Instability of Equilibria for the 2D Euler Equations on the torus
aa r X i v : . [ m a t h . D S ] M a y INSTABILITY OF EQUILIBRIA FOR THE 2D EULEREQUATIONS ON THE TORUS
HOLGER DULLIN, ROBERT MARANGELL AND JOACHIMWORTHINGTON
School of Mathematics and Statistics, Carslaw Building (F07),The University of Sydney, NSW 2006Email: [email protected] (corresponding author),[email protected], [email protected].
Abstract.
We consider the hydrodynamics of an incompressible fluid on a 2Dperiodic domain. There exists a family of stationary solutions with vorticitygiven by Ω ∗ = α cos( p · x )+ β sin( p · x ). This situation can be approximated asa structure preserving finite dimensional Hamiltonian system by a truncationintroduced by [24, 26] or by the more standard Galerkin style finite elementmethod. We use these two truncations to analyse the linear stability of thesesolutions and analytical and numerical results are compared. Following themethods used by [17] the problem is divided into subsystems and we provethat most subsystems are linearly stable. We derive a sufficient condition fora subsystem to be linearly unstable and derive an explicit lower bound for theassociated real eigenvalues independent of the truncation size N . Then weshow that the corresponding eigenvectors are in ℓ . This together with knownstability results for the 2D periodic Euler equations allows us to concludethat most of these stationary solutions are nonlinearly unstable. We confirmour results with a numerical computation of the spectrum for a large, finitetruncation. Finally we discuss the essential spectrum of the full problem asthe limit of the truncated problem. Introduction
In terms of the vorticity Ω( x , t ) : ( T × R + ) → R , the 2D incompressible Eulerequations are (see [1] Appendix 2 for an overview)(1.1) ∂ Ω ∂t + u ∂ Ω ∂x + u ∂ Ω ∂x = 0 , ∂u ∂x + ∂u ∂x = 0 . Here x = ( x , x ) T and u , u are the velocity components in the x and x directions respectively. We impose periodic boundary conditions Ω( π, x , t ) =Ω( − π, x , t ) and Ω( x , π, t ) = Ω( x , − π, t ). There is a family of stationary solu-tions given by(1.2) Ω ∗ = α cos( p · x ) + β sin( p · x )for α, β ∈ R and p ∈ Z .The study of stability of certain solutions of the planar Euler equations wasinitiated by the seminal work by [3] on the Lie-Poisson structure of the Euler equations where he invented the Energy-Casimir method to prove stability. This isrevisited in [2], in particular Section II.4. Arnol’d discusses a slightly more generalproblem where the torus has dimensions X × π and p = (0 , T , and shows thatthis solution is non-linearly stable when X ≤ π . In [19] it was shown that anyequilibrium with p = (0 , T and X > π is linearly unstable for the viscousproblem using linear stability analysis and infinite continued fractions. That paperalso shows linear instability for p = (0 , m ) T , m > X . The linearinstability of the inviscid problem for p = (0 , m ) T , m > m = m + m for any positive integers m , m , it was shown in [10] that the steady state for p = (0 , m ) T is nonlinearlyunstable.In [17] it is shown how to block-diagonalise the linearisation about the equi-librium with general p into so-called ‘classes’, and using this approach he againshowed that p = (0 , T is Lyapunov stable. This is used in [16] where the essen-tial and discrete spectrum of the linearisation of (1.1) are studied at the steadystate (1.2). They studied the full infinite system, approaching the problem froma functional analytic perspective. They found an upper bound on the number ofnon-imaginary isolated eigenvalues, and described the essential spectrum. Further-more, they showed that the spectrum of the linearised operator is the union of thespectrum coming from each of the classes from [17] and that the spectral mappingtheorem holds for the Euler operator linearised about Ω ∗ (Theorem 2 in [16]).In the viscous problem solutions e − νm t cos( mx ) are called bar states in [4]. Theyshow that for non-zero viscosity ν , and m = 1 these bar states are ‘quasi-stationary’,in that they decay on a slow timescale depending on the viscosity.We combine the block-diagonalisation used by [17] with the structure preservingfinite-dimensional sine truncation [24] and the Galerkin (see [18]) finite elementtruncation to prove that for a large class of p the stationary solutions (1.2) arenonlinearly unstable. Zeitlin’s sine truncation leads to a finite dimensional Poissonstructure and the Hamiltonian structure of the original PDE and its Casimirs arepreserved in this finite-dimensional truncation. The Galerkin truncation does notpreserve these Casimirs. See [2] and [14] for a discussion of the use of Poissonbrackets in hydrodynamics. The theoretical background of Zeitlin’s sine truncation(and a related truncation for a spherical domain) is discussed in [12], and theconcept of a “limit” of this algebra is discussed in depth in [6].In Section 2, the problem and the associated notation are introduced. The systemis first decomposed into Fourier modes which are described by a non-canonicalinfinite dimensional Hamiltonian system. Then truncation is taken to reduce to afinite-mode approximation. We linearise around the steady state, which decouplesthe problem into subsystems.In Section 3, we reproduce the “stable disc theorem” from [17] in the truncatedsetting. This theorem states that for classes whose mode numbers a satisfy | a | > | p | the spectrum is stable. Thus most class subsystems do not contribute unstablemodes to the spectrum of the full operator. Then we prove our “unstable disctheorem” 3.1, which states that if exactly one mode number of a given class is insidethe unstable disc then for sufficiently large N there is a positive real eigenvalue. Inour fundamental Theorem 3.5 we show that with certain additional assumptionsthis real eigenvalue is bounded away from zero when N → ∞ . Furthermore in NSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS 3
Lemma 3.6 we show that the corresponding eigenvector of the infinite dimensionalsystem is in ℓ .Section 4 provides the main Theorem 4.3 demonstrating non-linear instability ofthe stationary solution (1.2) for all choices of p but a few exceptions. In Lemma 4.2we establish when the conditions for the lower bound needed in Section 3 are met.Zeitlin’s truncation requires some care when proving results for both stable andunstable classes. Specifically it is not clear that the intersections between ourclasses and the disc | a | < | p | behave in the way we expect. In Lemma 4.1 we showthat for most choices of p , there is an appropriate truncation size N to control theZeitlin truncation so that Theorem 3.5 can be applied. The other cases of p can betreated using the Galerkin trunction.Section 5 provides some numerical results. The numerical efficiency and accu-racy of Zeitlin’s truncation is compared favourably to the Galerkin truncation. Aconnection is made between the nature of the subsystems and the number and typeof non-imaginary eigenvalues. A discussion of the number of non-imaginary eigen-values is included, and the accuracy of our calculated lower bound is assessed. Abrief section on the pure imaginary spectrum of our finite mode systems is included,replicating the results in [16] via a very different method. We can show that thepure imaginary spectrum of our finite dimensional approximation approaches theessential spectrum of the full system. As a result we can naturally define a densityof eigenvalues in the essential spectrum.2. Vorticity Evolution in Fourier Space, Truncation, Linearisation
Hamiltonian Formulation.
