Canonical scale separation in two-dimensional incompressible hydrodynamics
CCanonical scale separation in two-dimensionalincompressible hydrodynamics
Klas Modin and Milo Viviani Department of Mathematical Sciences, Chalmers University ofTechnology and University of Gothenburg Scuola Normale Superiore, Pisa
February 3, 2021
Abstract
Characterization of the long-time behaviour of an invicid incompressiblefluid evolving on a two-dimensional domain is a long-standing problem inphysics. The motion is described by Euler’s equations: a non-linear systemwith infinitely many conservations laws, yet non-integrable dynamics. Inboth experiments and numerical simulations, coherent vortex structures, orblobs, typically form after some stage of initial mixing. These formationsdominate the slow, large-scale dynamics. Nevertheless, fast, small-scale dy-namics also persist. Kraichnan, in his classical work, qualitatively describesa direct cascade of enstrophy into smaller scales and a backward cascadeof energy into larger scales. Previous attempts to quantitatively model thisdouble cascade are based on filtering-like techniques that enforce separationfrom the outset. Here we show that Euler’s equations posess a natural, intrin-sic splitting of the vorticity function. This canonical splitting is remarkablein four ways: (i) it is defined only in terms of the Poisson bracket and theHamiltonian, (ii) it characterizes steady flows (equilibria), (iii) it genuinly,without imposition, evolves into a separation of scales, thus enabling thequantitative dynamics behind Kraichnan’s qualitative description, and (iv) itaccounts for the “broken line” in the power law for the energy spectrum (ob-served in both experiments and numerical simulations). The splitting orig-inates from a quantized version of Euler’s equations in combination with astandard quantum-tool: the spectral decomposition of Hermitian matrices.In addition to theoretical insight, the canonical scale separation dynamicsmight be used as a foundation for stochastic model reduction, where the smallscales are modelled by suitable multiplicative noise. a r X i v : . [ m a t h - ph ] F e b Introduction
Consider Euler’s equations for a homogeneous, inviscid, and incompressible fluidconfined to a two-dimensional surface S . We shall, in fact, take S to be the unitsphere S as it makes our arguments more explicit and enables numerical simula-tions. However, most concepts are transferable to arbitrary compact surfaces.In vorticity formulation, Euler’s equations on S are˙ ω = { ψ, ω } ∆ ψ = ω, (1)where ω is the vorticity function of the fluid (related to the fluid velocity v via ω = curl v ), and ψ is the stream function, related to the vorticity function via theLaplace-Beltrami operator ∆ . The equations (1) constitute an infinite-dimensionalHamiltonian system with an infinite number of conservation laws: total energy H = − (cid:90) S ψω dS , linear momentum M = (cid:90) S n ω dS , for n the normal vector to S , and Casimir functions C f ( ω ) = (cid:90) S f ( ω ) dS , for any smooth real function f : R → R . These conservation laws are fundamentalin describing the long-time behaviour of the fluid, with linear momentum M , en-ergy H , and enstrophy C : = (cid:82) ω best understood (see, e.g., Herbert [8]). Usingthese quadratic invariants, Kraichnan [11, 12] showed that 2D fluids exhibit a dou-ble cascade: one towards the small scales, called direct enstrophy cascade, an othertowards the large scales, called inverse energy cascade. This phenomenon is crucialin the formation of stable large coherent vortices after an initial turbulent regime.Before Kraichnan, the formation of these condensates was explained by Onsager[16] working with a point vortex model and observing that fluid parcels with samevorticity sign tend to merge and condensate in large blobs. This formations oc-cur because they correspond to states of maximal entrophy. Several experimentalvalidations of the theories of Onsager and Kraichnan have been obtained in thelast decades [17]. In particular, a major e ff ort has been dedicated in deriving thepredicted energy profiles of a 2D fluid for long time simulations [2]. In [4], thecharacteristic broken line energy profile of a 2D fluid in a periodic box has been2btained using the wavelet-based vorticity splitting of proposed in [6]. This vortic-ity splitting characterizes large and small scales in terms of wavelet basis elements,providing a solid strategy in understanding the characteristic energy profile.However, one of the main limits