Canonical scale separation in two-dimensional incompressible hydrodynamics

Abstract The rules that govern a two-dimensional inviscid incompressible fluid are simple. Yet, to characterise the long-time behaviour is a knotty problem. The fluid fulfils Euler's equations: a nonlinear Hamiltonian system with an infinite number of conservation laws. In both experiments and numerical simulations, coherent vortex structures emerge after an initial stage. These formations dominate the large-scale dynamics, but small scales also emerge and persist. The resulting scale separation resembles Kraichnan's theory of forward and backward cascades of enstrophy and energy. Previous attempts to model the double cascade use filtering techniques that enforce separation from the outset. Here, we show that Euler's equations possess an intrinsic, canonical splitting of the vorticity function. The splitting is remarkable in four ways: (i) it is defined solely by the Poisson bracket and the Hamiltonian; (ii) it characterises steady flows; (iii) it innately separates scales, enabling the dynamics behind Kraichnan's qualitative description; and (iv) it accounts for ‘broken line’ energy spectra observed in both experiments and numerical simulations. The splitting originates from Zeitlin's truncated model of Euler's equations in combination with a standard quantum tool: the spectral decomposition of Hermitian matrices. In addition to theoretical insight, the scale separation dynamics enables stochastic model reduction, where multiplicative noise models small scales.


Introduction
Two-dimensional turbulence is the study of incompressible hydrodynamics at large (including infinite) Reynolds numbers.It is a vibrant field of both mathematics and physics that began with Euler [12], who derived the basic equations of motion.Turbulent flows in two space dimensions do not exist as classical fluids in nature.Rather, they constitute basic models of intermediate-scale flows in "almost" two-dimensional (thin) domains [31,30,9,40,10,49].The conditions of 2D turbulence can be emulated in experiments.One setup is a soap film flowing rapidly through a fine comb [8].Another is a conducting fluid confined to a thin layer and driven into turbulence by a temporally varying magnetic field [44].When such a "quasi-2D" flow is released it self-organises into blob-like condensates.The progression is depicted in Figure 1 for a spherical domain.Heuristically, the mechanism is driven by merging of equally signed vorticity regions.This large scale fusion is balanced by fine scale dissipation.In many ways 2D turbulence is propelled by the quest to understand the resulting scale separation.
To make theoretical progress, Onsager [39] applied statistical mechanics to a large but finite number of point vortices.They are weak (distributional) solutions of Euler's equations where vorticity is a sum of weighted Dirac pulses.Onsager realised that a fixed number of positive and negative point vortices, confined to a bounded domain, can have energies from −∞ to ∞.The phase volume function therefore has an inflection point at some finite energy.At this energy the thermodynamical temperature is zero.If the energy exceeds this point, so the temperature is negative, then equally signed vortices should cluster according to thermodynamics.This theory of statistical hydrodynamics is a prominent, although lesser known, part of Onsager's legacy [13].On the mathematical side, Caglioti et al. [5,6] and Kiessling [22] gave rigorous results on clustering of point vortices in the negative temperature regime.Their work has fostered much theoretical progress (see Marchioro and Pulvirenti [32] and references therein).On the experimental side, seventy years after Onsager presented his theory, the conditions of negative temperature point vortex dynamics were experimentally realised in planar Bose-Einstein condensates [16,21].As predicted, persisting vortex clusters emerge.
Onsager's theory cannot be applied to continuous vorticity fields (corresponding to smooth solutions).Consequently, it is natural to look for a statistical theory of continua.One approach is to expand the vorticity field in a Fourier series and then truncate it [26].The truncated, finite-dimensional system preserves phase volume and quadratic invariants, but not higher order invariants (Casimir functions).To account also for those invariants, another approach is to maximise the entropy of a probability distribution of coarse-grained vorticity fields [29,34,41].
All theories based on statistical mechanics assume ergodic dynamics.Rigorous results on ergodicity are available for 2D Navier-Stokes on a doubly periodic domain (flat torus) with added regular-in-space noise proportional to the square root of the viscosity .In this setting there exists a unique stationary measure   [27,28].As  → 0 one obtains a stationary measure  0 for the 2D Euler equations, but it is not expected to be unique [28].
Statistical mechanics is not the only approach.Another possibility is to study energy and enstrophy spectra.The inspiration comes from Kolmogorov's (1941) theory of 3D turbulence.Notably, Kraichnan [25] argued that viscous 2D fluids in forced equilibrium, where energy at an intermediate scale is fed into the system, exhibit a forward cascade of enstrophy into fine scales and a simultaneous backward cascade of energy into large scales.Direct numerical simulations typically support Kraichnan's theory [45, and references therein].
For two-dimensional systems with no energy dissipation at the large scale (so that vortex condensation occurs), numerical simulations develop a "broken line" energy spectrum with a slope of  −3 at the large scale (where  is the wave-number) and then a swift switch at an intermediate scale to a slope between  −5/3 and  −1 [3].An approximate broken line energy spectrum is also observed in zonal and meridional wind measurements on Earth over the scales 3-10,000 km [38].
To better understand characteristic energy spectra it is natural to impose a splitting of the vorticity field  =   +   into a large-scale component   and a small-scale component   .A wavelet-based vorticity splitting is proposed by Farge et al. [15] and applied to numerical simulation on the doubly periodic square (where condensation occurs) by Chertkov et al. [7].The results [7, fig. 1f] show an energy spectrum slope of  −3 for the large scale component and of  −1 for the intermediate-to-small scale component.It is a powerful technique to analyse energy spectra, but it depends on a choice of wavelet basis and on parameters identifying the different scales.Therefore the method cannot give insights on the mechanisms behind vortex condensation or broken line energy spectra.
In this paper we give a new, canonical decomposition of vorticity.With "canonical" we mean parameter-free, determined solely in terms of the data for the two-dimensional Euler equations: the Poisson bracket and the Hamiltonian function.The decomposition has the following properties: 1.The vorticity field  =   +   is a steady state if and only if   = 0.
2. Under numerical simulation of Euler's equations,   and   evolve into a separation of scales.The component   traps large-scale condensates whereas the component   captures small-scale fluctuations.
3. After a short transient time, the energy spectrum slope of   is about  −3 and of   is between  −5/3 and  −1 .
4. Over time, the component   displays an average enstrophy increase (quantifying the forward enstrophy cascade) and an average energy decrease (quantifying the backward energy cascade).
The coupled equations governing the dynamics of   and   embody a new line of attack for 2D turbulence.Our standpoint is that a detailed study of these equations may explain the mechanisms behind vortex condensation and broken line energy spectra, or at least give deep insights.The numerical simulations we present suggest so.