The stream function Ψ is defined through itsrelation to the fluid velocities by(2.1) u = + ∂ Ψ ∂x , u = − ∂ Ψ ∂x . The relationship between the stream function and the vorticity is(2.2) Ω = −∇ Ψand hence the PDE can be written as(2.3) ∂ Ω ∂t = ∂ Ω ∂x ∂ Ψ ∂x − ∂ Ω ∂x ∂ Ψ ∂x . For a fixed p ∈ Z and Γ ∈ R we wish to analyse the steady state Ω ∗ = α cos( p · x ) + β sin( p · x ).Note that we can write Ω ∗ = 2Γ cos( p · x + θ ), where θ = ± tan − (cid:16) − βα (cid:17) andΓ = ± √ α + β . The signs of θ and Γ will depend on the signs of α and β . If α = 0, then take θ = π . Define c = θ | p | p , so Ω ∗ = 2Γ cos( p · ( x + c )). Thusby taking the translation ˜ x = x + c we can instead just consider the steady stateΩ ∗ = 2Γ cos( p · ˜ x ) by a change of origin. Therefore for the remainder of this paperwe drop the tilde and simply consider the steady stateΩ ∗ = 2Γ cos( p · x ) . Expand Ω into a Fourier series with coefficients ω k ( t ) asΩ( x , t ) = X k ∈ Z ω k ( t ) e i k · x INSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS and combine (2.2) and (2.3). Then the Fourier coefficients are governed by theODEs(2.4) ˙ ω k ( t ) = X l ∈ Z \{ } k × l | l | ω − l ω k + l (where x × y = x y − x y for x , y ∈ R , and ˙ ω k := dd t ( ω k )). The condition ω k = ¯ ω − k is necessary for Ω to be real.Define the ‘ideal fluid’ Poisson Bracket in Fourier Space as(2.5) { f, g } = X k , l ∂f∂ω k ∂g∂ω l ( k × l ) ω k + l . The corresponding infinite dimensional Poisson structure matrix is J k , l = ( k × l ) ω k + l . Then (2.4) is a non-canonical Hamiltonian system with correspondingHamiltonian(2.6) H = X k ∈ Z \{ } ω + k ω − k | k | = 12 X k ∈ Z \{ } ω + k | k | . The Hamiltonian is obtained from the Kinetic energy H = 12 Z || u || d x = − Z ΩΨd x . Galerkin Truncation.
We now truncate to a finite mode approximation andstudy the spectrum of the equilibrium corresponding to Ω ∗ . We will present twoapproaches to this: a Galerkin-style finite element truncation, and a more sophis-ticated Poisson structure truncation by Zeitlin.First consider the Galerkin-style truncation (see [21]). Define the domain for ourtruncated Fourier modes(2.7) D = [ − N, N ] ∩ Z . Now set ω k = 0 , ˙ ω k = 0 for all k . Then the differential equations (2.4)define a finite set of ODEs, but not a Poisson system.2.3. Zeitlin’s Truncation.
An alternative truncation is that described by [24](see also [20], [12]). Restrict to the set of Fourier modes to(2.8) ω k , k ∈ D , and whenever a mode is referenced that is outside the domain D it is mapped backinto D . For this we have the notation ˆ k , which for any k denotes a mode ˆ k ∈ D for which the difference k − ˆ k = (2 N + 1)( a, b ) t for some integers a, b .Zeitlin gave the following Poisson bracket on the domain D : { f, g } = X k , l ∈D sin( ε k × l ) ε ∂f∂ω k ∂g∂ω l ω [ k + l , (2.9) J k , l = 1 ε sin( ε k × l ) ω [ k + l (2.10)where k , l ∈ D , and ε = π N +1 . The corresponding truncation of (2.6) is theHamiltonian(2.11) H = 12 X k ∈D\{ } ω + k ω − k | k | , NSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS 5 where only the domain of summation has changed.The vector field under the Zeitlin truncation is thus given by˙ ω k = ( J ∇H ) k = X l ∈ D J k , l ∇ H l (2.12) = 1 ε X l ∈ D sin( ε k × l ) ω [ k + l ω − l | l | . (2.13)The primary theoretical advantage of the Zeitlin truncation over the Galerkintruncation is that 2 N + 1 of the Casimirs present in the full system are preserved inthe Zeitlin truncated system. The disadvantage of the Zeitlin truncation is that wemust take some care when making arguments based on it (see Section 4). The twotruncations will be compared both numerically and analytically in later sections.For details of the construction and a description of the Casimirs see [24, 26].2.4. The Linearised system.
For a fixed p ∈ D \ { } the equilibrium point ofthe PDE Ω ∗ = 2Γ cos( p · x ) is an equilibrium point of the truncated ODE given by(2.14) ω ∗ l = (cid:26) Γ if l = ± p x ↔ − x , y ↔ − y , and x ↔ y , let p = ( p , p ) T with p ≥ p ≥ p > J a , b = ( a = b or b = (cid:16) | b | − | b − a | (cid:17) b × a ω a − b . The Jacobian of the Zeitlin truncated vector field (2.13) is(2.16) J ′ a , b = a = b or b = ε (cid:18) | b | sin( ε b × a ) + | \ ( b − a ) | sin( ε a × \ ( b − a )) (cid:19) ω \ ( a − b ) . Evaluating these at the equilibrium (2.14) gives the linearised systems(2.17) ˙ ω k = Γ (cid:18) | p | − | k + p | (cid:19) k × p ω k + p − Γ (cid:18) | p | − | k − p | (cid:19) k × p ω k − p for the Galerkin truncation and(2.18) ˙ ω k = Γ ε | \ k − p | sin( ε ( \ k − p ) × k ) + 1 | p | sin( ε p × k ) ! ω [ k − p + Γ2 ε | \ k + p | sin( ε ( \ k + p ) × k ) + 1 | p | sin( ε k × p ) ! ω [ k + p for the Zeitlin truncation.2.5. Decoupling into Classes.
The key observation is that ˙ ω k depends only ω k ± p . Thus the linearised systems can be block-diagonalised. This block-diagonalisationis analogous to the construction in [17]. Following Li we call the individual blocksclasses, which leads to the following definition: INSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS AB p = (3, 1)(0, 0) Figure 2.1.
