansion, in the Korteweg-de Vries model the O(a,~)-

terms are in the right ratio only at the "magic" depth e = jll/12 , and for this depth (4. 18) is a linear combira tion of a fifth, third and first order KdV equation. For more general e , one must fudge the condition (5.1) slightly to obtain complete integrability.

Nevertheless, this leads to an intriguing speculation. Does condition

(5.1) often hold in the perturbational derivation of model equations from con-



248 PETER OLVER

servative physical systems? If true, it would provide a good explanation of

the connnon feature of many systems that in the zeroth order perturbation one

has linear equations, and in the first order perturbation the equations are

nonlinear, but ccmpletely integrable so1iton equations. Presumably the

second order expansion leads to nonintegrable models with some chaotic com-

ponents. A good place to check this is in Zakharov' s derivation of the non-

linear Schrodinger equation as the modulational equation for periodic water

waves, [18].

REFERENCES

1. T.B. Benjamin, J.E. Bona and J.J. Mahony, "Model equations for long waves in nonlinear dispersive systems", Phil. Trans Roy. Soc. London A 272 (1972) 47-78.

2. T.B. Benjamin and P.J. Olver, "Hamiltonian structure, symmetries and conservation laws for water waves", J. Fluid Mech. 125 (1982) 137-185.

3· J.L. Bona and R. Smith, "A model for the two-way propagation of water waves in a channel," Math. Proc. Carob. Phil. Soc. 79 (1976) 167-182.

4. L.J.F. Broer, "Apprximate equations for long water waves", Appl. Sci. Res. 31 (1975) 377-395·

5. I.M. Gelfand and I. Ya. Dorfman, "Hamiltonian operators and related algebraic structures", Func. Anal. Appl. 13 (1979) 13-30.

6. J. Kevorkian am J.D. Cole, Perturbation Methods in Applied Mathena tics, Springer-Verlag, New York, 1981.

7. N. Lebovitz, "Perturbation expansions on perturbed domains", SIAM Rev. 24 ( 1982) 381-400.

8. A.LichnerDY~icz, "Les varietes de Poisson et leurs algebres de Lie Associees" J. Diff. Geom. 12 (1977) 253-300.

9· F. Magri, "A simple model of the integrable Hamiltonian equation", J. Math. Phys. 19 (1978) 1156-1162.

10. A. Nijenhuis, "Jacobi-type identities for bilinear differential concomitants of certain tensor fields", Proc. Kon. Ned. Akad. Wet. 58A (1955) 390-4(0

ll. P.J. Olver, "On the Hamiltonian structure of evolution equations", Math. Proc. Carob. Phil. Soc. 88 (1980) 71-88.

12. P. J. Olver, "Conservation laws of free boundary problems and the classification of conservation laws for water waves", Trans. Amer. Math. Soc. 277 (1983) 353-380.

13. P.J. Olver, "Hamiltonian and non-Hamiltonian models for water waves", Proceedings, 5th Symposium on Trends in Applications of Pure Mathematics to Mechanics, to appear.

14. H. Segur, "Solitons and the inverse scattering transform", Topics in Ocean Physics So (1982) 235-277-



HAMILTONIAN PERTURBATION THEORY AND WATER WAVES

15. C.L. Siegel and J.K. Moser, Lectures on Celestial Mechanics, Springer-Verlag, New York, 1971.

16. A. Weinstein, "The local structure of Poisson manifolds", CPAM preprint #97, Berkeley, 1982.

17. G.B. Whithwm, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974.

18. V.E. Zakharov, "Stability of periodic waves of finite wmplitude on the surface of a deep fluid", J. Appl. Mech. Tech. Phys. 2 (1968} 19o-194.

SCHOOL OF MATHEMATICS UNIVERSITY OF MINNESOTA MINNEAPOLIS) MN 55455

249







RESULTS ON EXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE VLASOV EOUATION

Stephen Wollman

ABSTRACT. A summary of results on the existence and uniqueness of solutions to the Vlasov equation is given. Both the Vlasov-Poisson system and the Vlasov-Maxwell system are considered. For the Vlasov-Poisson system the global--in-time classical solvability of the problem is discussed. For the Vlasov-Maxwell system a local existence and unique-ness theorem is stated.

0. INTRODUCTION. The purpose of the present paper is to give a summary of results on the existence and uniqueness of solutions to the Vlasov equation. Both the Vlasov-Poisson system and the Vlasov-Maxwell system are considered. Section 1 deals with the Vlasov-Poisson system. Here we are concerned with the solvability of the problem in classical en spaces. A general theorem is formulated which gives conditions for the solvability of the problem. We then show how the general theorem applies in various cases where global-in-time solutions to the problem are obtained. Section 2 deals with the Vlasov-Maxwell system. Here a local-in-time existence and uniqueness theorem is stated which gives solutions in classes of Sobolev spaces on Rn· In both cases we assume that the initial data has compact support and as a part of the proofs bounds on the support of the solution as a function of time are obtained.

1. THE VLASOV-POISSON SYSTEM

The Vlasov-Poisson system of equations describes the collective motion of particles in the presence of their own self generated electric (or gravitational) field. The system under consideration is 1980 Mathematics Subject Classification. 35020.

251





252

a)

b)

STEPHEN WOLLMAN

f(x,v,O)=g(x,v) 2

~<P=-4n J(f-b)dv

(x,v) - a point in six dimensional phase space.

f(x,v,t) - the phase space distribution function.

b(x,v) - a fixed background distribution.

( 1.1)

We assume g,b,£ C 1 o(R6), i.e.,continuously differentiable

functions with bounded support. We first state the general

theorem which gives conditions for the solvability of system

(l.l). The proof of the theorem is contained in [19). In the

statement of the theorem and in the applications the following

notation is used.

The

six dimensional phase space

The space of continuous functions with

continuous bounded derivatives on R6 which have bounded support in RG

C 1 (R6x[O,T])- The space of continuous functions with

continuous hounded derivatives on

RGX[O,T], T-time.

term "solution" refers to a function that is of class c 1 '

satisfies (1.1), and which is integrable over phase space in

such a way that appropriate conservation laws (like mass,

momentum, energy) can hold. The theorem is stated for spacial

dimension three~ however, we also consider its application in

lower dimensions.

THEOREM (1.1): Let g,b he functions in C 1 0 (R6). The

solution to (l.la,b) exists and is unique as an element of

C1 (R 6x[O,T) if and only if the system admits an a priori hound

on the support of C 1 solutions fort in the interval [O,T].

Proof: see [19 pp.l4fi-l48]

This theorem gives a criterion to determine whether the solution to (l.la,b) exists and is unique for a particular



SOLUTIONS TO THE VLASOV EQUATIONS

initial configuration. To use the theorem let us make the assumption that a C1 solution to the problem exists on the interval of time [O,T]. The theorem tells us that a C1

solution to the problem necessarily has bounded support in phase space. we then try to compute a priori the bound on the support of such a solution as a function of time. If it is possible to compute such an a priori bound for a solution then the theorem tell us that the C1 solution to the system in fact exists, is unique, and of course has the bound on support already computed. The important bound to compute is on velocity.

253

For initial data with bounded support the statement that the support of the solution remains bounded is equivalent to the statement that the electric field remains bounded. Therefore an alternative way of stating the general theorem is as follows: a C1 distribution function that is a solution to (1.1} cannot give rise to an unbounded electric field in a finite time. To prove existence and uniqueness of the solution to (1.1} of class C1 (R 6x[O,T] it is sufficient to show "a priori" that the electric field remains bounded for te[O,T].

For treatments of the classical existence and uniqueness problem for the Vlasov-Poisson system that apply for more general classes of initial data see [2], [8], [11], [15].

We now apply Theorem (1.1} to various cases in which one obtains a global-in-time solution to the problem.

i) The one dimensional problem:

Given Theorem (1.1) the one dimensional problem for general C1 initial data with compact support is easy to solve. In one space dimension the system (1.1} reduces to

af +vaf + E(x,t)af =0 -at ax -av

f(O}=g(x,v}

X oo

E( x ,t) = f_ .. f_ .. < f-b) dv

(a}

( 1.2)

(b)



254 STEPHEN WOLLMAN

We prove global-in-time solvability for this system by appealing to the alternative phrasing of the general theorem anrl produce an a priori bound for the electric field for all time.

X !_00 !_ 00 ( lfl+lbl )dvdx

Here we use the conservation of mass, that is, the fact that the L 1 norm of the solution is a constant of the motion. Thus the electric field is hounded for all time by the sum of the L 1

norms of the initial distribution and background distribution. Hence from the general theorem the solution to (1.2) 0f class C1 (Rzx[O,T]) exists for all time T.

A quite different proof for the one dimensional problem is given in ( ll] .

ii) The two dimensional system:

In two space dimensions the system (1.1) becomes

3f +v.V f+V ~.V f=O (a) TI X X v

ll¢'=2r.j(f-b)dv (1.3)

f(O)=g (h)

vx=(_a_, a ax 1 ax 2

ll=a2 + a2 -2 2 ax 1 ax 2

1 g,h,£ c 0 (R 4 )

Our goal is to produce an a priori hound on the support of a c 1 solution to (1.3) as a function of time. Por this we need

a hound on the field Vx<P·

q,(x,t)=-f h(x,t)ln(r)dx R2



SOLUTIONS TO THE VLASOV EQUATION

- 2 - 2 l/2 r=((x 1-x 1 ) +(x 2-x 2 l )

h(x,t) = f (f-b)dv v

For the sake of simplicity we will let b=O. Assume

lfj<M 0 , lh(x,t)I<K llh(x,t) n1= N

For a parameter R.>O

and

Thus

1/2 Let R.=(~ )

K and

h a ax. ln(r)dx 1

- f h a ln(rldx r~R. axi

1 f h _a _ln ( r) dx 1 ~ r< R. ax.

- 1

J lhl !xi-xi! dx <211KR. r~R. r2

I J h a ln(r)dx ~~ r>R. ax.-

1

1/R. Jlhldx =N/R.

l<~>x.I<211KHN/R. 1

I <Px. I <211 ( KN) 112+( KN) 11 2=311 ( KN) 1/2 1

To get an a priori bound on support we integrate the characteristic equations

dxi = v. dt 1

dv i i.L i=l,2 dt a xi

i.e. v (t)=v. (0)+ ft<P (s)ds

1 x. 0 1

255



256 STEPHEN WOLLMAN

Let us define the function

and let

where

2 E(x,v)=L !v·j

i=J. 1

g ( t) = SUp E (X, V) rl(t)

n(t)=supp f(t), the support of f(x,v,t) in R4 at time t.

Thus g(t) is a bound on the support of the solution in velocity space at time t. In terms of the function g(t) we have that

jh(x,t) j=l~fdv,<M 0 (2g(t)) 2

~~X. (t) 1<6n(MON)l/2g(t) 1

t lvi(tlj<lvi(O)j+ f 6n(M 0 N) 1/ 2g(s)ds

0

( 1. 4)

We note that llfii 1=N is a constant known from the initial data. Summing over i and taking the supremum of the left side of (1.4) over all trajectories (x(t), v(t)) such that (x(O),v(O))e:rl(O) (the support of g(x,v)), we get an in-equality for g(t)

the solution is

t

g(t)<g(O)+l2n(M N) 1/ 2 jg(s)ds 0 0

t <R+l2n(M N) 1/ 2J g(s)ds - 0

0

g(t)<Rexp{l2n(M N) 112 t} 0

This then gives us the required a priori bound for all t. Hence from Theorem (1.1) it follows that the classical solution to (1.3) exists and is unique for all time.




For a more detailed study of the two-dimensional

Vlasov-Poisson system see [15], [17].

iii) The three dimensional spherically symmetric problem:

257

Under spherical symmetry the electric field is a function only of radius, r, and time, t. The radial vector to a point in space and the velocity vector at that point form a plane (or

a line) going through the origin. Particles travelling in this

plane at time t=O remain in the plane for all time and the

phase space distribution function is invariant with respect to all planes going through the origin. Initial data of the form

g(x,v) = G(r,u,~) ( l. 5)

give rise to spherically symmetric solutions. A convenient set of phase space variables to use in analyzing trajectories are

r,u and a, the angle between x=(xl,xz,x3) and v=(vl,vz,v3)

(O~a~n). In this set of variables the system (1.1) reduces to

3F + cos( a)u aF -cos( a) 6(t,r) aF (a) at ar: au:

+(~- ~)sin( a) if =0 ( 1.6) u r a a

6(t,r)= -4n Jr - -2 -h(t,r)r dr (b) r2 0

h( t, r)=2rr J"" Jrr 0 0

(F-H)u 2 sin adadu

The characteristic equations for this system are

. r u cosa 1J - ~ ( t, r) cos a ( l. 7) a= (6/u-u/r)sina

A global-in-time existence and uniqueness theorem for the spherically symmetric problem was first proved by Batt in [2].

The proof in (2] is only valid, however, for the gravitational problem where the field always points inward along a radius. A different proof of this result is given in [18] which is also

valid for the electrostatic case. The result in [ 18] is presented in some more detail in [20].



258 STEPHEN WOLLMAN

To prove the result of global-in-time solvability of (1.1)

with data of the form (1.5) we show that solutions to (1.7)

with initial points in the support of the initial data are

bounded in time. We consider trajectories p(t)=(r(t),u(t),a(t))

frir which O<a(t)<n, It is a result of conservation of angular

momentum that if a(O)iO,n, then a(t)iO,n for all t. We can

extend results to particles travelling along a radius (ali)=O,u) by continuity,

We assume for the solution that

and that

O<F<D

oo oo n

flu 2 J J J IF-H,sinadau 2dur 2dr<M 0 0 0

and we let the constant A(T) be a bound on the charge density h

for te:[O,T)

lhl<2nf!F-Hjsinadau 2 du~A(T) The bounds D and M are known "a priori" from the initial data,

The bound A(T) gets computed as a part of the proof.

Let p(t) be a trajectory for which a(O)iO,n and let A=A(T)=4/3nA(T)

It can be shown that along a trajectory u(t) satisfies the

estimate

( 1.8)

The proof of this result is given in [ lR) and [ 20] •

From (1,8) we get the a priori bound on the support of solutions to (1,6) as follows.

Q(t)= supp F(t), the support ofF in r,u,a space for

each t. p= (r,u,a), a point in r,u,a space.

E(p)=u, the projection of p on the u-axis.

n < T 1 = un 1 t 1 , t e: r o , T J •

P(T)= supE(p), a bound on the support ofF in velocity pdi (T) space up to time T.




In terms of P=P(T) the charge density h has the bound

00 11 2 lhl<211j J jF-H,sinadau du

0 0

< 411 DP 3+B< 411 (D+B)P 3 3 3

< 411 c P3 3

00 11 2 211 f f I H I s in an au du < B

0 0 C = D + R

Therefore if we let

and set A=A(T)= 411 A(T)=(~) 2 cP 3 (T)

3 3

(411) 2 CP 3 3

it follows from (l.B) that along a trajectory

Since P=supu ( t), tE [0 ,T]

p(O)t:ll(O) an inequality for P is

P2<P2+GM2/3( 411 )2/3Cl/3p - 0 3

A bound for P is

P 0 = supE( p) pt:ll(O)

259

This hound is independent of T and is therefore uniform for all T. The general theorem now gives the result of global-in-time solvability of the system (1.1) with initial data of the form (1.5).

iv) The cylindrically symmetric Vlasov-Poisson system:

In the examples that have been solved so far, i.e., the lower dimensional problems for general initial data and the three dimensional spherically symmetric case, one obtains the



260 STEPHEN WOLLMAN

requisite a priori houno on the support of the solution using

only the property of conservation of mass. To get the solution

to the cylindrically symmetric problem the conservation of

energy is also utilized to get the nPcessary estimates. The

cylindrically symmetric proble~ was first solved hy Horst

[n] ,[9]. To get this result Horst obtains the following

interesting bound which is stated here in a lemma

LEMMA: If f is a solution to (l.la,b) then

l/l/fdv)sdxl 1/s<K l.$_s,$_5/3

K - constant

Proof: From the conservation of energy one obtains

f L'Vil 2 f<E (constant) x,v

(even in the gravitational case [4,p.23]).

A generalization of conservation of mass is

llsing these two a priori estimates one obtains the result of

the lemma. Por the complete proof see [9, p.22].

In the cylindrically symmetric case there is rotational symmetry around the z axis and angular momentum around the z

axis is conserved, i.e., a trajectory (x(t), v(t)) of (l.la,b)

satisfies

Let 2 2 1/2 r=(x 1 +x2 l ,

n=(x 1v 1+x 2v 2 ), l;=(x 1v 2-x 2v 1 )

z=x3' "z=v3.

Initial data of the form

g(x,v)= G(r,n,l;,z,vz) ( 1.9)

give rise to cylindrically symmetric solutions to (l.la,b). A convenient set of variables to use in analyzing trajectories in

this case are




r=<xf+x~)l/2, u=<vf+v~)l/2 z =x3 , vz=v 3

a, the angle between (x 1 ,x 2) and (v1 ,v2 ) (-11_$a_$11),

In terms of this set of variables the solution is a function of the form

261

f(x,v,t)=F(r,u,a,z,vz,t) ( 1.10)

where F satisfies the transport equation given in [19,p.l52]. The characteristic equations we wish to solve are

r = UCOS a 1 U= Cl 1jl COS a ar: a=[-Cl1jl /u -u/r]sina ar:

z =v z, v z

Here 1jl is the field which satisfies

a2w + 1/r Clljl + a2w =-411 2fFududadv Clr2 ar- Clz2 z

(We have set the background distribution to zero for simplicity).

( L 11)

To compute a priori bounds we need estimates for the field

Let

h(r,z,t)=fFududadv z

and assume the bounds on h suplni<A=A(T), te[O,T] R3

(fns rdrdz)l/s~K l~s~5/3

In terms of these bounds the required estimates for the field for te[O,T] are



262 STEPHEN WOLLMAN

I <a K5/9A4/9 rs_.e=(x)S/9

<(aKS/~Al/ 6 )/ 1/2 r>.e - r a - constant

[9,p.31] [19,p.l62].

( 1.12)

To get an a priori bound on the support of solutions to (l.la,b) with data of the form (1.9) we use estimates (1.12)

to produce bounds on the solutions to (1.11) for initial points in the support of the initial data. We consider

trajectories for which a(t)*O,±n, and extend results by continuity to trajectories travelling in planes containing the

z-axis (a(t)=O,±n). This computation is carried out in [1<)].

As a result we obtain the following bounds on the support of the solution. Let

rl( t) supp f( • ,t), the support of f (of the form (l.lO))in R6 at time t.

ii ( T) u n ( t) for tE [ 0 , T l t

u(x,v)= (v2l+v22)1/2, vz(x,v)=v3

and define the bounds

p =-sup I v (X' v) I , fl(T) z

Q= sup(u(x,v) ) 2 Q(T)

P0 , Q0 the bounds at T=O.

A set of inequalities sntisfied by P and 0 are

A solution is

Q<Q +9 S(PQ)l/6(R +Ql/2T)l/2 - 0 0

P<P +13P4/9Q4/9T - 0

(8, R0 - constants) [19,p. 169].




This is a requisite a priori bound on the support (in velocity space) of the solution to (1.1) with initial data of

the form (1.9) for tE[O,T]. The result of global-in-time

classical solvability of the problem then follows from the

general theorem.

263

The problem of global-in-time classical solvability of

system (1.1) in three dimensions for a general class of initial data is still an open problem. For some remarks on this

subject see [10].

2. THE VLASOV-MAXWELL SYSTEM

The Vlasov-Maxwell system is a nonlinear integra-differential system of equations that describes the motion of

particles in the presence of their own self generated electro-

magnetic field. This system of equations is one of the fundamental set of equations underlying the kinetic theory of plasma. We consider the following initial value problem

at +v.Vxf-(E+vxR) •Vvf=O (a) TI

f(O) = g aE - VxB=41Tjvfdv as + VxE=O (b) at at ( 2 .1) E(O) = E B(O) = B

where o' 0

V.E =-41Tjadv, V.B =0 (c) 0 - 0

E=(E1 ,E 2 ,E 3 ) , B=(B1 ,B 2 ,B 3 ) x=Cx 1 ,x 2 ,x 3 l , v=(v 1 ,v 2 ,v3 )

Given initial conditions (2.lc) it follows easily that the solution to (2.1) satisfies the additional equations

V•E=-41Tjfdv, V•B=O for t>O.

The main purpose of this section is to describe the local existence and uniqueness theorem for system (2.1) which is given in [20], [211. The system written here is for a single

charged species, say electrons; however, the result given in

this paper can easily be modified to accommodate a background

distribution or to deal with several interacting charged species. The local existence and uniqueness theorem for (2.1)

is obtained by generalizing a theorem of Kato [12, Theorem II,



264 STEPHEN WOLLMAN

p.l95]. In [ 12] Kato proves a theorem for symmetric hyperbolic systems with certain operator coefficients which act pointwise in t hut non-locally in X£Rn• With a small amount of

modification of this theorem we obtain a similar result for a

system with operator coefficients which are nonlocal in both x

and t. In this form the theorem can he applied to prove local existence and uniqueness of solutions to (2.1). The assumption

is that the initial data has compact support. Solutions are obtained as continuous mappings of t into classes of Sobolev

spaces on Rn· As a part of the proof and in order to apply

the general theorem bounds on the support of the solution as a function of time are obtained.

As far as we know the existence and uniqueness theorem stated here is the only such result for the fully nonlinear,

three dimensional equation. Several results have been proved

for the system in lower dimensions and with various additional modifications and restrictions [4], [5), [14].

We consider the system m

au + 2: "IT j=l

G.[u(t)]au =0 J ax-. ( 2 • 2)

J

"The functions u are of class Hs(Rm,Pl for each t where HS(Rm,P) is the standard Soholev space of functions defined

on Rm valued in a Hilbert space P, and having s L2 deriva-tives. In this paper P is identified with Rp for some integer p. The quantities Gj[•] are nonlinear operators that carry functions valued in L 1 [0,T; Hs(Rm,P)] into functions valued in L1 [0,T; HsuR.<Rm,R(P))]. Here the subscript uR. stands for a uniformly local Sobolev space as defined in [12] and B(P) is the set of hounded linear operators

on P. In this context elements of B(P) are pxp matrices." 1 We use the following notation

u . R the norm on Hs(R ,P} s ,m m (or Hs ( Rm, B ( P) ) )

I a the s . norm on HuR. ( Rm ,P) sR. ,m (or H\(R ,B(P))) u m

We state the theorem for (2.2) as it is given in [20], [21]. The proof is similar to that for [12, Theorem II]. 1 [21 , reprinted by permission of John Wiley and Sons,

publishers] •




THEOREM 2.1: suppose s>m/2+1. Let D be a bounded open

subset of Hs(Rm,P), Tan arbitrary positive number, and

A(T) the set L1 [O,T;DJnc[O,T; Hs-l(Rm,P)J., Let Gj( •) be

nonlinear operators on A(T) to Ll(O,T;Hsu~(Rm,B(P))]. Assume the following conditions

hold: The quantities Gj[h] are bounded (by a constant A) in the

( 2. 3)

H~~- norm for h£A(T), uniformly in t£[0,T].

For each j and t£[0,T] the map h(t)+ Gj [h(t)] satisfies the condition (2.4)

sup UG. [h(r)] - G. [h(r) II o r<t J J o .. ,m

s_ ll(T) sup Uh( r)-h( r) II r<t o,m

ll(T) - constant depending on T.

The maps t+Gj[h(t)] are continuous in the H 0 u~-norm

for each h£A(T). (2.5)

For each h£A(T), Gj[h](x)£R(P) is symmetric for each

t, X£ [0 ,T] XRm,j=l,,,, ,m, ( 2.6)

( 2. 7)

Then there is a unique solution of (2.1) defined on [O,T'], where T'<T such that

U£ C[O,T' ;D](lC1 [0,T': Hs-l(Rm,P)]

Proof. See [20] , [21] •

( 2. 8)

265

Theorem (2.1) is applied to give a local existence and uniqueness theorem for the Vlasov-Maxwell system. The solution

ohtained is a function with bounded support in R6 • By taking account of the bound on the support of the solution as a

function of time the system is put into a form where the coefficients in the equation are of class Hsu~· Theorem

(2.1) can then be applied to prove the local existence and uniqueness theorem for the system.



266 STEPHEN WOLLMAN

For the initial data to (2.la,b) it is assumed that

s E ,B EH (R 3 ) s>S 0 0 -

( 2.9)

llgll 5 , 6 ~ M

liE II 3 , IIR II 3 < N 0 s, 0 s, R,M,N - constants

(We suppress reference to the dimension of the range space of

the functions in this notation). The following existence and

uniqueness theorem for the Vlasov-Maxwell system is stated.

THEOREM ( 2 • 2 ) • Let s>5 and the initial conditions to (2.la,b) satisfy (2.9). There is a positive number T0 and a

unique function

such that f is a solution to (2.la,b).

Proof: It is shown that for the time interval [O,T0 ) the system (2.11 is of the type (2.2) and that Theorem (2.1) applies. The proof is given in [20], [21].

By Sobolev's Lemma the solution obtained in Theorem (2.2) is a strict solution to system (2.1).

BIBLIOGRAPHY

1. Arsenev, A.A., Existence and uniqueness in the small for the classical solution of a system of Vlasov equations,

Akad, Nauk SSSR Dokl. 28 (1974), 11-12. 2. Batt, J., Global symmetric solution of the initial

value problem of stellar dynamics, J. Differential Equations 25 (1977), 342-3fi4.

3. Ratt, ,J., Recent developments in the mathematical investigation of the initial value problem of stellar dynamics and plasma, Ann. Nuclear F.nergy, to appear.

4. Cooper. J., Klimas, A., Boundary value problems for the Vlasov-Maxwell equation in one dimension, J. Math • Anal. Appl. 75 (1980), 306-329.




5. ouniec, J., On an initial value problem for a non-linear system of Vlasov-Maxwell equations, Bull. Acad. Polar. sci. Ser. Sci. Tech. 21 (3) (1973), pp. 97-102.

267

6. Horst, E., "Zur Existenz Globaler Klassischer Losungen

desAnfangswertproblems der Stellardynamik", Disseration,

Munchen, 1979. 7. Horst, E., On the existence of global classical

solution of stellar dynamics, in "Mathematical Problems in the Kinetic Theory of Gases" (D.C. Pack and H. Neunzert, Eds.), Frankurt, 1980.

B. Horst, E., On the classical solution of the initial value problem for the unmodified non-linear Vlasov equation I, Math. Meth. in the Appl. Sci. 3 (1981), 229-248.

9. Horst, E., On the classical solution of the initial value problem for the unmodified non-linear Vlasov equation II, math. Meth. in the Appl. Sci. 4 (1982), 19-32.

10. Horst, E., New results and open problems in the theory of the Vlasov equation, Prog. Nucl., E.8 (1981), 185-189.

11. Iordanskii, s.v., The cauchy problem for the kinetic equation of plasma, Amer. Math. Soc. Transl. Ser. 2 35 (1964), 351-3fi3.

12. Kato, T., The cauchy problem for quasi-linear symmetric hyperbolic system, Arch. Rational Mech. Anal. 58 ( 1975), 181-205.

13. Kurth, R., Das Anfangswertproblem der stellar dynamik, z. Astrophys. 30 (1952), 213-229.

14. Neunzert, H., Petry, K.H., Ein existenzsatz fur die Vlasov-Gleichung mit selbstkonsistentem magnetfeld, Math. Met h. in the Appl. Sci .2 ( 1980), pp. 429-444.

15. Ukai, s. and Okabe, T., On classical solution in the large in time of two-dimensional Vlasov's equation, Osaka, J. Math. 15 (1978), 245-261.

16. Wollman, s., "Classical Solutions to the Vlasov-Poisson System of Equations", Technical Report No. 329, Department of Mathematics, University of New Mexico, 1977.

17. Wollman, s., Global-in-time solutions of the two-dimensional Vlasov-Poisson system, Comm. Pure Appl. Math. 33 (1980).

18. Wollman, s., The spherically symmetric Vlasov-Poisson system, J. Differential Equations 35 (1) (1980), 30-35 •

19. Wollman, s., Existence and uniqueness theory of the Vlasov-Poisson system with application to the problem with cylindrical symmetry, J. Math. Anal. Appl. 90 (1982), 138-170.

20. Wollman, s., "Existence and Uniqueness Theory of the Vlasov Equation", Technical Report MF 100 Courant Institute-MFD Division, 1982.

21. Wollman, s., An existence and uniqueness theorem for the Vlasov-Maxwell system, to appear in Comm. Pure Appl. Math.

Department of Mathematics, Baruch College, CUNY, New York, NY







I. INTRODUCTION

REMARKS ON COLLISIONLESS PLASMAS*

Robert T. Glasseyt and Walter A. Strauss

The classical equations for a collisionless plasma are the Vlasov-Maxwell equations, here called (VM). The unknowns are the density in phase space f(x, v, t) (x e IR3, v e IR3, t> 0) and the Maxwell electric and magnetic fields E(x, t), B(x, t) which satisfy

ft + v • vxf + l(E + v x B)

Et = 'V X B - j

(VM) Bt = - 'V X E

l v · B = 0

'V·E=p

'V f = 0 v

The charge density p and the current density j are given by

( 1) p = !fdv

(2) j = Jvfdv,

and l = ± 1.

A special case of (VM) is obtained by formally putting B = 0. This is the Vlasov-Poisson equation (VP), which takes the form

where

(3) E = - vxu' u = I pdy . 'V - + 1 TX-YT' I -- •

*Research supported in part by NSF MCS 81-20599 and NSF MCS 81-24187. tAlfred P. Sloan Research Fellow.

269





270 ROBERT T. GLASSEY AND WALTER A. STRAUSS

There are actually two equations here: r = +1 is the plasma physics equa-tion, where E represents a "repulsive" force, whereas r = -1 is the stellar dynamics case ([3]), where E represents an "attractive" force.

There is one other system of interest, namely, the special rela-tivistic version (RVM) (cf. [9]). Here, the Boltzmann equation (the equation for f in (VM)) is replaced by

(RVM)

The other equations are the same as in (VM), but the current density is now taken to be

j - J-;::=v==:; f dv .A + I v 12

Our interest is to study the Cauchy problem, and to determine if there is a solution in the large (i.e., for all time). Since this problem seems exceptionally difficult and is not yet solved, we concentrate here on certain asymptotic properties, assuming that a solution exists. In special cases, of course, global solutions are known to exist.

It is no doubt instructive to give a brief survey of known results. There is no global theorem avai !able for (VM) (or (RVM)) in more than one dimension, although we discuss here a special case in three space dimensions. Local existence has just been treated by Wollman [11]. In the case of one space dimension, there are the papers of Iordanski [8], Cooper and Klimas [4] and Duniec [5].

For the (VP) system the situation is much better, although the general Cauchy problem is not yet solved for classical solutions. The two-dimensional problem was solved by Wollman [12]. In three dimensions, glob-al weak solutions were shown to exist by Arsen'ev [1] when r = +1, and it follows from Horst's work [7] that this result is valid for r = -1 too. As for classical solvability, there is the incisive work of Batt [2] and Horst [7], from which it follows that the Cauchy problem for (VP) has global solutions if the Boltzmann data f(x, v, 0) is cylindrically symmetric, nonnegative, and of compact support in v. Batt's theorem was originally given for .., = -1; Wollman [10] extended this to r = +1 (still in the case of "radial" data). Finally, Horst [7] shows that, for



REMARKS ON COLLISIONLESS PLASMAS 271

r = -1 and space dimension four or more, there is a large class of solu-tions to (VP) which do not exist for all time.

Actually, the Vlasov-Maxwell system seems to be vastly more diffi-cult than the Vlasov-Poisson system. This is clearly seen when one con-siders weak solutions. The nonlinear term in the Boltzmann equation is governed by a solution to an elliptic equation (Poisson's equation) in the case of (VP). However, for (VM), the nonlinear term (the Lorentz force) in the Boltzmann equation is given by a solution of a hyperbolic sys-tem (Maxwell's equations) and can therefore be expected to be troublesome. Now, regarding weak solutions, it is known from conservation of energy (to be discussed below) that p e L"'(IR, L5/ 3(IR3)). (This is valid for (VM) as well as for (VP)). In (VP), one has by the definition (3), E = -vu, where t.u = -4llp. Thus E e w1•5/ 3(IR3) for fixed t, and this provides the compactness needed to pass to the weak limit in the equation (after, of course, suitable regularization of the potential u). This is fundamentally why there is an existence theorem for (VP). Such reasoning clearly fails for (VM) and, in fact, the existence of weak global solutions is still unsettled.

When discussing (VM) or (RVM), we will always take r = +1 here.

II. CONSERVATION LAWS

Consider a smooth solution of (VM) which is small at infinity. If we define the energy density e by

(4)

then the "energy identity" is

(5) ae I3 a 1J 2 or= ax-E(B X E)k - 2'" lvl vkfdv] k=1 k .

which leads formally to the conservation of energy

J edx = const. IR3

If we further define the momentum p by




(6) p = B x E - Jvfdv, then

fonnally again r

jpax = canst.,

and the angular momentum

J x x pdx

is conserved as well. Finally, we mention the dilation identity

(7) :r J<te + x-p)dx + ~ JJ!v 12fdvdx = 0

This is computed using the energy identity above, as well as the "momentum conservation law," which can be expressed as

(8) a:i =~a~. (!EI2 + IBI 2) + l a~ [Jvjvkfdv - (EjEk + BjBk)] J k=1 k

(j=1,2,3).

The (RVM) system possesses similar conservation laws. For example, the energy density eRVM is given by

etc. One can now easily show that, for (RVM), we have

and that f(·, v, t) has compact support, if this is true at t = 0. Similar laws have been written down by Horst [7] for (VP).

Here, and in what follows, we will assume that the Boltzmann data f(x, v, 0) is nonnegative and of compact support in x and v.

III. TIME DECAY FOR SOLUTIONS OF (VP)

When -y = +1 in (VP), the force E is repulsive in na-ture, and there is some inherent time decay, as shown in the following re-sult.



REMARKS ON COLLISIONLESS PLASMAS

THEOREM 1: Let 'Y = +1 in (VP) and consider a classical solution of finite energy. Then as t .... "',

b) for any a , 0 ::_a < 1,

We remark that, for e.g., cylindrically symmetric data f(x, v, 0) classical solutions exist in the large [2], [7].

PROOF: a) Define the quantities

F(t) = Jfix! 2fdvdx

J(t) = Jf I v! 2fdvdx

From conservation of energy, J(t) is uniformly bounded. Integrating the (VP) equation in v, we obtain

Pt + 'I X • j = 0.

273

When we multiply this by lxi 2 and integrate over x and v, we get

( 9)

from which it follows that

(9 I )

where we have used the Schwarz inequality. P=-fvfdv=-j, since B=O in (VP).

( 10) tJedx + Jx . pdx = I x t - 0

By (9, 9' ), we have the estimate

Now in the dilation identity Thus




However, the definition of the energy e in (4) shows that the left-hand side of (10) is greater than

and this proves a).

To establish b), recall that, by definition (3), E = -vu where t.u = -41!p (.y = + 1 here). Thus

-I pdy d u - IX - y I an

Jlvul 2dx = JIEI 2dx = 41!Jpudx

For u we have the obvious bound

Hence

u > I pdy > 1 I dy ly 1..::: IX I I X - y I - ZTXT I I I lp y :5_ X

= J.(r r p(rw, t)dwdrfn2 f p(nw'' t)dw'dn o lwr=1 o lw' 1=1

= i [~ a~[ r pdyidr o IYr:::r

1 , -2 [ I ]2 1f -2 I ]2 = 4Jnr pdy dr ~ 4 r [ pdy dr 0 IY 1.2T ta IY l~r

q.e.d.




(Here we have used the fact that

Ip(x, t)dx

is conserved (or, at least, bounded)).

IV. DECAY OF THE LOCAL ENERGY FOR (VM) AND (RVM)

The full (VM) and (RVM) systems also possess some inherent time decay.

The following result, hyperbolic in nature, demonstrates this.

THEOREM 2: Consider a classical solution of (VM) or of (RVM) of finite energy. Let R > 0 be fixed, and define the local energy by

E R(t) = I edx I xi~R

(resp. I eRVMdx). Then IXI~R

a) For (VM), there exists a sequence of times

ER(tn)- 0 as n-"'

b) For ( RVM), 1 im ER(t) = 0. t-m

t -"' n such that

PROOF: We sketch the proof only, since it is quite complicated. First, consider (VM) and the momentum identity (8). Let r = lxl. Multi-

3 plying (8) by x/r, summing on j and integrating over IR , we get

By the definition of p, and the fact that the L1-norm of f over IR6 is bounded, we have

< const. by conservation of energy. Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.



It follows that

( 11)

(We have used here the fact that f ~ 0, which follows whenever the data f(x, v, 0) ~ 0).

Next, we repeat the procedure, this time multiplying (8) by x. i t(r), where

1 t(r) = 1 - 2"\r+fT

Thus ~ ~ t(r) ~ 1 for all r.

Using the result (11) at the end of this computation, we obtain in this way

from which it follows that

and hence the result

{ 12)

I..,J e dxdt < .., OIR 3 ~ '

i r edxdt < ... Jo lx·bR

as well as the statement a).

The exact same procedure works for {RVM). We omit the details. Thus we have directly

{13)

The energy identity for (RVM) (the analogue of (5)) is



REMARKS ON COLLISIONLESS PLASMAS

Integrate this with respect to x tegrate with respect to R over

over the set lxl ~ R, and then in-0 < R 1 ~ R ~ ~ < "" to get

Multiplying this by t- t 1, and integrating with respect to t over the interval t 1 ~ t ~ ~· we have

Now choose t2 = t, t 1 = t- 1. The left-hand side is then at least as large as

The first term on the right-hand side then clearly tends to zero as t ~ ~

by (13). The second term there is dominated by

and this tends to zero also by (13). This proves b). Moreover, we see at once why only the weaker result a) is true for (VM). Indeed, on the right-hand side of the energy identity (5) for (VM), the integral of the term

cannot be dominated by Jedx.

277




V. A CLASS OF GLOBAL SOLUTIONS TO (VM)

Let f(x, v, t), E(x, t), B(x, t) be a classical solution of the (VM) system. Let U be an arbitrary rotation on m3 with det U = 1. Consider the functions

~

f(x, v, t) = f(Ux, Uv, t)

E(x, t) = u- 1E(Ux, t) ~

B(x, t) = u- 1B(Ux, t)

~ ~ ~

A long calculation, which we omit, shows that (f, E, B) is then also a solution of (VM). Now we assume that, at t = 0, we have f = f,

A A

E = E and B = B for all such U. (Thus we are imposing spherical symmetry.) Then by local uniqueness we have f = f, E = E, B = B for all t > 0. Now the condition E = E means that

E(Ux, t) = UE(x, t) for all such U.

It follows [6] that E (and hence also B) is of the form

where k depends on tial gradient, and so is static, and (VM)

E ( x , t ) = xk (! x 1, t )

x only through jxj. Therefore E is a spa-v x E = 0. From the Maxwell equations, then, B reduces to the single equation

ft + v • V/ + (E(x, t) + v x B(x, 0)) . V/ = 0

where E = -vu, 1 u = r * p.

This is essentially (VP) again, except for the additional term v x B(x, 0). However, this term is in the form of a cross product in v, and is static as well. Thus the existence considerations of Batt [2], Horst [7] and Wollman [10] can be adapted, and we obtain a class of symmet-ric global solutions.




References

1. Arsen 1ev, A., "Global existence of a weak solution of Vlasov 1 s system of equations," U.S.S.R. Comp. Math. and MaUl. Phys. 15 (1975), 131-143.

