INVERSE CASCADE on HYPERBOLIC PLANE

Transcription

1 INVERSE CASCADE on HYPERBOLIC PLANE Krzysztof Gawe,dzki based on work in progress with Grisha Falkovich Nordita, October 2011 to Uriel No road is long with good company turkish proverb

2 V.I. Arnold, Ann. Inst. Fourier (1966): incompressible Euler equation describes geodesics on the group of area-preserving diffeomorphisms can be written on any space with Riemannian metric (g ij ) in the form t v i +v k k v i = g ij j p ր Levi Civita covariant derivative տ inverse meric with incompressibility condition i ( gv i ) = 0 for g det(gij ) Noether symmetries of the Euler equation are given by currents (J 0,J i ) that are conserved: t ( gj 0 ) + i ( gj i ) = 0 In 2D incompressibility implies the existence of stream function ψ such that v j = ǫ ij 1 g i ψ In this case the scalar vorticity ω = ǫij g i v j = g ij i j ψ evolves by t ω = v i i ω = 1 g ǫ ij ( i ω)( j ψ)

3 Upon addition of the viscous dissipation and a source one obtains the forced Navier-Stokes equation with t v i +v k k v i ν( v) i = g ij j p+f i տ Laplace Beltrami operator ( v) i = g jk j k v i + g ij ( j k k j )v k In 2D we shall assume random force f i that is a Gaussian process with covariance f i (t 1,x 1 )f j (t 2,x 2 ) = δ(t 1 t 2 ) ǫ ki ǫ lj g(x1 ) g(x 2 ) x k 1 x l C ( ρ(x 1,x 2 )) 2 l f for C( ) a fast decreasing function, ρ(x 1,x 2 ) the geodesic distance, and l f the forcing scale

4 Arnold considered general geometries to study topological properties of flows: Arnold-Khesin, Topological Methods in Hydrodynamics, Springer 1998 Our motivation: search of conformal symmetry in 2D inverse cascasde signaled by numerical discovery of SLE statistics of 0-vorticity lines in Bernard-Boffetta-Celani-Falkovich, Nature Physics 2 (2006) Main idea: conformal symmetry may be easier to find in different 2D geometries supporting inverse cascade This appeared to be the case for the NS flows on the hyperbolic plane but it did not yet throw light on the (conjectured) SLE statistics

5 Why hyperbolic plane? it is a 2D space with 3-dimensional symmetry group (as the flat plane) and a constant negative curvature 2R 2 the 2D sphere with constant positive curvature 2R 2 also has a 3-dimensional symmetry group but no space to develope inverse cascade the hyperbolic plane has more room at large scales than the flat space: the circumference of the circle of radius ρ is equal here to 2πRsinh ρ R no geometric mechanism that would block the development of inverse cascade

6 Hyperbolic (upper half-)plane Lobachevsky-Bolyai plane Poincaré disc Upper hyperboloid H R in 3D Minkowski space M 3 with signature (+,+, ) H R = { (X 1,X 2,X 3 ) X 2 1 +X 3 2 X 2 3 = R 2, X 3 > 0 } X 3 X 2 X 1

7 Isometry group of H R = 3D Lorentz group = SL(2,R)/{±1} Convenient parametrization of H R : X 1 = rcosϕ, X 2 = rsinϕ, X 3 = R 2 +r 2 In terms of stream function ψ such that v r = R 2 +r 2 Rr ϕ ψ, v ϕ = R 2 +r 2 Rr r ψ and vorticity ω = ( R 2 +r 2 Rr r r R 2 +r 2 R ) r + 1 r 2 2 ϕ ψ the Euler equation on H R becomes t ω = R 2 +r 2 Rr ( ) ( r ω)( ϕ ψ) ( ϕ ω)( r ψ)

8 Noether symmetries of Euler equation on H R correspond to conserved currents (J 0, J r, J ϕ ) time translation invariance gives J 0 E = 1 2 ( R 2 R 2 +r 2 (vr ) 2 +r 2 (v ϕ ) 2), տ energy density J r,ϕ E = (J0 E +p)vr,ϕ 3D Lorentz group invariance gives for J 0 X = (v rx r +v ϕ X ϕ ), տ momentum density X r = { R 2 +r 2 sinϕ (R 2 +r 2 )cosϕ, 0 X ϕ = J r,ϕ X = J0 X vr,ϕ px r,ϕ { 1+r 1 R 2 +r 2 cosϕ r 1 R 2 +r 2 sinϕ 2 no analogue of Galilean invariance!