of this approach is that it is not a canoni-cal choice of what should be considered either large or small scales. Moreover,this choice is determined a priori only considering the enstrophy and the trunca-tion level of the wavelet expansion. Therefore, not much information on the fluidinvariants is contained in this choice.In this paper, we introduce an alternative canonical choice to determine the dif-ferent fluid scales. Our work is based on the Zeitlin discrete truncation [19] of the2D fluid on a sphere. We remark that the results here presented can be generalizedto any compact surface, considering the analogous Zeitlin model (see [18] for theflat 2-torus, and [3] for the general quantization theory of Poisson bracket on com-pact manifolds). The advantage in working with the Zeitlin discrete model is thatcomplex topological issues of the Euler equations can be described in terms of lin-ear algebras. Our splitting of the discrete vorticity is both natural and explicit, andit only relies of the eigenvalues decomposition of the discrete stream function. In-deed, we state that the vorticity which describes the large scales is the one on whichthe action of the stream function is trivial. This splitting has a precise geometricmeaning in terms of Lie algebras but also a dynamical interpretation as the steadyand unsteady vorticity components. Our scale separation conjecture is demon-strated numerically with a long time simulations. A neat separation of scales in thevorticity energy profile is completely described by our splitting. Furthermore, wedetermine the dynamics equations for the large and small scales, providing a con-crete base for a stochastic model reduction [10]. Finally, we show how to translateour results to the original Euler equations, interpreting the matrix-based results intofluid dynamical ones. In this section, we introduce a discrete model for the Euler equations, first in-troduced by Vladimir Zeitlin [18, 19]. The model, sometimes called consistenttruncation or quantized Euler equations , relies on the approximation of the infinitedimensional Poisson algebra of the smooth functions on the sphere C ∞ ( S ), wherethe Poisson bracket {· , ·} is determined by the volume form on S . More explicitly,we have that for any p ∈ S and f , g ∈ C ∞ ( S ): { f , g } p = p · ( ∇ f × ∇ g ) .
3t has been shown in [3], that there exists a basis of the Lie algebra su ( N ), such thatits structure constant converges to those of C ∞ ( S ), expressed in terms of sphericalharmonics. More specifically, let f , g ∈ C ∞ ( S ) and let Y lm , for l = , . . . , ∞ , m = − l , . . . , l , be the usual spherical harmonics. Then, for any N >
1, there exists abasis of su ( N ), denoted with T Nlm for l = , . . . , N , m = − l , . . . , l and a projection p N : C ∞ ( S ) → su ( N ) such that, in the operator norm: • if p N f − p N g → f = g • p N { f , g } = N / (cid:2) p N f , p N g (cid:3) + O (1 / N ) • p N Y lm = T Nlm , for any l < N .Using the projections p N , we can write the quantized Euler equations as:˙ W = [ P , W ] ∆ N P = W , (2)for P , W ∈ su ( N ), where P = p N ψ , W = p N ω and ∆ N is the discrete Laplaciandefined in [9]. Equations (2) have been studied in various contexts [1, 14, 15, 18,19], mainly for their formulation on the 2D flat torus. Their numerical integrationhas been a classical problem [14, 15], being an example of Lie–Poisson system[7]. The main feature of equations (2) is that they possess an increasing numberof conservation laws, proportional to N . In particular, we have that are conservedin time the total energy H ( W ) = / PW ), the Casimirs C k ( W ) = Tr( W k ), for k = , . . . N and the linear momentum L = ( W x , W y , W z ), as defined in [15].Equations (2) have been extensively numerically studied in [15]. In that work,it has been shown how the conservation of the discrete first integrals is determinantin the characterization of the finite state of the fluid. However, it is not clear the roleplayed by each of those. Moreover, equations (2), together with the conservativenumerical scheme employed, seem to show a persistent unsteadiness of the fluidfor long time, analogously to what observed in [5].In order to better understand the long-time behaviour of equations (2), and inparticular determine whether a steady state is eventually reached, the idea is toanalyse the commutator [ P , W ]. As it is shown in the next section, the idea is tosplit the vorticity in a dynamical passive and a dynamical active part. These twocan be determined looking at