2. Batt, J., "Global symmetric solutions of the initial-value problem of stellar dynamics," J. Diff. Eqns. 25 (1977), 342-364.

3. , "Recent Developments in the mathematical investigation of the imtial value problem of stellar dynamics and plasma physics," Ann. of Nucl. Energy 7 (1980), 213-217.

4. Cooper, J., and Klimas, A., "Boundary value problems for the Vlasov-Maxwell equation in one dimension," J. Math. Anal. Appl. 75 (1980), 306-329.

5. Duniec, J., "On an initial-value problem for a nonlinear system of Vlasov-Maxwell equations," Bull. Acad. Polon. Sci. Ser. Sci. Tech. 21 (1973), 97-102.

6. Handbuch der Physik, Vol. 3: "Nonlinear field theories of mechanics," p. 35.

7. Horst, E., "On the Classical solutions of the initial value problem for the unmodified non-linear Vlasov equation, Parts I and II," Math. Meth. in theAppl. Sci. 3 (1981), 229-248, and 4 (1982), 19-32.--

8. Iordanski, S., "The Cauchy problem for the kinetic equation of plasma," Trans!. Ser. 2, Vol. 35, Amer. Math. Soc. (1964), 351-363.

9. Weibel, E., "L 1equation de Vlasov dans la theorie speciale de la rela-tivite," Plasma Phys. 9 (1967), 665-670.

10. Wollman, S., "The spherically symmetric Vlasov-Poisson system," J. Diff. Eqns. 35 (1980), 30-35.

11. , "An existence and uniqueness theorem for the Vlasov-Maxwell system," preprint.

12. , "Global-in-time solutions of the two dimensional Vlasov-Poisson systems," Comm. Pure Appl. Math. 33 (1980), 173-197.

ROBERT T. GLASSEY DEPARTMENT OF MATHEMATICS INDIANA UNIVERSITY BLOOMINGTON, INDIANA 47405

WALTER A. STRAUSS DEPARTMENT OF MATHEMATICS BROWN UNIVERSITY PROVIDENCE, RHODE ISLAND 02912







TOWARD A NEW KINETIC THEORY FOR RESONANT TRIADS

Harvey Segur 1*

Abstract. Resonantly coupled triads of nearly linear wave modes are the dominant nonlinear interactions in many physical situations. We present here a new kinetic model of resonantly interacting wave packets. This model applies when the wave packets are "dilute" in the sense that they participate in triads only occasionally. It differs from previous statistical models in that no "random phase approximation" is ever made. Phase variations during interactions are computed explicitly, and they are not random. The derivation of the new model is analogous to that of Boltzmann's model of a dilute gas of molecules. However, the appropriate Stosszahlansatz for the wave packets is more complicated than Boltzni2mri's-s0 the final kinetic model differs significantly from Boltzmann's equation.

I. INTRODUCTION. Resonant triads are the simplest possible nonlinear

coupling of small amplitude (nearly linear) waves in a dispersive, nonlinear

system. They are thought to play a fundamental role in transferring energy

between wave modes in many physical contexts, including phonons in a crystal

lattice [ 1], nonlinear optics [2], internal waves in a rotating and/ or stratified ocean [3] and weak turbulence in plasmas [4];

applications are discussed in Ref. 5 and 6. some other

Because of the fundamental importance of triads in these problems, a number of approximate models have been developed to predict the effects of triad interact ions. Depending on the physical context and on the tastes of the developer, both deterministic [5, 7] and statistical [1 ,8] models are available. Unfortunately the regions of validity of these models are not well established, and occasionally the same question has been answered both "yes" and "no", depending on whether a deterministic or a statistical model was used (e.g., compare Refs. 7 and 9).

The purpose of this paper is to present a new kinetic theory of resonant

triads. Unlike earlier statistical models, the theory presented here does not

rely on the "random phase approximation" either directly or indirectly. In

*1980 Mathematics Subject Classification. 45K05, 73899, 82A05. 1supported by U.S. Army Research Office and by Office of Naval Research

281





282 HARVEY SEGUR

fact, phase variations during interactions are specifically included in the

present model. The fundamental concept of the proposed model is that of a

large collection of localized, nearly monochromatic wave packets, that are "dilute" in the sense that they interact in triads only occassionally. By

exploiting the analogy between this dilute gas of wave packets and Boltzmann's

dilute gas of molecules with repulsive potentials, we derive a kinetic theory

for the wave packets which is similar in spirit to Boltzmann's kinetic theory

for the molecules. HowPver, the mathematical structure of this model differs

significantly from Boltzmann's equation; in particular, equilibrium

distributions do not play the central role here that they do in Boltzmann's

theory.

The outline of this paper is as follows. The basic concepts of triad

interactions are reviewed in · ?: their physical der·i vat ion, the important

special case of a single triad, and the use of statistical theories when many

triads coexist. The review in . 2 contains only background material, and it

may be skipped by experts. The new kinetic model is developed in _r-. 3, beginning with the deterministic system to which the eventual kinetic theory

will apply: a Hamiltonian system containing a large number of nearly

monochromatic wave packets that interact in triads. Once this Hamiltonian

system is defined, the analogy with Boltzmann's problems of N bodies with

repulsive potentials can be drawn. Next we outline a BBGKY-hierarchy that

applies when the wave packets are dilute. Its truncation in the usual

fashion, coupled with an appropriate Stosszah!~~~atz, leads to the new kinetic model for triads, given by eqn. (49). The consequences of this model are

largely unexplored at this time, but a few preliminary results are presented.

2. REVIEW OF TRIAD CONCEPTS A. DERIVATION OF THE EQUATIONS. We begin by sketching a typical derivation

of the triad equations from a physical problem. More specific examples may be

found in Ref. 1-11 . Let

N(ljl) 0 ( 1 )

represent an energy-conserving, nonlinear system, whose solution is

represented by ljl( t t). (The requirement that the system be conservative is not essential for triads, but it will be essential for the kinetic theory in question. Therefore, we assume that any non-conservative effects are small

enough that they can be neglected over the time-scale on which the triad

interactions dominate the dynamics.)



TOWARD A NEW KINETIC THEORY FOR RESONANT TRIADS 283

Let '.II = 0 represent an equilibrium point that is not unstable.

Infinitesimal departures from this equilibrium are found by linearizing (1):

L ljJ1=o (2)

We seek the normal modes of the linear operator in (2). If it has constant

coefficients, this amounts to seeking solutions of the form

-> ljJ1 "' exp ( i k. x- iwt) .

Substituting this ansatz into (2) yields the 91spersion relation of the

linearized problem,

w = (3)

As a simple example, for the Boussinesq equation,

(4)

the dispersion relation of the linearized problem is

(5)

For triads to be relevant, the dispersion relation of the linearized problem should satisfy several requirements. (i) w(k) exists. Then for fixed k, w(k) has only a countable set of possible values (ordinarily only a finite set, as in (5)). This excludes problems like Vlasov's equations in plasma physics, for which w(k) ranges over a continuum, even for fixed k. (ii) w(k) is real for real k. This is assured, because the original problem is conservative and because ljJ=O is neutrally stable. In this sense, (5) is not a prototype unless we can enforce jkj~l.

(iii) w(k) admits resonant triads: triples of (k,w) that satisfy a resonance condition,

(6a,b)

where oi=~1. If w(-k)= -w(k), we may take o2=o 3= +1 without loss. To simplify the presentation we now assume



284 HARVEY SEGUR

(6 c)

Given w(k), a simple geometric procedure determines whether solutions of (6) exist [10]. For example, in (5) every wavenumber with jkj<1 participates in exactly one triad:

(7)

(iv) The wave modes of the linearized problem are dispersive [12]:

( 2 ) ~i.O a~. a~.

1 J

(8)

Otherwise (6) becomes trivial, because effectively (6a) ~ (6b).

Now we return to (1), and seek small but finite departures from equilibrium by means of a formal power series (O<E<<1):

00

~c~.t;E) =~En~n(x,t;E). 1

(9)

At leading order in E, ~ 1 satisfies (2), so ~ 1 can be represented as a linear superposition of the linearized wave modes. In the simplest possible version, we choose ~ 1 to be the sum of three uniform wave trains,

3 3 ~ 1 -~an exp(ikn.x-iw(kn)t) -~an exp(ien). ( 1 0)

If~ should be real, we keep only the real part of (10).

The first nonlinear effects appear at the next order in E:

( 11 )

where the right side denotes all of the quadratic interactions of ~ 1 with itself that are implied by (1). If these do not vanish identically, and if the three wavenumbers in (10) satisfy (6), then quadratic products of the form




exp(ie1 )*exp(ia2) are equivalent to exp(-ia3), which forces w2 to grow

linearly in time. It follows that the expansion in (9) becomes disordered at

least by the time £t=0(1).

This breakdown of the formal series in (9) can be corrected in a variety of ways. Here we use the method of multiple time-scales [13], in which

t = £t,

(10) is replaced by

3 ljJ1 =I:an(t) exp(ian), ( 12)

and the t-dependencies of the complex amplitudes, an ( t), are chosen to eliminate the secular terms in (11), so that ljJ2 remains bounded in time. The

result is three complex ordinary differential equations, (the triad equations):

( 1 3)

I ( ) () * n 13 , denotes complex conjugate, and Y1 ,Y2 ,Y3 ("interaction coefficients") are fixed real parameters determined by ( 1 ) . Because ljJ=O is stable, the (Yn)'s do not all have the same sign ("decay instability"). In many problems

( 1 4)

which we now assume. Then (6) assures that the (Yn)'s have different signs. In any case, in the triad model the solution of (1) consists of a set of

linear wave modes, slowly modulated by weak nonlinear effects according to ( 1 3).



286 HARVEY SEGUR

It is evident from this derivation that ~ 2 will grow disastrously even if

the resonance conditions, (6), are satisfied only approximately. This

suggests a generalization, in which both slow space and slow time variables

are introduced:

.. X

.. EX, t=Et. ( 15)

Now (10) is replaced by

3 ljl1 ="[,an<x,t)exp(ien),

and a similar analysis leads not to (13), but to

( 1 6)

.. th In (16), (Yn) was defined above, IZ<lx, and en is the group velocity of the n mode,

Both (13) and (16) are called "triad equations" or "three-wave equations".

They describe the interaction of exactly three waves, interacting in one

triad.

The choice of variables in (15) is appropriate if each of the wave

packets in question is nearly monochromatic. We should emphasize, however,

that other choices are also possible, and that these lead to models other than

(16). As discussed below, one derivation of the ususal statistical theory is

obtained by taking x = ~. t = Et, instead of (15).




B. INTEGRABILITY OF A SINGLE TRIAD. Either model of a single triad, (13) or

(16), is integrable. For (13), this may be demonstrated by means of three

independent integrals [14]:

( 17)

The system (13) is Hamiltonian, with conjugate variables

n=1 , 2,3,

with the ordinary Poisson bracket, and with H as the Hamiltonian. Because the integrals in ( 17) are in involution, and because c1 and c2 make the phase space compact, a theorem of Liouville assures the complete integrability of (13). [15] Alternatively, the general solution of (13) may be written directly in terms of elliptic functions.

In many problems the integrals in ( 17) have simple physical interpretations. Define

( 18)

which is positive definite by (14). Let represent the spatially integrated form of the physical energy. Then the expansion that leads to (13) also gives

Thus the Hamiltonian for ( 13) is not the physical energy, although they are related.



288 HARVEY SEGUR

The integrability of (16) is more involved, because its phase space is infinite-dimensional. Its complete integrability was established by solving (16) exactly as an initial value problem, using the Inverse Scattering Transform (IST). This feat was accomplished in a series of steps [16-20], but the work of Kaup [20] is comprehensive. A consequence of this formulation is an infinite set of conservation laws, including a Hamiltonian for (16),

( 19)

Another consequence is the following qualitative picture of how three compact wave packets interact (see Figure 1, and see [20] for a detailed analysis). Imagine three wave packets, each spatially localized and each having a dominant wave number (kn) and frequency (w(kn)). The detailed shape

+ and position of each wave packet are given by an\x,t). If these wave packets are separated in the distant past ( t -> -co), then each packet propagates with its own group velocity and without change of form, according to ( 16). When two or more of the wave packets overlap, they interact. The interaction lasts a finite time, after which the three wave packets separate, each traveling with its own group velocity once again. The dominant wave number and frequency of each wave packet is unaffected by the interaction, but all other characteristics may change. Even so, these changes are completely predictable because (16) is solvable by IST.

Region




Because both ( 13) and ( 16) are integrable, no randomness is introduced

into their solutions by the dynamics, even for arbitrarily long times ( '[-++"').

The only possible uncertainty in the solutions is that introduced by the initial data. exponentially.

Neighboring trajectories may diverge linearly, but not

C. STATISTICAL MODELS. Unfortunately, the integrable models of a single

triad are not general enough for many physical applications. When a

linearized dispersion relation admits a resonant triad, it often admits several simultaneously. (The ex:1mple in (5) is anomalous in this respect).

In the more common case, when (k1 ,k2 ,k3) form a triad, then so do (k1,k4,k5), .. .. .. and (k2 ,k6 ,k7), etc. In this case if (10) is replaced by

N

ljl1 = I:an(-r) exp(ien)

for some large N, then (13) is replaced by

a'ta1 . * * * * n 1 a2a3 + io 1 a4a5 + •••

a'ta2 = iY2 * * * * a3a1 + ie:2a6a7 ..• (20)

a'ta3 iY3 * * a1a2 + •••

a'ta4 * * io 4a5a1 + •••

a'ta5 = * * io5a1a4 + •••

Here N complex equations are coupled through M interlocked, simultaneous triads (M;::(N-1)/2). Except in very special cases [17], these equations are believed not to be integrable.



290 HARVEY SEGUR

Statistical models have been introduced at this point to simplify systems like (20). The derivation of most of these statistical models seem to have followed one of two quite different lines, which we now describe. (For a more careful derivation, see Ref. 1, ~or 8.) First, imagine an entire ensemble of copies of (20), in which each realization may begin with different initial conditions. Then define the mean energy of the nth mode:

where <•> denotes an ensemble average. Next assert a random phase approximation and obtain from (20) a statistical model of the form

o1 Fn 1E: F~Fm T(~,m,n) 6(k~+km+kn) o(w~+wm+wn), m,~

(21)

where T( ~, m, n) is a transfer function obtained from the interaction coefficients in (20).

The practical value of these statistical models under appropriate conditions has been established [ 1, ~]. However, the validity of the random

phase approximation in such a derivation has not been established, so the precise meaning of "appropriate" in the previous sentence remains obscure.

There are both theoretical and practical objections to this derivation of (21 ) •

( i) Ablowi tz & Haberman [ 17] found a whole sequence of multi -triad systems that are completely integrable. The random phase approximation is simply wrong for any of their systems. ( ii) As discussed in .\?3. systems like (20) often are Hamiltonian, with Hamiltonians of the form

with one term for each of the M triads in (20). But H involves the phases explicitly, so its time-independence guarantees that all of the phases cannot be random. Moreover, if (20) admits other phase-dependent integrals in addition to H, then each additional integral further constraints the phases of the waves. (iii) In Ref. 7 and 9, two independent computations for essentially the same




problem are reported, one with [9] and one without [7] the random nhase

approximation. Their numerical results differ so significantly that the

authors reached opposite conclusions ("yes" [7] and "no" [9]) to the question

that motivated both studies. This does not mean that one of the studies

necessarily is wrong, but it does suggest that the random phase approximation

is not as benign as it is sometimes claimed to be.

Benney, Saffman and Newell [21 ,221 gave quite a different derivation of

the statistical theories that: (i) does not assume the random phase

approximation; (ii) apparently demonstrates its validity under the hypotheses

they require. In their argument, (10) is replaced with a Fourier transform,

Then they derive statistical transfer equations like (21), making essential

use of the smoothness of A(k). Now assumptions about the smoothness of A(k)

are equivalent to assumptions about the spatial scale (x) in (15). In

particular, if A(k) varies over llk=0(1), then an(x,-rl in (15) varies over

llx=0(1), by the uncertainty principle, sox=~ in (15). On the other hand, for

the uniform wave trains that led to (13), A(k) is a sum of a-functions, so it

is not smooth. For the wave packets in (16), A(k) varies over regions of

width £ (by (15)), so ClA/Clk=0(£-1A), and A is not smooth as £->0. Thus the

Benney, Saffman and Newell argument does not apply where either ( 13) or ( 16)

are relevant.

Because the linear problem is dispersive, by (8), making A(k) smoother

amounts to increasing the importance of linear dispersion in the evolution of

~. The faster the waves disperse, of course, the less time they have to share energy through resonant coupling. Thus, Benney, Saffman and Newell justify a statistical theory in a regime where linear dispersion dominates nonlinear coupling, whereas (13) and (16) require just the opposite.

To summarize, the deterministic triad models like (13), (16) or (20)

apply to collections of small amplitude waves, each of which is either nearly

or exactly monochromatic, and for which the weak nonlinearity dominates the

even weaker linear dispersion. For exactly three waves, the triad model is

integrable. For a large number of waves, coupled through many triads, the

equations are presumably not integrable except in very special cases. Under

these circumstances (large number of interlocking tr lads, nonlinearity » linear dispersion) where the Benney, Saffman and Newell argument does not apply, the validity of the existing statistical models is dubious.



292 HARVEY SEGUR

This brings us to the main point of this paper, a new statistical theory proposed in 53. It differs from earlier models in that the phase of the waves during interactions is computed explicitly, and it is not random.

3. AN ANALOGY BETWEEN WAVE PACKETS AND MOLECULES A. THE DYNAMICAL EQUATIONS. Let N(1ji)=O represent a nonlinear Hamiltonian system, and let 1J!(t t)=O represent an equilibrium solution that is not unstable. Let the linearized problem (about 1ji=O) have a non-trivial dispersion relation, w(k), and let w(k) admit several simultaneous triads (i.e., solutions of (6)), so that a given wave number, k, participates in more than one triad. Consider initial data for N(1ji)=O consisting of a finite collection of spatially localized, nearly monochromatic, small-amplitude wave packets, each with its own dominant wave number and frequency. [These initial data might come from a collect ion of uncorrelated sources, each with its own characteristic frequency and each of which occasionally generates wave packets at or near its own frequency.] Let £«1 measure both the strength of the nonlinearity and the relative variations of wavenumbers within each wave packet. We want to replace N(lji)=O with a simpler set of equations that are accurate at least out to times t=D(£-1).

To do so, we construct a resonating set,

as follows. S contains the wavenumbers (and frequencies) of the sources, plus all of the wavenumbers that can resonate with them according to (6), plus all of the wavenumbers that can resonate with these, etc. To keep S finite, it may be necessary to impose some cutoff as I k , ..... , and we assume that this can be done in a natural way. Moreover, we may assume that that every wavenumber in S is connected through a sequence of resonance conditions, ( 6), to every other wavenumber in s. If not, then S may be decomposed into two or more disjoint sets, and the construction which follows can be repeated for each disjoint set.

Under the circumstances stated above, we may represent 1jl(~,t:£) as

J E~[aj(x,t)exp(iej)+a;(x,t)exp(-iej)J + 0(£2) (22)




where x .. d, •=e:t, ej-~j·~-w(kj)t, and the sum is taken over S. (Here we

assume that w is to be real.) Some of these wave envelopes may vanish at ••0. In any case, the methods outlined in 9 2 lead to equations of the form

+ ••• (23)

+ •••

+ •••

where V=Clx, dj=Clw/Clkl ~j, and the interaction coefficients on the right side are real parameters such that within each triad, Yjwj (or <Sjwj or e:jwj• etc,) is of one sign.

As t.he generality of this derivation suggests, a great many nondissipative physical systems can be approximated by (23) [6]. It applies equally well to internal waves in the ocean or to drift waves in a plasma. At the level of (23), the only difference between these two problems is a change in the numerical values of the parameters. In this sense, we regard (23) as a fundamental system for study. The kinetic theory to be developed below is an approximation to (23) that will be valid for appropriate initial data.

The system (23) describes the evolution of J complex wave modes that are coupled through M triads, so there are 3M terms on the right-hand side of (23). How are M and J related? The smallest number of triads that will couple all J wave modes in M•(J-1 )12. Here the coupling is so tenuous that annihilating any triple of interaction coefficients (e.g., o1 ,o4, o5 in (23))



294 HARVEY SEGUR

splits the system into two uncoupled dynamical systems. At the other extreme, the integrable systems in [17] are much more highly coupled. They have

J = p(p=.!_)_ 2

M p(p-1U.e.~ 3!

for some p. These two extreme cases coincide only for p=J=3, M=1.

Because (23) is the end result of a systematic approximation of a Hamiltonian system, it is reasonable to expect that it might itself be Hamiltonian. In the case of minimal coupling, M=(J-1 )/2, one can show that (23) is always Hamiltonian. There exist conjugate variables of the form

(24)

for j =1 , .•• , J, where the ( ll j) 's are normalizing constants that depend on the interaction coefficients in (23). The Poisson bracket of any two functionals on the phase space is defined to be

jA,B I J E /dx [JA. ..?~- ~ _§~] oq. op. op. oq. ·

j=1 J J J J (25)

Then the Hamiltonian for (23) has the form

where ok=~1, and Ak is a constant depending on the interaction coefficients.

If the coupling in (23) is more than minimal (M>(J-1 )/2), then an integral of the form (26) exists only if the interaction coefficients satisfy

certain compatibility conditions. In this paper we require that (23) be Hamiltonian, so its interaction coefficients may not be entirely arbitrary.

In (22)' We now distinguish wave packets from wave modes. describes the complex envelope of the j th wave mode. Each wave mode may

At any fixed time, a wave packet from the j th contain several wave packets. .. mode is an isolated, connected region of physical space in which aj(X,T) does not vanish. The number of packets in the j th mode ( N j) is therefol'e the




number of isolated regions where a j (x, 1) is nonzero. According to (23) , the

number of wave modes is always J, but the total number of wave packets (N=

2:Nj) may change in time.

Wave packets are the fundamental objects in the analogy that follows, so it will be convenient to recast (23) in terms of packets. This is done by

splitting each wave mode into its constituent wave packets and writing a

separate evolution equation for each packet. Formally this is

straight-forward, but in practice it is somewhat artificial because new wave

packets can be created. Thus it is necessary to have equations for packets

that do not even exist at <=0, but which will exist later in the evolution of (23). (The use of these "ghost" packets should be considered a formal device that is used here only because the alternatives seem even clumsier.) The equations for the wave packets also have the form (23), but the number of such

equations is N, where N:::J; in the problems of interest N»J. Henceforth, we will assume that (23) and (26) refer to wave packets, rather than wave modes.

The packets always can be recombined into wave modes if desired.

B. THE FUNDAMENTAL ANALOGY. Consider a gas consisting of N point particles

in 3-space, and let mj,Pj(<) and qj(<) denote respectively the mass, momentum and position of the jth particle. If the force between any two particles is repulsive and depends only on the distance between the particles, then the Newtonian laws of motion are represented by the well-known Hamiltonian,

H (27)

If all the particles are identical, then mj=m and ~jk=~ for all j,k. If the gas is a mixture of several species, then the particles are identical within each species, but different for different species.

There is an analogy between N wave packets interacting in triads, and N point particles with repulsive potentials.

(i) The two problems are Hamiltonian, with Hamiltonians given by (26) and (27). In each case, the Hamiltonian is the sum of N free-streaming terms, plus interactions.

( ii) Every particle of the j th species is characterized by mj; every

packet in the j th wave mode is characterized by ( kj, w(k j), c j). These labels are time-independent.



296 HARVEY SEGUR

(iii) The evolution of the nth particle is described by [~n(t),~n(t)], in a six-dimensional phase space (for each particle). The evolution of the nth

-> wave packet is described by an ( x, t), in an infinite-dimensional phase space. This· difference in dimensions is one of the major disparities between the two problems.

(iv) When the nth particle is spatially distant from all of the other particles, it achieves straight-line motion, according to

When the wave packets with which the nth packet resonates are not spatially overlapping, then the nth wave packet achieves straight-line motion, according to

(28)

(v) The simplest interaction of particles is a binary collision, with N=2 in (27). The simplest interaction of wave packets is a single triad, with N=3 in (26). Both are integrable. Both may be considered "scattering problems" in the sense that the final states (t++ oo) are completely determined by the initial states (t+- oo),

It is well known that Boltzmann's kinetic theory of gases is derived from the N-body problem, (27), when the gas is dilute. Using this analogy between particles and wave packets, we now construct a kinetic theory of wave packets that parallels Boltzmann's theory for particles.

c. DILUTENESS. The additional assumption that we must impose on the Hamiltonian system, (23), is that the wave packets are dilute. Let An denote the set of all nonzero wavepackets with which packet /In interacts directly, according to (23). Let ~n(t) denote the fraction of physical space covered by one or more of the nonzero wave packets in An at time t. (More generally, thi~ fraction should be computed over any sufficiently large volume of space, but we ignore this embellishment here.) We say that the wave packets are dilute at time t if




The point here is that in a dilute system, each wave packet spends the

vast majority of its time in free flight, (28), occasionally interrupted by

triad interactions. The likelihood of its participating in a triad interaction at any given time is small. The likelihood of its participating in two or more triads simultaneously can be ignored. Thus in a dilute system, virtually all interactions occur as a sequence of individual triad interactions.

Notice that the requirement for diluteness is not that wave packets

rarely meet, but only that wave packets that can resonate with each other according to (23) rarely meet. Thus one can imagine a system that is rather

noisy in the sense that nonzero wave packets are observed nearly everywhere, and yet is dilute in the sense used here.

According to this definition of diluteness, not only may we disregard all multi-triad interactions in a dilute system, but even the single triad interactions must be of a special type. A typical interaction begins when two resonant wave packets first overlap. Because of diluteness, it is unlikely that a wave packet from the third member of that particular triad will be co-present. Once the interaction starts, however, all three packets share the available energy, according to (16). Thus, a typical interaction is a dilute

system is like the one shown in Figure 2: two nonzero wave packets come in, three nonzero packets go out.

o', a',

Interaction Region

o, o,

Figure 2. In a dilute system, a typical triad interaction begins with one wave packet having zero energy.



298 HARVEY SEGUR

The special nature of triad interact ions in a dilute gas ( 2 in + 3 out)

has two important consequences.

(i) Virtually every interaction increases the number of nonzero wave packets

and it makes the gas of wave packets slightly less dilute. It follows that

the assumption of diluteness must eventually fail, so the kinetic theory

developed here is relevant only on a certain time-scale. As a practical

matter, the eventual failure of this theory is not as crippling as one might

fear: the approximation scheme leading to (23) already has imposed one

restriction on time-scales; diluteness now imposes another one.

( ii) A critical assumption in Boltzmann's kinetic theory is his

Stosszahlansatz. Our Stosszahlansatz is equally important, and it must

carry the information that a typical triad interaction in a dilute system is

( 2 in + 3 out) .

D. THE BBGKY HIERARCHY. A systematic derivation of Boltzmann's equation from

the Hamiltonian N-body problem, (27), was developed independently by

Bogoliubov, Born, Green, Kirkwood & Yvon, and is now known as the BBGKY method

[23]. We use here a generalization of that method, beginning with the

Hamiltonian system (23). However, the phase space for (23) is

infinite-dimensional, so a volume element in the phase space is not defined a

priori. Until the volume element has been defined precisely, the derivation

which follows should be considered only formal. A method is proposed in

Appendix A to define an infinite-dimensional volume element as the limit of a

convergent sequence of finite-dimensional volume elements, but the details

have not yet been carried out.

At any fixed time, T, the phase space for (23) is coordinatized by N sets

of conjugate variables (pn(x,T), qn(XoT)) defined by (24). Let zn=(pn,qn).

Among other things, the coordinates of the nth wave packet specify its instantaneous location is physical space. Because (23) is hyperbolic, the

packets move with finite speeds, so the region of physical space covered by

these packets remains finite over any finite time- interval. Let V denote the

physical volume of a ball that is big enough to contain the entire interaction

process over the interval in question.

Consider an ensemble of copies of (23), each of which may have different

initial conditions. Every set of initial conditions is one point in phase

space. The ensemble of solutions of (23) that evolves from this ensemble of points is represented by a probability distribution functional,

FN(z1 ,z2 , ... ,zN;T), which satisfies F N ( z 1, ••. , ZN; T) 0, (29)




Moreover, FN satisfies Liouville's equation,

where {•,•}and Hare defined by (25) and (26), respectively. There is no loss of information in this description: Liouville's equation is a hyperbolic system, whose characteristics are (23). The deterministic problem is recovered by making FN a o-function on the phase space.

Define

f1 Cz1 ;r) is the probability density of finding wave packet #1 at the point z1 in phase space at time t. The one-packet distribution functional for each of the n wave packets is defined analogously:

(31)

dzn missing

These are normalized so that

(32)

Following the usual derivation of Boltzmann's equation, we now assume that FN is symmetric with respect to interchanges of packets within the same wave mode. Then all the packets corresponding to a single wave mode (i.e., a single "species"), are represented by the same one-packet distribution functional. If (23) describes the interaction of J wave modes, there will be J such functionals.

We also need three-packet distribution functionals. For j<k<i, define

(33)

dzj dzk dz~ missing



300 HARVEY SEGUR

Five-packet distribution functionals are defined similarly. No other partial distribution functionals are needed here.

The evolution equation for each of the J one-packet distribution functionals is obtained by integrating Liouville's equation over the other

coordinates. For fj (zj 0r), we integrate (30) over dz1. Hdzj-l .dzj+l .•. d~, requiring that FN vanish on the boundaries of the domain of each zk, and that it vanish as lxl~ m The result may be written as

0

In (23), wave mode #j participates in Mj triads so

The sum in (34) is taken over these Mj triads, and each such model triad is counted Nk times, because there are Nk packets in the k th mode. Fjk~(zj,zk,z~; T) represents the appropriate three-packet distribution functional for each of the triads in question. The J equations like (34) are the first level in the BBGKY hierarchy.

We also need evolution equations for each of the three-packet distribution functionals in (34). These are also obtained by integrating over (30) appropriately. The result for F123 (z1,z2,z3;t) may be written as

(35)



TOWARD A NE;l KINETIC THEORY FOR RESOl'IA.NT TRIADS 301

Here H123 is the Hamiltonian for the single triad,

the sums in (35) involve every other triad in which wave packets #1 , 2 or 3

participate, and F 123rs is the appropriate five-packet distribution

functional, linking the two triads, (123) and (jrs), where j=1,2 or 3. There

is an evolution equation like (35) for every three-packet distribution

functional.

No approximations have been made to this point, beyond those required to

derive (23). Liouville's equation, (30), is equivalent to (23), while (34)

and (35) are exact consequences of (30). In the case of precisely one triad,

N=3, the last term in (35) vanishes, and (35) and (30) become identical.

E. TRUNCATING THE HIERARCHY. Next we introduce two assumptions, only one of

which is essential. The nonessential assumption is that the medium is

spatially homogeneous, so that spatial gradients vanish in (34),(35) and (36).

Now (34) reduces to

M

a, fj(zj,t)+V-2 t Nk [ctzkdz2, { Fjk2.•Hjk2.}=o, k

(37)

for j=1,2, ... ,J. The purposes of this assumption are to simplify the presentation and to obtain the simplest kinetic theory possible.

The assumption which is essential to obtain the kinetic theory is that N is large, V is also large, and the system is dilute. (Readers familiar with the usual kinetic theory of gases will recognize

as the diluteness parameter in the BBGKY formalism. If this parameter is

multiplied by the volume occupied by a typical wave packet, then the product

represents the fraction of physical space occupied by the wave packets with



302 HARVEY SEGUR

which a packet from mode j can interact. It may seem puzzling at this stage that the coefficient of the integrals in (35) and (37) is apparently (V-2)

rather than (V-1), but this factor works out correctly in the final model, (49)). Because of diluteness, the integral terms the integral terms in (35) and (37) are small, and may be neglected at leading order. (Here is where "course-graining" enters, where we lose information, and where the

deterministic problem (23) is reduced to a probabilistic one.) Now (35)

becomes approximately

(38)

i.e., at leading order, each three-packet distribution functional is affected

only by the triad interaction of those three packets, while multi-triad interactions are ignored.

At the same leading order, (37) becomes trivial. We must proceed to the

next order to obtain a nontrivial kinetic theory. (In effect, therefore, we now introduce a new, longer time-scale.) To do so, it is convenient to rewrite

(37) as

Mj

a,fj(zj,T)+ v-2 LNkfzkdz'Ldzm O(zm-zj) {Fjk'L'Hjk'L}=o, (39) k

so that each integral is taken over the entire phase space of a single triad. But the equations for a single triad are jntegrable, as discussed in §2, and this permits further analysis of (39).

To this point the integrals over phase space have been only formal. In Appendix A, a method is proposed to use the integrability of (16), the equations for a single triad, to define precisely the volume elements in (39).

The basic concept in Appendix A is that an integrable Hamiltonian system has a complete set of action-angles variables, which may be used conveniently to coordinatize the phase space.

Consider next a slight variation of the usual action-angle variables (which are assumed to exist and to be countable, as discussed in Appendix A). Because the Hamiltonian, (36), is a nontrivial constant of the motion, we may specify that it be one of the action variables. Denote its conjugate variable by T, so that



It follows that

TOWARD A NEW KINETIC THEORY FOR RESONANT TRIADS

{T,H}=L

dT .. 1. dt

303

(40)

Now construct a complete set of "action" and "angle" variables, using H and T as the first elements of each set. All the rest of these variables are in involution with H by construction, so they must all be time-independent. Lump these time-independent variables into 11. Now the magnitude of each volume element in (39) may be written as

(41)

The Poisson brackets in (39) also simplify in these variables:

(42)

It follows that in every integral in (39), the T-integral may be performed:

(43)

Note that a sign(!:) was lost in (41). It was recovered in (43), but without justification. The simplest justification of this choice is that the resulting kinetic equations, (45), are sensible with this choice and senseless with other choice.

Now T=t+T0, because of (40), sot+~ w when T+~ w, Therefore

FjkR. (zm,zk,ZR.;t) I .. FjkR. (zm,zk,zR.;+ w) i T=+ w

i.e., this is a three-packet distribution functional, evaluated after the



304 HARVEY SEGUR

completion of a triad interaction. But (16) is integrable, so there is a

unique point (z~,zi<,zi) in phase space such that the trajectory that began at

(z~,zk,zp is the distant past will end at (zm,zk,z2.) in the distant future. (This special point may be found by running the IST solution of (16)

backwards, so that final data~ initial data.) Because of (38),

(44)

Substituting (43) and (44) into (39) yields the N evolution equations for

the one-packet distribution functionals in a dilute system. For j=1, ••• ,J,

M.

a./j(zj,-r) = V-2 t Nk foHjk2.dlljk2. o(zm-zj)* k

(45)

where (z~,zk,zl) are defined to be the unique initial data for that triad that

result in (zm,zk,z2.) as final data. The interpretation of (43) is completely

analogous to that for Boltzmann's equation. It says that the probability

density of finding a wave packet from a mode #j at the point zj in phase space is affected only by two kinds of triad interactions. Those interactions that originate somewhere and terminate at (zj ,zk,z2.) increase the probability of finding mode #j at zj, while those that originate at (zj,zk,z2.) and terminate elsewhere decrease this probability.

F. THE STOSSZAHLANSATZ. To this point, the derivation given here parallels

as closely as possible the usual BBGKY derivation of Boltzmann's equation.

The final step in the derivation is to make an assumption about the

probability density of three packets that are about to participate in a triad

interaction; i.e., a Stosszahlansatz. Following Boltzmann, we assume that

the three packets are statistically independent before they begin to interact:

( 46)

As discussed in Appendix B, if the one-packet distribution functionals are not singular, then this assumption leads to a kinetic theory for the resonant triads that is quite similar to Boltzmann's kinetic theory for J species of




molecules. In particular, there is a Maxwell-Boltzmann type of equilibrium

solution for the system, and there is an H -theorem which asserts that

non-equilibrium distributions that are not singular are always driven to

equilibrium.

Unfortunately, ( 46) is untenable for a dilute system of wave packets

interacting in triads. Recall that in a dilute system, a typical interaction

begins when two nonzero wave packets first overlap; the third member of the

triad typically has zero amplitude initially. But (46) does not reflect this

fact. It assumes that all initial conditions for a triad are equally likely, whereas the typical initial conditions in a dilute system have one wave packet with zero amplitude.

The assumption of diluteness has been used once, to truncate the BBGKY hierarchy. It now enters again, to restrict the form of the one-packet

distribution functionals allowed in (45) and (46). Recall that fj(zj) is the probability density that a "typical" wave packet from the j th mode will be

found at zj. Because there is a finite likelihood that the wave packet in question will have zero amplitude in a dilute system, we set

( 47)

Wh r >o 1 '>o d e e g j - , :JJ - an

( 48)

Moreover, the integral in (48) is smaller than gj by a factor on the order of the (small) diluteness parameter.

The artifice of using "ghost" packets at zjzO requires that (48) be interpreted with care. The number of ghost packets included in our original description determines gn(t=O), and this is somewhat arbitrary. Consequently, the probability density of the real packets, 6n<zn, t), carries an arbitrary parameter, gn (0). The resulting evolution equation, however, is independent of this arbitrary parameter.

Substitute (46) and (47) into (45). The result is an equation that involves terms that are cubic in gn(t), others that quadratic in gn(t), others that are linear in gn(t) and two terms that are independent of gn(t). Some of these terms cancel identically. Note that



306 HARVEY SEGUR

i.e., if all three packets have no energy after the interaction, then they must-have had none before the interaction, so there was no real interaction at

all. Consequently the two terms under the integral that are cubic in gn ( t) cancel identically. Similarly,

for the same reason: this "interaction" involves only one nonzero wave packet

(~), so it is not a real interaction at all. Thus, all of the terms that are quadratic in gri(t) also cancel in pairs.

The two terms under the integral that are independent of gn(t) represent

triad interactions with all three incoming wave packets nonzero. We may neglect these terms because of diluteness. Finally, using diluteness again,

we have from (48) that gn(t)-v in the terms that remain.

After all of these simplifications have been made, and the explicit o-function in (45) has been evaluated, the final result is a kinetic model for a dilute system of J wave modes, interacting in resonant triads in a medium

that is spatially homogeneous. For j=1, ••• ,J

M

a1 6j<zj,t) = ~(~k) ~ Hjki ct ojki [oj6k6i +

k

M.

ddt gj{T) =- t(;~) fdHjki dlljk2. 6k(zk,t)6i(zi,t). k

(49)

(50)

Here 6j=1j(zj,t), 6j=6j<zj,t), oj=o(zj>. etc., 6j(zj,t) is the nonnormalized probability density of finding a wave packet from mode flj at the point zj in phase space at time t,(zj,zk,zi) is the unique initial point in phase space for which (zj,zk,zi) is the final point, the volume element is defined by (41), and the sum is takes over the Mj triads in which wave mode flj




participates. Note that the diluteness parameter in the final model is

(v-1LNk), as anticipated. The kinetic model defined by (49) is the main

result of this paper.

The system of equations (49) determinesthe evolution of the probability

densities of the nonzero wave packets in the various modes. Once these are

known, then (50) determines the rate at which the ghost packets (at zj=O) are being depleted. These subsidiary equations, (50), serve two purposes.

(i) Eqn. (50) shows that gj(1) decreases monotonically, so the process is

irreversible. In a sense, therefore, (50) is a kind of H-theorem.

(ii) The model defined by (49) remains valid only so long as the system

remains dilute, and gj(1)•N. Integrating (50) provides an estimate of when

the assumption of diluteness fails.

