Localized instabilities in fluids

Susan FriedlanderDepartment of Mathematics, Statistics, and Computer Science

The University of Illinois at Chicago851 S. Morgan Street

Chicago, IL 60607-7045

Alexander Lipton-Lifschitz∗

Global Modelling and Analytics GroupCredit Suisse First BostonEleven Madison Avenue

New York, NY 10010-3629

June 1, 2003

Abstract

We study the effects of localized instabilities on the behavior of invis-cid fluids. We place geometric optics techniques independently developedby Friedlander - Vishik and Lifschitz - Hameiri in the framework of thegeneral stability / instability problem of fluid dynamics. We show that,broadly put, all laminar flows are unstable with respect to localized per-turbations.

All animals are equal but some animals are more equal than others.George Orwell, “Animal Farm”

1 Introduction

The foundations of hydrodynamic stability theory were laid down by Helmholtz,Kelvin, Lyapunov, Poincare, Rayleigh, Reynolds, and Stokes in the nineteenthcentury. For more than a hundred years this subject attracted the attention of agreat number of researchers. A vast body of literature on this subject exists andencompasses the work of mathematicians, physicists, engineers, astrophysicists,geophysicists, meteorologists, etc. Much background material on the classicalapproach to fluid stability can be found in the substantive general texts byLin [128], Chandrasekhar [22], Joseph [85], Drazin and Reid [38], Swinney and

∗CSFB does not necessarily agree with or endorse the content of this article.

1

Gollub [157], as well as several more specialized monographs, reviews, and col-lections of papers, such as Yudovich [179], Holm et al. [83], Arnold and Khesin[7], Godreche and Manneville [72], Dikii [33], etc.

In addition to hydrodynamic stability, the twentieth century saw the birthof the sister discipline of magnetohydrodynamic stability which was developedin order to address important practical questions occurring, in particular, inthermonuclear fusion, astrophysics and dynamo theory, see, for example, Frie-man and Rottenberg [68] and Lifschitz [115] The beautiful synergy and cross-fertilization between these topics was instrumental in accelerating advances inboth of them.

The key question of hydrodynamic stability theory can be formulated asfollows: what happens to a given fluid flow under the influence of small distur-bances. If the flow is robust under the influence of all small disturbances it iscalled stable and can be expected to occur in nature. If there are perturbationswhich start to grow we call the flow unstable and expect it to break up or other-wise change its character. These possibilities were experimentally demonstratedby Reynolds [143].

The topic of stability is important both mathematically and for practicalconsiderations. Generally, it is much easier to establish the instability of agiven flow rather than to prove its stability. In fact, in this article we will arguethat, in a certain sense, all nontrivial inviscid flows are unstable.

There are two well-established techniques for the analysis of hydrodynamicstability / instability, namely, spectral methods (normal modes) (see, for, ex-ample, Chandrasekhar [22], Drazin and Reid [38]) and energy methods (see,for example, Arnold [4], Holm et al. [83], Arnold and Khesin [7], Vladimirov[171]). Recently, a modification of the spectral method called the pseudospec-tral method was proposed by Trefethen [163], Trefethen et al. [164], Waleffe[174] and others. These conventional techniques proved to be very useful forstudying the stability / instability of some steady flows with relatively simplestructure but their applicability to more complicated flows has proved to belimited. In this paper we emphasize a third technique, namely, the geometricaloptics method which is capable of probing the instability of general classes ofthree-dimensional inviscid flows. (By its very nature this method cannot provestability.) In contrast to spectral and energy methods, the geometrical opticsmethod is specifically designed for studying highly localized short-wave pertur-bations of an arbitrary background flows. These perturbations are localizedwave envelopes moving along the trajectories of fluid elements. The evolutionof a particular envelope is governed by a characteristic system of ordinary equa-tions along the relevant trajectory. In the language of geometrical optics thecharacteristic equations consist of the eikonal equation for the wave vector andthe transport equation for the velocity amplitude. Needless to say, the sys-tem of ordinary differential equations is more tractable than the full system ofpartial differential equations governing the dynamics of general perturbations.Broadly speaking, the flow is unstable if the magnitude of the perturbed veloc-ity amplitude grows in time without bound along at least one trajectory. As wewill describe later, this observation produces an effective tool for detecting fluid

2

instabilities.Four review articles by the present authors covering some of the material

discussed below are Lifschitz [120], Friedlander [57], Friedlander and Yudovich[67], and Friedlander and Shnirelman [60].

3

2 Formulation

2.1 Background

There are two complementary ways for describing fluid motion. We can eitheranalyze the distribution of the velocity, the pressure, and the density at everypoint in space; or we can investigate the history of every particle, Lamb [92].Accordingly, there are two complementary descriptions, namely, the Euleriandescription and the Lagrangian description, and two sets of governing equations:the Euler equations and the Lagrange equations. Euler and Lagrange publishedtheir investigations in mid-eighteenth century, see Euler [46], [47], Lagrange[91]. It is interesting to note that Euler actually developed both space-basedand particle-based approaches to describing fluid motions.

2.2 The Eulerian Equations

Inviscid incompressible fluid motions whose density is normalized to unity aregoverned by the Euler equations describing the distribution of the velocity Vand the pressure P in space:

DV

Dt+∇P = 0, (1)

∇ ·V =0, (2)

V (x,0) = V0 (x) , (3)

where V (x,t) is the velocity, and P (x,t) is the pressure of a general flow in R3,and

D

Dt=

∂

∂t+V · ∇, (4)

is the advective derivative along the velocity field V. If the flow is consideredin a finite (or semi-finite) domain D, equations 1, 2 are augmented by theimpenetrability condition on the boundary ∂D,

n ·V|∂D = 0, (5)

where n is the unit vector normal to ∂D. Other physically plausible and math-ematically appropriate boundary conditions are periodicity conditions or thefree space problem with suitable decay at infinity. The initial velocity field V0naturally satisfies the same conditions. The vorticity of the flow is given byΩ = ∇×V. The vorticity equation is

DΩ

Dt− Ω · ∇V =0. (6)

4

The basic flow is denoted by U (x,t), P0 (x,t), Ω (x,t) = ∇×U (x,t), its per-turbation is denoted by v (x,t), p (x,t), ω (x,t). The evolution of a perturbationis governed by

∂v

∂t+U · ∇v + v · ∇U+ v · ∇v +∇p = 0, (7)

∇ · v =0, (8)

v (x,0) = v0, ∇ · v0=0. (9)

If, for example, the problem is considered in a finite domain, the impenetrabilitycondition is imposed,

n · v|∂D = 0. (10)

The linearized equations for the perturbation are

∂v

∂t+U · ∇v + v · ∇U+∇p = 0, (11)

∇ · v =0, (12)

v (x,0) = v0, ∇ · v0=0, (13)

plus the appropriate boundary condition. Taking the curl of equation (11) givesthe equation for the evolution of the perturbation vorticity ω ≡ ∇× v:

∂ω

∂t+U · ∇ω − ω · ∇U+ v · ∇Ω− Ω · ∇v = 0, (14)

or, equivalently,

∂ω

∂t− U, ω+ v,Ω = 0, (15)

where , denotes the Poisson bracket of two vector fields, i.e.

A,B = B · ∇A−A · ∇B. (16)

We can introduce the linearized operator:

Lv = −Π U · ∇v + v · ∇U , (17)

where Π is the projection operator on the space of divergence-free vector fieldssatisfying the appropriate boundary conditions. In this notation the governinglinear equation assumes the form

∂v

∂t−Lv = 0, (18)

5

v (0) = v0. (19)

When the basic flowU is stationary, we can introduce the corresponding spectralproblem

λv−Lv = 0. (20)

A considerable portion of this paper will be concerned with the spectrum of theoperator L.

2.3 The Lagrangian Equations

The Eulerian equations introduced in the previous subsection are the most pop-ular but not the only set of equations governing the evolution of fluid motions.Lagrange discovered a complementary set of equations which emphasizes thelocal aspects of fluid motion. Let a = (a1, a2, a3) be the labels of a fluid parti-cle, and x = (x1, x2, x3) its Cartesian coordinates. Lagrange considered x as afunction of a and t and wrote the equations for x (a, t) in the form

x1,ttx1,a1+ x2,ttx2,a1

+ x3,ttx3,a1+ Pa1

= 0,x1,ttx1,a2

+ x2,ttx2,a2+ x3,ttx3,a2

+ Pa2= 0,

x1,ttx1,a3+ x2,ttx2,a3

+ x3,ttx3,a3+ Pa3

= 0,(21)

∂ (x1, x2, x3)

∂ (a1, a2, a3)=∂ (x10, x20, x30)

∂ (a1, a2, a3), (22)

where x0 = (x10, x20, x30) are the initial coordinates of a fluid particle at timet = 0, and we use the notation ∂ (x) /∂ (a) for the Jacobian of the mappinga→ x. The first set of equations follows from the second law of Newton forfluid particles, while the second equation is a consequence of the incompress-ibility. When the initial positions of the fluid particle x10, x20, x30 are used asits Lagrangian labels, equation 22 simplifies to

∂ (x1, x2, x3)

∂ (x10, x20, x30)= 1. (23)

The Eulerian velocityV (x, t) can be recovered by inverting the mapping a→ x,

V (x, t) =∂x (a (x, t) , t)

∂t. (24)

By eliminating the pressure from equations (21) and integrating them overtime Cauchy presented them in the form

x1,ta2x1,a3

− x1,ta3x1,a2

+ x2,ta2x2,a3

− x2,ta3x2,a2

+ x3,ta2x3,a3

− x3,ta3x3,a2

= C1,x1,ta3

x1,a1− x1,ta1

x1,a3+ x2,ta3

x2,a1− x2,ta1

x2,a3+ x3,ta3

x3,a1− x3,ta1

x3,a3= C2,

x1,ta1x1,a2

− x1,ta2x1,a1

+ x2,ta1x2,a2

− x2,ta2x2,a1

+ x3,ta1x3,a2

− x3,ta2x3,a1

= C3,

(25)

6

where the vector C = (C1, C2, C3) is an integral of motion, Ci = Ci (a1, a2, a3)which is closely related to the vorticity of the flow.

Recently, Abrashkin et al. [3] discovered an elegant way of representing theLagrange-Cauchy equations. They introduced two matrices

R =

x1,a1x1,a2

x1,a3

x2,a1x2,a2

x2,a3

x3,a1x3,a2

x3,a3

, C =

0 C3 −C2−C3 0 C1C2 −C1 0

, (26)

and represented the equations of motion in the matrix form

RTt R−RTRt = C, (27)

detR = detR0, (28)

where the subscripts and superscripts T denote the differentiation and matrixtransposition, respectively. These equations have to be augmented with theintegrability conditions which express the fact that R is a Jacobian matrix,

∂Rij∂ak

=∂Rik∂aj

. (29)

The vorticity of the flow has the form

Ω =1

detR0RC. (30)

It is clear that some aspects of the fluid motion are easier to understand inthe Eulerian framework while others are easier to describe in the Lagrangianframework. In particular, as we will see below, some interesting classes of exactsolutions of the equations of motion have a very natural description in terms ofR and C.

2.4 Stability and Instability

In this paper we adopt the general definition of stability developed by Lyapunovin the late nineteen century and used by Yudovich [179], Holm et al. [83], Holm[82], among many others. We specialize it to problems at hand as need occurs.

To start with, we define the spectral stability of the basic flow (which onlymakes sense for steady flows U, U = U (x)) as follows. We say that a steadybasic flow is spectrally stable provided that the discrete spectrum of the operatorL defined in an appropriate function space X has no strictly positive real part.

We also define the linearized stability of the basic flow. We choose a functionspaceX of vector fields with a norm ||.||X where problem (18), (19) is well-posed.The basic flow U is linearly stable provided that for every ε > 0 there is a δ > 0such that the inequality ||v0||X < δ implies the inequality ||v (t) ||X < ε. To putit differently, if there exists an initial condition v0 ∈ X such that ||v (t) ||X isunbounded on the whole t-axis, we call the basic flow linearly unstable, otherwise

7

we call it stable. Under such a definition, the basic flow U is not restricted tobe steady, but the issue of stability / instability of time dependent flows is moredelicate since the question arises as to what is a change in the basic flow andwhat is growth in the perturbation. The matter is clear if the basic flow issteady, periodic or quasi-periodic.

Finally, we define the nonlinear stability of the basic flow. As before, wechoose a function space X with a norm ||.||X such that the problem (7), (9) iswell-posed. We call the basic flowU nonlinearly stable if for every ε > 0 there isa δ > 0 such that the inequality ||v0||X < δ implies the inequality ||v (t) ||X < ε,otherwise we call it stable. We give a more precise definition in subsection 7.2below.

The following links between different stability definitions are interesting toobserve. Linearized stability implies spectral stability but not vice versa. Spec-tral stability is necessary but not sufficient for nonlinear stability. Linearizedstability is neither necessary nor sufficient for nonlinear stability.

We emphasize that the stability of a given flow is crucially dependent onthe choice of the function space X , and, even more importantly, the norm onthis space. A detailed discussion can be found in the book by Yudovich [179].Several representative examples are considered below.

The thrust of the present paper is the subject of instability which is theconverse of the above definitions of stability.

We now give a brief qualitative description of two classical instabilities,namely, Kelvin-Helmholtz and Rayleigh-Taylor instabilities which involve rathersimple basic flow patterns, i.e., particles moving in straight lines or circles. Theseinstabilities are caused by two complementary physical mechanisms which arethe ultimate causes of the vast majority of hydrodynamic instabilities.

2.5 Kelvin-Helmholtz instability

Kelvin-Helmholtz instabilities were qualitatively described by Helmholtz [79]and quantitatively analyzed by Kelvin [87]. Instabilities of this kind frequentlyoccur in nature, two such manifestations are the so-called “wind-over-water”and “clear air turbulence” instabilities.1 Consider two parallel streams of in-viscid incompressible fluids superposed one above the other and assume thatthe upper and lower streams have positive and negative velocities, respectively.Since at the interface there is a discontinuity in the velocity, the vorticity is anonzero delta-function-like distribution which can be modeled as a vortex sheet.Consider a small sinusoidal perturbation of this sheet. For the two-dimensionalflow in question the vorticity is conserved under the motion of the fluid parti-cles. Thus the vorticity has to induce a velocity in the positive (or negative)direction in parts of the sheet displaced upwards (or downwards). At the undis-turbed points of the sine wave, the vorticity induces a rotational velocity whichamplifies the wave and causes the instability to grow. As a result, the vor-ticity sheet develops vortex rolls and eventually breaks in a turbulent fashion.

1On occasion, these instabilities can have tragic consequences for travellers by boat orairplane.

8

Kelvin-Helmholtz instability is global in nature and flow specific. Since the ba-sic mechanism is strongly affected by specific features of the underlying flowsuch as its shear, as well physical forces such as gravity, viscosity, etc., it is verydifficult, if not impossible, to decribe Kelvin-Helmholtz instabilities of genericflows.

2.6 Rayleigh-Taylor instability

Rayleigh-Taylor instabilities were first theoretically described by Rayleigh [142]whose work was subsequently complemented by theoretical and experimentalanalysis by Taylor [160]. In a typical Taylor-Couette experimental apparatusa fluid moves between two concentric cylinders rotating with different angularvelocities. In steady motion the centrifugal force at any radius is balanced bythe pressure gradient. Consider a small radial displacement of a fluid ring.Conservation of angular momentum causes a change in the angular velocity,which now may or may not be sufficient to offset the pressure force. If thebalance is maintained, the ring moves back to its original position thus initiatingan oscillatory pattern. If the balance is violated, the ring moves away from itsoriginal position, the initial disturbance grows and the instability develops. Thusa necessary and sufficient condition for instability with respect to axisymmetricdisturbances can be established in terms of the radial derivative of the angularvelocity Ω (the so-called Rayleigh discriminant),

d

dr

(r4Ω2

)> 0. (31)

Experiments in which the differential of the angular velocity of the corotatingcylinders is gradually increased show that above a certain threshold a pattern ofsmall counterrotating Taylor vortices is set up. This pattern subsequently bifur-cates into azimuthal travelling waves, twisting regimes, quasiperiodic regimes,and so on, until the flow becomes turbulent. Bifurcations from primary to sec-ondary instabilities and beyond are studied in by Chossat and Iooss [25]. TheRayleigh-Taylor instability is local in nature and very robust, it is weakly af-fected by the specific features of the underlying flow. As we will see belowRayleigh-Taylor instability can be observed for generic three-dimensional flowsin the form of localized geometrical-optics type instabilities.

3 Equilibria and Other Exact Solutions

3.1 Background

The problem of finding equilibria is notoriously difficult, accordingly, very fewnontrivial equilibria in more than one dimension are known. However, it is easyto find an equilibrium depending on one spatial variable, for example, plane-parallel shear flows and differentially rotating cylinders. Equilibria dependingon two spatial variables are more difficult to find, still, there are several classes

9

of examples. Fully three-dimensional equilibria are exceedingly difficult to de-scribe. The same observation is true for time-dependent exact solutions of theequations of motion.

The governing equations for a steady state solution U (x) of the Euler equa-tions have the form

U · ∇U+∇P = 0, ∇ ·U =0, (32)

or, alternatively,

Ω×U+∇H = 0, ∇ ·U = 0, (33)

where the scalar fieldH = U2/2+P is the so-called Bernoulli function. Equation(59) shows that both streamlines and vorticity lines lie on the level surfaces ofH . Very little is known about general solutions of equation (32). However, aninteresting observation was made by Arnold [5] who proved that for equation(59) all compact noncritical level surfaces of H (i.e., surfaces which do notcontain zeroes of ∇H) are diffeomorphic to a 2D torus.

3.2 One-dimensional Equilibria

The simplest solution of the above equations is plane parallel flow of the form

U (x) = (U1 (x2) , 0, 0) , (34)

where U1 is an arbitrary function of its argument. A simple generalization ofthis flow known as plane parallel shear flow is

U (x) = (U1 (x2) , 0, U3 (x2)) . (35)

The axisymmetric analogue of the plane parallel shear flow is known asrotational shear flow. In cylindrical coordinates r, θ, z this flow is given by

U (r, θ, z) = (0, U (r) , 0) . (36)

As before, this flow has a two-component generalization

U (r, θ, z) = (0, V (r) ,W (r)) . (37)

3.3 Two- and Three-dimesional Equilibria and Exact So-

lutions

We now turn to strictly 2D flows and assume that the velocity field has the form

U (x1, x2) = (U1 (x1, x2) , U2 (x1, x2)) . (38)

It is clear that the vorticity of the flow which is perpendicular to the 2D plane hasa scalar magnitude Ω (x1, x2). By virtue of incompressibility, we can introducethe so-called stream function Ψ (x1, x2), such that

U1 = −∂Ψ

∂x2, U2 =

∂Ψ

∂x1, Ω (x1, x2) = ∆Ψ(x1, x2) . (39)

10

The name obviously comes from the fact that stream lines of the flow are thelevel curves of the function Ψ. Simple manipulations show that Ψ satisfies thePDE

(Ψx2

∂

∂x1−Ψx1

∂

∂x2

)∆Ψ = 0. (40)

A special class of solutions is such that the Bernoulli function depends only onΨ, H = H (Ψ), and Ψ satisfies the PDE of the form

∆Ψ = H ′ (Ψ) , (41)

and appropriate boundary conditions. In general, this PDE is nonlinear (andhas to be solved numerically), however, in some special cases, its analyticalsolution can be found. The simplest case occurs when H ′ is a linear function ofΨ, H ′ = −λΨ, so that equation (41) becomes the eigenvalue problem

∆Ψ = −λΨ, (42)

plus appropriate boundary conditions. For instance, if Ψ is bounded at infinity,the corresponding solution describing a cellular flow has the form

Ψ (x1, x2) = sin (k1x1) sin (k2x2) , λ = k21 + k22 . (43)

A less symmetric variant

Ψ = cos(x1 +mx2) + a cos(x1 −mx2) (44)

is illustrated in Figure 1.

