Understanding Complex Systems

Founding Editor

Prof. Dr. J.A. Scott KelsoCenter for Complex Systems & Brain SciencesFlorida Atlantic UniversityBoca Raton FL, USAE-mail: kelso@walt.ccs.fau.edu

Editorial and Programme Advisory Board

Dan BrahaNew England Complex Systems, Institute and University of Massachusetts, Dartmouth

Péter ÉrdiCenter for Complex Systems Studies, Kalamazoo College, USA and Hungarian Academy ofSciences, Budapest, Hungary

Karl FristonInstitute of Cognitive Neuroscience, University College London, London, UK

Hermann HakenCenter of Synergetics, University of Stuttgart, Stuttgart, Germany

Viktor JirsaCentre National de la Recherche Scientifique (CNRS), Université de la Méditerranée, Marseille,France

Janusz KacprzykSystem Research, Polish Academy of Sciences, Warsaw, Poland

Kunihiko KanekoResearch Center for Complex Systems Biology, The University of Tokyo, Tokyo, Japan

Scott KelsoCenter for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, USA

Markus KirkilionisMathematics Institute and Centre for Complex Systems, University of Warwick, Coventry, UK

Jürgen KurthsPotsdam Institute for Climate Impact Research (PIK), Potsdam, Germany

Andrzej NowakDepartment of Psychology, Warsaw University, Poland

Linda ReichlCenter for Complex Quantum Systems, University of Texas, Austin, USA

Peter SchusterTheoretical Chemistry and Structural Biology, University of Vienna, Vienna, Austria

Frank SchweitzerSystem Design, ETH Zürich, Zürich, Switzerland

Didier SornetteEntrepreneurial Risk, ETH Zürich, Zürich, Switzerland

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Understanding Complex Systems

Future scientific and technological developments in many fields will necessarily depend upon comingto grips with complex systems. Such systems are complex in both their composition - typically manydifferent kinds of components interacting simultaneously and nonlinearly with each other and their envi-ronments on multiple levels - and in the rich diversity of behavior of which they are capable.

The Springer Series in Understanding Complex Systems series (UCS) promotes new strategies andparadigms for understanding and realizing applications of complex systems research in a wide variety offields and endeavors. UCS is explicitly transdisciplinary. It has three main goals: First, to elaborate theconcepts, methods and tools of complex systems at all levels of description and in all scientific fields,especially newly emerging areas within the life, social, behavioral, economic, neuro and cognitive sci-ences (and derivatives thereof); second, to encourage novel applications of these ideas in various fieldsof engineering and computation such as robotics, nano-technology and informatics; third, to provide asingle forum within which commonalities and differences in the workings of complex systems may bediscerned, hence leading to deeper insight and understanding.

UCS will publish monographs, lecture notes and selected edited contributions aimed at communicat-ing new findings to a large multidisciplinary audience.

Springer ComplexitySpringer Complexity is an interdisciplinary program publishing the best research and academic-levelteaching on both fundamental and applied aspects of complex systems - cutting across all traditional dis-ciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and com-puter science.

Complex Systems are systems that comprise many interacting parts with the ability to generate a newquality of macroscopic collective behavior the manifestations of which are the spontaneous formation ofdistinctive temporal, spatial or functional structures. Models of such systems can be successfully mappedonto quite diverse “real-life” situations like the climate, the coherent emission of light from lasers, chem-ical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of theinternet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opin-ions in social systems, to name just some of the popular applications.

Although their scope and methodologies overlap somewhat, one can distinguish the following mainconcepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems,catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptivesystems, genetic algorithms and computational intelligence.

The two major book publication platforms of the Springer Complexity program are the monographseries “Understanding Complex Systems” focusing on the various applications of complexity, and the“Springer Series in Synergetics”, which is devoted to the quantitative theoretical and methodological foun-dations. In addition to the books in these two core series, the program also incorporates individual titlesranging from textbooks to major reference works.