9 In the Navier-Stokes equation with the viscous dissipation and random Gaussian forcing (f r,f ϕ ) as given before one has the energy balance t J 0 E = ν ω 2 dissipation rate ε + ( l f 2 C (0) ) injection rate ι Flat space inverse cascade scenario of Kraichnan (1967) and Batchelor (1969) - well substantiated by theory, simulations, and experiments Scales: l ν dissipative scale l f forcing scale ρ running scale At scales ρ and long times energy flows into a condensate mode In terms of stream functions: ψ(t,x1 )ψ(t,x 2 ) 1 2 ιρ2 t + const.ι 2/3 ρ 8/ with ρ = x 1 x 2 and... not contributing to velocity 2-pt function

10 Inverse cascade - condensation scenario on H R We postulate that ψ(t,x1 )ψ(t,x 2 ) Ψ 0 (x)t + Ψ st (x) +... for x cosh( ρ R ) with ρ the hyperbolic distance, cosh( ρ R )= R 2 +r 1 2 R 2 +r 2 2 r 1 r 2 cos(ϕ 1 ϕ 2 ) R 2 One more length-scale present: R l f! For l f ρ R this expression should agree with the flat space one, in particular, we should have R 2 Ψ 0 (1) = ι տ energy injection rate We shall try to find the form of modes Ψ 0, Ψ st for R ρ using scaling arguments The scenario seems self-consistent but, at the end, its credibility should be tested numerically!

11 Scaling theory For equal-time velocity correlation n-pt functions F n,m H R,ν,C,l f (t;r 1,ϕ 1,...,r n,ϕ n ) = one has a tautological scaling relation m j=1 v r (t;r j,ϕ j ) n j=m+1 λ 2 3 n m F n,m H R,ν,C,l f (λ 2 3t;λr 1,ϕ 1 ;...;λr n,ϕ n ) = F n,m H R/λ,λ 4/3 ν,λ 2 C,l f /λ (t;r 1,ϕ 1 ;...;r n,ϕ n ) v ϕ (t;r j,ϕ j ) The forcing on both sides corresponds to the same energy injection rate ι Scaling limit λ of RHS should describe stochastic Euler equation on the (light-)cone H 0 in 3D Minkowski space yielding the long time - large distance asymptotics of the inverse cascade on H R A geometric effect: far away H R looks like H 0!

12 Euler equation on H 0 = { (X 1,X 2,X 3 ) X 2 1 +X2 2 = } X2 3 X 3 Parametrization of H 0 : X 1 = rcosϕ, X 2 = rsinϕ, X 3 = r X 2 X 1 H 0 inherits from 3D Minkowski space a degenerate metric Isometry group of H 0 = Diff(S 1 ) = 1D conformal group (= half of 2D conformal group) 3D Lorentz group Conformal symmetry arises similarly as in the AdS-CFT correspondence! In terms of stream function ψ such that v r = ϕ ψ, v ϕ = r ψ and the vorticity ω = r (r 2 v ϕ ) = r (r 2 r ψ) the Euler equation takes the standard looking form t ω = (v r r +v ϕ ϕ )ω = ( r ω)( ϕ ψ) ( ϕ ω)( r ψ)