The differential equations governing the set ofFourier Coefficients decouple into ‘classes’ when linearised. For a ∈ Z , ˙ ω a depends only on ω a + p and ω a − p . Extending this ideawe get a subset of coefficients that only depend on each other, theclass led by a . These coefficients all lie on a straight line with di-rection p . Classes that do not have an intersection with the discindicated are stable. Clases that intersect the shaded region indi-cated at a lattice point have a pair of real eigenvalues (Theorem3.1). Classes that intersect the disc but do not have intersect theshaded region at a lattice point lead to either a complex quadrupletor two pairs of real eigenvalues. Definition 2.1 (Classes) . For some a ∈ D (and p fixed by the choice of equilib-rium), the class Σ a ⊂ D is defined for the Galerkin truncation by(2.19) Σ a = { a + k p ∈ D | k ∈ Z } . or equivalently for the Zeitlin truncation(2.20) Σ ′ a = { \ a + k p ∈ D | k ∈ Z } . Figure 2.1 illustrates this idea. Note that the Zeitlin truncated classes Σ ′ a ‘wraparound’ the domain D .[17] makes the analogous definition for the non-truncated system. In that pa-per, the classes are infinitely large, and there are infinitely many classes. In ourdefinition, there are finitely many classes of finite size, which depend on the trun-cation size N . As D is finite, Σ a and Σ ′ a are both finite. Write p = ( p , p ) T , and κ = gcd( p , p ). Then(2.21) | Σ a | ≤ (cid:22) N + 1max( p , p ) (cid:23) NSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS 7 and(2.22) | Σ ′ a | = 2 N + 1gcd(2 N + 1 , κ ) . Note | Σ ′ a | does not depend on a , and is odd for all choices of N and p . We use thisfact many times later. For gcd( p , p ) = 1, | Σ ′ a | = 2 N + 1.For Σ a we can make a “canonical” choice of a by selecting a in a length | p | strip. For Σ ′ a a canonical choice for a is found by restricting a to be in a | p | by | Σ ′ a || p | rectangle centred around oriented so the sides of length | p | are parallel to p . Forthe Zeitlin truncation a unique choice is not possible if κ >
1. See Section 4 and4.1 for details.Fixing p and a canonical choice of a we now restrict our attention to the asso-ciated subsystems Σ a and Σ ′ a . Introduce new notation(2.23) ω k = ( ω a + k p if a + k p ∈D , ω ′ k = ω \ a + k p , (2.24) ρ k = 1 | p | − | \ a + k p | for k = ∈ Z , (2.25) α = Γ a × p ∈ R , α ′ = Γ sin( ε ( \ a + k p ) × p ) ε ∈ R . The value of the coefficient ρ k is related to the distance of the lattice point \ a + k p from the boundary of the disc of radius | p | , where negative values of ρ k correspondto lattice points inside the disc. Note that α does not depend on k . Also note that(2.26) lim N →∞ α ′ = lim ε → α ′ = Γ a × p = α. Noting that sin( ǫ ( \ k + p ) × k ) = sin( ǫ p × k ) and rewriting (2.17) and (2.18) withthe new notation gives a compact form of the linear systems˙ ω k = α ( ρ k +1 ω k +1 − ρ k − ω k − ) , (2.27) ˙ ω ′ k = α ′ ( ρ k +1 ω ′ k +1 − ρ k − ω ′ k − ) . (2.28)All indices in (2.28) are written modulo | Σ ′ a | .Let m , m ∈ Z be such that a − m p ∈ D , a − ( m + 1) p and a + m p ∈ D , a + ( m + 1) p . If we write ω = (cid:0) ω − m , ω − m +1 , ..., ω − , ω , ω +1 , ..., ω m (cid:1) T ,then ˙ ω = αA ω where(2.29) A = ρ − m +1 · · · − ρ − m ρ − m +2 · · · − ρ − m +1 · · · · · · ρ m −
00 0 0 · · · − ρ m − ρ m · · · − ρ m − . INSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS
If we write ω ′ = (cid:0) ω ′ , ω ′ , ..., ω ′ n − (cid:1) T , then ˙ ω ′ = α ′ A ′ ω ′ where(2.30) A ′ = ρ · · · − ρ n − − ρ ρ · · · − ρ ρ · · · − ρ · · · · · · ρ n − + ρ · · · − ρ n − . If a = or a is parallel to p , α = α ′ = 0. Thus the associated class onlycontributes zero eigenvalues and will not contribute to the linear instability of thesystem. We can thus ignore the classes with α = α ′ = 0.Note that A can be written as A = JS where(2.31) J = · · · − · · · − · · · · · · · · · − , S = ρ − m · · · ρ − m +1 · · ·
00 0 ρ − m +2 · · · · · · ρ m . A similar construction can be made for A ′ = J ′ S ′ . As J and J ′ are skew-symmetric and S and S ′ are symmetric, these are (non-canonical) Hamiltoniansystems. For both systems, H = P k ρ k ω k . From this it follows that if λ is aneigenvalue of A or A ′ then − λ , ¯ λ and − ¯ λ are also eigenvalues. Note that det( J ′ ) = 0as J ′ has odd size, and therefore J ′ is not symplectic with a one-dimensional kernel. J is symplectic if and only if | Σ a | is even.We now focus on the behaviour of the eigenvalues of A and A ′ as a function ofthe ρ k values. Note that there is a symmetry in a . For every class Σ a or Σ ′ a , theclass Σ − a or Σ ′− a generates the same set of eigenvalues. It is worth noting that α ( − a ) = − α ( a ) and α ′ ( − a ) = − α ′ ( a ), but as all eigenvalues occur in ± pairs, thisdoes not affect the spectrum. Thus for the full system rather than a particular classall eigenvalues occur with even multiplicity. Definition 2.2 (The Unstable Disc) . Introduce the disc D p (2.32) D p = { x ∈ D | | x | < | p |} . This disc is shown in figure 2.1. A simple but important observation is
Lemma 2.3.
A lattice point is inside the unstable disc if and only if the corre-sponding ρ is negative: (2.33) \ a + k p ∈ D p ⇐⇒ ρ k < . Proof.
This is true as ρ k < | a + k p | < | p | from (2.24), which isexactly the condition that \ a + k p ∈ D p . (cid:3) As J ′ is circulant, one could write down its eigensystem explicitly by applying a discreteFourier transform (see [13]) and therefore find a set of canonical coordinates (see [8]). This willnot be used in this paper, but may be of interest. NSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS 9
This is illustrated in figure 2.1. The point a inside the disc corresponds to ρ < ρ k >
0. Also note that \ a + k p ∈ ∂D p ⇐⇒ ρ k = 0 . Stability and Instability of Classes
Stable Classes.
The matrices A and A ′ are similar to skew-symmetric ma-trices by conjugation.(3.1) T = √ ρ − m · · · √ ρ − m +1 · · ·
00 0 √ ρ − m +2 · · · · · · √ ρ m , (3.2) T AT − = √ ρ − m ρ − m +1 · · · −√ ρ − m ρ − m +1 √ ρ − m +1 ρ − m +2 · · · −√ ρ − m +1 ρ − m +2 · · · · · · . A very similar construction exists for A ′ .If ρ k > k , this transformation is real and thus A and A ′ are similarto real skew-symmetric matrices. Thus all eigenvalues are purely imaginary and A and A ′ can be diagonalised and so the class is linearly stable. By (2.33) thiscondition is true exactly if(3.3) Σ a ∩ D p = ∅ . This is the finite-dimensional analogue of Li’s
Unstable Disc Theorem (TheoremIII.1) in [17], though the method of proof used in that paper is naturally verydifferent. A discussion of the details such as choice of p and N required for (3.3)to hold follows in Section 4. Because of this result, only classes with a ∈ D p cancontribute linear instability. Also a = implies α = α ′ = 0 and so this class cannotcontribute linear instability.3.2. Unstable Classes.