The system (49) admits a trivial equilibrium solution,

j=1 •••• ,J. (51)

This solution is also unstable. It seems unlikely that (49) admits any stable

equilibria, because (49) describes the evolution of a dilute system as it

becomes less and less dilute. Hence the natural equilibria should lie in the

nondilute limit, but in this limit (49) is no longer valid.

A more difficult question is whether the transient process described by (49) achieves a limiting form as 1~"'. If so, then this limiting behavior

might be as useful in applications as the Maxwell-Boltzmann distribution has

proved to be in Boltzmann's problem. Whether such an asymptotic form exists for (49) is open.

To summarize, (49) is a new kinetic model of a dilute gas of isolated

wave packets that interact in triads. It models a transient process in which the gas of wave packets becomes less and less dilute. If the solutions of (49) tend to a limiting form as 1~"'. then this limiting behavior could have important physical applications. Whether such asymptotic behavior exists for wide classes of initial data is open.

ACKNOWLEDGEMENTS. The author is grateful for many helpful conservations with

Martin Kruskal, Dave McLaughlin, Guido Sandri and Ian Sykes.



308 HARVEY SEGUR

APPENDIX A - AN INFINITE DIMENSIONAL VOLUME ELEMENT

According to (38) and (39), in a dilute system of wave packets the

important infinite-dimensional space is the phase space of a single triad,

( 16). Here we propose a method to define precisely a volume element in this

space. The definition is only a proposal, because it depends on assumptions

that have not yet been verified.

The triad equations (16), are integrable in the sense of being solvable

by IST [20]. Virtually all of the equations that are integrable in this sense

are also integrable in the sense of having a complete set of action-angle

variables [24]. These are canonical variables with the properties that the

action variables are time-independent, while the corresponding angle variables

change linearly in time. Such a formulation has not yet been given for the

triad equations, ( 1 6), but presumably there is no fundamental difficulty in

doing so. In any case we now assume that ( 16) has a complete set of

action-angle variables, which coordinatize the phase space.

Kaup [20] showed that (16) has a countably infinite set of conservation

laws. In other problems solvable by IST (e.g., the KdV equation), there is a

countable set of motion constants that are obtained from the conservation

laws, that are in involution, and that are functionally independent.

Furthermore, there is a meaningful class of initial data for which this

countable set of motion constants can be used as the action variables in an action-angle formulation. We now assume that for (16), there is a meaningful

(e.g., dense) set of initial data such that for any initial data from this

class, a countable set of action-angle variables exists. Moreover, we assume

that these variables can be constructed explicitly, using the infinite set of

conservation laws.

Subject to these assumptions, we may define a volume element for the

infinite-dimensional phase space of (16).

( i) Any finite truncation of the infinite set of action-angle variables

provides coordinates for a finite-dimensional phase space, corresponding to an

extremely restricted class of initial data for (16). A volume element in this

finite-dimensional phase space is well defined.

( i i) A volume element in the infinite-dimensional phase space of ( 1 6) is

defined to be the limit of a sequence of volume elements from these



TOWARD A KINETIC THEORY FOR RESONANT TRIADS 309

finite-dimensional spaces. Because there is a meaningful class of initial

data for which this sequence of truncated conservation laws converges, this

limit should exist for appropriate initial data.

APPENDIX B - THE BOLTZMANN STOSSZAHLANSATZ

Assume that the three-packet distribution functional before an

interaction is given by (46). Substitute this into (45), and obtain a closed

set of kinetic equations. For n;1, ••• ,J,

Mn

v-2 LNk /dHk'X.n dOk'X.n Ui/if~-fkf'X.fn] k

(B1 )

where fk=fk(zk,1), fk=f(zk,T), etc., (zk,z2_,z~) are the unique initial data

that evolve into (zk,z'X.,zn) as the final data, the surr in (B1) is taken over the Mn triads in which wave mode #n participates, and the integration over the

6-function in (45) has been performed. We show here that the consequences of (B1) are quite analogous to those of Boltzmann's equations for a multi-species

system of particles.

Consider first the equilibrium solutions of (B1). One way to achieve

equilibrium is by detailed balancing, i.e., by requiring that within each integral in (B1),

(B2)

Precise knowledge of the regions of integration in (B1) is unimportant for these equilibria, because the integrand vanishes everywhere. suggestive form of (82) is

A more

Now it is clear that (83) may be solved by any functional of the solution of a single triad that: (i) is additive; (ii) is symmetric in the three wave

modes involved in the triad; and (iii) attains the same constant value before and after a collision.



310 HARVEY SEGUR

The simplest solution of (83) is the constant solution: for any m,

constant. (84)

Other solutions of (83) come from the conservation laws for a single

triad. The number of conservation laws is infinite [20], so the number of

solutions of (83) may also be infinite. In any case, we present here

solutions of (83) obtained at the lowest two levels of the conservation laws.

At the lowest level, there is an "energy integral," corresponding to (18):

Here we have reverted to the less abstract notation of (23). Note that every

term is (85) is positive, by (14). It follows that (83) is satisfied if we

choose

2 , l+ wmlaml -Blx _r_m_, m=k,i,or n, (86)

where B is an arbitrary positive constant that eventually plays the role of an

inverse temperature. This solution is analogous to the Maxwell-Boltzmann

distribution for dilute gas of particles.

The appropriate constant at the next level is the Hamiltonian, (26). The non-additive interact ion terms in (26) vanish before and after the

interaction, so another solution of (83) is

(87)

All of these solutions may be superposed, of course, because (83) is linear.

A pattern is suggested by writing (86) and (87) in terms of the Fourier

transform, am<k), of am<x):

(B6')



TOWARD A KINETIC THEORY FOR RESONANT TRIADS 311

(B7')

Whether this pattern persists at the higher level conservation laws is

unknown.

For completeness, one other equilibrium solution of (B1) should be

mentioned. For every n, let

(B8)

i.e., wave mode /In has zero amplitude (everywhere!) with a probability of 1.

Since all three modes have zero energy before the collision, there is no

energy to share, so they also have no energy after the collision, and

z' k z' Q, z' n 0

Thus (B8) represents a trivial solution of (B1) when there is no energy.

[There is a corresponding trivial solution of Boltzmann's equation, which

is easiest to envision when the molecules have short-range potentials. If the

molecules are widely separated and there is no motion, then there are no

actual collisions, Pi i)2 = 0, and f(p)=o{p) is an equilibrium

solution of Boltzmann's equation. Note that this distribution is not driven

to a Maxwell-Boltzmann distribution].

Next we show that (B1) has an H-theorem which is the direct analogue of Boltzmann's famous theorem. Define the (negative) entropy of the system to be

H(-r) N

~ j(ctznfn(zn,<) Q.n fn(zn,<). n=1

By direct computation, using (32),

dH -=

(89)



312 HARVEY SEGUR

Now use (B1), and the usual arguments of symmetry of the cell is ion terms with respect to interchange of arguments [23]. The final result is

N -tE (B10) n=1

Now the right-hand side is negative unless (B2) is satisfied. Then, because His bounded below, one shows by the usual argument that every probability distribution for which H is finite is driven by (B1) to an equilibrium distribution satisfying (B2).

Bibliography

·1. Peierls, R.E., Quantum Theory of Solids, Clarendon Press, Oxford, 1955.

2. Bloembergen, N., Nobel Prize Lecture, Dec. 8, 1981, Stockholm, reprinted in Science, Vol. 216, 1982, pp. 1057-1064.

3. Phillips, O.M., The Dynamics of the Upper Ocean, 2nd ed., Cambridge Univ. Press, London, 1977.

4. Davidson, R.C., Methods in Nonlinear Plasma Theory, Academic Press, New York, 1972.

Tsytovich, V.N., Nonlinear Effects in Plasmas, (translated), Plenum Press, New York, 1970.

5. Kaup, D.J., A. Reiman & A. Bers, Rev. Mod. Phys., Vol. 51, 1979, pp. 275-310.

6. Benney, D.J. and A.C. Newell, J. Math. and Phys. (Stu. App. Math.), Vol. 46, 1967, pp. 133-139.

1. Watson, K.M., B.J. West & B.I. Cohen, J. Fluid Mech., Vol. 77, 1876, pp. 185-208.

8. Hasselmann, K., Rev. Geophysics, Vol. 4, 1966, pp. 1-32.

9. Olbers, D.J. & K. Herterich, J. Fluid Mech., Vol. 349-379.

92' 1979, pp.

10. Ablowitz, M,J. & H. Segur, />4.2 of Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, PA, 1981.

11. Simmons, W.F., Proc. 551-575.

Royal Soc. London, A, Vol. 309, 1969, pp.

12. Whitham, G., Linear & Nonlinear Waves, Wiley-Interscience, New York, 1974.




13. Cole, J.D. & J. Kevorkian, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, 1981.

14. Armstrong, J.A., N. Bloembergen, J. Ducuing & P.S. Pershan, Phys. Rev., Vol. 127, 1962, pp. 1918-1939.

15. Arnold, V.I., Mathematical Methods in Classical Mechanics, Springer-Verlag, N.Y, 1978.

16. Zakharov, V.E. & S.V. Manakov, JETP Letters, Vol. 243-245.

18, 1973, pp.

17. Ablowitz, M.J. & R. Haberman, J. Math. Phys., Vol. 16, 1975, pp. 2301-2305.

18. Cornille, H., J. Math. Phys., Vol. 20, 1979, pp. 1653-1666.

19. Niznik, L.P., Ukranian Math. J., Vol. 24, 1973, p. 110 (in Russian), ---,Naukora Dumka, Kiev, 182 pp., 1973.

20. Kaup, D.J., Physica, Vol. 1D, 1980, pp. 45-67.

21. Benney, D.J. & P.G. Saffman, Proc. Royal Soc. London, A, Vol. 289, 1966, pp. 301-320.

22. Benney, D.J. & A.C. Newell, Stud. App. Math., Vol. 48, 1969, pp. 29-53~

23. Grad, H., "Principles of the Kinetic Theory of Gases," Handbuch der Physik, Springer Verlag, Berlin, Vol. 12, 1958, pp. 205-294

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Aeronautical Research Associates of Princeton 50 Washington Road, P.O. Box 2229 Princeton, New Jersey 08540







A SPECTRAL METHOD FOR EXTERNAL VISCOUS FLOWS

Philippe R. Spalart1

ABSTRACT. A spectral method is presented and applied to the numeri-cal solution of the Navier-Stokes equations over a flat plate. The general approach for the treatment of the far-field could be applied to other, relatively simple, external geometries. The flat plate is treated with Fourier series in the horizontal directions and an exponential mapping, combined with Jacobi polynomials, in the verti-cal direction. Leray's weak formulation of the incompressible Navier-Stokes equations is used; it eliminates the pressure term. In one- and two-dimensional tests the method proved very accurate; exponential convergence was observed and the error level for a given number of terms was low. The three-dimensional version of the pro-gram will be used to simulate the transition of a boundary layer to turbulence at moderate Reynolds numbers.

1. INTRODUCTION. Spectral methods are now commonly used for the numerical

simulation of viscous fluid flows, provided that the geometry is simple. Their

most successful application is in the simulation of turbulent flows, especially

homogeneous turbulence for which periodic boundary conditions are physically

acceptable. Because of mathematical difficulties (1), the application to more

realistic, nonperiodic geometries has progressed slowly. This paper describes one attempt to apply a spectral method to the flow over a flat plate. Periodic conditions are retained in the directions parallel to the plate. The treatment of the semi-infinite y-direction is new.

The main advantage of a spectral method is that the derivatives of a func-tion can be evaluated more accurately by differentiating a spectral expansion term by term than by taking, for instance, finite differences (2). Thus, for the same number of degrees of freedom, the rapidly varying components of the solution are better represented. A common estimate is that twice as many

degrees of freedom would be required, in each direction, for a finite-difference simulation to achieve the same accuracy as the spectral simulation (this dif-ference might be reduced by the use of more modern Fade differences.) A well-

designed spectral method also achieves "exponential" convergence, provided that

1980 Mathematics Subject Classification: 35-plO. 1Supported by the National Science Foundation.

315

© 1984 American Mathematical Society 0271·4132/84 $1.00 + $.25 per page




316 SPALART

the exact solution itself is infinitely differentiable. The formal accuracy is

much superior to the accuracy of finite-difference methods. This advantage is

rarely used in practice.

One weakness of spectral methods is that it is often difficult to find

expansion functions that match a given domain; finite-difference methods,

finite-element methods and, a fortiori, Lagrangian "grid-free" methods are much

more versatile. The usual way to undertake a new type of domain is to map it

into one that is more amenable; for instance, polar coordinates map a circle

into a rectangle (1). However, the derivatives are then multiplied by metric

coefficients that depend on the mapping and vary L< space, and the behavior o;

these coefficients strongly influences the efficiency of the resulting method.

A spectral method makes the derivative computed at one point depend on the

values of the function over the whole domain, which is not always natural.

This global character of the expansion has some numerical disadvantages: in

some cases it prevents the use of an implicit time-advance scheme, especially

when nonlinear terms are involved. The brief review that follows is limited to

the simulation of incompressible viscous flows. These flows are a good applica-

tion, first because the solutions are regular, which promotes the accuracy of

spectral methods, and also because the whole flow field is physically coupled

by the instantaneous pressure interactions. Thus, the global nature of the

expansion is more natural than it would be for a compressible flow, in which

finite domains of dependence exist.

Fourier spectral methods are now the standard tool when periodic conditions

are used. Fourier series are accurate, convenient, and only slightly more

costly than the other methods. Spectral methods are also well established for

the treatment of bounded intervals; most often, the solution is expanded in

terms of Chebyshev polynomials. Chebyshev polynomials can resolve boundary

layers at the ends of the interval very well ("boundary layer" is taken in the

mathematical sense here.) Also the transforms from spectral space to physical

space can be reduced to Fourier transforms, so that they are done in

O(N log(N)) operations. Chebyshev methods have been applied to the flow in

channels, between straight walls (3), and, more recently, between curved

walls (4,5).

Few other flows have heen treated: the flow in a circular pipe is treated

by Orszag and Patera (3) and by Leonard and Wray (6), with rather different

methods. The flow in a boundary layer (here taken in Prandtl's sense of the

viscous shear layer of a fluid along a wall) was treated by Orszag (7) with an

algebraic mapping and Chebyshev polynomials, but not in as much detail as the

other flows. Cain et al. treated a time-developing mixing layer (8). These

are the only two unbounded inhomogeneous flows that have been simulated by

fully spectral methods so far. Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.


EXTERNAL VISCOUS FLOWS 317

The problem of transition and turbulence in a boundary layer is of major

interest in aeronautics and is far from being solved. Extensive direct or

"large-eddy" simulations will answer some of the questions; how soon these simu-

lations will be possible remains to be determined. Only incompressible simula-

tions are planned. The minimum Reynolds number at which turbulence can be

sustained is also unknown.

Time-growing boundary layers will be simulated. Spatially growing boundary

layers are more realistic, but much harder to simulate numerically. Wray and

Hussaini showed that a time-growing simulation, with periodicity in space, can

give results very close to the spatially-growing ·experiments, which were peri-

odic in time (9). The nonlinear breakdown of a boundary layer seems less sensi-

tive to the type of growth than the linear stability of small disturbances. Wray and Hussaini could only simulate the early stages of breakdown, after which

the numerical resolution became too coarse. It is planned to use grids at least

twice as fine as those used by Wray and Hussaini.

Section 2 describes a one-dimensional example that will introduce the key

ideas to be applied to the treatment of unbounded domains. Section 3 presents

Leray's weak formulation of the equation; section 4 describes the treatment of

the infinite domain for solution of the Navier-Stokes equations; and Section 5

describes some numerical aspects of the method. Section 6 contains the two-

dimensional results and section 7 presents the conclusion.

2. A ONE-DIMENSIONAL EXERCISE WITH CHEBYSHEV POLYNOMIALS. This section

describes a method for the solution of the heat equation over [Q,oo] when the

disturbances are confined near the wall and decay quickly as y + oo This situ-

ation resembles a boundary layer. The equation to be solved is

with boundary and initial conditions

u(O,t) = uw(t) u(y,O) = uo(Y)

It is assumed that the initial value uo(y) decays faster than any exponential of y:

for any n

1

2

3

Using the fact that the fundamental solution of the heat equation is a Gaussian,

it can be shown that the solution for all times also decays faster than any exponential:

u(y,t) a(e-ny) for any n 4



318 SPALART

This point will be essential to the accuracy of the method.

Orthogonal polynomials are convenient as "building blocks" for the con-

struction of basis functions. They allow one to minimize the coupling between

different modes, have convenient recurrence and differentiation relationships,

and are inexpensive to evaluate on the computer. Many different families are

known, depending on the weight functions, and the user can choose among them

according to the type of functions that are to be approximated.

However, if an unbounded interval is considered, a straightforward use of

polynomials is not recommended (2); all polynomials tend to as y + "', in

contrast with the decay to 0 that is expected from the solution. This is why a

domain like [O,oo] is almost always mapped into a finite interval (7). Poly-

nomials in terms of the transformed variable are then used, and they behave

well. The exponential mapping

n exp(-.1L) Yo

will be used here. The parameter Yo is the length scale in the y-direction.

The metric coefficient is

dn dy

n Yo

it has a very simple expression in terms of n, which reduces the coupling

between modes.

The mapping is necessarily singular at n + 0, y + "'· As a result, a function that is regular (for instance c.,) in terms of y might not be regu-lar in terms of n near n = 0. On the other hand, it has been shown that for exponential convergence, using polynomials, the solution must be c., in terms of n (2). Therefore, it is essential to examine the regularity of u(n). The general argument that follows was used by Orszag, although his mapping was different (7).

Equation 4 can be rewritten as

-n u(n, t) = o(n )

that is, u(n) is smaller than any power of its derivatives tend to 0 as n + 0, so that closed interval [0,1]. The expansion of u

therefore, result in exponential convergence.

for any n

n near n = 0. u(n) is actually

Therefore, all C over the "'

in terms of polynomials will,

Orszag actually mapped [O,oo] into [-1,1], used Chebyshev polynomials in the usual manner, and obtained rapid convergence for an eigenvalue. However it will now be shown that a more efficient procedure exists. Figure 1 shows a typical distribution of u in terms of n. The function is extremely smooth

5

6

7




near one end of the interval, since all its derivatives are 0. In contrast, the

Chebyshev polynomials as used by Orszag concentrate resolution at both ends of

the interval, as indicated by the density of zeros of the polynomials (Fig. 2).

This resolution is wasted in the neighborhood of -1. An alternative is to use only the odd Chebyshev polynomials, and to take

them only over half of the usual interval ([0,1] instead of [-1,1]). The

family of polynomials thus defined is still orthogonal. The first few of these polynomials are plotted in Fig. 3. From the position of the zeros, it is clear

that resolution is now concentrated only near n = 1, that is, at the wall. No

unnecessary resolution is "spent' on the very smooth part of the function, so

the same accuracy can be obtained with roughly half the number of terms.

To show that exponential convergence will still be obtained although a

smaller family of polynomials is used, one need only consider the function being

approximated as the restriction to [0,1] of an odd function defined on

[-1,+1]. This odd function is itself C00 over [-1,1]: all its derivatives

match at n = 0, since they are all 0. It is then clear that the odd Chebyshev polynomials form a complete set for the type of functions that are considered.

The even Chebyshev polynomials could also have been used, instead of the

odd ones. The completeness argument would be same, and in both cases it is

possible to reduce the transforms to a Fourier transform, so that a fast

Fourier transform can be used.

To check the method, the heat equation was solved with uo(Y) = 0 and

u (t) = 1. The exact solution is known analytically; it is the error function. w

The convergence of the solution after a given time, and with very fine time-

integration, is shown in Fig. 4. The error is plotted versus the number of

polynomials used. The convergence is clearly exponential, which validates the arguments presented in this section.

In summary, it is both possible and desirable, by carefully choosing the mapping and the family of polynomials, to fit the expansion functions to the expected behavior of the solution and use as small a family as possible. To retain fast convergence, the completeness of the set of polynomials used must be ascertained, and this is generally easier in terms of the transformed variable.

3. WEAK FORMULATION OF THE EQUATIONS. The equations to be solved are the incompressible Navier-Stokes equations. The independent variables are the coordinates X= (x,y,z), the dependent variables are the velocity vector

U = (u,v,w) and the pressure p. The differential equations are

au Clt + Q.VQ = -Vp + v~U 8



320 SPALART

.0 ~ 0.5 /

~ 1.0

Figure 1. Function of Transformed Variable

Figure 2. Chebyshev Polynomials

Figure 3. Odd Chebyshev Polynomials




'iJ.U = 0

The boundary conditions are

u 0 at solid walls

u + u at "' -<o

The far-field condition, Eq. 11, will be made more precise later. No boundary conditions are provided for the pressure. The role of the pressure term is to enforce the incompressibility condition (the pressure can be interpreted as a Lagrange multiplier associated with the incompressibility constraint, Eq. 9).

9

10

11

It does not introduce any infot.uation; thus, it would seem natural to eliminate it. This can be achieved either by taking the curl of Eq. 8 and working with the vorticity, or by solving the equations inside the subspace of velocity fields that already satisfy incompressibility and using a weak formulation. The second procedure is more efficient for numerical solution by a spectral method, mostly because the primitive variables are retained in the boundary conditions. This was recently shown by Leonard and Wray (6); the formulation used here is slightly different from theirs.

The weak formulation is due to Leray and is described by Ladyzhenskaya (10) and Temam (11). It is used for the mathematical theory of the Navier-Stokes equations. This theory is well developed for bounded domains; difficulties appear when the domain extends to infinity. For the rest of this chapter the domain will be considered as bounded; the only boundary conditions are of the type of Eq. 10. Unbounded domains will be considered later.

"' 'o .... 'b .... ... 'o .....

1-o 0 t'b

l'l:l,.....

.. 'o ....

0

i'o .... ~

b .... 0 10 20 30 40 50 60

Number of Terms

Fig. 4. Solution of the Heat Equation Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.


322 SPALART

The subspace V is the space of velocity fields that satisfy the incom-

pressibility condition, Eq. 9, and the boundary conditions, Eq. 10. It is a

Hilbert space; several inner products are used, one of which is

< ~.y > = f f f ~·Y dxdydz 12

If the solution is sought inside V , Eqs. 9 and 10 will obviously be satis-

fied and only the momentum equation, Eq. 8, remains. If the inner product of

Eq. 8 and a weight function V in V is taken, the result is

d dt < ~.y > + = -v < v~,vy > 13

The gradient of the total pressure has disappeared from the equation, because it

is orthogonal to y, for the inner product of Eq. 12 (this involves integrations

by parts). Since the weight function V can be any element of V, Eq. 13 is

sufficient to define d~/dt inside V and will replace Eq. 8 (10,11). Note

that there is a choice to keep or exclude any gradient term, for instance

17(~ 2 /2). It was decided to keep only the "U x w" part of the convection term

because this part is 0 in the irrotational region, which will make the quadra-

ture of easier.

In Leonard's formulation, the weight functions are taken in a slightly

larger space than V ; they satisfy only the normal velocity boundary condition.

This is sufficient to obtain Eq. 13 (except that Leonard did not integrate the

viscous term by parts). The two formulations are mathematically equivalent,

since the space V is dense inside Leonard's weight-function space (for the topology induced by Eq. 12), but there are significant numerical differences.

He has more freedom in choosing the weight functions; in particular, the weight

functions need not be as regular as the expansion functions. Moser et al. (5)

included the Chebyshev orthogonality weight function in the weight functions, so

that they are not C00 at the walls, and their results did not suffer. On the other hand, Leray's formulation has more symmetry (see below).

The formulation in terms of divergence-free vector fields has several

numerical advantages. When the usual formulation, Eqs. 8 and 9, is used, the

incompressibility condition is generally applied as a correction after the con-

vection terms and the treatment of the two boundary conditions (normal and tan-

gential) is delicate. With the divergense-free formulation. both boundary

conditions are treated simultaneously and it is easy to use an implicit time-

advance scheme for the Stokes terms. In addition, the memory requirements are

lower. Only two words per grid point are needed to store all the information. because of the divergence-free condition.

Leray's weak formulation also emphasizes the natural symmetry of the Stokes

equations. It involves two symmetric positive-definite bilinear forms: < ~.1,/ >




and < V~,V~ >. This symmetry is retained by the numerical method; the corre-

sponding matrices will be real, symmetric, and positive-definite. This guaran-

tees that the numerical Stokes eigenvalues will be real and negative, as are

the exact ones. Other formulations do not always yield real negative eigen-

values for the Stokes problem. There are other numerical advantages in having

such matrices: lower storage, stability of the Cholesky decomposition, and the

ability to diagonalize two matrices simultaneously.

4. TREATMENT OF THE INFINITE DOMAIN. The treatment of an infinite domain pre-

sents additional mathematical difficulties: some of the theorems used for the

existence and uniqueness proofs no longer hold; some function spaces that were

identical in the bounded case are no longer identical; and still other diffi-

culties stem from the fact that integrals like the energy might not be finite

(11,12). The problems are less severe, however, with time-dependent flows than

with steady flows. For instance, the energy of a flow cannot grow with time; if it is finite at time 0 it will remain finite.

For this study, all the functions will decay exponentially as y + 00 •

Thus, all the integrals will be finite. The question is: Are such functions

general enough to represent the flows under consideration? If they are not, Gibbs-type phenomena will be observed, and the convergence will be slow.

The velocity field is composed of a mean flow which is only a function of

y and has a boundary-layer character, and of disturbances that are periodic in

the x- and z- directions. "Boundary-layer character" is taken to mean that

the disturbances tend to 0 as y + oo and, in particular, that the vorticity

decays faster than any eponential of y. This condition is satisfied, for

instance, by the error-function solution that was obtained in section 2: the derivative is a Gaussian.

As time progresses, the vorticity will still have a Gaussian-type decay. This is due to the Gaussian shape of the fundamental solution of the heat equa-tion; the convection of vorticity occurs only on finite distances and does not influence the asymptotic behavior (we also assume that the boundary term and stretching term are bounded). Thus, the velocity field, away from the wall, is effectively irrotational. It is also divergence-free; therefore, it satisfies Laplace's equation.

The solution to Laplace's equation that is a pure Fourier component in the x- and z-directions, with wave-numbers y + oo is:

k and X

k2 , and is bounded as

i (k x + k z) X Z e 14



324 SPALART

This function decays exponentially in terms of y. In addition, if the velocity

is decomposed into a

u(x,y,z)

harmonic component uo and a vertical component

-y(kx2 + k 22 )~ '(k + k ) ~ xx zZ [u 0e + u 1 (y)] e

u0 can be chosen so that u1 , like w, decays faster than any exponential:

for any n

15

16

Therefore, the vertical component u1, like the heat-equation solution of

section 2, can be accurately represented by an exponential mapping and by poly-

nomials on the finite interval. Note that the fu-~tion representing the har-

monic part is not regular in terms of n near n = 0; some of its derivatives

are infinite (7). This does not prevent the spectral method from differenti-

ating it accurately. The only functions that have to be integrated numerically

are components of . These functions decay fast (faster than any power

of n) since w itself decays fast and since U is bounded. Therefore, they

are regular enough in terms of n for the Gauss quadrature to be accurate.

The basis functions used to represent u1 are n-times combinations of

shifted Jacobi polynomials. The orthogonality weight function of the poly-

nomials is n on [0,1], so that the basis functions are orthogonal with

respect to the weight function 1/n, which is the metric coefficient (Eq. 6).

Thus, they are orthogonal for the inner product, Eq. 12 (uniform weight in ter~s

of y). These polynomials were the most convenient and yield a satisfactory

distribution of points in the y-direction.

Numerically, the splitting given by Eqs. 15 and 16 is advantageous, because

the two components have different length scales in the y-direction. The har-monic component, driven by pressure forces, extends much farther (~ A, the

period in the x-direction, typically 206*) than the vortical component, which is

driven by viscous forces (~ 26*). Thus it is more efficient to have separate

functions. This advantage will be enhanced if larger A periods are used. In

Fig. 5, the first few basis functions are plotted, including the one that incor-

porates the harmonic component. The disparity of scales is evident. This

figure also indicates that Yo should be ~ 6* for the vertical component

(which is effectively 0 beyond 36*) to be well represented.

The exponential mapping minimizes the coupling of the derivatives of the

basis functions; thus, it is optimal in terms of matrix bandwidth. Similar

arguments were used by Cain et al. for the choice of their mapping (8). The

distribution of points in the y-direction also seems more favorable than when an algebraic mapping is used.

The operation count for transforming from wave space to physical space by

a Jacobi transform is O(N2 ), where N is the number of points in the y y




8

Figure 5. Basis Functions

y-direction. Chebyshev polynomials, which allow faster transforms, could not

be used with the symmetric formulation because their weight function is infinite

at the wall (in sec. 2 the heat equation was not solved with a symmetric formu-

lation). In addition, they seem to cluster too many points near the wall for a

boundary-layer transition study; fine resolution is needed at distances from

the wall of the order of 26*.

This concludes the description of the far-field treatment over a flat

plate. The same general procedure could be applied to the flow past a finite

body, assuming that the flow started from rest. Again, the velocity field away

from the body would be irrotational; in this case, however, the solution to

Laplace's equation behaves like negative powers of r as r ~ oo. A natural choice would then be to take polynomials in terms of 1/r as basis functions.

5. NUMERICAL ASPECTS. One can distinguish the spatial discretization and the

temporal discretization, although the two interact during the design of an

algorithm. Some aspects of the spatial treatment have already been described:

the representation is by Fourier series in the x- and z-directions and by

Jacobi polynomials in the y-direction. The plus and minus modes are used, as

described by Moser et al. (5). The plus modes are aligned with the wave-vector and the minus modes are orthogonal to it; the plus and minus modes are ortho-

gonal to each other, in the sense of Eq. 12. Moser's formalism, with h and

g functions satisfying appropriate boundary conditions, is used.

The solution of the Stokes part of the equation is done entirely in

spectral space. Various matrices are involved; their elements are inner pro-ducts of h or g functions and their derivatives. As mentioned, all the



326 SPALART

matrices are symmetric. The integrals are computed analytically and stored. In

addition, the h functions for the minus modes are orthogonalized with respect

to the products of h and the products of dh/dy; thus, all the matrices

involved are diagonal. This saves time and also allows the use of more accurate

techniques for the integration in time. The plus matrices are not diagonalized;

they depend on the wave number, and too much memory would be needed to store all

the eigenvectors. These plus matrices have nine bands; they are reconstructed

and solved by the Cholesky method at each step.

The time-advance scheme is hybrid: the Stokes term and the convection

term are treated differently. The Stokes term has large, real, negative eigen-

values (see sec. 6) and for such a stiff term an implicit scheme is necessary.

As mentioned, all the viscous matrices are diagonal, except in the y-direction

for the plus modes. This term is treated by the Crank-Nicolson scheme, which

is second-order accurate and A-stable. All the other terms are treated by the

technique of integrating factors, which is more accurate and has the advantage

of damping the fast-decaying components better than the Crank-Nicolson scheme.

An implicit scheme would also be desirable for the convection term. to

improve the stability. Unfortunately, no straightforward algorithm is available

that can treat this term implicitly without solving a full linear system,

involving all the terms of the expansion. Solving such a large system is

impractical. The matrix would be full because with a spectral method the non-

linear term couples all the basis functions.

For the convection term the Adams-Bashforth second-order scheme and a Runge-Kutta third-order scheme designed by Leonard were tried. The Runge-Kutta

scheme is explicit, third-order accurate. requires three evaluations of the

derivative, and is equivalent to the classical Runge-Kutta scheme for linear

equations. It has proved to be stable up to Courant numbers of about 1.7. so

the linear stability limit (/3) seems to apply.

Both schemes can be programmed with only four variables per grid point if

the program uses only the core memory of the computer, and seven variables if it

uses a disk. The Runge-Kutta scheme was chosen; it is at least as stable

(especially at high Reynolds numbers) and more accurate for the same cost. The

overall accuracy of the time-integration is only second order, since the inte-

gration of the viscous term is second order. The evaluation of the nonlinear term is done by collocation, and aliasing

is removed in all directions by the 2/3 rule. Simulations at high Reynolds

numbers are intended, and a strict control of the energy is desirable. For runs with 64 3 grid points, the code (written in Vectoral) takes about

13 sec per complete Runge-Kutta step on a Cray-1. The O(N2 ) transform is

several times slower than the fast Fourier transform and uses about half of the




central processing unit time. The solution of the Stokes term is much faster

then the evaluation of the convection term. In production runs there will be

less points in the y-direction than in the x- and z-directions, so that the

penalty for having a slow transform will be lessened. A better optimization of

the O(N2) transform also seems to be possible; the fast Fourier transform used

is very well optimized on the Cray-1.

6. TWO-DIMENSIONAL RESULTS. Two types of results will be presented: some

eigenvalues of the Stokes and of the Orr-Sommerfeld linear equations, and the

results of simulations of the full nonlinear Navier-Stokes equations.

The computation of eigenvalues is often used as a test for new numerical

methods. It allows the detection of major errors and of Gibbs phenomena, which

result in slow convergence. Thus, it is a test of the "healthy" character of

the system of expansion functions that was chosen. In many cases the exact

eigenvalues are available, or can be computed by solving a one-dimensional problem.

The eigenvalues are computed using as much material from the Navier-Stokes

code as possible: the same basis functions and the same collocation points.

The convergence of the numerical eigenvalues will be described as a function of

N, the number of collocation points used in the y-direction. The number of

polynomials is 2N/3. The value of the wave number in the x-direction is

kx = 0.3012, and the eigenvalues have been divided by kx, so that they have the dimension of a velocity.

As was mentioned earlier, with the formulation used here, the Stokes eigen-

values are guaranteed to be real and negative. Selected eigenvalues are plotted in Fig. 6, in absolute value.

In general, one expects the eigenvalues that are close to 0 to correspond to smooth eigenfunctions, so that the smallest eigenvalues should be the first ones to be computed accurately. The numerical results in Fig. 6 show that the

smallest eigenvalue decreases slowly as more degrees of freedom are used. The corresponding eigenfunctions are plotted in Fig. 7 for N = 40 and N = 140. It shows that the least-damped eigenfunction is very smooth in the y-direction and has its amplitude so far from the solid wall that it interacts very little

with it, so that is is damped only by the viscous term in the x-direction. The limiting value for the eigenvalue should then be -vkx. The value of vkx is

-3 0.58 x 10 , so that the convergence is very far from complete, even with 140 points. A comparison of Fig. 7 with Fig. 5 shows the reason for this slow con-

vergence: the expansion functions were not designed to approximate functions

that are still large at y/yo ~ 8 or more. The last collocation point is included in Fig. 7. It is at 6.1 for N = 40 and at 8.6 for N = 140; it



328 SPALART

"b .......

b .......

'b ....... Q)

::l'b (ij ...... > Q)"'b ~ ....... ...... ~00

.O"""' <~

'o ....... .. 'o .......

"' -'o .... 20

1

+ + +

1 LEGEND

o = Smallest Eigenvalue o = v/Ayl. " = Largest Eigenvalue + =Complete spectra

+ + + + +

40 60 60 100 120 Number of Points

Figure 6. Stokes Eigenvalues

LEGEND

4 Y/Yo

o = 40 points o = 140 points

6

Figure 7. Stokes Equation, Slow Decay

+ + +

+ + +

140

8

tends to infinity very slowly, and this accounts for the slow convergence of the

smallest eigenvalue.

Figure 6 also shows the largest negative eigenvalue, and the value of v/~y 2 , where ~y is the smallest interval between two points. They tend to

infinity at the same rate, which was to be expected since the most-damped eigen-

function corresponds to the shortest oscillations, and these have a wavelength

of the order of ~y. The corresponding eigenfunctions are plotted in Fig. 8. Finally, Fig. 6 shows the complete spectra for N 40 and N = 140. It

can be seen that the spectrum, while increasing in range towards zero and



ExTERNAL VISCOUS FLOWS 329

infinity, also becomes denser. It suggests that the numerical method is

attempting to reproduce a continuous spectrum. Actually, the exact spectrum is

known to be continuous.

In general, the results regarding the Stokes eigenvalues illustrate the

difference between an unbounded domain and a bounded domain; here the spectrum

is continuous, and there is no isolated eigenvalue for the numerical eigenvalues

to approach. This can give the appearance of slow convergence.

The same problem is present for some of the Orr-Sommerfeld eigenvalues.

The Orr-Sommerfeld equation is well known; it governs the stability of small

disturbances (13). Time-ampli:'' ~ modes were computed, that is, the wave-number

is real and the frequency is complex. The undisturbed state was defined by a

Blasius profile, a Reynolds number of Re = 520, and a wave-number of 0.3012,

the displacement thickness being 1. This is approximately the critical condi-

tion, at which disturbances first become unstable (14).

Figure 9 shows the spectrum for several values of N, including only the

eigenvalues with imaginary parts between -2 and 0. The eigenvalues were divided

by kx to provide the complex velocity. There are now a few isolated eigen-

values to which the numerical eigenvalues converge. Figure 9b indicates that

there are at least four discrete eigenvalues at this Reynolds number. The

values for the least-damped eigenvalue are the following:

N cr ci 30 0.3959183327 0.0000347207 50 0.3959190834 0.0000354410

100 0.3959190841 0.0000354410 150 0.3959190845 0.0000354407 200 0.3959190845 Q, Q0QQ354LIQ1

LEGEND o = 40 points o = 140 points

Figure 8. Stokes Equation, Fast Decay Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.


330

0 0

on 0

_.,I r... ltl A.

f::'o ltl ... t:1 '6h ltl

Eon j'

0 N I

0 c:i

0 N I

0.0

0.0

SPALART

0 [Jd'

0 0 0

0 0 0

p 0

·u

0

0

0.2 0.4 0.6 0.8 1.0 Real part

(a) N = 50

0 01 0 cflO 0

...n 0 0 p

[

0

0

0 0 0

0

0.2 0.4 0.6 0.8 1.0 Real part

(b) N = 200

Figure 9. Complex Velocity




The convergence of this eigenvalue is exponential. The reason is that this

eigenfunction has its amplitude at distances from the wall of order 1. It is

plotted in Fig. 10. Being located close to the wall, this eigenfunction is well

represented, and fast convergence follows. Convergence also seems to be sig-

nificantly faster than was obtained by Orszag, with an algebraic mapping and

Chebyshev polynomials (7). The spectrum also shows a "string" of eigenvalues that originates near the

point Cr = 1, ci = 0. As N increases, the string slowly converges to the

cr = 1 axis. Again, these modes are located far from the wall, in the uniform

free stream; thus, they effectively obey the Stokes equations, except that they

are transported at a velocity of 1. The exact spectrum is known to be continu-

ous in the Cr = 1 axis, and the corresponding eigenfunctions have a sinusoidal

behavior away from the wall (15). Such disturbances are not included by the

method since they do not decay to 0 as y ->- "'· These functions might play a

role if transition is influenced by periodic noise in the free stream, but such

a case is not considered here.

The critical condition is given as Re = 520, kx = 0.3012 by Jordinson

(14), and Re = 519.060, kx = 0.30377 by Davey (quoted in Ref. 16). The

results from the spectral code indicate Re = 519.0638 and kx = 0.303773, in

very good agreement with Davey.

Results from the solution of the full Navier-Stokes equations will now be

described; these results are mostly qualitative. Two cases were treated; in

both cases, a small initial disturbance was added to a Blasius boundary layer.