the part of the vorticity which does and does notcommute with the matrix P . In this section, we present and discuss a new vorticity splitting for the quantizedEuler equations. As mentioned above, the idea is to derive a dynamical passive,4enoted by W s , and a dynamical active, denoted by W r , part for the discrete vortic-ity W . The splitting we present is natural in the sense that it does not require any adhoc choice to define it. What we only need is to introduce the projections onto thestabilizer of the discrete stream function P and onto its orthogonal complement.Finally, we derive and discuss the dynamical equations for W s and W r .Let us consider the quantized Euler equations:˙ W = [ P , W ] ∆ N P = W , for P , W ∈ su ( N ). Let us assume P to be generic , i.e. it has all the eigenvaluesdistinct. Introducing the Frobenious inner product on su ( N ), we can define thedecomposition of W = W s + W r , where W s is the orthogonal projection of W ontothe stabilizer algebra of P :stab P = { A ∈ su ( N ) s.t. [ A , P ] = } , and W r is its orthogonal complement. We notice that stab P can also be defined as:stab P = { A ∈ su ( N ) s.t. A , P simultaniously diagonalizible } . Hence, more explicitly, W s can be construct by finding E unitary which diagonal-izes P , E † PE = Λ , and then defining Π s : su ( N ) → stab P , such that: W s : = Π s ( W ) = E diag( E † W E ) E † . Hence, the quantized Euler equations can be written as:˙ W = [ ∆ − N ( W s + W r ) , W r ] . W s and W r Consider a general matrix flow on su ( N ) of the form˙ P = F ( P ) , (3)for P ∈ su ( N ) and F : su ( N ) → su ( N ) smooth. Let E and Λ be an eigenbasis andthe corresponding eigenvalues of P . We want to determine the evolution of E and Λ . It is known that su ( N ) is foliated into (co-adjoint) orbits defined as: O P = { Q = U PU ∗ for U ∈ S U ( N ) } . The tangent space at P ∈ su ( N ) to the orbit O P is spanned by { e k e † l } k (cid:44) l , where E = [ e , . . . , e N ] is an orthonormal eigenbasis of P . If we take the normal directions Here we assume P generic, i.e., with all the eigenvalues distinct. { e k e † k } of the tangent spaces (w.r.t. the canonical bi-invariant inner product),we obtain the linear subspace of matrices in su ( N ) sharing the same eigenbasis(simultaneously diagonalizable). In the language of Lie algebras, it corresponds tothe infinitesimal isotropy algebra corresponding to P . Thus, if Π s : su ( N ) → su ( N )denotes orthogonal projection onto span { e k e † k } , then Π r = Id − Π s . Numerically, Π s X is computed as Π s X = E diag( E † XE ) E † . Notice, as expected, that neither Π s nor Π r depend on the eigenvalues of P , only the eigenbasis. This means that wecan write equation (3) as ˙ P = Π r F ( P ) + Π s F ( P )The first part of the flow, ˙ P = Π r F ( P ), changes the eigenbasis but not the eigenval-ues and vice versa. The question is: what is the generator of P (cid:55)→ Π r F ( P )? Sinceit is isospectral it should be of the form P (cid:55)→ [ B ( P ) , P ], but what is B ( P )? Letus denote X = F ( P ). It is then straightforward to check that if all the eigenvalues p , . . . , p N of P are di ff erent, then Π r X = (cid:88) k (cid:44) l x kl e k e † l = (cid:88) k (cid:44) l ( p k − p l ) b kl e k e † l = [ (cid:88) k (cid:44) l b kl e k e † l , P ] = [ B , P ]where x kl are the components of X in the basis E , and b kl (cid:66) x kl / ( p k − p l ). Thus, inthe generic case, with all eigenvalues di ff erent, we can construct the generator B ( P )from the eigenvalues p , . . . , p N and the eigenbasis e , . . . , e N of P . This allows usto write the equation (3) in terms of the eigenvalues and eigenbasis of P as˙ p k = e † k Fe k , F = F (cid:16) N (cid:88) k = p k e k e † k (cid:17) ˙ e k = Be k , B = (cid:88) k (cid:44) l e † k Fe l p k − p l e k e † l Remark 1.
The evolution of the eigenbasis of P plays a major role in the variationof the energy metric H ( P ) = / ∆ N PP ). Using that ∆ N P = [ X , [ X , P ] +
12 [ X − , [ X + , P ]] +
12 [ X + , [ X − , P ]]we see that E † ∆ N PE = [ Y , [ Y , Λ P ]] +
12 [ Y − , [ Y + , Λ P ]] +
12 [ Y + , [ Y − , Λ P ]]where Y i = E † X i E , i.e., the matrices X i expressed in the eigenbasis E . Thus, the Y i are also transported by the generator B , so they can be thought of as advectedquantities ˙ Y i = ± [ B , Y i ] . Y i somehow corresponds to deforming the metric via a symplectomor-phism. Notice, however, that the Y i are complicated from the start (since E iscomplicated from the start). Remark 2.
