Statistics of weather fronts and modern mathematics

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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
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Euler equation in 2d describes transport of
vorticity
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Family of transport-type equations
m2 Navier-Stokes m1 Surface
quasi-geostrophic model, m-2 Charney-Hasegawa-Mi
ma model
Electrostatic analogy Coulomb law in d4-m
dimensions
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This system describes geodesics on an
infinitely-dimensional Riemannian manifold of the
area-preserving diffeomorfisms. On a torus,
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Add force and dissipation to provide for
turbulence
()
lhs of () conserves
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Kraichnans double cascade picture
Q
P
k
pumping
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Inverse Q-cascade
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Small-scale forcing inverse cascades
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Locality scale invariance ? conformal
invariance ?
Polyakov 1993
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_____________

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• Boundary
• Frontier
• Cut points

perimeter P
Bernard, Boffetta, Celani GF, Nature Physics
2006, PRL2007
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Vorticity clusters
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Schramm-Loewner Evolution (SLE)
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What it has to do with turbulence?
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C?(t)
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m
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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

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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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