Valery I. Klyatskin

Stochastic Equations:Theory and Applications inAcoustics, Hydrodynamics,Magnetohydrodynamics,and Radiophysics, Volume 2

Coherent Phenomena inStochastic Dynamic Systems

Translated from Russianby A. Vinogradov

ABC

Valery I. KlyatskinA.M. Obukhov Institute of

Atmospheric PhysicsRussian Academy of SciencesMoscowRussia

ISSN 1860-0832 ISSN 1860-0840 (electronic)ISBN 978-3-319-07589-1 ISBN 978-3-319-07590-7 (eBook)DOI 10.1007/978-3-319-07590-7Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014941082

c© Springer International Publishing Switzerland 201This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other physical way, and transmission or informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed. Exempted from this legal reservation are brief excerpts in connectionwith reviews or scholarly analysis or material supplied specifically for the purpose of being enteredand executed on a computer system, for exclusive use by the purchaser of the work. Duplication ofthis publication or parts thereof is permitted only under the provisions of the Copyright Law of thePublisher’s location, in its current version, and permission for use must always be obtained from Springer.Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violationsare liable to prosecution under the respective Copyright Law.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.While the advice and information in this book are believed to be true and accurate at the date of pub-lication, neither the authors nor the editors nor the publisher can accept any legal responsibility for anyerrors or omissions that may be made. The publisher makes no warranty, express or implied, with respectto the material contained herein.

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Springer is part of Springer Science+Business Media (www.springer.com)

5

To Sonya Klyatskina

Preface

This monograph is revised and more comprehensive translation of my Rus-sian monographs [154] and in 2 volumes [155, 156]. For reasons of usability,the material is divided into two practically independent volumes. In Volume2, general methods developed in Volume 1 are used to consider the the-ory of coherent phenomena occurring in stochastic dynamic systems with aprobability one, i.e., almost in all realizations of a random process. Amongthe examined phenomena are clustering of particles and passive scalar tracerfield (density field) in random velocity field, passive vector tracer field (mag-netic field) in a random velocity field, dynamic localization of plane wavesin layered random media, propagation of monochromatic waves in randommedia, and formation of caustic structures in different-nature wavefields prop-agating in multidimensional random media (under the assumption that wavepropagation is described in terms of the scalar parabolic equation).

Diffusion and clustering of both inertialess and low-inertia passive scalartracer (density field) in different media and under different conditions areconsidered here in significantly greater detail on the basis of monograph [154].In addition, I added consideration of diffusion and clustering of magnetic fieldenergy in random media.

Working at this edition, I tried to take into account remarks and wishes ofreaders about both style of the text and choice of specific problems. Differ-ent mistakes and misprints are corrected. The book is destined for scientistsdealing with stochastic dynamic systems in different areas, such as hydro-dynamics, magnetohydrodynamics, acoustics, radiophysics, theoretical andmathematical physics, and applied mathematics, and can be useful for seniorand postgraduate students.

Volume 2 consists of five Parts and one Appendix.Part I of this volume deals with the problem of stochastic structure forma-

tion in random hydrodynamic flows. In particular, starting from an analysisof the steady-state probability density, it considers coherent structures of vor-tex formation (vortex genesis) in stochastic quasi-geostrophic flows, which arerelated to rotation and random topography of the bottom. This Part includes

VIII Preface

also challenges of analyzing the possibility of formation of anomalously largewaves (rogue waves) on the sea surface in terms of stochastic topography.

Part II deals with diffusion and clustering of particles and passive scalartracer density (density field) in random hydrodynamic velocity field.

Part III deals with diffusion and clustering of passive vector field (mag-netic) in random magnetohydrodynamic velocity field.

Part IV considers the phenomenon of dynamic localization accompanyingplane wave propagation in layered random media, and

Part V considers statistical description of wavefields propagating in ran-dom multidimensional media including the problem of caustic structure for-mation. These problems are formulated in terms of both ordinary and partialdifferential equations, and every of them can be divided into the number ofseparate problems of independent physical interest.

