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