Remarks on 2D inverse cascade

Transcription

1 Remarks on 2D inverse cascade Franco Flandoli, Scuola Normale Superiore Newton Institute, Dec Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

2 Inverse cascade in 2D turbulence It is observed in 2D fluids that injection of energy at small scale produces subsequent activation of larger scales (a flux of energy from smaller to larger scales). It is called inverse cascade. We consider the case with dissipation at large scale, typically a friction (due to the 2D dimensionality). A stationary regime is established. An inertial range is formed where a special scaling law for the energy spectrum is observed: E (k) C ε 2/3 k 5/3 where ε = energy flux, scale independent. If u r denotes a typical value of (relative) velocity at scale r, then u r C ɛ 1/3 r 1/3. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

3 Dimensional analysis Dimensional analysis is essentially the only tool which allows one to identify this law. Assumptions: 1 E (k) depends only on k and on ε = energy flux 2 a scaling law holds: E (k) = C ε α k β. Then, since k has dimension [L] 1, E = [L]2 E 0 (k) dk has dimension, [T ] 2 hence E (k) has dimension [L]3 [T ] 2, ε has dimension [L]2 [T ] 3 (because it is de dt ), ( ) [L] 3 [T ] 2 = [L] 2 α ( [T ] 3 [L] 1) β Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

4 Dimensional analysis ( ) [L] 3 [T ] 2 = [L] 2 α ( [T ] 3 [L] 1) β α = 2 3 β = 3 2α = 5 3. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

5 Vorticity field Numerical simulation (from Boffetta-Ecke 12) A rigorous mathematical theory explaining the existence of inverse cascade is still missing, not to speak of E (k) C ε 2/3 k 5/3. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

6 Advantages and limitations of dimensional analysis Simple and fast; correct results in 2D Same argument in 3D gives a partially wrong result Based on assumptions which are plausible but not directly observable No equations of fluid motion, no explanation of the mechanism. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

7 The fluid dynamic equations Generally accepted model: the Navier-Stokes equations with friction and some forcing term: t ω + u ω + αω = ν ω + η The equation here is written in vorticity form. Here: u = velocity ω = vorticity η = force α = friction coeffi cient ν = viscosity coeffi cient. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

8 Small viscosity, simplified equations Usually viscosity is very small. Many aspects should be incorporated in the Euler equations t ω + u ω + αω = η. Viscosity, although small, could play a role in the so called anomalous viscous dissipation: a quantity is strongly dissipated in spite of infinitely small dissipation constant. In 3D there is anomalous energy dissipation. In 2D, thanks to a priori bounds on the enstrophy ω 2 (t, x) dx, energy is conserved. In 2D, there could be anomalous viscous dissipation of enstrophy. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

9 Invariant measures Major problem: identify invariant measures of t ω + u ω + αω = η which incorporate features of inverse cascade and energy spectrum. In the case of Navier-Stokes equations with noise forcing, under appropriate conditions the invariant measure is unique (Hairer-Mattingly; Kuksin, Sinai, Kupiainen and many others). The unique measure presumably contains the informations we are looking for but it is not explicit and almost nothing concrete is known for it. When viscosity ν goes to zero, there are limit points µ of the unique invariant measures µ ν, and such µ s (potentially non unique) are solutions of the Euler equation above. But this route remained abstract until now. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

10 Explicit invariant measures for Euler equation Let us simplify further and discuss first invariant measures for the Euler equations t ω + u ω = 0 Essentially two classes are known: 1 Measures concentrated on point vortex configurations; 2 Measures equal to, or based on, 2D white noise. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

11 Explicit invariant measures for Euler equation In rough terms, all these explicit measures are variations on the following idea: µ α,β (dω) = Z 1 α,β exp ( βh (ω) αe (ω)) dω where H (ω) = energy = u (x) 2 dx (also = G (x, y) ω (x) ω (y) dxdy, G (x, y) = Green function in the domain D R 2 of the fluid) E (ω) = enstrophy = ω 2 (x) dx which are two invariants of Euler motion, for regular solutions. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

12 Point vortices Special vorticity fields are the distributional ones ω N (t) = N Γ i δ Xi (t) i=1 (the intensities Γ i have the meaning of circulations). They are solution, in a very weak sense, of Euler equations, when d dt X i (t) = Γ j K (X i (t) X j (t)) j =i where K (x, y) = x G (x, y) is the interaction kernel (Biot-Savart), asymptotically given by K (x, y) (x y) x y 2 for small x y. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