The Reynolds number was 1400, which is close to the conditions of the experiment done at Flow Research (17). The minimum wavenumber is 0.35. In the first case,

the free-stream velocity was kept at a value of 1. In the second case, it was

LEGEND

4 Y/Yo

o = 40 points

6

Figure 10. Orr-Sommerfeld Equation

6



332 SPALART

reduced linearly to a value of 0.8 in 500 time-units (based on the initial

velocity and displacement thickness) and then kept constant; this is one of the

cases treated by the experiment. The deceleration of the free stream strongly

modifies the velocity profile of the boundary layer; it introduces an inflection

point (the velocity profile is plotted on the right of Fig. 11) and makes it much more unstable, which is reflected by the larger growth rates of the dis-

turbances, or by the lower critical Reynolds number (17).

In the first case the disturbance never grows enough to be seen on plots.

Under the effect of viscosity the velocity profile tends to relax to an error-

function; since the error-function boundary layer Jq more stable than the

Blasius layer, the disturbance then decays. In the second case, the disturbance

is seen to grow to an amplitude of order 1 under the effect of the deceleration

and eventually decay again after the plate returns to uniform velocity. The

spanwise vorticity at different times is plotted in Fig. 11. As was expected

from a two-dimensional simulation, transition to a turbulent state with fine-

grained motion does not occur. Whenthe third dimension is included, the finite

amplitude two-dimensional disturbance will be unstable to three-dimensional

disturbance, and the stretching of vorticity will generate fine structures.

Preliminary three-dimensional results show this transition, including the

appearance of vorticity spikes.

7. CONCLUSIONS. A spectral numerical method has been designed specifically for

the solution of the Navier-Stokes equations over a flat plate, assuming the flow has the character of a boundary layer. A weak formulation of the equations is used. Periodicity is assumed in the horizontal directions, and special atten-tion is paid to the far-field behavior as y ~ oo, to obtain the best possible accuracy.

The one-dimensional heat equation on [O,+oo] has been solved accurately and efficiently using an exponential mapping and odd Chebyshev polynomials. Exponential convergence was observed.

A Navier-Stokes code has been written for simulations in two and three

dimensions. The separation of the harmonic component and the vortical component is expected to improve the spatial efficiency and allow the use of fewer points

in the y-direction. Results for the eigenvalues of the Orr-Sommerfeld equation show that this was achieved in terms of the convergence to smooth solutions.

Three-dimensional tests with sharp gradients are needed to confirm the potential

of the method for turbulent simulations. The study of the eigenvalues of the linearized equations revealed that the

Stokes eigenfunctions, and some of the Orr-Sommerfeld eigenfunctions, are out-side the class of functions that was chosen for the expansion of the solution.



EXTERNAL VISCOUS FLOWS

:;.... l') f-------0.100 0.100 N_)-------0.200 0.200

0.400 0.400

0~~~~~~~~===;==~==~~~~ d 0 2 4 6 8 10 12 14 16 18

X

(a) Beginning of Deceleration.

(b) End of Deceleration.

J------0.100---------- -------:;.... l') 0.100

1------0.200-- --N a soo o.2oo-. => <::::::::

0.300-----

0 2 4 6 8 10 12 14 16 18

X

(c) After Deceleration.

_)------0.100--------------0.100--:;,..."' N_)------0.200 ------------0.200--

o+===~===r===r===r--~--~---.-=~~~~~ 0 2 4 6 8 10

X 12 14

(d) After a long period of time.

16 18

Figure 11. Two-Dimensional Boundary Layer;-~= Velocity Profile, --= Vorticity Contours.

333



334 SPALART

These eigenfunctions do not have a boundary-layer character; they remain finite

as y + oo, The class of exponentially decaying functions is considered appro-priate for the simulation of a boundary layer.

From a numerical point of view, the formulation that was adopted results in

small memory requirements and allows a straightforward and accurate time-advance

scheme. The transforms in the y-direction are rather expensive: the cost is

O(N2). For a problem of typical size the penalty is moderate compared with the

total computational work. In addition, a better use of the points in the

y-direction reduces the number of transforms needed in the x- and z-directions.

The three-dimensional code is now being used to simulate boundary-layer transition on a flat plate, with grids of the order of 1283. The development

of a strongly three-dimensional turbulent state from predominantly two-

dimensional small disturbances will be investigated. A deceleration can be used

to control the process. The simulation of a fully turbulent, statistically

steady boundary layer is being considered; a body force could be applied to

provide momentum and prevent the growth of the boundary-layer thickness. Tests

will show whether such a simulation is possible with available computers.

Acknowledgments. The author had very helpful discussions 1vith A. Leonard,

P. Moin, R. Moser, and A. Wray at NASA Ames Research Center.

BIBLIOGRAPHY

1. Orszag, S. A., "Spectral methods for problems in complex geometries." J. Comp. Phys., 37 (1980), 70-92.

2. Gottlieb, D., and Orszag, S. A., "Numerical analysis of spectral methods," NSF-CMBS Monograph No. 26, Soc. Ind. and Appl. Math .• Phila-delphia, 1977.

3. Orszag, S. A., and Patera, A. T., "Secondary instability of wall-bounded shear flows," J. Fluid Mech., 128 (1983), 347-385.

4. Marcus, P. S., Orszag, S. A., and Patera, A. T., "Simulation of cylin-drical Couette flow," Proc. 8th Intl. Conf. on Num. Meth. in Fluid Dyn., Springer-Verlag, New York, 1982.

5. Moser, R. D., Moin, P., and Leonard, A .• "A spectral numerical method for the Navier-Stokes equations with applications to Taylor-Couette flow." to appear in J. Comp. Phys.

6. Leonard, A., and Wray, A., "A new numerical method for the simulation of three-dimensional flow in a pipe," NASA TM-84267, 1982.

7. Orszag, S. A., "Numerical simulation of turbulent flow over a compliant boundary," Report No. 63, Flow Research Inc., Kent, Wash., 1976.

8. Cain, A. B., Reynolds, W. C., and Ferziger, J. H •• "A three-dimensional simulation of transition and early turbulence in a time-developing mixing layer," Report TF-14, Thermosciences Div., Dept. of Mech. Eng., Stanford U., Stanford, Calif., 1981.




9. Wray, A., and Hussaini, M. Y., "Numerical e.:zperiments in boundary-layer stability," AIM Paper 80-0275, Pasadena, Calif., 1980.

10. Ladyzhenskaya, 0. A., The mathematical theory of viscous incompres-sible flow, 2nd ed., Gordon and Breach, New York, 1969.

11. Temam, R., "Navier-Stokes equations and nonlinear functional anal-ysis," NSF-CMBS Monograph No. 41, Soc. Ind. and Appl. Math., Philadelphia, 1983.

12. Heywood, J. G., "On uniqueness questions in the theory of viscous flows," Acta Math., 136 (1976), 61-102.

13. Schlichting, H., Boundary-layer theory, 7th ed., McGraw-Hill, New York, 1979.

14. Jordinson, R., "The flat plate boundary layer. Part l. Numerical integration of the Orr-Sommerfeld equation," J. Fluid Mech., 43 (1970), part 4. 801-811.

15. Grosch, C. E., and Salwen, H., "The continuous spectrum of the Orr-Sommerfeld equation. Part 1. The spectrum and the eigenfunctions," J. Fluid Mech., 87 (1978), part 1, 33-54.

16. Drazin, P. G., and Reid, W. H., Hydrodynamic stability, Cambridge University Press, New York, 1981.

17. Gad-el-hak, M., and Davis, S. H., "Study of effect of acceleration/ deceleration on turbulence and transition," Report No. 217. Flow Research. Inc., Kent, Wash., 1982.

C. F. D. BRANCH. MAIL STOP 202-A-1 NASA AMES RESEARCH CENTER MOFFETT FIELD, CALIFORNIA 94035







FORECASTING THE OCEAN'S WEATHER: NLMERICAL MODELS FOR APPLICATION TO OCEANOGRAPHIC DATA

Robert N. Mi ll er1

ABSTRACT. M:>del ing of i ntPnsive oceanographic data sets on synoptic scales 1 eads to interesting problens in techniques for boundary value problems. Intensive data sets collected in 1 imited areas make open boundary models necessary, and theoretical questions renain about the well posedness of such problens and about the nunerical approximation schenes used to solve then. Here the nunerical bound-ary value schene used in the Harvard Open Ocean Model is presented in detail and stability and accuracy results are stated. The problen of posedness of the boundary value problen is illustrated through a simple example. Finally, a sample calculation using the model on a nonlinear problen is presented.

l. INTRODUCTION. Oceanic mesoscale variability, i.e. variability of deep ocean currents on spatial scales of tens to hundreds of kilometers and tempo-ral scales of weeks to months have been the subject of increasing interest in the ocean science community for the past two decades. Such motions were first discovered unequivocally with the developnent of instrunents capable of long term direct measurenents of deep currents. After the first studies with neutrally buoyant floats tracked from a nearby vessel, nearly a decade elapsed while a new generation of instrunents capable of the accuracy and high resolu-tion necessary for the study of these motions was developed. The Mid-Ocean Dynamics Experiment {MODE; see MODE group [13]) was an intensive experiment undertaken in the early 7O's to investigate mid-ocean eddies. From this and other experiments, including the later US-USSR POLYMODE experiment {see Robinson [16]) 1 arge intensive data sets have been acquired, containing data on currents of these scales. These mesoscale eddies are the dynamical analog of the highs and lows of normally observed mid-latitude weather, and therefore the basic model physics is simi .ar, but because of the difference in the rele-vant scales, new modeling techniques are necessary in order to deal with these data sets in mid-ocean regions which do not have natural boundaries. Typical model regions are squares 300 to 500 km. on a side. One such model, currently

1980 Mathenatics Subject Classification. 65N30, 76U05, 76V05, 861\05. 1The author gratefully acknowledges the support of Office of Naval Research Contract nunber NOOOr4-R3-K-0488.

337





338 RORERT N. MILLER

in use at several 1 aboratories in the U.S. is the Harvard Open Ocean fv'odel. nesign philosophy, testing and calibration of the model can be found in Miller, Robinson and Haidvogel [11]. Here we present in detail the method used in that model, and some of the mathematical questions that arise in connection with its construction. Some test results are also presented.

2. FEATURES OF THE MODEL. The relevant dynamical model is the baroclinic quasigeostrophic model; for details of the model equations, see, e.g., Pedlosky [15]. The model is derived from the three dimensional Navi er-Stokes equations with the vertical momentun equation replaced by the hydrostatic equation; these are the so-called primitive equations of meteoroloqy. The derivation of the quasigeostrophic model proceeds by expanding the relevant quantities as a perturbation expansion in the Rossby nunber R0=V 0/{f 0L) , where v0 is the scale speed, L is the scale length, and f 0 is the angular speed of rotation of the fluid at the center of the model region. The fluid is assuned to obey the Boussinesq approximation, the flow is assuned to be adiabatic and the fluid is assuned to be stably stratified with local variation in density assuned to be O{R0). The model equations are written in terms of the quasi-geostrophic streamfunction q, and the potential vorticity r: , with the horizontal velocity components (u ,v) given by:

u = -c!J y

This sign convention is opposite to that used in most fluid dynamics because in this case the streamfunction q, is identified with the pressure, and the standard convention waul d have q, proportional to negative pressure.

The resulting equations in dimensionless form are:

1. 0 1 [ hl + EJ( r)>, •) ] (I; + C f) = 0

2. M + r 2 (a cVz ) z = r: where: J is the Jacobian J{q,,¢)= c!Jx<lly- cVy<llx

f = f0 + ~Y is the coriolis parameter

E is the beta-Rossby nUTiber v 0 ;{~L 2 ). The dimensionless constant r and the dimensionless function a(z) , where the depth z is scaled by the depth of the main thermocline, describe the properties of the water colunn. The open lateral boundary conditions are those described by Charney,

Fjortoft and von Neunann [4] {hereafter CFvN), i.e. q, is specified everywhere on the boundary and r: is specified at inflow points.



MODELING THE OCEAN'S WEATHER 339

3. IMPLEMENTATION OF OPEN BOUNDARY CONDITIONS. The problan of the impl anenta-tion of open boundary conditions arises in a number of different contexts, and there is a wide variety of methods in the literature. The following summary is not intended to be exhaustive. In the case of gridded methods, the method itself requires values of the vorticity at the outflow in order to evaluate derivatives at points adjacent to the boundary. Care is needed in specifying such variables to avoid potential instabilities due to overspecification. Basically there have been two approaches to this problan. One approach is to specify boundary conditions so that there are no artificial reflections at the open boundary. Engquist and Majda [5] did extensive theoretical work on the problan in a general context. Recently, a number of schanes following this approach have been proposed for mesoscale meteorological models in which gravity waves are assumed to propagate out of the top of the model region (see Rougeaul t [3], Kl anp and Our ran [9] and others. It is important to note the unfortunate fact that the tenn "mesoscale" means two different dynamical regimes to oceanographers and meteorologists.) These latter methods, along with that proposed by Bennett [1] deal with the fact that the waves in question are dispersive by the explicit use of the dispersion relation and integral transfonns in the tangential direction at the open boundary. Such nonlocal methods would be difficult to apply to the open boundary problan in a rectangular region. A local method was proposed by Orlanski [14].

The other approach is to make the approximate solution as accurate as possibe at the boundary, and leave open the possibility of tolerating some spurious reflection of outgoing waves. This is the approach we take here; Gustafsson and Kreiss [7] took this approach in a different problan which was not dispersive.

Our method uses finite elements to solve (1). At inflow boundaries, we prescribe the potential vorticity. At outflow boundaries the potential vorticity is predicted. The method using 2-D bilinear elements is presented by Haidvogel et al. [il], along with a scheme for prediction of potential vorticity at outflow points. Haidvogel et al. do not present a detailed analysis of the scheme; such an analysis of this two dimensional scheme would be very complex. Here the qualitative and quantitative attributes of the method are examined in detail for a simplified one dimensional problan.

In order to analyze the scheme, we use the linearized zonal barotropic vorticity equation:

3. ; + Ul;x + <Vx 0

4. <Vx x = C '

0 < X < L.



340 ROBERT N. MILLER

He prescribe initial values for <V and C , and boundary values for q, at all times. For U > 0, we specify c on the left boundary only. CUr approximate solution uses standard piecewise 1 inear finite elements on a uniform mesh width h. The left and right elements are one sided, supported in the intervals [O,h] and [L-h,L] respectively. In keeping with standard finite element techniques, we write:

N N <V = \ <V; ; c y Ci <l>i

i=O i=O where:

-r - X /h ' 0 .;; X .;; h <~>o

0 otherwise

{(x- (i-1)h)/h, (i-1)h "x.;; ih

<l>i,i=1 ,N_ 1 = ((i+1)h- x)/h ih..; x..; (i+1)h

0 otherwise

¢N = { ( x - ( N0-1 ) h ) /h , ( N-1 ) h < x < Nh

otherwise

The resulting equations are:

-U 1 = -z( t;1 - Col - -z( '~1 - t~>o l

= ~(C:l - Col - ~ (cJ'2 -<Vol

h ' 2h ' fiCN- 2 + PN-1 h ' + ()C:N

-U = -z(t:N -

1 c:N-2 l - -z(q,~J - 'vN- 2 l

h ' fiCN-1 +~eN = ~ (eN -1

CN-1) - z(q,N - <VN-1 )

In this case, we know that c0 is to be specified, so we could simply specify c0 (tl , discard the first equation and solve the remaining N equations inN unknowns. Here, we use the fact that the entire region, including boundaries, is governed by the basic dynamical model to eliminate ctJ between the first tv,Q equations and c;J between the 1 ast tv,Q equations and replace the second and N - 1st equations with the results, i.e. we derive the following set of prognostic equations for the interior region a 1 one:




-U 1 1 1 1 1 = 2(1;2 - 21:1 - 2co) - 2(q,2 - ~1 - ~o)

= ~ ( 1;3 - 1;1 ) - ~( q,3 - q,1)

h 1 2h 1

6~-3+ PN-2 -U 1 = 2(/;N-1 - I;N-3) - 2(q,N-1 - q,N-3)

h 1

6/;N-2 -U1 1 11 1

= 2(2/;N + 2/;N-1 - l:N-2) - 2(21'N + 2q,N-l - q,N-2) •

This same trick works in two space dimensions; the formulas are given in [8], where it is called the "7 /2 matrix method". In this 1-D example, we could refer to this decoupling as the '7/12 matrix method". This method can be shown to be stable by the energy method. We now investigate the accuracy of the method by comparing wavelike solutions to the finite elenent equations to wavelike solutions of (3) and (4). The model system (3), (4) has wavelike solutions of the form

q, = exp[i(kx + wt)]

with the dispersion relation: 1 5. w = k - Uk •

These are the Rossby waves.

1; = -k2exp[i (kx + wt)]

The scheTie also has Rossby wave-1 ike solutions of the form

q,j = exp[i (jk h + wt)] , 1;j = Bq,j , 2 where B comes from the approximation of equation (4), i.e. B .. -k. From the

interior equations, we find that wavelike solutions of the schene satisfy the dispersion relation:

(6) 3 sin kh -(1 + UB) w = ""'2_+_C;:_O_S -:-k~h • h • B

which approximates (5) to fourth order in h, assUTiing B = -k 2+ O(h 4). We choose the example of a model of a wave incident from the 1 eft of our

open region. If we prescribe r;0 (t) = B cos(wt) , direct calculation shows that in the interior the "tran9Tli tted" wave has the form q,. = a cos(kjh + wt), j = 1, ••• ,N-1 , where k is r::lated to w through (6) an~ a= 1 - O(h 2). So in this case, truncation error in the open boundary schene takes the form of dissipation, and there is no phase shift. At the right hand end of our simpl Hied model region the condition that ~- 1 = a cos[ (N - 1)kh + wt] and 1;N_1 = Ba cos[ (N - l)kh + wt] implies that q,N must be specified with greater than unit amplitude and with some phase shift. These errors are O(k 2h2). The potential vorticity also contains second order amplitude and phase errors.



342 ROBERT N. MILLER

4. THE POSEDNESS OF THE OPEN BOUNDARY PROBLEM: AN EXAMPLE. Next, we examine the question of whether the original problen with the boundary conditions described is well posed. Since the original example in [4] which was used to demonstrate the appropriateness of the boundary conditions was done with a non-simply connected region on the surface of acyl inder, the question of what happens in a simply connected region, where there must be points at which the fl rM is tangent to the region, remains. Sundstrom [18] proposed a proof of well posedness, but, as Bennett and Kloeden [2] noted, Sundstr·om's proof depends on a bound for the tangential derivative of the vorticity, and it is not clear that such a derivative exists at all. Bennett and Kloeden argue that the problem is i 11 posed because it is equivalent to specifying initial data for a hyperbolic equation on a characteristic surface. Workers who have investigated appropriate boundary conditions for hyperbolic equations (see esp. Maj da and Osher [1 0]) exclude this case, and their methods are not directly applicable here. A simple example, intended to shed light on the posedness problem, follows.

Consider the equation:

where U is a positive constant on the domain consisting of the unit disk x

[O,oo). The analog of the CFvN boundary conditions is to prescribe c initially and impose C on the left hand semicircle for all time. The value of C at any given point in this domain can be identified with a boundary or an initial value, so for any fixed point (x,y,t) in the solution region, c(x,y,t) depends continuously on the data. However, c can have unbounded derivat~ve in the inter·ior. If ~1e parameterize the boundary by the angle e, the inflow set is given by y < e < 3 ~. If we prescribe the boundary

condition c = ei(k cos 8 + wt) , the interior solution will eventually become: c exp {iC~x + wt t (k + ul j 1 - iJ} which is not differentiable at y = ±1 , the points at which the flow is tangent to the model region, unless k + w/U=O. This is exactly the condition for a function of the form exp[i(kx + wt)] to be a solution of (7). In this example, we see explicitly that even though the vorticity at any given point is a continuous function of the initial and boundary conditions, its y derivative is not a uniformly continuous function of the boundary conditions. But this is an extremely weak singularity, and it is reasonable to ask about the behavior of practical nunerical schemes. In the spirit of the problem at hand, the boundary and initial conditions waul d be expected to come from some exterior fl rM

consistent with the model, i.e., for the correct value of U, the boundary and




initial conditions actually would satisfy k + w/U = 0. The major manifesta-tions of truncation error in schanes for problans of this type is phase error, i.e. an n-th order schane would have simple harmonic solutions with k + w/U = O(hn), where h is the grid spacing. Therefore, for y = 1- h, /1 - / = O(h112 ) , and thus as h approaches zero, the (k + w/U) ;/1 - y2

in the derivative waul d be O(hn-1/ 2 ). A reasonably accurate scheme waul d

therefore converge smoothly to the consistent solution. However, in models such as ours, the phase error may not be controlled by the horizontal grid spacing h. In a baroclinic model with sparse vertical resolution, signifi-cant phase error can result from vertical truncation error. In the case of application of open ocean models to real data, uncertainty in the internal deformation radii can result in systematic errors in the phase speed of disturbances. In these cases it is plausible that the normal derivative of the computed solutions near the boundary in the vicinity of a tangent point might diverge as h approaches 0. In a nonlinear problem, it is also plausible that large vorticity gradients could be swept downstream and possibly be advected into the interior where instabilities may result. This phenomenon has not been observed directly. In fact, in all trials so far, the derivative of the vorticity normal to the boundary in a neighborhood of a tangent point appears to converge smoothly to a finite value as h approaches zero. This could be a consequence of the dissipation imposed implicitly by many practical numerical methods; but, implicit numerical dissipation tends to vanish as h approaches zero. At this point, numerical manifestation of this phenomenon is elusive.

5. A COMPUTATIONAL EXAt-PLE FOR A NONLINEAR PROBLEM. We present here a computational example of the performance of the scheme applied to a well known exact solution of the barotropic vorticity equation on the beta plane. Figure 1 shows an example in which a barotropic modon {see, e.g. Fl ierl et al. [6] for details of this dipole-1 ike solution)propagates out of the domain without incident. One caul d imagine that the camp 1 i cated flow pat terns that results when the center of the modon crosses the eastern boundary of the model region would cause trouble; but, in this case, it does not.

These theoretical questions ranain, but the model has performed well in a variety of circumstances. Results from trials of the earlier barotropic version of the Harvard Open Ocean i"bdel with simulated data appeared in [17]. A forecast study with baroclinic simulated data is now being completed. Pre 1 imi nary results wi 11 appear in [1 2].



344 ROBERT N. f4I LLER

----- ~---·~ . .. , ' , ' , ...... --...... \

I ,' ,. --- ""' \ t I , -- ', \ t I 1 I 1 ' ....... '\ , ' f

\ ' ~ ' ... ".E.J ,. ' : ,' ' .. ~ ... ::=::::~-"' ... ' 1------

~ CONTOUR INTERVAL .3

A

I I

1

, , ,

CONTOUR INTERVAL= .3

c

.· ' II ,*" _.- •.,.' " , , ... --... "'

f I ' .,. - "\ \ f ,,,-fl .. ,'\\' I t \' \.J f ,J t

~ ..... "' .... ~:::-:.":.: ... ', ....... -:.~-:. ~- .. ..

CON TOUR INTERVAL .3

B

CONTOUR INTERVAL .1

D

I I

.· • I •

I t' , I I I I t I I

\ t :II I \ ~ II: ' \ \' \,

\ ...... '~~

Figure 1. St reanfunction maps of open boundary simulation of a barotropic modon. Dashed contours are negative. Nondimensional size of the basin is 5 with origin at the center. The modon radius is 1. Initial position is x = .5, x = .5, y = 0. This medon has speed 1, i.e. it moves a distance equal to its radius in unit time. Time step= .005. A fourth order Shapiro filter (see [11]) is applied every fourth time step. Normalized RMS streamfunction error remains under 4% throughout the 3 time unit simulation. A. Initial. B. T = 1. C. T = 2. 0. T = 3. Note change in contour interval.




R IBLIOGRAPHY

1. A. F. Rennett, "Open Boundary Conditions for Dispersive \·Javes", J. Atrnos. Sc i . 3 3 (1 97 6) , 1 7 6-182.

2. A. F. Bennett and P. E. Kloeden, "Boundary Conditions for Limited Area Forecasts", J. Atrnos. Sci. 35 (1978), 990-996.

3. P. Bougeault, "A Non-Reflective Upper Roundary Condition for Limited-Height Hydrostatic r-bdels", r-bn. Wea. Rev. 111 {1983), 420-429.

4. J. r,, Charney, R. Fjortoft and J. von Nei.ITlann, "Numerical Integration of the Barotropic Vorticity Equation", Tellus 2 (1950), 217-254.

s. B. Engquist and A. Majda, "Absorbing Boundary Conditions for the Numerical Simulation of Waves", Math. Comput. 31 (1977) 629-651.

6. G. R. Fl ierl, V. 0. Larichev, J. C. McWilliams and G. M. Reznik, "The Dynamics of Baroclinic and Barotropic Solitary Eddies", Dyn. Atrnos. Oceans 5 (1 980), 1-41.

7. B. Gustafsson and H. n. Kreiss, "Boundary Conditions for Time Dependent Problems with an Artificial Boundary", J. Comput. Phys. 30 (1979), 333-351.

8. O.B. Haidvogel, A. R. Robinson and E. E. Schulman, "The Accuracy, Efficiency and Stability of Three Numerical Models with Application to Open Ocean Problems", J. Comput. Phys. 34 (1980), 1-53.

9, J. B. Klemp and fl. R. nurran, "An Upper Boundary Condition Permitting Internal Gravity Wave Radiation in Numerical f\4esoscale Models", Mon. Wea. Rev. 111 (1983), 430-444.

1n. A. Majda and s. Osher, "Initial-Boundary Value Problems for Hyperbolic Equations with Uniformly Characteristic Boundary", Comm. Pure Appl. Math. 2 R (1 97 5) , 6 07 -67 5.

11. R.N. Miller, A. R. Robinson, and D. B. Haidvogel, "A Baroclinic Ouasigeostrophic Open Ocean Model", J. Cornput. Phys. 50 (1983), 38-70.

12. R.N. Miller and A. R. Robinson, "~namical Forecast Experiments with a Baroclinic Quasigeostrophic Open Ocean r-bdel", in "Predictability of Fluid Motions", G. Holloway and B. J. West, eds., llmerican Institute of Physics Conference Proceedings (1983), to appear.

13. The MODE group, "The Mid-Ocean flynarnics Experiment", Deep-Sea Res. 25 (1978), 859-910.

14. I. Orlanski, "A Simple Boundary Condition for Unbounded Hyperbolic Flows", J. Comput. Phys. 21 (197fi), 251-269.

15. J. Pedlosky,"Geophysical Fluid Dynamics", Springer-Verlag, New York, 1979. 16. A.R. Robinson, "Qynamics of Ocean Currents and Circulation: Results of

POLYMODE and Related Investigations", in "Proceedings of the International School of Physics, Summer Institute, Varenna, Italy, 1980", preprinted as POLYMOOE Report, necernber 1980, USPMOC-MIT.

17. A. R. Robinson and n. R. Haidvogel, flynamical Forecast Experiments with a Barotropic Quasigeostrophic Open Ocean ~bdel ", J. Phys. Oceanogr. 1 0 (1 980)' 1909-1 928.

18. A. Sundstrom, "Stability Theorems for the Barotropic Vorticity Equation", i~on. We a. Rev. 97 (1969), 340-345.

DEPARTMENT OF MATHEMATICS TULANE UNIVERSITY NEW ORLEANS, LOUISIANA 70118






Part Ill. Bifurcation and Dynamical Systems







GEOMETRY AND DYNAMICS IN EXPERIMENTS ON CHAOTIC SYSTEMS

Harry L. Swi nne.v1

ABSTRACT. Phase space portraits, Poincar~ sections, and nans have '1een obtained from exper'iments on a nonequilihrium chemical reaction anrl a fluirl flow. For some naramet~r values thP. rlynamical behavior is describP.rl by chaotic (stranqe) attractors, that is, ti1P. larqest Lvapunov exponent is positive. Several distinct routes to chaos have been observed, i ncludi nq period rloubl i nq, intermittency (near a tanqP.nt bifurcation), anrl a quasioeriodic-c!Jaotic trans it ion.

The hvrlrorlvnatTJic anrl chemical systems sturlie1 in our laboratory experiments will be described first, anrl then some of the exrerimP.ntal results •1ill f)e brieflv summarizer!. Full reports of the results have dPneared e 1 Sf'vlhf>re.l-11

THE PHYSICAL SYSTEMS The Relousov-Zhaf)otinskii reaction has "leen sturlierl in a stirred

fixerl-volume flow reactor. The system can be mai"ltainerl indefinitely in a well-rlefined noneC]uilibrium state by fixinq the flow rate of the chemicals th rouqh the reactor. Transit ions between rlifferP.nt dynami ca 1 reqi mes can be studied by chanqinq the flow rat~; thus the flow rate is the hifurcation parameter for these P.xperifT1ents. The reaction is stirred vigorously so that rliffusion processes are neqliqi!)le; hence, to a qoorJ approximation, the experi,qents are describer! by coupled first order ordinary differential

t . 1 equa.1ons, (i,j=l, ... ,N)

w~ere the X.(t) are the concentrations of the N different chemical snecies in 1

the reactor (N = 30+), the functions fi (Xi) contain quadratic nonlinearities, anti r is the flow rate of the chP.micals; only four chemicals have nonzero innut concP.ntrations X~. 1\ four variable rmdel extracted from the full er]uations rlP.scrihes at lP.ast some of the experimental observatinns. 4

ThP. seconrl svstem sturlied is a classic one in fluid dvnamics, the Couette-Taylor (circular Couett.e) svstam, ~1hich consists n" a fluid between inrlependentlv rntatinrJ cylinders. In the experiments to bP. rlescribed here the

1980 ~:athematics Subject Classification. 76-05, 76E30, /6U05 lsupported by National Science Foundation Grant MEA8206889

349





350 H. L. Sv!I NNEY

inner cylinder rotates and the outer cylinder is at rest; the behavior ~lith

bottl cvlinders rotatinq is much more co1nplicat~ct. 10 • 11 When the inner cylinder is at rest, the first instability that occurs is one in v1hich toroidal vortices form, encirclinq the inner cylinder anri stackerl in the axial dir,~ction. In 19?.3 Ci. I. Taylor ohserved thPse vortices and calculated (in a linear strtbility analysis) the critici!l Revnolds number Rc for the instability. 1\t the next instability reacherl with increasinq Reynolds numhr>r, azimuthal travelinC] waves appear on tl1e wavy vortices ("wi!VY vortex flow"). •\t liJrqer larqer R, there is ilnother well-defi neri transition, to doubly periodic flow where the velocity nower spectrum cnnsists of two fundamental fre(wencies and their combinations; the fundamental frequencies corresooni to t~10 sets of trc~velinq azimuthal 11aves that have ciifferent wave speeds. 8 The space-time symmetries observed for this douhly periodic flow have been found to '">e in accord 1vith those predicted usinq dynamical systems concents and the rotational invariance of tile boundary conMtions and the Navier-Stokes equation. 7•12 Finally, with a further increase in R, there is a transition to nonperiodic flow where the velocity power spectrum contains broadband noise as well as sharp frequency components. .1\s

discusc;ed helow, this nonperiodic behavior is described l)y a low-dimensional stranqe attrJ'Ictor.

PHASE PORTRAITS, STRANGE ATTRACTORS, ANO niMENSION Phase portraits can he constructed fr0m measure1~ents of a sinqle dynamical

observable, V(tk), obtained in measur~rnents in successive time intervals tk = kt:.t [k=l, 2, •.• ,oo]. (In our exoeriments the number of measure111ents k is typicnlly 3271)8). Phase portraits of riirnension rn Ciln be constructed frofll the vectors {V(tk), V(tk+T), ••• , V(tk+(rn-l)T}. This procedure is justified by emhedrlinq theorems if m is sufficiently larqe. 13

Phase portraits ohtained from di!t.a for the chemical reaction anrl for l.ouette-Tavlor no1~ are shown in Fiqures l(a) anrl 2(a) respectively. The limit sets in these phase portraits ilre attractors--that. is, the orbits ranidlv return to the limit sets followinq small perturbations. 3

The attractor in Fi'1ure l(a) and the attractors f0r R/Rc = 12.0 and 15.2 in Fiqure 1(b) are stranqe--thilt is, the larqest Lvaounov exponent is positive. This exponent, which charilcterizes the rate of separation of nearby orbits, was deterr~ined directly from a five-ciirnPnsional ohase portrait for the l.ouette-Tavlor dat~ 5 ,1i and frofll a one-dimensi0nal map [Fiqure 1(c)] for the Relousov-lhahotinskii data.3

The Poincar~ section in Figure l(h) indicates that the attractor in Fiqure 1(a) is essentiallv two-1ir~ensional, althouqh the difllension must be sliqhtly qreater than two since the nttrnctor is str-1nge. r:oordinate values for successive points 0f the Poincar~ section nrnvide a sequence {Xn} which defines



GEOMETRY AND DYNAMICS IN EXPERIMENTS ON CHAOTIC SYSTEMS 351

(a) (b) / . • , ,

# ~ - ,/ .... I' N , + . - #

M .. ~ -Lt') c:c , .. + . . .. .. - .

ctl ./ , ~

B(ti+T)

Fi qure 1. flata obtai ~e~ for a nonperiodic state observed in the Belousov-Zhahotinskii reaction. ' (a) A two-dimensional phase portrait, ~(tk + r) vs. B(tk), constructed from measurements of the time-dependence B(tk) of one of the chemicals; r = 53 sec., while the average periorf of an orbit is about 130 sec. (b) A Poincar~ section qiven by the intersection of orbits in a three dimensional phase portrait ~~ith a plane norfTlill to tile paper oassinq through the dashed line in (a); the thirrl axis, wi1ich is normal to the paper, is given by the ci1emical concentration at t1vice the basic time delay, B(tk + 2 x 53 sec). (c) A one-dimensional map constructed by plottinq.as ordered pairs (Xn,Xn+1), the successive values of the ordinate of tra.iectories when ti1ey cross the rlasi1ed line in (a).(from ref.2) ·

a one-rlimensional man, Xn+l = f(Xn), as shown in Figure l(c). The points fall on a sinqle-valued curve, indicatinq that the system is deterministic.

Th~ attractor for Couette-Tavlor flow at R/Rc = 10.1 is a two-torus, as can be seen frof'l Figure 2(b). At R/Rc = 12.0, which is beyond the onset of 'lonperiodic flow, the torus has hecome fuzzy, and at R/Rc = 15.2 the torus is no lonCJer apparent.



352

(b)

·•. -.

·. '·

H.L. SWINNEY

:.i:' ·-.J ... -.·

·. -· . "' -:...· .

. ;: : .. -:

.. ::~ .. . ···'·

·.-.:-: ...

. .. .···'·.

.-··· .. . .. · . . · -: ~ . . : ........

._.:· . . . . .. ...

Fiqure 2. Data obtained for ':nuette-Taylor flow'i,.S at Reynolds numbers below and above the onset of nonperiorlic flow at R/Rc = 11.4. (a) Two-dir~ensional phase portraits, V(tk+<) ~· V(tk), obtained from velocity mf'asurernP.nts (•=130 msec). (b) Poincnr~ sections given by the intersection of orbits in three-dimensio'l<'tl ohase nortraits ["'ith the f:l1ird axis normal to the paoer qiven hv V(t~+2<)]1vith planes normal to the dasherl lines in (a). (from ref. 6)

TIJere are a number of different rlefinitions of rlimension, inclurlinq capacity, Hausdorff, fractal, information, Ly;wunov (Knolan-Yorke), a11d the dimension of the natural measure, which have beE>n discussed anrt compared by Farmer et a1. 14 The basic idea behind several of these methods is that if a small ball of rtimension d is in a Euclidean space f dimension m (md), then the mass in the ball is proportional to £rt; hence the rlirnension of the ball can be determined bv the scalinq of dimension with radius. Sirnilarly, the dimension of an experimentally observed at tractor can he rJetermi ned fror~ the scaling with £ of the number of points N(E) inside a ball of rildius £: the slope of loq N(£) vs. log £ yielrts the rtimension of the attractor if the dimension m of the reconstructed phase portrait is sufficiently larqe. The dimension computerl in this way by Farmer anrl Jen for the attractors observect in the Couette-Taylor experiment is 2 for the douhly periodic flow, and it increases continuously with increasinq Reynolds numher to a value of about 4 at a Reynolds number 10% above the onset of chaos. 6




l•le conclude from these analyses that the no'lperiodic regimes observed in the Relousov-Zhahotinskii reaction anrl in CouP.tte-Tnvlor flo1v at low Reynolds nur~her are rlescrihed by low dimensional strange attractors. Th,Js, althouq~ the formal limitinq nrocesses required in the definitions of Lvapunov exronents anrl di1oension inevitably breakdown in any experiment at some SJllall (noise) scale, it is sti 11 rossible to approximate the exponents anti the dimension for laboratory svstef'lS since the qeometrical form of an attractor persists in the nresence of s~all amounts of noise.

OTHER REMARKS Period douhlinq and the Universal sequence. The mathematical theory of

one-riiJ'Iensinnal maos of the interval is hiqhl.v rlevelooed. 15 Mans with a single extre~Jm, such as in Figure 1(c), are oredicted to exhibit (as a function of bifurcation narameter) an infinite period doublinq sequence followed hy a universal (i.e., mao-inrlependent) sequence of oeriorlic intervals. The first few doublinqs of the period doubling sequence (periods 1, 2, 4, anrl 8) and ten of the predi cterl subsequent neriodi c intervals have he en ohse rved in the Relousov-Zhahotinskii experiment. The order of the periodic intervals anrl the map iteration patterns ohserverl in the experiments arr. in accord with the tf1eorv for one-dimensional maos." An arldii:ional prediction of tiJe theory is that each periodic interval should appear at a tangent bifurcation and enrl in a oeriorl rlouhlinq sequence; this is observer! for the widest periodic interval, neriorl 3.

Quasiperiodic-chaotic transition. Two groups16,17 have r!eveloped inrlependentl.v a renormalization qroup theory for the breakdown of a b'/0-torus at the onset of chaos. This theory rloes not aool.v to the Couette-Ta.vlor system since the analysis rloes not take into account the rotational symmetry. 12

Comparison with ~ 1 avier-Stokes equation. Marcus9•18 has determined in a numerical solution of the Navier-Stokes equation the speed of the travelinq azimuthal wave~ in wavy Couette-Taylor flow. ll. pseudo-soectral technique was userl with 32 axial Fourier nnrles (for one Taylor vortex oai r), 32 azimuthal Fourier 1~rles, anrl 33 rar!ial Chebyshev nolynomials. Have sneerls were measured anrJ comouted for the same parameter values for three cases and the aqreemPnt was within 0.1% in each case.

SIJM11ARY The exnerifllP.nt'> rlescribed here ar~> exaMples of the many on the dynamics of

nonequilibrium systems t"lat are being conrlucted in biological, chenical, fluid dvnanlic, and solid state lal:>oratories throughout the world. 19 Fe"' other areas of science todav, if anv, interact so strongly with current research in mat. hematic s.



354 H. L. SWINNEY

ACK ~HML F.I1G~~ENTS. This research University of Texas nonlinear Marcus, l.J,lil anrl J. 11. Farmer, F..

l•tas conducted in collaboration with dynamics C]roup,l-ll David Rand,?,l2 P.

,Jrn, and J.P. Cr,.tchfielrJ. 6

This research was partially supported by NSF Grant MEA82-06889.

BIRLIIlCRAPHY

the s.

1. H. L. Swinney, "Observations of orrler and c'laos in nonlinrar systems," Physica 70 {1983), 3-15.

?.. R. H. Sirnoyi, A. Half, and H. L. Swinney, "One-dimensional dynamics in a multico•TJnonent chemical reaction," Phys. Rev. Lett. 49 (1982), 245-248.

