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

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Presentation on theme: "G. Falkovich February 2006 Conformal invariance in 2d turbulence."— Presentation transcript:

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

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

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5 2d Navier-Stokes equations In fully developed turbulence limit, Re=UL  -> ∞ (i.e. ->0): (because dZ/dt≤0 and Z(t) ≤Z(0)) 

6 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.

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9 _____________ =

10 P  Boundary  Frontier  Cut points  Boundary  Frontier  Cut points

11 Schramm-Loewner Evolution (SLE)

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14 C=ξ(t)

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16 Vorticity clusters

17 Phase randomizedOriginal

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20 Possible generalizations Ultimate Norway

21 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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