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Statistics of weather fronts and modern mathematics
Gregory FalkovichWeizmann Institute of Science
Exeter, March 31, 2009
D. Bernard, A. Celani,G. Boffetta, S. Musacchio
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Euler equation in 2d describes transport of vorticity
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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
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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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pumping
kQ
Kraichnan’s double cascade picture
P
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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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perimeter P
Boundary Frontier Cut points
Boundary Frontier Cut points
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=ξ)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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