3. ,J. C. Raux, R. H. Sirnoyi, anrJ H. L. Swinney, "Observation of a strange attractor," Phvsica 8D (1 983), 257-266.

4. J. S. Turner, ,J. ~. Roux, 1·1. 11, McCor•~ick, anrl H. L. Swinney, "Alt£~rnating periodic and chaotic regimes in a che"lical reaction: experiment and theory," Phys. Lett. il51~ (1981), 9-12.

5. A. iirandstater, ,J, Swift, H. L. Swinney, and A. \blf, "A stranqe c~ttractor in a Couette-Tavlor experiment," in Turbulence and Chaotic Phenomena ~Fluids, rd. bv T. Tat.sumi (North-Holland, 1\msterda•~, 1983).

6. 1\. Brandstater, ,J, S11ift, H. L. Swinney, .ll,. 1Iolf, ,J.D. Farmer, F.. ,Jen, and J. P, Crutcflfield, "Low-dimensi0nal chaos in a system with Avogadro's number of deqrees of freedorn," Phys. Rev. Lett. 51 (19il3).

7. M, Gorman, H. L. Sv1innev, and 11. A. ~and, "11oublv perioctic circular Couette flow: experiments co•TJ[lared with nrerlictions from dynamics and symmetry," Phys. Rev. Lett. 46 (1981), 992-995.

8. R. Shavt, C. n. Anderrck, I_, A. Reith, anrJ H. L. St~innev, "Sunerposition of travelinC] wavf's in the circular Couette systefll," Phys. Rev. Lett. 48 (1982)' 1172-1175.

g, r;, P. KinCJ, Y. l.i, l·l. Le~~. '-1. L. S1~inney, and P. s. Marcus, " 1 ~ave soeerfs in w~vv Taylor vortPx flow," J. Fluid ~~ech., to apoear (19H3).

10. C. D. 1\nr!ereck, r<.. llickmc:n, and H. L. S1~inney, "New flows in a circular Couett.e system with co-rotatinq cylinrlers," Phys. Fluids L6 (1983), 139S-14fll.

11. C. 11. /'lndereck, S. S. Liu, anr! H. L. S1~inney, "Flow between indepenrlentlv rot.atinq concentric cvlinders," in nre!1aratio'l.

12. n. A. Rand, "Dynamics and svmmetry: predictions for modulated waves in rotatinq fluids," .Arch. Rat. t1ech. ll,nal. 79 (1982), 1-37.

13. F. Takens, "Oetectinq stranqe attractors in turbulence," in Lecture Notes in Mathe1natics, Vol. il9R, erl. by 11. A. Ranrl and l.. S. Younq (Springer, flerl in-;-1YH1), !1.%6-381.

14. J. n. Farmer, F.. Ott, and ,1, Yorke, "Dimension of chaotic attractors," Physicil 711 (1983), 153-180.

15. P. IDllet a11rl J. P. Eckmann, Ituated MaDs of the Interval as Dynamical Systems (Birkhauser, Roston, lCJHO)-.---- ----

16. s. Ostlund, n. Ranri, ,J, Sethna, and E. Siqqia, "llniversal properties of the transition fro1~ quasioeriorlicitv to chaos in dissipative systems," Phvsica 8n (1983), 303-342.

17. r1. J. FeiqPnhilum, L. P. Karianoff, anrf S. ,J. Shenker, "Duasineriodicitv in rlissioative systems," Physica sn (1982) 370,

18. P. S. 11arcus, "Sir:1ulation of Tavlor-Couette flrm, '"tmerical r1ethods; numerical results," submitte;i to J. Fluid ~1ech.




19. See, e.q., N. Abraham, ,J. P. Gollub, and H. L. Swinney, "Testing nonlinear dynamics," Physica D, to appear.

OEPARTMENT OF PHYSICS IINIVERSITY OF TEXAS MJSTIN, TEXAS 78712







DIMENSION ESTIMATES FOR ATTRACTORS

John Guckenheimer

Numerical and theoretical studies of three dimensional flows and one and two dimensional iterations have yielded a coherent picture of "chaotic" dynamics in its simplest forms. This body of knowledge is relevant for the experimental and analytical study of fluid dynamics in regimes which represent the transition to "turbulence" . Here turbulence is used loosely as referring to aperiodic flow with a continuous power spectrum. This paper is a discussion and review of aspects of dynamical systems theory which appear to be useful in the interpretation of experi-mental observations together with some new remarks about the statistical problems of estimating the Hausdorff dimensions of attractors. The methods I describe are of more general applicability than just to fluid experiments, but I have restricted myself to procedures which appear feasible with the amount of data which is readily available from work with fluids.

The issues which I address involve determining whether the state of a fluid can be represented by a "reduced" model with few degrees of freedom. If one assumes that the system is behaving in a deterministic fashion, then one would like a dimension estimate for the attractors which occur in the state space of the fluid. Once transients in the fluid system have decayed, the observed system follows a trajectory in state space whose closure is called an attractor of the system. For chaotic systems , attractors typically have a frightful topological structure which makes even the definition of dimension problematic. Given the lack of clear under-standing of the finest details of the chaotic motion in simple models such as the forced Duffing equation or the Henon mapping [7] , it is unreasonable that any statistical method can be proved to give accurate estimates of dimension for all attractors . The best that one can hope for is a procedure which is reliable for classes of well under-stood examples and that the scope of these classes of examples might be enlarged in the future. Here I concentrate on a simple, computationally inexpensive statistical method and discuss a few examples, each of whose analysis contains only some of

Research partially sponsored by Air Force Office of Scientific Research Grant AFOSR 83-0143 and National Science Foundation Grant MCS 82-002260. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes not withstanding any copyright notation thereon.

357





358 JOHN GUCKENHEIMER

the difficulties of the general problem. Hopefully, these heuristic considerations

will be helpful in understanding the practical efforts to compute the dimension of

attractors from experimental data.

1. DIMENSIONS AND ESTIMATES. Farmer, Ott and Yorke [3] have reviewed a

number of related concepts of fractional dimensions and their application to attrac-

tors. The reader should consult this paper for additional discussion and background

about dimensions.

Definitions of dimension which depend upon a specific measure rather than

just a geometric set of points in state space are relevant for the study of attractors.

The most reasonable view of this matter (based on experience with numerical simula-

tions) appears to be the following. Initial conditions for a flow are interpreted as

avoiding sets of zero Lebesgue measure with special properties. Properties which

hold in sets of positive Lebesgue measure are observed in watching long trajectories.

Different trajectories in a chaotic attractor wqy howe r1ifferent asymptotic properties, but I assume that there is a unique, ergodic probability measure upon the attractor

A which represents the asymptotic properties of almost all trajectories in its basin of attraction. This means that

1 T lim T J

0 f(<p(x,t))dt = J fdw

11 for almost all x in the basin of attraction of 11 and w-integrable functions f .

The existence and uniqueness of \.l are proved for hyperbolic attractors by Sinai,

Bowen, and Ruelle [2] and for a large class of one dimensional mappings by Jakobsen [9] . Sets of \.l-measure zero should be excluded from dimension calcula-tions, even if they have larger Hausdorff dimension than a set of full \.l-measure. This is evident in the generalized baker's transformation discussed below.

The choice of calculation strategy is an important one. Procedures based upon "box counting" techniques [5] require large amounts of computational time, and the computational time grows rapidly with the dimension of the set being described. The procedure which I analyze here is called the pointwise dimension by Farmer et. al.

For an attractor 1\ , x E 11 , and a probability measure \.l supported on 11 ,

one defines d (x) =lim log!J(B(r,x))

P r+O logr

where B (r, x) is the ball of radius r centered at x in the state space of the flow. If dp (x) exists and is independent of x for IJ-almost all x £II. , then this is defined to be the pointwise dimension of i\ . For simplicity, I describe the pro-cedures in terms of discrete time. One way think in terms of a discrete sampling of a continuous system or in terms of a Poincare section of a continuous system for the purposes of these measurements.

Estimation of dp (x) from a trajectory of length n can be performed effi-



DIMENSION ESTIMATES FOR ATTRACTORS 359

ciently. The technique is to first compute the distances o (t) = I x- ¢ t(y) I from x

to each point ¢ t(y) in the trajectory bc.3ed at y . The assumption of the existence of an asymptotic measure implies that if N (r) is the number of o (t) with O(t) <r ,thenN(r) isagoodapproximationto ll(B(x,r))when n islarge. The

n numbers N (r) are easily calculated by sorting the n numbers o (t) . One can then plot log N (r) versus logr to estimate the limit value of log ll(B (x ,r)) /logr as r + 0 . The number of operations needed to calculate the o (t) grows like n · k , k being the dimension of phase space. The number of operations needed to sort the o (t) is of order nlogn . Thus the computational time has a total order of magnitude which is n (k + logn) . This is significantly smaller than the computa-tional time required by other methods [ 4 J and is the only dimension estimate which appears feasible on a minicomputer at this time. The practical questions in the implementation of the method involve (1) the size of the statistical fluctuations which one should expect as one varies n and (2) the optimal way to extrapolate to the limit r + 0 . The first of these issues is considered here in terms of examples, following a review of some of the properties of order statistics.

2. ORDER STATISTICS [W] . If X. ,i=l, · · · ,n are independent random variables 1

with the same continuous cumulative distribution function F(x) , then almost surely

Xi :f Xj for i :f j . If the Xi's are ordered to yield X(l) < x(2) < · · • < X (n) with X(i) e fx1, · · · ,Xn} , then the X(i) are called the order statistics of the Xi . The order statistics of { F (X.)} are independent of the distribution function F since

1 each F (X.) is a uniform random variable on the interval [ 0, l] . Therefore, it is

1 possible to study the order statistics F (X(i)) in a setting which is completely non-parametric (not dependent upon a distribution) .

The theory of order statistics plays an important role in our estimates of dimension because the distances 8(t) are sorted (i.e., arranged in numerical order) as part of the computational process. One may view the 8 (t) as random variables whose distribution is given by V (r) = \1 (B (r, x)) . In other words, the probability that 8(t) is smaller than r is the )1-volume of the ball of radius r centered at x . The order statistic~ 8 (i) are used to estimate the distribution V(r) by assumingthat V(o(i))"" ~ (with n thenumberofcomputed distances). Built into this procedure is the implicit assumption that the 8 (t) are independent of one another. If the sampling interval is sufficiently long, then sensitivity to initial

* conditions suggests that this is a reasonable condition for chootic attrac1Drs . The theory of order statistics then describes the distribution of the quantities

v (i) = V ( 8 (i)) for varying realizations of the random variables ~ t (y) . I do not go into much detail concerning order statistics here, but note that the

* Some continuous-time attractors appear to have a long time "phase coherence" that must be accounted for here. For quasiperiodic attractors, the ~ t (y) will not be independent, but they will typically be uniformly distributed.




distribution of the ith observed volume v(') is given by the Beta distribution . . n! i-1 n-i ~ i .

Be(1 ,n-1+1) = (i-1)! (n-i)! v (1-v) which has mean n+l and variance

i(n;i+l) . If i = k- j , then Be(i ,n-i+l) also gives the distribution of (n+l) (n+2)

v (k) - v (j) . Wilks [10] also includes information concerning the joint distribution

of different order statistics. As an illustration of these ideas, I discuss the estima-tion of the dimension of a torus and a cube of dimension d using these methods.

Consider a torus Td = S 1 x · • · x S 1 with distance function o ( e , 1jJ) =

max 6(9.-ljJ.). Here cS(9.,1jJ.) isthelengthoftheshortestarcon s1 joining 1_0~d J J J J

e . and l/1 • : cS( e . , l/J.) = min < I e . - $. I , 21r - < e . - $ . ) I ) if o < e . , w . < 2 '![ • The J J J J J J J J -] J

simplicity of this example for dimension estimates lies in the fact that the open ball of radius 'lf centered at g_ E Td covers almost all of Td and the volume of the ball of radius r , 0 < r < 'lf is (!:.) d (with the volume of Td normalized to be 1 ) .

'![

Thus the function logV vs. logr to be estimated is a straight line of slope d . From a random sample of n points 1jJ. on Td , an estimate for d can be obtained from

I the order statistics cS(') of o. = cS ( e, l/J.) • One need only pick two fixed values

1 I I

of ~ , ! and estimate log (k) - log ( 9.) as the value of d . The variance in n n log o (k) - log o ( 9.)

the estimates of V will be order Jn . This implies that, for fixed ~ and ~ and n large, the variance in the estimat~ of d will be of order ,ftn . A :Ore com;lete statistical analysis of the variance in the estimate for d is possible. Note, however, that the relative precision of this estimate is independent of d . Figure 1 illustrates the results of a numerical computation of logV vs. logr for n = 5000 and d = 2 , 3 , 5, 10, 25, 50, and 99.

50

logr

2

logV

Figure I




As a second illustration of these dimension estimates, I consider an example in which the estimates depend upon the choice of reference point x . Denote the unit cube Id = ( [ 0,1]) d in d -dimensional Euclidean space with the sup norm o (x ,y) = max ( I x. -y ·I ) . For the reference point x chosen as the center or one

- - b.j ~d J J vertex of the cube, the analysis of this example is essentially the same as the toral example discussed above. However, if x is chosen randomly in Id , then there are

. new complications in the dimension estimates due to the fact that x will be closer to faces of the cube normal to some coordinate axes than to others. In particular if e (l) , · · · ,e (d) are the order statistics of ei = min (xi ,1-xi) , then one can explicitly

compute that V(r) = '1

·,j· 'd r~l::(j) :: :~) :a~~l- •a>

( if 1 - e (j) 2.. r

Consequently, logV will be a piecewise smooth function of logr which is linear with slope d only for r < e (l) (assuming ~ is in the interior of the cube). In this situation, estimating dimension from the order statistics o (i) leads to an under-estimate of d due to the presence of boundaries of Id . The geometric structure of the support of the measure whose dimension one is trying to estimate can bias these procedures. To obtain accurate estimates of dimension, one needs information about the shortest length scales for which these geometric effects play a role in the dimen-sion estimates. Similar effects can be seen in dimension estimates for a rectangular solid with different edge lengths but reference point in the center. If the edge lengths of a rectangular solid are 2e1 < 2e2 < • • • < 2ed , then logV is a linear function of logr of slope d-j in the range ej < r < ej+l

3. CANTOR SETS, FRACTALS, AND ATTRACTORS. The final ingredient which affects the estimation of Hausdorff dimensions for attractors is their complicated "fractal" structure. This results in a situation for which the volume function V (r)

X is likely to lose smoothness, even though it can be expected to vanish like rd as r ..,. 0 . I shall present a simple illustration of this phenomenon and then discuss two examples.

Consider the standard Cantor set C c [O ,1] defined by C = { ~ a.3-i I i=l 1

a. = 0 or 2 for all i } . C supports an invariant probability measure J.l for which 1 "" -i 1 each of the sets Cb , .. · , bk = { E a. 3 I a. = b. for 1 < i < k} has volume - .

1 i=l 1 1 1 -- - 2n The volume function V (r) corresponding to l.l is easily computed for arbitrary x;

X I take x = 0 for the sake of elegance. Explicitly, V 0 (r) is given by the Cantor

. a. . function defined first on C by V 0 ( E ai 3 - 1) = E f 2 -I

be constant on each component of the complement of C and then by extending V 0 to . See Figure 2. The function




v

r

Figure 2

v0 (r) has zero derivative almost everywhere, but lim l~gV((r; still exists and is r-+-0 og r

easily calculated to be log 2/log 3 , the Hausdorff dimension of C . A simple estimate of the type of fluctuations in the di!Tlension esti!Tlates which

can be expected due to the lack of smoothness in V 0 (r) can be obtained from the -n -n -n values V0 (3 ) = v0 (2·3 ) = 2 . Choosing these two values of r, we obtain the

values ~~~ ; and 10) 2 log 3 - n log 2

= ~ (1 + .!.. ~) . Thus, fluctuations in the log 3 n log 3

dimension estimates which are associated with the lack of ;.!Tloothness of V (r) decrease logarithmically with the values of r for which there are good volume esti-mates. Consequently, an exponentially growing number of points in a random sample are necessary for increased accuracy in dimension estimates constructed from the values of V(r) for two different values of r . I have not explored the possi-bility that other techniques for estimating lim l~gV((r; will yield fluctuations which r->-Q og r decrease more rapidly than logarithmic with sample size, but call attention to this issue which may represent a fundamental limitation on the accuracy of numerical Pstimates of dimension.

Let me turn finally to a discussion of the types of fractal structures which one expects to find in attractors. Since there is no detailed understanding of the fine structure of a "typical" attractor, it is only possible to describe the structure of individual examples and restricted classes of attractors. The issues involved in this discussion are an active area of research, particularly with regard to the conjec-tures of Yorke et. al. [ 3] about the relationship between Liapunov exponents and




dimensions. Here, I confine attention to two examples, a higher dimensional version

of the generalized baker's transformation discussed by Farmer, Ott and Yorke [ 3]

and a quadratic function mapping an interval to itself. Let r3 be the unit cube in JR 3 and pick numbers a,b1,b2 ,b3 ,b 4 € (0,1) .

Define a mapping F: r3 + 13 by requiring that (l)

(2)

(3)

F is continuous except on the plane x1 = a . DF is the diagonal matrix with eigenvalues (a-1,b1,b2) for

-1 x1 < a and ((l-a ,b 3,b 4) for x1 > a. 3 F has fixed points at the vertices (0,0,0) and (l,l,l) of I

See Figure 3.

a 1-a

Figure 3

The image of Fn has 2n components which are rectangular solids, and the · DFn . d' 1 . . h . ( -k(l )n-k b k n-k b k n-k) mappmg IS a wgona matrix wit entries a -a , 1-o 3 , 2--b 4

for trajectories which have k iterates satisfying x1 < a . The attractor 1\ of F is the product of an interval (in the x1 direction) with a Cantor set (in the x2 ,x3 directions) .

The asymptotic measure for 1\ (which describes the asymptotic behavior of almost all trajectories in 13 with respect to Lebesgue measure) concentrates on a small fraction of the components of Fn (1 3) when n is large and a # ~ . Almost all trajectories spend a proportion approximately a of their iterates in the region 0 < x1 < a because Lebesgue measure in the x1 direction is preserved by F .




Thus the asymptotic measure \J is sensitive primarily to components of Fn which have ~ "' a , k being the number of iterates in the region 0 < x1 < a . The measure \J is therefore concentrated in rectangular solids whose side lengths are

approximately (bt~ 3 n(l-a) ,b 2 n~ 4 n(l-a)) in the (x2,x3) directions. The

number of such rectangular solids grows like (a -a(l-a)- (l-a) )n . Following Farmer et.al. [3] , one can give a complete analysis of the distribution of sizes for the

n 3 components of F (I ) .

Clearly, the dimension of \J will be one plus the dimension of the Cantor set C obtained by intersecting A with a plane parallel to the (x2 ,x3) coordinate plane. To study C we have the following construction which yields both C and the measure v whose product with d~ gives \J • Take the square S = [ 0,1] x [ 0,1] andconstructtworectangles R0 = [O,b1]x[o,b2] and R1 = [l-b3 ,I]x [I-b4] .

The v measure of R1 is a and the v-measure of R2 is (1-a) . Recursively define the Cantor set C and measure v such that C is contained in 2n rectangles

n n R0 , .. · ,R n-l and 2

1 (l) R. = R. 1 1

(2) If A~ is the affine transformation mapping S onto R~ , 1 1

preserving the coordinate directions with their orientations , ~ n ~ n · then R. = A. (R0) and R =A. (R1) . 1 1 i+2n 1

(3) v(R~+l) =a v (RIJ and v (Rn+l) = (1-a)v(R~) 1 1 i+2n 1

See Figure 4.

I +t-Rl I 2 I Rl 1

I R;

1 R2 I Ro

0

Figure 4




One can do explicit calculations of the volume functions V (r) from this X

description of the Cantor set C . For instance, note that the lower left vertex of Rj

( n d. (i-1-d.) n d. (i-1-d.) )

is located at the point l: c.b 3Ib1 I (l-b 3) , .l.: c.b 4Ib2 1 (l-b 4) where i=l 1 1=1 1

d (n-d ) are b nb n and 3 1

. t; 2i-l i-1 n J = u c. and d. = l.: c. . The side lengths of R. i=l I 1 k=l 1 J

d (n-d ) d (n-d ) b 4 nb 2 n and its v-measure is (1-a) n a n . Rather than carrying these

ca.lculations farther, I make two remarks: First, the values of V x (r) will depend strongly on the choice of x E C . Farmer et. al. [ 3] conjecture that as r -+ 0 , V (r) will have a log-normal distribution in x . This dependence of V (r) on x

X X must be dealt with sensibly in computing dimension estimates for v . The second

a (l-a) a (1-a) a (l-a) remark is that if a (1-a) > b 3 b 2 > b 4 b 2 then the measure will concentrate on rectancrles R:t:I whose horizontal and vertical coordinates do not

t J overlap. It follows that the dimension of v in this case will be independent of b 2

b b and b4: d =a log a3 + (1-a) log (l-~) since (a-a(l-a) -(1-a))n disjoint balls of

radius (b 3ab1 (1-a))n will each have measure approximately (aa(l-a) (1-a))n . This

calculation and the corresponding calculation for the case b 3 8 b1 (l-a) > aa(l-a) (l-a) >

b /b2 (l-a) are consistent with the conjectures of Yorke et.al. concerning the rela-tionship between the dimension of \) and the Liapunov exponents of the attractor A.

The second example of an attractor introduced here is a quadratic function f(x) =a- x2 which maps the interval [ -a,a] into itself provided that 0 < a~ 2 It is known that there is a set A c (0, 2] of positive Lebesgue measure such that if a E A , then f has an al)solutf'ly continuous invariant measure \.1 whose dimension is 1 [ 9]. More is known about 11 • In particular, P has a singular

1

density which blows up like (± (x-c)) ":! to one side of each point c of the form fi(O) , i > 0 . These singularities will cause large fluctuations in the function V x(r) as x varies, but the structure of these fluctuations is apparently different from the fluctuations of V (r) in the generalized baker's transformation discussed

X above. The invariant measures of the quadratic transformation are the best avail-able model for the structure that one might expect along expanding directions inside a nonhyperbolic attractor. Since nonhyperbolic attractors are hardly known to exist, only vague speculations about them are possible. Further progress on these matters would be greatly facilitated by additional numerical investigation of examples like the Henon mapping [ 8] .

4. DISCUSSION: FLUID ATTRACTORS. The time has come to consider the analysis of experimental data and the implications of the dimension estimates for a




physical understanding of the dynamics seen in an experiment. A motivating factor

is the ability to clearly distinguish physical systems whose dynamics can be described by a low dimensional strange attractor from systems for which such a description is not possible. There are two issues which are intertwined in this problem, namely dimension and determinism. Strange attractors are deterministic: increased precision in one's knowledge of initial state allows one to make predictions about the evaluation of a trajectory for- increased time.

It is generally difficult to prepare a physical system so that a specified initial

state on a strange attractor can be obtained allowing a direct test for determinism [ 6] . Instead, one prepares an initial state in the basin of attraction for an attractor but must then wait for transients to die and for the system itself to then follow the

attractor until one comes close to the specified state. The sensitivity to initial condi-tions within the attractor implies that the time at which this will happen is unpredict-able and that repetitions of the same experiment will not p:ive reproducible results in this regard. One strategy for obtaining similar initial states is to conduct a long experiment in which a trajectory on an attractor returns close to a state which has already occurred during the experiment. This strategy is reasonable only if the recurrence time is experimentally realistic. If it is too long, then it will be impos-sible to test whether the systems evolves in a predictable fashion from specified initial conditions for a specified period of time.

The issue of dimension intrudes itself directly into the problem of estimating recurrence times. If one has a reference point x for which one wants to estimate recurrences within a distance r , then the expected recurrence time will be propor-tional to (V (r)) -l . In a d-dimensional attractor V (r) is of order rd , so that

X X the recurrence time grows exponentially with dimension. For attractors of moderate dimension (say 10) , returns within distances of order of 1% should simply not be realizable. Thus an estimate of dimension can be useful in preventing one from attempting the impossible.

Laboratory fluid experiments have been a fertile ground for testing ideas about nonlinear dynamics because one passes from regular to irregular dynamical behavior in regimes where it is difficult to construct a reduced model with a strange attractor from the underlying fluid equations. The experimental results have produced a rich phenomenology, only some of which appears to have a direct analogy with the bifur-cations found in low dimensional dynamical systems. Efficient techniques for obtain-ing crude estimates of dimensions should be a useful probe for studying these issues.

I have tried to separate in this paper those features of the fluctuations in pointwise diiPension estimates which are dependent solely upon sampling errors from those which depend upon the complicated geometric structure of attractors. This analysis leads me to assert that the dimension estimates are a practical way of distin-



DIJVIENSION ESTIMATES FOR ATTRACTORS 367

guishing (1) situations in which a physical system is quasiperiodic or aperiodic due

to weak interactions of several modes that have been excited with comparable ampli-tude from (2) situations in which there is a low dirrensional strange attractor which can be described in terms of few modes. Such a distinction is relevant to experi-ments such as Rayleigh-Benard convection where the experimental results depend strongly upon the aspect ratio of the container. Experiments with large aspect ratio present a situation in which the fluid instability occurs initially with a large number of modes of the linearized fluid equations being near marginal stability. In this regime, low dimensional attractors have not yet provided good models for the fluid dynamics. Dimension estimates have the potential for demonstrating conclusively that low dimensional models are inappropriate for these fluid regimes.

BIBLIOGRAPHY

l. R. Bowen, Equilibrium states and ergodic theory of Anosov diffeomor-phisms, Springer Lecture Notes in Mathematics (1975), 470.

2. R. Bowen and D. Ruelle, Ergodic theory of axiom A flows, Inventiones Math. 29 (1975), 181-202.

3. J.D. Farmer, E. Ott and J. A. Yorke, The dimension of chaotic attractors, Physica 7D, 1983.

4. P. Grass berger and I. Procaccia, Characterization of strange attractors, Phys. Rev. Letters 50 (1983) , 346-349.

5. H. Greenside, A. Wolf, J. Swift and T. Pignataro, Impracticality of a box-counting algorithm for calculating the dimensionality of strange attractors, Phys. Rev. A. 25 (1982) , 3453-3456.

6. J. Guckenheimer, Noise in Chaotic Systems, Nature 298 (1982), 358-361. 7. J. Guckenheimer and P. Holmes, Forced Oscillations, Dynamical Systems,

and Bifurcation Theory, Springer-Verlag, 1983. 8. M. Henon, A two dimensional mapping with a strange attractor, Comm.

Math. Phys. 50 (1976), 69-78. 9. M. Jakobson, Absolutely continuous invariant measures for one-parameter

families ofone-dimensional maps, Comm. Math. Phys. 81 (1981), 39-88. 10. S.S. Wilks, Order statistics, Bull. Am. Math. Soc. 54 (1948), 6-50.

DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA SANTA CRUZ, CALIFORNIA 95064







SOLITARY WAVES AS FIXED POINTS OF INFINITE-DIMENSIONAL MAPS IN AN OPTICAL BISTABLE RING CAVITY

D. W. Mclaughlin, J. V. Moloney, and A. C. Newell

Phase-locked solitary waves are shown to be the stable fixed points of an infinite-dimensional map obtained from a bistable optical ring cavity.

PACS numbers: 02.50. + s, 05.40. + j, 42.65.- k

The ultimate goal of these studies is to find methods for analyzing nonlinear systems which exhibit coherent spatial structure and temporal chaos and give every indication of lying on low-dimensional chaotic attractors. Efforts to date have followed the prescription of projecting the field vari-able into some finite-dimensional Galerkin basis thereby obtaining a finite set of nonlinear ordinary differential equations. The equations are then analyzed by direct integration or by converting the continuous time variable into appropriate discrete steps and obtaining a corresponding iterative map. 1 The trouble with the approach is that the truncation associated with the pro-jection is rarely justified and the outcome is often sensitive to the dimen-sion of the basis chosen to approximate the field variable. A key step, therefore, in the analysis of these situations is to find an appropriate ba-sis in which the original infinite-dimensional system is almost separable and in which a few modes can capture its essential low-dimensional character. Solitary waves and solitons, although traditionally associated with inte-grable systems and not with the kind of chaotic dynamics often encountered in maps, are prime candidates for such a basis. Certainly this is a rea-sonable hope when the physics in question can be essentially modeled by an integrable system under external influences. One would then expect there to be certain parallels to the breakup of Kolmogorov-Arnold-Moser surfaces

Reprinted, by permission, from Physical Review Letters~ (1983), 75-78, ©1983 The American Physical Society.

369




370 D. W. MCLAUGHLIN, J. V. MOLONEY, AND A. C. NEWELL

and homoclinic orbits in finite-dimensional systems such as the driven damped pendulum.2

In this Letter we make a start in the right direction by showing that solitary waves can be the fixed points of infinite-dimensional maps. In particular, our approach allows us to explain the spatial rings found by Moloney and Gibbs3 in their numerical model of a bistable optical ring cavity. We anticipate that in wider parameter ranges we will find that these rings exhibit temporal chaos while yet maintaining a spatially coherent structure.

Consider Fig. 1(a). A unidirectional polarized input laser beam, propagating the z direction, with an input profile which is Gaussian in one transverse direction, enters a nonlinear medium at point H, z = 0 at t = 0. After propagating through the medium to point I, z = L1, the beam is then reflected through four mirrors (two of which reduce its intensity) so as to re-enter the nonlinear medium at time t 0 = (L1 + ~)/c, and reinforce the original pump field. The signal continues to circulate around the cavity and our goal is to predict the output after many passes.

The input laser beam signal is Ein(x,z,t) = 2RE{A{x)exp[i{kz-wt)]}r, k = w/c, where A(x) has a prescribed shape. In the nonlinear medium

- -E{x,z,t) = 2Re{B(x,z,t)exp[i(kz - wt}[}r, ( 1 }

where the envelope B satisfies

{2}

Here a 0 is the linear absorption per unit length, 6 is the laser-atom detuning normalized to the dipole decay width, and F, which we assume to be large, is the Fresnel number measuring the transverse diffraction of the beam. To describe the evolution of the field through the nonlinear medium by Eq. (2}, we have assumed (see Ref. 3 for details} that (a} the nonlinear medium inversion and dipole relaxation times are short compared to the cavity roundtrip time t0 and that (b) there is no significant steepening in the propagation direction. In the linear return medium IJKH, we simply assume that the field satisfies the linear wave equation,

E{x,z = L,t} = E{x,z = L1't - ~/c), L = L1 + ~·



SOLITARY WAVES AS FIXED POiNTS

The infinite-dimensional map is obtained as follows. Consider the field E(x,O,t) at z = 0 in the various time intervals In, (n- 1)t0 < t < nt0•

Lz --- Lr

a (a) (b)

21 2.1

0 064 X

(c) (d)

FIG. 1. (a) Unidirectional passive ring cavity containing a non-linear (two-level atom) saturable medium of length L1. (b) Schematic of the plane-wave bistable loop obtained from the steady-state solution (fixed point) of Eq. (7), for R = 0.9, p = 2, and kl = 0.4 rad. When transverse effects are in-cluded [solve Eq. (5) with boundary condition (6) and F= 200] stationary N-solitary-wave trains appear on the high-transmission branch at equal increments of a(O), the peak input Gaussian field amplitude (N = 1, 7 are indicated by upward arrows). The same parameter values are used to generate all subsequent results. (c) Initiation of the seven-solitary-wave train from the sharp gradients on the outer edges of the transverse laser beam (intracavity field on the 23rd pass). The final steady-state transverse pulses are confined to the interval (x ,x ) + -= (0.61, - 0.61). The lower trace in this figure shows the much broader Gaussian profile after propagatipg through the non-linear medium once. (d) Steady-state-seven-solitary-wave train at a(O) = 0.194 evolving from the transient in (c) after 200 cavity passes.

371



372 D. W. MCLAUGHLIN, J. V. MOLONEY, AND .A. C. NEWELL

~ iwt ~ For o < t < o0, E(x~O,t) = 2Re[../TA(x)e Jr. For t 0 < t < 2t0, E(x,O,t) = 2Re[../TA(x)e-lwt]r + RE(x,L1,t- ~/c). Here R is there-flection coefficient at the mirrors at H,I (a !urge fraction R of the intensity is reflected) and T the transmission coefficient. Ob-serve that B is a function of t only through ~ = t - z/c and Eq. (2) tells us how B evolves as a function of z for fixed ~.

The dependence of B on ~ is determined by the data at z = 0. But A(x) is independent of t and therefore, on the first pass, ·

-iwt B is. Similarly in the nth interval In, the envelope of e at z = 0 is independent of t. Therefore w~ can simply replace the dependence of B on t by stating in which interval it has been gene-rated. On .the nth pass the initial condition at z = 0 is

and so we can write

80 = 0. Our task is to determine the limit of Bn(x,z = 0) as n ~ m

if it exists. It is useful to introduce the new variables

Y = .Jfx,

whence (2) and (3) become

. aGn fGn Gn 21- + --::--2: - = 0,

a~; ay 1 + 2GnGn

ikL Gn(y,O) = a(y) + Re Gn_ 1(y,p),

n > 1, G0 = 0. In (5) we have omitted the attenuation and other terms of order 1/~ as they are assumed to be small. Equation (6), which acts as the initial data for the solution of (5) during the nth pass through the nonlinear medium, is the infinite-dimensional map of interest.

(3)

(4)

(5)

(6)



SOLITARY WAVES AS FIXED POINTS 373

In order to understand the dynamics in the case when f >> 1 and the input transverse profile is Gaussian-like, it is first necessary to understand what happens when a(y) and therefore G are independent of the transverse coordinate. This is called the plane-wave case. Then the solution of (5) is trivial and one finds

where g(IGI) = (1 + 2IGI2)-1. In this case (6) is a one-dimensional com-plex map from one member of the sequence [Gn{y,O) = gn]n=O to the next:

Its fixed points as functions of the input field a are shown in Fig. 1(b). For a range of a and p the map exhibits bistable be-havior. As the input amplitude is increased past the point a1, the output field jumps to the upper branch at U. [As the parameter ranges are broadened, a wide variety of behavior is possible. For example, the graph in Fig. 1(b) can have more than one S bend leading to multiple fixed points. Also for a fixed a and increasing p or vice versa, period doubling sequences leading to chaotic behavior are observed4; but this is not the phenomenon we focus our attention on here.]

(7)

Now look at an input field whose transverse profile is Gaussian-like. If f >> 1, then the effects of diffraction are initially negligible and at each y the beam behaves as if it were a uniform plane wave at that amplitude. But, from Fig. 1(b), we see that those points of the Gaussian profile for which a(y) > a1 will switch up to the upper branch and those parts for which a(y) < a1 will stay on the lower branch. Therefore at x± where a(y±) = a1 the derivative of the response field is very large. At these locations, diffraction is important and beginning at the edges x+ and x narrow pulses of width ~x = 0{1/Jf) are generated which eventually fill out the region between x and and become the steady-state response of the system. This indeed situation observed numerically and shown in Figs. 1(c) and 1(d).

X+' is the What we

now show is that these pulses are the solitary waves of Eq. (5). What is particularly new in this study is that the solitary-wave parameters (ampli-tude and phase) are not determined by initial conditions but by the stable fixed points of the map (6). We develop a theory that predicts their values and our theoretical results are in excellent agreement with the results obtained by numerical experiment. Further, the number of pulses is




a function of the transient shape realized after a few passes [Fig. 1(c)] and is proportional to ~f and the maximum input field amplitude.

Taking advantage of the symmetry in y, the solitary wave of (5) is

Gs(y,z;;) = P(>.y,>.) exp[i(>-2 - 1)r,;/2 +ir],

wheretheshape P(e,>.) satisfies (e=>-Y)

[If we had used a Kerr nonlinearity obtained by taking the small-P limit of (9), the last term in (5) would be -Gn + 2~G~ and P = >. sech(e), the soliton of the nonlinear Schrodinger equation.]

2.1

IGnl

·..,____ 0 l__--'----'--..L.C..:o.' ........

-0.1 0 01 -0.1 X

(a)

0 X

(b)

0.1

FIG. 2. (a) Comparison of the single-solitary-wave shape (N = 1) at a(O) = 0.1 predicted from the fixed-point equa-tion (10) (dashed line) with the numerical solution of Eq. (5) (over 200 cavity passes) with boundary condition (6) (solid line). The slight discrepancy in height (-1.3%) may be due to the back-ground radiation evident in the full numerical solution but ignored in the fixed-point equation. (b) The central peak of the seven-solitary-wave train [see Fig. 1(d)] compared to the shape predicted by the fixed-point equation [a(O) = 0.194]. The discrepancy in fit (-5%) is consistent with perturbative estimates of changes in shape due to interactions with nearby solitary waves.

We now sketch the basic ideas of our theory. If the equation (5) of profile evolution in the nonlinear medium were indeed the nonlinear Schrodinger equation and therefore integrable, we could decompose Gn(y,O) into its soliton and continuous spectrum basis (in which basis the equation

(8)

(9)



SOLITARY WAVES AS FIXED POINTS 375

is separable) and, using (6), find the soliton and continuous spectrum parameters of the field envelope on the nth pass in terms of those on the (n- 1)st pass. This would give a representation of the infinite-dimensional map in a separable basis. In most cases, we expect the soliton content of the data to dominate that of the continous spectrum and therefore the map would naturally reduce to a finite-dimensional one. Further, if the solitons were widely separated [as the solitary wave pulses of G are in Fig. 1(c)], then to a good approximation, one could obtain a map (, , ) to (' , ) for each soliton individually. An-1,An-1 An•An

Our problem with a saturable nonlinearity is not integrable and we cannot separate the equation in a soliton basis; nevertheless, we can ask what solitary wave P(e,>.n)e~r would emerge on the nth pass from the initial conditions

Gn(y,O) = a(y) + RP{e,>.n_1Jexp[irn_ 1 + ikL + {i/2)p(>-~_ 1 - 1)].

Expressing this idea mathematically defines a map M from (An_ 1,rn_ 1) to (>. rn). (The underlying assumption, which is not necessary in the

n' integrable case, is that the change 6>.n_ 1,6rn_ 1 on each pass is not too great.) This Letter does not afford us the space to go into the mathemati-cal analysis. in preparation. be written

This and other details will be given in a longer paper The condition that M has a fixed point (>-,r) can

cosr<P,a(e/>.)> = {1 - R cos [kL + ~p(>- 2 - 1)]}<P,P>.

In (10), <P,Q> = J~ P(e)Q(e)de. -(I)

This is the principal analytic

( 10)

result of the Letter. We solve {10) numerically for a number of cases with different parameter values and compare our predicted solitary-wave shapes and amplitudes with those obtained by numerical experiment. These results are summarized in Figs. 2(a) and 2(b).

We emphasize that these solitary waves, which are the transverse nonlinear normal modes of the cavity, are the infinite-dimensional analogs of the upper-branch plane-wave fixed points given in Fig. 1(b). By analogy with that case and the low-Fresnel-number results of Ref. 5, we expect that, as the parameter ranges are broadened, a rich variety of behavior will occur. This should include period doubling and other routes to states




of temporal chaos in which the solitary wave amplitudes flutter chaotically but in which states an overall spatial coherence is maintained.

We thank the U. S. Army Research Office, the U. S. Air Force Office for Scientific Research, and the National Science Foundation for support of these studies.

1see A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer, Berlin, 1983), p. 446.

2A. R. Bishop, K. Fesser, P. S. Lomdahl, W. C. Kerr, M. B. Williams, and S. E. Trullinger, Phys. Rev. Lett. 50, 1095 (1983); J. C. Eilbeck, P. S. Lomdahl, and A. C. Newell, Phys. Lett. 87A, 1 (1981).