Place Figure1 nearhere. Symmetric and asymmetric “cats-eye” flows.

This flow has a “cats-eye” type pattern with a periodic cellular structure thatcontains hyperbolic points and elliptic points as well as oscillatory regimes.

A somewhat more difficult but still tractable case occurs whenH ′ is piecewiseconstant, so that the equilibrium equation can be written as

∆Ψ = −λχD(t), (45)

where χD(t) is the characteristic function of a moving bounded domain D (t),which is equal to one when (x1, x2) ∈ D (t), and zero otherwise. If D (t) is anellipse rotating with constant angular velocity, then the corresponding solutionis given by Kirchhoff and Lamb [92].

An interesting generalization of this solution is given by Kida [88], see alsoNeu [134], Bayly, Holm and Lifschitz [12], and others. Kida’s solution describesan elliptic vorticity patch superimposed on a field consisting of pure strain andpure rotation. The background field has the form

U1 =1

2(sx1 − ωx2) , U2 =

1

2(ωx1 − sx2) . (46)

11

The patch remains elliptic for all time but its aspect ratio and orientation changein time. The area of the patch and its vorticity remain constant, and can benormalized to π and unity respectively. The geometric characteristics of thepatch, namely, its aspect ratio (ratio of the major to the minor axis) η (t) andthe angle θ (t) between the major axis and the x1 axis, are governed by a systemof ODEs:

dη

dt= sη cos (2θ) ,

dθ

dt=

η

(η + 1)2 −

1

2sη2 − 1

η2 + 1sin (2θ) +

1

2ω.

η (0) = η0, θ (0) = θ0. (47)

Inside the patch the combined velocity field is linear,(U1U2

)=

1

2

(s− δ sin (2θ) − (1 + ω) + δ cos (2θ)

(1 + ω) + δ cos (2θ) −s+ δ sin (2θ)

)(x1x2

). (48)

Kirchhoff’s solution can be obtained from Kida’s solution by putting s and ωto zero, η = η0, θ = η0t/ (η0 + 1)

2+ θ0. The full dynamics of Kida’s vortices

are fairly complex; the actual details can be found in the references above.A natural generalization of purely 2D flows are cylindrically symmetric flows

of the form

U (r, θ, z) = (U (r, z) , 0,W (r, z)) . (49)

As before, we can introduce the stream function Ψ (r, z) such that

U = −1

r

∂Ψ

∂z, W =

1

r

∂Ψ

∂r, (50)

Ω =

(0,

(1

r

∂

∂r

(1

r

∂Ψ

∂r

)+

1

r2∂2Ψ

∂z2

), 0

). (51)

The corresponding equilibrium condition is

1

r

∂

∂r

(1

r

∂Ψ

∂r

)+

1

r2∂2Ψ

∂z2= H ′ (Ψ) . (52)

The best known solution of this equation is called Hill’s spherical vortex, Hill[81], Lamb [92]. For such vortex

Ψ (r, z) =

− 34W0

(1−

(r2 + z2

)/a2)r2,

(r2 + z2 ≤ a2

),

12W0

(1− a3/

(r2 + z2

)3/2)r2,

(r2 + z2 ≥ a2

).

(53)

Here a is the spherical radius of the vortex core and W0 is the uniform velocityat infinity. The corresponding H is piecewise linear and can be found if needoccurs. The stream function Ψ (r, z) is shown in Figure 2.

12

Place Figure 2 near here. Hill’s vortex.

It is natural to generalize the above flows by introducing the velocity com-ponent V (r, z) in the θ direction. It is easy to show that

V (r, z) =1

rf (Ψ) , (54)

where f is an arbitrary function of Ψ. The corresponding equilibrium equationhas the form

1

r

∂

∂r

(1

r

∂Ψ

∂r

)+

1

r2∂2Ψ

∂z2= H ′ (Ψ)− 1

r2f (Ψ) f ′ (Ψ) . (55)

In the fluid dynamics context this equation was derived by Bragg and Hawthorne[17]. Independently, it was introduced to describe plasma equilibria by Gradand Rubin [74], and Shafranov [152]. Qualitative properties of this equationhave received much attention in the MHD literature, see, for instance, a detailedaccount in Lifschitz [115]. Its quantitative analysis is very complex and dependson a particular judicious choice of profile functions H , f , see, for example,Lifschitz, Suters and Beale [127] who took H and f as powers of Ψ in order toexploit the intrinsic symmetry properties of equation (55) and describe a richfamily of fluid equilibria that are called vortex rings with swirl. A typical vortexring belonging to this family is illustrated in Figure 3.

Place Figure 3 near here. A typical vortex ring with swirl.

Among other things, this Figure shows that, in agreement with Arnold’s ob-servation, all compact noncritical level surfaces of H are toroidal. Generically,there are critical surfaces outside of the vortex core, which separate surfacesdiffeomorphic to torii, and surfaces diffeomorphic to cylinders. For general dis-cussion of vortex rings (without and with swirl) see Moffatt [133], Turkington[165], Sharif and Leonard [153] and Saffman [149].

Some authors prefer to write the equilibrium equation (55) in the form

2y∂2Ψ

∂y2+∂2Ψ

∂z2= 2yH ′ (Ψ)− f (Ψ) f ′ (Ψ) , (56)

where y = r2/2.It is possible to construct flow equilibria in the whole space with physically

acceptable properties. In the MHD context this was recently done by Bogoy-avlenskij [16]. His results translate directly to the Euler equilibria. We assumethat H ′ = pΨ, p > 0, f = qΨ, and write the equilibrium equation as

2y∂2Ψ

∂y2+∂2Ψ

∂z2=(2py − q2

)Ψ. (57)

13

Separation of variables, Ψ = ψ (y) [a cos (ωz) + b sin (ωz)], yields the followingODE for ψ:

2y∂2ψ

∂y2−(2py + ω2 − q2

)ψ = 0, (58)

which can be recognized as a generalized Laguerre equation. For “resonance”q2 = 4n

√p+ ω2 equation (58) has a solution of the form

ψ = 2√pyL(1)n (2

√py) e−

√py =

√pr2L(1)n

(√pr2)e−√pr2/2, (59)

where L(1)n (.) is a generalized Laguerre polynomial, see Abramowitz and Stegun

[1], equation 22.6.17. The corresponding solution of the equilibrium equationwhich is rapidly decaying in the r direction and is periodic in the z direction,has the form

Ψ (r, z) =√pr2L(1)n

(√pr2)e−√pr2/2 [a cos (ωz) + b sin (ωz)] . (60)

A typical “resonance” equilibrium is show in Figure 4.

Place Figure 4 near here. A typical Bogoyavlenskij flow.

By virtue of linearity, we can superimpose these solutions (while keeping p, qfixed), and construct equilibria, quasi-periodic in the z direction. Most of thecorresponding stream surfaces are diffeomorphic to cylinders.

An important class of exact solutions of the Euler equations constitute theso-called linear flows with the velocity and pressure fields which are linear andquadratic in the spacial coordinates, respectively,

U (x, t) = K (t)x, P0 (x, t) =1

2M (t)x · x, (61)

whereM (t) is a symmetric matrix. For this velocity and pressure to satisfy theEuler equation (1) the 3× 3 matrices K (t),M (t) have to satisfy the followingconditions

d

dtK (t) +K2 (t) +M (t) = 0, trK (t) = 0. (62)

Among linear flows two-dimensional elliptical flows of the form

(U1, U2) = $

(−a1a2x2,

a2a1x1

), P0 = p0 +

1

2$(x21 + x22

), (63)

a1 ≥ a2 > 0, and hyperbolic flow of the form

(U1, U2) = $

(a1a2x2,

a2a1x1

), P0 = p0 −

1

2$(x21 + x22

), (64)

14

play a special role since they encapsulate important local features of more gen-eral flows having points of stagnation. We empasize that for flows (64) thepressure is negative at infinity but this difficulty can be disregarded for mostcases of interest. The ellipticity of the flow (63) is characterized by a nondimen-sional parameter δ =

(a21 − a22

)/(a21 + a22

). In the limiting cases δ = 0, δ = 1,

this flow reduces to the circular and linear flows, respectively.An interesting class of two-dimensional rotational Ptolemaeus flows was dis-

covered by Abrashkin and Yakubovich [2]. In Lagrangian variables these flowshave the form

x1 + ix2 = F (a1 + ia2) eiµt +G (a1 − ia2) eiνt, (65)

where F (.) , G (.) are analytical functions of their arguments, and µ, ν are realconstants. For these flows trajectories of fluid elements (which are either epicy-cloids or hypocycloids,) are the same as trajectories of planets in the Ptolemaeusframework. In principle, these flows can be considered in bounded domains andmatched with appropriate irrotational flows. In particular, Kirchhoff vorticescan be viewed as a special type of Ptolemaeus flows. The Eulerian descriptionof these flows is implicit and very complex,

F−1(e−iµt [ν (x1 + ix2) + i (V1 + iV2)]

(ν − µ)

)

= G−1(e−iνt [µ (x1 + ix2)− i (V1 + iV2)]

(µ− ν)

), (66)

where F−1, G−1 are the inverse functions of F,G, and the overbar denotes thecomplex conjugate.

Subsequently, Abrashkin et al. [3] found a three-dimensional generalizationof Ptolemaeus flows by effectively separating variables in the matrix equationsof motion (27) - (29). A typical solution has the form

x1 + ix2 =[(2γ − ω) (a1 + ia2) e

−iωt + (2γ + ω) (a1 − ia2) eiωt + h (a3)]e−iγt,

x3 = a3,

(67)

where h (a3) is an arbitrary function such that 1/h′ (a3) 6= 0. For h (a3) = 0,flows (67) become Ptolemaeus flows. There is an intricate relation betweenlinear flows (61) and flows discovered by Abrashkin et al. [3].

3.4 Beltrami Flows

A class of solutions which are complementary to integrable flows such as thevortex rings described above, are the celebrated Beltrami flows for which thevelocity and vorticity fields are proportional, i.e.,

∇×U = λU, λ = const, (68)

15

see, for example, Yoshida [176] and references therein. For such flows theBernoulli function H is constant. Beltrami fields U can be interpreted as eigen-vectors of the curl operator ∇× supplied with appropriate boundary conditions.In general, Beltrami flows have regions where stream lines are chaotic.

An important example of Beltrami flow is the so-called Arnold-Beltrami-Childress (ABC) flow of the form

U (x1, x2, x3) = (A sinx3 + C cosx2, B sinx1 +A cosx3, C sinx2 +B cosx1) .(69)

Henon [80] investigated numerically a particular case A = B = C = 1 andfound that the Lagrangian trajectories appear to fill some open domains of thethree-dimensional torus. Following this indication that ABC flows exhibit thephenomenon of Lagrangian chaos, such flows became the subject of consider-able study. Dombre et al. [37] investigated ABC flows both analytically andnumerically. They showed that for certain parameter values resonances occurwhich disrupt the so-called KAM surfaces and the remaining space is occupiedby the chaotic particle paths. Stagnation points may occur and when they dothere is numerical evidence that they are connected by a web of heteroclinicstreamlines.

4 Linearized Euler Equations

4.1 The Spectral Problem

We now return to the linearized Euler equations given by (18), (19). For a givenequilibrium velocity U (x) the classical approach to linear stability is based onan investigation of the spectrum of the operator L given by (17) in a functionspace X of vector fields where (18), (19) are well posed. However the operatorL is a degenerate, non-self adjoint, non-elliptic, non-local operator and for anarbitrary steady flow U (x) the structure of spectrum σ is remarkably littleunderstood.

Definition 1 We adopt the following definition of spectral points for anyBanach space B and operator T ∈ α(B). A point z ∈ σ(T ) is called a point ofdiscrete spectrum if it satisfies the following conditions: z is an isolated pointin σ(T ); z has finite multiplicity; the range of (z − T ) is closed, which impliesthere is a subspace Q ⊂ B where (z − T ) is invertible.

On the contrary, if z ∈ σ(T ) does not satisfy the above conditions, it is calleda point of essential spectrum. For the Euler operator L we have, as we willdiscuss in subsection 5.6, partial information concerning the essential spectrumfor general classes of flows U. However the existence of discrete eigenvaluesis at present too difficult for any general results. Much of the discussion insuch standard texts as Chandrasekhar [22] or Drazin and Reid [38] concernsproperties of eigenvalues of L, but only in cases of specific, relatively simpleflows U. Because of its “non-standard” nature, there are no general theoremsthat may be applied to prove the existence of unstable discrete eigenvalues for

16

L, i.e. discrete points z ∈ σ(L) with Rez > 0. However in certain rather specialexamples it is possible to show that unstable eigenvalues exist.

4.2 Some 2D and Examples of Unstable Eigenvalues

The spectral problem for the linearized Euler operator is considerably simplerin 2 dimensions rather than in 3 dimensions. In particular, in 2 dimensions wecan define a scalar stream function to replace the divergence free vector field.We write

U = e3 ×∇Ψ(x1, x2), v = e3 ×∇φ(x1, x2, t). (70)

Hence,

Ω = ∇×U = ∆Ψe3 = Ωe3, ω = ∇× v = ∆φe3 = ωe3, (71)

Here e3 is the unit vector perpendicular to the 2-dimensional plane with Carte-sian coordinates (x1, x2). The 2-dimensional equilibrium equations will be satis-fied when Ψ satisfies an elliptic equation of the form (41). In general the secondPoisson bracket on the RHS of (15) is very difficult to analyze. However in 2dimensions the problem greatly simplifies because e3 · ∇ ≡ 0. The vorticityequation (15) gives the spectral problem

λω + (U · ∇)ω + (v · ∇)Ω = 0, (72)

where ∂ω/∂t is replaced by λω. We consider the eigenfunction φ and the eigen-value λ for equation (72); after substituting (70), (71) into (72) we obtain

λ∆φ =

(Ψx2

∂

∂x1−Ψx1

∂

∂x2

)(∆φ −H ′′(Ψ)φ). (73)

We take the boundary conditions to be 2π-periodicity in (x1, x2).A simple and very classical example that has received much attention in the

literature of the past 100 years is plane parallel shear flow (see, for example,Chandrasekhar [22], Drazin and Reid [38]). In this case U = (U(x2), 0) and(73) becomes the so-called Rayleigh equation:

(λ

ik+ U(x2)

)(d2

dx22− k2

)Φ(x2)− U ′′(x2)Φ(x2) = 0, (74)

where we have written

φ(x1, x2, t) = Φ(x2)eikx1eλt. (75)

The celebrated Rayleigh stability criterion [141] says that a sufficient conditionfor stability is the absence of an inflection point in the profile U(x2). Thiscriterion follows from a simple application of the so-called “Energy method” for

17

stability in which an energy integral is constructed by integrating (74) multipliedby the complex conjugate of Φ(x2) to give:

Reλ∫ 2π

0

U ′′ |Φ|2

|U − iλ/k|2dx2 = 0. (76)

A related sufficient condition for linear stability was discovered by Fjortoft [53].In fact a stronger stability result, namely nonlinear stability, follows from a

method developed by Arnold [4] sometimes called the “Energy-Casimir method”which is based on the existence of two different integrals of the motion describedby the nonlinear equations of motion. Arnold’s methods prove that for planeparallel shear flow in 2D the Rayleigh criterion guarantees not only spectralstability but also nonlinear stability in a space J1 where the norm

‖v‖J1= ‖v‖L2 + ‖∇ × v‖L2 , (77)

is finite (see Arnold and Khesin [7] for more details). In the case of plane parallelflow this sufficient condition becomes the definiteness of the quadratic form

∫

Π

[(∆Ψ)2

U

U ′′+ |∇Ψ|2

]dx1dx2. (78)

Here Ψ (x1, x2) is an arbitrary 2π/k periodic in x1 and 2π periodic in x2 func-tion having generalized second derivatives that are square integrable over therectangle [0, 2π/k]× [0, 2π].

The classical treatment of the instability problem for the Rayleigh equationwas based on formal asymptotics of a perturbation of a special “neutral mode”(see Tollmein [161], Lin [128]. However the formal treatment does not givecomplete information about the asymptotic behavior and does not exclude thepossibility that the neutral mode is isolated (see Drazin and Reid [38]). Howard[84] proved that for continuous profiles with inflection points there were nomore unstable eigenvalues than the number of inflection points. Rosenbluth andSimon [146] used an alternative perturbation approach to establish a sufficientcondition for instability, however no explicit profiles were exhibited for whichthis condition is satisfied. The first rigorous proof of the existence of unstableeigenvalues was given by Faddeev [50] for monotonic profiles with inflectionpoints. In the case of monotonic profiles, Faddeev observed that the spectraloperator could be written in terms of an operator A belonging to a class knownas the Friedrichs’ model. He remarked that the problem for non monotonicprofiles is more difficult since the spectrum of A becomes multiple.

Meshalkin and Sinai [130], followed by Yudovich [177], Frenkel [55], andZhang and Frenkel [180] investigated the instability of a viscous shear flowU(x2) = sinmx2 using techniques of continued fractions. More recently Fried-lander et al. [61], [15], [59] showed that these techniques could be used forthe inviscid equation (74) with U(x2) = sinmx2 (so called Kolmogorov flows).Eigenfunctions are constructed in terms of Fourier series that converge to C∞-smooth functions for eigenvalues λ that satisfy the characteristic equation. We

18

write

Φ(x2) =∞∑

n=−∞ane

inx2 . (79)

The recurrence relation equivalent to (74) yields the following tridiagonal infinitealgebraic system

dn+m = βndn + dn−m, n ∈ Z, (80)

where

βn (z) =2z(n2 + k2

)

(n2 + k2 −m2) , (81)

dn = an(n2 + k2 −m2

). (82)

For each integer j = 0, 1, ..., [m/2], we construct a sequence dj+pm, p ∈ Z, thatsatisfies the recurrence relation (80). Let

dj = 1,

dj+pm = ρj+mρj+2m...ρj+pm,

dj−pm = ρj−mρj−2m...ρj−(p−1)m, (83)

where

ρj+pm ≡ − 1[βj+pm, βj+(p+1)m, ...

