Conformal invariance in two-dimensional turbulence

2. Guido Boffetta Dipartimento di Fisica Generale University of Torino Conformal invariance in two-dimensional turbulence D.Bernard, G.Boffetta, A.Celani, G. Falkovich, Nature Physics, 2 124 (2006) www.ph.unito.it/~boffetta

3. A physical motivation for two-dimensional turbulence 2D Navier-Stokes equation are a simple model for large scale motion of atmosphere and oceans: thin layers of fluid in which stratification and rotation supress vertical motions.

4. 2d Navier-Stokes equations Two inviscid quadratic invariants: Energy/enstrophy balance in viscous flows: (palinstrophy) In fully developed turbulence limit, Re=UL/n -> ∞ (i.e. n->0): (because dZ/dt≤0 and Z(t) ≤Z(0)) no dissipative anomaly for energy in 2d: no energy cascade to small scales !

5. 1/L kF kd The double cascade In the limit Re->∞, 2d turbulence shows a direct enstrophy cascade to small scales at rate z. Energy flows to large scales at rate e generating the inverse cascade. (Kraichnan 1967) • Two inertial range of scales: • energy inertial range 1/L < k < kF • (with constant e) • enstrophy inertial range kF < k < kd • (with constant z) Two power-law self similar spectra in the inertial ranges. The double cascade scenario is typical of 2d flows, e.g. plasmas and other geophysical models.

6. Exact results Kolmogorov’s 4/5 law (1941) Following the derivation obtained by Kolmogorov for 3d turbulence (Kolmogorov 4/5 law) is it possible to obtain for 2d cascades two exact results: inverse energy cascade: direct enstrophy cascade:

7. k-5/3 Lenght (km) Geophysical data Mesoscale wind variability (radar and balloon): k-5/3 K.S. Gage, J.Atmos.Sciences 36 (1979) GASP aircraft dataset: k-5/3 for wavelenghts 10-300 km Nastrom, Gage, Jasperson, Nature 310 (1984)

8. Early laboratory experiments Thin layer of mercury with electrical forcing in a uniform magnetic field suppressing vertical motions (linear friction due to Hartmann layer). J.Sommeria, JFM 170, 139 (1986) Energy spectrum

9. Laboratory experiments: soap films (Y. Couder, W. Goldburg, H. Kellay, M.A. Rutgers, M. Rivera, R.E. Ecke) interferometry, LDV, PIV M.A. Rutgers, PRL 81, 2244 (1998)

10. Laboratory experiments: electrolyte cell (P. Tabeling, J. Gollub, A. Cenedese) J. Paret, P.Tabeling, PRL 79 4162 (1997)

11. G.Boffetta, A.Celani and M.Vergassola, Phys. Rev. E 61, R29 (2000): 20482 U.Frisch, P.L. Sulem, Phys. Fluids 27, 1921 (1984): 2562 S7(r) S5(r) Direct numerical simulations of 2d turbulence Kolmogorov scaling: no intermittency

12. Direct numerical simulations of 2d turbulence (G. Boffetta and A. Celani, 2005) Set of simulations at high resolutions with a parallel pseudo spectral code. Simultaneous observation of direct and inverse cascade y w Energy/enstrophy fluxes in spectral space Energy spectra k-5/3 k-3

13. Conformal invariance in 2d statistical physics Under broad conditions: homogeneity + isotropy + scale invariance = invariance under conformal transformations (local combination of translation, rotation and dilatation, preserve angles) There are counterexamples (e.g. elasticity in 2d, Riva and Cardy 2005) Is there conformal invariance in two-dimensional turbulence? First attempt by Polyakov (enstrophy direct cascade, 1993) Conformal invariance for the inverse cascade: geometrical properties (vorticity domains) stochastic Loewner equation

14. tip tt H trace gt K hull xt tt 0 Conformal mapping is a powerful tool for characterizing shapes in 2D by means of analytic functions. Conformal mapping Consider a curve gtH starting from the origin (t parameterizes the curve) The complement of the hull K (the set of points which cannot be reached from infinity together with g) is simply connected, thus  analytic function g : H\K  H g(z) maps the hull K on the real axis (and the growing tip t on a point zR) This map is unique if we fix normalization, e.g. g(z)~z+O(1/z) as z Example: a vertical segment 0  z  i a=t Introducing the “time” t=a2/4 , gt(z)z+2t/z, for a vertical segment starting from xR:

15. tt+dt tt E.g: the map for the vertical segment is solution to LE with xt=x=const xt Loewner Equation The growth of the curve gt can be mapped on the evolution of the conformal mapping gt(z) For a trace growing in the upper half plane H from 0 to ∞ Loewner equation (1923) g0(z) = z * The trace gt is univocally (i.e. no branching) generated by the (continuous) driving xt which is at any time the map of the tip g(tt)= xt * Conversely, given gtwe can determine the hull Ktand thus the map gt(z) and the driving xt=gt(tt) trace  driving