For classes with a ∈ D p , there are two primary possibili-ties to consider:i) There is exactly one intersection between the class and the disc (ie, Σ a ∩ D p = { a } ). This can only occur when a is chosen to be in the shaded area indicatedin figure 2.1.ii) There are exactly two consecutive intersections between the class and the disc(ie,Σ a ∩ D p = { a , a + p } or Σ a ∩ D p = { a , a − p } ). This occur when a ∈ D p ischosen outside the shaded area indicated in figure 2.1.For the Zeitlin style truncation, there is also a third possibility we must consider:iii) There are at least two non-consecutive intersections between the class andthe disc (ie, a ∈ Σ ′ a ∩ D p and \ a + k p ∈ Σ a ∩ D p for some k = − , , Note that points on the boundary are treated as being outside the disc. Also notethat it is not possible for three consecutive lattice points in a class to be in theunstable disc. If a , a − p and a + p were are all in D p , they would lie along adiameter as D p has diameter 2 | p | and the distance from a − p to a + p is 2 | p | .Therefore a = (0 ,
0) and a ± p ∈ ∂D p . This is the only possibility to have threeconsecutive lattice point in D p , and hence D p can at most contain two consecutivelattice points. Figure 2.1 makes this idea clear.From our numerical and analytical results we can categorise the spectrum of theclass in these three cases:i) The spectrum has a single pair of real eigenvalues and all other eigenvalues onthe imaginary axis. This is proved in Theorem 3.1.ii) The spectrum typically corresponds to a quartet of complex eigenvalues ± α ± βi , and all other eigenvalues on the imaginary axis. It can also correspond totwo pairs of real eigenvalues, though seems to be less common.iii) This corresponds to the class ‘wrapping around’ the truncated domain of latticepoints and intersecting the disc again; see 4.1. The spectrum is a combinationof case (i) and case (ii) according to how successive intersections with the discoccur.This last case is atypical and does not occur with the Galerkin truncation. Usuallythis case can be avoided by a proper choice of N , however when both entries of p are even, it cannot be avoided. This is discussed in detail in Section 4, particularlyLemma 4.1.All our numerical evidence is consistent with the result in [16] that the number ofeigenvalues with non-zero real part is ≤ | D p | (twice the number of interior latticepoints in the unstable disc), and our observation is that this is the exact number ofhyperbolic eigenvalues.For case (i), Theorem 3.1 proves that this case always leads to non-zero realeigenvalues. For some p and a leading to case (i), Theorem 3.5 describes an explicitlower bound which is independent of the truncation size N for the real eigenvalues.This agrees with numerically observed results, eg the ǫ = 0 inviscid result shown inFigure 2 of [15]. It should be noted that our methods do not preclude the possibilitythat there are other eigenvalues with non-zero real part unaccounted for; we simplyassert that there is at least one eigenvalue with positive real part. This togetherwith the results from [10], [16], and [22] is sufficient to conclude nonlinear instabilityfor the whole system.For the following section, we consider a general set of parameters ( a , a , ..., a n − )instead of ρ k . Introduce a tridiagonal matrix(3.4) T βα = a α +1 · · · − a α a α +2 · · · − a α +1 a α +3 · · · − a α +2 · · · − a α +3 · · · · · · a β · · · − a β − , NSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS 11 and its characteristic polynomial(3.5) T βα ( x ) = det( xI − T βα )for some integers 0 ≤ α < β ≤ n −
1. Then T βα ( x ) can be recursively defined byexpansion from top left to bottom right T αα ( x ) = 1 , T α +1 α ( x ) = x + a α +1 a α , T βα ( x ) = x T β − α ( x ) + a β a β − T β − α ( x ) . (3.6)or by expansion from bottom right to top left T ββ ( x ) = 1 , T ββ − ( x ) = x + a β − a β , T βα ( x ) = x T βα +1 ( x ) + a α a α +1 T βα +2 ( x ) . (3.7)Note that(3.8) a k > α ≤ k ≤ β = ⇒ T βα ( x ) > x > . This can be seen by the recursive definitions; all terms are positive. The following is also useful:(3.9) T βα (0) = (Q βk = α a k if β − α is odd,0 if β − α is even.(3.10) dd x T βα ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x =0 = β − α is odd, P β − α k =0 (cid:16)Q βj = α ; j = α +2 k a j (cid:17) if β − α is even.These can be proved by simple induction arguments.Introduce similar notation for (2.30)(3.11) A ′ = a · · · − a n − − a a · · · − a a · · · − a · · · · · · a n − + a · · · − a n − , with the characteristic polynomial(3.12) A ( x ) = det( xI − A ) . Then, if n is odd, A ( x ) = x T n − ( x ) + a n − a n − T n − ( x ) + a n − a T n − ( x )= T n − ( x ) + a a n − T n − ( x )(3.13)This can be demonstrated by expanding by minors along the last row and columnof xI − A . Recall that n is odd for the relevant problem ( n = | Σ a | from equation(2.22)). Note that the (3.6) and (3.7) satisfy the condition for Favard’s theorem (see [9]) . Thus thepolynomials T ( x ; a , ...a j ) are orthogonal for j = 1 , , , ... with respect to an inner product withsome weight function (see [23]). However, as the α k terms may be negative this weight functionwill not always be positive. This will not be used here, but may be useful in future work fordescribing the imaginary part of the spectrum. We first show that in case (i) there is some non-zero real eigenvalue. Becauseof the Hamiltonian nature of the system this means there is a plus/minus pair ofeigenvalues and so there is linear instability. This is then extended to show thatunder certain conditions there is pair of real eigenvalues with an explicit lowerbound independent of N . Theorem 3.1 (Real Eigenvalues in case (i)) . If ρ < , and ρ k ≥ for all k =1 , , ..., n − and ρ k = 0 for at most one of k = 1 , , ..., n − , then for sufficientlylarge N , (2.30) has a non-zero real eigenvalue.Proof. The characteristic polynomial A for odd n has leading term x n and constantterm 0. By combining (3.13) and (3.10) (noting that n is odd, so n − n − A d x (cid:12)(cid:12)(cid:12)(cid:12) x =0 = dd x T n − ( x ) (cid:12)(cid:12)(cid:12)(cid:12) + a a n − T n − ( x ) (cid:12)(cid:12) = n − X k =0 n − Y j =0; j =2 k a j + n − X k =0 n − Y j =1; j =1+2 k a j = n − X k =0 n − Y j =0; j = k a j (3.14) = n − Y j =0 a j n − X k =0 a k ! . (3.15)Note that (3.15) is only valid for a k = 0 for all k , but (3.14) is always valid . Nowlet a k = ρ k . First assume ρ k > k = 0. As ρ k → | p | > | \ a + k p | → ∞ and the size n of the classes grows linearly with N , P n − j =0 1 a j = P n − j =0 1 ρ j > N (where ρ k = a k ). However Q n − j =0 a j < a < a j > j = 1 , .., n −
1, and hence the linear coefficient of the characteristic polynomialis less than zero.If ρ k = 0 for exactly one k , then (3.14) consists of only one term, Q n − j =0; j = k ρ j .This is less than zero as ρ < ρ j > j = 0 , k .If ρ k = 0 for more than one value of k , then the linear term is zero and we cannotapply this argument.As the constant term is zero, and the linear term is non-zero, then the lowestorder non-zero coefficient of the polynomial is negative.We now argue by contradiction. Assume all roots of the polynomial are imag-inary (say iω k ) or complex ( γ j + iδ j ) or zero. Then because eigenvalues occur in The expression in equation (3.14) is the so-called n -1 st elementary symmetric polynomial inthe variables a j NSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS 13 positive and negative pairs as well as conjugate pairs the polynomial has the form A ( x ) =(3.16) x n n Y k =1 ( x − iω k )( x + iω k ) n Y j =1 [( x − γ j − iδ j )( x − γ j + iδ j )( x + γ j − iδ j )( x + γ j + iδ j )]= x n n Y k =1 ( x + ω k ) n Y j =1 (cid:0) x − x ( γ j − δ j ) + ( γ j + δ j ) (cid:1) . (3.17)The lowest order non-zero coefficient (of x n ) is Q n k =1 ( ω k ) Q n j =1 (cid:0) ( γ j + δ j ) (cid:1) > (cid:3) For the Galerkin case (3.4), although the result is the same we cannot apply thesame proof. Since we were not able to show that these eigenvalues remain boundedaway from the imaginary axis when N → ∞ we now show in a different way thatfor both truncations there are eigenvalues whose real part is bounded away fromzero for N → ∞ . These proofs have stricter conditions on the ρ k then Theorem 3.1,but we will see that they can be met for most p . Lemma 3.2 (Lower Bound for Real Eigenvalue (Zeitlin)) . If a < , and a k > for all k = 0 , and a + a < , then (3.18) A ( p − a ( a + a )) < . Proof.