] ,

ρj−pm ≡ − 1[βj−(p+1)m, βj−(p+2)m, ...

] . (84)

The infinite continued fractions in (84), which are functions of z, converge toanalytic functions for z ∈ D = z |Rez > 0,−1 < Imz < 1. Furthermore

ρ∞ ≡ limp→∞

ρj+pm = z −√z2 + 1, (85)

hence for z ∈ D we have |ρ∞| < 1. Thus the sequence of coefficients dj+pm givenby (84) decay to zero exponentially as p → ∞. Matching the algebraic system(80) across n = j with the coefficients given by (83) leads to the characteristicequation relating z and the wave numbers k, j, namely,

(βj +

1

[βj+m, ...]

)(βj−m +

1

[βj−2m, ...]

)+ 1 = 0. (86)

Thus for each eigenvalue z satisfying (86) we have constructed a C∞-smootheigenmode Φ given by (79). In [15] properties of the continued fractions are usedto prove the existence of roots z ∈ D for wave numbers such that k2+ j2 < m2.

19

This approach constructs all the unstable modes of the Rayleigh equation witha sinusoidal profile. However, it cannot be used in order to construct stableeigenmodes for which Rez = 0, since such values of z lie outside D.

The existence of unstable eigenvalues for shear flows with a general rapidlyoscillating profile U(mx2), mÀ 1, was demonstrated in [15] using homogeniza-tion techniques to compute the spectral asymptotics. Gordin [73] has solvednumerically an interesting problem of finding a “maximally unstable” profileU(x2), provided its enstrophy

∫|U ′(x2)|2 dx2 is fixed.

We note that for the Rayleigh equation with smooth profiles there is no un-stable continuous spectrum. Results concerning the stable continuous spectruminclude those of Dikii [32], Case [20], Rosencrans and Sattinger [147].

There are just a few results concerning unstable eigenvalues for flows withmore spatial structure than shear flows. Friedlander et al. [66] examined 2Dflows that are a natural extension of Kolmogorov flows to allow for oscillatorystructure in more than one dimension. A formal asymptotic construction of abranch of unstable eigenvalues is given for a flow with stream function (44) withmÀ 1 and 0 < |a| < 1. The eigenvalue problem (73) is a PDE with oscillatorycoefficients where the basic operator is a product of a skew symmetric operatorand a symmetrizable operator. In general, the infinite set of Fourier coefficientsare non separable and it is not possible to obtain the characteristic equationas in the case of shear flow. However when there is one “fast” variable (i.e.m À 1), homogenization techniques lead to an “averaged” ODE whose spec-trum contains unstable eigenvalues with eigenfunctions in L2. The “averagedequation” associated with the homogenization procedure is

λ0

(d2

dx21− j2

)Φ0 +

(1 + a2

2+ a cos 2x1

)(d2

dx21− j2

)Φ0x1x1

− 4a sin 2x1

(d2

dx21− j2

)Φ0x1

− 4a cos 2x1Φ0x1x1= 0, (87)

where Φ0 (x1) is 2π-periodic. For the modes with j = 0 the “averaged” equation(87) becomes

d2

dx21

((1 + a2

2+ a cos 2x1

)d2

dx21Φ0

)+ λ0

d2Φ0dx21

= 0. (88)

Provided that |a| 6= 1, this equation can be written in the standard Sturm-Liouville form as

d2Φ0dx21

+2λ0

1 + a2 + 2a cos 2x1Φ0 = 0, (89)

Φ0 =(1 + a2 + 2a cos 2x1

) d2Φ0dx21

. (90)

This spectral problem has a complete family of orthonormal, square-integrable

functionsΦ0N

and corresponding eigenvalues λ0N of multiplicity no greater

20

than two. The corresponding Rayleigh quotient is

λ0N =

∫ 2π0

∣∣∣d Φ0N/dx1∣∣∣2

dx1

2∫ 2π0 (1 + a2 + 2a cos 2x1)

−1∣∣∣ Φ0N

∣∣∣2

dx1

. (91)

Thus the eigenvalues for the j = 0 modes are purely real.In a recent paper Li [110] considers a similar problem for the 2D extension

of the Kolomogorov problem. He uses a Galerkin approximation to truncatethe infinite dimensional system that arises from equation (73) and examinesthe eigenvalues that for this approximate model. However, there is no rigorousjustification for the connection between the eigenvalues of the truncated systemand unstable eigenvalues for the Euler equations.

For some very special flows, such as linear flows in ellipsoidal domains, onecan analyze the spectrum of the linearized operator L in great detail. Flows ofthis kind attracted a lot of attention, see for example, Greenspan [75], Gledzerand Ponomarev [71], Vladimirov and Il’in [170], Vladimirov and Vostretsov[172]. Consider, for example, a simple basic flow of the form (63), which can beconsidered as a steady solution of the Euler equations in the ellipsoidal domainD defined by the following condition

D =x : x21/a

21 + x22/a

22 + x23/a

23 < 1

. (92)

It turns out that the spectral problem for the corresponding operatorL is exactlyreducible to a finite-dimensional matrix spectral problem. To accomplish thisreduction we choose a basis of incompressible vector fields tangential to theboundary ∂D of the ellipsoidal domain D in the form of vector polynomials

ζ = ∇× (Θpe) , (93)

where

Θ = $2(1− x21/a21 − x22/a22 − x23/a23

), (94)

p is a monomial in x1, x2, x3, and e is a unit coordinate vector, see, e.g. Lebovitz[99], [100]. It can be shown that if ζ is a vector polynomial of degree n, thenso is Lζ. Thus, vector subspaces S(n) spanned by vector polynomials of degreen of the form (93) are invariant with respect to the action of the operator L.Accordingly, the usual Galerkin truncation of the spectral problem is exact. Thetruncated spectral problem for the subspace S(n) has the form

λM(n)c−N (n)c = 0, (95)

where

M(n)i,j =

⟨ζ(n)i , ζ

(n)j

⟩, N (n)i,j =

⟨Lζ(n)i , ζ

(n)j

⟩, (96)

21

and 〈., .〉 denotes the scalar product defined by

〈ζ, ζ′〉 =∫

D

(ζ1ζ

′1 + ζ2ζ

′2 + ζ3ζ

′3

)dx1dx2dx3. (97)

Even though the corresponding matrix problem cannot be solved analytically,it can easily be solved numerically. Figure 5 shows the union of spectra of thelow-n truncated spectral problems, 0 ≤ n ≤ 10, for a typical ellipsoidal domain.

Place Figure 5 near here. The discrete spectrum of a linear flow in an elliptoid.

In summary, the problem of the existence of “strong” instabilities associatedwith discrete unstable eigenvalues of the operator L given by (17) is almostuntouched in two dimensions and at present completely inaccessible for anyfully three-dimensional flow except for linear ones.

5 Localized Instabilities

5.1 Background

In contrast with the paucity of examples we possess showing the existence ofunstable discrete eigenvalues, there exist many examples corresponding to asomewhat different type of instability, namely instability to localized perturba-tions which can be viewed as high frequency wavelets. The asymptotic methodsfor investigating such instabilities are analogous to geometrical optics in thetheory of light rays.

It is widely believed that such short wavelength instabilities are responsiblefor the transition from large scale coherent structures to

spatial chaos; (cf. a review by Bayly et al. [13] and references therein). Inorder to describe such instabilities one can use solutions with complicated timedependence. The idea of using these solutions in hydrodynamic stability can betraced back to Kelvin [86] and Orr [135]. For many years little attention was paidto this idea until geometrical optics was introduced in the study of compressiblefluids by F.G. Friedlander [56] in 1958 and Ludwig [129] in 1960. Later Eckhoff[40], [41] and Eckhoff and Storesletten [43], [44] studied the stability of azimuthalshear flows and more general symmetric hyperbolic systems using an approachbased on a generalized progressive wave expansion. Eckhoff showed that localinstability problems for hyperbolic systems can be essentially reduced to a localanalysis involving ODEs. We note that the incompressible linearized Eulerequations (18) do not form a strictly hyperbolic system and the results of Eckhoffcan not be directly applied to these equations. Craik and Criminale [30], Craik[28], Craik and Allen [29], Forster and Craik [54] revisited the ideas of Kelvin forthe incompressible Euler equations and exhibited instabilities associated withspecial exact solutions of these equations. General localized instabilities of thistype are known under the name of broadband instabilities. They have beenused by Bayly [9] in order to confirm numerical results of Pierrehumbert [136]

22

describing the behavior of quasi-two-dimensional flows with elliptic streamlines.A similar technique was used by Lagnado et al. [90] to investigate the stabilityof flows with hyperbolic streamlines.

5.2 Kelvin Modes

We briefly summarize the ideas developed by Kelvin and his followers. To thisend we consider a linear flow of the form (61). The corresponding linearizedEuler equations have the form

∂v

∂t+ (K (t)x·∇)v+K (t)v +∇p= 0, ∇ · v = 0. (98)

It turns out that these equations have a family of solutions of the form

v (x, t) = a (t) eik(t)·x, (99)

which are generalized Fourier modes. A direct calculation shows that the equa-tions describing the temporal evolution of the pair k (t), a (t) have the form

dk

dt= −KTk, k (0) = k0, (100)

da

dt= −Ka+ 2

Ka · k|k|2

k, a (0) = a0, (101)

with k0 · a0 = 0. The evolution of this pair strongly depends on the choice ofthe matrix K which is independent of x but may or may not be a function of t.Below we discuss several specific examples where these equations can be solvedexplicitly. The general heuristic conclusion is that the underlying linear flow isunstable provided that equation (101) has a solution which grows with time.

Chandrasekhar made the following interesting observation: the superposi-tion of the linear background flow and a single perturbation of the form (99),i.e.

V(x, t) = K(t)x + a(t)eik(t)·x (102)

where K(t) is a spatially independent matrix satisfying equation (62) and a(t)and k(t) are spatially independent vectors satisfying the ODE system (100),(101) actually satisfies the full nonlinear Euler equations (7). This is due tothe fact that in this very special situation the nonlinear term v · ∇v vanishesidentically due to incompressibility.

More generally, we can consider solutions of the form

v (x, t) = a (t) eik(t)·(x−x(t)), (103)

which represent generalized modes centered at x (t). A simple calculation yieldsthe following equation for x (t):

dx

dt= Kx, x (0) = x0. (104)

23

It is easy to see that equations (100), (104) preserve the scalar product k (t) ·x (t) = k0 · x0, so that solutions (103) can be obtained from solutions (99) by asimple phase shift. As we will see later, in general, this relation does not hold.

5.3 Rapid Distortion Theory

In a different context of rapid distortion theory (RDT) of turbulence theseideas were independently discussed by Prandtl [140], Taylor [159], Batchelor andProudman [14], Townsend [162], Cambon et al. [18], and others. An excellentaccount of RDT is given by Pope [139]. Let us briefly summarize their findings.We start with the Reynolds decomposition of the velocity field V (x, t) into itsmean 〈V (x, t)〉 and fluctuation v (x, t), the pressure P (x, t) is decomposed ina similar way. In is well known that the mean velocity field and pressure aregoverned by the Reynolds, or mean momentum, equations:

D 〈V〉Dt

− ν∆ 〈V〉+∇〈P 〉+∇s = 0, ∇ · 〈V〉 = 0, (105)

∆ 〈P 〉+ ∂ 〈Vi〉∂xj

∂ 〈Vj〉∂xi

+∂2sij∂xi∂xj

= 0, (106)

where

D

Dt=

∂

∂t+ 〈V〉 · ∇, (107)

is the advective derivative along the mean velocity field 〈V〉.Here ν is the kine-matic viscosity and s is the celebrated Reynolds stress tensor or the matrix ofvelocity covariances, sij = 〈vivj〉. In homogeneous turbulence, the fluctuatingvelocity and pressure are governed by the following equations

D 〈v〉Dt

+ v·∇ 〈V〉+ v·∇v − ν∆v +∇p = 0, ∇ · v = 0, (108)

∆p+ 2∂ 〈Vi〉∂xj

∂vj∂xi

+∂2vivj∂xi∂xj

= 0, (109)

where the terms v·∇ 〈V〉 and 2 (∂ 〈Vi〉 /∂xj) (∂vj/∂xi) represent interactionsbetween the turbulence field v and mean velocity gradients ∇〈V〉. RDT as-sumes that these gradients are spatially uniform (i.e., that the mean velocityfield is linear in the spacial coordinates, cf. equation (61),) and that the meanrate-of-strain tensor

〈S〉ij =1

2

(∂ 〈Vi〉∂xj

+∂ 〈Vj〉∂xi

), (110)

24

is so large that terms the terms representing interactions between v and ∇〈V〉dominate all others. The resulting RDT equations have the form

D 〈v〉Dt

+ v·∇ 〈V〉+∇p = 0, ∇ · v = 0, (111)

∆p+ 2∂ 〈Vi〉∂xj

∂vj∂xi

= 0. (112)

They are clearly equivalent to equation (98). Batchelor and Proudman [14]were the first to construct solutions of RDT equations in the form of Fourier-Kelvin modes. Needless to say that their solutions are equivalent to Kelvin’s.However, the emphasis of RDT is not on the behavior of an individual modeand instabilities associated with such a mode, but rather on the evolution ofthe spectrum of a velocity field composed of many independent stable modes.We don’t discuss the latter problem here and refer the reader to the referencesquoted above.

5.4 Eckhoff’s Approach

In a seminal paper, Eckhoff showed that local instability problems for hyperbolicsystems can be essentially reduced to a local analysis involving ODEs. Considera linear symmetric hyperbolic system of the form

vt +

N∑

n=1

Anvxn+ Bv = 0, (113)

supplied with the initial condition

v (x, t) = v0 (x) , (114)

and appropriate regularity conditions, or, symbolically,

vt −Lv = 0. (115)

In order to address the question of instability of its zero solution v (x, t) ≡ 0,we consider a family of rapidly oscillating solutions of the form

vε (x, t) = a0 (x, t) eiS(x,t)/ε + εv(1) (x, t; ε) , (116)

and analyze their asymptotics for ε→ 0. It is easy to derive the eikonal equationfor the phase function S (x, t):

det

ISt +

N∑

n=1

AnSxn

= 0, (117)

25

which is a family of N PDEs of order 1 and, as such, can be considered asODEs along the corresponding characteristics, cf., for instance, Courant [27].Individual members of this family have the form

St +Θ(x, t, Sx1, ..., SxN

) = 0, (118)

where Θ (x, t, Sx1, ..., SxN

) is an eigenvalue of the symmetric matrix∑N

n=1AnSxn.

Let Θ be a particular eigenvalue of fixed multiplicity µ, and E1, ...,Eµ the cor-responding orthonormal system of eigenvectors:

(N∑

n=1

Ankn)Ep = ΘEp, Ep ·Eq = δpq (119)

where 1 ≤ p, q ≤ µ, or, equivalently,N∑

n=1

ankn = Θ, (120)

where anpq = AnEp·Eq . Here and below we use the notation k = ∇S, kn = Sxn

.By using the fact that Θ is a homogeneous function of k of degree 1, we canshow that

anpq =

∂Θ

∂knδpq. (121)

The amplitude a0 can be written as

a0 =∑

1≤p≤µαpEp, (122)

where the scalar functions αp are to be determined. The corresponding transportequation governing the evolution of αp has the form

αpt +

N∑

n=1

∂Θ

∂knαpxn

+ bpqαq = 0, (123)

where

bpq = (∂Ep/∂t−LEp) · Eq. (124)

It is a system of PDEs of order 1, or, equivalently, a system of ODEs and isrelatively easy to study. Eckhoff [40] proved that zero solution of the originalsystem of PDEs is unstable when, for a certain choice of Θ, PDE (113) hasa solution which grows in time. We can replace equations (118), (123) by anequivalent system of ODEs of the form

dx

dt=∂Θ

∂k, x (0) = x0, (125)

26

dk

dt= −∂Θ

∂x, k (0) = k0, (126)

dαpdt

= −bpqαq , α (0) = α0, (127)

provided that these ODEs do not break up in finite time, or, in other words,caustics do not exist. (Fortunately, in many physically interesting situationsthey do not.) It is clear that Θ plays the role of the Hamiltonian governingthe evolution of the pair (x,k). The situation becomes particularly transpar-ent when Θ is a linear function of k, Θ (x,k) = $ (x) · k. In this case thecorresponding Hamiltonian system for (x,k) is

dx

dt= $ (x) , (128)

dk

dt= −∂$ (x)

∂x· k, (129)

and no caustics occur.We note that the incompressible Euler equations (18) do not form a strictly

hyperbolic system and the Eckhoff’s results can not be directly applied to theseequations. However, his results point one in the right direction.

5.5 The Geometrical Optics Approach

In a series of papers that we will summarize below Friedlander and Vishik [62]- [64], [169], and Lifschitz and Hameiri [78], [124], and Lifschitz [116] - [120]pursued complementary approaches to geometrical optics techniques as appliedto questions of instability of the incompressible Euler equations. Using WKBasymptotics based on short wave length perturbations it was shown that a ge-ometric quantity Λ which can be considered as a “fluid Lyapunov exponent”carries considerable information concerning the instability of an Euler flow.This exponent is defined as follows. We consider an initial localized waveletdisturbance of the form

v(x, 0) = a0 (x) eik0·x/ε, ε¿ 1. (130)

The disturbance as it evolves under the linearized Euler equations (18) has theform

v(x, t) = a (x, t) eiS(x,t)/ε +O(ε) (131)

P (x, t) = εq(x, t)eiS(x,t)/ε +O(ε2). (132)

The evolution of the localized disturbance is graphically shown in Figure 6.

Place Figure 6 near here. The evolution of a localized fluid blob.

27

The leading order terms in the asymptotics give the “geometrical optics” equa-tions for the wave number vector k ≡ ∇S and the amplitude vector a:

dx

dt= U(x, t), x(0) = x0, (133)

dk

dt= −

(∂U

∂x

)Tk, k(0) = k0, (134)

da

dt= −

(∂U

∂x

)a+ 2

(∂U/∂x) a · k|k|2

k, a(0) = a0. (135)

where

k0 · a0 = 0, |k0| = |a0| = 1, (136)

and

d

dt=

∂

∂t+U · ∇,

(∂U

∂x

)=

(∂Ui∂xj

). (137)

The condition a0 · k0 = 0 is a consequence of ∇ · v = 0 (incompressibility). Indynamical systems terminology, (134) is the cotangent equation and (135) thebicharacteristic amplitude equation over a trajectory of the flow U.

Clearly equations (133) - (135) reduce to Kelvin’s equations (104), (100),(101) in the special case when the background flow U (x,t) is linear in x. Ofcourse, these equations are closely related to the system of Eckhoff’s equations.