The matrices E kk = ie k e † k , for k = , . . . , N form an orthonormal basisfor stab P . In particular, these elements can have an interpretation as moving frame,which varies accordingly to the evolution of P . Furthermore, by plotting them,it is remarkable that they correspond to some kind of wavelet basis. Indeed, forthe Euler equations, the interpretation of the spectral decomposition of P is suchthat its eigenvalues correspond to the values that the respective stream functiontakes, whereas its eigenvectors (or the matrices E kk ) correspond to the connectedcomponents of the level curves of the respective stream function (see Figure 1).Let us now consider the matrix flow for W s and W r . By the definition of W s wehave that: ˙ W s = ddt (cid:16) E diag( E † W E ) E † (cid:17) = [ ˙ EE † , W s ] − Π s ([ ˙ EE † , W r ]) , where we have used the fact that Π s ( ˙ W ) = EE † = − E ˙ E † . This dynamics isvery close to the one of P : ˙ P = [ ˙ EE † , P ] + E ˙ D P E † . Hence, it is crucial to determine a formula for ˙ EE † . To this end, we know that thedynamics of P can be orthogonally decomposed as:˙ P = Π r ( ∆ − N [ P , ∆ P ]) + Π s ( ∆ − N [ P , ∆ P ]) . Hence, it is clear that: [ ˙ EE † , P ] = Π r ( ∆ − N [ P , ∆ P ]) . We notice that ˙ EE † can be taken in stab ⊥ P . In fact, the dynamics of W s remains thesame for any ˙ EE † + S , where S ∈ stab P . The map:[ · , P ] : stab ⊥ P → stab ⊥ P is invertible and ˙ EE † is uniquely determined in stab ⊥ P . Hence, in conclusion wehave the following theorem for the dynamics of W s , W r : Theorem 1.
Let W be a solution to equations (2) and W s , W r respectively be theorthogonal projections of W onto stab P and its orthogonal complement. Then, W s and W r satisfy the following system of equations: ˙ W s = [ B , W s ] − Π s [ B , W r ]˙ W r = − [ B , W s ] + Π s [ B , W r ] + [ P , W r ][ B , P ] = Π r ∆ − N [ P , W r ] , a) E (b) E (c) E (d) E (e) E (f) E (g) E (h) E Figure 1: Some basis element of stab P at t = t end for the first numerical simulationof Section 4. 8 here B is the unique solution to the third equation in stab ⊥ P and P = ∆ − N ( W s + W r ) . From theorem 1, we can deduce some facts about the W s , W r dynamics. Firstof all, we notice that if W r = B =
0. In fact, if W r =
0, we get that B ∈ stab P ∩ stab ⊥ P . Hence B =
0. Conversely, if B =
0, we get that ˙ W s = W r = [ P , W r ]. Hence, in that case W s plays the role of a fixed topography for W r ,which satisfies a Euler-type equation. From the third equation, we deduce that itmust be Tr( ∆ − N W s [ ∆ − N W r , W r ]) = . (4)Equation (4) and B = H ( W r ) is conserved and provides afull separation of scales, for which the large scales are in equilibrium, whereas thesmall scales have still some dynamics left, which does not a ff ect the large ones.An other observation is that if [ B , W s ] =
0, then B = W s =
0. Viceversa, if Π s [ B , W r ] =
0, then again we deduce that Tr( ∆ − N W s [ ∆ − N W r , W r ]) = P without any change of the eigenvalues, but not vice versa. Let us now look in how the energy and the enstrophy are related to the splitting W s and W r . By defintion, the energy H can be written as: H ( W ) =
12 Tr( ∆ − N ( W s + W r ) W s ) =
12 Tr( ∆ − N ( W s ) W s ) −
12 Tr( ∆ − N ( W r ) W r )and the enstrophy E ( W ) = − Tr( W s ) − Tr( W r ) . Hence, we have the interesting fact that H ( W ) = H ( W s ) − H ( W r ) E ( W ) = E ( W s ) + E ( W r ) . (5)We notice that W s = W =
0, being √ H ( · ) a norm. Moreover, if W (cid:44)
0, 0 < H ( W ) ≤ H ( W s ) < E ( W s ) ≤ E ( W ) . The fact that the energy of W s is larger than the total energy H ( W ) can be inter-preted as the fact that the large scales W s are slowed down by the small scales W r which acts as a dissipative field, moving the energy towards the small scales (seefigure 4). 9t is possible to represent the energy-enstrophy splitting (5) in a more geometricway. The first equation of (5) tells that W and W r are orthogonal in the energy norm,whereas the second equation says that W S and W r are orthogonal in the L norm.Let us take the W , W r -plane, and let r : = √ H ( W r ) and H = √ H ( W ). Then, for W r = (0 , r ) and W = ( H , R . Wewant to express the L norm with respect the energy norm. Let us first observe that W s = ( H , − r ). Then, the L norm can be defined as G = C ∗ C , where C : R → R is such that C (0 , r ) T = (0 , E sin α ) T and C ( H , T = E (cos α, sin α ) T , where α is the angle between W and W s in the L norm and E = √ E ( W ). Then, we havethat: G = E H E rH sin α E rH sin α E r sin α , in the W , W r -plane. We can derive the following inequalities: r = √ H ( Wr ) ≤ √ E ( Wr ) = E | sin α | N r = N √ H ( Wr ) ≥ √ E ( Wr ) = E | sin α | . Hence, if sin α (cid:44)