Part VI – Appendices discusses in detail the derivation of equations ofthe imbedding method that offers a possibility of reformulating boundary-value wave problems (linear and nonlinear, stationary and nonstationary)into initial-value problems with respect to auxiliary variables.

It is worth noting that purely mathematical and physical papers devotedto considered issues run into thousands. It would be physically impossibleto give an exhaustive bibliography. Therefore, in this book I confine myselfto referencing the papers used or discussed in this book, recent reviews andpapers with extensive bibliography on the subject.

Moscow Valery I. Klyatskin

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV

Part I: Stochastic Structure Formations in RandomHydrodynamic Flows

1 Equilibrium Distributions for Hydrodynamic Flows . . . . . . 31.1 Two-Dimensional Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Rogue Waves as an Object of Statistical Topography . . . . 152.1 Statistical Topography of Random Field ξ(R, t) . . . . . . . . . . . 21

Part II: Density Field Diffusion and Clustering in RandomHydrodynamic Flows

3 Main Features of the Problem and DeterminingEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1 Low-Inertia Tracer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Inertialess Tracer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 Relationship between the Lagrangian and EulerianDescriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Statistical Description of Inertialess Tracer Diffusionand Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Approximation of the Delta-Correlated (in Time) Velocity

Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.1 Lagrangian Description (Particle Diffusion) . . . . . . . . . 424.2.2 Eulerian Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 Additional Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3.1 Plane-Parallel Mean Shear . . . . . . . . . . . . . . . . . . . . . . . . 60

X Contents

4.3.2 Effect of Molecular Diffusion . . . . . . . . . . . . . . . . . . . . . . 624.3.3 Consideration of Finite Temporal Correlation

Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.4 Diffusion Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4 Features of Tracer Diffusion in Fast Random Wave Fields . . . 704.4.1 Eulerian Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5 Integral One-Point Statistical Characteristics ofDensity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.1 Spatial Correlation Function of Density Field . . . . . . . . . . . . . 805.2 Spatial Correlation Tensor of Density Field Gradient and

Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2.1 Extension to the Case of Inhomogeneous Initial

Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 Tracer Diffusion and Clustering in RandomNondivergent Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.1 Diffusion and Clustering of the Buoyant Inertialess

Tracer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.1.1 Buoyant Tracer in Random Surface z (R, t) . . . . . . . . . 91

6.2 Diffusion and Clustering of Low-Inertia Tracer . . . . . . . . . . . . 936.2.1 A Feature of Low-Inertia Particle Diffusion (The

Lagrangian Description) . . . . . . . . . . . . . . . . . . . . . . . . . . 946.2.2 Low-Inertia Tracer Diffusion (The Eulerian

Description) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.2.3 Spatial Correlations of Field V (r, t) . . . . . . . . . . . . . . . . 996.2.4 Correlation Tensor of Spatial Derivatives of Field

V (r, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.2.5 Temporal Correlation Tensor of Field V (r, t) . . . . . . . . 1046.2.6 Conditions of Applicability of the Obtained

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.3 Diffusion and Clustering of Low-Inertia Tracer . . . . . . . . . . . . 107

6.3.1 Spatial Correlations of Field V (r, t) . . . . . . . . . . . . . . . . 1086.3.2 Temporal Correlation Tensor of Field V (r, t) . . . . . . . . 111

7 Diffusion and Clustering of Settling Tracer in RandomFlows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.1 State of Art and Main Equation of the Problem . . . . . . . . . . . 115

7.1.1 Particle Diffusion (Lagrangian Description) . . . . . . . . . 1167.1.2 Eulerian Description of the Tracer Density Field . . . . 118

7.2 Diffusion and Clustering of the Density Field . . . . . . . . . . . . . 1197.3 Low-Inertia Settling Tracer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.3.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.3.2 Diffusion Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 130

Contents XI

7.3.3 Space–Time Correlation Tensor of Field v(r, t) . . . . . . 1317.3.4 Space–Time Correlation Tensor of Field div v(r, t). . . . 133

Part III: Magnetic Field Diffusion and Clustering in RandomMagnetohydrodynamics Flows