On every trajectory U of the basic flow we associate an exponent Λ where

Λ(x0,k0, a0) = limt→∞

1

tlog |a(x0,k0, a0, t)| . (138)

We define

Λmax = supx0,k0,a0

Λ, (139)

and we call Λmax a fluid Lyapunov exponent.A parallel theory was developed in the past few decades in connection with

instabilities in MHD that are localized near magnetic field lines. This topic isdiscussed in detail in Lifschitz [115]. It can be shown that, in the framework ofthe ideal MHD model, magnetic lines of force form a family of rays for so called“ballooning” modes which are considered to cause the most dangerous instabili-ties (cf. Connor et al. [26], Dewar and Glasser [31], Hameiri [77], Lifschitz [114],[115], Eckhoff [42]). More recently Vishik and Friedlander [65] gave a rigoroustreatment of geometrical optics applied to MHD. They showed that the growthrate of a localized instability is bounded from below by the growth rate of anoperator given by a system of local hyperbolic PDEs along each magnetic lineof force.

28

5.6 The Unstable Essential Spectrum

Let the unperturbed flowU be a steady Euler equilibria. Friedlander and Vishik[62] - [64], [169] used WKB asymptotics to develop a useful tool for studyingthe unstable essential spectrum of the Euler operator L linearized about U (seealso Lifschitz [113], [116]). One of the main ideas in these papers is to replacethe study of the spectrum of L by the study of the spectrum of the evolutionoperator etL (i.e. the Green’s function for (2.8)-(2.9)). This permitted the de-velopment of an explicit formula for the growth rate of a small perturbation dueto the essential spectrum. Roughly speaking, “high” frequency perturbationscan produce instability in the essential (or continuous) spectrum and “low” fre-quency perturbations are associated with discrete eigenvalues. The followingtheorem proved by Vishik [168] gives an expression for the essential spectralradius ress(e

tL) (i.e. the maximum growth rate in the essential spectrum) interms of the maximum fluid Lyapunov exponent Λ. The results are proved forfree space or periodic boundary conditions and are valid in any spatial dimen-sion although, of course, dimensions 2 and 3 are physically the most relevant.We consider perturbations with v0 ∈ L2, ∇ · v0 = 0 and assume that U isC∞-smooth.

Theorem 1

ress(etL) = etΛmax , (140)

where Λmax is defined via (138) - (139).This theorem is proved by writing the evolution operator etL as a product

of a pseudo-differential operator and a shift operator along the trajectory ofthe equilibrium flow U. This allows the growth of the evolution operator to bestudied to precise exponential asymptotics.

The result of Theorem 1 gives one piece of information concerning the stabil-ity spectrum for inviscid flows, namely the maximum growth rate of instabilityin the essential spectrum. Moreover it implies that any point z in the spectrumσ(etL) such that |z| > eΛmaxt is necessarily an eigenvalue of finite multiplicity.A positive lower bound for the value of the Lyapunov exponent Λ can be explic-itly computed in many examples. Furthermore, Theorem 1 provides an effectivesufficient condition for instability of large classes of inviscid fluid flows. Sinceexpression (140) involves the supremum over initial conditions (x0,k0, a0), it isonly necessary to show there exists one set of initial conditions for which Λ givenby (138) is positive to conclude that Λmax > 0 and hence the unstable essentialspectrum is nonempty. Thus the existence of an exponentially growing ampli-tude vector a is an effective device for detecting L2 instability in the essentialspectrum of etL where L is the Euler operator associated with the flow U. Wenote that the connection between exponential stretching and fluid instabilitywas first observed by Arnold [6]. In the following sections we will describe manyclasses of flows for which an analysis of the ODE system (133)-(136) shows thereexists a positive fluid exponent Λ.

Friedlander and Vishik [62] considered the effect of norms with higher deriva-tives. They showed that for initial conditions such that v0 is in the Sobolev space

29

Hs, the analogous fluid Lyapunov exponent Λs is

Λs = limt→∞

1

tlog∣∣∣(1 + k2)s/2a

∣∣∣ . (141)

Thus there are different “degrees” of instability for different degrees of smooth-ness of the initial data.

As Lifschitz [116] observed, the system of ODE (133)-(136) on each trajec-tory of the flow U can also be used to obtain information about the spectralbound of the operator L itself as opposed to the evolution operator etL. Definefrom (133)-(136)

Λ⊥ = supx0,k0,a0,k0·U=0

Λ (142)

i.e. Λ⊥ is the maximal exponent subject to the restriction that k0 is perpendic-ular to U on a given trajectory. Latushkin and Vishik [94] prove that

Λ⊥ ≤ S(L) (143)

where S(L) = sup Rez : z ∈ σ(L). This result is proved using an elemen-tary dynamical system construction to relate the spectral bound S(L) to therestricted fluid Lyapunov exponent Λ⊥.

Sela and Goldhirsch [151] described the essential spectrum for unboundedelliptical flows (63) directly, while Lifschitz [121], [122] analyzed their spectrum,as well as the spectrum for unbounded hyperbolic flows, via geometrical opticstechnique. Here we briefly summarize Lifschitz results. First we consider ellip-tical flows, [121]. The Euler equations linearized in the vicinity of such a flowhave the form (98), with

K =

0 −$a1/a2 0$a2/a1 0 0

0 0 0

. (144)

It is convenient to use nondimensional variables

x′i =xiai, t′ = $t, v′i =

vi$ai

, p′ =p

$2a23, (145)

where a3 =√(a21 + a22) /2, and rewrite equations (98) in the nondimensional

form

∂v

∂t+ (J x·∇) v+J v + G−1∇p= 0, ∇ · v = 0, (146)

where

J =

0 −1 01 0 00 0 0

, G =

1 + δ 0 00 1− δ 00 0 0

, (147)

30

and δ is the ellipticity parameter. Here and below primes are omitted for brevity.The corresponding spectral problem has the form

[λ+ (J x·∇)+J ]v + G−1∇p= 0, ∇ · v = 0. (148)

The spectum of this problem is denoted by σ (δ). By separating variables in thex3 direction,

v (x1, x2, x3) = exp (ik3x3)v (x1, x2) ,

p (x1, x2, x3) = exp (ik3x3) p (x1, x2) , (149)

we obtain the spectral problem in the form

[λ+ (J x·∇) +J ]v + G−1∇k3p= 0, ∇k3 · v = 0, (150)

where ∇k3 = (∂x1, ∂x2

, k3), we denote the corresponding spectrum by σk3 (δ).It is clear that

σ (δ) = ∪−∞<k3<∞

σk3 (δ) . (151)

A simple rescaling suggests that σk3 (δ) is independent of k3 provided that k3 6= 0(the case k3 = 0 can be studied separately). Thus, the spectrum of problem(148) is infinitely degenerate and, consequently, essential. Without loss of gen-erality we put k3 = 1. In order to solve spectral problem (150) we use theFourier transform with respect to x1, x2. A tedious algebra yields the followingFourier-transformed spectral problem written in terms of v⊥ (k1, k2):

[λ+ (J⊥x·∇)+J⊥] v⊥−2G−1⊥ k⊥·k⊥G−1⊥ k⊥·k⊥ + 1

J⊥v⊥ · k⊥ = 0, (152)

where the subscripts ⊥ denote projections on the (x1, x2) plane. The corre-sponding v3 = −v⊥ · k⊥. We introduce polar coordinates (ρ, ψ) in the (k1, k2)plane and rewrite problem (152) as

(λ+

d

dψ+ 2N

)(vρvψ

)= 0, (153)

where N is a periodic matrix function of ψ,

N =

(0 − 1−δ2

ρ2(1−δ cos 2ψ)+1−δ2

1 δρ2 sin 2ψρ2(1−δ cos 2ψ)+1−δ2

). (154)

This matrix depends on ρ parametrically, so that we can fix ρ = ρ0 and studythe corresponding spectral problems separately. It can be shown that theirspectra, which are denoted by σ1,ρ0 (δ), consist of two series of eigenvalues ofthe form

λ±,ρ0,n = ±νρ0 + in, n = 0,±1,±2, ..., (155)

31

where

νρ0 =1

2πLn

[1

2

(trΛρ0 +

√(trΛρ0)

2 − 4

)], (156)

Λρ0 is the monodromy matrix for equation (153) with λ = 0, and Ln is the prin-cipal branch of the logarithm. When ρ0 varies from 0 to ∞ the correspondingλ±,ρ0,0 cover the cross in the complex plane described by the following conditions

Reλ = 0, − 12 ≤ Imλ ≤ 1

2 ,−λ (δ) ≤ Reλ ≤ λ (δ) , Imλ = 0,

(157)

where λ (δ) = max0≤ρ0≤∞Reνρ0 . The entire spectrum σ1 (δ) is obtained fromthis cross via shifts by in in the complex plane. It has a rather exotic “skeleton”structure.

The spectrum of hyperbolic flows is even more exotic. In [122] it is shownthat this spectrum occupies a strip in the complex plane along the imaginaryaxis which is defined by the condition

−1 ≤ Reλ ≤ 1. (158)

This spectrum is independent of the geometry of the basic flow.

5.7 Growing Perturbations

In the previous subsection we discussed results concerning the spectral prob-lem associated with the linearized Euler equations that can be obtained usingthe ODE (133)-(136). Another complementary approach to geometrical opticstechniques applied to the linearized Euler equations was taken by Lifschitz andHameiri [124] and Lifschitz [117] - [120]. This is based on a measure of thegrowth of a perturbation itself as a solution of the linearized Euler equations.The growth may be exponential or algebraic in time in a norm that has phys-ical meaning (e.g. growth in the energy or the enstrophy of a perturbation).Lifschitz and Hameiri [124] used energy inequalities to show that the formalWKB asymptotic solutions for v(x, t) constructed via (130)-(136) are close tothe actual solutions of the linearized Euler equations (18). Hence the growthrate of the actual solutions to these equations with initial data (19) that is suf-ficiently smooth and nonzero only in some ball, could be estimated in terms ofthe behavior of the leading order terms of the asymptotic solutions.

Theorem 2 The fastest growth rate of the velocity perturbation v (x, t)bounded from below by the fastest growth rate of the amplitudes a. The flowis unstable in the energy norm if

limt→∞

sup|a0|=1,|k0|=1,a0·k0=0

|a (x0,k0, a0, t)| → ∞. (159)

If the exponent Λ defined by (138) is positive, then this instability is associ-ated with exponential growth in time.

32

We note that, from (131), the leading order amplitude of the perturbationvorticity ω is a vector b where

b = k× a. (160)

Thus the existence of vectors x0, k0, a0 such that

limt→∞

|k× a| → ∞ (161)

implies “enstrophy” instability. In the following subsection we discuss relationsbetween growth in the energy norm and growth in the enstrophy norm due tohigh frequency perturbations.

5.8 General Properties of the Amplitude Equations

As we discussed, localized instabilities of a basic Euler flow U are associatedwith growing solutions of the amplitude equations (134)-(135) on a trajectoryof U. These equations have a rich geometric structure and certain generalproperties can be used to prove that large classes of Euler flows are unstable,see section 6 below. We summarize the main general properties:

(A) It is immediate from (134), (135), (136) that

k · a = 0, t ≥ 0 (162)

(this is consequence of the fact that the flow is incompressible).(B) In 2D it follows from (134), (135) and (162) that

d

dt(k × a) = 0, (163)

or, equivalently,

d

dt(|k| |a|) = 0. (164)

Hence the product |k| |a| is constant on a trajectory of U. Thus in 2D there ex-ists a growing amplitude a if and only if these exists a decaying cotangent vectork. In particular, we conclude that the only nondegenerate, steady 2D flows forwhich there is an exponentially growing a are flows with a hyperbolic stagnationpoint. It follows from (163) that in 2D the classical Lyapunov exponent (i.e. theexponential growth rate of a tangent-vector) and the fluid Lyapunov exponentΛ are equal.

As we stated in (160), the leading order amplitude b of a high frequency per-turbation of the vorticity is given by k× a. Hence equation (163) shows that in2D there is never high frequency growth in the vorticity. This is an obvious con-sequence of the conservation of vorticity in 2D. Applying the vorticity equationanalogue of Theorem 1, it follows that the unstable essential spectrum of theevolution operator of the vorticity equation in 2D is always empty. Hence any

33

instability in the 2D vorticity equation must come from unstable eigenvalues.As we now discuss, the situation in is very different.

(C) In equations (133)-(135) plus volume conservation imply

d

dt|k · a1 × a2| = 0 (165)

where a1 and a2 are two linearly independent vectors satisfying (135) for a givencotangent vector k. Thus we conclude that the existence of a decaying cotan-gent vector k is sufficient to imply the existence of a growing amplitude a andhence instability. Furthermore (165), together with volume conservation andthe duality of tangent vectors and cotangent vectors, proves that the existenceof a positive classical Lyapunov exponent (i.e. an exponentially growing tangentvector) implies the existence of a positive fluid Lyapunov exponent Λ. In thiscondition is sufficient for exponential fluid instability but not necessary, as wewill discuss later.

(D) In the vorticity amplitude b is not constant: it satisfies the ODE

db

dt=

(∂U

∂x

)b− (k·Ω)k× b

|k|2. (166)

As Lifschitz [119], [120] observes equation (166) is closely connected to thetangent equation,

dη

dt=

(∂U

∂x

)η. (167)

A particular pair of solutions to (166) with k·Ω = 0 is

b1 = Ω, b2 =k× Ω

|Ω|2 + α(t)Ω, (168)

with the corresponding velocity amplitudes

a1 =k× Ω

|k|2, a2 =

Ω

|Ω|2− α(t)k × Ω

|k|2, (169)

where

α(t) = 2

∫ t

0

k· (Ω× (∂U/∂x) Ω)

|Ω|4dτ. (170)

It follows from (168) that the existence of a growing cotangent vector k withk·Ω = 0 implies the existence of a growing b. Since the flow is volume preservinga decaying k must be matched by a growing k. It follows from either (165) or(169) that the existence of a decaying k implies the existence of a growing a.Hence the existence of a decaying k implies at least weak (algebraic) instabilityin the Lyapunov sense in both the velocity and the vorticity norms.

34

For flows U with exponential stretching, volume preservation plus dualitybetween tangent and cotangent vectors implies the existence of exponentiallygrowing and decaying vectors k. Hence in this case the exponential growth ratesof instabilities in both the velocity and vorticity norms are positive. However themaximum rates are not necessarily the same. A simple example that illustratesthis is a flow with a hyperbolic fixed point at which the eigenvalues of the matrix(∂U/∂x) are (1,−1/2,−1/2). In this case the maximal exponent for the growthin |a| is 1/2 but the maximal exponent for the growth in |b| is 1. In the case ofsteady flowsU with exponential stretching, we have additional information fromTheorem 1 (and its analogue for the vorticity equation), namely the maximalexponential growth rates determine the essential spectral radii of the velocityand vorticity evolution operators.

5.9 Viscous instabilities

It is interesting to investigate the impact of viscosity on short wavelength insta-bilities. The Navier-Stokes equations describing the evolution of a viscous fluidhave the form

DV

Dt− ν∆V +∇P = F, (171)

∇ ·V = 0, (172)

V (x,0) = V0 (x) . (173)

Here ν is the kinematic viscosity of the fluid, and F is the forcing term. Wefollow Lifschitz [117] and assume that viscosity is vanishingly small, so that

ν = ε2ν, (174)

where ε is a small parameter and ν is the normalized viscosity chosen is such away that the corresponding Reynolds number is of order unity. (It is clear thatthe decomposition (174) is not unique.) Let

(U(x,t; ε2

), P0

(x,t; ε2

))=(U (x,t) , P0 (x,t)

)+O

(ε2), (175)

be an exact solution of the Navier-Stokes equations whose stability we want toanalyze. Here the leading order terms describe a flow of an inviscid fluid. Weassume that on any fixed time interval [0, T ] the viscous flow converges to theinviscid flow uniformly in ε. The linearized Navier-Stokes equations have theform

∂v

∂t+ U · ∇v + v · ∇ U−ε2ν∆v +∇p = 0, (176)

∇ · v =0, (177)

35

v (x,0) = v0, ∇ · v0=0. (178)

We choose v0 in the form (130). We emphasize that in contrast to the inviscidcase, the magnitude of the small parameter ε cannot be chosen arbitrarily andis determined by the magnitude of viscosity. On a fixed time interval [0, T ] thesolution of the linearized problem has the form (131), (132). It is easy to showthat the eikonal equations (133), (134) are unaffected by viscosity while theamplitude equation (135) has the form

da

dt= −

(∂U

∂x

)a+ 2

(∂U/∂x) a · k|k|2

k− ν |k|2 a, a(0) = a0. (179)

By introducing a new variable

a = a exp

∫ t

0

ν |k|2 dt′, (180)

we can rewrite the viscous amplitude equation (179) in the inviscid form (135).Thus, the viscous flow is unstable in the velocity norm provided that

limt→∞

sup|a0|=1,|k0|=1,a0·k0=0

exp

−∫ t

0

ν |k (x0,k0, t)|2 dt′|a (x0,k0, a0, t)| → ∞.

(181)

It is unstable in the vorticity norm provided that

limt→∞

sup|a0|=1,|k0|=1,a0·k0=0

exp

−∫ t

0

ν |k (x0,k0, t)|2 dt′

× |k (x0,k0, t)|2 |a (x0,k0, a0, t)| → ∞. (182)

We emphasize that ideal instabilities are can be stabilized by viscosity for somevalues ν but remain unstable for other values of this parameter. A completeviscosity stabilization occurs only when the growth rate of the wave vector kis sufficiently large. The above stability conditions are given by Lifschitz [117],see also Dobrokhotov and Shafarevich [35], [36], they are in agreement with thenecessary conditions for the so-called fast fluid instability given by Friedlanderand Vishik [62], who use geometrical optics techniques to prove that a necessarycondition for instability in the Navier Stokes equation as epsilon goes to zerois an instability in the underlying Euler equation. Landman and Saffman [93]analyzed the impact of viscosity on the stability of elliptical flows. There areinteresting interactions between the above analysis and dynamo theory, see, e.g.,Vishik [167], Bayly [10], Dobrokhotov et al. [34], and the article by Gilbert [70]which appears in this volume.

6 Examples

6.1 Flows with stagnation points

It is natural to apply the geometrical optics stability theory to the analysis offlows with stagnation (or fixed) points. Consider a background flow U(x,t) with

36

a stagnation point x0 such thatU(x0,t) = 0. The stability equations (133)-(135)at this point reduce to Kelvin’s equations (100), (101). When the backgroundflow is steady these equations have constant coefficients.

We already know from subsection 5.8 that flows with hyperbolic stagnationpoints (which implies exponential stretching at this point) are unstable. Now weconsider elliptic points. Under generic conditions the presence of an elliptic pointimplies the existence of a hyperbolic point and hence flows with elliptic pointsare unstable by the same token. However, for the purpose of illustration and toput our analysis into historical prospective, it is useful to discuss instabilitiesrelated to stagnation points in some detail. Furthermore the nature of theinstabilities associated with hyperbolic and elliptic points can be somewhatdifferent.