By expanding (3.13) using (3.7), A ( x ) =( x + ( a a + a a ) x ) T n − ( x )(3.19) + ( x + a a ) a a T n − ( x )+ a a n − T n − ( x ) . Thus A ( p − a ( a + a )) = − a a a T n − ( p − a ( a + a ))(3.20) + a a n − T n − ( p − a ( a + a )) . As a , a , a , a n − > a < T terms are positive by (3.8) the resultfollows. (cid:3) We make a similar case for the Galerkin truncation
Lemma 3.3 (Lower Bound for Real Eigenvalue (Galerkin)) . If β > , a < , and a k > for all k = 0 , and a + a < , then (3.21) T βα ( p − a ( a + a )) < . Proof.
Begin by noting that(3.22) T γα ( p − a ( a + a )) ≥ for all γ < T α ( x ) = ( x + a a ) T − α ( x ) + a a − x T − α ( x )(3.23) T α ( p − a ( a + a )) = − a a T − α ( p − a ( a + a ))(3.24) + a a − p − a ( a + a ) T − α ( p − a ( a + a )) . As a < a k > k = 0, and T − α , T − α take positive values for positivearguments, T α ( p − a ( a + a )) < T α ( p − a ( a + a )) =( x + a ( a + a )) x T − α ( p − a ( a + a ))(3.25) + ( x + a a ) a a − T − α ( p − a ( a + a ))= − a a a − T − α ( p − a ( a + a )) . As a , a − > a < T terms are positive by (3.8) it follows that T α ( p − a ( a + a )) < . Now if γ >
2, we can make a recursive argument:(3.26) T γα ( x ) = x T γ − α ( x ) + a γ a γ − T γ − α ( x ) . Now x = p − a ( a + a ) > a γ a γ − > γ > T α < , T α <
0, then T γα < γ >
2. Therefore T βα ( p − a ( a + a )) < (cid:3) We can also make a similar construction if a < a k ≥ k = 0 and a + a n − <
0. In this case, A ( p − a n − ( a + a n − )) ≤ Lemma 3.4.
Assuming the same conditions as Lemma 3.2 and Lemma 3.3, thereexists some x ∗ , x ∗ > p − a ( a + a ) such that T βα ( x ∗ ) = 0 , A ( x ∗ ) = 0 .Proof. The leading order term of A ( x ) is always x n regardless of a , ..., a n − , and A ( x ) is real for real x . Thus lim x →∞ A ( x ) >
0. But by Lemma 3.2 A ( p − a ( a + a )) <
0, and the result follows by the intermediate value theorem.The same argument can be applied to T βα ( x ), with the equivalent result. (cid:3) We now turn our attention back to the context of our problem.
Theorem 3.5 (Lower Bound for Real Eigenvalues in case (i)) . If a ∈ D p and a + k p ¯ D p (equivalently \ a + k p ¯ D p ) for k = 1 , , .., n − and (3.27) λ † = p − ρ ( ρ + ρ ) is real, there exist λ , λ > λ † such that λ is an eigenvalue of (2.29) and λ is aneigenvalue of (2.30) .Similarly if λ † = p − ρ n − ( ρ + ρ n − ) is real there exist λ , λ > λ † such that λ is an eigenvalue of (2.29) and λ is an eigenvalue of (2.30) .Proof. This follows from the previous three lemmas, letting ρ i = a i and makingnote of (2.33). (cid:3) NSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS 15
Section 4 clarifies under what conditions λ † is real, and Theorem 3.5 holds. Forexample, when p = (1 , T then a = (0 , T leads to a case (i) class, so Theorem 3.1holds, but the reality conditions in Theorem 3.5 are not satisfied. It is also possiblethat λ † = 0, so the real eigenvalue could be zero, so we also require that λ † >
0. Asufficient condition on a for this to hold is given in Lemma 4.2.3.3. Associated Eigenvector.
We must also consider whether this eigenvalue is“valid” in the sense that the corresponding eigenfunction in the full system is infact a function in L ( T ), the square integrable function on the torus. For this weneed to show that the Fourier coefficients are in ℓ . In this section we are going towork with the infinite dimensional system. The reason for this is that although thetruncations are useful in the calculation of eigenvalues, it is simpler to analyse thecorresponding eigenvectors in the full system. Thus our approach is to use the finitedimensional Zeitlin truncation to compute the eigenvalues, then take the limit as N → ∞ , and then study the decay of the corresponding eigenvector.Consider the linearised matrix for the full system(3.28) M = . . . ... ... ... ... . . . · · · ρ · · ·· · · − ρ − ρ · · ·· · · − ρ ρ · · ·· · · − ρ · · · . . . ... ... ... ... . . . . This is the limit in some sense of the matrices (2.29) and (2.30) which correspondsto the limit of the Hamiltonian systems A = JS and A ′ = J ′ S ′ with Hamiltonian H ( ω ) = P k ρ k ω k . Lemma 3.6.
The infinite dimensional linearised system given by (3.28) has apositive real eigenvalue λ under the same conditions as Theorem 3.5. Furthermore,the associated eigenvector is in ℓ .Proof. According to Theorem 3.5, there exist positive real eigenvalues with somelower bound (independent of N ) of (2.30) (and (2.29)) for any N and either choiceof truncation. By taking N → ∞ we can conclude there exists some positive realeigenvalue λ of (3.28).Now consider an eigenvector associated with this eigenvalue, v = ( ..., v − , v , v , v , ... ) T .For this to correspond to a real L eigenfunction of the full problem (that is, for theFourier series to converge), we need these Fourier coefficients to decay sufficientlyfast: they need to be a sequence in ℓ .The entries of the (infinite dimensional) eigenvector v k of (3.28) correspondingto eigenvalues λ satisfy the recursion relation(3.29) λv k = ρ k +1 v k +1 − ρ k − v k − . Since all ρ k = 0 this can be rewritten as(3.30) v k +1 = λρ k +1 v k + ρ k − ρ k +1 v k − . Consider the limiting behaviour as k → ∞ . then ρ k → | p | . In this limitsolutions to (3.30) behave like solutions to(3.31) v k +1 = λ | p | v k + v k − , see, e.g., [11].This linear recurrence has the general solution(3.32) v k = C µ k + C µ k where C , C ∈ R are constants and µ , µ are solutions to µ − λ | p | µ − µ µ = − | µ | < | µ | > | µ | = | µ | = 1 as λ | p | = 0).Now as v is an eigenvector associated with a real eigenvalue, the span of theeigenvector is an invariant subspace of the Hamiltonian system with Hamiltonian H ( ω ) = P k ρ k ω k . In fact, let ω (0) = v , then ω ( t ) = e λt v . As the Hamiltonianis an integral of the motion, H ( v ) = H ( e λt v ). By taking the limit t → −∞ , H ( v ) = H (0) = 0.Therefore,(3.33) H ( v ) = X k ρ k v k = 0;(3.34) X k =0 ρ k v k = − ρ v . Now, if C = 0,(3.35) X k =0 ρ k v k ∼ X k =0 ρ k ( C µ k ) → ∞ , recalling that ρ k → | p | and ρ k > k = 0. But | ρ | < v k is finite,so there is a contradiction. Thus C = 0 and v k = C µ k in the asymptotic limit,where | µ | <
1. This is exponential decay, which is sufficient for the Fourier seriesto converge.Similarly for k → −∞ , the limiting behaviour is governed by(3.36) v k − = − λ | p | v k + v k − . Again, this means v k is asymptotic to C µ k + C µ k for | µ | < | µ | >
1. Bythe same argument as above, we can conclude that C = 0 and so v k = C µ k as k → ∞ . Thus the Fourier coefficients decay exponentially on both sides with | k | ,and hence v is in ℓ . (cid:3) Instability of Equilibria
To prove instability of an equilibrium for a given p we need to find at least oneunstable class Σ a / Σ ′ a . A necessary condition for a lattice point a to lead to anunstable class is to be inside the unstable disc. More precisely we desire a latticepoint a that leads to a class of case (i) as this is the simplest situation for us todeal with. Theorem 3.5 asserts that there is a real eigenvalue with an explicit lowerbound under some certain conditions. The goal now is to determine for which p there exists a lattice point a such that the conditions of Theorem 3.5 are satisfied.There are a number of other considerations when we take Zeitlin’s truncation.Although this preserves the geometric structure and the Casimirs of the originalproblem, it introduces a problem that is not present in [17]. This has already beenmentioned in the classification of classes; it is the appearance of case (iii), in which aclass intersects the unstable disc at non-consecutive points. This may occur becausenow we have periodic boundary conditions not only in physical space, but also in NSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS 17 N +1 | p | a \ a + k p a \ a + ( k + 1) p a + p Figure 4.1.