Two-dimensional Euler equations
Our starting point is Euler's equations for an inviscid, incompressible fluid on a twodimensional closed surface.Throughout the paper we take the surface to be the unit sphere S 2 ⊂ R 3 .It makes our arguments more explicit and enables numerics.Most concepts are readily transferable to arbitrary closed surfaces (in particular to the flat torus, which is the most studied example in the literature albeit less relevant than the sphere in applications).
In vorticity formulation, Euler's equations on S 2 are The Hamiltonian (total energy) for the vorticity equation ( 1) is .
In addition to total energy, there is an infinite number of conservation laws: total angular momentum n , n unit normal on S 2 , and Casimir functions (), for any smooth  : R → R.
These conservation laws are fundamental for the long-time behaviour.In particular, the presence of infinitely many Casimir functions sets apart 2D from 3D fluids.

Overview of the paper
Zeitlin's truncated model of Euler's equations originate from the vorticity formulation (1).In the spirit of quantization theory, the space of vorticity functions is replaced by the space () of skew-hermitian complex matrices, while the Poisson bracket is replaced by the matrix commutator.The size  of the matrices is the spatial discretization parameter, related to 'Planck's constant' in quantum theory via ℏ = 1/.We present an overview of how functions and matrices are related in section 2. There are at least two advantages of Zeitlin's model.First, it yields a spatial discretization that preserve all the underlying geometry of the Euler equations: the Hamiltonian structure and the conservation laws.Combined with a symplectic timeintegration scheme one can obtain fully structure preserving numerical methods [35].Second, complicated topological or geometrical properties of the Euler equations can be described in terms of standard tools from linear algebra and matrix Lie groups.Indeed, our splitting naturally arise from the standard spectral decomposition of Hermitian matrices applied to the stream matrix.The splitting has a precise geometric meaning in terms of Lie algebras, but also a dynamical interpretation as the steady and unsteady vorticity components.The details are given in section 3.
Numerical experiments for the canonical splitting are presented and discussed in section 4. We demonstrate that the components in the splitting convergence into a separation of scales.In addition, they capture broken line energy spectra.
In section 5 we translate our results about Zeitlin's model to the continuous Euler equations.All matrix concepts used for the canonical splitting have classical, fluid dynamical counterparts.But to define them rigorously requires prudence.It is essential to use a weak formulation.The section sets the foundation for an  ∞ -based theory of canonical vorticity splitting, independent from Zeitlin's model.It is the most mathematical part of the paper.Even so, a heuristic explanation is straightforward.Let us give it here.For a smooth vorticity function , the aim is to construct a splitting  =   +   .Let () be a closed level curve of .We define   , restricted to , to take the constant value In other words,   is obtained via averaging of  along the level curves, or streamlines, of .The mathematical difficulties arise where the level curves contain bifurcations.
In our context, quantization means to find a projection from smooth functions  ∞ (S 2 ) to complex skew-Hermitian matrices () such that the Poisson bracket under this map is approximated by the matrix commutator.
Quantization on the sphere is explicit.Indeed, Hoppe and Yau [19] gave an operator Δ  on () that, up to the truncation index , has the same spectrum as the Laplace-Beltrami operator on the sphere.The eigenvectors of Δ  are thereby quantized versions    of the spherical harmonics   .By computing the eigenbasis    , which is fast due to the diagonal structure of the Hoppe-Yau Laplacian, an explicit quantization scheme for the sphere is obtained.It yields a projection map Π  :  ∞ (S 2 ) → () relating functions to matrices via For more details and explicit formulae we refer to Hoppe and Yau [19], Zeitlin [48], Modin and Viviani [35].
where  ∈ () is the vorticity matrix and  ∈ () is the stream matrix.The condition correspond to vanishing total circulation ∫ S 2  = 0.The Euler-Zeitlin equations (2) have been studied in various contexts, primarily on the flat torus [47,33,1], but more recently also on the sphere [48,35].Their main feature is that they preserve the rich geometry in phase space of the original equations (1), namely the Lie-Poisson structure (see Modin and Viviani [37,35] for details).In turn, this implies conservation of total energy  () = Tr()/2, (quantized) Casimirs   () = Tr(  ), and angular momentum L = (  ,   ,   ).No conventional spatial discretization of the Euler equations preserve all these properties.Remark 2.1.Although the spherical setting is more complicated to work with, the Euler-Zeitlin equations are actually more accurate on the sphere than on the torus.There is a deep geometrical reason for this: quantization on the sphere exactly preserves the SO(3) symmetry, whereas the corresponding translational symmetry in the quantization of the torus is only approximately captured.This difference is evident in the definition of the discrete Laplace operator on the sphere and the torus.
In our previous work [37,35] we develop a Lie-Poisson preserving numerical method for the Euler-Zeitlin equations on the sphere and we use it to study the longtime behaviour.As reported also by Dritschel et al. [11], the numerical results give strong evidence against the predictions of statistical mechanics theories, derived for the sphere by Herbert [17]; see also Bouchet and Venaille [4].Rather, the results suggest the existence of near-integrable parts of phase space that act as barriers for the statistical predictions to be reached.Those near integrable solutions take the form of interacting vortex blobs (3 or 4 depending on the total angular momentum), perfectly reflecting integrability results for Hamiltonian blob dynamics on the sphere [36].
During our work with Zeitlin's model a new point of view emerged.More than a spatial discretisation, the Euler-Zeitlin equations themselves provide tools for studying the long-time behaviour of 2D hydrodynamics.Those tools include, in particular, Lie theory for (), which is exceptionally well understood, for example from the point of quantum theory, representation theory, and linear algebra.In particular, looking through the lens of Lie algebra theory, it is natural to split the vorticity matrix  by orthogonal projection onto the stabilizer of the stream matrix ; the underpinning of this paper.
By simulating the Euler-Zeitlin equations (2) using our Lie-Poisson integrator, and then for each output compute the canonical splitting, we see that it captures the dynamics of vortex condensation and scale separation, directly related to the theory of Kraichnan [25] for an inverse energy cascade.

Canonical splitting of the vorticity matrix
In this section we present and discuss canonical vorticity splitting for Zeitlin's model."Canonical" means the splitting only depends on the Lie algebra structure (and, of course, on the vorticity matrix  and the stream matrix  governing the dynamics).It does not require any ad hoc choice of scale as previous methods do.The separation of scales results from the dynamics.
Consider again the Euler-Zeitlin equations (2).Equip () with the Frobenius inner product.The canonical splitting of the vorticity matrix is obtained by taking   to be the orthogonal projection of  onto the stabilizer of the stream matrix If  is generic, so all its eigenvalues are distinct, then stab  is equivalently given by In this case   is obtained via the spectral decomposition: first find  ∈ SU() which diagonalizes , i.e.,  †  = Λ, then set Π  : () → stab  as Remark 3.1.Relative to the splitting (3), the Euler-Zeitlin equations (2) can be written Thus,  =   +   is a steady solution (equilibrium) if and only if   = 0, so the dynamics is "driven by" the residual part   .

