Dresden, May 2010 Introduction to turbulence theory Gregory Falkovich

Plan Lecture 1 (one hour): General Introduction. Wave turbulence, weak and strong. Direct and inverse cascades. Lecture 2 (two hours): Incompressible fluid turbulence. Direct energy cascade at 3d and at large d. General flux relations. 2d turbulence. Passive scalar and passive vector in smooth random flows, small-scale kinematic magnetic dynamo. Lecture 3 (two hours): Passive scalar in non-smooth flows, zero modes and statistical conservation laws. Inverse cascades, conformal invariance. Turbulence and a large-scale flow. Condensates, universal 2d vortex.

 L Figure 1

Waves of small amplitude

Energy conservation and flux constancy in the inertial interval Kinetic equation Scale-invariant medium

Waves on deep water Short (capillallary) waves Long (gravity) waves Direct energy cascade Inverse action cascade

Plasma turbulence of Langmuir waves non-decay dispersion law – four-wave processes Direct energy cascades Inverse action cascades Interaction via ion sound in non-isothermal plasmaElectronic interaction

Strong wave turbulence Weak turbulence is determined by Strong turbulence depends on the sign of T For gravity waves on water

Burgers turbulence

Incompressible fluid turbulence

?

General flux relations

Examples

Kolmogorov relation exploits the momentum conservation

Conclusion The Kolmogorov flux relation is a particular case of the general relation on the current-density correlation function. Using that, one can derive new exact relations for compressible turbulence. We derived an exact relation for the pressure-velocity correlation function in incompressible turbulence We argued that in the limit of large space dimensionality the new relations suggest Burgers scaling.

2d turbulence two cascades

The double cascade Kraichnan 1967 The double cascade scenario is typical of 2d flows, e.g. plasmas and geophysical flows. kFkF Two inertial range of scales: energy inertial range 1/L<k<k F (with constant  ) enstrophy inertial range k F <k<k d (with constant  ) Two power-law self similar spectra in the inertial ranges.

Passive scalar turbulence Pumping correlation length L Typical velocity gradient Diffusion scale Turbulence - flux constancy

Smooth velocity (Batchelor regime)

2d squared vorticity cascade by analogy between vorticity and passive scalar

Small-scale magnetic dynamo Can the presence of a finite resistance (diffusivity) stop the growth at long times?

Lecture 3. Non-smooth velocity: direct and inverse cascades ? ?

Anomalies (symmetry remains broken when symmetry breaking factor goes to zero) can be traced to conserved quantities. Anomalous scaling is due to statistical conservation laws. G. Falkovich and k. Sreenivasan, Physics Today 59, 43 (2006)

Family of transport-type equations m=2 Navier-Stokes m=1 Surface quasi-geostrophic model, m=-2 Charney-Hasegawa-Mima model Kraichnan’s double cascade picture pumping k

Inverse energy cascade in 2d

Small-scale forcing – inverse cascades

Inverse cascade seems to be scale-invariant

Locality + scale invariance → conformal invariance ? Polyakov 1993

Conformal transformation rescale non-uniformly but preserve angles z

perimeter P  Boundary  Frontier  Cut points  Boundary  Frontier  Cut points Bernard, Boffetta, Celani &GF, Nature Physics 2006, PRL2007

Vorticity clusters

Connaughton, Chertkov, Lebedev, Kolokolov, Xia, Shats, Falkovich

Conclusion Turbulence statistics is time-irreversible. Weak turbulence is scale invariant and universal. Strong turbulence: Direct cascades have scale invariance broken. That can be alternatively explained in terms of either structures or statistical conservation laws. Inverse cascades may be not only scale invariant but also conformal invariant. Spectral condensates of universal forms can coexist with turbulence.

Turbulence statistics is always time-irreversible. Weak turbulence is scale invariant and universal (determined solely by flux value). It is generally not conformal invariant. Strong turbulence: Direct cascades often have symmetries broken by pumping (scale invariance, isotropy) non-restored in the inertial interval. In other words, statistics at however small scales is sensitive to other characteristics of pumping besides the flux. That can be alternatively explained in terms of either structures or statistical conservation laws (zero modes). Inverse cascades in systems with strong interaction may be not only scale invariant but also conformal invariant. For Lagrangian invariants, we are able to explain the difference between direct and inverse cascades in terms of separation or clustering of fluid particles. Generally, it seems natural that the statistics within the pumping correlation scale (direct cascade) is more sensitive to the details of the pumping statistics than the statistics at much larger scales (inverse cascade).

How decoupling depends on d? Pressure is an intermittency killer Robert Kraichnan, 1991

It is again the problem of zero modes