We begin with hyperbolic points. The simplest way for a flow to have expo-nential stretching (i.e. positive classical Lyapunov exponent) is for this to occurat one point, i.e. a point xs at which U(xs) = 0 and the matrix (∂U/∂x) atxs has an eigenvalue with positive real part α. As we discussed in section 5.8,the fluid Lyapunov exponent is then positive and the exponential growth rateof an instability is bounded from below by α. An explicit example of a 2D flowwith a hyperbolic point is a cellular flow with stream function (43), or the lesssymmetric variant (44). Many of the equilibria discussed in section 3 containhyperbolic (and elliptic) points. Hills spherical vortex (see equation (53)) hashyperbolic points at z 6= 0, r = ±a. The so called ABC flow given by equation(69) has hyperbolic stagnation points for all values of A ≥ B ≥ C such thatB2 + C2 ≥ A2.

We note that in , as opposed to 2D, flows may exhibit exponential stretch-ing without having a stagnation point. For example, Friedlander et al. [58]and Chicone [24] proved the existence of hyperbolic closed trajectories for ABCflows in certain parameter ranges where there are no stagnation points. Fur-thermore conceptual, rather than physical flows, such as Anosov flows have beenconstructed to have the property of exponential stretching on every Lagrangiantrajectory. Clearly from our previous discussion, all such flows have a positivefluid Lyapunov exponent and are exponentially unstable.

We now turn to instabilities associated with elliptic points. For strictly2D steady flows elliptic stagnation points (i.e. the eigenvalues of (∂U/∂x) arepure imaginary) give rise to no exponentially growing instabilities, althoughany shear in U implies algebraically growing instabilities. However, if we allowperturbations, the mechanism of “vortex tube stretching” permits exponentialinstabilities. The most classical of these is the Rayleigh centrifugal instabilityto perturbations of a circularly symmetric rotating flow where the angular mo-mentum decreases with radius. This prototypical instability was generalized toelliptic columnar vortex structures by Pierrehumbert and Widnall [137], Lei-bovich and Stewartson [108], and others. In this context Bayly [9], [11] usedFloquet analysis to study the growth of a localized perturbation transportedon an elliptic streamline. He reduced the problem to the consideration of asystem of ODE closely related to the system given by equations (133)-(135) andobtained sufficient conditions for “elliptic” instabilities. Under the condition

37

that the magnitude of the circulation decreases outwards, instability with expo-nential growth is a generic property regardless of the symmetry or asymmetryof the flow U . Bayly’s work was extended by many authors, see, for example,Landman and Saffman [93], Waleffe [173] and Fukumoto and Miyazaki [69]. Lif-schitz and Hameiri [124] exploited the geometrical optics treatment for detectinglocalized instabilities to obtain sufficient conditions for exponential instabilitiesof general steady flows U with elliptic points. They used the Floquet methodto analyze the ODE (133)-(135) with time periodic coefficients. They obtainedsufficient conditions for the corresponding monodromy matrix to have eigenval-ues larger than unity. Numerical calculations indicated that elliptic stagnationpoints in flows are unstable. For nonsteady flows analogous computations canbe made of classical Lyapunov exponents in the neighborhood of an elliptic pointto demonstrate instability.

Here we briefly summarize the corresponding results in the form which isconsistent with our analysis of the spectrum of elliptical flows. Consider anelliptical flow (63). In addition to nondimensional variables given by equation(145) we introduce the nondimensional wave vector k′ = (a1k1, a2k2, a3k3), andwrite the geometrical optics (or Kelvin’s) equations (100), (101) in the form

dk⊥dt

= J⊥k⊥,dk‖dt

= 0, (183)

da⊥dt

= −J⊥a⊥ +2J⊥a⊥ · k⊥

G−1⊥ k⊥ · k⊥ + k‖ · k‖G−1⊥ k⊥, (184)

da‖dt

=2J⊥a⊥ · k⊥

G−1⊥ k⊥ · k⊥ + k‖ · k‖k‖,

where the subscripts ‖ and ⊥ refer to vector and matrix components parallelto the x3 axis and the (x1, x2) plane, respectively, and matrices J , G are givenby (147). It is clear that

k =(√

1− µ2 cos t,√1− µ2 sin t, µ

), (185)

where µ is the cosine of the angle between k and e3 at t = 0. It is obvious thatwe can concentrate on the first equation (184). We introduce new variablesc = (c1, c2), where

c1 = a⊥ · k⊥, c2 = a⊥ · J⊥k⊥, (186)

and after some algebra represent the first equation (184) in the form

dc

dt= 2N (t, µ, δ) c, (187)

where

N (t, µ, δ) =

0

µ2(1−δ2)(1−µ2)(1−δ cos 2t)+µ2(1−δ2)

−1 − (1−µ2)δ sin 2t(1−µ2)(1−δ cos 2t)+µ2(1−δ2)

. (188)

38

It is clear that 2×2 matrices N given by (154) and N given by (188) are closely

related. The matrix N is π-periodic so we can use the standard Floquet theoryfor solving the stability problem. We introduce the monodromy matrix M, i.e.,the value of the fundamental matrix solution at t = π. It is easy to show thatdetM = 1, so that the eigenvalues of M satisfy the equation

ν2 −∆(δ, µ) ν + 1 = 0, (189)

where ∆ = TrM is an analytic function of δ, µ. Thus, one of the roots is realand exceeds unity in absolute value when |∆| > 2 which is the necessary andsufficient condition for exponential instability. When |∆| < 2 we have stability.

Finally, when |∆| = 2 we can either have stability (when M is diagonal), oralgebraic instability. It is very easy to compute ∆ numerically and to show thatfor every δ > 0 there exists an interval µmin < µ < µmax such that |∆(δ, µ)| > 2,so that all elliptical flows are unstable.

Possible mechanisms for suppression of elliptic instabilities attracted muchattention. In particular, it was realized that effects of rotation can have strongstabilizing impact, see Craik [28], Bayly et al. [12], Lebovitz and Lifschitz [104],Leblanc [95], and Leblanc and Cambon [97], [98]. The Euler-Coriolis equationsof motion written in a coordinate frame rotating, with respect to inertial frame,with angular velocity ϑ have the form

DV

Dt+ 2ϑ×V +∇P = 0, ∇ ·V =0. (190)

The linearized equations for the perturbation (v, p) of a basic flow (U, P0) are

Dv

Dt+v · ∇U + 2ϑ× v +∇p = 0, ∇ · v =0. (191)

The corresponding geometrical optics equations consist of the eikonal equation(134) and the amplitude equation of the form

da

dt= −∂U

∂xa− 2ϑ× a+ 2

[(∂U/∂x) a+ ϑ× a] · k|k|2

k, a(0) = a0. (192)

For linear flow the above equation reduces to the Kelvin’s form

da

dt= −Ka− 2ϑ× a+ 2

[Ka + ϑ× a] · k|k|2

k, a(0) = a0. (193)

see Lifschitz [120].Consider an elliptical flow (63) in a coordinate frame rotating with angular

velocity ω around the x3 axis. The general solution of the corresponding eikonalequation has the form (185). The amplitude equation reduces to the form (188)

with the coefficient matrix N which is a π-periodic function of t and parametersδ, µ and f , where

f =$

ϑ

(a1a2

+a2a1

). (194)

39

This stability problem can be solved via the standard Floquet method. Thestability diagram for a rotating elliptical flow shows that for f ∼ −2 ellipticalflows are stable.

For time-dependent elliptical flows, for instance, for Kirchhoff-Kida’s vor-tices, the stability problem is much more complex because it requires solvingODEs with quasiperiodic coefficients. Bayly et al. [12] developed a practicalmethod for solving the stability problem based on its reduction to , see alsoForster and Craik [54]. In particular, Bayly et al. [12] showed that Kirchhoff-Kida’s vortices are typically unstable except when the interior vorticity is ap-proximately the negative of the background vorticity. Their results are in agree-ment with the results obtained by Robinson and Saffman [145], Miyazaki andFukumoto [131], Le Dizes et al. [106], Le Dizes and Eloy [105], Eloy and LeDizes [45], Leweke and Williamson [109].

6.2 Integrable Flows

The best known class of integrable flows consists of vortex rings with swirl.Recalling the equilibria equation (59), we observe two extreme cases. Firstlythe so-called Beltrami flows where ∇×U = λU and ∇H ≡ 0. In general sucha “chaotic” flow is not integrable and analysis of explicit flows, such as ABCflows, strongly suggests that all Beltrami flows are exponentially unstable. Onthe other extreme we have integrable flows where ∇H 6= 0 and the surfacesH = H0 are integrals of the motion. As we remarked, Arnold [5] proved thatsuch compact surfaces are nested tori and these fluid equilibria correspond tosteady vortex rings with swirl.

The classical Lyapunov exponents for an integrable flow

Ω×U+∇H = 0, ∇H 6= 0 (195)

are all zero. Hence we can not invoke the general result of section 5.8 to concludethat vortex rings with swirl are unstable. A more detailed analysis of the systemof ODE (133)-(135) is necessary to prove instability results in this situation.There are a number of papers in which such analysis is given. Friedlanderand Vishik [64] used the system of ODE to study instabilities of axisymmetrictoroidal equilibria of the form (195). They obtained a sufficient condition for aFloquet exponent of the monodromy matrix associated with the system to begreater than unity. This produced the following (nonsharp) geometric sufficientcondition for exponential instability of a vortex ring with swirl, namely

∫ T

0

(Kn · ∇H − τg

U · (∇×U)

|∇H |2)dt ≥ 0 (196)

where K is the curvature, τg the geodesic torsion and n the principal normal ofa helical streamline as it wraps around a toroidal surface H = H0 with period T .

A similar approach to the problem was taken by Lifschitz and Hameiri [125]who used the system (133)-(135) to obtain a simple necessary condition forthe stability of the core of a vortex ring with swirl. For general vortex rings

40

the transport equations either have periodic coefficients or coefficients thatare asymptotically periodic. Accordingly, two types of instability were distin-guished:

(A) Instabilities having Floquet behavior and exponential growth rate.(B) Instabilities growing algebraically in time.In general, computation of the Floquet exponents can only be done numeri-

cally but in the simpler situation of vortex rings without swirl the analysis leadsto sufficient condition for stability that is a generalized Rayleigh criterion andis consistent with (196).

These results were extended by Lifschitz el al. [127] where the growth ratesof localized disturbances predicted by the WKB-geometrical optics method werecompared with numerical solutions of the full time dependent Euler equationsthat simulated the evolution of a vortex ring. The solutions to the perturbedproblem were obtained using a vortex method in which the flow is represented bya collection of Lagrangian vortex elements moving according to their inducedvelocity. It was found that the WKB analysis did a reasonably good job ofpredicting the growth of instabilities in comparison with the vortex methodcalculations on the full Euler equations. The growth rates coming from theWKB analysis were approximately twice the growth rates from the treatmentof the full equations. This can be explained by noting that the asymptoticanalysis predicts the maximum possible growth rate while the vortex methodcalculation produces a rate for a particular disturbance not chosen necessarilyto represent the maximum. It is also important to note that the WKB analysisexcludes nonlocal interactions, whereas the vortex method calculations includesthe full nonlinear effects but introduces a discretization error.

6.3 Secondary instabilities

As we know, the sum of a linear flow and a Kelvin mode is an exact solutionof the Euler equations. Thus, it makes sense to analyze the stability of such asum with respect to localized short wavelength instabilities. Since Kelvin modescan be viewed as primary perturbations of the linear flow, the correspondinginstabilities can be viewed as secondary. The analysis of secondary instabilitiesof linear flows was initiated by Lifschitz and Fabijonas [123], and continued byFabijonas et al. [49], Fabijonas [48], Lifschitz et al. [126], Miyazaki and Lifschitz[132], and others. Here we briefly summarize the corresponding results.

Let U be a primary linear flow of the form (63). We introduce nondimen-sional variables (145) and write the Euler equations in the form (146). Next,we consider the same equations in a rotating coordinate system and write

t′ = t′′, x′ = Sx′′, V′ = S (V′′ + J x′′) , P ′ = P ′′ +1

2G2x′′⊥ · x′′⊥, (197)

where S solves the equation dS/dt = SJ , S (0) = I, and G2 = STGS is the

41

metric tensor in the rotating system,

S =

ct −st 0st ct 00 0 1

, G2 =

1 + δc2t −s2t 0−s2t 1− δc2t 0

0 0 1

, (198)

where cτ , sτ denote cos τ , sin τ . Omitting primes for the sake of brevity, wewrite the governing equations in the form

DV

Dt+ 2JV + G−12 ∇P = 0, ∇ ·V = 0. (199)

In the rotating coordinate system the equilibrium solution is trivial by con-struction, U = 0, P0 = const. For δ = 0 the governing equations (199) arethe standard Euler-Coriolis equations while for δ 6= 0 they can be considered asgeneralized Euler-Coriolis equations with time-dependent metric tensor.

Kelvin modes which are exact solutions of equations (199) have the form

U = ΥA (t) sink · x, P0 = Υα (t) cosk · x, (200)

where k = (sθ, 0, cθ)T

is the wave vector of the standing wave (which is timeindependent), A (t) is the normalized amplitude, and Υ is the scaling factor.The corresponding Kelvin equations can be written as

dA

dt= −2JA+ 2

JA · kG−12 k · kG

−12 k, A0 = (cθ, 0,−sθ)T

α (t) = 2JA · kG−12 k · k . (201)

Following the same logic as before, we rotate the coordinate system around theunit vector e2 in such a way that A0 and k turn into the unit vectors e1 and e3,respectively. In the rotated coordinates the governing equations (199) assumethe form

DV

Dt+ 2J3V + G−13 ∇P = 0, ∇ ·V = 0, (202)

where

J3 =

0 −cθ 0cθ 0 sθ0 sθ 0

, G3 =

1 + δc2θc2t −δcθs2t δsθcθc2t−δcθs2t 1− δc2t −δsθs2tδsθcθc2t −δsθs2t 1 + δs2θc2t

.

(203)

The corresponding standing wave has the form

U = ΥA (t) sinx3, P0 = Υα (t) cosx3, (204)

where A3 = 0, α = 2(1− δ2

)sθA2/

(1− δs2θc2t − δ2c2θ

), and (A1, A2) solve a

2×2 system of periodic ODEs similar to (187). Thus, in the rotated coordinate

42

system the flow is independent of x1, x2. Since A is a solution of a Floquetproblem, it can be either periodic, or quasi-periodic, or growing in time.

We now apply the geometrical optics technique to study the stability of theKelvin mode (204). The geometrical optics equations have the form

dx

dt= ΥA (t) sinx3, x (0) = x0, (205)

dξ

dt= −KT ξ, ξ (0) = ξ0, (206)

dc

dt= − (K+ 2J3) c+ 2

(K + J3) c · ξG−13 ξ · ξ G−13 ξ = 0, c (0) = c0. (207)

A simple algebra reduces equations (205) - (207) to a 2× 2 system of ODEs ofthe form (187). However, the coefficient matrix can be either periodic, or quasi-periodic, or exponentially growing. The stability problem has to be solved forvarious parameter values. The corresponding solution is very difficult and timeconsuming. In general, secondary instabilities are always present.

7 Nonlinear Instability

7.1 Background

Problems connected with stability and instability of the full nonlinear Eulerequations (1)-(4) are even more complex than those related to the linearizedEuler equations that we discussed in sections 5 and 6. Hence many questionsremain open. However there have been “small steps of progress” which illumi-nate the challenges of the nonlinear problem and we will now describe some ofthese results.

Loosely speaking a flow is called nonlinearly stable if every disturbance thatis initially “small” generates a solution to the nonlinear Euler equation whichstays “close” to the original flow for all time. There are several natural precisedefinitions of nonlinear stability and its converse, nonlinear instability. Thesedefinitions reflect the crucial dependence of a stable or unstable state on thenorm in which growth with time of disturbances is to be measured. As weremarked and illustrated with a simple example, even for a linear problem theanswer as to whether or not a solution to a PDE is stable or unstable can dependon the norm.

There are very few known explicit solutions for the time dependent, nonlinearEuler equations. There are even, as we discussed in section 3, only a limitedselection of known explicit steady equilibria. Hence it is worthwhile noting thefollowing two explicit examples of “growing” Euler flows:

(A) As we remarked in subsection 5.2, Chandrasekhar has observed thatthe sum of a “linear” flow and a single Kelvin mode of the form (102) is an

43

exact solution to the nonlinear Euler equations. Hence a basic flow of the formU = K(t)x is nonlinearly unstable to a Kelvin mode perturbation provided thatthe ODE system admits a growing amplitude a(t). In section 6 we discussed howthis can occur. In particular, if K is constant, then the existence of eigenvaluesof K with positive real part implies the existence of a vector a(t) that growsexponentially. Of course a drawback of the flow (102) is that in general it doesnot satisfy physically appropriate boundary conditions. However this examplemotivated the work on secondary localized instabilities for the linear problemdiscussed in subsection 6.3.

(B) Yudovich [177], [178], observed that there exists a class of exact solutionsto the nonlinear Euler equations which imply that all non constant 2D steadyshear flows are unstable with respect to perturbations in any norm which in-cludes the maximum of the vorticity modulus. Consider the plane parallel shearflow U = (f(x2), 0, 0) with x2 ∈ [0, 2π]. From Rayleigh’s classical result thisflow is linearly (spectrally) stable in L2 if there are no inflection points in theprofile f(x2). It is easy to check that the following is an exact solution to thenonlinear Euler equations (1) for any smooth functions f and W :

V(x, t) = (f (x2) , 0,W (x1 − tf(x2))) . (208)

The corresponding vorticity is

Ω = − (tf ′(x2)W′ (x1 − tf(x2)) ,W ′ (x1 − tf(x2)) , f ′(x2)) . (209)

Hence no matter how small the magnitude of the initial vorticity, the magni-tude of the vorticity of the flow (208) grows (linearly) with time provided onlyf ′(x2) 6= 0 and W ′ 6= 0. This set of exact solutions to the nonlinear Euler equa-tion can be easily generalized to suitable x3-independent perturbations of any2D steady flow. However, again, flows of the form (208) do not satisfy physicalboundary conditions.

7.2 A Nonlinear Instability Theorem

One reason why proving nonlinear stability is a very difficult proposition isthat to date there are no results for existence and uniqueness for all time forthe nonlinear Euler equations with appropriate boundary conditions and ini-tial conditions in a suitable function space. It is mathematically reasonableto consider a definition of nonlinear stability / instability in function spaces forwhich there is at least local in time existence and uniqueness. Claims of stabilitywould still run into the major obstruction of the existence in of global in timesolutions. However the proof of the converse, i.e. instability, is not so restrictedsince finite time “blow up” would be one special case of instability under thefollowing definition which we formulate for a general nonlinear evolution PDE.