0, we get that: E N ≤ r | sin α | ≤ E . (6)Therefore, we see that in the limit for N → ∞ the ratio | sin α | r is potentially un-bounded. Indeed, it could happen that the L norm of W r is always far from beingzero, whereas its energy norm goes to zero. This corresponds to the case when W r is shifted more and more towards the small frequencies keeping its enstrophy large.Because of the inequality (6), we notice that in the finite dimensional case this cannever occur. In this section, we want to present some results concerning the splitting of W s and W r . Our observations are based on some numerical evidences and heuristicconsiderations. One main motivation for studying the vorticity splitting in the W s and W r component is that the unsteadiness of the fluid can be precisely understoodin terms of non-vanishing of W r . Moreover, in order to derive W s and W r , it is notrequired any knowledge of neither the Fourier decomposition nor the values of W .From an analytical point of view, W s represents a projection of W onto a smoother10ubspace. Indeed, the relation via the Laplace operator between P and W saysthat P admits, in general, two more spatial derivatives than W . Hence, since W s isrelated to P via a polynomial relationship, W s is in general more regular than W .Vice versa, W r contains the rougher part of W . The tempting interpretation of thevorticity splitting above presented is that W s represents the low dimensional largescale dynamics, whereas W r represents the noisy small scale dynamics.In order to assess this conjecture, we consider the numerical simulation studiedin [5, 15]. In Figure 2 the three vorticity fields W , W s , W r are shown at the times t = t and t = t end ∼ s . As expected, at the end of the simulation, the largescales of the vorticity are all contained in the smooth field W s , whereas W r is almosteverywhere white noise. In Figure 3, we have plotted the field B at t = t end and therelation between the values taken by P and W s , in their spatial plot. It is quite clearthat far form the center of the blobs the level curves of B and W s are very closeone to the other, even though they do not commute. A possible interpretation ofthis fact is that the most of the dynamics is concentrated around the blobs, wherethe mixing of the vorticity continues for very long (infinite) time. Furthermore, wenotice that B has exactly two saddle points and those correspond to the bifurcationsappearing in the P − W s plot. The branches in the P − W s plot correspond to thefact that four main blobs emerge. Clearly, the fact that the plot is not a graph of afunction implies that a possible functional relation between W s and P can only belocal in space.The scale separation of the vorticity is even more clear looking at the energyspectrum of W . In Figure 4, we have plotted in log-log scale the energy diagramsof the three vorticty fields W , W s , W r at t = t end . The energy level H ( l ), correspond-ing to the wave-number l = , . . . , N , encodes the energy of the modes with respectto the harmonics Y lm , for m = − l , ..., l . We notice that the energy spectrum of W is analogous to the one obtain in [2]. We can see that the two slopes in the energyspectrum of W depend on the fact that the first l − slope comes from the energyspectrum of W s , whereas the l − slope comes from the energy spectrum of W r . Inthis sense, we can say that the vorticity splitting is a scale separation of the vorticityfield.We consider another numerical simulation, where the three blob formationemerges, as predicted in [15]. In Figure 5 the three vorticity fields W , W s , W r areshown at the times t = t and t = t end ∼ s . As expected, at the end of thesimulation, the large scales of the vorticity are all contained in the smooth field W s , whereas W r is almost everywhere white noise. In Figure 6, we have plottedthe field B at t = t end and the relation between the values taken by P and W s , intheir spatial plot. In contrast with the one of Figure 3, we notice that there is onlyone saddle point, which corresponds to the bifurcation appearing in the P − W s plot. The branches in the P − W s plot correspond to the fact that three main blobs11 a) Vorticity field W ( t ) (b) Vorticity field W ( t end )(c) Vorticity field W s ( t ) (d) Vorticity field W s ( t end )(e) Vorticity field W r ( t ) (f) Vorticity field W r ( t end ) Figure 2: The three vorticity fields W , W s , W r at t = t and t = t end .12 a) B field at t = t end (b) P − W s values at t = t end W s P Figure 3 H ( l ) l Figure 4: Energy spectrum in log-log scale of the three vorticity fields W dashedblack, W s dashed-dot red, W r dashed-dot green at t = t and t = t end . In blue arereported the slopes l − and l − . 