Dynamics of 𝑊 𝑠 and 𝑊 𝑟
For insight on the splitting (3) let us express the dynamics of the Euler-Zeitlin equations in the variables   and   .Consider first a general flow on () of the form for some smooth vector field  : () → ().Let  ∈ SU() and Λ ∈ diag  denote an eigenbasis and corresponding eigenvalues for .Given (4) we first determine the evolution of  and Λ.The Lie algebra () is foliated in orbits (cf.Kirillov [23]) given by In the generic case (when all eigenvalues are distinct), the tangent space where   are the components of  in the basis , and     /(   −   ).Thus, in the generic case we can construct the generator () from the eigenvalues  1 , . . .,   and the eigenbasis  1 , . . .,   of .This allows us to write equation (4) in terms of the eigenvalues and eigenbasis of  as The matrices    = i   †  ∈ () forming the eigenbasis of  are quantized analogues of the level curves of the stream function .
We now apply the Lie theory machinery to obtain the dynamics of   and   .From the definition of   we have where we used Π  ( ) = 0 and   † = −  † .The dynamics for  is similar Hence, a formula for   † is needed.But we know that the dynamics of  can be orthogonally decomposed as ).Notice that   † can be taken in stab ⊥  .In fact, the dynamics of   remains the same for any   † + , where  ∈ stab  .The map In conclusion we have derived the following result for the dynamics of   and   .Theorem 3.1.Let  =  () and  = () be the vorticity and stream matrix for a solution to the Euler-Zeitlin equations (2).Let   and   respectively be the orthogonal projections of  onto stab  and its orthogonal complement as in (3).Then   and   satisfy the following system of equations where  = Δ −1  (  +   ) and  is the unique solution in stab ⊥  to From Theorem 3.1 we can deduce properties of   and   .First, if   = 0 then  ∈ stab  ∩ stab ⊥  so  = 0. Conversely, if  = 0 then   = 0 and Hence, in that case   plays the role of a fixed topography, and   satisfies the Euler-Zeitlin equation with forcing (6).From equation ( 5) we deduce that  = 0 also implies Another observation is that if [,   ] = 0 then  = 0 so   = 0, and vice versa if Π  [,   ] = 0 then again equation (7) holds.This means that it is possible to have an evolution of the eigenvectors of  without any change of the eigenvalues, but not the other way around.
Remark 3.2.The projection Π  : () → stab  has rank  in the generic case.Hence, the dynamics of   in the moving frame    can be described by only  components, i.e., its eigenvalues.Therefore, the vorticity splitting can be interpreted as reduced dynamics.As we shall see in section 5 below, the projection Π  is a quantized version of a mixing operator.Such operators was used by Shnirelman [42] to characterise stationary flows.

Energy and enstrophy splitting
Let us now study how energy and enstrophy relate to the canonical splitting (3).Since Tr(  ) = 0, the energy, corresponding to the energy norm, fulfils Tr Yet, the enstrophy, corresponding to the enstrophy norm, fulfils ).This gives the interesting relations Notice that  (  ) is always larger than  () and that the energy of   and   have to increase or decrease at the same rate.On the other hand, if the enstrophy of   decrease with a rate then the enstrophy of   must increase with the same rate.The canonical splitting thus coheres with Kraichnan's (1967) description of an inverse energy cascade and a forward enstrophy cascade.
The energy-enstrophy splitting (8) has a geometric interpretation.It says that  and   are orthogonal in the energy norm, but   and   are orthogonal in the enstrophy norm.Consider the plane spanned by  and   , and let   =  (  ) and  0 =  ().Then, since   = (0, √   ) and  = ( √  0 , 0) in this plane, energy correspond to the Euclidean norm on R 2 .We want to express the enstrophy norm relative to the energy norm.First observe that   = ( √  0 , − √   ).The positive definite matrix  for the enstrophy inner product restricted to the (,   )-plane can be written  =  , where the matrix  ∈ R 2×2 is determined by  • (0, Here,  is the angle between  and   in the enstrophy norm, and  0 =  ().Then we have Moreover, with notation as above Proof.First notice that   = 0 if and only if  = 0 since √︁  (•) is a norm.The inequalities (9) then follow from (8) and the fact that the enstrophy is always larger than the energy.To get the second inequality of (10), we use that the largest eigenvalue of the discrete Laplacian Δ  is ( − 1).
Remark 3.3.In the limit  → ∞ the ratio sin 2 /  in (11) is potentially unbounded.It could happen, and in fact does happen, that the enstrophy norm of   is far from zero, while its energy norm tends to zero.This corresponds to   being shifted towards small scales while not decreasing its enstrophy.It is a manifestation of Kraichnan's theory of forward enstrophy and inverse energy cascades.

Dynamically emerging scale separation
In this section we give numerical evidence that the canonical vorticity splitting (3) capture the dynamics of scale separation.We also provide some theoretical arguments in support.To give a complete mathematical proof that fully explains the numerical observations is a challenge not addressed in this paper.
A principal motivation for studying the vorticity splitting  =   +   is that unsteadiness precisely means non-vanishing of   .From an analytical point of view,   represents a projection of  onto a smoother subspace.Indeed, the relation via the Laplace operator between  and  says that  admits two more spatial derivatives  1, 2, 4. The energy   decays almost to zero, so that most of the energy is contained in   (reflecting the inverse energy cascade).On the other hand, the enstrophy   increases over time (reflecting the forward enstrophy cascade).than .Hence, since   is related to  via a polynomial relationship,   is in general more regular than .Vice versa,   contains the rougher part of .The tempting conjecture is that   represents the low-dimensional, large-scale dynamics, whereas   represents the noisy, small-scale dynamics.To assess this conjecture, we present two numerical simulation, both with randomly generated, smooth initial data.These two simulations represent the two generic behaviours described by Modin and Viviani [35]: formation of either 3 or 4 coherent vortex blob formations, strongly correlated to the momentum-enstrophy ratio.

