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Stochastic geometry of turbulence. Gregory Falkovich Weizmann Institute. D. Bernard , G. Boffetta, Celani, S . Musacchio , K. Turitsyn,M . Vucelja. APS meeting, 28 February 2012. Fractals, multi-fractals and God knows what. depends neither on q nor on r - fractal.

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Stochastic geometry of turbulence

Gregory Falkovich

Weizmann Institute

D. Bernard, G. Boffetta,

Celani, S. Musacchio,

K. Turitsyn,M. Vucelja

APS meeting, 28 February 2012

fractals multi fractals and god knows what
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

What about a single isoline?

Random Gaussian Surfaces


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,



  • Frontier
  • Cut points

perimeter P

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




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)



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.