8 Probabilistic Description of Magnetic Field in RandomVelocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398.2 Statistical Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

9 Probabilistic Description of Magnetic Energy inRandom Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1459.1 Delta-Correlated Random Velocity Field Approximation . . . . 1459.2 Stochastic Dynamo in Critical Situations . . . . . . . . . . . . . . . . 150

9.2.1 Features of Magnetic Field Diffusion in CriticalSituations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

9.2.2 The Main Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1549.2.3 Pseudoequilibrium Velocity Field . . . . . . . . . . . . . . . . . . 1639.2.4 Random Acoustic Velocity Field . . . . . . . . . . . . . . . . . . . 1679.2.5 Equilibrium Thermal Velocity Field . . . . . . . . . . . . . . . . 171

10 Integral One-Point Statistical Characteristics ofMagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17310.1 Spatial Correlation Function of Magnetic Field . . . . . . . . . . . . 17310.2 On the Magnetic Field Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . 17610.3 On the Magnetic Field Dissipation . . . . . . . . . . . . . . . . . . . . . . . 179

Part IV: Wave Localization in Randomly Layered Media

11 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18511.1 Wave Incidence on an Inhomogeneous Layer . . . . . . . . . . . . . . 18511.2 Source Inside an Inhomogeneous Layer . . . . . . . . . . . . . . . . . . . 188

12 Statistics of Scattered Field at Layer Boundaries . . . . . . . . 19112.1 Reflection and Transmission Coefficients . . . . . . . . . . . . . . . . . . 191

12.1.1 Nondissipative Medium (Normal Wave Incidence) . . . 19312.1.2 Nondissipative Medium (Oblique Wave Incidence) . . . 19612.1.3 Dissipative Medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

12.2 Source Inside the Medium Layer . . . . . . . . . . . . . . . . . . . . . . . . . 20212.3 Statistical Localization of Energy . . . . . . . . . . . . . . . . . . . . . . . . 20312.4 Diffusion Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

12.4.1 Unmatched Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20512.4.2 Matched Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

XII Contents

13 Statistical Description of a Wavefield in RandomMedium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21313.1 Normal Wave Incidence on the Layer of Random Media . . . . 213

13.1.1 Nondissipative Medium (Stochastic WaveParametric Resonance and Dynamic WaveLocalization) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

13.1.2 Dissipative Medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22513.2 Plane Wave Source Located in Random Medium . . . . . . . . . . 229

13.2.1 Half-Space of Random Medium . . . . . . . . . . . . . . . . . . . 23313.2.2 Asymptotic Case of Small Dissipation . . . . . . . . . . . . . . 235

13.3 Peculiarity of Statistical Description of Acoustic Field . . . . . . 23813.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

13.4.1 Wave Incident on the Medium Layer . . . . . . . . . . . . . . . 24513.4.2 Plane Wave Source in the Medium Layer . . . . . . . . . . . 24613.4.3 Nonlinear Problem on Wave Self-action in Random

Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

14 Eigenvalue and Eigenfunction Statistics . . . . . . . . . . . . . . . . . . 25314.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25314.2 Statistical Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

15 Multidimensional Wave Problems in Layered RandomMedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26115.1 Nonstationary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

15.1.1 Formulation of Boundary-Value Wave Problems . . . . . 26115.1.2 Statistical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

15.2 Point Source in Randomly Layered Medium. . . . . . . . . . . . . . . 26815.2.1 Factorization of the Wave Equation in Layered

Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26815.2.2 Parabolic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27015.2.3 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

16 Two-Layer Model of the Medium . . . . . . . . . . . . . . . . . . . . . . . . 27716.1 Formulation of Boundary-Value Problems . . . . . . . . . . . . . . . . 27716.2 Statistical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Part V: Wave Propagation in Random Media

17 Method of Stochastic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 28917.1 Input Stochastic Equations and Their Implications . . . . . . . . 28917.2 Delta-Correlated Approximation for Medium Parameters . . . 293

17.2.1 Estimation of Depolarization Phenomena inRandom Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

17.3 The Delta-Correlated Approximation and the DiffusionApproximation for Wavefield . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30917.3.1 Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