13 Noether symmetries of Euler equation on H 0 correspond to conserved currents J 0 E = 1 2 r2 (v ϕ ) 2, J r,ϕ E = (J0 E +p)vr,ϕ J 0 ζ = r2 v ϕ ζ, J r ζ = J0 ζ vr +rpζ, J ϕ ζ = J0 ζ vϕ pζ for any periodic function ζ(ϕ) (integral of r 2 v ϕ along each light-ray in H 0 is separately conserved) Diff(S 1 ) symmetry is spontaneously broken to the Lorentz one in the λ scaling limit of the stochastic NS equation on H R! The precise nature of this breaking remains to be understood The scaling limit has a tautological scale invariance: λ 2 3 n m F n,m H 0 (λ 2 3t;λr 1,ϕ 1 ;...;λr n,ϕ n ) = F n,m H 0 (t;r 1,ϕ 1 ;...;r n,ϕ n )

14 For the velocity 2-pt functions on H 0 there are 2 scale invariant solutions that in terms stream functions have the form ψ(t,x1 )ψ(t,x 2 ) = ( A(lnx) 2 +Blnx ) t 6A(lnx)tlnt +... ψ(t,x1 )ψ(t,x 2 ) = Cx 1/ for x = r 1 r 2 ( ) 1 cos(ϕ 1 ϕ 2 ) L 2 with an arbitrary length scale L In the inverse cascade-condensate scenario for stochastic NS equation on H R they imply for the condensate and stationary modes the behavior: Ψ 0 (x) Ψ st (x) x 1 ιr 2( (lnx) 2 lnx ) +... const.ι 2/3 R 8/3 x 1/ x 1 (t ln t term absorbes a logarithmic divergence of the rescaled 2-pt function λ 2/3 F 2,2 H R,ν,C,l f (λ 2/3 t;λr 1,ϕ 1 ;λr 2,ϕ 2 ))

15 X 3 For the Lorentz-invariant 2-pt function using Minkowskian scalar product v(t,x 1 ) v(t,x 2 ) of vectors tangent to H R v(t,x1 ) v(t,x 2 ) = R 2 x (x 2 1) x ( Ψ0 (x)t + Ψ st (x) ) v X 1. x 1 x 2. v X 2 where x = cosh ( ρ R) this implies that the condensate contribution t and equal to 2ιt for ρ R decreases linearly in hyperbolic distance ρ for ρ R the stationary contribution ι 2/3 ρ 2/3 for ρ R decreases exponentially R 2/3 ι 2/3 e ρ/(3r) for ρ R no obvious contradiction but what s the physics behind? Spectral interpretation via SL(R, 2)-related Fourier analysis on H R is possible

16 Condensate contribution to the invariant velocity 2-pt function

17 Flux relation In the flat space the inverse cascade scenario implies the flux relation Θ ( x 2 x 1 x 2 x 1 v(t,x 1) )( v(t,x 1 ) v(t,x 2 ) = 1 2 ιρ On H R the flux relation takes the form Θ ( e(x 1,x 2 ) v(t,x 1 ) )( v(t,x 1 ) v(t,x 2 ) ) ( = 2R 1 sinh ρ ) ( Ψ R 0 cosh( ρ )) R where e(x 1,x 2 ) is the unit vector tangent at x 1 to the geodesic curve joining x 1 to x 2 This agrees with the flat space ezpression for ρ R since R 2 Ψ 0 (1) = ι but behaves like 1 2 ιρ for ρ R

18 What is the physics of the inversion of sign of ρ Θ around ρ = R?

19 Conclusions Our analysis confirms the inverse cascade- condensation scenario in stochastic NS equation on the hyperbolic plane but questions about physical interpretation of the result remain Asymptotic behavior of the condensate and stationary modes at distances ρ R were determined by a scaling limit that lives on the cone H 0 Precise way in which this limit breaks the Diff(S 1 ) symmetry of the Euler dynamics on H 0 remains to be understood It may be a clue to an eventual link between this asymptotic symmetry and the SLE statistics of 0-vorticity lines Numerical simulations of forced NS flows on H R would be welcome Experimental realizations of such flows are difficult since H R cannot be embedded isometrically into the 3D Euclidian space! In particular, soap films would not do but they may provide indication on the effect of negative curvature on the inverse cascade

20 Croyez que tout mortel a besoin d indulgence Fénelon