Vanishing momentum simulation
This simulation starts from smooth, randomly generated initial data.Each spherical harmonic coefficient   in the range 2 ≤  ≤ 10 was drawn from the standard normal distribution.Remaining coefficients were set to zero.The data was then transformed to a vorticity matrix of size  = 512 (see Modin and Viviani [35] for details and explicit formulae).The so generated initial conditions are available in the supplementary material for reproducibility.
For time discretization of the Euler-Zeitlin equations (2) we use the second order isospectral midpoint method [37].The time step length is 1.2239 • 10 −4 s.This corresponds to 0.2 time units in Zeitlin's model which scales time by 4 √ / 3/2 .To visualise the fluid motion stages, we sample at initial time ( = 0), at intermediate time during mixing ( = 13 s), and at long time well after the large scale condensates are formed ( = 318 s).The vorticity matrices at these outputs are then transformed to functions in azimuth-elevation coordinates.The result is shown in Figure 1.
Due to vanishing momentum ( 1 = 0 in the initial data), integrability theory for Hamiltonian dynamics on the sphere suggests that 4 vortex blobs should appear [35,36].Indeed they do appear, and then pass into quasi-periodic orbits.The first We have run many more simulation with randomly generated initial conditions.The two simulations presented here capture the universal behaviour in all simulations.movie of the supplementary material captures the entire process.On top of the large scale condensates, a noisy, fine scale structure emerge.In essence we see a separation of scales.Figure 2 displays azimuth-elevation fields for the stream matrix  and the vorticity matrix components   and   at the same sampled output times.After some time of initial mixing the large scales of the vorticity are all contained in   , whereas   collects the small-scale fluctuations.The long time   state resembles noise, but not completely uniform.In fact, the non-uniform character captures the quasi-periodic dynamics since  = [,   ].An open problem is to model the noise by a stochastic process.
Figure 3 shows the evolution of the energy and enstrophy of   and   .Over time   increases whereas   decreases.Also notice that   ,   ,   , and   fulfil the energy and enstrophy relations (8).The residual vorticity   decreases in energy norm but increases in enstrophy norm.This is a quantification of Kraichnan's (1967) qualitative description.
The scale separation of the vorticity is even more clear in the spectral domain.Figure 4 contains energy spectra for ,   , and   at the sampled output times.The energy level  (), corresponding to the wave-number  = 1, . . ., , contains the energy of the modes for the spherical harmonics   with  = −, . . ., .We notice that the energy spectrum of  is similar in nature to that described by Boffetta and Ecke [3].Indeed, the "broken line" slope in the energy spectrum of  originate from an  −3 slope of   and an  −1 slope of   .Thus, the vorticity splitting yields a scale separation of the vorticity field that exactly reflect the broken line spectra previously observed in numerical simulations and empirical observations.The broken line spectrum is not reached at the intermediate time, before mixing has settled.

Non-vanishing momentum simulation
In this simulation the initial data was generated much like before, but now the range of non-zero spherical harmonics coefficients  is 1 ≤  ≤ 10.This means total angular momentum is no longer zero.For reproducibility also of this simulation, the generated initial conditions are available in the supplementary material.Time discretization, step size lengths, etcetera are selected as in the previous simulation.
Figure 5 shows azimuth-elevation fields corresponding to the stream matrix , the vorticity matrix , and the components   and   , sampled at initial time ( = 0), intermediate time ( = 13s), and long time ( = 344s).The entire motion is captured in the second movie of the supplementary material.Three vortex blobs emerge.The formation of these, from initial data with non-vanishing momentum, is again predicted and demonstrated by Modin and Viviani [35].It reflects integrability of low-dimensional Hamiltonian dynamics on the sphere.As before, the large scale vorticity patterns are contained in the   component.The   component swiftly develop noisy fluctuations.At long time it is less uniform than in the vanishing momentum simulation.The reason is that the 3 blobs here move faster than the 4 blobs in the previous simulation.
The scale separation of the vorticity is again evident from the energy spectrum of . Figure 6 gives energy spectra for the three vorticity fields ,   , and   .The results are analogous to those in Figure 4.

Stream function versus vorticity branching and blobs
In the literature on 2D turbulence steady solutions are often characterised by a functional relation between stream function and vorticity.Branching in such relations has been used as a measure of unsteadiness [11].
Since  =   if and only if  is a steady solution, it is natural to consider scatter plots between values of  and   .Such diagrams are given at the initial, intermediate, and long times, in Figure 7 for the vanishing momentum simulation, and in Figure 8 for the non-vanishing momentum simulation.
The following interpretation of branches is primal over interpretations related to unsteadiness.Each branch represents and characterises a specific blob in   .Upward branches represent blobs with positive values.Vice versa for downward branches.It is particularly clear at the long times, where there are fewer blobs.But the interpretation is valid also at the initial and intermediated times.This is revealed by carefully comparing the branching diagrams with the values of  and   in Figure 2 and Figure 5.The end of each specific branch (sometimes they overlap) is then readily identified with a specific blob.How steep the branch is corresponds to   /.This can be used to determine the shape of the blob, assuming axi-symmetry.For example, in Figure 8 at long time, the steeper of the two upward branches correspond to the left-most, sharper of the two positive blob in Figure 5.The two negative blobs at long time in Figure 2 are almost indistinguishable, which is reflected as overlapping downward branches in Figure 7.