16. tip tt H trace gt Example: a vertical segment of length a starting from the origin: K gt ia hull 0 xt Example: The solution to LE with xt=x=const with g0(z) = z and driving xt=gt(tt) i.e. a segment of length a=2√t Loewner equation (Loewner, 1923) A curve gtH starting from the origin defines a analytic function which maps the complement of the hull K to H: g : H\K  H g(z) maps the hull K on the real axis (and the growing tip t on a point xR) The growth of the curve gt can be mapped on the evolution of the conformal mapping gt(z) (t parameterizes the curve): The trace gt is univocally generated by the (continuous) driving xt which is at any time the map of the tip g(tt)= xtand conversely, gtdetermines gt(z) and thus xt trace  driving

17. An example of Loewner evolution (from driving to trace) driving trace for other examples see e.g. Kager, Nienhuis and Kadanoff, J. Stat. Phys. 115, 805 (2004)

18. 0 + - - + A - + - + + - - - - - + + + + + - - + - + - + - - - - - + + + + + - - - - - + + + + + 0 0 Stochastic Loewner Equation (O.Schramm, 2000) For applications in statistical mechanics we are interested in random curves gt: Loewner equation with a random driving xt Markov property conformal invariance then (assuming reflection symmetry and continuity) xt is proportional to a random walk: diffusion coefficient k parameterizes different universality classes of critical behavior. Problems in 2d critical systems reduced to problems in 1d Brownian motion (see Cardy, SLE for theoretical physicists, Ann.Phys. 318, 81 (2005)

19. trace frontier Phases of SLE (Rohde & Schramm, 2001): The shape of the trace depends on the value of k: increasing k the trace turns more frequently * 0 < k < 4 simple curve * 4 < k < 8 non-simple curve ( intersections) * k > 8 space filling Fractal dimension of SLE traces: DF=1+k/8 (Beffara, 2002) SLE duality For k > 4, the external frontier of the hull (i.e. the boundary of H\Kt) is a simple curve described by SLEk’ with k’=16/k (thus D’F=1+2/k) Duplantier (2000); proven by Beffara (2002) for k=6

20. Brownian motion • Old conjecture by Mandelbrot (1982): • the frontier of BM is a SAW with D=4/3 • Lawler, Schramm & Werner, 2000 (via SLE): • pioneer points: D=7/4 (SLE6) • frontier: D=4/3 (SLE8/3) • cut points D=3/4 SLEk and critical systems • k=2 loop-erased random walk • k=8/3 self avoiding random walk • k=3 cluster boundaries in Ising • k=6 cluster boundaries in percolation • k=8 uniform spanning trees

21. Vorticity clusters in the inverse cascade of 2d turbulence

22. Single vorticity cluster

23. Fractal dimensions of vorticity clusters • Boundary • Frontier • Cut points L=side of square covering the cluster k=6, k’=8/3 as in critical percolation H.Saleur and B.Duplantier, PRL 58, 2325 (1987)

24. Probability distribution of vorticity clusters • Size • Boundary __ prediction SLE6 see Cardy and Ziff, J.Stat. Phys. 110, 1 (2003) Vorticity isoline as SLE6 traces ? size s= # connected sites of same sign boundary t= # connected sites adjacent to opposite sign

25. Are vorticity isoline compatible with SLE traces ? From traces to driving functions * Generate isolines from vorticity field * Numerical inversion of SLE for obtaining associate driving functions * Compute statistical properties of driving functions (Brownian ?, k ?)

26. z=tt gdt xt-1 Inversion of SLE as composition of discrete slit maps over dt gt+dt = gdtº gtwith gdt solution to LE with xconstant from t and t+dt: with xt = Re(tt) and dt = Im2(tt)/4 O(N2) algorithm Deterministic example of slit maps inversion xt = t sin(t)

27. How we can apply to NS simulation in a periodic domain without boundaries ? SLE is defined for traces from two points on the boundary of a domain gt The problem of boundary conditions Locality: For k=6 the trace does not feel boundaries until it doesn’t hit them (obvious for percolation)

28. H\A gt g’t gt A H ht h0 H H\A g ’t A’ xt x ‘t Locality For k=6 the trace does not feel boundaries until it doesn’t hit them

29. Unrolling a vorticity isoline

30. Driving functions

31. Driving x(t) is Brownian motion zero-vorticity lines are SLEk k = 5.9  0.3

32. * Independent percolation: short correlated * Correlated percolation: * For H>3/4 same universality class of percolation (Harris, 1974) * For vorticity in inverse cascade i.e. H = 2/3 < 3/4 * In principle, different class from percolation (but maybe close) Vorticity clusters and percolation Is the inverse cascade just a complicate way to generate a percolation field ? Comparison with a Gaussian field with same Fourier spectrum (phase randomization): check of the importance of dynamics

34. Phase randomized Original

35. Gaussian field is not SLE

36. z q Calculating with SLE: Schramm’s formula Probability that the trace gpasses to the left of a point z (for k=6) (Schramm, 2001)

37. Calculating with SLE: Crossing formulae • Probability that in a rectangle of • aspect ratio r=y/x: • - a cluster crosses from top to bottom • four-legged cluster connects 4 sides Cardy (1992), Watts (1996) CFT Smirnov (2001), Dubeat (2004) SLE

38. Statistical mechanics of two-dimensional turbulent inverse cascade Zero-vorticity isoline are conformally invariant random curves They are compatible with SLE6 What about other 2D turbulent systems ? Is conformal invariance a general property of inverse cascade ? Is it always k=6 (percolation-like) ? ... see next talk !