Contents XIII

17.3.2 Diffusion Approximation for the Wavefield . . . . . . . . . . 31217.4 Wavefield Amplitude-Phase Fluctuations . . . . . . . . . . . . . . . . . 317

17.4.1 Random Phase Screen (Δx� x) . . . . . . . . . . . . . . . . . . 32117.4.2 Continuous Medium (Δx = x) . . . . . . . . . . . . . . . . . . . . 321

18 Geometrical Optics Approximation in RandomlyInhomogeneous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32518.1 Ray Diffusion in Random Media (The Lagrangian

Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32518.2 Formation of Caustics in Randomly Inhomogeneous

Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32918.3 Wavefield Amplitude-Phase Fluctuations (The Eulerian

Description) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

19 Method of Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34319.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34319.2 Statistical Description of Wavefield . . . . . . . . . . . . . . . . . . . . . . 34719.3 Asymptotic Analysis of Plane Wave Intensity

Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35119.3.1 Random Phase Screen . . . . . . . . . . . . . . . . . . . . . . . . . . . 35319.3.2 Continuous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

20 Caustic Structure of Wavefield in Random Media . . . . . . . . 36320.1 Elements of Statistical Topography of Random Intensity

Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36420.2 Weak Intensity Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36520.3 Strong Intensity Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Appendices: Appendices Imbedding Method in Boundary-ValueWave Problems

General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

A Stationary Boundary-Value Wave Problems . . . . . . . . . . . . . 377A.1 One-Dimensional Stationary Boundary-Value Wave

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377A.1.1 Helmholtz Equation With Unmatched Boundary . . . . 377A.1.2 Helmholtz Equation with Matched Boundary . . . . . . . 391A.1.3 Acoustic Waves in Variable-Density Media and

Electromagnetic Waves in Layered InhomogeneousMedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

A.1.4 Acoustic-Gravity Waves in Layered Ocean . . . . . . . . . . 403A.2 Waves in Periodically Inhomogeneous Media . . . . . . . . . . . . . . 412

A.2.1 Wave Incident on the Layer of PeriodicallyInhomogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . 413

A.2.2 Bragg Resonance in Inhomogeneous Media . . . . . . . . . 417

XIV Contents

A.3 Boundary-Value Stationary Nonlinear Wave Problem onSelf-Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419A.3.1 General Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419A.3.2 Wave Incidence on a Half-Space of Nonlinear

Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426A.3.3 Examples of Wavefield Calculations in Nonlinear

Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430A.4 Stationary Multidimensional Boundary-Value Problem . . . . . 438

A.4.1 Stationary Nonlinear MultidimensionalBoundary-Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . 447

B One-Dimensional Nonstationary Boundary-Value WaveProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455B.1 Nonsteady Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

B.1.1 Problem on a Wave Incident on Medium Layer . . . . . . 457B.2 Steady Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460

B.2.1 Inverse Problem Solution . . . . . . . . . . . . . . . . . . . . . . . . 465B.3 One-Dimensional Nonlinear Wave Problem . . . . . . . . . . . . . . . 468

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

Introduction XV

. . .Chaos is the place which servesto contain all things; for if thishad not subsisted neither earth norwater nor the rest of the elements,nor the Universe as a whole, couldhave been constructed. . . .

Sextus Empiricus, Against thePhysics, against the Ethicists, R.G.Bury, p. 217, Harvard UniversityPress, 1997.

Introduction

In recent time, the interest of both theoreticians and experimenters has beenattracted to the relation between the behavior of average statistical characte-ristics of a problem solution and the behavior of the solution in certain hap-penings (realizations). This is especially important for geophysical problemsrelated to the atmosphere and ocean where, generally speaking, a respectiveaveraging ensemble is absent and experimenters, as a rule, have to do withindividual observations.

