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Introduction to turbulence theory

A comprehensive introduction to turbulence theory by Gregory Falkovich, covering wave turbulence, incompressible fluid turbulence, passive scalar dynamics, inverse cascades, and more. The lectures delve into the energy cascades, flux relations, 2D turbulence, and the interaction of waves in various mediums. The text explores the general flux relations, the Kolmogorov relation, double cascades in 2D flows, and the behavior of turbulence in both weak and strong regimes. Concluding with a discussion on statistical conservation laws, anomalies in turbulence, and the coexistence of spectral condensates with turbulent flows.

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Introduction to turbulence theory

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  1. Introduction to turbulence theory Gregory Falkovich http://www.weizmann.ac.il/home/fnfal/ Dresden, May 2010

  2. 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.

  3. W L Figure 1

  4. Waves of small amplitude

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

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

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

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

  9. Burgers turbulence

  10. Incompressible fluid turbulence

  11. ?

  12. General flux relations

  13. Examples

  14. Kolmogorov relation exploits the momentum conservation

  15. 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.

  16. 2d turbulence two cascades

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

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

  19. Smooth velocity (Batchelor regime)

  20. 2d squared vorticity cascade by analogy between vorticity and passive scalar

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

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

  23. 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)

  24. 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 k pumping

  25. Inverse energy cascade in 2d

  26. Small-scale forcing – inverse cascades

  27. Inverse cascade seems to be scale-invariant

  28. Locality + scale invariance → conformal invariance ? Polyakov 1993

  29. Conformal transformation rescale non-uniformly but preserve angles z

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

  31. Vorticity clusters

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

  33. 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.

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