Canonical splitting of the vorticity function
In this section we map our results for the Euler-Zeitlin equations to the original Euler equations (1).Indeed, all the concepts needed in the canonical splitting for the Euler-Zeitlin equations have classical counter-parts; some are listed in Table 1.However, one has to be careful to rigorously define these concepts: the transition from the quantized to the classical equations is valid only in a weak sense.Mathematically, the correct framework is  ∞ weak-star convergence.Formally, we may nevertheless proceed as follows, keeping in mind that concepts are transferable only in the weak sense.The key ingredient is that the projection Π  onto the stabilizer of  corresponds to averaging along the level-sets of the stream function .This gives the canonical splitting for the vorticity function via the projector Π  as The projector Π  is time dependent, since the level curves of  change with time.
Let us now proceed with more mathematical details on this constructions.First, recall again Euler's equations in vorticity formulation To define the continuous analogue of the vorticity matrix splitting, we have to understand the equations (12) in the weak sense.Indeed, in general the projections Π  and Π ⊥  cannot preserve the smoothness of .But for any  ∈ [1, ∞] they are continuous operators from  0 (S 2 ) to   (S 2 ) with operator norm one.Since continuous functions are dense in   (S 2 ), we can extend Π  and Π ⊥  to continuous operators on  ∞ (S 2 ).This result fits well with the global well-posedness of equations (12), which precisely requires a vorticity function in  ∞ [46,32].
To show this, let us first give equations (12) in the weak sense.For any  ≥ 2, if  ∈   (S 2 ) then  ∈  2,  (S 2 ) ⊂  1 (S 2 ).We define the weak Poisson bracket as {, }, for any test function  ∈  ∞ (S 2 ).Hence, we define the stabilizer of  as Next we define the  2 orthogonal projection Π  onto stab  .
for  → ∞.We want to show that  ∈ stab  .Let  be a function in  1 (S 2 ), then we get for  → ∞.As anticipated above, the operator Π  has an explicit form when evaluated on continuous functions.To state it, we first make the following assumption on the critical points of .Assumption 1.Let  ∈  1 (S 2 ) be the stream function.Then the critical points of  define a set of zero Lebesgue measure on S 2 , such that it is never dense in any neighbourhood of one of its points.
We say that  is generic whenever it satisfies Assumption 1.Consider now some  ∈  1 (S 2 ).Then  ∈ stab  if and only ∇  and ∇ are parallel.Since we take  to be generic, the points where { | ∇() = 0} lie on a set of zero measure, nowhere dense.Therefore, since  is continuous,  ∈ stab  if it is constant on the connected components of the level curves of .Then the projection of  onto stab  can be defined by evaluating  on the level curves of , i.e., the streamlines.Let  be a connected component of a streamline, then define the projection Π  :  1 (S 2 ) → stab  as In the limit when  approaches a single point, clearly Π  () | =  ().
The operator Π  does not, in general, preserve continuity of  .Indeed, consider a bifurcation saddle point  ∈ S 2 , i.e.  is a saddle point of  such that the streamline passing through  contains a bifurcation point.We then have the following result.Proposition 5.2.Let  be generic and Π  the projector as defined in (13).Then, if  ∈ S 2 is a bifurcation saddle point for , there exists  ∈  1 (S 2 ), such that Π  (  ) is discontinuous at the streamline passing through .Vice versa, given  ∈  1 (S 2 ), if Π  (  ) is discontinuous at some point  ∈ S 2 , then the streamline passing through  contains a bifurcation saddle point.