13 Invariant measures for point vortices Lebesgue measure is invariant for point vortex dynamics. Define the interaction energy H int (ω N ) = N i,j=1 i =j Γ i Γ j G (X i, X j ), G (X i, X j ) log X i X j which is finite as soon as the points X 1,..., X N are different. The measure µ β,n (dx 1,..., dx N ) = Z 1 β,n exp ( βh int (ω N )) dx 1 dx N Z β,n = exp ( βh int (ω N )) dx 1 dx N is also invariant for the point vortex dynamics. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

14 Invariant measures: point vortices > vorticity fields Invariant measures for point vortex dynamics give rise to invariant measures for Euler equations. Define H 1 (D) : = ɛ>0 H 1 ɛ (D) M (D) : = space of signed measures H 1 (D) The measure µ β,n (dx 1,..., dx N ) on D N is mapped into a Borel measure µ β,n (dω) on H 1 (D) by the map N (X 1,..., X N ) Γ i δ Xi. i=1 Here for simplicity the case (Γ i ) constant is considered, but one can generalize to random circulations. The measure µ β,n (dω) is invariant for Euler equation in the sense it is the time-marginal of a stationary solution. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

15 Negative temperatures Onsager (around 1949) indicated the role of these measures, for β < 0 as relevant to describe aggregated vortex structures. From µ β,n (dω N ) = Z 1 β,n exp ( βh int (ω N )) dx 1 dx N for β < 0 configurations with high interaction energy H int (ω N ) are favoured. For equal sign vortices, namely Γ i Γ j > 0, high values of H int (ω N ) = N i,j=1 i =j Γ i Γ j log X i X j are obtained when vortices are close each other, aggregated. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

16 The mean field limit In suitable domains D, when β < 0 (in a suitable range) it is proved (Caglioti-Lions-Marchioro-Pulvirenti, and others) that µ β,n, as a measure on H 1 (D), under appropriate scaling of the circulations Γ N i, converges to δ ρr : µ β,n δ ρr where ρ r H 1 (D) is in fact smooth, it is a time-invariant solution of Euler equation, approximately equal to: ρ r (x) = r 2 ρ ( r 1 (x x 0 ) ), ρ (x) = 1 (1 + x 2) 2. The value of the scaling parameter r is related to β: β "large" small r small negative temperature high aggregation. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

17 The mean field profile ρ (x) = 1 (1 + x 2) 2 y This shape is a sort of universal vortex patch profile. x Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

18 The global mean field profile in decaying turbulence Large scale structures with profile ρ (x) are observed in certain real situations, typically in decaying turbulence, where the full system goes to a global equilibrium (although rescaled due to exponentially decaying intensity) Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

19 Another extreme: white noise vorticity The previous mean field profile is obtained when circulations Γ N i as 1 N. If we take Γ N i = ±1 N scale (with uniform random choice of ±1) and we choose the positions X i uniformly and independently in D, then ω N = N i=1 ±1 N δ Xi converges weakly to white noise. White noise is formally given by µ (dω) = Z 1 exp ( E (ω)) dω, E (ω) = ω 2 (x) dx. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

20 White noise vorticity White noise here is also called "entrophy measure" µ (dω) = Z 1 exp ( E (ω)) dω, E (ω) = ω 2 (x) dx. This measure was introduced in 2D fluid mechanics by Abeverio, Høegh-Krohn and coworkers around Albeverio-Cruzeiro 90 prove existence of a stationary solution of Euler equation with such marginal. The construction mentioned above as limit of point vortex dynamics is in F., to appear on CPDE. This measure has attracting features: the associated velocity field is a solenoidal analog of the Gaussian Free Field, conformal invariant (cf. Falkovich et al.) Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

21 White noise vorticity Simulations may look plausible (2000 vortices versus Rivera-Ecke 05) But the spectrum is wrong and aggregation is very weak. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

22 Energy-Enstrophy measures On H 1 (D) there are other invariant measures, of the form µ β (dω) = Z 1 β exp ( βh (ω) E (ω)) dω. A technical problem is the definition of the kinetic energy H (ω), infinite for elements of class H 1 (D). Called µ the enstrophy measure, the following limit exists in L 2 ( H 1 (D), µ ) : : H (ω) := L 2 µ lim N (H N (ω) E µ [H N ]) called renormalized kinetic energy. Then the measure µ β (dω) = Z 1 β exp ( β : H (ω) : E (ω)) dω is well defined (Albeverio and co.) as well as microcanonical analogs (F. Cipriano) µ [E 0,E 1 ] (A) = µ (A (: H (ω) : [E 0, E 1 ])). µ (: H (ω) : [E 0, E 1 ]) Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