For fixed p and a , there are some concerns with thewrapping of the Zeitlin truncation and the way this affects theintersection Σ a ∩ D p . These situations are discussed in Lemma4.1. Left: the shortest distance between a and some non-consecutive \ a + k p is at least N +1 | p | . As we are interested in the limit N → ∞ ,this distance can be made arbitrarily large, so that N +1 | p | > | p | .This ensures that a and \ a + k p cannot both be in the disc D p . Right:
The situation where there exists k ∈ R such that \ a + k p lies on the line segment between a and a + p . This causes problemsfor our values of ρ . For \ a + k p to lie at a lattice point on the linesegment, gcd( p , p ) >
1. If gcd( p , p ) is odd, we can avoid thissituation by choosing N per Equation (4.1); if p and p are botheven this situation is unavoidable.Fourier space. There are two distinct problems caused by this, as illustrated infigure 4.1.The lattice points of a class lie on parallel line segments with direction vector p inthe domain of Fourier modes. In the Zeitlin truncation there is more than one suchline segment in the domain. The first problem appears when the distance betweenthese line segments is so small that more than one line segment intersects theunstable disc. This can be fixed by making N sufficiently large. The second problemoccurs when non-consecutive lattice points lie on the same line segment intersectingthe unstable disc. If gcd( p , p ) is not even this can be fixed by choosing N per(4.1). Note that for our purposes, gcd( p ,
0) = p and gcd( p , p ) = gcd( | p | , | p | ). Lemma 4.1 (Correct choices of N for the Zeitlin truncation) . For all p = ( p , p ) T such that κ = gcd( p , p ) is not even, there exists a sequence of N which increaseswithout bound such that for all choices of a any two non-consecutive lattice pointsin Σ ′ a cannot both be in the unstable disc.Proof. Let(4.1) N = (2 ˜ N + 1) κ − for A ∈ N . Thus 2 N + 1 = (2 ˜ N + 1) κ . If κ is not even, then such an N is a positiveinteger and thus a valid grid size. Select as a lower bound ˜ N > | p | − κ κ so that N > | p | − . p is fixed and finite so this lower bound is always finite. We can thusfind an infinite sequence of N that increases without bound by letting ˜ N increasewithout bound.If x ∈ Σ a then x = \ a + k p for some k ∈ N . Thus x lies on the line parallel tothe vector p that passes through the point a + ∆ x (2 N + 1) for some ∆ x ∈ Z (notethat this point will be outside the domain D ).Similarly if y ∈ Σ a then it lies on the line parallel to p that passes through a + ∆ y (2 N + 1) for some ∆ y ∈ Z . Then the distance between these two lines is(4.2) d = | (( a + ∆ x (2 N + 1)) − ( a + ∆ y (2 N + 1))) × p || p | = (2 N + 1) | ( ∆ x − ∆ y ) × p || p | . ( ∆ x − ∆ y ) × p ∈ Z , so d = 0 or d ≥ N +1 | p | . If d ≥ N +1 | p | , this corresponds to x and y lying on different line segments, as in figure 4.1. Thus the distance betweentwo points on different line segments is at least N +1 | p | > | p | for our choice of N .If d = 0, then x and y must lie on the same line segment. Thus y lies on theline parallel to p passing through x .Write p = κ q so q = ( q , q ) T where gcd( q , q ) = 1. Then as x , y ∈ Z , y = x + k q for some k ∈ Z . But x , y ∈ Σ a so y = x + j p + (2 N + 1) ∆ for some ∆ ∈ Z .Thus k q = j p + (2 N + 1) ∆ . So(4.3) k q = jκ q + (2 ˜ N + 1) κ ∆ = κ ( j q + (2 ˜ N + 1) ∆ ) . Thus k q is divisible by κ , but the elements of q both be divided by κ by definition,so κ | k . Thus y = x + κβ q for some β ∈ Z , and so y = x + β p . Then if | y − x | < | p | (the necessary condition for x , y both in D p ) this implies β = 0 or β = ±
1, andthe result follows. (cid:3)
The outstanding issue with Zeitlin’s truncation is that there is no appropriatechoice of N when gcd( p , p ) is even. If κ > κ, N + 1) = 1, then p will generate all multiples of q due to the wrapping operation. Thus classes thatintersect the disc can return after leaving the disc and intersect the disc again,breaking the assumption of 3.2 that ρ k < k . This behaviourcontinues for all values of N with gcd( κ, N + 1) = 1. If κ is even, this is true forany N ; if κ is odd, we select appropriate N to avoid this.It is important to note that this is not an error per se, but merely a failureof Lemma 4.1 for our proof. The wrapping of the Zeitlin truncation associatesmodes in an artificial way, but still generates correct results. For instance if p =(6 , T = 2(3 , T , the class led by a = (1 , T intersects the unstable disc again at( − , T = (1 , T − (3 , T for any finite truncation size. If we compare the non-imaginary eigenvalues of the class led by a = (1 , T with those of the Galerkin-truncated systems for a = (1 , T and a = ( − , T , the same eigenvalues aregenerated, with similar convergence to Figure 5.2.Fortunately, this problem does not arise with the Galerkin truncation as thewrapping operation is omitted, and so the proof goes through without issue. Forboth truncations it is still necessary to establish whether for a given p there is an a such that the conditions of Theorem 3.5 are met. We address that now. NSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS 19 ( √ − ) | p | p (0 , Figure 4.2. If a is inside the dashed blue circle centred at theorigin, then ρ + ρ < ρ + ρ n − <
0. This circle has centre(0 ,
0) and radius ( √ − | p | . The shaded region here shows theoverlap of this condition and the shaded region in figure 2.1. Toshow that there is at least one lattice point in the shaded regionswe show the disc D c inscribed in this region (indicated by the smallshaded red circles) has radius larger than / √ . Lemma 4.2.