13 a) Vorticity field W ( t ) (b) Vorticity field W ( t end )(c) Vorticity field W s ( t ) (d) Vorticity field W s ( t end )(e) Vorticity field W r ( t ) (f) Vorticity field W r ( t end ) Figure 5: The three vorticity fields W , W s , W r at t = t and t = t end .14 a) B field at t = t end (b) P − W s values at t = t end W s P Figure 6emerge. Hence, looking at Figure 3-6, we see a clear connection between saddlepoints of B and the number of blobs.The scale separation of the vorticity is again evident looking at the energyspectrum of W . In Figure 4, we have plotted in log-log scale the energy diagramsof the three vorticty fields W , W s , W r at t = t end . We notice that the energy spectrumof W is analogous to the one in Figure 4.Finally, we remark that the projection Π s : su ( N ) → stab P has rank N . Hence,the dynamics of W s in the moving frame E kk can be described by only N compo-nents, i.e. its eigenvalues. Therefore, the vorticity splitting can be also interpretedas a reduced dynamics. In this section we consider the Euler equations for the vorticity:˙ ω = { ψ, ω } ∆ ψ = ω. (7)where ω ∈ C ([0 , ∞ ) , C k ( S )), for some k ≥
1. In order to define the analogous ofthe finite dimensional splitting, we have to understand equations (7) in the weaksense. Indeed, we will show that in general the analogous of the projections Π s , Π r cannot preserve the smoothness of ω . However, they are still continuous withoperator norm one, from C k ( S ) to L p ( S ), for any k ∈ [0 , ∞ ] , p ∈ [1 , ∞ ]. Hence,by density of the smooth functions in L p ( S ), p ∈ [1 , ∞ ) we can extend Π s , Π r tocontinuous operator to L ∞ ( S ). This result fits very well with the well-posednessof equations (7), which for finite domains precisely requires a vorticity in L ∞ , [13,Chp. 2.3]. 15 ( l ) l Figure 7: Energy spectrum in log-log scale of the three vorticity fields W dashedblack, W s dashed-dot red, W r dashed-dot green at t = t and t = t end . In blue arereported the slopes l − and l − .Let us first define the right hand side of equations (7) in the weak sense. Forany p ≥
2, let ω ∈ L p ( S ). Then, ψ ∈ W , p ( S ) ⊂ C ( S ). Finally, we define theweak Poisson bracket as: (cid:90) S { ψ, ω } φ dS = − (cid:90) S ω { ψ, φ } dS , for any test function φ ∈ C ∞ ( S ). We can now define the stabilizer of ψ as:stab ψ : = { f ∈ L ( S ) s.t. { f , ψ } = } . We want now define the projection Π s onto stab ψ . One issue is that the stabilizerof ψ is not closed in L strong topology. However, for continuous functions it ispossible to define a projector, which minimizes the L distance from stab ψ . Wefirst take the following assumption on the critical points of ψ Assumption 1.
Let ψ ∈ C ( S ) be the stream function. Then, the critical points of ψ define a set of zero Lebesgue measure on S , such that it is never dense in anyneighbourhood of one of its points.We say that ψ is generic , whenever it satisfies Assumption 1. Let us nowconsider some f ∈ C ( S ). We see from this definition that f ∈ stab ψ if and only ∇ f and ∇ ψ are parallel. Since we are taking ψ generic, the points where ∇ ψ = f continuous, f ∈ stab ψ if it is constant on the connected components of the level curves of ψ . Then theprojection of f onto stab ψ can be defined evaluating f on the level curves of ψ , i.e.the streamlines. Let γ be a connected component of a streamline, then we definethe projector Π s : C ( S ) → stab ψ as: Π s ( f ) | γ = γ ) (cid:90) γ f ds . (8)In the limit case, when γ is a point, clearly Π s ( ω ) | γ = f ( γ ). We notice that theoperator Π s does not preserve in general the continuity of f . Indeed, let us calla point p ∈ S a bifurcation saddle point if p is a saddle point of ψ such thatthe streamline passing through p contains a bifurcation point. We then have thefollowing result: Proposition 1.
