G. Falkovich February 2006 Conformal invariance in 2d turbulence

Simplicity of fundamental physical laws manifests itself in fundamental symmetries. Strong fluctuations - infinitely many strongly interacting degrees of freedom → scale invariance. Locality + scale invariance → conformal invariance

Conformal transformation rescale non-uniformly but preserve angles z

2d Navier-Stokes equations In fully developed turbulence limit, Re=UL  -> ∞ (i.e. ->0): (because dZ/dt≤0 and Z(t) ≤Z(0)) 

The double cascade Kraichnan 1967 The double cascade scenario is typical of 2d flows, e.g. plasmas and geophysical flows. kFkF 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.

_____________ =

P  Boundary  Frontier  Cut points  Boundary  Frontier  Cut points

Schramm-Loewner Evolution (SLE)

C=ξ(t)

Vorticity clusters

Phase randomizedOriginal

Possible generalizations Ultimate Norway

Conclusion Within experimental acuracy, zero-vorticity lines in the 2d inverse cascade have conformally invariant statistics equivalent to that of critical percolation. Isolines in other turbulent problems may be conformally invariant as well.