For all p = ( p , p ) T except (1 , T , (1 , T , (1 , T (and permutationsand sign changes thereof ) there exists a choice of a such that the reality conditions ofTheorem 3.5 are satisfied for an appropriate choice of N in the Galerkin truncation.Furthermore, if κ = gcd( p , p ) is not even there is also a choice of a such that theconditions of Theorem 3.5 are satisfied for the Zeitlin truncation.Proof. For the bounds given in Theorems 3.3 and 3.2 to be real, positive, andhence a valid bound, we require ρ < ρ k ≥ k = 0, and ρ + ρ < ρ + ρ n − < ρ < , ρ , ρ − >
0, then | a | < | p | and | a ± p | > | p | . This is true if and onlyif a is in the shaded region figure 2.1. In the Galerkin truncation, this is sufficientto show that ρ k > k = 0. By Lemma 4.1, in the Zeitlin truncation wecan find an unbounded sequence of choices of N such that this is sufficient to show k = 0 , ρ k >
0. Thus for an appropriate choice of N we only need to prove thatthere exists an a such that | a ± p | > | p | .As λ † = p − ρ ( ρ + ρ ) (or equivalently λ † = p − ρ − ( ρ + ρ − )) is required tobe real and non-zero, and ρ − >
0, then ρ + ρ < ρ + ρ − < If | a | < ( √ − | p | , then | a ± p | ≤ | a | + 2 | p | < ( √ | p | . So ρ + ρ ± < | p | − | ( √ − p | − | ( √ p | (4.4) = 0 . We thus need to show there exists some lattice point a such that | a | < ( √ − | p | and | a ± p | > | p | . These two conditions are illustrated in figure 4.2 in the shadedregion. Note that this is sufficient but not necessary for Theorem 3.5 to hold.The idea now is to specify that a is in the disc inscribed by the shaded regionin figure 4.2, which we call D c . This disc is tangent to the circles with radii | p | and centres ± p and the circle centred at the origin with radius ( √ − | p | . It isa simple geometric exercise to show that such a circle has centre ± √ (cid:18) − p p (cid:19) andradius (cid:16) √ − (cid:17) | p | .If a ∈ D c , it is outside the circle with centre − p and radius | p | . Thus a + p isoutside D p and ρ >
0. Similarly, ρ − >
0. As D c is inside the disc with centreorigin and radius (cid:16) √ − (cid:17) | p | clearly ρ < ρ + ρ ± <
0. Ifwe take a Zeitlin truncation, choose appropriate N such that ρ k ≥ k = 0by Lemma 4. Then the conditions of Theorem 3.5 are satisfied and the resultingbound λ † is real and positive.All that remains is to show that there exists an integer lattice point a ∈ D c .Any disc with a radius greater than √ must contain some integer lattice point (asit wholly contains a square of side length 1). Thus if(4.5) | p | > √ √ − √ ≈ . . then the D c has radius greater than √ p .Checking the small number of p values with | p | < .
57 and κ odd there are ap-propriate lattice points a for most such p . The following table shows an appropriatevalue for a for most such p , and “None” where no such a exists.(4.6) p (4 , T (3 , T (3 , T (3 , T (3 , T (2 , T a (1 , − T (1 , − T (1 , − T (1 , − T (0 , T ( − , T For reflections/rotations of these values of p the corresponding reflection/rotationof a is an appropriate choice.Thus for all p = ( p , p ) T such that κ = gcd( p , p ) except except (1 , T , (1 , T , (1 , T and reflections and rotations of these, there is a choice of a so that the conditionsof Theorem 3.5 is satisfied when N = (2 ˜ N +1) κ − for any ˜ N > | p | − κ κ . (cid:3) Now we can combine the results about eigenvalues and eigenvectors from thelast section and the conditions on p when they are applicable in our main Theorem 4.3.
The steady state Ω ∗ = α cos( p · x ) + β sin( p · x ) is nonlinearlyunstable for all p = ( p , p ) T except p = (cid:0) ± , (cid:1) T , p = (cid:0) , ± (cid:1) T and possibly p = (cid:0) ± , ± (cid:1) T , p = (cid:0) ± , ± (cid:1) T , p = (cid:0) ± , ± (cid:1) T . NSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS 21
Proof.
By Lemma 4.2, for all p except those listed above there exists some a suchthat ρ < ρ + ρ < ρ + ρ n − <
0) for an appropriate choice of N . Thus by Theorem 3.5 there exists a real positive eigenvalue λ . Moreover, theeigenvalue is greater than p − ρ ( ρ + ρ ) (or p − ρ n − ( ρ + ρ n − )) which is bothpositive and independent of the choice of truncation size N . The truncation size N can be increased without bound, by Lemma 4.1. Hence there is a hyperboliceigenvalue in the limit N → ∞ and the spectrum of the PDE is unstable. Nowrecall that any steady state Ω ∗ = α cos( p · x ) + 2 β sin( p · x ) can be rewritten asΩ ∗ = 2Γ cos( p · ˜ x ) and so the full result follows.By Lemma 3.6, the eigenvector associated with the eigenvalue λ is in ℓ . Theclasses led by a and − a have the same eigenvalue, and the the corresponding eigen-vectors can be combined to construct coefficients ω k of a real eigenfunction Ω λ corresponding to λ . Since the eigenvectors v are in ℓ the periodic function Ω λ is in L . Together with the result in [16], which shows that the spectral mappingtheorem holds, establishes linear instability. To conclude nonlinear instability werefer to the work of [10] and [22]. In [10] it was shown that sufficient conditions fornonlinear instability are linear instability together with a ‘spectral gap’ condition.In [22] it was shown that the essential spectrum of the linearised Euler operator inthe cases we are considering is i R . Because of the presence of a point of discretespectrum bounded away from the imaginary axis, we have a spectral gap, and hencenonlinear instability. (cid:3) Note that this does not preclude the possibility that the values of p listed asexceptions do not also lead to a linearly unstable steady state Ω ∗ . In fact, for p = (cid:0) , (cid:1) T , (cid:0) , (cid:1) T and reflections/rotations thereof numerical results find non-zero non-imaginary eigenvalues. For p = (cid:0) , (cid:1) T there is one complex quadrupletof eigenvalues, ± . ± . i to five decimal places, calculated with N = 1500and Γ = 1. For p = (cid:0) , (cid:1) T there are two real pairs and two complex quadrupletsof eigenvalues.Theorem 4.3 together with the numerical results mentioned and the spectralmapping theorem shown in [8] indicate that the only linearly stable equilibriumof type (1.2) is p = (cid:0) ± , ± (cid:1) T , (cid:0) ± , ± (cid:1) T , because only these have the singlelattice point a = (0 , T inside the unstable disc. This exceptional case leads onlyto zero eigenvalues. All lattice points outside the unstable disc (including thoseon the boundary) do not contribute to instability (see section 3.1) and hence thisequilibrium is spectrally stable. By [14] it is then linearly stable, and by [2] and[17] it is also Lyapunov stable.5. Some Numerical Results
The Unstable Spectrum.
The Zeitlin class decomposition means that wenow typically compute the eigenvalues of (2 N + 1) matrices, each of size (2 N +1) × (2 N + 1). Without the class decomposition, the eigenvalues of one (2 N + 1) × (2 N + 1) matrix need to be computed. So the class decomposition results in anextremely significant saving of computation time. For a Galerkin truncation, thiscomputational saving is even more pronounced. However, this is at the expense ofaccuracy (see figure 5.2). Figure 5.1.
All eigenvalues for the case p = (5 , T , Γ = .The Zeitlin truncation is used with N = 200. Note there are 200(or 100 plus-minus pairs) eigenvalues with non-zero real part, andthere are 100 interior lattice points in D p \ { } . This confirmsthe result from [16]. Of the non-imaginary eigenvalues, 56 are realand 144 are complex. The number of interior points that satisfy ρ < ρ , ρ − > ρ < ρ < ρ − <
0. This usually createsa complex quadruplet but in a few cases corresponds to two realpairs instead. Increasing N does not change the number of non-imaginary eigenvalues.Figure 5.1 shows all the eigenvalues associated with a fixed value of p and N .There are exactly twice the number of interior lattice points in D p \ . This agreeswith the result in [16] that the discrete spectrum of the corresponding operator hasat most 2 | D p | − p tested there is equality.Figure 5.2 shows the values at which the calculated eigenvalues converge as afunction of the size N of our truncation domain D . Compared to the Galerkin trun-cation the eigenvalue converges for much smaller values of N when using Zeitlin’smethod.Figure 5.3 shows the correspondence between the location of values of a and thetypes of eigenvalues of the class Σ ′ a . This corresponds to the results of Section 3.Compare the positioning of the Fourier modes with figure 2.1.5.2. The Stable Spectrum.