Definition 2 We define nonlinear stability for a general evolution equationof the form

du

dt= Lu+N (u), u(0) = u0, (210)

44

where L and N are respectively the linear and nonlinear terms. Let X and Zbe a fixed pair of Banach spaces with X ⊂ Z being a dense embedding. Weassume that for any u0 ∈ X there exists a T > 0 and a unique solution u(t) to(210) with

u(t) ∈ L∞ ((0, t); X)⋂C ([0, T ], Z) (211)

in the sense that for any φ ∈ D(0, T )

T∫

0

u(τ)φ′(τ) + (Lu(τ) +N (u(τ))) φ(τ) dτ = 0. (212)

The initial condition is assumed in the sense of strong convergence in Z:

limτ→0+

‖u(τ)− u0‖Z = 0. (213)

The trivial solution u0 = 0 of (210) is called nonlinearly stable in X (i.e. Lya-punov stable) if for all ε > 0 there exists δ > 0 such that ‖u0‖X < δ implies thatwe can choose T in (211) to be T =∞; and ‖u (t)‖X < ε for a.e. t ∈ [0,∞).

The trivial solution is called nonlinearly unstable in X if it does not satisfythe conditions stated below.

In the context of the Euler equations the natural function space X is theSobolev space Hs with s > n/2 + 1. It is well known (cf. Wolibner [175],Lichtenstein [111]) that solutions to the Euler equations exist locally in time insuch spaces and the local property will be sufficient for an instability result. Wefirst formulate the relevant theorem concerning nonlinear instability in a generalsetting.

We consider the stability of the zero solution of an evolution equation (210),where L and N are respectively the linearized and nonlinear parts of the gov-erning equation. Once the spectrum of the linear part L is analyzed and shownto have an unstable component (i.e. the zero solution is linearly unstable) thenthe question arises whether the zero solution is nonlinearly unstable. It is wellknown (see, for example, Lichtenberg and Lieberman [112]) that the linear in-stability implies nonlinear instability in the finite-dimensional case (i.e. if (210)is an ODE). In the infinite-dimensional case (PDE) such a general result is notknown, although for some particular types of evolution PDE’s it has been shownthat linear instability implies nonlinear instability (e.g. such a result for the in-compressible Navier-Stokes equations in a bounded domain has been proved byYudovich [179]). Difficulties with deriving the nonlinear instability from thelinear one usually appear whenever the essential spectrum of L is non-empty.As we discussed in subsection 5.6, this is generally the case for the linearizedEuler operator.

Friedlander et al. [61] proved the following abstract nonlinear instabilitytheorem under a spectral gap condition.

Fix a pair of Banach spaces X → Z with a dense embedding. Let equation(210) admit a local existence theorem in X. Let N and L satisfy the followingconditions:

45

(A)

‖N (u)‖Z ≤ C0 ‖u‖X ‖u‖Z , (214)

for u ∈X with ‖u‖X < ρ for some ρ > 0.(B) A spectral “gap” condition, i.e.

σ(etL) = σ+ ∪ σ− with σ+ 6= φ, (215)

where

σ+ ⊂z ∈ C

∣∣eMt < |z| < eγt, σ− ⊂

z ∈ C

∣∣eβt < |z| < eαt, (216)

with

−∞ < β < α < M < γ <∞ and M > 0. (217)

Then the trivial solution u = 0 to equation (210) is nonlinearly unstable.The main idea of the proof of this theorem is as follows. We assume the

contrary, namely that the trivial solution u = 0 is nonlinearly stable. Let ε > 0sufficiently small be given: it will be specified later. From the definition ofnonlinear stability it follows that there exists a global solution u(t), t ∈ [0,∞)such that ‖u (t)‖X < ε provided‖u (0)‖X < δ(ε).

We project u (t) onto two subspaces using the spectral gap condition (216),(217). We denote by P± the Riesz projection corresponding to the partition ofthe spectrum created by the gap and introduce a new norm ||| · ||| on Z. Forany x ∈ Z let

|||x||| = |||P+x|||+ |||P−x||| (218)

=

∞∫

0

‖e−τLP+x‖ZeτMdτ +∞∫

0

‖eτLP−x‖Ze−ταdτ.

The norm ||| · ||| is equivalent to ‖ · ‖Z , i.e. there exists C > 0 such that

C−1‖x‖Z ≤ |||X ||| ≤ C‖x‖Z . (219)

Since u(t) is a solution to (214) it can be shown that

(|||P+u(t)||| − |||P−u(t)|||)|t2t1 ≥t2∫

t1

M |||P+u(τ)||| − α|||P−u(τ)||| (220)

+C−1|||u(τ)||| − |||N (u(τ)) |||dτ,

for any interval 0 ≤ t1 ≤ t2.We choose the initial condition u0 = δw0, where w0 ∈ X is an arbitrary

vector satisfying

|||P+w0||| > |||P−w0|||, ‖w0‖X < 1. (221)

46

Since ‖u0‖X > δ our assumption of nonlinear stability implies

‖u (t)‖X < ε for a.e. t ∈ [0,∞), (222)

and from condition (221)

|||N (u(t)) ||| ≤ Cc0 ‖u (t)‖Z ≤ C2c0ε|||u(t)||| for a.e. t ∈ [0,∞). (223)

Now the inequalities (222)-(223) plus Gronwall’s inequality give

|||P+u(t)||| − |||P−u(t)||| ≥ δ (|||P+w0||| − |||P−w0|||) eMT , t ∈ [0,∞),(224)

provided ε chosen so that ε < min(C−3c−10 , ρ). Since M > 0, for sufficientlylarge t the inequality (224) contradicts our assumption that ‖u (t)‖X < ε. Hencethe trivial solution to (223) is nonlinearly unstable in X .

We now consider Theorem 3 in the context of the Euler equations (1). Wewrite

V = U+ v,

Lv = −(U · ∇)v − (v · ∇)U−∇p,N (v) = −(v · ∇)v −∇q; (225)

thus the notation of the general theorem applies to instability of the steady flowU. The local existence requirement and condition (A) of Theorem 3 are easyto satisfy by making the natural choice for the spaces X and Z, namely

X = Hs, s >n

2+ 1 and Z = L2 (226)

with the restriction to vector fields that are divergence free and satisfy appro-priate boundary conditions. However the spectral gap condition is much moredifficult to verify for a given steady solution because, as we have discussed insubsection 5.6, the essential spectrum of etL is generally non-empty but its exactstructure is not known.

One piece of information we have about the structure of the spectrum is theessential spectral radius theorem discussed in subsection 5.6. In some examplesthe “fluid Lyapunov exponent” Λ can be explicitly calculated. Also Theorem1 implies, in particular, that any z ∈ σ(etL) with |z| > eΛt is a point of thediscrete spectrum (i.e. an isolated point with finite multiplicity where the rangeof (z− etL) is closed). Any accumulation point of σdisc(e

tL) necessarily belongsto σess(e

tL). Thus if

σ(etL)⋂|z| > eΛt

6= ∅, (227)

then there exists a partition

σ(etL) = σ+⋃σ− (228)

47

satisfying the gap condition (216).There are several examples of 2-dimensional flows where Λ can be computed

and discrete unstable eigenvalues calculated to show that (216) holds. Theseare the examples of discrete unstable eigenvalues discussed in subsection 4.3.As we remarked, in 2D the fluid Lyapunov exponent and the classical Lyapunovexponent are equal. Hence Λ = 0 for any plane-parallel shear flow. It thereforefollows from Theorem 3, plus the result of [15] that there exist unstable discreteeigenvalues for any shear flow with a rapidly oscillating profile, that all suchshear flows are nonlinearly unstable in Hs with s > 2.

Other recent results concerning nonlinear instability of 2-dimensional shearflows include the work of Grenier [76] who proves nonlinear instability in L∞

for piecewise linear profiles. Koch [89] proves in 2 dimensions that nonlinearstability in C1,α requires uniform boundedness of the derivatives of the flowmap which implies that all steady shear flows are nonlinearly unstable in C1,α.

A more general 2-dimensional flow than parallel shear flow that can be shownto be nonlinearly unstable is the “cats-eye” flow studied in Friedlander et al.[66], see Figure 1. In this case the existence of hyperbolic stagnation pointsimplies that Λ > 0. The exact value of Λ can be calculated as the positiveeigenvalue of the gradient matrix of U at the hyperbolic point. The results ofFriedlander et al. [66] show that there exist discrete unstable eigenvalues withreal part greater than Λ, hence again we can invoke Theorem 3 to prove thatsuch “cats-eye” flows are nonlinearly unstable.

We recall the remark in Section 5.8 that in 2D there is no unstable essentialspectrum for the linearized vorticity equation. Hence any L2 instability inthe 2D vorticity equation must arise from discrete unstable eigenvalues. ThusTheorem 3 can be immediately applied in this context to conclude that linearinstability implies nonlinear instability in Hs for the 2D vorticity equation. Arecent paper of Bardos et al. [8] proves a stronger result, namely that theexistence of an unstable eigenvalue for the vorticity equation with real partgreater than the Lyapunov exponent for U implies nonlinear instability in L2

for the 2D vorticity equation.To date we do not have enough information about the structure of the spec-

trum of the linearized Euler operator for general flows, to apply Theorem 3or other known techniques to prove nonlinear instability in the most “natural”function spaces for the equation.

There is also a considerable body of literature concerning nonlinear stability/ instability in spaces which are not the “correct” spaces for the Euler equationin the sense that the possibility of finite time blow up in such spaces has not beenruled out. This includes the celebrated stability results developed by Arnold [4]and applied by many followers. As we remarked in subsection 4.3, Arnold’smethods apply to stability in a function space we denote by J1. More precisely,he showed that the steady solutions of the Euler equations are vector fields on acertain infinite dimensional manifold M that are the critical points of an energyfunctional E restricted to M . If the critical point is a strict local maximumor minimum of E, then the steady flow is nonlinearly stable in a space J1whose elements are divergence free vectors v having a finite norm (77). The

48

theory for proving nonlinear stability is mathematically elegant but has limitedapplicability to fluid flows for several reasons in addition to the fact that thespace J1 is not a “correct” space for the Euler equations. Firstly, there may beno critical points of E at all on the manifold M . Secondly, the natural way toprove that a critical point is a strict local maximum or minimum is to provethat the second variation of E is negative or positive definite at the criticalpoint. However it was shown that for the Euler equations this quantity is neverdefinite (see Sadun and Vishik [148]).

Another approach to nonlinear stability / instability with respect to growthin the space J1 was given by Shnirelman [155] who introduced so called “min-imal” flows where the vorticity of the flow is effectively mixed through trans-portation via the Euler equations so that further mixing is impossible. Shnirelmanconjectures that all generic 2D Euler flows have a similar asymptotic behavioras t → ∞ and every flow tends to some minimal flow. Such minimal flows arenot stable but rather “compactly unstable”.

It is an open and challenging questions as to the relationship between Lya-punov stable flows, Arnold stable flows, minimal flows and spectrally stableflows (i.e. no unstable eigenvalues in Spec L). It is conjectured that those flowswhich are neither Arnold stable nor spectrally unstable, are nonlinearly unsta-ble in the space J1, but the nature of their instability is different from that oflinearly unstable flows.

8 Astrophysical applications

8.1 Background

Hydrodynamic stability questions naturally arise in the astrophysical contextand had attracted a lot of attention of mathematicians and astrophysicist alike.In order to build an adequate astrophysical stability theory, one needs to incor-porate effects of rotation, compressibility and gravity into the model. There isa vast and growing literature on the subject which can be traced back to theworks of Riemann [144], Poincare [138], Lyapunov and others. Limitations ofspace make it impossible to give even a brief overview of all the relevant astro-physical stability problems and their solution methods. Accordingly, below werestrict ourselves to two representative examples. For a general discussion werefer the reader to the well-known books and review articles by Cox [21], Schutz[150], Tassoul [158], and Unno et al. [166].

8.2 Differentially rotating stars

Axisymmetric differentially rotating stars play an important role in astrophysics.In this section we briefly summarize their stability analysis presented by Lebovitzand Lifschitz [101], [102], and subsequently extended by Faierman et al. [51],and Faierman and Moller [52].

49

General equations governing the evolution of a compressible star occupyinga finite domain Dt can be written in the form

DV

Dt+∇Pρ−∇Ψ = 0, (229)

Dρ

Dt+ ρ∇ ·V = 0, (230)

DP

Dt+ γP∇ ·V = 0, (231)

Ψ (x,t) = Γ

∫

Dt

ρ (x′,t)

|x− x′|dx′. (232)

Here V,ρ, P,Ψ are the fluid velocity, density, pressure, and gravitational po-tential, respectively; γ is the adiabatic exponent, and Γ is the universal grav-itational constant. These equations are supplemented with the usual initialconditions for V,ρ, P , and appropriate boundary conditions on the free bound-ary ∂Dt. The exact nature of the boundary conditions is not known. In manycases of interest we can use the simple impenetrability boundary condition ofthe form

V (x,t) · n (x, t) = 0 on ∂Dt. (233)

Steady axisymmetric differentially rotating stars are described by solutionsof equations (229) - (232) of the form:

U = Ω(r, z)eθ, ρ = R (r, z) , P0 = P0 (r, z) , Ψ = Ψ(r, z) , (234)

where (r, θ, z) are the standard cylindrical coordinates. The corresponding equi-librium conditions are:

−rΩ2 + 1

RP0,r −Ψr = 0,

1

RP0,z −Ψz = 0. (235)

Closed-form solutions of equations (235) are available only in exceptional cases;in general, they have to be solved numerically. Below we assume that a certainequilibrium solution, representing a differentially rotating star, is given andconcentrate on studying its stability.

To analyze the stability of a differentially rotating star we have to investi-gate the evolution of the small perturbations v,ρ, p, ψ of the velocity, density,pressure, and gravitational potential. However, rather than doing so directly,in is more convenient to express ρ, p in terms of the so-called Eckart variablesm and n:

ρ =R (m+ n)

C, p = RCn, (236)

50

where C =√γP0/R is the local speed of sound (see Eckart [39]). The linearized

equations for v,m, n, ψ are

Dv

Dt+ v · ∇V − C∇κ1m+ C∇n+ C∇κ2n−∇ψ = 0, (237)

Dm

Dt+ v · C∇κ3 = 0, (238)

Dm

Dt+ C∇ · v + v · C∇κ1 = 0, (239)

ψ = K (m+ n) , (240)

where D/Dt = ∂/∂t+Ω∂/∂θ is the convective derivative along the equilibriumvelocity field Ω (r, z) eθ, while the functions κi are given by

κ1 = ln(P1/γ0

), κ2 = ln

(γP

(γ−1)/γ0 /C

), κ3 = ln

(R/P

1/γ0

), (241)

and K is the integral operator of the form

Kg (x) = Γ

∫

D

R (x′) g (x′)

|x− x′|C (x′)dx′. (242)

The corresponding initial and boundary conditions are

v (x,0) = v0 (x) , m (x,0) = m0 (x) , n (x,0) = n0 (x) , (243)

v (x,t) · n (x) = 0 on ∂D. (244)

Symbolically we can write the linearized problem in the form

∂e

∂t−Le = 0, (245)

e (0) = e0, (246)

where e is a five-component vector function, e =(v,m, n) =(vr, vθ, vz,m, n

),

and L is a symmetric hyperbolic integro-differential operator defined by thedifferential expression of the form

L = Ar∂/∂r +Aθ∂/∂θ +Az∂/∂z +B, (247)

and boundary conditions (244). Here

Ar = −

0 0 0 0 C0 0 0 0 00 0 0 0 00 0 0 0 0C 0 0 0 0

, (248)

51

Aθ = −

Ω 0 0 0 00 Ω 0 0 C/r2

0 0 Ω 0 00 0 0 Ω 00 C 0 0 Ω

, (249)

Az = −

0 0 0 0 00 0 0 0 00 0 0 0 C0 0 0 0 00 0 C 0 0

, (250)

B =

0 2rΩ 0 Cκ1,r +K,r −Cκ2,r +K,r

−2Ω/r − Ω,r 0 −Ω,z K,θ/r2 K,θ/r

2

0 0 0 Cκ1,z +K,z −Cκ2,z +K,z

−Cκ3,r 0 −Cκ3,z 0 0−C (κ1,r + 1/r) 0 −Cκ1,z 0 0

.

(251)

The apparent asymmetry of the matrix Aθ is due to the fact that we use con-travariant components

(vr, vθ, vz

)of the velocity field v.

Since the coefficients of the evolution operator and the boundary conditionsare time and angle independent, we can consider particular solutions of the form

e (x, t) = eλt+ikθe′ (y) , (252)

where y =(r, z) are coordinates in the poloidal cross-section Dp of the domainD occupied by the star, and k = 0,±1,±2, ... .2 For brevity, below we omitprimes. The corresponding spectral problem for e (y) has the form

λe (y)−Lke (y) = 0, (253)

where Lk is an integro-differential operator defined by the differential expressionof the form

L = Ar∂/∂r +Az∂/∂z +Bk, (254)

Bk =

−ikΩ 2rΩ 0 Cκ1,r +Kk,r −Cκ2,r +Kk,r

−2Ω/r − Ω,r −ikΩ −Ω,z ikKk/r2 Kk/r

2 − ikC/r20 0 −ikΩ Cκ1,z +Kk,z −Cκ2,z +Kk,z

−Cκ3,r 0 −Cκ3,z −ikΩ 0−C (κ1,r + 1/r) −ikC −Cκ1,z 0 −ikΩ

,

(255)

2Lebovitz and Lifschitz [101] use iω rather than λ as a spectral parameter.

52

and boundary conditions (244).Although a complete spectral analysis of the operator Lk is far beyond

our current technical capabilities, we can find some points belonging to itsessential spectrum by constructing the corresponding Weyl sequences of quasi-eigenfunctions (cf. Lifschitz [115] for a detailed discussion of the Weyl method).Let y0 = (r0, z0) be a point in the poloidal domain Dp, and l = (cos ν, sin ν) bea unit poloidal vector. We define the 2× 2 symmetric matrix of the form

M =

(P,zκ3,z/R −rΩΩ,z − (P,rκ3,z + P,zκ3,r) /2R

−rΩΩ,z − (P,rκ3,z + P,zκ3,r) /2R P,rκ3,r/R+(r4Ω2

),r/r3

).

(256)

By constructing a properly oriented Weyl sequence localized in the vicinity ofy0, it can be shown that the following dispersion relations

λ = −ikΩ (y0) , λ = −ikΩ (y0)± i√M (y0) l · l (257)

determine non-isolated spectral points λ associated with y0, l. In general, thesespectral points can be complex. We introduce the following notation

µmin y0= min

νM (y0) l · l, µmax y0

= maxν

M (y0) l · l, (258)

and distinguish the following cases:(A) The purely real case, the matrix M is non-negative, 0 ≤ µmin y0

;(B) The purely imaginary case, the matrix M is non-positive, µmax y0

≤ 0;(C) The mixed case, the matrix M is neither non-negative nor non-positive,

µmin y0< 0 < µmax y0

.It is clear that the spectrum of the problem (253) can be purely imaginary

only if the matrix M (y0) is non-negative for all points y0 in Dp. By using theenergy estimate derived in Lebovitz and Lifschitz [101], one can prove that thenon-negativity of the matrix function M is necessary for stability. By usingthe classical Sylvester criterion we can represent the corresponding necessarystability condition in the form of two inequalities

P,zκ3,zR

≥ 0, (259)

(r4Ω2

),r

r3≥ κ3,rκ3,z

(r4Ω2

),z

r3. (260)

The latter condition is the generalization of the classical Rayleigh stability con-dition (31) for differentially rotating fluid masses with explicit z dependence. Ifat least one of the conditions (259), ((260) is violated, the star is unstable withrespect to localized perturbations growing exponentially fast.