3J. V. Moloney and H. M. Gibbs, Phys. Rev. Lett. 48, 1607 (1982).

4An enormous literature exists on optical bistability and associated instabilities. A comprehensive review of the subject may be found in Optical Bistability, edited by C. M. Bowden, M. Ciftan, and H. R. Rob! (Plenum, New York, 1981). The first prediction of chaos in a ring cavity was by K. Ikeda, Opt. Commun. 30, 257 (1979) [see also K. Ikeda, H. Daido, and A. Akimoto, Phys. Rev. Let~ 45, 709 (1980)].

5J. V. Moloney, F. A. Hopf, and H. M. Gibbs, Phys. Rev. A 25, 3442 (1982). This work predicts that the transverse profile will undergo transi-tions to chaos via periodic, quasiperiodic, and frequency locking when the Fresnel number is small.

D. W. MCLAUGHLIN, J. V. MOLONEY, AND A. C. NEWELL APPLIED MATHEMATICS AND OPTICAL SCIENCES CENTER UNIVERSITY OF ARIZONA TUCSON, ARIZONA 87521




HOPF BIFURCATION AND THE BEAM-PLASMA INSTABILITY*

John David Crawfordt

ABSTRACT. For finite mode instabilities in dissipative systems, invariant manifolds methods allow the bifurcation analysis to be reduced to the locally attracting center manifold. In a kinetic model of electron plasma dynamics, these methods are applied to the one mode beam-plasma instability which occurs via Hopf bifurcation. The instability results in a nonlinear oscillation, and the amplitude equation can be solved to describe the time asymptotic state.

Introduction

I shall describe results on a simple plasma model which exhibits a beam-plasma instability as a Hopf bifurcation. The analysis is geometric in that it utilizes the notion of an invariant center manifold to reduce the problem to two dimensions. Since Hopf bifurcation can be analyzed by other techniques as well, 1 it is worthwhile to rev.ew those aspects of the geometric methods which are particularly valuable.

In dissipative systems, even in infinite dimensions, when an equilibrium becomes unstable due to a finite number of unstable linear modes, the nonlinear dynamics of the instability is finite dimensional, i.e. the dynamics may be described by a finite dimensional vector field. An appreciation of the stable and center manifolds associated with the equilibrium makes this fact simple and intuitive.

In addition to this conceptual virtue, the geometric point of view offers practical advantages as well. By deriving (approximately) the vector field on the center manifold, one obtains the finite dimensional dynamics which captures all the (local) qualitative behavior exhibited by the instability. Although

AMS Subject Classification: 54Fl4, 76X05. *work supported by the Director, Office of Energy Research, Basic Energy Sciences Division of the U.S. Department of Energy, under contract DE-AC03-76SF00098.

+Current address: Physics Department, University of California, San Diego, La Jolla, CA 92037.

377





378 J. 0. CRA~·JFORD

these results are essentially perturbative and therefore only apply near the threshold for the instability, there is no requirement that a specific parameter be chosen as the bifurcation parameter, and the relevant finite dimensional system may be derived without a priori assumptions about the form of the solutions, e.g., fixed points, periodic orbits, tori, strange attractors, etc. This is very helpful when the instability involves more than one or two d. . 2 1mens10ns.

Invariant Manifolds for Fixed Points

The general setting assumes an evolution equation

X E M ( 1 )

defined on some phase space M. £11 is a 1 i near operator and N11 ( · ) is a smooth nonlinear operator satisfying N)Ol = 0. Both £11 and N11 may depend on physical parameters 11· M may be a finite dimensional manifold, or an appropriately chosen function space for infinite dimensional problems. (The technical issues involved when proving the existence of center manifolds for partial differential equations are discussed by Ho.lmes and Marsden. 3)

The state x = 0 is an equil ibritAm and its linear stability is governed by the spectrum of £11 , denoted o(£11 ). Emphasizing here the finite dimensional case for illustration, suppose o(£\1) contains ns eigenvalues in the left half plane, nc eigenvalues on the imaginary axis, and nu eigenvalues in the right half plane, as shown in Figure la. Barring the occurrence of degeneracy,

ns + nc + nu dim M

and the associated eigenvectors span the stable subspace Es, the center

lm I

I ®

I

® f ---0----®-- --+------0-------Re

l ® ® ~I

Figure 1. (a) A tl]pical spectrum for £\1 consisting of real and conjugate pairs of eigenvalues. In infinite dimensional problems, £\1 may also have continuous spectrum. (b) The linear invariant subspaces Es, Ec, and Ell determine the structure of the linearized flow.

XBL 838-3011



HOPF BIFURCATION & THE BEAM PLASMA INSTABILITY 379

subspace Ec, and the unstable subspace Eu, respectively. These subspaces are invariant under the linearized dynamics

dx = £ x dt l.l

and determine the qualitative features of the linearized flow, as shown in Figure lb.

When the nonlinear effects represented by Nl.l(x) are restored, the dynamics of linear eigenvectors are coupled, and the linear subspaces Es, Ec, and Eu are no longer invariant. There are, however, nonlinear analogues of the linear subspaces. Intuitively, the nonlinear effects distort the solutions of the linear eigenspaces so that the flat linear eigenspaces are "warped" into curved surfaces or manifolds. These manifolds organize the dynamics of the nonlinear problem just as the linear eigenspaces serve to structure the linear dynamics.

Thus, associated with the linear subspaces are the local stable, center, and unstable manifolds, denoted Ws, We, and Wu, respectively (see Figure 2a).4

X = /

/

/ /

----Eu ~ u w

a(L ): Hopf JJ

Conjugate eigenvalues

Re

Figure 2. (a) The invariant manifolds ws' we and ~~u are the nonlinear analogues of the linear subspaces. (b) At criticality for a nondegenerate Hopf bifurca-tion, the spectrum of £l.l has a simple conjugate pair of eigenvalues on the imagi-nary axis.



380 J. D. CRAWFORD

Each local manifold contains x = 0 and is tangent at x = 0 to the appropriate linear subspace. Thus each manifold has the same dimension as its associated linear subspace. Furthermore, each local manifold is invariant with respect to the fuU nonlinear dynamics: if an initial condition x(O) belongs to W5 ,

We, or Wu, then for O<t<T, the solution x(t) to (1) lies within the manifold containing x(O).

The dynamics of solution curves in Ws or Wu is trivial, at least near x=O. As t-+oo solution curves in Ws approach x=O, and as t-+ -co solutions in Wu approach x=O; in both cases the asymptotic rate of approach is exponential since the linearized dynamics dominates. No such simple charac-terization is possible for the dynamics in We; at x=O the linear stability is neutral and nonlinear effects remainsessential. ~Jhen the dimension of we is greater than two, the center manifold dynamics may encompass all the complex dynamics studied in dynamical systems theory: aperiodic motion, chaotic recurrence, Smale horseshoes, strange attractors, etc.

Hopf Bifurcation

Suppose for J.l < 0 the equilibrium x=O is stable (nu = nc = 0, n5 = dim M) but as J.l increases through J.l = 0 a simple conjugate pair of eigenvalues >.,>. crosses into the right half plane, then we typically have a Hopf bifurcation; see Figure 2b for the spectrum at J.l = 0. Denote by ljJ and ~ the eigenvectors of this pair, sv

For the application considered here, at bifurcation a stable periodic orbit emerges from the equilibrium; physically one observes a time independent state yield to a single frequency oscillation. In Couette flow a beautiful example of this phenomenon is the transition from the Taylor vortex state to the flow with a rotating wave. 5

For the spectrum of Figure 2b, the fixed point x=O will have a two-dimensional center manifold and a stable manifold of dimension (dim M)- 2; see Figure 3a. As J.l increases above J.l = 0, the ce11ter manifold is replaced by a two-dimensional unstable manifold (assuming no other spectral elements cross the axis). Let WJ.l denote this one parameter family of two dimensional invariant manifolds, i.e. WJ.l=O = We and W].J>O = Wu; for J.l near 0. WJ.l is ZocaZZy attracting because of the strong contraction provided by W5 in the "transverse" directions. 6 Specifically, there is a neighborhood of x=O such that all solutions remaining in that neighborhood as t--roo will approach ~/J.l'




/ tp/

~----tP /

/ /

t I ~X= 0

jl

/' Es I I I

~w--___ ~~ -:--&A•!• h (A, A)]

/ I I ,.. / I , / l I ,' /,:: ______ ----•'-

tp/ I (A, A, 0) I I

XBL 838·3005A

Figgre 3. (a) The center manifold W for nondegenerate Hopf bifurcation is two dimensional. (b) Near the fixed point, the invariant manifold w~ may be described as the graph of a function h(A,A).

Thus the flow on Wll, which is given by a two dimensional vector field, describes the time asymptotic motion. This reduction in dimension as t ~ oo

greatly simplifies the problem, and suggests the following strategy: "project out" the two dimensional vector field which describes the flow on Wll, then analyze that flow to understand the asymptotic development of the instability.

The Model

Consider a one dimensional plasma of finite length L and periodic boundary conditions. For the study of high frequency electrostatic waves, the ion dynamics may be neglected, and the electron distribution function F(x,v,t) evolves according to

.££. + .££. + ~ .£1 aF at v ax m ax av C(F)

_ti 00 = 41Ten 0 [1- J dv F(x,v,t)] ax 2 -"'

(2)

where (x,v) are the electron position and velocity, n0 = N/L is the mean density, and -elm is the electron charge/mass ratio. 7 Normalize F such that



382 J. D. CRAWFORD

nor dv F(x,v,t) n(x,t) density of electrons and -oo

L r J dx n(x,t) N

0

For the collision model C(F), take the Krook operator

where 'J > 0, and c

C(F) = \l (F -F) c eq

F (x v) " !lJ~~-tJ g . (v) eq ' n 0 eq

This collision model conserves the particle density since lf)

f dv C (F) 0 J

-00

but not energy or momentum. 8

For an initially homogeneous, nonequilibrium state, geq(v), the self-consistent electric field is zero,

F ( x v t=O) 1 ' '

~-±2_ ,. 0 ax t=,O

and the dynamics reduces to

with solution

F (x,v,t) 1

= g (v)+{F(v)-g (v)) ·eq o eq -v t e c

Since C(F) conserves n(x,t), no spatial dependence develops, so

8¢, _. = 0 ()X

for t?>O.

If, however, the initial condition is perturbed so that

F (x v tcoQ) = F (v) + f(x,v,t=O) 2 ' ' 0

then, depending on the shape of F0 , the resulting electric fields may drive the growth of unstable waves. To study this one writes an equation for '\ f = '\F2 - all ,



HOPF BIFURCATION & THE BEAt4 PLASMA INSTABILITY

= -v c [ f - geq J oo dv' f ( x, v' , t)] -oo

00

= 4'!fen 0 J dv f(x,v,t) -oo

aF1 aF where the approximation av ~ a: has been used in the nonlinear term to obtain an autonomous description of f(x,v,t) . 7 By introducing the Fourier expansions,

\' ) ikx f(x,v,t) = L fk(v,t e k

383

(3)

where ~k = 21f/L, and eliminating ~f' then (3) yields an evolution equation for f,

where

with

af = £f + N(f) at

The linear operator in (4) is defined by

L f = -v f 0 0 c 0

Lkfk -[(ikv+vc)fk + iknk r dv' fk(x,v',t)], k~O -"'

and the nonlinear operator in (4) is

N(f) = \' eHx iw~ afi-k I"' dv' L f - fk(x,v' ,t) R. k~O k av

-oo

(4)

Here w~ = 4rre 2 n0 /m is the plasma frequency. In (4), the point f=O is a stationary solution; physically, it corresponds to the distribution function F (v) .

0



384

S..VI .,.... Q)

<0:::3 / o..--<0

Q) > ..,c

lmA

J. D. CRAWFORD

4--0

VI ..,IV Q) :::3

.-- .--

im;>,

o(L): •c=O

<OQ) 0> 0>

------------r------------ReA g- ~ --------+------------ ReA ·2·;v \ 0~ ~ uo

0

t~~ c:rcv

\Continuous Spectrum 'Continuous Spectrum

Figure 4. (a) Typical spectrum of the linear operator in (4) for Vc > 0. The continuum is always present. The roots of the dispersion function determine the discrete eigenvalues. (b) For Vc = 0, o(£) reflects the Hamiltonian structure of the dynamics. At criticality, the eigenvalue quadruplet collapses to a degenerate conjugate pair embedded in the continuum.

Linear Stability

XBl 637-471

The calculation of o(£) follows Case closely and will be described elsewhere. 7•9 The results are illustrated in Figure 4a. There is a line of continuous spectrum at ReA = -vc with eigenfunctions which are distributions. There may also be discrete eigenvalues; these are determined by the roots of the dispersion function,

Z E C (5)

0 and o(£) contains a conjugate eigenvalue pair

with eigenfunctions

ikx(~) e v- z ' 0

e -i kx ( -n~ ) v- z0

satisfying

£lj!

In the limit vc-+ 0, J\(z) is even in k, and the eigenvalues of £ form



HOPF BIFURCATION & THE BEAM PLASMA INSTABILITY

quadruplets, as shown in Figure 4b. The fundamental reason for this is the Hamiltonian structure of the Vlasov-Poisson system. 10

Following Case, define the inner product L oo

< ¢,1)!> = I dx I ¢(x,v)1jJ(x,v) 0 00

and the adjoint opera tor £ + ,

The eigenfunctions of £+ and £ satisfy biorthogonality relations analogous to those found by Case for the vc = 0 problem. 9 In particular, the adjoint eigenvectors for 1.. 1 and 1.. 1 ,

+-£ 1jJ

385

satisfy <1)!,1)!> = <~.~> = 1, and are orthogonal to all other eigenvectors of£.

When the initial state F0 (v) is sufficiently close to geq(v), then cr(£) is contained within the left half plane, as shown in Figure 4a. 7 If F0 (v) is distorted, for example by adding a component of electrons at high velocity ("the beam"), then it is possible for a complex conjugate pair of eigenvalues 1.. 1 ,A. 1 to cross the imaginary axis signa1 ing the onset of growing electrostatic waves. For suitably chosen length L, the first unstable mode corresponds to the minimum wave number k = 2n/L. The remainder of this discussion concerns the analysis of this one mode instability.

Center Manifold Dynamics

As the procedure for calculating the center manifold vector field is fully discussed in recent texts, I will simply outline the calculation. 4 •6

By defining the complex amplitude

A(t) = < ~.f>

the perturbation may be decomposed as

f(x,v,t) = A(t)1jJ(x,v) + A(t)~(x,v) + S{x,v,t) (6)

where < 1jJ,S> <~,S> = 0. The evolution equation (4) is then rewritten

A(t) = AlA + ( ~. N(f)) (7)

~i = £5 + N(f) - <~, N(f)>- <~. N(f)>. Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.


386 J. D. CRAWFORD

To restrict (7) to W~, the two dimensional invariant manifold associated with the bifurcation ( ~ now denotes the physical parameters in the distribution function), introduce local coordinates on W~ near the fixed point; see Figure 3b. The function h(A,A) measures the deviation of W~ from the (1/J,~)-plane. An element of W~ near the fixed point therefore has the form

c - - -f (x,v,t) = A(t)l)J(x,v) + A(t)l)J(x,v) + h(x,v, A(t), A(t)) (8)

Inserting (8) into (7) yields

which is an autonomous equation for A(t); this determines the evolution of fc in (8).

To make practical use of (9) requires an explicit representation for h(A,A). An equation for h follows from the observation that for solutions fc in (8) there are two ways to compute the rate of change of Sc(x,v,t) = h(x,v,A,A). From (7) directly:

( 10)

or from (8)

( 11)

Equating the right-hand sides of (10) and (11) provides the desired equation for h which is then solved using a power series in A,A. To lowest nontrivial order, this power series has the form

where

e i2kx h~~(v)

h( 1 )(v) and h( 2 )(v) are explicitly computable functions of velocity. 7 2k -

Knowing h(A,A) to quadratic order in A determines (9) to cubic order




where

B

H rdv hW(v) -00

lj!zo = ( v--n~o) In polar variables, A = p(t) eiB(t), the amplitude equation (12) becomes

B = Im ;\. 1 + (Im S)p2 + ...

When the conjugate pair associated with the instability has just entered the right half plane, then 0 < Re ;\. 1 « 1, and the instability will saturate at small amplitude, i.e., p=O for O<p«l, only if ReS< 0. The saturated amplitude is

Results for Lorentzian Beams and Plasmas

For simplicity, let the equilibrium distribution be

and take as the initial distribution (corresponding to f=O),

where nPFP is the plasma distribution, nbFb is the beam distribution, a is the plasma temperature, 6 is the beam temperature, and u is the beam velocity. Consider the case of a low-density cool beam; specifically, choose

E = \!clwe = 0.001

nb/n 0 = 0.05

6/a = 0.5 ,



388

u

J. D. CRAWFORD

2.0 £ = 0.001 k = 0.17 //

0 = 0.5 nb= 0.05

1.0

w Unstabl\ Stable

\ la 0 \ I ' ~~!A

-0.10 0.5 1.0 1.5

ku XBL838-3007

Figure 5. (a) Bifurcation surface for a cool low-density beam (solid line). Also shown is the bifurcation surface for (dashed line); their intersection marks a double Hopf bifurca-tion. Points A and B for k = 0. 17 are the selected points of low and high growth rate. (b) The four roots of the dispersion function for k = 0. 17. The real part of the frequency w = Re k z0 , is plotted against the drift frequency ku in units of we. Only on the indicated branch is condition (15) satisfied.

The roots of the dispersion function vary with the wave number k and the beam velocity u. Since 1. 1 = -vc- ikz 0 , points in the (k,u)-plane for which Imz 0 =~elk correspond to criticality. This surface is the solid curve in Figure 5a. For k = 0.17 the real parts of the dispersion roots appear in Figure 5b as a function of u. The condition for instability:

vc >T

is fulfilled along the indicated branch of the dispersion function.

( 15)

The cubic coefficient 13 has a negative real part along the bifurcation 7 surface in Figure 5a, and also along the vertical line marking k = 0.17.

We may, therefore, evaluate the approximate expression for the saturated distribution,

where

F2 (x,v,t) ~ F0 (v) + As(t)w(x,v) + As(t)w(x,v)

+ A2 h(1)(v) ei2kx + lA 12 h(2)(v) + A2 h(!) -i2kx s 2k s o s 2k e + ···

e(t)

ie{t) Ps e

[Im 1. 1 + (Im 13)p~ + ... ]t + 8(0)




0.3r-----Wave frame

1 point A

I I

I I i

0.1 r- I I

1

./// !

o L_ _ __.__...__J.__.__j _ _j,~l -7-+- I

0.2

F (v' + v ) \ 0 w

~-------------,-···-· -·-------···-----·-

4 x 104 1 · Wave frame : : : ::l po;nt A 1

-: : :; ~-- h~'l(v' + •: __ ]

I I 1o4 r-· I 104 ~ I J L_ l _ _j_J_ ___ _

-2 X

-3 X

-3-2-101234 -2-1 0 1 2 v' v'

XBL838·2977

Figure 6. (a) Initial velocity distribution at point A as seen in the wave frame. (b) The lowest order correction to the spatially homogeneous component of the distribution function. Shown for point A.

Of greatest interest is the spatially homogeneous component

( 16)

as seen from the frame of reference moving at the phase velocity of the wave The coordinates of this frame are (x' ,v'),

X' X-V t v, = vw = 1 de w v- vw - k Cit

Figure 6a shows the unperturbed distribution corresponding to point A in Figure 5a where the growth rate y of the instability is relatively small. Figure 6b shows the shape of the lowest order correction to the unperturbed distribution. The basic effect is to slow down the resonant particles which move at v' ~ 0. The saturated component of (16) for point A appears in Figure 7.

At point B in Figure 5a the linear growt~ rate is approximately two

vw.

orders of magnitude larger. The shape of h~ 2 ) in Figure 8a now reveals some broadening of the plasma distribution as well as a strong acceleration of the resonant particles. Figure 8b shows the unperturbed and saturated distributions.

Fi na 1 Remarks

1) Although I have concentrated on the one mode instability, this model exhibits a codimension two double Hopf bifurcation. The dashed curve in Figure Sa is the bifurcation surface for the mode at 2k. For the parameters



390 J. D. CRAWFORD

' Wave frame I point A 1

y = 9.4 X 10- 3

yh = 9.4 p:= 2.9 X 10-B

F0 (v' + vw) + p~h~l(v' + v w)

''-~""~

\\,

I I

~~----'----"----~__[_ __ ! _ _1__~---===----===

-2-1 0 1 2

v' XBL 838-2974

0.1

Figure 7. For point , the homogeneous component of the saturated distribution function showing the effect of the lowest order correc-tion. y = ReA is the linear growth rate.

F0 (v' + v) y = 0.085 yh = 85.0 e! = 8.59 x 10-s

2 4 6 8 10

v' XBL 8310-12143 Figure 8. (a) For point B, the lowest order correction to the

spatially homogeneous component of the distribution function. (b) For point B, the initial velocity distribution (dotted line) and the homogeneous component of the saturated distribution (solid line) showing the effect of the lowest order correction.

at which the two bifurcation surfaces cross, there are two conjugate pairs of eigenvalues on the imaginary axis simultaneously. Here the dynamics of the instability is four dimensional and one expects much more complicated phenomena. 4

2) ~Jhen vc + 0, the expansion coefficients h(l) and h(z) in the power series for h become singular at v = vw. The presence of these singular



HOPF BIFURCATION & THE BEAM PLASMA HJ'TABILITY 391

resonance dominators is intimately related to the Hamiltonian structure of the collisionless model, and to the fact that at criticality the eigenvalues are now embedded in the continuum; see Figure 4b. Whether the collisionless instability has finite dimensional dynamics is not known. Previous treatments have assumed that the continuum does not affect the dynamics. 11 In light of the delicacy of Hamiltonian bifurcations, this assumption needs specific justification. An analysis of the collisionless instability which will extend the geometric approach discussed here, and explicitly acknowledge the Hamilton-ian structure of Vlasov-Poisson dynamics is underway.

Acknowledgments

I am grateful to H.Abarbanel, A.Kaufman, and J.Marsden for their advice and support.

REFERENCES

1. G.Iooss and D.D.Joseph, Elementary Stability and Bifurcation Theory, Springer-Verlag, Berlin, 1980.

2. J.Guckenheimer, "Multiple bifurcation problems of codimension two," Siam. J. Math. Analysis, in press, 1984.

3. P.J.Holmes and J.~1arsden, "Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis," Automatica 14 (1978), 367. -

4. J.Guckenheimer and P.Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer Applied Math. Sciences 43 (1983). -

5. H.Swinney, these proceedings; R.C.DiPrima and H.Swinney, "Instabil-ities and transition in flow between concentric rotating cylinders," in H drod namic Instabilities and the Transition to Turbulence, Topics in Applied Physics, 45 1981 , Springer-Verlag, Berlin; also D.Rand, "Dynamics and symmetry:predictions for modulated waves in rotating fluids," Arch. Rational Mech. Anal. 21 (1982), 1.

6. The one parameter family of manifolds W is the three dimensional center manifold of the "suspended" system. See reference 4 and B.D.Hassard, N.D.Kazarinoff, and Y.-H.Wan, Theor and A lications of Ho f Bifurcation, London Math. Society Lee. Notes~. Cambridge Univ. Press, Cambridge 1981).

7. B.Small, J.D.Crawford and H.Abarbanel, Effects of Collisions on Longitudinal Plasma Oscillations, preprint (1983); J.D.Crawford, Hopf Bifur-cation and Plasma Instabilities, Ph.D. Thesis, University of California, Berkeley (1983); J.D.Crawford and H.Abarbanel, Hopf Bifurcation in Plasma Kinetic Theory, preprint (1983).

8. P.L.Bhatnager, E.P.Gross and M.Krook, "A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-componentsystems," Phys. Rev. 94 (1954), 5ll; N.A.Krall and A.W.Trivel-piece, Principles of Plasma Physics-;-McGraw Hill, New York (1973).

9. K.Case, "Plasma oscillations," Ann. Phys. ]__ (1959), 349. 10. R.Abraham and J.Marsden, Foundation of Mechanics, Addison-~Jesley,

Reading, MA (1978); P.Morrison, "The Maxwell-Vlasov equations as a continuous Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.


392 J. D. CRAWFORD

Hamiltonian system," Phys. Lett. BOA (1980), 383; J.Marsden and A.Weinstein, "The Hamiltonian structure of the Maxwell-Vlasov equations," Physica 4D (1982), 394. ---

11. A.Simon and M.Rosenbluth, "Single mode saturation of the bump-on-tail instability: immobile ions," Phys. F.luids ]1 (1976), 1567.

LAWRENCE BERKELEY LABORATORY and PHYSICS DEPARTMENT UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CALIFORNIA 94720




SOME REMARKS ON CHAOTIC PARTICLE PATHS IN TIME-PERIODIC, THREE DIMENSIONAL SWIRLING FLOWS

Philip Holmes1

ABSTRACT. We outline potential applications of global bifurcation theory and Melnikov methods to the study of three dimensional, time periodic velocity fields in incompressible fluid flows. We concen-trate on the example of vortex breakdown in swirling flows and the description of particle paths in the fluid.

1. INTRODUCTION. In this brief note I wish to suggest an area in which low dimensional dynamical systems theory might profitably be applied to the study

of fluid dynamical problems. The basic idea is to obtain a fuller under-

standing of the complex fluid motions which often occur in 'pre turbulent'

velocity fields. This idea is not new; previous versions have been suggested both by fluid dynamicists (Cantwell [1981], Cantwell et al. [1978], Aref

[1983]) and by dynamical systems theorists (Broer [1980]). Aref's paper in

particular provides a conceptually simple example of a two dimensional flow

subject to time periodic perturbations in which chaotic advection of fluid

particles occurs. The situation here is slightly different, however, in

that we intend to apply the methods to time periodic, non-axisymmetric three dimensional incompressible flows. An interesting class of such flows arizes in vortex breakdown (Leibovich [1983]). There is extensive experimental information as well as theoretical modelling available for such flows, and, while these notes are preliminary and speculative in nature and many of the details remain to be filled in, I feel that the ideas rest on firm ground. In particular, they should provide a useful framework in which to interpret visualization results for unsteady flows.

The basic idea is to take an axisymmetric, three dimensional flow which contains a recirculating region, with stagnation points, and study the

effects of non-axisymmetric, time periodic perturbations on particle paths for the fluid in the neighborhood of that structure. In terms of dynamical

1980 Mathematics Subject Classification: 34C35, 34Dl0, 76Exx. lsupported by NSF under grant No. CME 79-19817

393





394 PHILIP HOLMES

systems theory, certain degenerate invariant manifoldsof homoclinic or

heteroclinic orbits connecting the stagnation points can be expected to break

up under arbitrarily small perturbations. Hence a (weakly) growing non-

axi~ymmetric mode might be expected to cause major qualitative changes to the

fluid flow, since it destroys stream surfaces which previ01~sly separated flow regions, even if this unstable mode is itself a "simple" time periodic one.

The main result is that time periodic flows will often have chaotic particle

paths. The tools to be used are classical hydrodynamic stability analysis

(Chandrasekhar [1961]), to establish the existence of such linearly unstable

growing modes, and Melnikov theory (Melnikov [1963], Guckenheimer and Holmes [1983, Chapter 4], Holmes and Marsden [1982a,b,l983])- to study the

effects of such a mode on integrated fluid motions in the basic axisymmetric

velocity field. For background in fluid flow visualization work, see Stuart et al. [1963].

Guckenheimer and Holmes [1983] provide an introduction to the qualitative

theory of dynamical systems and the Melnikov method.

2. BUBBLE TYPE VORTEX BREAKDOWN. Leibovich [1978] [1983] provides reviews of the vortex breakdown process of interest. He describes experimental and

theoretical evidence for a "bubble type", predominantly axisymmetric distur-bance which can·become established in a slowly diverging vortex tube carrying

a swirling flow, as sketched in Figure 1. Here we show "ideal" or averaged

stream surfaces on which fluid particles travel in helical paths. (Leibovich [1978]). In fact streak lines from dye injected upstream at the tube center show

Figure 1. Axisymmetric bubble type vortex breakdown (after Leibovich [1978]).



CHAOTIC PARTICLE PATHS IN SWIRLING FLOWS 395

that the spherical bubble surface is not invariant, and that fluid is exchanged

between the main flow and the "interior" reverse flow region in a complicated

manner. However, laser doppler velocity measurements both inside the bubble

and near its surface indicate that the pointwise velocity is predominantly

time-periodic, and although the velocity field is clearly non axisymmetric,

it is evidently not chaotic in the usual sense of that word (Faler and

Leibovich [1977], Leibovich [1978]), Before we can describe how such a simple time periodic velocity field can

give rise to the complex particle paths implicit in the flow visualizations

outlined above, we outline a model for the axisymmetric breakdown sketched in Figure 1. 1\Te start with a columnar flow in cylindrical coordinates of the

form

(1)

the stream surfaces of which are cylinders. A solution to the Euler equations

which fits the experimental measurements quite well except in the neighborhood

of the bubble is provided by the Burgers-Rott vortex: 2

vo(r) qlol(l-e-r )/r (2)

where q > 0 and o > 0 in a jet-like approach flow (and o < 0 in the wake-

like flow downstream of the bubble). Such a vortex can support axisymmetric

dispersive waves, and it turns out that such waves can propagate upstream in

some situations (Leibovich [1970], Randall and Leibovich [1973]). The columnar

solution (1-2) can be derived from a stream function ~ 0 (r) and a circulation function r0(r) (Lamb [1932]). In terms of the stream function, the flow carry-ing such a long wavelength axisymmetric wave can be written

(3)

where A is determined by a Korteweg de-Vries (K-dV) equation and the struc-ture function ~(r) is an eigenfunction of a second order equation (the eigenvalue c0 is the phase speed of the propagating wave). In the multiple scales perturbation method used to obtain (3) the amplitude of the wave is

assumed to be 0(£) and the wavelength 0(£2). As £ increases, the

"rotating solitons" thus found introduce the two stagnation points of Figure 1.

This is seen most easily if we pass to a coordinate system travelling with the

~-Tave, z -+ z + ct , to obtain

(4)



396 PHILIP HOLMES

In (4) we have neglected the higher order terms, and we note that it is

necessary to take s - 0.7-0.8 to obtain the bubble structure (Leibovich

[1970]). Thus the model cannot be rigorously justified, but it does match the

axisymmetric aspects of the experimental data quite well.

In the experiment, the wave does not move up and out of the tube, but rather

equilibrates and remains stationary. Leibovich [1983] describes a plausible scenario for this behavior. A (small) downstream disturbance triggers the

axisymmetric soliton, which begins to move up the tube. As it moves, it

encounters a narrower cross section due to the slightly flared tube walls, and

therefore steepens and speeds up (Randall and Leibovich [1973]). It then

becomes unstable to a periodic non axisymmetric mode, perhaps in a Hopf bifur-

cation. The non axisymmetric mode acts as an "effective viscosity" or energy

drain on the axisymmetric motion, thereby causing the bubble to slow down and

equilibrate.

To investigate the validity of this conjectural scenario, we are currently

computing the three dimensional linearized stability of rotating flows of the 2 form (4), where ~ 0 (r) generates the columnar flow (1-2), A1 (z) = sech z

in the (normalized) soliton solution of the Kd-V equation, and ~(r) is a

numerically obtained structure function which vanishes at r = 1 so that the

solution fits into a circular cylinder. Using a Fourier decomposition to

separate azimuthal modes, we are left with a nonseparable eigenvalue problem

in r and z, which we are solving using Galerkin methods with Fourier and

Chebyshev polynomial modes in z and r respectively, after first transforming onto a finite z-interval and considering only those modes which decay exponen-tially at z = ±00 • The bifurcation parameter is s, the amplitude of the bubble. We conjecture that, as s increases, one or more non axisymmetric modes will become linearly unstable and exhibit exponentially growing oscilla-

i(n6-+wt) tions. Letting ~(r,z,6,t) = e y2(r,z) denote such a mode, the velocity field in the fluid, close to bifurcation, will be reasonably well

approximated by

u ( ) U ( ) + U ( ) i(n6-+wt) ~O r + s 1 r, z \l 2 r, z e

for \l « 1.

(5)

Moreover, if a supercricial Hopf bifurcation does occur (Marsden and

McCracken [1976], Guckenheimer and Holmes [1983]) at s = s0 , then the amplitude \l of the 'long term' oscillatory mode is locally of 0(~); i.e., it grows relatively rapidly with s. However, there are problems with this analogy, since the model is inviscid and we expect all eigenvalues of the linear system to be purely imaginary for s small. Instability presumably

arises when two complex conjugate pairs meet and split: one pair acquiring




positive real part and the other acquiring negative real part. Such a

Hamiltonian bifurcation is generally much more complex than the Hopf bifurca-tion, but the addition of small viscosity would presumably perturb it into a Hopf bifurcation.

3. THE EFFECTS OF NON AXISYMMETRIC, TIME PERIODIC PERTURBATIONS ON PARTICLE PATHS. The experimental and theoretical evidence outlined in §2 suggests that the velocity field of the fluid in bubble type vortex breakdown can be written in the form

U(r,z,e,t) ( 0 ) ( u1 (r,z)) (u2(r,z,e;t))

- ::::: +' ::::::: +" ::::::::::: (6)

where 0 < 1.1 << e:

t for some 0 < T and u2,v2,w2

For fixed are

£

27f periodic in (perhaps 0(1))

e and T-periodic in and 1.1 0, the

axisymmetric velocity field p0 + e:Y1 has two stagnation points at ~ = (O,·,a) and b = (Q,.,b) which are connected by a spherical stream sur-face and the interval (a,b) of the z-axis, Figure 1. We will indicate how Melnikov theory can be used to compute the effects of the doubly periodic perturbation 1.1U2 on such stream surfaces, and the particle trajectories near them, for 1.1 << e:.

In the usual Melnikov method (Melnikov [1963], Greenspan and Holmes [1983], Guckenheimer and Holmes [1983, Chapter 4]) one considers T-periodic perturba-tions of two dimensional Hamiltonian vector field of the form

(7)

y = f 2(x,y) + J.lg2(x,y,t),

where f 1 = 3F/3y, f 2 = -aF/ax and the gi may or may not similarly derive from a Hamiltonian G. If for 1.1 = 0 the unperturbed system has a homoclinic orbit x 0 (t) = (x 0 (t),y0 (t)) to a hyperbolic saddle p0 as sketched in Figure 2, then for 1.1 << 1 the Melnikov function

M(t0) = f <~A !l) (~ 0 (t),t + t 0)dt -oo

provides a good measure of the distance

between the perturbed stable and unstable manifolds of the corresponding saddle point p of the perturbed Poincar~ map (Figure 2(b)). If the 1.1 g-perturbation is Hamiltonian, then (8) be be written as

(8)

(9)



398 PHILIP HOLMES

(a) fL=O (b) fL >0

Figure 2. Melnikov theory for periodically forced planar system.

M(t0) =- J {F,G}(x 0 (t),t+t0)dt, (10) -oo

where {·,·} denotes the Poisson bracket. The final result is that, if

M(t0) has simple zeros, then for ~ ~ 0 sufficiently small, the stable and

unstable manifolds of intersect transversally. Generalizations of the

method to n-dimensions and ton- and co-many degree of freedom Hamiltonian systems are available (Gruendler [1983], Holmes and Marsden [1981) [1982a,b)

[1983)). The present problem differs from (7) in that the unperturbed system is an

axisymmetric three dimensional divergence-free vector field of the form

r cu1 (r,z)

z wo(r) + Ew1 (r,z) (11)

8 vo(r) + E:v1 (r,z),

1 a () 1 d where wo = --- '!' (r) ul = ~ '!'1 (r,z) and wl =- r ar 'l'l (r,z), where r ar 0 ' '¥1 (r,z) = cp1 (r)A1 (z), and VO' vl are obtained from circulation functions r ..

If we assume that the stagnation points are hyperbolic, i.e., the eigen-

values of the linearization of the vector field of (11) have non-zero real

parts, then, as in the planar case, the implicit function theorem may be used

to show that the stagnation points ~ and b persist as small 0(~)

T-periodic orbits for the T-periodic perturbed vector field. Selecting a

Poincare cross section

the induced Poincare map to P will have hyperbolic fixed points ~

~



CHAOTIC PARTICLE PATHS IN SWJRL[.NG FLOWS 399

a =a +O(y), b = b + O(y). Moreover, perturbation theory for invariant -]J - -y manifolds (Hirsch et al. [1977]) shows that these fixed points have stable and

unstable manifolds E-close to those of the unperturbed flow. Thus we can

expect the two dimensional unstable manifold of a -v and the two dimensional

stable manifold of b -]J

to intersect transversely, somewhat as sketched in

Figure 4, below. If this occurs, then the stream surface which originally

bounded the bubble is broken and fluid exchange can occur, with fluid from the

main stream being entrained into the bubble at the back and fluid returning

from the bubble to the main stream. The bubble will pulse and rotate periodi-

cally in time.

Before we attempt a fuller description of the flow around the perturbed

bubble, we outline how Melnikov theory might be adapted to prove that specific

velocity field perturbations, obtained either theoretically as in §2 or by

fitting experimental data, cause the bubble to burst. We first note that

since the flow is incompressible and there are no sources or sinks in the

bubble, the invariant manifolds cannot lie wholly one within the other as

sketched in Figure 3. Thus the manifolds either coincide, as in Figure 1, or

Figure 3. Inadmissible stream surfaces for the perturbed bubble.

intersect in distinct solution curves (particle paths) as indicated in Figure 4 (generically, but not necessarily, the intersections will be transverse).

(a) ( b) Figure 4. The bubble is burst: the invariant manifolds intersect transversely. (a) cut away view of the broken surface, (b) longitudinal and meridional cross sections: fluid entrained from upstream shown shaded. Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.


400 PHILIP HOLMES

We require a three dimensional generalization of the planar Melnikov theory.

In fact ann-dimensional generalization has been provided by Gruendler [1983],

who replaces the ,-product or 2-form ~' ~ = ~(f,&) with an n form

~(a 1 , ... ,a) = det[a1 , ... ,a ]. ~ ~n ~ ~n

Writing the perturbed vector field (6) as

and denoting the fundamental solution matrix of the homogeneous part of the first variational equation

~ = ~~(~(t-t 0 ))~ + ~(~(t-t 0 ),t),

[~ 1 (t-t 0 ), ... ,~n(t-tJ],detjXj = 1, Gruendler computes the

(vectorial) distance between the manifolds in terms of a set of n-forms

(13)

(14)

(15)

i 1 i-1 i+l n Jt m (t0) = ~(~ , •••• ~ ,g,~ , •..• ~ )(y(t-t0 ,t)exp(- (V·f)(y(y-t0))dy,(l6) 0

where y is a solution on the homoclinic manifold. In the present case n = 3

and, rather than a homoclinic manifold, we have a two dimensional heteroclinic

manifold connecting a ~~

and b • ~~

(The fact that the connection is heteroclinic does not really matter.) A single scalar quantity therefore suffices to mea-

sure the distance between the perturbed manifolds at a particular point on the to

unperturbed bubble on a chosen cross section L (think of the distance

between the manifolds along the normal to the unperturbed bubble surface). The Gruendler theory then implies that this distance is characterized by the Melnikov function:

where 00 1 2 to

M(t0 ,e0) j[~(g,~ .~ )(y(t-t0),t)exp(- J (V·f)(y(T-t0))dT]dt. ~ 0

(17)

Since the flow is incompressible, V·f = 0 and the formula simplifies to

Joo 1 2 M(t0 ,e0) = ~(g.~ .~ )(y(t),t+t0)dt, (18)

-oo

where we have changed the integration variable t + t + t 0 . Here the unper-

turbed solution is based at the point y(O) = (y(O),e 0 ,z(O)) on the bubble

and the entries ~ 1 .~ 2 are two linearly independent solutions of the homo-

geneous part of (15) which remain in the two dimensional tangent space of the homoclinic bubble surface. Since the bubble is axisymmetric, these solutions

are invariant under changes in initial angle eo and thus both to and eo enter only in the perturbation vector field in the form g(r(t),e(t)+e0 , z(t), t+t0). Hence M(t0 ,e0) is automatically 2rr periodic in e0 and T-periodic in t 0 •




The final result is that, as in the planar case, if M(t0 ,e0) has simple

zeros as a function of and eo· then the invariant manifolds for the three dimensional Poincare map intersect transversely along curves as shown in Figure 4. Since we expect the perturbation to be of the form ~ 2 (r,z)ei(ne+wt) (§2), the integral (18) can be reduced to an integral in t

multiplied by a doubly periodic sinusoidal function of eo and to, and then all we need to do is check that the integral is nonzero, for which a numerical evaluation will suffice. Analytic evaluation as in Holmes and Marsden [1981, 1982a,b,l983] will probably be impossible since even the unperturbed flow is only known numerically.