Figure 5.4 shows the density of the imaginary partsof the spectrum for a = ( − , T , p = (7 , T . There are also non-imaginaryeigenvalues but these are not shown on the figure.For any ε >
0, we can choose sufficiently large N so that there exists some a such that | b | < ε for all b ∈ Σ ′ a . So the imaginary spectrum of this class can NSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS 23
20 40 60 80 100 N λ Zeitlin TruncationGalerkin Truncation
10 20 30 40 50 N λ Zeitlin TruncationGalerkin Truncation
Figure 5.2.
Numerically computed real eigenvalues vs Fouriermode domain size N .For these figure a = (0 , T and p = (3 , T .The red dashed lines shows the eigenvalues computed by theGalerkin truncation in equation (2.29), and the black solid linesshow the eigenvalues computed by the Zeitlin truncation in equa-tion (2.30). For the left figure, the same truncation domain (2.7) isused for the Zeitlin truncation and the Galerkin truncation, mean-ing a larger matrix is computer for the Zeitlin truncation. For theright figure, a Galerkin truncation with 2 N + 1 modes was chosenso that the same number of Fourier modes are included in bothcalculations. The convergence of the eigenvalue as a function of N computed with the Zeitlin truncation is significantly better ineither case. These plots omit the factor of α/α ′ for clarity.be approximated by taking ρ k ≈ | p | . The resulting matrix A from (2.30) is nowcirculant. A circulant matrix is diagonalised by a discrete Fourier transform (see[13]). Thus the eigenvalues of A are then found to be(5.1) λ j = 2 i | p | sin (cid:18) πjn (cid:19) for j = 0 , ..., n − j is the size of A. See [13] for details of this calculation.Thus the approximate imaginary spectrum of Σ a for sufficiently large | a | lies inthe interval i | p | [ −| α | , | α | ] on the imaginary axis. Taking the limit N → ∞ (and so n → ∞ ), for each x ∈ [0 ,
1] there is a correspondence with an eigenvalue λ x where x = π sin − (cid:16) λ x | p | i (cid:17) . Differentiating this gives the density function(5.2) F ( x ) = | p | π p α − | p | x . That is, the proportion of the eigenvalues lying between c i and c i on theimaginary axis is R c c F ( x )d x for c , c ∈ | p | [ −| α | , | α | ]. This curve is also plotted in5.4, and it agrees well with the numerically calculated eigenvalues. This is surprisingas the value of | a | is not particularly high. We can conclude that equation (5.2)gives a reasonable approximation of the imaginary spectrum for many choices of a . −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−0.2−0.15−0.1−0.0500.050.10.150.2 a=(1,0)a=(1,0) a=(1,0)a=(1,0) a=(0,−1)a=(0,−1)a=(0,−1)a=(0,−1)a=(1,−1) a=(1,−1)a=(0,−2) a=(0,−2)a=(0,0)Eigenvalues with non−zero real part for p =(2,1) −3 −2 −1 0 1 2 3−2.5−2−1.5−1−0.500.511.522.5 Types of eigenvalues corresponding to various values of a for p =(2,1) Figure 5.3.
Non-imaginary eigenvalues (left) and the correspond-ing lattice points a in the unstable disc (right) for the equilibriumwith p = (2 , T . At the top, the eigenvalues with non-zero realpart for every unstable class Σ a are shown. Eigenvalues with zeroimaginary part are marked with a red × , complex eigenvalues aremarked with a black +, and the zero eigenvalues are marked witha blue dot. In the bottom figure we see the values of a that corre-spond to these classes. The zero class led by a = (0 , T gives onlyzero eigenvalues. Compare the locations of the classes correspond-ing to real eigenvalues to the shaded region in figure 2.1. For thesefigures, Γ = , so Ω ∗ = cos( x · p ), and a Zeitlin truncation is usedfor the approximation.[16, 22] describe the essential spectrum of the linearised operator that coincideswith our limit N → ∞ . The essential spectrum for the class led by a is given inthat paper as(5.3) σ ess = i [ −| β | , | β | ] , where β = 2 | p | ( a × p )Γ . In the limit N → ∞ , sin( ε a × p ) ε → a × p , so | p | | α | → | β | . Thus our approximationfor large N reproduces the essential spectrum of a single class calculated in [16].Note that this is the essential spectrum associated with a single subsystem of thelinearised problem. The essential spectrum of the full system is the superpositionof all these essential spectra. It was shown in [22] that this is i R , which followsfrom considering the superposition of (5.3) for all possible values of a .6. Conclusion
We have demonstrated the non-linear instability of the stationary solutions withvorticity Ω ∗ = 2Γ cos( p · x ) in the Euler equations for almost all values of p suchthat p = (1 , T (or rotations and reflections thereof). We started out by usingthe Zeitlin truncation, and observe the numerical approximation of eigenvalues forfinite N obtained from Zeitlin’s truncation converge much faster with N . However, NSTABILITY OF EQUILIBRIA FOR THE 2D EULER EQUATIONS ON THE TORUS 25 −1.5 −1 −0.5 0 0.5 1 1.500.20.40.60.811.21.41.61.8 i λ R e l a t i v e F r equen cy Calculated eigenvalue densityApproximated eigenvalue density
Figure 5.4.
The density of the imaginary part of the spectrum,for the class p = (3 , a = (1 , − N = 1000, and Γ = 0 . N = 1000. The thick red line shows the approximatedensity computed by taking the approximation ρ k → | p | . For thisfigure a Zeitlin truncation is used for the approximation.the Zeitlin truncation does not behave well when gcd( p , p ) is even. Because ofthis, we developed much of our theory for both truncations.In addition, we have recreated and extended a number of results described by[17], [16], and [22]. Specifically, we have shown that the “unstable disc theorem”presented in [17] still holds true in the current context of finite dimensional approx-imation. Moreover, we have shown that for almost all p we can use the unstabledisc to prove instability (as opposed to the existing stability results developed byLi and others). We have also numerically verified the bound on the number ofnon-imaginary eigenvalues and the essential spectrum of an individual class in [16]and [22]. We used very different approaches and arguments to those papers.There are obvious extensions of this work. The first is a complete descriptionof the non-imaginary spectrum. This would require first showing that for two con-secutive negative values of ρ , the corresponding subsystem has four non-imaginaryeigenvalues, either two real pairs or a complex quadruplet. This would be a steptowards proving that the bound from [16] is sharp. Another extension would beto see if any of the methods used in this paper could be applied to more complexsteady states, for instance Ω ∗ = sin( p x ) sin( p y ).A similar analysis of the Euler fluid equations on a three dimensional torus wouldbe interesting. A Zeitlin-style structure preserving truncation for the 3D case is notpossible, as the Casimirs that make such a truncation useful are not present in the3D problem[24]. Similar stability results may still be possible using a Galerkin styletruncation instead. There is also a discussion of a structure-preserving truncation that includes aviscosity term in [26]. A comparison of Zeitlin’s truncation to standard truncationswith a viscosity term included as in Figure 5.2 would be valuable.A significant extension of this material would be an application of the samemethods to the Euler problem on a sphere. There similar structure preservingtruncation also due to Zeitlin for the sphere [25] and so there is some hope ofsimilar results in that setting.
Acknowledgements
RM gratefully acknowledges M. Beck and Y. Latushkin for extremely helpfuldiscussions on the known stability results on the 2D Euler equations.
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