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Stochastic geometry of turbulence
Gregory FalkovichWeizmann Institute
November 2014
D. Bernard, G. Boffetta, A. Celani, S. Musacchio, K. Turitsyn, M. Vucelja
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Fractals, multi-fractals and God knows what
depends neither on q nor on r - fractal
depends on q – multi-fractal
depends on r - God knows what
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Turbulence is a state of a physical system with many degrees of freedom
deviated far from equilibrium. It is irregular both in time and in space.
Energy cascade and Kolmogorov scaling
Transported scalar (Lagrangian invariant)
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Full level set is fractal with D = 2 - ζ
Random Gaussian Surfaces
What about a single isoline?
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3d is a mess
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Schramm-Loewner Evolution - SLE
2d is a paradise
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What it has to do with turbulence?
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C=ξ(t)
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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
ζ
m
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Small-scale forcing – inverse cascades
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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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Scalar exponents ζ of the scalar field (circles) and stream function (triangles), and universality class κ for different m
ζ κ
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Inverse cascade versus Direct cascade
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M Vucelja , G Falkovich & K S Turitsyn Fractal iso-contours of passive scalar in two-dimensional smooth random flows. J Stat Phys 147 : 424–435 (2012)
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Smooth velocity, locally anisotropic contours
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Within experimental accuracy, isolines of advected quantities are conformal invariant (SLE) in turbulent inverse cascades. Why?
Vorticity isolines in the direct cascade are multi-fractal.
Isolines of passive scalar in the Batchelor regime continue to change on a time scale vastly exceeding the saturation time of the bulk scalar field.Why?
Conclusion
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