Let ψ be generic and Π s the projector as defined in (8) . Then, ifp ∈ S is a bifurcation saddle point for ψ , there exists f ∈ C ( S ) , such that Π s ( f ) is discontinuous at the streamline passing through p. Viceversa, given f ∈ C ( S ) ,if Π s ( f ) is discontinuous at some point p ∈ S , then the streamline passing throughp contains a bifurcation saddle point.Proof. [ S ketch ] The issue about the continuity of f can be treated locally. Hence,let us work in Cartesian coordinates x , y . Let p ∈ S be a bifurcation saddle pointfor ψ and γ the streamline passing through p . Then, let β be a curve intersecting γ only in p and let f be a smooth function positive at one side of β and negativeat the other one, such that (cid:82) γ f ds =
0. Then being p a bifurcation point, for anyneighbourhood U of p , there exist streamlines totally contained in one or anotherside of β . Then, the average of f on those streamlines is either strictly positive ornegative, creating a discontinuity at γ .Viceversa, let f ∈ C ( S ), such that Π s ( f ) is discontinuous at some point p ∈ S . Then, the streamline passing through p cannot be homomorphic to any ofthose in some tubular neighbourhood. Hence, the streamline passing through p must contain a critical point for ψ , which also is a bifurcation saddle point. (cid:3) However, we have the following regularity for Π s . The operator Π s is a boundedoperator with unitary norm between C ( S ) and L p ( S ), for every p ∈ [1 , ∞ ]. Since C ( S ) is dense in any L p ( S ), for p ∈ [1 , ∞ ), it is possible to extend Π s to any L p ( S ), for every p ∈ [1 , ∞ ). By the continuity of the L p norm with respect to p ,we conclude that we can also extend Π s to a bounded operator on L ∞ ( S ).Hence, from now on, let us consider equations (7) in the weak form, for ω ∈ L ∞ ( S ). It is clear that Π s = Π s . Moreover, we can formally define the operator17 s via the kernel K ( x , y ) = length ( γ x ) δ γ x ( y ), for any x , y ∈ S , where γ x is theconnected component of the streamline passing in x . In this way we get that Π s isself-adjoint with respect to the L inner product: (cid:90) S f Π s gdS = (cid:90) S f ( x ) (cid:90) S K ( x , y ) g ( y ) dS ( y ) dS ( x ) = (cid:90) S g ( y ) (cid:90) S K ( x , y ) f ( x ) dS ( x ) dS ( y ) = (cid:90) S g Π s f dS , for any f , g ∈ C ( S ). Then, by extension we get the same result in L ∞ ( S ).Then, we have the following proposition: Proposition 2.
Let f ∈ L ∞ ( S ) and ψ be generic. Then f ∈ stab ψ if and only if Π s f = f .Proof. Let us prove the result for f ∈ C ( S ), and conclude by extension. Let f ∈ stab ψ . Then, ∇ f is parallel to ∇ ψ almost everywhere. Hence, the gradientof f along any streamline must vanish, and so on each connected component itis constant. By continuity of f we deduce that f must be constant also on thestreamlines containing some critical points. Therefore, Π s ( f ) = f .Let us now assume that Π s ( f ) = f . Then, f must be constant on each connectedcomponent of a streamline. Hence, ∇ f is orthogonal to the streamlines. Since, ∇ ψ is also orthogonal to the streamlines, we conclude that { f , ψ } = f is instab ψ . (cid:3) Hence, we can derive the analogous results of Section 3 for the continuouscase. First of all, let us recall that the stream function ψ satisfies the equation:˙ ψ = ∆ − { ψ, ∆ ψ } . (9)Equation (9) is not Hamiltonian, but we can split the right hand side into the Hamil-tonian and non-Hamiltonian part, using the projection Π s :˙ ψ = Π r ∆ − { ψ, ∆ ψ } + Π s ∆ − { ψ, ∆ ψ } . Analogously to the finite dimensonal case, we would like to find a generator forpart Π r ∆ − { ψ, ∆ ψ } . In particular, we would like to find a function b , such that { b , ψ } = Π r ∆ − { ψ, ∆ ψ } . (10)18t is clear that a necessary condition for the equation { b , ψ } = f to have a solution b , is that f ∈ stab ⊥ ψ . Indeed, we have that for any b ∈ C ( S ), { b , ψ } ∈ stab ⊥ ψ , as it iseasily checked: (cid:82) S { b , ψ } gdS = − (cid:82) S { g , ψ } bdS = , for any g ∈ stab ψ . However, in general equation (10) can be solved only where ∇ ψ (cid:44)
0. Around the critical points of ψ the gradient of b is potentially unbounded.Moreover, the right hand side in equation (10) can be discontinuous at the levelcurves of ψ containing saddle points of ψ . Hence, b can only be defined almosteverywhere. Furthermore, we have the following: Proposition 3.