Proof.(Sketch) The issue about the continuity of  can be treated locally.Hence, let us work in Cartesian coordinates.Let  ∈ S 2 be a bifurcation saddle point for  and  the streamline passing through .Then, let  be a curve intersecting  only in  and let  be a smooth function positive at one side of  and negative at the other one, such that ∫    = 0. Then being  a bifurcation point, for any neighbourhood  of , there exist streamlines totally contained in one or another side of .Then, the average of  on those streamlines is either strictly positive or negative, creating a discontinuity at . Vice versa, let  ∈  1 (S 2 ), such that Π  (  ) is discontinuous at some point  ∈ S 2 .Then, the streamline passing through  cannot be homeomorphic to any of those in some tubular neighbourhood.Hence, the streamline passing through  must contain a critical point for , which also is a bifurcation saddle point.
However, we have the following regularity for Π  .Proposition 5.3.For any  ∈ [1, ∞], the operator Π  can be extended to a bounded operator with norm one on   (S 2 ).
Proof.Let us first notice that Π  is defined from  0 (S 2 ) to  1 (S 2 ), and satisfies Π    1 ≤   1 , for any  ∈  0 (S 2 ).Since  0 (S 2 ) is dense in  1 (S 2 ), it is possible to extend Π  to a bounded operator on  1 (S 2 ).Secondly, Π  is well defined from the space of simple functions to  ∞ (S 2 ), and satisfies Π    ∞ ≤   ∞ , for any  simple.Since the space of simple functions is dense in  ∞ (S 2 ), it is possible to extend Π  to a bounded operator on  ∞ (S 2 ).Furthermore, since Π  fixes the constant functions, it must be that Proof.We prove the result for  ∈  1 (S 2 ), then conclude by extension.Let  ∈ stab  .Then, ∇ is parallel to ∇ almost everywhere.Hence, the gradient of  along any streamline must vanish, and so on each connected component it is constant.By continuity of  we deduce that  must be constant also on the streamlines containing critical points.Therefore, Π   = .Assume now that Π   = .Then  must be constant on each connected component of a streamline.Hence, ∇ is orthogonal to the streamlines.Since ∇ is also orthogonal to the streamlines, we conclude that {, } = 0, i.e.  ∈ stab  .
We are now in position to derive continuous analogues of the results in section 3 (which, remember, are based on Lie theory for matrices).First, the stream function  satisfies the equation  = Δ −1 {, Δ}.This equation is not Hamiltonian.But we can split the right hand side into a Hamiltonian and non-Hamiltonian part via the projection Analogous to the quantized case, we seek a generator for It is clear that a necessary condition for the equation {, } =  to have a solution  is that {, } = 0 for any  ∈ stab  .
However, in general equation ( 14) can be solved only where ∇ ≠ 0. Around the critical points of  the gradient of  is potentially unbounded.Moreover, the right hand side in equation ( 14) can be discontinuous at the level curves of  containing saddle points of .Hence,  can only be defined almost everywhere.Furthermore, we have the following: Proposition 5.5.Let  ∈  0 (S 2 ) and  be generic.Then  ∈ stab ⊥  if and only if there exists  almost everywhere smooth, such that {, } =  on the set ∇ ≠ 0.
Proof.The if part is clear.Let instead take  ∈ stab ⊥  .Then, for any point  ∈ S 2 , we have to solve the PDE for : where ∇ ⊥  =  × ∇.If ∇ does not vanish, equation ( 15) can be solved by integration in the direction of ∇ ⊥ .In the points where ∇ does not vanishes, ∇ is not defined by equation (15), and it can be unbounded around those points.Hence, the field  is almost everywhere smooth and satisfies {, } =  , where ∇ ≠ 0.