23 Energy-Enstrophy measures as limit of point vortices Again, also for the measures µ β (dω) = Z 1 β exp ( β : H (ω) : E (ω)) dω µ [E 0,E 1 ] (A) = µ (A (: H (ω) : [E 0, E 1 ])). µ (: H (ω) : [E 0, E 1 ]) there are results claiming they are limit of point vortices: µ β as limit of µ β,n under the rescaling Γ N i = ±1 N (Benfatto-Picco-Pulvirenti, Bodineau-Guionnet, Grotto-Romito) µ [E 0,E 1 ] as limit of microcanonical analogs for point vortices (F.-Luo), same rescaling. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

24 Energy input and friction dissipation Concerning the more realistic model t ω + u ω + αω = η (1) F. Grotto has a work in preparation on invariant measures and point vortex approximation. Vortices are injected as a PPP and their intensity is dissipated by friction. For fields roughly speaking of the form one has the equation ω N (t) = N ±e αt δ Xi (t) i=1 N t ω N + u N ω N + αω N = δ (t t k ) ±1 δ Xi (t k N k ) and the limit N is investigated, proving convergence to a white noise solution of (1). Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

25 Back to 2D inverse cascade turbulence Problem: the previous measures do not have a clear multiscale structure similar to inverse cascade (only a weak form, the so called thick points, C. Antonucci and F. Grotto) and the spectrum is always very different from E (k) C ε 2/3 k 5/3. However, based on some of the previous elements plus additional ingredients, it is possible to give a picture of inverse cascade which may be in the right direction. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

26 From point vortices to vortex structures The first basic step is replacing point vortices by vortex structures. Identified structures in turbulence include blobs and filaments. I will use only blobs. I will support the idea of a system with a special form of local equilibrium and thus I will invoke Onsager structures ρ r (x) = r 2 ρ ( r 1 (x x 0 ) ) as preferential. ρ (x) = (1 + x 2) 2 Moreover, I think the correct results are obtained by a mechanical statistic viewpoint: each blob is made of a large number of vortex points. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

27 Configuration (initial phase) As in F. Grotto model, new structures are added by an external mechanism of vortex creation. At the beginning we have only a few small structures, quite separated. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

28 Distant subsystems From a statistical viewpoint, the subsystems are almost isolated when the distance is relatively large. The centers of mass move as point vortices, thus the blobs "dance" together, but the internal structure is relatively insensitive of the presence of the other blobs. This is rigorously described by t ω i + u ω i = 0 ω = N ω i, i=1 t ω + u ω = 0 u = 1 ω There are theorems stating that if ω i (0) δ Xi (0) and X i (t) remain at finite distance, then ω i (t) δ Xi (t). Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

29 Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

30 Subsystems close to each other Completely different is the behavior when two clusters are close enough. They undergo intense stretching, filamentation, and merging. This process is fundamental, and I presume the building brick of the inverse cascade. Unfortunately it is still poorly understood and quantified. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

31 Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

32 Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

33 Phenomenology If the two clusters are suffi ciently close to each other, they behave like a single system which starts outside equilibrium and tends to it. The equlibrium, as predicted by Onsager theory, is a single blob with shape approximately ρ r ( x x 0 ). The time of convergence to equilibrium is macroscopic, opposite to classical equilibrium statistical mechanics. It is of the order of one turnover time, where turnover is measured at distance r from the center. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

34 An example of non-equilibrium statistical mechanics Thus we have preliminarily identified a few main features of this "non-equilibrium" statistical mechanic system: locally the system is at equilibrium, but this concept is non-homogeneous in space, it applies to local portions of space that are quasi-isolated by "local" we do not mean arbitrarily small (from the macroscopic view-point); locality is somewhat quantized the quantized equilibrium configurations have a scaling structure convergence to equilibrium of these local subsystems requires a macroscopic time. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

35 Some mathematics Define the interaction energy between two Onsager structures at distance d 1 g (d) = log x y ρ (x) ρ (y + de 1) y x Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