The above stability conditions can also be derived (and, in fact, strength-ened) via the geometrical optics stability method. We refer the reader toLebovitz and Lifschitz [101] for further discussion.

53

Instabilities described in this section are partly due to the interplay of theeffects of compressibility and gravity. For further discussion of the compress-ibility effects in the purely hydrodynamic context, see Leblanc [96], Le Duc andLeblanc [107], among others.

8.3 Riemann ellipsoids

The Riemann ellipsoids are the only known family of incompressible rotatingstars that depart from axial symmetry. Due to this fact they played, and con-tinue to play, a consistently important role in the theory of rotating stars. Theirstability had been the subject of many investigations by Riemann [144], Poincare[138], Cartan [19], Chandrasekhar [23], Lebovitz [99], [100], and others. In thissection we briefly summarize recent findings of Lebovitz and Lifschitz [103],[104], who extended the classical normal mode stability analysis of the Riemannellipsoids to ellipsoidal harmonics of order five, and complemented it with thegeometrical optics stability analysis of these ellipsoids.

General equations governing the evolution of an incompressible star occu-pying a finite domain Dt can be written in a coordinate frame rotating, withrespect to an inertial frame, with angular velocity ϑ, in the form

DV

Dt+ 2ϑ×V +

∇Pρ−∇

(Ψ+

1

2|ϑ× x|2

)= 0, (261)

∇ ·V = 0, (262)

Ψ (x,t) = Γ

∫

Dt

ρ

|x− x′|dx′. (263)

These equations are supplemented with the usual initial conditions forV,P , andkinematic and physical boundary conditions on the free boundary ∂Dt. Thekinematic condition requires that particles which are initially on the bound-ary remain on the boundary; the physical condition requires that the pressurevanishes there:

V (x,t) · n (x, t) = 0 and P (x,t) = 0 on ∂Dt. (264)

The S-type Riemann ellipsoids form a family of steady solutions of equations(261) - (264) for which the domain D is an ellipsoid with the semiaxes a1, a2, a3,the velocity U is a linear function of x,

U = $

(a1x2a2

,−a2x1a1

, 0

)= Kx, (265)

the pressure P0 is a quadratic function of x,

P0 = Π0

(1− x21

a21− x22a22− x23a23

), (266)

54

and the angular velocity ϑ is directed along the x3 axis, ϑ = ϑe3.3 The nondi-

mensional ratios (a2/a1, a3/a1) define the constants ϑ,$,Π0 up to an inter-change and/or a simultaneous sign change of ϑ and $. The interchange ϑ,$ →$,ϑ replaces an equilibrium by its adjoint equilibrium which is characterizedby the same shape but a different velocity field. Configurations with |$/ϑ| < 1,|$/ϑ| > 1, |$/ϑ| = 1, are called direct (Riemann), adjoint (Dedekind), andself-adjoint, respectively. Without loss of generality we may assume that ϑ ≥ 0.Not all choices of semiaxes ratios correspond to an S-type ellipsoid, the ad-missible configurations occupy a horn-shaped domain in the unit square 0 <a2/a1 ≤ 1, 0 < a3/a1 ≤ 1. A detailed description of S-type ellipsoids, includingan extensive bibliography, is given by Chandrasekhar [23].

The stability properties of the Riemann ellipsoids are governed by the lin-earized equations of the form

Dv

Dt+Kv + 2ϑ× v +∇ (p− ψ) = 0, (267)

∇ · v = 0, (268)

ψ (x,t) =1

π

∫

∂D

ξn|x− y|dσy. (269)

Here v and p are the usual Eulerian perturbations of the velocity and pressure,while ξn is the normal component of the Lagrangian displacement ξ. For con-

venience, the unit of time is chosen as (πGρ)−1/2

. By necessity, the linearizedequations involve the Lagrangian displacement. In principle, it is possible toeliminate v in favor of ξ via the relation

Dξ

Dt−Kξ = v, (270)

see Lebovitz [100]. The boundary condition of vanishing pressure has the form

p|∂D = |∇P0| ξn|∂D . (271)

The geometrical optics stability method augmented with an appropriate en-ergy estimate is powerful enough to handle problem (267) - (271). In fact, theLagrangian term does not cause serious practical difficulties. The actual calcu-lations follow the general pattern of subsection 6.1; details are given in Lebovitzand Lifschitz [104]. In addition, the normal mode analysis which we used insubsection 4.2 for studying the stability of linear flows in ellipsoidal domainscan be generalized for studying the stability of Riemann ellipsoids, although theactual technical details are very involved, see Lebovitz and Lifschitz [103].

3Lebovitz and Lifschitz [103], [104], use the notation ω, λ, P0 rather than ϑ,$,Π0.

55

9 Conclusion and open problems

In this article we have discussed how WKB asymptotics related to geometri-cal optics techniques can be used to detect instabilities in ideal fluid motion.We have described how this approach, which is based on localized high spatialfrequency disturbances, fits into the classical framework of spectral and energymethods applied to fluid stability theory. These methods complement each otherand make it possible to claim that in some appropriate sense “all Euler flowsare unstable”. It is clear that there are a number of different kinds of insta-bility. In fact the differences are so big that one could argue they deserve tobe regarded as different phenomena although they all satisfy the basic conceptof instability, namely the ultimate growth in some norm of a disturbance thatis initially small. We have emphasized that the norm in which such growth ismeasured is a very important ingredient in evaluating the instability.

When there exist discrete unstable eigenvalues in the spectrum of the Eulerequation linearized about a given steady flow, there is “fast” linear instabilityand a disturbance grows exponentially with time. The non-standard nature ofthe linearized Euler operator for general flows (namely, it is degenerate, non-elliptic and non-selfadjoint) rules out the application of standard general theo-rems to deduce the existence of eigenvalues. Rather the problem must be tackledon a case by case basis for particular flows. To date there are only very fewexamples in 2D and none for fully flows where the existence of discrete unstableeigenvalues has been exhibited. Clearly it would be important to obtain suchresults for more examples, particularly those that model physically realistic fluidbehavior.

On the other hand, geometrical optics techniques reduce the instability prob-lem to the consideration of the growth rate of an “amplitude” determined bya system of ODE. As such the problem becomes considerably more tractablethan the original problem posed in terms of PDE. This growth rate detects“slower” instabilities associated with the linearized Euler equation. The growthrate may be exponential or may be “very slow” algebraic growth. A positiveexponential growth rate implies that the unstable continuous spectrum of thelinearized Euler equation is nonempty. Such a positive exponential growth rateoccurs in rather generic flows (e.g. any flow with a hyperbolic point). In thiscase the spectrum of the evolution operator fills an annulus. For each point ofthe spectrum we can construct a solution of the linearized Euler equation whichgrows in time but not necessarily monotonically; rather it may have “bursts”at some moments in time and yet be small much of the time. Hence such aninstability is a different, and weaker phenomenon than an eigenvalue instability.

It is natural to ask if instability in the linearized Euler equation implies in-stability in the full nonlinear Euler equation. We have described results thatgive a positive answer to this question when there exist discrete eigenvalues , i.e.the case of “fast” linear instability. However it is an open and challenging ques-tion when the instability is of the weaker type associated with the continuousspectrum and detectable by geometrical optics methods.

Localised instabilities can also be used to investigate the growth of a per-

56

turbation of an unsteady state; although, except in the context of time periodicbasic states, the issue arises as to how to differentiate between an evolving basicstate and the growing perturbation. We have discussed certain specific relatedproblems including the connections between the “amplitude” ODE and Kelvinmodes, rapid distortion theory and secondary instabilities. These are furtherindications of the power of geometrical optics techniques to probe the intricatebehavior of ideal fluids.

The stability / instability of fluid flows is a classical subject with manyimpressive achievements and well developed methods. However there remainsignificant unsolved problems. The very different types of instability discussedin this article suggest that the whole concept of instability should be revisitedand physical intuition incorporated to classify a “graduated scale” of instabilitiesand to examine their implications for fluid behavior.

57

References

[1] M. Abramowitz and E.A. Stegun, Handbook of Mathematical Functions,Dover, New York, 1972.

[2] A.A. Abrashkin and E.I. Yakubovich, Planar rotational flows of an idealfluid, Sov. Phys. Dokl. 29 (1984), 370-371.

[3] A.A. Abrashkin, D.A. Zenkovich and E.I. Yakubovich, Matrix formulationof hydrodynamics and extension of Ptolemaeus flows to three-dimensionalmotions, Izv. VUZov. Radiophis. 39 (1996), 783-796 (in Russian).

[4] V.I. Arnold, Conditions for nonlinear stability of stationary plane curvi-linear flows of an ideal fluid, Dokl. Acad. Nauk SSSR 162(5) (1965),773-777.

[5] V.I. Arnold, Sur la topologie des ecoulements stationnaires des fluidesparfaits, C. R. Acad. Sci. Paris 261 (1965), 17-20.

[6] V.I. Arnold, Notes on the three-dimensional flow pattern of a perfect fluidin the presence of a small perturbation of the initial velocity field, Appl.Math. Mech. 36(2) (1972), 236-242.

[7] V.I. Arnold and B. Khesin, Topological Methods in Hydrodynamics,Springer, New York, 1998.

[8] C. Bardos, Y.Guo and W.Strauss, Stable and unstable ideal plane flows,submitted for publication.

[9] B.J. Bayly, Three dimensional instability of elliptical flow, Phys. Rev. Lett.57 (1986), 2160-2163.

[10] B.J. Bayly, Fast magnetic dynamos in chaotic flows, Phys. Rev. Lett. 57(1986), 2800-2803.

[11] B.J. Bayly, Three-dimensional centrifugal-type instabilities in inviscidtwo-dimensional flows, Phys. Fluids 31 (1988), 56-64.

[12] B.J. Bayly, D.D. Holm and A. Lifschitz, Three-dimensional stability ofelliptical vortex columns in external strain flows, Phil. Trans. Roy. Soc.Lond. A 354 (1996), 1-32.

[13] B.J. Bayly, S.A. Orszag and T. Herbert, Instability mechanisms in shear-flow transition, Ann. Rev. Fluid Mech. 20 (1988), 359-391.

[14] G.K. Batchelor and I. Proudman, The effect of rapid distortion of a fluidin a turbulent motion, Quart. J. Mech. Appl. Math. 7 (1954), 83-103.

[15] L. Belenkaya, S. Friedlander and V.I. Yudovich, The unstable spectrumof oscillating shear flows, SIAM J. Appl. Math. 59 (1999), 1701-1715.

58

[16] O. Bogoyavlenskij, Exact global plasma equilibria, Uspekhi Mat. Nauk 55(2000), 63-102, in Russian; translation in Russian Math. Surveys 55 (2000), 463-500.

[17] S.L. Bragg and W.R. Hawthorne, Some exact solutions of the flow throughannular cascade actuator discs, J. Aero. Sci. 17 (1950), 243-249.

[18] C. Cambon, J.-P. Benoit, L. Shao and L. Jacquin, Stability analysis andlarge-eddy simulation of rotating turbulence with organized eddies, J.Fluid Mech. 278 (1994), 175.

[19] H. Cartan, Sur les petites oscillations d’une masse fluide, Bull. Sci. Math.46 (1922), 317-352, 356-369.

[20] K.M. Case, Stability of inviscid plane Couette flow, Phys. Fluids 3 (1960),143-148.

[21] J.P. Cox, Theory of Stellar Pulsations, Princeton Univ. Press, Princeton,1980.

[22] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, OxfordUniv. Press, Oxford, 1961.

[23] S. Chandrasekhar, Ellipsoidal Figures of Equilibrium, Dover, New York,1987.

[24] C. Chicone, A geometric approach to regular perturbation theory with ap-plications to hydrodynamics, Trans. Amer. Math. Soc. 347 (1995), 4559-4598.

[25] P. Chossat and G. Iooss, The Couette-Taylor Problem, Springer, Berlin,1994.

[26] J.W. Connor, R.J. Hastie and J.B. Taylor, High mode number stabilityof an axisymmetric toroidal plasma, Proc. Roy. Soc. Lond. A 365 (1979),1-17.

[27] R. Courant, Partial Differential Equations, Interscience, New York, 1962.

[28] A.D.D. Craik, The stability of unbounded two- and three-dimensionalflows subject to body forces: some exact solutions, J. Fluid Mech. 198(1989), 275-295.

[29] A.D.D. Craik and H.R. Allen, The stability of three-dimensional time-periodic flows with spatially uniform strain rates, J. Fluid Mech. 234(1992), 613-628.

[30] A.D.D. Craik and W.O. Criminale, Evolution of wavelike disturbances inshear flows: a class of exact solutions of the Navier-Stokes equations, Proc.R. Soc. Lond. A 406 (1986), 13-26.

59

[31] R.L. Dewar and A.H. Glasser, Ballooning mode spectrum in generaltoroidal systems, Phys. Fluids 26 (1983), 3038-3052.

[32] L.A. Dikii, Stability of plane parallel flows of an ideal fluid, Sov. Phys.Dokl. 5 (1960), 1179-1182.

[33] L.A. Dikii, Hydrodynamic Stability and the Dynamics of the Atmosphere,Gidrometeoizdat, Leningrad, 1976 (in Russian).

[34] S. Dobrokhotov, V.M. Olive, A. Ruzmaikin and A. Shafarevich, Magneticfield asymptotic in a well conducting fluid, Geophys. Astrophys. Fluid Dyn.82, no. 3-4 (1996), 255-280.

[35] S. Dobrokhotov and A. Shafarevich, Parametrix and asymptotics of local-ized solutions of the Navier-Stokes equations in R3 linearized on a smoothflow, Math. Notes 51 (1992), 47-54.

[36] S. Dobrokhotov and A. Shafarevich, Asymptotic solutions of linearizedNavier-Stokes equations, Math. Notes 53 (1993), 19-26.

[37] T. Dombre, U. Frish, J.M. Greene, M. Henon, A. Mehr and A.M. Soward,Chaotic streamlines in the ABC flows, J. Fluid Mech. 167 (1986), 353-391.

[38] P.G. Drazin and W.H. Reid, Hydrodynamic Stability, Cambridge Univ.Press, Cambridge, 1981.

[39] C. Eckart, Hydrodynamics of Oceans and Atmospheres, Pergamon, Ox-ford, 1960.

[40] K.S. Eckhoff, On stability for symmetric hyperbolic systems, I, J. Diff.Eqns. 40 (1981), 94-115.

[41] K.S. Eckhoff, On stability for symmetric hyperbolic systems, II, J. Diff.Eqns. 43 (1982), 281-304.

[42] K.S. Eckhoff, Linear waves and stability in ideal magnetohydrodynamics,Phys Fluids 30 (1987), 3673-3685.

[43] K.S. Eckhoff and L. Storesletten, A note on the stability of steady inviscidhelical gas flows, J. Fluid. Mech. 89 (1978), 401-411.

[44] K.S. Eckhoff and L. Storesletten, On the stability of rotating compressibleand inviscid fluids, J. Fluid. Mech. 99 (1980), 433-448.

[45] C. Eloy and S. Le Dizes, Three-dimensional instability of Burgers andLamb-Oseen vortices in a strain field, J. Fluid. Mech. 378 (1999), 145-166.

[46] L. Euler, Principes generaux du mouvement des fluides, Hist. de l’Acad.de Berlin (1755).

60

[47] L. Euler, De principiis motus fluidorum, Novi Comm. Acad. Petrop. xiv(1759).

[48] B. Fabijonas, Secondary instabilities of linear flows with elliptic stream-lines, PhD Thesis, Univ. of Illinois, Chicago, 1997.

[49] B. Fabijonas, D.D. Holm and A. Lifschitz, Secondary instabilities of flowswith elliptic streamlines, Phys. Rev. Lett. 78 (1997), 1900-1903.

[50] L.D. Fadeev, On the theory of the stability of stationary plane-parallelflows of an ideal fluid, Zapiski Nauchnykh Seminarov LOMI 21 (1971),164-172.

[51] M. Faierman, A. Lifschitz, R. Mennicken and M. Moeller, On the essentialspectrum of a linearized Euler operator, Z. Angew. Math. Mech. 79 (1999),739-755.

[52] M. Faierman and M. Moeller, On the essential spectrum of a differentiallyrotating star in the axisymmetric case, Proc. Roy. Soc. Edinburgh A 130(2000), 1-23.

[53] R. Fjortoft, Application of integral theorems in deriving criteria of sta-bility for the baroklinic circular vortex, Geofys. Publ. Oslo 17 (1950), no.6.

[54] G.K. Forster and A.D.D. Craik, The stability of three-dimensional time-periodic flows with ellipsoidal stream surfaces,

J. Fluid Mech. 324 (1996), 379-391.

[55] A.L. Frenkel, Stability of an oscillating Kolmogorov flow, Phys. Fluids A3 (1991), 1718-1729.

[56] F.G. Friedlander, Sound Pulses, Cambridge Univ. Press, Cambridge, 1958.

[57] S. Friedlander, Lectures on stability and instability of an ideal fluid,IAS/Park City Math. Series 5 (1999), 229-304.

[58] S. Friedlander, A.D. Gilbert and M.M. Vishik, Hydrodynamic instabilityfor certain ABC flows, Geophys. Astrophys. Fluid Dyn. 73 (1993), 97-107.

[59] S. Friedlander and L.N. Howard, Instability in parallel flows revisited,Studies Appl. Math. 101 (1998), 1-21.

[60] S. Friedlander and A. Shnirelman, Instability of steady flows of an idealincompressible fluid, to appear in Advances in Mathematical Fluid Dy-namics, Birkhauser, 2002.

[61] S. Friedlander, W. Strauss and M.M. Vishik, Nonlinear instability in anideal fluid, Annales I.H.P., J. Nonlineare 14 (1997), 187-209.

61

[62] S. Friedlander and M.M. Vishik, Dynamo theory, vorticity generation, andexponential stretching, Chaos 1 (1991), 198-205.

[63] S. Friedlander and M.M. Vishik, Instability criteria for the flow of aninviscid incompressible fluid, Phys. Rev. Lett. 66 (1991), 2204-2206.

[64] S. Friedlander and M.M. Vishik, Instability criteria for steady flows of aperfect fluid, Chaos 2 (1992), 455-460.