We remark that the use of Melnikov or equivalent methods will be essential in order to prove that the bubble surface is broken: one cannot rely on naive 'genericity' arguments in problems in which symmetry groups act. Here there is an obvious s1 symmetry imposed by the cylindrical geometry of the apparatus. In the Taylor-Couette experiment, a similar symmetry leads to the appearance

i(ne+wt) of rotating waves with (S,t) dependence of the form e regardless of amplitude (i.e. not just in the small amplitude linear approximation of ( 5). (Rand [1982]). Now such a rotating wave might leave the spherical material sur-face of the bubble invariant (albeit wrinkled). In fact some experimentalists believe that the stream surfaces between neighboring Taylor cells remain invariant after the appearance of waves, although this is not implied by the discrete ( 2 ~) rotational symmetry of the velocity field.

n We now briefly comment on the implications of transverse homoclinic inter-

sections for the fluid motion. Referring to Figure 4, which shows a cut away drawing of the punctured bubble, it should be clear that thin fingers of fluid can come from upstream outside the unstable manifold of a but inside the

~\l

stable manifold of b , thus becoming entrained into the bubble, where they ~\l

must circulate at least once. (In terms of Figure 3, both situations occur, but at different angles e and times t). Similarly, fluid originating inside the bubble, just downstream of a , can lie outside the stable manifold of b

~\l ~\l

and hence an escape from the bubble. The complicated intersection structure implicit in transverse homoclinic orbits (Guckenheimer and Holmes [1983]) implies that there exist solution curves (fluid particle motions) originating upstream which enter the bubble, circulate arbitrarily many times, and finally exit, to continue downstream. This is the sense in which the flow is spatially chaotic. In addition to such qualitative information, quantitative calcula-tions are also feasible: for example, it should be possible to compute typical "residence times" for particles entering the bubble in terms of the perturbation strengh \l· (Standard Melnikov theory immediately implies that the splitting of



402 PHILIP HOLMES

the bubble surface is of 0(~).) See Holmes and Marsden [1982a, Appendix B] for information on the number of iterates of the map necessary to guarantee recurrent motions.

4. ·oTHER PROBLEMS. Several other well known axisymmetric flows are known to become unstable to non axisymmetric, time periodic perturbations. The steady Taylor cell structure of circular Couette flow, for example, undergoes a Hopf

bifurcation to wavy Taylor cells somewhat above the critical Taylor number for onset of the cells themselves. In this situation, Melnikov theory might be applied to determine whether the material stream surfaces or boundaries separating the cells are broken, allowing fluid to pass up and down the cylin-ders. In fact recent experimental work (Mobbs and Brindley [1983]) indicates that non axisYmmetric structures similar to those of vortex breakdown appear within individual cells considerably before the appearance of wavy cells in wide gap experiments. Recent numerical simulations of Marcus [1983] show similar non axisymmetric effects.

Aref [1983] did analytical and numerical work on a two dimensional flow with time periodic perturbations and demonstrated very clearly the kind of mixing

attendance upon the breakdown of stream surfaces described here. The standard planar Melnikov theory applies essentially directly to his situation of two point stirrers (vortices) in a two dimensional circular disc of fluid.

The methods sketched in this paper should not be confused with somewhat similar attempts to interpret experimental data in terms of two dimensional velocity fields by Cantwell et al. [1978]. In that work a coordinate change was proposed to enable the observer to follow a developing boundary layer and thereby 'freeze' the fluid particle paths. In the present case, a fully three dimensional and time dependent flow is treated, and the spatial chaos clearly cannot be removed by coordinate changes. However, the concepts of invariant manifolds (stream surfaces) which intersect in homoclinic a~d heteroclinic orbits (particle paths) should be useful in the interpretation of flow visualization data from experiments. The s1 symmetry of the vortex tube apparatus should also be used as in Rand [1982] to make predictions on the structure of the velocity fields which appear at various stages of vortex breakdown.

BIBLIOGRAPHY

H. Aref, "Stirring by chaotic advection", J. Fluid Mech. [1983] (in press). H. Broer, "Quasi periodic flow near a codimension one singularity of a

divergence free vector field in dimension three", Dyn.<tmical Systems and Turbulence, [1981], pp. 75-89, ed. D.A. Rand and L.-S. Young, Springer Lecture ~otes in Mathematics No. 898.



CHAOTIC PARTICLE PATHS IN SWIRLING FLOWS

B.J. Cantwell, "Organized Motion in Turbulent Flow:' Ann. Rev. Fluid Mech., 13 [1981], 457-517.

B.J. Cantwell, D. Coles and P. Dimotakis, "Structure and entrainment in the plane of symmetry of a turbulence spot", J. Fluid Mech., 84 [1978], 641-72.

403

s. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability Theory, Oxford University Press, 1961.

J.H. Faler and S. Leibovich,"Disrupted states of vortex flow and vortex breakdown", Phys. Fluids, 20 [1977], 1385-1400.

B.D. Greenspan and P.J. Holmes, '~omoclinic orbits, subharmonics, and global bifurcations in forced oscillations", Nonlinear Dynamics and Turbulence, Chapter 10 [1983], 172-214, ed. G. Barenblatt, G. Iooss and D.D. Joseph, Pitman, London.

J. Gruendler, "A generalization of the method of Melnikov to arbitrary dimension" [1983], (submitted for publication).

J. Guckenheimer and P.J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983.

M.W. Hirsch, C.C. Pugh and M. Shub, Invariant Manifolds, Springer Lecture Notes in Mathematics, No. 583, Springer Verlag, New York, Heidelberg, Berlin, 1977.

P.J. Holmes and J.E. Marsden, "A partial differential equaton with infin-itely many periodic orbits: chaotic oscillations of a forced beam", Archive for Rational Mechanics and Analysis, 76 [1981], 135-166.

P.J. Holmes and J.E. Marsden, "Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom", Comm. Math. Phys, 82 [1982a], 523-544.

P.J. Holmes and J.E. Marsden, "Melnikov's method and Arnold diffusion for perturbations of integrable Hamiltonian systems", J. Math. Phys., 23 [1982b], 689-675.

P.J. Holmes and J.E. Marsden, "Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups", Indiana U. Math. J., 32 [1983], 273-310.

H. Lamb, Hydrodynamics, Cambridge University Press, 1932. S. Leibovich, "Weakly nonlinear waves in rotating fluids", J. Fluid Mech. ,

42 [1970], 803-822. S. Leibovich, "The structure of vortex breakdown", Ann. Rev. Fluid Mech.,

10 [1978], 221-246. S. Leibovich, "Vortex stability and breakdown: Survey and extension"

AIAA J. (to appear).

P.S. Marcus, "Simulation of Taylor-Couette flow- numerical results for wavy-vortex flow with one travelling wave~ [1983] (submitted for publication).

J.E. Marsden and M. McCracken, The Hopf Bifurcation and its Applications, Springer Verlag, New York, 1976.

V.K. Melnikov, "On the stability of the center for time periodic perturba-tions", Trans. Moscow Math. Soc., 12 [1963], 1-57.

F. Mobbs and J. Brindley, Personal communications, [1983]. D.A. Rand, "Dynamics and symmetry: predictions for modulated waves in

rotating fluids", Arch. Rat. Mech. Anal., 79, [1982] 1-37.



404 PHILIP HOLMES

J.D. Randall and S. Leibovich, "The critical state: a trapped wave model of vortex breakdown", J, Fluid Mech., 5l [1973], 495-515.

J.T. Stuart, R.C. Pankhurst and D.W. Bryer, "Particle paths, filament lines and streamlines", N.P.L. Aero Report, 1057, Teddington, U.K. [1963].

Department of Theoretical and Applied Mechanics and Center for Applied Mathematics

Cornell University, Ithaca, NY 14853




A UNIVERSAL TRANSITION FROM QUASI-PERIODICITY TO CHAOS (ABSTRACT)

Eric D. Siggia

A common route to chaos in dissipative systems proceeds from periodic to quasi-periodic flow (with two independent frequencies). Then, in the absence of rotational symmetry, the system generally mode locks before becoming turbulent. Beyond these qualitative features, the numerous experiments that have examined this regime differ in detail. Dynamical system theory had made the occurrence of the above transitions plausible but has provided no nontrivial quantitative and model independent information.

This situation, on the theoretical side, has recently changed with a proposal on how to modify the experiments so as to make the transition to chaos occur in a quantitatively universal manner. 1•2 The essence of our proposal follows from K. A. M. theory which is the weak coupling limit of the strong coupling problem relevant to the turbulent transition. In addition to the Rayleigh number, the experimenter must control a second parameter so as to maintain the frequency ratio in the quasi-periodic state at a fixed irrational value p. The golden mean, (~- 1)/2, is the optimal ratio experimentally.

The universality, which is restricted to the low frequencies in the time series, is obtained under the above circumstances because the transition to chaos is continuous. In particular the singular low frequency structure in the spectrum develops continuously as R ~ RT from below or as the frequency ratio approaches (~- 1)/2 at RT. These asser-tions are established by a renormalization group analysis that resembles the one developed by Feigenbaum to account for the universal features of period doubling. The renormalization group may be applied at any fixed irrational frequency ratio provided only that the integers in the continued fraction do not grow too rapidly.

405





406 ERIC D. SIGGIA

We again stress that all the low frequency complex amplitudes obtained from either a fluid experiment or a forced nonlinear oscillator at the quasi-periodic to turbulent transition depend only on p. At present, the theoretical predictions are most easily derived numerically by iterating the map

~p 1 = 'P + w0 - fn sin (211~p)

for a = 1 (corresponding to R = RT) and adjusting w0 to achieve the desired rotation number, p. While it is impossible to predict where in parameter space one will see this transition for a general system; once one is sufficiently close at fixed p the full low frequency singularity should be seen as the "Reynolds number" increases.

Similar ideas have recently been applied to the structure of the spectrum and eigenfunctions of the Schrodinger equation with a quasi-periodic potentia 1. 3

REFERENCES

1) D. Rand, S. Ostlund, J. Sethna, E. Siggia, Phys. Rev. Lett. 49, 132, 1982 and Physica 80, 303 (1983).

2) S. Shenker, Physica 50, 405 (1982), and M. Feigenbaum, L. Kadanoff, S. Shenker, Physica "SU, 370 (1982).

3) M. Kohmoto, L. P. Kadanoff and C. Tang, Phys. Rev. Lett. 50, 1870 (1983) and S. Ostlund, R. Pandit, D. Rand, A. J. Schellahuber, ana E. Siggia, Phys. Rev. Lett. 50, 1873 (1983).

ERIC D. SIGGIA DEPARTMENT OF PHYSICS CORNELL UNIVERSITY ITHACA, NEW YORK 14853




ON THE NONPATHELOGICAL BEHAVIOR OF NEWTON'S METHOD

* * James H. Curry and C. Eugene Wayne

ABSTRACT

Numerical analysis is, in general, concerned with the success of the algorithms which it studies. There are, however, other vantage points from which these algorithms can be viewed. For the specific algorithm known as Newton's Method, the dynamical system theory viewpoint allows the posing of questions which the classical numerical analyst has not pursued. This article asks some of those questions.

1. INTRODUCTION. Newton's method, applied to polynomials of the complex variable, z, leads naturally to the study of iteration of rational maps, R(z), of the complex plane, C, into itself. Two cases of interest are, a description of the asymptotic behavior of different trajectories for a fixed rational function, and a determination of how these trajectories change as some parameter is varied. Both types of behavior are illustrated in the accompanying figures.

In the next few pages relevant material on the theory related to the first of these situations is given. Specifically, the behavior of trajectories in the neighborhood of periodic orbits of R is examined. Most of the results we present are quite old, originating in the work of P. Fatou [10] and F. Julia [14]. The presentation will follow the recent review of P. Blanchard [3], which the reader may consult for further details.

2. PRELIMINARIES: ITERATION OF RATIONAL FUNCTIONS. Let R be a rational function R:C + C, C = C U {m} is the extended complex plane. Assume that deg R ~ 2, deg R = max (degree of numerator and denominator of R). There

*The National Center for Atmospheric Research is sponsored by the National Science Foundation. AMS Subject Classifications: 58Fl3, 65D99.

407





408 Curry and Wayne

exist two disjoint sets, J(R), the Julia set, and F(R) the Fatou set such that C = J U F. Roughly speaking the Fatou set consists of those points which possess a neighborhood that is "well behaved" under iteration with R. Precisely, define F(R) = {zECI there exists an open neighborhood U with Z£U such that the set of functions Rnju• n = 1,2, .•. form a normal family}, Rn denotes the n-fold composition of R with itself, and the requirement that Rnju be a normal family implies that there is a subsequence Rnjlu, which converges uniformly on compact subsets of U. The Julia set is defined by J(R) = C\F(R).

The following results add insight to the situation which concerns us:

Proposition 1: J(R) is non-empty. The proof follows from a degree counting argument [3].

Proposition 2: Let :zo,z1 ,z2 , ••• ,zn_1 be an attractive orbit having period n for R(z). Then Zi€ F(R) for i = 0,1, ... ,n-l.

Proposition 3: J(R) is the closure of the set of repelling periodic points.

Theorem: If R(z) has an attracting periodic point then the orbit of at least one critical point will converge to it.

The above theorem suggests a strategy for searching for attractive periodic solutions. If an attractive periodic solution exists then it must attract a critical point.

3. NEWTON'S METHOD. We will refer to Nf(z) as the Newton Transformation associated with the function f(z). Specifically,

Nf(z) z - .!i!L f' ( z)

(3.1)

Transformation (3.1) is much studied by numerical mathematicians whose primary concern is the choice of a z0 such that the Newton sequence

(3.2)

converges to a solution of f(z) = 0. In general the complementary question: how can the sequence (3.2) fail to converge to a solution of f(z) = 0, has not, in general, been asked. However, certain results are known for the class of polynomials having all real roots.



On the Nonpathelogical Behavior of Newton's Method 409

In [1], [2] B. Barna considers the Newton transformation associated with a polynomial, f(x) having all real roots. He finds that if deg (f) ~ 4 then the set of initial conditions for which (3.2) fails to produce a sequence which converges to a solution of f(x) = 0 is homeomorphic to a zero measure Cantor set. However, a subtle question still remains, within Barna's context: given an initial condition which lies between the two extreme roots, to which solution to f(x) = 0 will the sequence produced by (3.2) converge.

In what follows Newton's method associated with the one parameter family

f (z) = z3 + (a- 1) z - a a (3.3)

will concern us, a and z are assumed to be complex. The polynomials fa(z) are the monic cubics whose roots sum to zero and have z = 1 as a zero.

In the theorem stated in the previous section, a central role is played by critical points. The Newton transformation associated with (3.3) has four critical points, located at the roots of (3.3) and at the point z = 0. The critical points at the zeros of fa(z) play no role in the quest for attractive periodic cycles, they are already attractive periodic cycles of period one. Hence the free critical point located at the origin is the only one available to converge to an attractive periodic cycle, if such a nontrivial cycle exists.

4. NUMERICAL RESULTS: PARAMETER SPACE. In order to determine whether a stable attracting periodic solution exists for the Newton transformation associated with fa(z). The theorem cited in section 2 can be used. Specifically, since z = 0 is a critical point for Nfa(z) that point was iterated as many as 200 times while varing the complex parameter a on a 700 x 700 grid. Figure 4.1 is an example of the results of the numerical experiment just outlined.

In Figures 4.1, 4.2 and 4.3 the horizontal and vertical axes are the real and imaginary parts of the parameter a. In the black area iterates of the free critical point, produced by the Newton transformation, converge to the real root z = 1. Two types of white regions are present in each of these figures. Parameter ranges where the orbit of z = 0 converge to one of the complex roots of fa(z) and a region where the Newton transformation exhibits periodic behavior.

Figure 4.2 is an enlargement of one of the regions where periodic behavior is obtained, the scale should be noted. In the central most part of this figure the oribit of the free critical point is attracted to a periodic orbit having period three, e.g. a= (1.0199866540, .9580782418). If the



410

1 .25

1 .20

1.15

"" 1 .10 :1: .....

1 .05

1 .00

.95

.80

Curry and Wayne

WHERE CRITICAL POINT TENDS TO ONE

.85 .90 .95 1. 00 1 .05 1 .10 1.15

REI AI

Figure 4.1




WHERE CRITICAL POINT TENDS TO ONE .99

.98

.97

.96

<(

.95

.94

.93

.92 ~--~--~--~--~--~--~--~--~--~--~~~~~ .98 1 .00 1 .01 1 . 02 1. 03 1 .04 1 . 05

REI AI Figure 4.2



412 Curry and Wayne

WHERE CRITICAL POINT TENDS TO ONE

1.105

1 . 100

1 .095

1 .090

1 .085

< - 1 .080 :1:

1 .075

1 .070

1 .065

1 .060

1 .055

1 .050 1.060 1.065 1.070 1.075 1.080 1.085 1.090 1.095 1.100 1.105 1.110 1.115 1.120

REI AI

Figure 4.3



.5

.4

.3

.2

.1

~ 0

-.1

-.2

-.3

- 4 t ,_ .5

-.5

On the Nonpathelogical Behavior of Newton's Method

PARAMS

-.4 -.3

0.9200000000

·.('·

.. ... ~ ... ;.. . -o:.";:

.-':·:

-.2 -.1

+

0

X

. .:,•· .... ":~l~

Figure 5.1

' ··:.. .

.1

1.0200000000

.2 .3

413

.4 .5



414

PARAMS .5 I I I

.4 -

.3 -

.2

.1

>- 0

-.1

-.2

-. 3 1-

-.4 -

-.5 1 -.5 -.4 -.3 -.2

Curry and Wayne

1 .0000000000 I I I

+

I I I

-. 1 0 .1

X

Figure 5.2

1 .0500000000 I

·~ ~~ r .• ·"?··.·• .. .. · ..

I

.2

I

I

.3

I

-

-

-

-

-

-

-

-

I

.4 .5




PARAMS 1 .0199866540 0.9580782418 .5 I I I I I I I I I ,, .4 ,.. ~ { -

' ...... .,. At .. •

·~.~ · . ._,

.3 r- ...• .. 11 -;_~ .. ~ · . .-;, .~1

·~ ... r~ •. Jt/• ... '

.2 !-- ... \)..,. -•.A v;, f': .• ''.JI ";~'=:

,, !;~ -.. ~ ....

~ r ... '"-'t; ., • .1 ~

-~~--

~f ·.··

>- 0 - + --·~ . ry

,.'i ~··-., . -~~ -.1 - ·~·4iil0>. -~~ ~ :.1 ~/..... ,l) .•

-.2 r-, ... $ .. ~., ,;.

·:·.'; .. ~~- ..•. ; . ... . ..~ .. ~~.;?:~ -

,u." ... . .,!• \ ...... :·,,._·-: ... ~'

-.3 ~

. .,.,. ··l·. -.,; llP• ... .• 1 .

.......

-.4 -

-.5 I I 1 1 I I I I I

-.5 -.4 -.3 -.2 -.1 0 .1 .2 .3 .4 .5 X

Figure 5.3



416 Curry and Wayne

region is exited in the direction of a = (1.01172699403, .9758981555) period doubling bifurcations are observed. In particular for the last value of a presented, Newton's method has a superstable period six. The "periodic set" is called the Mandelbrot set and is similar to the one obtained in [14].

If we return to Figure 4.1, other structures can also be identified, for example the parameter value a= (.8495576691, 1.2330725099) is an approxima-tion to the a value where the orbit of the critical point hits infinity in three iterations. In any arbitrarily small neighborhood of this value there are attracting periodic solutions having large period. Figure 4.3 is an enlargement of a structure which is similar to Figure 4.1. Associated with this figure there is also a Mandelbrot set whose fundamental period is five; this behavior occurs for a in a neighborhood of a = (1.0922524896, 1.0624735186).

The period five cycle also period doubles and there is a superstable period ten for a = (1.0925197313, 1.0628229142). We speculate the "vertex" of Figure 4.3 corresponds to a parameter value where z = 0 hits infinity in five iterations.

In general we conjecture that if the Mandelbrot set has a period n in its central most section, then the "almond" in which it is embedded has vertex which corresponds to the orbit of the critical point hitting, the repulsive periodic point, infinity in n iterations. And in the neighborhood of any such vertex point there are parameter values where the critical point converges to periodic orbits having an arbitrarily large period.

5. JULIA SETS As mentioned in the introduction, J(fa)• the Julia set is nonempty.

We end this note by presenting three examples of Julia sets, Figures 5.1 through 5.3. Though these figures appear to be "similar" they are not. Indeed, the recent work of R. Mane, P. Sad and D. Sullivan [15] proves that Julia sets in different components of parameter space are topologically dissimilar. In the remaining figures the horizontal scale labeled by x is the real part of the complex variable z and the y-axis is the imaginary part of z. The associated regions in parameter space can be identified in Figure 4.1 and are provided at the top of each figure. The strategy for producing Julia sets has been described in [6] and we refer the interested reader to that article.

6. COMMENTS In this note we have presented a very short introduction to the behavior

one can expect from iterated rational functions and specifically Newton's




Method. The bibliography contains articles, some of which were not cited in the text, but which must be mentioned. In particular we single out the recent works of Oouady and Oouady-Hubbard who have done both theoretical and numeri-cal studies of Newton's Method. Their numerical work includes figures similar to 4.1, but in color. They also note the existence of the Mandelbrot set, but have no explanation of why it should be so remarkably like the one pictured in [16].

ACKNOWLEDGMENTS This work was carried out while J.H. Curry was a visitor at the National

Center for Atmospheric Research, Atmospheric Analysis and Prediction Division. He thanks Drs. R. Anthes and J. Herring for their hospitality and Eileen Boettner for typing and editing the manuscript. His work was supported in part by NSF Grant PRM 81 06833 and is gratefully acknowledged. C.E. Wayne was supported by the Institute for Mathematics and Its Application, University of Minnesota, Minneapolis.

BIBLIOGRAPHY

1. B. Barna, "Uber die Divergenzpunkte des Newtonschen Verfahren zur Bestimmany von Wurzeln algebraischer Gleichungen," I. Publicatione Mathematicae, Debrecen, 3 (1953), 109-118.

2. B. Barna, "Uber die Divergenzpunkte des Newtonschen Verfahren zur Bestimmany von Wurzeln algebraischer Gleichungen," II. Publicatione Mathematicae, Debrecen, 4 (1956), 384-397.

3. P. Blanchard, "Complex dynamics", Institute for Mathematics and Its Applications, University of Minneapolis, Minneapolis (Preprint, 1983).

4. P. Collet and J.-P. Ekmann, Iterated Maps of the Interval as Dynamical Systems, Birkh~user (1980).

5. P. Collet, J.-P. Ekmann and O.E. Lanford III, "Universal properties of maps on an interval", Comm. Math. Phys., 76 (1980), 211-254.

6. J.H. Curry, L. Garnett and D. Sullivan, "On the iteration of a rational function: Newton's Method", Comm. Math. Phys. (to appear, 1983).

7. B. Derrida, a. Gervais and Y. Pomeau, "Universal metric properties of endomorphisms", J. Phys. A. Math. Gen., 12 (1979), 269-296.

8. A. Douady, "Systems dynamiqu~ holomorphes", Seminire Bourbaki, No. 599 (1982/1983).

9. A. Douady and J. Hubbard, "Iteration des polynomes quadratique complexes", C. R. Acad. Sci. Paris, 294 (1982), 123-126.

10. P. Fatou, "Sur les equations fonctionelles", Bull. Soc. Math. France, 47 (1919).

11. M. Feigenbaum, "The universal metric properties of nonlinear transformations", J. Stat. Phys., 19 (1978), 25-52.

12. M. Feigenbaum, "Quantitative universality for a class of nonlinear transformations", J. Stat. Phys., 21 91979), 669-706.



418 Curry and Wayne

13. M. Hurley and C. Martin, "Roots of polynomials, Newton's algorithm, and chaotic dynamical systems", Case Western Reserve University {preprint, 1982).

14. G. Julia, "Memoire sur l'iteration des function rationelles", J. Math. Pures Appl., 4 (1918), 47-245.

15. R. Man~, P. Sad, and D. Sullivan, "On the dynamics of rational maps", preprint {1982).

16. B. Mandelbrot, "Fractal aspects of z + >..z{1-z) for complex >.. and a", Annals of NY Acad. Sci., 357 {180), 249-259.

17. R.B. May, "Simple mathematical models with very complicated dynamics", Nature, 261 {1976), 459-467.

18. A. Ostrowski, solutions of Equations in Euclidean and Banach Spaces, Academic Press, New York, 1973.

19. S.K. Wong, "Newton's Method and symbolic dynamics", Bernard Baruch College (CUNY) (preprint, 1982).

JAMES H. CURRY DEPARTMENT OF MATHEMATICS UNIVERSITY OF COLORADO BOULDER, CO 80309

Current Address: National Center for Atmospheric Research Box 3000 Boulder, Colorado 80307

C. EUGENE WAYNE DEPARTMENT OF MATHEMATICS THE PENNSYLVANIA STATE UNIVERSITY STATE COLLEGE, PA 16802

Current Address: Department of Mathematics University of Virginia Charlottesville, VA 22903




SUCCESSIVE BIFURCATIONS IN THE INTERACTION OF STEADY STATE AND HOPF BIFURCATIONS

Jurgen Scheurle 1

ABSTRACT. Bifurcations in a two parameter family of vector fields are analyzed. At criticality the vector field has a pair of com-plex conjugate eigenvalues as well as a simple real zero eigenvalue. Particular attention is paid to the existence of tori and to quasi-periodic motion on them. Explicit computable conditions are given. The relationship to the work of Langford, Guckenheimer, Herman and others is given.

1. INTRODUCTION. In this contribution we consider a two-parameter family of dynamical systems of the form (1.1) u = F(~,v,u) , wherethemap F:FxlRxX-+X issufficientlysmooth,and ~and v are parameters. The underlying Banach space may even be infinite-dimensional. We assume that u = 0 is an equilibrium point for all values of the parameters,i .e. (1.2) F(~,v,O) = 0 The linear operator DuF(~,v,O) is supposed to possess a simple real eigenvalue a(~,v) and a pair of simple complex conjugate eigenvalues B(~,v) ± iy(~,v)

which simultaneously cross the imaginary axis at ~ = 0 , v = 0, i.e. (1.3) a(O,O) = 0 , 8(0,0) = 0 , y(O,O) > 0 . The remaining spectrum is assumed to be strictly bounded away from the imaginary axis.

Such singularities occur quite naturally in two-parameter families of vec-tor fields (Arnold [1983], Guckenheimer [1982]). One is interested in qualita-tive changes of the dynamics, when the parameters vary near the critical point. It has been known for a long time that under certain non-degeneracy conditions on F , there are generic paths through the parameter space along which one has the following sequence of successive bifurcations. First a transcritical stationary bifurcation from the trivial solution occurs, where the steady states which bifurcate to the right attain stability from the trivial solution. These then lose stability again and bifurcate into periodic orbits. This Hopf bifur-cation is supercritical. So the periodic solutions acquire stability for a

1980 Mathematics Subject Classification 58Fl4, 58F27. 1 Research supported by "Deutsche Forschungsgemeinschaft" contract I3 Sche 233/2-1.

419

© !984 American Mathematical Society 0271-4132/84 $1.00 + $.25 per page




420 Jurgen Scheurle

small parameter range. Next a point is reached, where the Floquet exponents of the periodic orbits are purely imaginary which, in general, leads to a bifur-cation into invariant two-tori (Neimark-Sacker bifurcation). Finally, in a second Hopf bifurcation, the periodic orbits run back into the trivial solution (Langford [1979]).

It is not hard to write down nice conditions in terms of second-order Taylor coefficients of F which guarantee the primary and the secondary bi-furcation to occur. However, conditions for the tertiary bifurcation from per-iodic orbits to invariant two-tori have only been sketched by former authors. In what follows we shall gil'e complete and explicitly computed formulas for such conditions which in particular describe the direction of bifurcation and the stability properties of the tori. Contrary to other authors we do not assume any symmetry properties of F (cf. Broer [1981], Braaksma and Broer [1981], Chow and Hale [1982], Guckenheimer [1981], and Spirig [1983]). More-over, we shall prove that there are many parameter values where the tori are quasiperiodic. Of course, general principles for flows on torioidal manifolds suggest such a result (Arnold [1965], Herman [1977]), but in the present sit-uation a careful analysis is required to prove it, because the tori are nearly degenerate. Actually, it turns out that along certain curves in parameter space all the bifurcating tori are quasiperiodic. Thus one has an abundance of successive bufurcations which are the first part of a so-called Landau sequence.

These results have already been stated and proved in Scheurle and Marsden [1983]. As a matter of fact, proofs will only be outlined here. For more de-tails we refer to Scheurle and Marsden [1983], although we shall work out the argument that the invariant tori actually bifurcate from the periodic orbits in much more detail than it has been done there. We also refer to this paper for an application to the Brusselator equations. In this case F is only defined on some subset of X , but the theory goes through equally well.

Although Langford's paper formed the basis for the present work, our approach is more in the spirit of singularity theory and the work of Gucken-heimer [1982] and Holmes [1980] in that it uses normal forms and the idea of unfolding. First, an appropriate normal form is determined, and then the problem is solved for this normal form. Of course, there are now many papers in bifurcation theory using this approach, such as Golubitsky and Schaeffer [1979], Golubitsky and Langford [1981], Iooss and Langford [1980], and Langford [1981]. Some partial results similar to ours have been given by Broer [1982].

2. NORMAL FORM AND RESULTS. In order to determine a normal form for (1.1) the problem is reduced to three dimensions by the construction of a center mani-



Interaction of steady state and Hopf bifurcations 421

fold, and then the method of Po·incare and Birkhoff is used. Both methods are construcutve as far as lower order terms in the Taylor expansion of the vector field are concerned. A center manifold is a submanifold of X which is tan-gential to the eigenspace of D F(~,v,O) belonging to the eigenvalues a and u 8 ± iy , and locally invariant and attractive for the flow induced by (1.1). Itdependssmoothly on the parameters. In particular, a center manifold con-tains all solutions near u = 0 , which are bounded for all time. Thus the restriciton of the original system to a center manifold completely describes all the dynamics we are interested in. Because of the attractivety of center manifolds, also stability properties are preserved. For an existence proof see e.g. Marsden and McCracken [1976]. Of course, there is a problem with the smoothness of center manifolds. As a matter of fact we can not expect that the restricted vector field belongs to the category of analytic or Coo vector fields, no matter how smooth F is (Carr [1981]). But for any k , it is of class ck, if F is sufficiently smooth.

Up to terms of order four, the Poincare-Birkhoff normal form of the re-stricted system reads as follows in cylindrical coordinates:

r = 8r + arz + erz 2 + ... + 0(4} (2.1) z = az + bz 2 + cr 2 + fr 2 z + gz 3 + ... + 0(4)

6 = y + h 1 z + h 2 r 2 + h 3 z 2 + .. . + 0( 4) I r Her·e all coefficients are smooth functions of ~ and v , a , 8 and y are given by the eigenvalues of DuF(~,v,O) which cross the imaginary axis; a,b,c, and h1 only depend on terms of order less than three in the Taylor expansion of F , and the remaining coefficients only depend on terms of order less than four. The symbols o(4} stand for functions which are 2rr-periodic in e and of order 4 in r and z . Neglecting these terms the normal form obviously has a rotational symmetry with respect to the z-axis. This is a crucial point of our method, since it allows a simple construction of approximate solutions. Indeed, the flow of the approximate system is com-pletely determined by its projection on the (r,z)-plane. Stationary points correspond to stationary points in the (r,z)-plane with r = 0 , the periodic solutions we are looking for correspond to stationary points in the (r,z)-plane with r r 0 , and invariant two-tori correspond to invariant circles. Note, that a general higher order term in the e-equation has a singularity at r = 0 . This is due to the introduciton of cylindrical coordinates. It causes some trouble in the continuation of stationary points and periodic or-bits near a Hopf bifurcation to the full non-symmetric system. To this end it is therefore more convenient to work with Cartesian coordinates. But these problems have been completely solved already by Langford [1979]. So we will concentrate on the Neimark-Sacker bifurcation and the construction of the bifurcating two-tori. Here the singularity at r = 0 turns out to play no role.



422 ,Jurgen Scheurle

Let us now formulate the non-degeneracy conditions which we impose on F . They are written up in terms of the coefficients appearing in (2.1) ((~,v)

near (2.2)

(2.3)

(2.4)

{0,0)):

b

(2b-a) 2

a(a,i3) f 0 ~

ab < 0 , c ( b - a) > 0

~b(}- 1)d - 4 ~c e - 2bf + 6cg ] f 0

The first condition is a generalized "Hopf condition" and stems from Langford [1979]. Rot..s;hly speaking it says that the three eignevalues a and 13 ± iy cross the imaginary axis in a non-degenerate way. These conditions enable one to replace ~ and v by new parameters A and a via

(2.5) A a(~,v)

This follows by the implicit function theorem. (2.3) imposes a restriction on the second order terms of F and guarantees the transcritical stationary bi-furcation along a curve el and the Hopf bifurcations along curves e2 and e4 in the (A,cr)-plane. Setting

( 2. 6) a(O,O) = a , b(O,O) b

these curves are assymptotically given by

el A = o( a2)

( 2. 7) e2 A = bo bo-a o a

+ 0( a2 ) as a--+-0.

e4 A a+O(a 2 )

The stability properties mentioned in the introduction, refer to any path through the upper (A,a)-half plane which crosses these curves transversally from left to right, each in one point. In the lower half plane the trivial solution looses its stability to the periodic solutions bifurcting from it along e4 .

For the torus bifurcation a third condition (2.4) is needed which includes third order terms. The reason is that the system in which terms of order greater than two are neglected possesses a first integral in the bifurcation point. This leads to a vertical torus bifurcation. But a hyperbolic structure is neccessary to continue the tori to the full system. In case of a vertical bifurcation the tori are not hyperbolic. By (2.4) a vertical bifurcation is excluded. The sign of !J determines the direction of bifurcation and the stability of the bifurcation tori. The usual principle of exchange of stability is valid. More precisely, we have the following theorem:




Theorem 2.1 Let the map F : R x R x X_.. X be sufficiently smooth and satisfy the assumptions (1.2), (1.3), (2.2), (2.3) and (2.4). Then there is a curve e3 ;., = ;.,*(a) through the origin in the (;.,,a)-plane, where the Floquet exponents of the periodic solutions of (1.1) are purely imaginary. The func-tion A* is assymptotically given by

(2.8) ;.,*(a) = 2bo a+ O(a 2 ) as a -• 0 ~

Moreover, there is a wedge-like domain

S ={(;.,,a) I!A-;.,*(a)l ~ a4 , lal ~ cro} , ao > 0

such that for all parameter values in S either to the right or to the left of e3 depending on the sign of Q ' the system (1.1) possesses an invariant two-torus. This torus depends continuously on the parameters. If Q < 0 it exists to the right of e3 and is locally attractive. If Q > 0 it exists to the left of e3 and is repelling. It contains the corresponding periodic orbit in its interior and shrinks to this as e3 is approached.

The claim that in a generic way through parameter space many of the bifur-cating tori are quasiperiodic,is based on the following theorem.

Theorem 2.2 Let the assumptions of Theorem 2.1 be satisfied. Denote that part of S , where the invariant tori of (1.1) exist by S* . Then there is a "Cantor set" of continuous curves in S* along which the flow on the tori is quasiperiodic with two basic frequencies w1 and w2 , i.e. in appropriate angle variables the flow is given by h = w1 t + const. , ¢ 2 = u1 2 t + const. One of the frequencies is very small, the other one lies near Yo= y(O,O) , and a KAM-condition holds:

(2.9)

for all integer vectors j (j1,j 2) f 0 with positive constants E,T . The frequency ratio w1/wz and so the Poincare rotation number is constant along each curve. These curves emenate at the neutral stability curve e3 of the periodic solutions and run all the way through S* up to the other boundary curve. The curves form a set of positive measure in parameter space.

Thus a path in parameter space generally meets a set of positive measure of quasiperiodic tori. Despite the fact that individual quasiperiodic flows are structurally unstable, their occurence in this bifurcation is stable and in fact their occurence along appropriate arcs in parameter space is an open con-dition. Proofs of the theorems will be given in the next sections.

Let us conclude this section with some remarks on open problems. Due to Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.


424 Jurgen Scheurle

results in Arnold [1965] and Herman [1977] one expects periodic tori to occur in open regions of parameter space (phase locking) (c.f. also Iooss and Joseph [1981]). But this phenomenon has not been rigorously proven yet in the present situation. Another question far from being completely answered is the disap-pearance of the invariant tori. Simple considerations show that this phenomenon must occur somewhere in between the curves e2 and e3 • In the rotational symmetric case it happens through a heteroclinic bifurcation (Chow and Hale [1982], Guckenheimer [1981]) but in the non-symmetric case one expects transversal hetero- and homoclinic orbits to appear which do not directly explain the disappearance of the tori. See Langford [1982] for another idea. On the other hand, it is very difficult to obtain exact verifiable results concerning transversal hetero- and homoclinic orbits, since the corresponding "Melnikov function" is exponentially small and does not give an answer to this problem at finite orders in verturbation theory (c.f. Broer and Vegter [1982], Holmes and ~1arsden [1982] and Sanders [1980]). It is planned to study this question in a forthcoming joint paper with P.Holmes and J.Marsden.

