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Statistics of weather fronts and modern mathematics

Statistics of weather fronts and modern mathematics. Gregory Falkovich Weizmann Institute of Science. D. Bernard, A. Celani, G. Boffetta, S. Musacchio. Exeter, March 31, 2009. Euler equation in 2d describes transport of vorticity. Family of transport-type equations. m=2 Navier-Stokes

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Statistics of weather fronts and modern mathematics

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

  2. Euler equation in 2d describes transport of vorticity

  3. 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

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

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

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

  7. Inverse Q-cascade

  8. Small-scale forcing – inverse cascades

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

  10. _____________ =

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

  12. Vorticity clusters

  13. Schramm-Loewner Evolution (SLE)

  14. What it has to do with turbulence?

  15. C=ξ(t)

  16. m

  17. 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

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

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