Conformal invariance in two-dimensional turbulence Guido Boffetta Dipartimento di Fisica Generale University of Torino D.Bernard, G.Boffetta, A.Celani, G. Falkovich, Nature Physics, (2006)

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.

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

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

Exact results 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: Kolmogorov’s 4/5 law (1941)

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 km Nastrom, Gage, Jasperson, Nature 310 (1984) Lenght (km) k -5/3

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

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)

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

Direct numerical simulations of 2d turbulence U.Frisch, P.L. Sulem, Phys. Fluids 27, 1921 (1984): G.Boffetta, A.Celani and M.Vergassola, Phys. Rev. E 61, R29 (2000): S 5 (r) S 7 (r) Kolmogorov scaling: no intermittency

Direct numerical simulations of 2d turbulence Set of simulations at high resolutions with a parallel pseudo spectral code. (G. Boffetta and A. Celani, 2005) N  L/r F r F /r   I  I 20482x x x x Energy/enstrophy fluxes in spectral space Energy spectra k -5/3 k -3 Simultaneous observation of direct and inverse cascade 

Conformal invariance in 2d statistical physics Conformal invariance for the inverse cascade: geometrical properties (vorticity domains) stochastic Loewner equation 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 mapping Conformal mapping is a powerful tool for characterizing shapes in 2D by means of analytic functions. Consider a curve  t  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  ) is simply connected, thus  analytic function g : H\K  H g(z) maps the hull K on the real axis (and the growing tip  on a point  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t 0 Introducing the “time” t=a 2 /4, g t (z)  z+2t/z, for a vertical segment starting from  R: K tt tt H tip trace hull tt

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

Loewner equation (Loewner, 1923) A curve  t  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  on a point  R) Example: a vertical segment of length a starting from the origin: ia 0 The growth of the curve  t can be mapped on the evolution of the conformal mapping g t (z) (t parameterizes the curve): with g 0 (z) = z and driving  t =g t (  t ) K tt tt H tip trace hull tt The trace  t is univocally generated by the (continuous) driving  t which is at any time the map of the tip g(  t )=  t and conversely,  t determines g t (z) and thus  t trace  driving Example: The solution to LE with  t =  =const i.e. a segment of length a=2√t gtgt

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)

Stochastic Loewner Equation For applications in statistical mechanics we are interested in random curves  t : Loewner equation with a random driving  t (O.Schramm, 2000) diffusion coefficient  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) then (assuming reflection symmetry and continuity)  t is proportional to a random walk: 0 A conformal invariance Markov property

Phases of SLE (Rohde & Schramm, 2001) : The shape of the trace depends on the value of  : increasing  the trace turns more frequently * 0 <  < 4simple curve * 4 <  < 8non-simple curve (  intersections) *  > 8space filling Fractal dimension of SLE traces: D F =1+  /8 (Beffara, 2002) trace frontier For  > 4, the external frontier of the hull (i.e. the boundary of H\K t ) is a simple curve described by SLE  ’ with k’=16/k (thus D’ F =1+2/  ) Duplantier (2000); proven by Beffara (2002) for  =6 SLE duality

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(SLE 6 ) frontier:D=4/3(SLE 8/3 ) cut pointsD=3/4 SLE  and critical systems   =2loop-erased random walk   =8/3self avoiding random walk   =3cluster boundaries in Ising   =6cluster boundaries in percolation   =8 uniform spanning trees

Vorticity clusters in the inverse cascade of 2d turbulence

Single vorticity cluster

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

Probability distribution of vorticity clusters  Size  Boundary __ prediction SLE 6  Size  Boundary __ prediction SLE 6 size s= # connected sites of same sign boundary t= # connected sites adjacent to opposite sign Vorticity isoline as SLE 6 traces ? see Cardy and Ziff, J.Stat. Phys. 110, 1 (2003)

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 ?,  ?)

Deterministic example of slit maps inversion g t+  t = g  t º g t with g  t solution to LE with  constant from t and t+  t:  t = t sin(t) Inversion of SLE as composition of discrete slit maps over  t  t-1 z=  t gtgt with  t = Re(  t ) and  t = Im 2 (  t )/4 O(N 2 ) algorithm

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

Locality For  =6 the trace does not feel boundaries until it doesn’t hit them tt A H\A gtgt A’ tt H\A htht  ‘ t H  ’ t H h0h0 g’ t

Unrolling a vorticity isoline

Driving functions

Driving  (t) is Brownian motion   zero-vorticity lines are SLE   = 5.9  0.3

Vorticity clusters and percolation * 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) * 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) Comparison with a Gaussian field with same Fourier spectrum (phase randomization): check of the importance of dynamics Comparison with a Gaussian field with same Fourier spectrum (phase randomization): check of the importance of dynamics Is the inverse cascade just a complicate way to generate a percolation field ?

Phase randomizedOriginal

Gaussian field is not SLE

z  (Schramm, 2001) Probability that the trace  passes to the left of a point z (for  =6) Calculating with SLE: Schramm’s formula

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

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