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G. Falkovich

G. Falkovich. Conformal invariance in 2d turbulence. February 2006. Simplicity of fundamental physical laws manifests itself in fundamental symmetries. Strong fluctuations - infinitely many strongly interacting degrees of freedom → scale invariance.

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G. Falkovich

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  1. G. Falkovich Conformal invariance in 2d turbulence February 2006

  2. Simplicity of fundamental physical laws manifests itself in fundamental symmetries. Strong fluctuations - infinitely many strongly interacting degrees of freedom → scale invariance. Locality + scale invariance → conformal invariance

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

  4. 2d Navier-Stokes equations a In fully developed turbulence limit, Re=UL/n -> ∞ (i.e. n->0): (because dZ/dt≤0 and Z(t) ≤Z(0))

  5. 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.

  6. _____________ =

  7. Boundary • Frontier • Cut points P

  8. Schramm-Loewner Evolution (SLE)

  9. C=ξ(t)

  10. Vorticity clusters

  11. Phase randomized Original

  12. Possible generalizations Ultimate Norway

  13. Conclusion Within experimental acuracy, zero-vorticity lines in the 2d inverse cascade have conformally invariant statistics equivalent to that of critical percolation. Isolines in other turbulent problems may be conformally invariant as well.

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