G. Falkovich

February 2006

Conformal invariance in 2d turbulence

Simplicity of fundamental physical laws manifests itself infundamental 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

E

1

2u

2d2x

Z

1

22d2x

In fully developed turbulence limit, Re=UL-> ∞ (i.e. ->0):

(because dZ/dt≤0 and Z(t) ≤Z(0))

u

t uu

p

2u u f

u0

u

t uu

p

2u u f

u0

The double cascade Kraichnan 1967

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

kF

Two inertial range of scales:•energy inertial range 1/L<k<kF

(with constant )•enstrophy inertial range kF<k<kd

(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 randomized Original

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.