1. Statistics of weather fronts and modern mathematics Gregory Falkovich Weizmann Institute of Science Exeter, March 31, 2009 D. Bernard, A. Celani, G. Boffetta, S. Musacchio

3. Euler equation in 2d describes transport of vorticity

4. Family of transport-type equations m=2 Navier-Stokes m=1 Surface quasi-geostrophic model, m=-2 Charney-Hasegawa-Mima model Electrostatic analogy: Coulomb law in d=4-m dimensions

6. This system describes geodesics on an infinitely-dimensional Riemannian manifold of the area-preserving diffeomorfisms. On a torus,

7. (*) Add force and dissipation to provide for turbulence lhs of (*) conserves

8. pumping k Q Kraichnan’s double cascade picture P

9. Inverse Q-cascade

10. Small-scale forcing – inverse cascades

11. Locality + scale invariance → conformal invariance ? Polyakov 1993

13. _____________ =

14. perimeter P  Boundary  Frontier  Cut points  Boundary  Frontier  Cut points Bernard, Boffetta, Celani &GF, Nature Physics 2006, PRL2007

15. Vorticity clusters

16. Schramm-Loewner Evolution (SLE)

18. What it has to do with turbulence?

19. C=ξ(t)

23. m

27. Different systems producing SLE Critical phenomena with local Hamiltonians Random walks, non necessarily local Inverse cascades in turbulence Nodal lines of wave functions in chaotic systems Spin glasses Rocky coastlines

28. Conclusion Inverse cascades seems to be scale invariant. Within experimental accuracy, isolines of advected quantities are conformal invariant (SLE) in turbulent inverse cascades. Why?