Dynamics of 𝜔 𝑠 and 𝜔 𝑟
To derive the dynamical equations for   , we cannot directly define the field  corresponding to the quantized field  above.Instead, we consider the volume preserving vector field  [] := Π ⊥  Δ −1 {, Δ}.We note that  corresponds to the infinitesimal action of a map   which transports  by deforming its level curves.Hence,   acts naturally on stab  .Let us write Π   for the projection onto stab  at time .Then, for any point  ∈ S where  is implicitly defined by the third equation and  = Δ −1 (  +   ).We notice that the equations of motion for   can also be written in a more compact form as where the square bracket is the commutator of operators.Finally, notice that the energy and enstrophy splitting is valid also in the classical setting  () =  (  ) −  (  )  () =  (  ) +  (  ).

Conclusions and outlook
Based on the Euler-Zeitlin equations we have reported on a new technique for studying 2D turbulence via canonical splitting of vorticity.In numerical simulations this splitting dynamically develops into a separation of scales.These numerical results are supported by some theoretical evidence.To develop a full mathematical understanding, even within the setting of Zeitlin's model, is an interesting open problem.We have further presented mathematical foundations for a weak  ∞ theory in the continuous setting, independent of Zeitlin's model.
As the numerical simulations so strikingly capture the scale separation process, and as the inverse relations of the corresponding energy-enstrophy splitting reflect the stationary theory of Kraichnan, it is likely that further numerical and theoretical studies of the canonical vorticity splitting shall unveil more details on the mechanism behind vortex condensation.Furthermore, the splitting into large scales   and small scales   suggests to use these variables as a basis for stochastic model reduction (cf.Jain et al. [20]) of the two-dimensional Euler equations, with   modelled as multiplicative noise.
Since the eigenvalues of  are constant in time, the dynamics of   ,   variables is simpler than the one of   ,   .In fact, we have that  =  and so ).To understand how the dynamics of the   ,   variables look, we consider the first numerical simulation of Section 4. In Figure 9 the three stream function fields ,   ,   are shown at the times  = 0 and  =   .We notice that from a very smooth , the projection Π  onto the stabilizer rougher field  produce a much coarser image.In particular, both   and   do not show any additional structure or scale separation unlike to   and   .