36 Some mathematics Then total kinetic energy of these two structures, when at scale r Γρ r ( x x 0 ) + Γρ r ( x y 0 ) is H initial = Γ 2 ( 4 log 1 r + 2g (0) + 2g ( y 0 x 0 )) If these two structures merge into an Onsager structure the final kinetic energy is Γρ r (x) H final = Γ 2 ( log 1 r + g (0) ). Below we shall see that Γ = 2Γ and then we get a relation between r and y 0 x 0. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

37 Some mathematics On the energy layer [h, ), for the motion of (X 1,..., X N ), consider the microcanonical measure Theorem µ [h, ) N = 1 {Hint [h, )}/Leb 2N (H int [h, )). Given [h, ) there is β (h) such that (for smooth test functions φ) lim dµ [h, ) N Γ N D N N N φ (x i ) Γ φρ β(h) dx i=1 D = 0 Putting together the two arguments, let us start from two point vortex clusters (X i ), (Y j ) Γ N N i=1 δ Xi + Γ N N ( δ Yj Γρ rini x x 0 ) ( + Γρ rini x y 0 ) j=1 ranco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

38 Some mathematics that has total energy h Γ 2 ( 4 log 1 r + 2g (0) + 2g ( y 0 x 0 )) we have a family (Z i ) = (X i, Y j ) Γ N 2N δ Zk = (2Γ) 1 2N k=1 2N δ Zk k=1 that lives around the energy layer {H int = h} and thus, with resepct to the microcanonical ensamble, we expect (2Γ) 1 2N 2N δ Zk (2Γ) ρ r. k=1 Here r could be computed by investigating the link β (h), or by energy conservation ( 4Γ 2 log 1 r ) + g (0) = Γ (4 2 log 1 r + 2g (0) + 2g ( y 0 x 0 )). Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

39 Anomalous dissipation of enstrophy When two sub-systems merge Γ N N i=1 δ Xi + Γ N N j=1 δ Yj Γ N 2N δ Zk k=1 Γρ rini ( x x 0 ) + Γρ rini ( x y 0 ) 2Γρ rfin ( x z 0 ) they preserve kinetic energy (for point vortices H int (ω N ) is conserved, and it is an approximation of the kinetic energy of the continuum limit) but they dissipate enstrophy: using the relation between r and r obtained by energy conservation, one can prove that ( ) 2 ( 2 2Γρ rfin (x) dx < 2 Γρ rini (x)) dx 2r 2 fin < r 2 ini r fin > 2r ini (numer. r fin 1.68 r ini ) Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

40 From local to global picture The previous one is the local picture. The system is moving through a sequence of meta-stable equilibrium states. There are clusters of point vortices of size r injection << 1 produced by some external mechanism. When the distance is less or around some thresold, pairs of clusters merge and form larger clusters. The process repeats, with larger and larger clusters arising. We observe a bath of clusters of different sizes. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

41 In order to conjecture a "structure" in this disorder of clusters, we idealize again and assume that: Hp. 1 merging between pairs of clusters of different sizes and simultaneous merging between more than three clusters are negligible events with respect to merging between clusters of the same size. In other words, in the sequel we shall only take into account merging between clusters of the same size. Consequence of this is that sizes are quantized: r injection = r linj <... < r l+1 < r l <... < r 0 and we have a vorticity field of the form ( ) ω (x) = Γ l ρ rl x x 0 i. l i Λ l Remark. Assumption Hp.1 is made here for simplicity; its validity or its irrelevance for the final result are under investigation. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

42 Hp. 2 Consider a two-blob configuration ρ ( x x1 0 ) ( ) + ρ x x 0 2 of unitary size and intensity and its point vortex approximation 1 N N i=1 δ Xi + 1 N N j=1 δ Yj. Assume x 0 1 x2 0 d. In a time of order one the point vortex field made of (X i (t), Y j (t)) approximates a blob of size α > 1. There are three elements in this assumption, inspired by the numerical simulations: a typical (average) coalescing distance d a typical (average) size α of the final structure, (numer. α 1.68) mixing time is of the order of one turnover time, hence order one from u = K ω. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

43 Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

44 Rescaling assumpion Hp.2 Corollary ( ) ( ) Given a two-blob configuration Γρ r x rx Γρr x rx 0 2 of size r and its point vortex approximation Γ N N i=1 δ (r ) X + Γ N N j=1 δ (r ) i Y, if j rx 1 0 rx0 2 d r, in a time of order τ = r 2 ( ) Γ the point vortex field made of X (r ) i (t), Y (r ) j (t) approximates a blob of size αr. Proof. The point vortices X i (t) : = r 1 X (r ) i Y j (t) : = r 1 Y (r ) j ( r 2 t/γ ) ( r 2 t/γ ) approximate a two-blob configuration ρ ( x x1 0 ) ( ) + ρ x x 0 2 of size one at time zero and satisfy the dynamics of point vortices. Since x 1 0 x 2 0 d, Hp.2 tells us that (X i ( (t), Y j (t)) approximates ) a blob of size α in a time of order one. Back to X (r ) i (t), Y (r ) j (t) gives the result. ranco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