3. PROOF OF THEOREM 2.1. As already mentioned in the introduction we only expose the general outlines of the proof. In particular, we will not ex-plicitly count the number of derivatives of F which are actually used. Starting with the normal form (2.1), where ~ and v are replaced by the new parameters A and o (see (2.5)), we scale as follows

( 3.1) ,\ =o ~ , r = or , z = oz

Since (e,r,z) are cylindrical coordinates, (2.1) is invariant under the transformation r -+-r and e-+ e + n . Therefore we can restrict ourselves to the case r > 0 for all o . By (2.1) we obtain the equations

r o{ {X-1 )r + arz + oerz 2 + O(o")} ( 3. 2) z oFz + bz" + cr 2 + ofr 2 z + ogz 3 +O(o 2 )}

8 y + ohlz + ozhzrz + o2 h3z 2 + o(o 3)/r

Now we neglect terms of order o2 and consider the reduced, rotational sym-metric system in the (r,z)-plane

( 3. 3) r o{(~-1)r + aorz} z o{Xz + b0z2 + c0r 2 }

with a0 , b0 and c0 as defined in (2.6). In order to obtain approximations for the periodic orbits of (3.2) we look for stationary solutions of (3.3) with r > 0 . An easy computation yields




(3.4) ru = {-_!_ (X(l-X) + ~ (1-Xlz)}~

co ao aij 1 -X

ao The corresponding of the matrix J

Floquet exponents are approximately given by the eigenvalues of the linearization of (3.3) around (r 0 ,zo)

1 (3.5) nf(A) = ~ tr J ± (i tr 2 J - det J) 2

2

So the neutral stability curve e3 is approximately obtained by solving the equation tr J = 0 for X . The solution is

(3.6) - 2bo A.o= 2b 0 -a 0

Set Re n°= ~ 0 and Im n°= ~ 0 • Then we have 1 2

~0 > 0

~ 0 (X 0 ) = _ 2aobo _. ~ 0 > 0 (2bo -ao )2

Next we do a perturbation analysis to compute the exact periodic solutions and the exact Floquet exponents. Set

(3.8)

Introducing this ansatz into (3.2) and writing x = (x 1 ,x 2 ) yields

x crA(~)x + crB(x,x) + o(cr 2 )

8 = y + crh1,oZo + crh1,oXz + O(cr 2 )

Here A(X) has the eigenvalues n~(X) ,

(3.10)

and the symbols O(cr 2 ) stand for functions which are 2n-periodic in e and of order a2 as a+ 0 uniformly with regard to all the other variables restricted to appropriate domains. In particular, lx 1 1 <r·0 • Note that r

"' is strictly bounded away from 0 if A. is restricted to a small interval around zero. But this is enough for our purposes since we are mainly inter-ested in the ~eimark-Sacker bifurcation. So the singularity in the a-equation no longer bothers us.

In order to continue the approximate periodic orbits to the full system and to construct the exact neutral stability curve e3 we use transformation techniques. We shall need the following notations. Let K be either Rn



426 Jurgen Scheurl e

or a:n. The space of k-times continuously differentiable functions from R to k k K , which are 2rr-periodic, is denoted by C2 1f (.IR,K). For f E C21f(R,K),

we define the continuous projection operators

1 J211 P: ft-t- z.rr O f(e)dB, Q = id- P.

k+ 1 k Moreover, we introduce the operator L: Czlf (.IR,K) --r Czrr(.IR,K): f f-+ f'y 0 •

It is easily seen that L is an isomorphism from QCk+l(lR,K) on QCk(R,K)

Lemma 3.1 If Ia! and 111 are sufficiently small, then there exists a coordinate transformation

x = x + X 0 (o,~,e) (3.11)

such that in the new coordinates, equation (3.9) has the following form:

(3.12)

The functions Xo and Go are 21T-eeri odic in 8 and of order a as a --+ 0 ; IY - Yl = O(a) too. Thus the periodic solutions are given by x = o . e = yt + cons t.

Proof: Inserting (3.11) into (3.9) and observing (3.12) we end up with the following equations for (X 0 ,0 0 ) and y :

(3.13)

k+1 k . Assume that the map G: .IR 3 xC 21f (R,.IR 3 ) -+ c21f(.IR,.IR 3 ) wrnch maps (o,~.~.X 0 ,8 0 ) to the left-hand side of (3.13) is sufficiently smooth. Note that (O,O,y 0 ,0,0) is a solution of G = 0 , and the correpsonding linearized operator with regard to (X 0 ,o 0 ) is just L So we can use a Liapunov-Schmidt method and split this equation into a P- and a Q-part. By the im-plicit function theorem the Q-part is easily solved for (QX 0 ,8 0 ) E QCk+1(JR,.IR 3)

as a function of a3,~ and PX 0 E lR . The solution is of order o2 as a -+ 0 What is left is to solve the P-part for ~ and PX 0 as functions of a and ~ . But this is a purely algebraic problem of the form

oA(1)PX 0 - aB(PX 0 ,PX 0 ) - PO(a 2 ) 0 (3.14)




After dividing the first equation through by a, again the implicit function theorem applies to solve it, provided that A(~) is regular. But A(O) is regular. So A(~) is regular for small 1~1 . It follows that PXo and y - y are of order a as a-+ 0 . Thus the lemma follows.

Now we consider the tranformed equation (3.12). If the linear part of the x-component did not depend on e , then the Floquet exponents n1

~

the periodic solutions would be given by the eigenvalues of the corresponding matrix. But as known from Floquet theory this can be achieved by a further change of coordinates. We also transform away the x-term in the a-component.

Lemma 3.2. _!i_ lal and 1~1 are sufficiently small, then there is a coord-inate tranformation (near ~ 0

(3.15)

such that in the new coordinates (3.12) has the form

(3.16) X= aA(a,X)x + aB(x,x) + O(a 2 ·lxl 2 )

e = y + o(a·lxl 2 )

The matrix valued functions xl and 01 are 2~-periodic in e , IXll O(a 2 )

and IA-AI=o(a) as a-+0

Proof: Inserting (3.15) into (3.12) and observing (3.16) we obtain the fol-lowing 1 inear equations for X1 , e1 and A :

(3.17) Xly + oX1A- oAX1 + oA- aA- o(a2 )(E + X1) 0

eE + aG1A ah1,oX1- ah1,o-O(a 2 )(E + Xd 0

:;::

Here the symbols o ( a2 ) stand for matrices which only depend on a , :\ , and ~ e , but not on X1 nor on e1 ; E is the unit matrix. Now we proceed analogously to the proof of Lemma 3.1. If we identify the space of pairs of matrices (X 1 ,e1 ) with R6 , then for a= 0 the operator corresponding to (3.17) is just L. So for sufficiently small lal, (X 1,Qed E QCk+1(R,R 6 }

is uniquely determined by the Q-component of (3.17). The solution is of order a2 as a-+ 0 . Finally A and Pe 1 are determined by

(3.18)

To solve the second equation for PG1 we use the fact that A is a regular



428 Jurgen Scheurle

mattrix. Note that Pe 1 = o(l) as a -+0. Therefore the transformation (3.15) might be singular for large I xi . From (3.16) we read off that the Floquet exponents n1 are given by the eigenvalues of aA . Since !A - AI = O(a)

'2 it follows

(3.19) n 1 = n ~ + o( a 2 ) 2 2

Set n 1 = ~± is Then e3 is given by the solution of the equation ~ = 0 . 2

But in view of (3.7) and (3.19) the implicit function theorem yields a unique solution ~ = ~*(a) = o(a) . So (2.8) is now proved.

For the construction of the invariant two-tori which bifurcate along e3

we perform another transformation. By this is achieved that most terms of order less than seven in the Taylor expansion of the vector field are eliminated except for some resonant ones which do not depend on the new 8-variable. So we do some kind of averaging.

Lemma 3. 3 If

~2 (p near

(3.20)

Ia! and 1~1 are sufficiently small, then there exist p, 0 and <1\ ,¢2 E lR) such that (3.16) reads as follows:

h,

Here the symbols o(pk) stand for functions of (a,tp,h.~ 2 ) which are 2rr-

periodic in ¢1 and ¢2 and of order pk ~ p -+ 0 ; ~ and s are defined as in (3.19), y as in (3.12) . The Mk are real quantities which only depend on a and ~ . In particular, we have

(3.21) o ~ (O o) + ~ (O O) a~* (O) ' aa ' a~ ' aa

where ~ 0 is equal to ~ defined in (2.4) evaluated at ~ v = 0 .

Proof: After a similarity transformation of the x-variable we can assume that aA is written in complex Jordan normal form, i.e.

aA ( n1 0 )

0 n2

Now let us consider a local coordinate transformation given by x = y + Yn(a,~, ¢,y) , ~ = ¢ + ~n(a,f,¢,y) , where Yn = (Y~,Y~) and ~n are (vector) homo-geneous polynomials of degree n in y (n ~ 2) and 2n-periodic in ¢. It is easily seen that terms of order less than n in the Taylor expansion of the vector field are not changed by this transformation and, if we require the n-th




order terms of the transformed vector field to vanish, then we obtain the fol-lowing equations

(3.22)

for Y n 3Y

and ~n

n --y + ()<j>

adi n --y +

()<j>

2 av IarnkYk

k=1 k crAYn - o(cr) 0

0

Here the O(cr) stand for the old n-th order terms. In terms of coefficients these equations read as follows (£ = 1,2)

av£,q -a%-- y + Y~'q(q·n) - n£Y~'q = O(cr)

a~q n - + ~q(q·n) = O(cr) a¢ Y n

where q is a double index with I q I = n ; So for cr y£,q n for

corresponding operator is again L , and the Q-projections of are uniquely determined in c~! 1 (JR,([), provided that lcrl < cr 0

= 0 the and ~q n

some con-stant cr 0 which might depend on n . Moreover, the P-projections are uniquely determined iff (q·n - ni)/cr F 0 and (q·n)/cr F 0 , respectively. The other P-projections are set equal to zero which, in general, leads to non-vanishing terms of type M(cr,~)yq in the new vector field, M = O(cr) . Since n~(O) -n~(O) (cf. (3.19)), the corresponding q have the property q£ = qi + 1 (£ F ~) and q1 = q2 , respectivley.

By successive transformations of this type with n running from 2 up through 6 , and the introduction of polar coordinates in the y-plane

(3.24) Y = pei<Pk (k 1 2) k = ' '

we end up.with an equation of type (3.20). The claimed properties of M1 fol-low by explicit computations up through order one in cr (see also Scheurle and Marsden [1982]). The property M1 (0,0) = 0 is due to the fact that the system (3.3) is integrable (Hamiltonian) for ~ = ~ 0 •

To find an approximation for the bufurcating tori we now look for equilib-rium points of the p-equation in (3.2),neglecting the o(p 7 )-term. Set ~ = ~*(cr) + A and scale as follows

(3.25)

Then we obtain



430 Jurgen Scheurle

Recall that s vanishes along the curve A ~*(o) By (3.7} it therefore follows s =so > 0 for o = 0 and A = 0 By (3.21) we have M1(0,0) = ~ 0 • Hence, by the implicit function theorem, (3.26) has a solution of the form

( 3. 27) p = ~ + O(Ao) , 0 \j M:

either for positive or negative A depending on the sign of ~ 0 If ~ 0 > 0 , then A is negative; if ~ 0 > 0 , then A is pesitive. Here we can assume IAI < Note that A measures the distance of a point (A,o) from e

3 Finally, set

(3.28)

With this ansatz, (3.20) becomes

R N1R + Nz(o,A,R,¢1,¢z) (3.29) ¢1 \11 + N3(o,A,R,¢1,¢z)

¢2 Vz + N4(o,A,R,¢1,¢z)

where

N1 o 4 IAI(2M1P~ + 4oiAIMzilci) (3.30) v1 1; + o3 jAI(M3il5 + o2 JAIM4i)6)

Vz Y + o3IAI(MsP 2 + o2 IAIM6il6)

The functions N2 ,N 3 and N4 are 2rr-periodic in ¢1 and ¢2 and of order O(A 2 o5 ) as A-+ 0 and a -• 0 . We can think of R,¢ 1 ,¢ 2 as action angle variables for the approximate torus, and of v1 and v2 as frequencies of the corresponding flow. By the definition of N1 this torus is stable if ~ 0 < 0 and unstable if ~ 0 > 0 . The crucial point is that the perturbation given by N2 ,N 3 and N4 is of order striclty less than the attraction (respectively repelling) rate which is of order o4ll\l as o-+0 and IAI-+0. This allows one to apply the degenerate case (ii) of Sacker's Theorem [1965] to obtain a smooth toroidal manifold R = T(o,A,¢1,¢ 2 ) for the full system (3.29). This torus has the same stability properties as the approximate one. The func-tion T and its derivatives are continuous functions of the parameters. Note, due to the scaling (3.25) the torus shrinks to the corresponding periodic orbit when A-+ 0 i.e. when e3 is approached. Thus Theorem 2.1 follows.

4. PROOF OF THEOREM 2.2. To prove quasiperiodicity of the flow on a torus one has to deal with the problem of small divisors. We use KAM-theory to overcome this difficulty. We only sketch the basic idea (cf. Scheurle and Marsden [1982]).



Interaction of steady state and Hopf bifurcations

The flow on the tori constructed in the previous section is governed by the equations

(4.1) h v1 + N3(a,J\,h,¢z,T) ¢2 Vz + N4(a,J\,¢1,¢2,1)

Let us introduce the additional parameter K-1 by scaling the time:

(4.2) h vdK + N3/K </>2 v2/K + N4/K

431

If we replace the items vk/K by frequencies vk which do not depend on the parameters, and which satisfy a KAM-condition (2.9), then there is a modified system

(4.3) </>1 v1 + 6v1 + N3/K </>2 Vz + 6v2 + N4/K

which has quasiperiodic flow with the two independent frequencies vk (see e.g. Zehnder [1975]). The modifying terms 6vk are real-valued functions of the parameters of order a5!\ 2 as a -+ 0 and J\ -+ 0. In order to find parameter values where the flow of (4.2) is quasiperiodic, it therefore remains to solve the algebraic equations

(4.4) vdK = v~ + 6vl(K,a,J\) vz/K = vg + 6vz(K,a,J\)

Formally this is easily done by the implicit function theorem. In fact, for v~ near 0 and vg near Yo we have the solution K = 1 , a = 0 for all !\ • By the definition of matrix with respect to K

\11

and and vz

0 is

C;oo

and ( 3. 7), the corresponding Jacobi an just

YOJ

so is regular. Therefore we can solve for K and o as functions of J\

thereby obtaining the desired curves of quasiperiodic solutions in parameter space. Here we can fix vg to be equal to Yo ' since in the original time scale we obtain the frequencies Wl = KV~ and W2 = K\1~

A rigorous proof, however, requires a careful analysis, since estimates strongly depend on the constant E in (2.9), and it is not obvious whether there are enough frequency vectors w0 = (vf,y 0 ) with v~ sufficiently close to 0 and (2.9) being satisfied for a given E One has to use the fact that the modifying terms 6vk are of order o 2 as o -+ 0 uniformly for



432 Jurgen Scheurle

as s-+ 0 . This implies that for each w0 with [v 0 [ < a 0 - const.a 2 which satisfies (2.9), equation (4.4)

1 - 0 has a unique solution a= a(w 0 ,II) , K = K(w 0 ,II) in [a[ ~Go and [K -1[~ Go But the Lebesgue measure of those v~ in any real interval for which (2.9) is not satisfied for a fixed second frequency is of order s as s -+ 0 uniformly for small intervals (Siegel and Moser [1971]). Therefore the relative measure of the set of v~ in [v£[ < Go - const.G 2 for which (2.9) is satisfied tends to 1 as s -+ 0 To prove that the curves G = G(w 0 ,11) form a set of positive measure in parameter space we use the fact that for /\ = O, (4.4)

has a solution for all w0 near (O,y 0 ) , since the 6vk vanish there. The function v£ ~--+ a(v~,y 0 ,0) is a local diffeomorphism, and therefore maps sets of positivE measure into sets of positive measure. Thus, this last claim follows by Fubini 's theorem, and the proof of Theorem 2.2 is now finished.

References

1. V.I. Arnold, "Small denomination I. Mappings of the circumference into itself", AMS Translations, Series 2, 46 (1965), 213-284.

2. V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, 1983.

3. B.L.J. Braaksma and H.W. Broer, "Quasiperiodic flow near a codimension one singularity of a divergence free vector field in dimension four," Preprint, (1981).

4. H. Broer, "Quasiperiodic flow near a codimension one singularity of a diver~ence free vector field in dimension three", Springer Lee. Notes in Math., 898 (1981), 75-89.

5. H. Broer, "Quasiperiodicity in local bifurcation theory", Preprint, ( 1982).

6. H. Broer and G. Vegter, "Subordinate Sil'nikov bifurcations near some singularities of vector fields having low codimension", Preprint, (1982).

7. J. Carr, Applications of center manifold theory , App. Math. Sci., Springer, 35 (1981).

8. N. Chow and J.K. Hale, Methods of Bifurcation Theory, Springer, (1982).

9. M. Golubitsky and D. Schaeffer, "A theory of imperfect bifurcation via singularity theory", Comm. Pure Appl. Math. 32, (1979), 21-98.

10. M. Golubitsky and B. Langford, "Classification and unflodings of de-generable Hopf bifurcations", J. Diff. Equat., 41 (1981), 375-415.

11. J. Guckenhei mer, "On a codimens ion two bifurcation", Springer Lee. Notes in Math., 898 (1981), 99-142.

12. J. Guckenheimer, "Multiple bifurcation problems of codimension two", Preprint, (1982).




13. R.M. Herman, "Mesure de Lebesgue et numbre de rotation", Springer Lee. Notes in Math. 597 (1977),271-293.

14. P. Holmes, "Unfolding a degenerate nonlinear oscillator: a codimen-sion two bifurcation", Ann. N.Y. Acad. Sci. 357 (1980), 473-488.

15. P. Holmes and J. Marsden, "Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom", Comm. Math. Phys. 82 (1982), 523-544.

16. G. Iooss and D.O. Joseph, "The behaviour of solutions lying on an in-variant 2-torus arising from the bifurcation of a periodic solution." in Appli-cations of Nonlinear Analysis in the Physical Sciences, H. Amann, N. Bazley and K. Kirchgassner edts., Pitman, 1981, 92-114.

17. G. loess and W.F. Langford, "Conjectures on routes to turbulence via bifurcations", Ann. of the New York Acad. Sci. 357 (1980), 489-505.

18. W. F. Langford, "Periodic and steady-state mode interactions 1 ead to tori", SIAM J. Appl. Math. 37 (1979), 22-48.

19. W. F. Langford, "Chaotic dynamics in the unfo 1 dings of degenerate bi-furcations", Proc. Int. Symp. on Appl. Math. and Inform. Sci., Kyoto Univ. (1982).

20. J.E. Marsden and M. McCracken, The Hopf bifurcation and its applica-tions , App. Math. Sci., Springer 19 (1976).

21. R.J. Sacker, "A new approach to the perturbation theory of invariant surfaces", Comm. Pure Appl. Math. 18 (1965), 717-732.

22. J.A. Sanders, "A note on the validity of Melnikov's method", Report 139, Wiskundig Seminarium, Vrije Universteit, Amsterdam, (1980).

23. J. Scheurle and J.E. Marsden, "Bifurcation to quasiperiodic tori in the interaction of steady state and Hopf bifurcations" to appear in SIAM J. Math. Anal ..

24. C.L. Siegel and J.K. Moser, Lectures On Celestial Mechanics, Springer, 1971.

25. F. Spirig, "Sequence of bifurcations in a three-dimensional system near a critical point", J. Appl. Math. and Phys. (ZAMPF) 34 (1982), 259-276.

26. E. Zehnder, "Generalized implicit function theorems with applications to some small divisor problems I", Comm. Pure Appl. Math. 28 (1975), 91-140.

Jurgen Scheurle Mathematisches Institut A Universitat Stuttgart D-700 Stuttgart 80 West Germany







CONVECTION IN A ROTATING FLUID LAYER

James W. Swift1

ABSTRACT. A model for convection in a rotating fluid layer, due to Busse and Clever [1], is modified to include non-Boussinesq effects. The three dimensional ODE of the new model has saddle-node, pitchfork, Hopf, and global bifurcations.

1. INTRODUCTION. Thermal convection in a fluid layer heated uniformly from

below and rotating about a vertical axis can undergo an unusual instability.

Kuppers and Lortz [2] have shown that, if the rotation rate exceeds a

critical value, any convection roll is unstable to a new roll oriented preferentially at 58° to the original roll, as measured in the direction of

rotation. The subsequent time evolution is somewhat puzzling since there are

no stable, steady solutions of small amplitude. Kuppers and Lortz argued

that the flow would become turbulent. They did not consider the possibility

of limit cycle behavior, perhaps because they considered parameters where the

convective instability is direct rather than oscillatory.

Busse and Clever (1] analyzed a system of three rolls, mutually oriented

at 60°. They found that the rolls cyclically replace each other due to the

Kuppers-Lortz instability, but the transition time from one roll to the next grows exponentially as time goes on. The reason is that there is a heteroclinic cycle connecting the rolls, which are saddle points in the three dimensional phase space of time dependent roll amplitudes. (See Fig. 3). This heteroclinic cycle is like an attracting limit cycle of ''infinite period" and the transition time from one roll orientation to the next increases as the trajectory approaches the heteroclinic cycle. Busse and

Clever predicted that experimental noise would keep the transition time

finite, fluctuating about some mean value depending on the noise level.

Heikes and Busse [3,4] performed experiments on a rotating system using

shadowgraph visualization and found that the rolls aligned themselves in

1980 Mathematics Subject Classification. 76E15. 1 Supported by the California Space Institute under Grant No. CS13-83

435





436 SWIIT

randomly oriented patches about S to 10 rolls wide. There is a beautiful

film of this experiment which shows one patch of rolls growing at the

expense of another, with the net effect of rotating the roll orientation at a

given point by approximately 60° as predicted.

The experiments of Heikes and Busse were done with methyl alcohol, which

satisfies the Boussinesq approximation (S] well, however this approximation

is poor in many geophysical and astrophysical examples of rotating

convection.

The purpose of this paper is to examine the model of Busse and Clever [1] when non-Boussinesq effects are included and there is no noise. The

Boussinesq approximation causes a symmetry in the system of ODE's which

forces the saddle connections of the heteroclinic cycle. When the Boussinesq

symmetry is broken, the saddle connections are broken to yield a long period limit cycle.

The three dimensional ODE with non-Boussinesq effects included has many

secondary bifurcations. The bifurcations of fixed points are computed

explicitly, but there is also a global bifurcation of a limit cycle which has not yet been analyzed completely, but which may indicate deterministic

chaotic dynamics.

In section 2 the three dimensional system of ODE's is derived from the

convection PDE's, and the symmetries of the two systems are related. In

section 3 the ODE's are analyzed and the bifurcation diagrams are drawn.

2. FROM PDE's TO ODE's. The equations describing convection in a rotating fluid layer, suitably non-dimensionalized, are [2]:

+ v u = 0

1 (a~ + <~ P at

ae + + 2 at + u ve = R u · z + v e .

(1)

The dependent variables are the fluid velocity ~. the deviation from a

linear temperature profile e. and a generalized pressure 7T which gives all

gradient forces, including the centrifugal force. The dimensionless parameters are the Prandtl number P = viK, the Taylor

number 1:2 = (4!22d4 )tv2 , and the Rayleigh number R = (agt.Td3)fvK, where\! is +

the kinematic viscosity, K is the thermal conductivity, 0 is the rotation rate about a vertical axis, d is the thickness of the fluid layer, a is the thermal expansion coeffiecient, AT is the temperature difference between the top and bottom plates, and -gz is the acceleration due to gravity.



ROTATING CONVECTION 437

The equations (1) have the symmetry of the proper Euclidean group in the

horizontal (x,y) plane, that is, the semi-direct product of proper rotations and translations 80(2) x R2 . The rotations are

(2a)

The translations are

( xxx + Dx) y y + Dy ; z z

~+~, e + e, n+n (2b)

Note that the Coriolis force term (ti x~£) is not symmetric under orientation reversing transformations in the horizontal plane.

In addition, when 8 = 0, the equations (1) have the Boussinesq Symmetry, which is a midplane reflection coupled to a temperature inversion.

Gh-D (~){~). ,._,. ··· (3)

This symmetry requires the validity of the Boussinesq approximation, which says that all material properties, such as V and K, are independent of temperature. For real fluids the Boussinesq symmetry is often broken because the viscosity depends on temperature, but the term proportional to 8 in (1)

also breaks the symmetry and is mathematically simpler. When the Prandtl number is greater than 1 the convective instability of

a roll is due to a single eigenvalue A passing through zero as the Rayleigh number is increased beyond Rc. As in Golubitsky et. al. [5], and Busse and Clever [1], assume that the bifurcating solutions to the linear problem are a linear combination of three rolls

3 El(;,t) =Re( E j=l

+ + a.(t)eikj.x f(z)),

J + x = (x,y,z)

+ where ai (i = 1,2,3) are complex amplitudes and ki are three critical wavevectors, mutually oriented at 1200 in the horizontal plane, so that ""+- ~ -+ -+ A

k1 + k2 + k3 = 0 and ki • z = 0.

With these assumptions, the Center Manifold Theorem [6] allows a reduction of the PDE's (1) to a system of ODE's for the amplitudes ai(t),

l.ai + higher order terms.



438 SWIFT

As a result of the rotational symmetry (2a) the vector field g commutes

with a cyclic permutation of the ai's (120° rotation in the x-y plane),

and the complex conjugation of the ai's (180° rotation in the x-y plane)

A reflection through a vertical plane, coupled with a reversal of the

rotation direction leaves (1) invariant, and the consequence for g is

however this is not a true symmetry unless ~ = 0.

The invariance of (1) under translations in the horizontal plane -+ -+ -+ x -+x + D implies that

Finally, the Boussinesq symmetry (3), if it holds, gives an inversion

symmetry

The most general ODE with these symmetries is

al = la1 + ea283 - a1 (1a1 12 + ala2 12 + ~la 3 1 2 > + O(a4 , a2l, l 2 ) (4}

plus cyclic permutations for a2 and a 3 , where e = 0 if the Boussinesq

symmetry holds, and a = ~ if there is no rotation.

One effect of the e term is to cause the ai's to become real. Let

¢ = arg(a1) + arg(a2) + arg(a 3 >. then

al al- al tl

2i la1 12

<P = -e

Therefore ¢ approaches 0 or n, depending on the sign of e, and we can choose -+

a displacement D in (2b) that makes all the ai's real. Much of Golubitslry

et. al. [5] concerned the subtle effects on the phase ¢ when e is perturbed

from zero. These complications also exist in the

the bifurcations that occur for A = O(e 2) and

rotating problem, however 3

iEl lai 12 = O(e 2 ) can be studied by letting ai be real and truncating the ODE to third order. Let




z, an. truncate (4) to obtain

i AX + eyz - x(x2 + ay2 + Pz2>

y = A.y + EZX - y(y2 + az2 + ,Px2) (5)

z = A.z + exy - z( z2 + ax2 + py2).

These equations have the symmetry of a tetrahedron T, a 12 element group

of proper rotations about the origin in R3 , generated as the semi-direct

product of

(x,y,z) + (y,z,x), (x,y,z) + (-x,-y,z), and (x,y,z) + (x,-y,-z) ( 6)

When there is no rotation of the fluid layer a = p and the equations (5)

have the full symmetry of a tetrahedron Td' a 24 element group generated by (6) plus a reflection through the x = y plane:

(x,y,z) + (y,x,z). (7)

The reflection symmetries severely limit the dynamic behavior possible

in (5) since a trajectory cannot pass through a hyperplane of reflection.

The case where a= p was studied by Buzano and Golubitsky [7], and Golubitsky

et. al. [5], where the parameter a is replaced by a = a/1-a.

When the Boussinesq symmetry holds e = 0 and the system (5) has the

symmetry Th' gene.rated by (6) and ( x, y, z) + ( -x, -y,- z) . (8)

The 24 element group Th includes 3 reflections through hyperplanes, such as (x,y,z) + (-x,y,z).

When a= p and e = 0 the equations (5) have the full symmetry of a cube

(or octagon), Oh' which is a 48 element group generated by (6), (7), and (8). Finally, when a = P = 1 and e = 0, the equations (5) have spherical

symmetry, 0(3).



440

I

I

"' ------ -~--.

SWIFT

~-· ~ , I

I

.-"

\• ""'. / ". /

I ·. ;

\ / .·""' Oh ;cx=~,t=O

Fig. 1 The symmetries of (5) as represented on a cube. Planes of reflection are indicated by dotted lines.

3. 1HE BIFURCATION DIAGRAMS. First consider the Boussinesq system (e = 0),

which was studied by May and Leonard [8] as a model of population dynamics.

This special case is important because it is an organizing center. For a

generic system the non-degeneracy condition e I 0 is likely to hold. In the

non-degenerate bifurcation, however, only the solutions which branch off the

conduction solution (i.e. x=y=z=O) are captured by a local analysis.

Therefore one studies the degenerate bifurcation, with 8 = 0, and the unfolding, where e is perturbed from zero. Secondary bifurcations can be analyzed when 8 I 0, and these bifurcations are guaranteed to occur when e is sufficiently small. In practice this often gives qualitatively correct behavior even when e is quite large.

The stationary solutions of (5), when e

Name

Conduction (C)

Roll (R)

Hexagon (H)

General Solution (G)

Multiplicity

1

6

8

12 x2

y2

z2

Eguation

x=y=z=O

A.(a-1) ap-1

~ ap-1 0

0, are of four types:

Eigenvalues

-A.,-A.,-A.

-2>.., 2A.(a-1)(6-1) aP-1

-A.[(a-1)(8-l)+(a-8) 2] ap-1




For rolls and general solutions there are additional solutions related

to those listed by the symmetry (see Fig. 1). The eigenvalues of the Jocobian matrix of (5), evaluated at the stationary solution, determine the linear stability of each solution type. A negative eigenvalue is stable.

The general solutions only exist when (a-1)(~-1) > 0. In the complementary parameter region there are instead the heteroclinic cycles (HC) mentioned in the introduction, which connect 3 rolls as shown in the phase portraits accompanying the bifurcation diagrams (see Fig. 3).

The qualitative behavior of the system (5) can be summarized by drawing bifurcation diagrams for various fixed values of the parameters a and ~. The bifurcation diagrams plot (x2 + y2 + z2) as a function of A. The quantity x2 + y2 + z2 is proportional to the convective heat flux (Nusselt # - 1) and A is proportional to the amount of temperature difference above critical (R - Rc). These bifurcation diagrams describe an experiment where the temperature difference is quasistatically increased.

2

1

' ' ' ' ' ' \ ' ' -~ ' ', '

I

N 1II -----. ___ )II

Figure 2. The a - ~ parameter space is divided into open regions by the lines a = ~. a= 1, ~ = 1, and a= ~. and a+~ =2. (Only a and ~positive are considered for simplicity). Within each region the behavior of the system (5) is qualitatively similar. A reversal of the rotation rate (~ +-~) interchanges a and ~. so only a > ~ need be considered. The dotted line summarizes the results of Kuppers and Lortz [2] for infinite Prandtl number and stress-free boundary conditions. Regions III and IV may be relevant for other convection systems.



442 SWIFT

:x.l. + 112 + r...-a. z.

L)_oc R-Rc R G

~ 1 I

H c

II HC H c

ill HC R c

Fig. 3 The bifurcation diagrams of (5) when & = 0. The Roman numerals correspond to the regions in the a- II plane shown in fig. 2. The solution types are: R =roll, H =hexagon, G =general, and HC = heteroclinic cycle. The stable solutions are drawn with a bold line, or solid dots. The phase portraits are schematically drawn for/.. > 0, where x, y, z are the time dependent roll amplitudes. The spheres represent attracting invariant surfaces. These surfaces are not smooth unless a + II = 2.



I

II

ill

N

ROTATING CONVECTION

+ t + \p >..H 'As

•• LC R •• -t •• H •• H c

H-t H- oo LC 00

00

0 00 R

c G R

c

443

)..>\H

\>\H

Fig. 4 The bifurcation diagrams of (5) for e > 0, along with phase portraits at selected values of A. The solution types are abbreviated as in fig. 3, except LC = limit cycle. In regions I and IV both A ) AH and A < A.H are possible, although only the latter is drawn. When l+ 0 the twg types of hexagons, H+ and H-, are inequivalent. The four H+ solutions, related by the symmetry (6), are situated on the vertices of a tetrahedron in phase space. In the original fluid layer H+ solutions have flow up in the center of the hexagon and down on the sides. The cell walls form a honeycomb pattern.



444 SHlFT

When t f 0, the stationary solution types are as follows:

Conduction (C)

Roll (R)

n+ if sgn(x))1

H- if sgn(x)<l

General (G)

Multiplicity

1

6

4,2

12

Equation

x=y=z=O

t±~ 2 +4A. ( 1 +a+!l) x=y=z= 2(1+a+!J)

f 2+(a+B-2)A. aj)-1

Eigenvalues

-A.,-1..,-A.

-21..-e

2A.-(a+!J+4)x2 +

i/3 (a-f3)x2

Unknown

The values of x2 ,y2, and z2 for the general solution are the three roots

of the cubic in x2 :

where r 0

[e2 + (a+B-2)A.] 1 4 + & (a!l - 1) (a!l-1)2

[(1+a+fl)s 2 + xA.] x2 y

1/2 [(a- !!) 2 + (a- 1) 2 + (!l- 1) 2 ] (a- 1){!!- 1) A.- e2

This expression was found by Ken Rimey, using Vaxima and some clever

tricks. Vaxima is a Berkeley variant of Macsyma, a computer program developed

at MIT for doing algebraic calculations. The eigenvalues of the general

solution have not been computed, but they are known when s 2 (( !..(same

as s = 0), and when the general solutions are created at the pitchfork

bifurcation of the rolls, to be discussed below.

There are many secondary bifurcations:

1) Saddle-node of hexagons. As A. is increased past A.sn two hexagons appear

with nonzero amplitude.

A.sn = 4(l+a+j)) xsn = 2(l+a+!J) E

2) Pitchfork of a roll, creating two general solutions at

(a-1)(!J-1)




The general solutions exist for A > Ap' and have the same stability near

the bifurcation as the rolls have for A < Ap·

3) Hopf of a hexagon, creating a limit cycle at

The sub- or supercriticality of the Hopf bifurcation can be computed relatively simply by exploiting the z3 symmetry about the x=y=z axis. Rotate

and translate to the coordinates (u,v,w), where the Hopf bifurcation is at

the origin and w is the axis of 3-fold symmetry. Let t; = u + iv, then the vector field has the symmetry t; + ei2n/3 t; , w + w, and consequently the Taylor

expansion about the fixed point at A = A8 , is:

t; i111 t; + cr~ 2 + 11t; It; 12 + v w t; + •••

w = -dw + el~l 2 + •••

where 111, d, and e are real, and cr, 11• and V are complex. The center manifold calculation is trivial due to the symmetry:

w = e/dlt;l 2. On the bowl shaped center manifold one can change coordinates

so that the radial coordinate satisfies

r = Re(J1 + Ve/d)r3 '= -ar3.

When a > 0 the bifurcation is su~ercritical, meaning that there is a stable

limit cycle created as A is varied (assuming d > 0). When a < 0 the bifurcation is subcritical, and there is an unstable limit cycle near the bifurcation [6].

For the Hopf bifurcation in (5) one finds

a = a_ 9

(a+B-2) (a+8+4) (a+jl+2)

The linear stability of the hexason and the sign of a determine the stability of the limit cycles, and indicate that the limit cycles exist for

A > AH.

4) A global bifurcation occurs when the limit cycle collides with three

seneral solutions. The period of the limit cycle increases until three

seneral solutions are connected in a heteroclinic cycle at As· As A is increased beyond AS the saddle connections are broken and there is no more limit cycle. Assuming the eisenvalues of the seneral solution are real, the



446 SWIFT

global bifurcation proceeds as shown in fig. 5. This is sure to be the case

near a = ~ since the trajectories in the neighborhood of the hexagon are

strongly attracted to the plane normal to the x:oy=z a:r.is.

-X < )..<).. H :J ).=). 3 A> A

3

0t\-' 'I\

~'fo* 5~t~Z 'i.Y / . \, / / """ "

/ J' ' ~ ~ ..... / .. - --- ~ __...._, 1( / ---.

\I I

.X <A H A::: AH ). >A H

Fig. 5. The global bifurcation when the eigenvalues of the general solution are real. When a=~ the three general solutions pass through the hexagon solution at AH and there is no Hopf bifurcation nor global bifurcation.

Since the ODE is actually three dimensional rather than two dimensional,

there is the possibility of chaotic dynamics. As the limit cycle grows it

could period double and become "strange". This would be particularly

likely if the two attracting eigenvalues of the general solution are

complex. Then the saddle connections, or heteroclinic cycle, would appear as

in figure 6. Given certain conditions on the eigenvalues [9] there would be

chaotic solutions near the heteroclinic cycle. This chaos is, however, very

subtle since most trajectories escape to one of the roll solutions.




Fig. 6. Possible heteroclinic cycle, an alternative to fig. 5.

4. CONCLUSION. The idea that turbulence may be described as a strange

attractor in tPe dynamical system of the fluid equations has been around for

two decades now [10]. and there is some experimental verification of this

notion [11], however this has never been rigorously demonstrated for any real

system. The transition to turbulence in a rotating fluid layer is well suited

to small amplitude investigation, although much work remains to be done. In

particular, the spatial dependence (i.e. patch structure) has been ignored.

Rotating convection has three properties which may be generally useful

in searching for systems where deterministic chaos is present in the center

manifold ODE's. First, there is a high degree of symmetry, which forces the

center manifold to be multidimensional. Second, when a portion of the

symmetry is broken the secondary bifurcations can be analyzed in the unfolding of the degenerate bifurcation. Finally, the symmetry does not

include hyperplanes of reflection which would severely limit the possible

dynamical behavior.

I would 1 ike to thank Edgar Knobloch for encouraging and supporting

this work, and Ken Rimey for performing or verifying many of the calculations using Vaxima.

BIBLIOGRAPHY

1. F.H. Busse and R.M. Clever, "Nonstationary Convection in a Rotating System," in Recent Developments in Theoretical and Experimental Fluid Mechanics, ed. U. Muller, K.G. Roesner, and B. Schmidt, Springer, Berlin, 1969, 376-385.

2. G. Kuppers and D. Lortz, "Transition from Laminar Convection to Thermal Turbulence in a Rotating Fluid Layer," J. Fluid Mech., 35 (1969), 609-620.



448 SWIFT

3. K.E. Heikes and F.H. Busse, "Weakly Nonlinear Turbulence in a Rotating Convection Layer,'' Ann. N.Y. Acad. Sciences, 357 (1980), 28-36.

4. F.H. Busse and K.E. Heikes, "Convection in a Rotating Layer: A Simple Case of Turbulence,'' Science, 208 (1980), 173-175.

5. M. Golubitsky, J.W. Swift, and E. Knobloch, "Symmetries and Pattern Selection in Rayleigh-Benard Convection, '' Physica D, to appear.

6. J. Marsden and M.C. McCracken, "The Hopf Bifurcation and its Applications," Appl. Math. Sci., 19, Springer-Verlag, New York, 1976.

7. E. Buzano and M. Golubitsky, "Bifurcation on the Hexagonal Lattice and the Planar Benard Problem," Phil. Trans. R. Soc. Lond. A, 308 (1983), 617-667.

8. R.M. May and W.J. Leonard, "Nonlinear Aspects of Competition between Three Species,'' SIAM J. Appl. Math., 29 (1975), 243-253.

9. A. Arneodo, P. Coullet, and C. Tresser, "Oscillators with Chaotic Behavior: An Illustration of a Theorem by Shil'nikov," J. Stat. Phys., 27 (1982), 171-182.

10. E.N. Lorenz, "Deterministic Non-Periodic Flows," J. Atmos. Sci., 20 (1963), 130-141.

11. H.L. Swinney, "Geometry and Dynamics in Experiments on Chaotic Systems,'' in this volume.

DEPARTMENT OF PHYSICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720

Note: A further reference should be included.

A. M. Soward, "Bifurcation and Stability of Finite Amplitude Convection in a Rotating Fluid Layer," preprint, University of Newcastle-upon-Tyne.

Dr. Soward also investigates the effect of vertical asymmetries and gets very similar results. Dr. Soward's preprint was in existance when my paper was

independently prepared. I thank him for allowing me to publish this anyway.

CDEFGHIJ-898 Licensed to North Carolina St Univ. Prepared on Thu Nov 7 20:09:43 EST 2013 for download from IP 152.14.136.96.

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