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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
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
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)
Full level set is fractal with D = 2 - ζ
Random Gaussian Surfaces
What about a single isoline?
3d is a mess
Schramm-Loewner Evolution - SLE
2d is a paradise
What it has to do with turbulence?
C=ξ(t)
Euler equation in 2d describes transport of vorticity
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
This system describes geodesics on an infinitely-dimensional Riemannian manifold of the area-preserving diffeomorfisms. On a torus,
(*)
Add force and dissipation to provide for turbulence
lhs of (*) conserves
pumping
kQ
Kraichnan’s double cascade picture
P
Inverse Q-cascade
ζ
m
Small-scale forcing – inverse cascades
perimeter P
Boundary Frontier Cut points
Boundary Frontier Cut points
Bernard, Boffetta, Celani &GF, Nature Physics 2006, PRL2007
Scalar exponents ζ of the scalar field (circles) and stream function (triangles), and universality class κ for different m
ζ κ
Inverse cascade versus Direct cascade
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)
Smooth velocity, locally anisotropic contours
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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