Conformal Invariance in Two-dimensional Turbulence

This talk is devoted to recent findings in the statistics of two-dimensional turbulence.
After a brief review of the classical phenomenology, on the basis of high resolution numerical simulations, I show that some features of two-dimensional turbulence display conformal invariance. In particular, the statistics of vorticity clusters in the inverse cascade is equivalent to that of critical percolation, one of the simplest universality classes of critical phenomena.Vorticity isolines are therefore fractal lines described by Stochastic Loewner Equation curves SLE6. This result is generalized to a class of 2d turbulent systems, including Surface Quasi-Geostrophic turbulence (which corresponds to SLE4) and Charney-Hasegawa-Mima turbulence.
The picture emerging from these results is that conformal invariance may be expected for inverse cascades in two-dimensions therefore opening new perspectives in our understanding of 2d turbulent flows.
References:

• D. Bernard, G. Boffetta, A. Celani, and G. Falkovich,
  Nature Physics 2, 124 (2006)
• G. Boffetta,
  J. Fluid Mech. 589, 253 (2007)
• D. Bernard, G. Boffetta, A. Celani, and G. Falkovich,
  Phys. Rev. Lett. 98, 024501 (2007)