Seeking solutions to dynamic problems for these specific realizations ofmedium parameters is almost hopeless due to extreme mathematical com-plexity of these problems. At the same time, researchers are interested inmain characteristics of these phenomena without much need to know spe-cific details. Therefore, the idea to use a well developed approach to randomprocesses and fields based on ensemble averages rather than separate obser-vations proved to be very fruitful. By way of example, almost all physicalproblems of atmosphere and ocean to some extent are treated by statisticalanalysis.

Randomness in medium parameters gives rise to a stochastic behavior ofphysical fields. Individual samples of scalar two-dimensional fields ρ (R, t),R = (x, y), say, recall a rough mountainous terrain with randomly scatteredpeaks, troughs, ridges and saddles. Common methods of statistical averaging(computing mean-type averages — 〈ρ (R, t)〉, space-time correlation function—

⟨

ρ (R, t) ρ(

R′, t′)⟩

etc., where 〈· · · 〉 implies averaging over an ensembleof random parameter samples) smooth the qualitative features of specificsamples. Frequently, these statistical characteristics have nothing in commonwith the behavior of specific samples, and at first glance may even seemto be at variance with them. For example, the statistical averaging over all

XVI Introduction

observations makes the field of average concentration of a passive tracer in arandom velocity field ever more smooth, whereas each its realization sampletends to be more irregular in space due to mixture of areas with substantiallydifferent concentrations.

Thus, these types of statistical average usually characterize "global" space-time dimensions of the area with stochastic processes but tell no details aboutthe process behavior inside the area. For this case, details heavily depend onthe velocity field pattern, specifically, on whether it is divergent or solenoidal.Thus, the first case will show with the probability one that clusters will beformed, i.e. compact areas of enhanced concentration of tracer surroundedby vast areas of low-concentration tracer. In the circumstances, all statisticalmoments of the distance between the particles will grow with time exponen-tially; that is, on average, a statistical recession of particles will take place.

In a similar way, in case of waves propagating in random media, an ex-ponential spread of the rays will take place on average; but simultaneously,with the probability one, caustics will form at finite distances. One more ex-ample to illustrate this point is the dynamic localization of plane waves inlayered randomly inhomogeneous media. In this phenomenon, the wavefieldintensity exponentially decays inward the medium with the probability onewhen the wave is incident on the half-space of such a medium, while all sta-tistical moments increase exponentially with distance from the boundary ofthe medium.

These physical processes and phenomena occurring with the probabilityone will be referred to as coherent processes and phenomena [166]. This typeof statistical coherence may be viewed as some organization of a complexdynamic system, and retrieval of its statistically stable characteristics is sim-ilar to the concept of coherence as self-organization of multicomponent sys-tems that evolve from the random interactions of their elements [265]. In thegeneral case, it is rather difficult to say whether or not the phenomenon occurswith the probability one. However, for a number of applications amenable totreatment with the simple models of fluctuating parameters, this may be han-dled by analytical means. In other cases, one may verify this by performingnumerical modeling experiments or analyzing experimental findings.

For these reasons, I interpret the main problem of the statistical analysisof stochastic dynamic systems as revealing the fundamental features of suchsystems, which appear with probability one, i.e., almost in all realizations ofthe dynamic systems under consideration, on the basis of the correspondingstatistical analysis.

I note a curious fact that nontrivial situations can be realized even inGaussian random fields! Such a situation is realized, for example, in the two-dimensional problems of geophysical hydrodynamics in rotating liquids withrandom bottom topography (see Part 1).

The complete statistics (say, the whole body of all n-point space-timemoment functions), would undoubtedly contain all the information about theinvestigated dynamic system. In practice, however, one may succeed only in

Introduction XVII

studying the simplest statistical characteristics associated mainly with one-time and one-point probability distributions. It would be reasonable to askhow with these statistics on hand one would look into the quantitative andqualitative behavior of some system happenings?

This question is answered by methods of statistical topography (see, forexample [126]). These methods were highlighted by Ziman [336], who seemsto had coined this term. Statistical topography yields a different philosophy ofstatistical analysis of dynamic stochastic systems, which may prove useful forexperimenters planning a statistical processing of experimental data. Theseissues are treated in depths in this book.