Figure 1 :
Figure 1: Evolution of vorticity for Euler's equations on the sphere.Vorticity regions of equal sign undergo merging to form stable, interacting vortex condensates.
where {•, •} is the Poisson bracket on S 2 ,  is the vorticity function of the fluid, related to the fluid velocity v via  = curl v, and  is the stream function, related to the vorticity function via the Laplace-Beltrami operator Δ. Geometrically, the equations (1) constitute an infinite-dimensional Lie-Poisson system [2, and references therein].The phase space consists of vorticity fields and is equipped with the following infinitedimensional Poisson bracket ≺ ,  ()

Figure 2 :
Figure 2: Vanishing momentum simulation.Progression of the stream matrix  and the components   and   for the same simulation as in Figure 1.Initially   and   are similar in nature, but they evolve so   traps the large-scale condensates whereas   captures the small-scale fluctuations.

elevationPFigure 5 :
Figure 5: Non-vanishing momentum simulation.Progression of the stream matrix  and the components   and   of the vorticity matrix .Initially   and   are similar in nature, but they evolve so that   contains the large-scale condensates whereas   contains the small-scale fluctuations.

Figure 7 :Figure 8 :
Figure 7: Vanishing momentum simulation.Values of  versus values of   .The end of each branch correspond to a blob in the   plot in Figure 2. Upward if the blob is positive, otherwise downward.For example, at intermediate time, the sharp upward branch close to the -axis matches the small positive blob of   in the lower right corner of the corresponding plot in Figure 2.
The vorticity formulation (1) uses only the Laplacian Δ and the Poisson bracket {•, •}.With the quantized analogues Δ  and [•, •] we thereby obtain the Euler-Zeitlin equations  = [, ], O  is spanned by {i   †  } ≠ , where  = [ 1 , . . .,   ] is an orthonormal eigenbasis of .The orthogonal directions   O ⊥  = span{i   †  } = stab  is the linear subspace of matrices in () sharing the same eigenbasis (they are simultaneously diagonalizable).Thus the two projections Π  : () → stab  and Π ⊥  Id − Π  : () → stab ⊥  correspond to decomposition in the basis {i   †  }  and {i   †  } ≠ .Notice, as expected, that neither Π  nor Π ⊥  depend on the eigenvalues of , only on the eigenbasis.We can now write equation (4) as  = Π   () + Π ⊥   ().The first part of the flow changes the eigenbasis but not the eigenvalues and vice versa.The question is: what is the generator of  ↦ → Π ⊥   ()?Since it is isospectral it should be of the form  ↦ → [(), ] for some () ∈ ().Let us denote  =  ().It is then straightforward to check that if all the eigenvalues  1 , . . .,   of  are different, then Figure 3: Vanishing momentum simulation.Evolution of the decomposed enstrophies   and   (left), and decomposed energies   and   (right).The dashed, vertical lines indicate the sample times in Figures Figure 4: Vanishing momentum simulation.Spectrum in log-log scale for the energies ,   , and   at the initial (left), intermediate (middle), and long time (right).The dashed lines indicate the slopes  −3 and  −1 .The slope of   is almost settled at the intermediate time.The slope of   take much longer to settle.At long time, the broken line spectrum of  is captured well by the components   and   , which themselves have almost the same average slope at each scale.
Figure 6: Non-vanishing momentum simulation.Spectrum in log-log scale for the energies ,   , and   at the initial (left), intermediate (middle), and long time (right).The dashed lines indicate the slopes  −3 and  −1 .The slope of   is almost settled at the intermediate time.The slope of   take much longer to settle.At long time, the broken line spectrum of  is captured well by the components   and   , which themselves have almost the same average slope at each scale.

Table 1 :
Dictionary between Euler's and Zeitlin's models of hydrodynamics.
[42,43]by the Riesz-Thorin theorem, we conclude that Π    = 1, for any  ∈[1, ∞].Hence, from now on, let us consider equations(12)in the weak form, for  ∈  ∞ (S 2 ).It is clear that Π 2  = Π  .Moreover, we can formally define the operator Π  via the kernel    (), for any ,  ∈ S 2 where   is the connected component of the streamline passing through .In this way we get that Π  is self-adjoint with respect to the  2 inner product, i.e., for any  ,  ∈  ∞ (S 2 )By extension these equalities are valid for  ,  ∈   (S 2 ) whenever  ∈ [1, ∞].We notice that the operator Π  , defined by the kernel  (, ), is a mixing operators (or polymorphisms or bistochastic operators), as introduced by Shnirelman[42,43].Such operators give rise to a partial ordering on  2 (S 2 ): for any  ,  ∈  2 (S 2 ),   if there exist a mixing operator, defined via the kernel , such that  =  * .In his work, Shnirelman shows that stationary flows are characterised as minimal elements of this ordering.In a way, our work shows that it is enough to consider mixing operators of the form Π  .Within this class,  is minimal if there exists a stream function such that Π   = .As we see next, this in turn implies that  is stationary.Let  ∈  1 (S 2 ) be generic.For  ∈  ∞ (S 2 ) we then have  ∈ stab  ⇐⇒ Π   = .