45 Summary Summarizing: we have a multiscale field with ( ) ω (x) = Γ l ρ rl x x 0 i l i Λ l r injection = r linj <... < r l+1 < r l <... < r 0 r l = αr l+1 and mixing time at level l of order τ l = r l 2. Γ l Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

46 Injection rate, energy flux, time-stationarity We now assume that (by some mechanism) structures of level l inj appear at random following a PPP in space-time with average number of events in unit of space-time given by λ linj. This provokes an input energy flux (per unit of space-time) ɛ = λ linj ɛ linj where ɛ linj is the kinetic energy of one structure of level l inj. We assume the system reaches stationarity. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

47 Energy flux at all scales The "linear structure" built above (clusters of size r l+1 which merge into clusters of size r l+1 etc.) implies (as in queuing networks) that the same energy flux ɛ (per unit of space-time) is observed at each scale. This is not precise since a frictional dissipation mechanism should be present, which dissipates uniformly at all scales. But locally in scales (especially at small scales) its effect is secondary if the friction constant is very small. Hence we have the same rule at every level: ɛ = λ l ɛ l where λ l is the average number of events at level l in unit of space-time (by event we mean either new arrivals or departures) and ɛ l is the kinetic energy of one structure of level l. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

48 Energy of individual clusters A blob Γρ r (x) has energy log x y Γρ r (x) Γρ r (y) dxdy = Γ 2 log r (x y) ρ (x) ρ (y) dxdy Γ 2 log 1 r Hence ɛ l Γ 2 l log 1 r l and we get ɛ = λ l ɛ l λ l Γ 2 l log 1 r l. It remains to compute λ l. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

49 The mean throughput In queuing theory λ l is the throughput, or average arrival/departure rate. (A version of) Little s law states λ l = n l (and we already know that τ l r l 2 τ l where n l is the average number of structures involved in potential merging events. By equilibrium considerations (n l rl 2 area occupied by merging structures) Collecting these facts we have n l C rl 2. ɛ = n l ɛ l C Γ3 l τ l rl 4 log 1. r l Γ l Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

50 Correct energy scaling It follows Γ l C ɛ 1/3 r 4/3 l up to logarithmic corrections in r l hence, denoting by u l = Γ l r l the typical velocity at level l, u l C ɛ 1/3 r 1/3 l which gives the correct scaling law. [Logarithmic corrections have been invoked in the literature; experiments do not clarify, due to few scales.] Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

51 Summary Numerical observation of cluster merging (+Onsager theory applied locally) support the idea of a sort of queuing network, in a stationary regime; in the simplest case of a linear queue we use the relation ɛ = n l τ l ɛ l. The relations τ l r l 2 Γ l and ɛ l Γ 2 l log 1 r l come from rigorous scaling arguments; but at level l = 0 they are both conjectured from numerical observations and Onsager theory. n l C r 2 l is based on equilibrium considerations. Together they produce ɛ Γ3 l r 4 l and the correct scaling u l C ɛ 1/3 r 1/3 l. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

52 Open questions Very diffi cult: define an explicit probability measure on vortex configurations which reflects these ideas build rigorous tools to deal with this kind of non-equilibrium systems (with quantized local equilibrium, macroscopic relaxation time), in analogy with the hydrodynamic limits based on local equilibrium Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

53 Open questions Partial steps: try to disprove the previous picture, in order to identify what should be understood better prove fragments of this scheme, like a quantification of the convergence to Onsager equilibrium, or a description of the onset of the cascade investigate simplified fluid dynamic models apt to rigorous results, like dyadic models for inverse 2D turbulence example: L.A. Bianchi and F. Morandin, following their work CMP 2018 on 3D structure function for dyadic models on trees, are proving that no anomalous scaling (no intermittency) holds for Obukov inverse cascade model on a tree. Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54

54 Thank you for your attention Franco Flandoli, Scuola Normale Superiore () 2D inverse cascade Newton Institute, Dec / 54