[65] S. Friedlander and M.M. Vishik, On stability and instability criteria formagnetohydrodynamics, Chaos 5 (1995), 416-423.

[66] S. Friedlander, M.M. Vishik and V.I. Yudovich, Unstable eigenvalues as-sociated with inviscid fluid flows, J. Math. Fluid Mech. 2 (2000), 365-380.

[67] S. Friedlander and V.I. Yudovich, Instabilities in fluid motion, NoticesAMS 46 (1999), 1358-1367.

[68] E. Frieman and M. Rottenberg, On hydromagnetic stability of stationaryequilibria, Rev. Mod. Phys. 32 (1960), 898.

[69] Y. Fukumoto and T. Miyazaki, Local stability of two-dimensional steadyirrotational solenoidal flows with closed streamlines, J. Phys. Soc. Japan65 (1996), 107-113.

[70] A. Gilbert, Dynamo theory, in Handbook of Mathematical Fluid Dynam-ics, vol. 2, S. Friedlander and D. Serre, Eds, Elsevier, Amsterdam, 2003.

[71] E.B. Gledzer and V.M. Ponomarev, Instability of bounded flows withelliptical streamlines, J. Fluid Mech. 240 (1992), 1-30.

[72] C. Godreche and P. Manneville, Hydrodynamics and Nonlinear Instabili-ties, Cambridge Univ. Press, Cambridge, 1998.

[73] A. Gordin, Private communication.

[74] H. Grad and H. Rubin, Hydromagnetic equilibria and force-free fields,Proc. Second UN Int. Conf. 31 (1958), 190-197.

[75] H. Greenspan, Theory of Rotating Fluids, Breukelen, Brookline, MA,1990.

[76] E. Grenier, On the nonlinear instability of the Euler and Prandtl equa-tions, Comm. Pure Appl. Math. 53 (2000), 1067-1091.

[77] E. Hameiri, On the essential spectrum of ideal magnetohydrodynamics,Comm. Pure Appl. Math. 38 (1985), 43-66.

[78] E. Hameiri and A. Lifschitz, Ballooning modes in plasmas and in classicalfluids, Proc. Sherwood Theory Conf. Williamsburg VA, 1990.

62

[79] H. Helmholtz, Uber diskontinuierliche Flussigkeitsbewegungen, Phil. Mag.36 (1868), 337-346.

[80] M. Henon, Sur la topologie des lignes de courant dans un cas particulier,C. R. Acad. Sci. Paris 262 (1966), 312.

[81] M.J.M. Hill, On a spherical vortex, Phil. Trans. Roy. Soc. Lond. A 185(1894), 213-245.

[82] D.D. Holm, Nonlinear stability of ideal fluid equilibria, in Nonlinear Topicsin Ocean Physics: Proceedings S.I.F., Course CIX, A.R. Osborne, Ed.,North-Holland, Amsterdam, 1991.

[83] D.D. Holm, J.E. Marsden, T. Ratiu, and A. Weinstein, Nonlinear stabilityof fluid and plasma equilibria, Phys. Rep. 123 (1985), 1-116.

[84] L.N. Howard, The number of unstable modes in hydrodynamic stabilityproblems, J. Mecanique 3 (1964), 433.

[85] D.D. Joseph, Stability of Fluid Motions, 2 volumes, Springer, Berlin, 1976.

[86] Lord Kelvin, Stability of fluid motion: rectilinear motion of viscous fluidbetween two parallel plates, Phil. Mag. 24 (1887), 188-196.

[87] Lord Kelvin, Hydrokinetic solutions and observations, Mathematical andPhysical Papers, iv, Hydrodynamics and General Dynamics, CambridgeUniv. Press, Cambridge, 1910.

[88] S. Kida, Motion of an elliptic vortex in a uniform shear flow, J. Phys. Soc.Japan 50 (1981), 3517-3520.

[89] H. Koch, Instability for incompressible and inviscid fluids, Chapman &Hall / CRC Res. Notes Math. 406 (2000), 240-247.

[90] R.R. Lagnado, N. Phan-Thien and L.G. Leal, The stability of two-dimensional linear flows, Phys. Fluids 27 (1984), 1094-1106.

[91] J. Lagrange, Memoire sur la theorie du mouvement des fluides, Nouv.Mem. de l’Acad. de Berlin (1781).

[92] H. Lamb, Hydrodynamics, Cambridge Univ. Press, Cambridge, sixth edi-tion, 1995.

[93] M.J. Landman and P.G. Saffman, Three-dimensional instability ofstrained vortices, Phys. Fluids 30 (1987), 2339-2342.

[94] Y. Latushkin and M.M. Vishik, Linear stability in an ideal fluid, Preprint.

[95] S. Leblanc, Stability of stagnation points in rotating flows, Phys. Fluids9 (1997), 3566-3569.

63

[96] S. Leblanc, Destabilization of a vortex by acoustic waves, J. Fluid Mech.414 (2000), 315-337.

[97] S. Leblanc and C. Cambon, On the three-dimensional instabilities of planeflows subjected to Coriolis force, Phys. Fluids 9 (1997), 1307-1316.

[98] S. Leblanc and C. Cambon, Effects of the Coriolis force on the stabilityof Stuart vortics, J. Fluid Mech. 356 (1998), 353-379.

[99] N.R. Lebovitz, The stability equations for rotating, inviscid fluids:Galerkin methods and orthonormal bases, Astrophys. Geophys. Fluid Dyn.46 (1989), 221-243.

[100] N.R. Lebovitz, Lagrangian perturbations of Riemann ellipsoids, Astro-phys. Geophys. Fluid Dyn. 47 (1989), 225-236.

[101] N.R. Lebovitz and A. Lifschitz, Short wavelength instabilities of rotatingcompressible fluid masses, Proc. Roy. Soc. Lond. A 433 (1992), 265-290.

[102] N.R. Lebovitz and A. Lifschitz, Local hydrodynamic instability of rotatingstars, Ap. J. 408 (1993), 603-614.

[103] N.R. Lebovitz and A. Lifschitz, New global instabilities of the Riemannellipsoids, Ap. J. 458 (1996), 699-713.

[104] N.R. Lebovitz and A. Lifschitz, Short wavelength instabilities of Riemannellipsoids, Phil. Trans. Roy. Soc. Lond. A 354 (1996), 33-56.

[105] S. Le Dizes and C. Eloy, Short-wavelength instability of a vortex in amultipolar strain field, Phys. Fluids 11 (1999), 500-502.

[106] S. Le Dizes, M. Rossi and H.K. Moffatt, On the three-dimensional insta-bility of elliptical vortex subjected to stretching, Phys. Fluids 8 (1996),2084-2090.

[107] A. Le Duc and S. Leblanc, A note on Rayleigh stability criterion forcompressible flows, Phys. Fluids 11 (1999), 3563-3566.

[108] S. Leibovich and K. Stewartson, A sufficient condition for the instabilityof columnar vortices, J. Fluid Mech. 126 (1983), 335-356.

[109] T. Leweke and C.H.K. Williamson, Cooperative elliptic instability of avortex pair, J. Fluid Mech. 360 (1998), 85.

[110] Y. Li, On 2D Euler equations: the Energy-Casimir instabilities and thespectra for linearized 2D Euler equations, to appear, J. Math. Physics.

[111] L. Lichtenstein, Uber Einige Existenzprobleme der Hydrodynamik ho-mogener unzusammendruckbarer reibunglosser Flussigkeiten und dieHelmholtzschen Wirbelsatze, zweite Abhandlung, Math. Zeit.

26 (1927), 196-323, 387-415, 725.

64

[112] A.J. Lichtenberg and M.A. Lieberman, Regular and Chaotic Dynamics,Springer, New York, 1992.

[113] A. Lifschitz, On the continuous spectrum in some problems of mathemat-ical physics, Sov. Phys. Dokl. 29 (1984), 625-627.

[114] A. Lifschitz, Continuous spectrum in general toroidal systems (ballooningand Alfven modes), Phys. Lett. A 122 (1987), 350-356.

[115] A. Lifschitz, Magnetohydrodynamics and Spectral Theory, Kluwer Aca-demic Publishers, Dordrecht, 1989.

[116] A. Lifschitz, Essential spectrum and local stability condition in hydrody-namics, Phys. Lett. A 152 (1991), 199-204.

[117] A. Lifschitz, Short wavelength instabilities of incompressible three-dimensional flows and generation of vorticity, Phys. Lett. A 157 (1991),481-486.

[118] A. Lifschitz, On the instability of three-dimensional flows of an ideal in-compressible fluid, Phys. Lett. A 167 (1992), 465-474.

[119] A. Lifschitz, On the instability of certain motions of an ideal incompress-ible fluid, Advances Appl. Math. 15 (1994), 404-436.

[120] A. Lifschitz, Instabilities of ideal fluids and related topics, Z. Angew. Math.Mech. 75 (1995), 411-422.

[121] A. Lifschitz, Exact description of the spectrum of elliptical vortices inhydrodynamics and magnetohydrodynamics, Phys. Fluids 7 (1995), 1626-1636.

[122] A. Lifschitz, On the spectrum of hyperbolic flows, Phys. Fluids 9 (1997),2864-2871.

[123] A. Lifschitz and B. Fabijonas, A new class of instabilities of rotating fluids,Phys. Fluids 8 (1996), 2239-2241.

[124] A. Lifschitz and E. Hameiri, Local stability conditions in fluid dynamics,Phys. Fluids A 34 (1991), 2644-2651.

[125] A. Lifschitz and E. Hameiri, Localized instabilities of vortex rings withswirl, Comm. Pure Appl. Math. 46 (1993), 1379-1408.

[126] A. Lifschitz, T. Miyazaki and B. Fabijonas, A new class of instabilities ofrotating flows, Eur. J. Mech. B / Fluids 17 (1998), 605-613.

[127] A. Lifschitz, H. Suters and T. Beale, The onset of instability in exactvortex rings with swirl, J. Comp. Phys. 129 (1996), 8-29.

[128] C.C. Lin, The Theory of Hydrodynamic Stability, Cambridge Univ. Press,Cambridge 1967.

65

[129] D. Ludwig, Exact and asymptotic solutions of the Cauchy problem,Comm. Pure Appl. Math. 13 (1960), 473-508.

[130] L. Meshalkin and Ya. Sinai, Investigation of stability for a system ofequations describing plane motion of a viscous incompressible fluid, Appl.Math. Mech. 25 (1961), 1140-1143.

[131] T. Miyazaki and Y. Fukumoto, Three-dimensional innstability of strainedvortices in a stably stratified fluid, Phys. Fluids A 4 (1992), 2515-2522.

[132] T. Miyazaki and A. Lifschitz, Three-dimensional instabilities of inertialwaves in rotating fluids, J. Phys. Soc. Japan 67 (1998), 1226-1233.

[133] H.K. Moffatt, Generalised vortex rigs with and without swirl, Fluid DynRe. 3 (1988), 22-30.

[134] J. C. Neu, The dynamics if a columnar vortex in an imposed strain, Phys.Fluids 27 (1984), 2397-2402.

[135] W.McF. Orr, The stability or instability of the stready motions of a perfectfluid, Proc. R. Irish Acad. A 27 (1907), 9-69.

[136] R.T. Pierrehumbert, Universal short-wave instability of two dimensionaleddies in an inviscid fluid, Phys. Rev. Lett. 57 (1986), 2157-2159.

[137] R.T. Pierrehumbert and S.E. Widnall, The two- and three-dimensionalinstabilities of a spatially periodic shear layer, J. Fluid Mech. 114 (1982),59-82.

[138] H. Poincare, Sur l’equilibre d’une masse fluide animee d’un mouvementde rotation, Acta. Math. 7 (1885), 259-380.

[139] S.B. Pope, Turbulent Flows, Cambridge Univ. Press, Cambridge, 2000.

[140] L. Prandtl, Attaining a steady air stream in wind tunnels, TechnicalManuscript 726, NACA (1933).

[141] Lord Rayleigh, On the stability or instability of certain fluid motions,Proc. Lond. Math. Soc. 9 (1880), 57-70.

[142] Lord Rayleigh, Investigation of the character of the equilibrium of an in-compressible heavy fluid of variable density, Scientific Papers, CambridgeUniv. Press, Cambridge, 1900.

[143] O. Reynolds, An experimental investigation of the circumstances whichdetermine whether the motion of water shall be direct or sinuous, and thelaw of resistance in parallel channels, Phil. Trans. Roy. Soc. Lond. A 174(1883), 935-982.

[144] B. Riemann, Ein Beitrag zu den Untersuchengen uber die Bewegung einesflussigen gleichartigen Ellipsoides, Gott. Abh.

IX (1860), 3-36.

66

[145] A.C. Robinson and P.G. Saffman, Three-dimensional stability of an ellip-tical vortex in a straining field, J. Fluid Mech. 142 (1984), 451-466.

[146] N.M. Rosenbluth and A. Simon, Necessary and sufficient condition for thestability of plane parallel inviscid flow, Phys. Fluids 7 (1964), 557-558.

[147] S.I. Rosencrans and D.M. Sattinger, On the spectrum of an operator oc-curring in the theory of hydrodynamic stability, J. Math. Phys. 45 (1966),289-300.

[148] L. Sadun and M.M. Vishik, The spectrum of the second variation for anideal incompressible fluid, Phys. Lett. A. 182 (1993), 394-398.

[149] P. Saffman, Vortex Dynamics, Cambridge Univ. Press, Cambridge, 1992.

[150] B.F. Schutz, Problems in astrophysical fluid dynamics, Lect. Appl. Math.20 (1983), 99-140.

[151] N. Sela and I. Goldhirsch, Stability theory of an elliptical vortex, Phys.Fluids A 4 (1992), 2252-2270.

[152] V.D. Shafranov, On equilibrium magnetohydrodynamic configurations,Sov. Phys. JETP 6 (1957).

[153] K. Sharif and A. Leonard, Vortex rings, Ann. Rev. Fluid Mech. 24 (1992),235-279.

[154] H. Schlichting, Boundary-Layer Theory, McGraw Hill, New York, 1979.

[155] A. Shnirelman, On the L2 instability and L2 controllability of steady flowsof an ideal incompressible fluid, in Journees des Equations aux DerivesPartielles, St Jean de Mont, 1999.

[156] D. Sipp and L. Jacquin, Three-dimensional centrifugal-type instabilitiesin two-dimensional flows in rotating systems, Phys. Fluids 12 (2000),1740-1748.

[157] H. Swinney and L. Gollub (eds.), Hydrodynamic Instabilities and Transi-tion to Turbulence, Springer, New York, 1985.

[158] J.-L. Tassoul, Theory of Rotating Stars, Princeton Univ. Press, Princeton,1978.

[159] G.I. Taylor, Turbulence in a contracting stream, Z. Angew. Math. Mech.15 (1935), 91-96.

[160] G.I. Taylor, The instability of liquid surfaces when accelerated in a direc-tion perpendicular to their planes. I, Proc. Roy. Soc. Lond. A 201 (1950),192-196.

67

[161] W. Tollmien, Ein allgemeines Kriterium der Instabilitat laminarerGeschwindigkeitsverteilungen, Nachr. Ges. Wiss. Math.-Phys. Kl. 50(1935), 79-114.

[162] A.A. Townsend, The Structure of Turbulent Shear Flow, Cambridge Univ.Press, Cambridge, 1976.

[163] L.N. Trefethen, Pseudospectra of linear operators, SIAM Review 39(1997), 383-406.

[164] L.N. Trefethen, A. E. Trefethen, S. C. Reddy and T. A. Driscoll, Hydro-dynamic stability without eigenvalues, Science 261 (1993), 578-584.

[165] B. Turkington, Vortex rings with swirl: axisymmetric solutions of theEuler equations with nonzero helicity, SIAM J. Math. Anal. 20 (1989),57-73.

[166] W. Unno, Y. Osaki, H. Ando and H. Shibahashi, Nonradial Oscillationsof Stars, Univ. of Tokyo Press, Tokyo, 1979.

[167] M.M. Vishik, Magnetic field generation by the motion of a highly con-ducting fluid, Geophys. Astrophys. Fluid Dyn. 48 (1989), 151-167.

[168] M.M. Vishik, Spectrum of small oscillations of an ideal fluid and Lyapunovexponents, J. Math. Pures Appl. 75 (1996), 531-557.

[169] M.M. Vishik and S. Friedlander, Dynamo theory methods for hydrody-namic stability, J. Math. Pures Appl. 72 (1993), 145-180.

[170] V.A. Vladimirov and K.I. Il’in, Three-dimensional instability of an ellipticKirchhoff vortex, Izv. Akad. Nauk SSSR, Mekh. Zidk. Gaza 9 (1988), 356-360.

[171] V.A. Vladimirov and K. I. Ilin, On Arnold’s variational principles in fluidmechanics , in: The Arnoldfest: Proceedings of a Conference in Honourof V. I. Arnold for his Sixtieth Birthday, E. Bierstone, B. Khesin, A. Kho-vanskii, J.E. Marsden, Eds, American Mathematical Society, Providence,1999.

[172] V.A. Vladimirov and D.G. Vostretsov, Instability of steady flows withconstant vorticity in vessels of elliptic cross-section, Prikl. Matem. Mekh.50 (1986), 369-377.

[173] F. Waleffe, On the three-dimensional instability of strained voritices, Phys.Fluids A 2 (1990), 76-80.

[174] F. Waleffe, Hydrodynamic stability and turbulence: beyond transients toa self-sustaning process, Stud. Appl. Math. 95 (1995), 319-343.

68

[175] W. Wolibner, Un theoreme sur l’xistence du mouvement plan d’un flu-ide parfait, homogene, incompressible, pendant un temps infiniment long,Math. Zam. 37 (1933), 698-726.

[176] Z. Yoshida, Beltrami functions: A paradigm of streamline chaos, Curr.Topics Phys. Fluids 1 (1994), 155-178.

[177] V.I. Yudovich, Example of secondary stationary flow or periodic flow ap-pearing while a laminar flow of a viscous incompressible fluid loses itsstability, Appl. Math. Mech. 29 (1965), 453-467.

[178] V.I. Yudovich, On loss of smoothness for the Euler equations, DinamikaSploshnoi Sredy 16 (1974), 71-78.

[179] V.I. Yudovich, The Linearization Method in Hydrodynamical StabilityTheory, American Mathematical Society, Providence, 1989.

[180] X. Zhang and A.L. Frenkel, Large-scale instability of generalized oscillat-ing Kolmogorov flows, SIAM J. Appl. Math.

58 (1998), 540-564.

69

10 Figure Captions

• Figure 1. A typical asymmetric “cats-eye” flow pattern.

• Figure 2. The meridional cross-section of a typical Hill’s vortex.

• Figure 3. A typical vortex ring with swirl. Reprinted with permissionfrom [127].

• Figure 4. The meridional cross-section of a typical Bogoyavlenskij flow.

• Figure 5. The discrete spectrum of a linear flow in an ellipsoid.

• Figure 6. The evolution of a localized fluid blob (after [57]).

70