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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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slide1

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

slide3

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)

slide4

Full level set is fractal with D = 2 - ζ

What about a single isoline?

Random Gaussian Surfaces

slide12

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

slide13

This system describes geodesics on an infinitely-dimensional Riemannian manifold of the area-preserving diffeomorfisms. On a torus,

slide19

Boundary

  • Frontier
  • Cut points

perimeter P

Bernard, Boffetta, Celani &GF, Nature Physics 2006, PRL2007

slide24
Scalar exponents ζ of the scalar field (circles) and stream function (triangles), and universality class κ for different m

ζ

κ

slide27

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)

slide36

Conclusion

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?