Let f ∈ C ( S ) and ψ generic. Then, f ∈ stab ⊥ ψ if and only if thereexists b almost everywhere smooth, such that { b , ψ } = f , where ∇ ψ (cid:44) .Proof. The if part is clear. Let instead take f ∈ stab ⊥ ψ . Then, for any point p ∈ S ,we have to solve the PDE for b : ∇ ψ ⊥ · ∇ b = f , (11)where ∇ ψ ⊥ = p × ∇ ψ . If ∇ ψ does not vanish, equation (11) can be solved byintegration in the direction of ∇ ψ ⊥ . In the points where ∇ ψ does not vanishes, ∇ b is not defined by equation (11), and it can be unbounded around those points.Hence, the field b is almost everywhere smooth and satisfies { b , ψ } = f , where ∇ ψ (cid:44) (cid:3) In order to derive the dynamical equations for ω s , we cannot use directly thefield b . Instead, we consider the volume preserving vector field X [ ψ ] : = Π r ∆ − { ψ, ∆ ψ } . We note that X corresponds to the infinitesimal action of a map ϕ t which transports ψ by deforming its level curves. Hence, ϕ t extends naturally to action on stab ψ .Let us write Π ts the projection onto stab ψ at time t . Then, for any point x ∈ S , let γ ( t ) = { y | ψ ( y ) = ψ ( x ) and y belongs to the same connected component of x } , and d ˆ s t the normalized Lebsgue measure on γ ( t ). We have then the formal identity: ω s ( t , x ) = Π ts ω ( t , x ) = (cid:82) γ ( t ) ω ( t , y ) d ˆ s t ( y ) = (cid:82) γ (0) ( ϕ ∗ t ω )( t , y ) d ˆ s ( y ) = Π s ( ϕ ∗ t ω )( t , ϕ − t ( x )) . φ ∈ C ∞ ( S ), we have ddt (cid:82) S ω s ( t , x ) φ ( x ) dS = ddt (cid:82) S Π s ( ϕ ∗ t ω )( t , ϕ − t ( x )) φ ( x ) dS = (cid:82) S (cid:16) Π s ( ϕ ∗ t L X ω )( t , ϕ − t x ) − L X Π s ( ϕ ∗ t ω )( t , ϕ − t x ) (cid:17) φ ( x ) dS = − (cid:82) S ω ( t , x ) L X Π ts φ ( x ) + L X Π ts ω ( t , x ) φ ( x ) dS where L X is the Lie derivative, which simply acts on functions as L X f = X [ f ]. Wenotice from the previous calculations that L X has to be evaluated only on elementsin stab ψ . Hence, the time derivative of ω s is well defined in the weak sense.Let us now formally denote X : = −{ b , ·} . Then, interpreting the Poisson bracketin the weak sense, we can write the dynamical system for ω s , ω r as:˙ ω s = { b , ω s } − Π s { b , ω r } ˙ ω r = −{ b , ω s } + Π s { b , ω r } + { ψ, ω r }{ b , ψ } = Π r ∆ − { ψ, ω r } , where b is implicitly defined by the third equation and ψ = ∆ − ( ω s + ω r ). We noticethat the equations of motion for ω s can also be written in a more compact form as:˙ ω s = [ Π s , L X ] ω, where the square bracket is a commutator of operators.Finally, we notice that the energy and enstrophy splitting still hold: H ( ω ) = H ( ω s ) − H ( ω r ) E ( ω ) = E ( ω s ) + E ( ω r ) . Appendix A: Stream function splitting
In this section, we present the analogous splitting of Section 3 for the stream func-tion P . Let stab W be the stabilizer of W in su ( N ). Then, we can consider thestream function splitting P = P s + P r , where P s is the orthogonal projection of P onto stab W , and P r the orthogonal complement. Similarly to what we have done inSection 3, P s can be constructed by finding U unitary which diagonalizes W : U ∗ WU = D , and then defining P s = U diag( U ∗ WU ) U ∗ . By definition the quantized Euler equa-tions for the stream function P can be written as:˙ P = ∆ − N [ P r , ∆ ( P s + P r )] , H ( P ) = Tr( ∆ N ( P s + P r ) P s ) = Tr( ∆ N ( P s ) P s ) − Tr( ∆ N ( P r ) P r ) andthe enstrophy E ( P ) = − Tr(( ∆ N ( P s + P r )) ). Notice that the enstrophy E ( P ) = − Tr(( ∆ N P ) ), in the P s , P s does not admit a splitting contrary to the energy H ( P ) = Tr( ∆ N PP ): H ( P ) = H ( P s ) − H ( P r ) E ( P ) (cid:44) E ( P s ) + E ( P r ) . Since the eigenvalues of W are constant in time, the dynamics of P s , P r variablesis simpler than the one of W s , W r . In fact, we have that ˙ UU = P and so˙ P s = [ ˙ UU , P s ] − Π s [ ˙ UU , P r ] + Π s ( ˙ P ) = [ P r , P s ] + Π s ( ∆ − N [ P r , ∆ N ( P s + P r )]) , and ˙ P r = − [ P r , P s ] + Π r ( ∆ − N [ P r , ∆ N ( P s + P r )]) . To understand how the dynamics of the P s , P r variables look, we consider the firstnumerical simulation of Section 4. In Figure 8 the three stream function fields P , P s , P r are shown at the times t = t and t = t end . We notice that from a verysmooth P , the projection Π s onto the stabilizer rougher field W produce a muchcoarser image. In particular, both P s and P r do not show any additional structureor scale separation unlike to W s and W r . References [1] R. V. Abramov and A. J. Majda, Statistically relevant conserved quantitiesfor truncated quasigeostrophic flow,
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