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Emerging Symmetries and Condensates in Turbulent Inverse Cascades

Lack of scale-invariance in direct turbulent cascades, Navier-Stokes equations, Charney-Hasegawa-Mima model, electrostatic analogy, condensation in 2D turbulence, vorticity clusters, Bose-Einstein condensation, atmospheric turbulence, condensates breaking symmetries.

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Emerging Symmetries and Condensates in Turbulent Inverse Cascades

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  1. Emerging symmetries and condensates in turbulent inverse cascades Gregory Falkovich Weizmann Institute of Science Cambridge, September 29, 2008 כט אלול תשס''ח

  2. Lack of scale-invariance in direct turbulent cascades

  3. 2d Navier-Stokes equations Kraichnan 1967

  4. 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 lhs of (*) conserves k pumping

  5. Small-scale forcing – inverse cascades

  6. Strong fluctuations – many interacting degrees of freedom → scale invariance. Locality + scale invariance → conformal invariance Polyakov 1993

  7. _____________ =

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

  9. Vorticity clusters

  10. Schramm-Loewner Evolution (SLE)

  11. C=ξ(t)

  12. Different systems producing SLE • Critical phenomena with local Hamiltonians • Random walks, non necessarily local • Inverse cascades in turbulence • Nodal lines of wave functions in chaotic systems • Spin glasses • Rocky coastlines

  13. Bose-Einstein condensation and optical turbulenceGross-Pitaevsky equation

  14. Condensation in two-dimensional turbulence M. G. Shats, H. Xia, H. Punzmann & GF, Suppression of Turbulence by Self-Generated and Imposed Mean Flows, Phys Rev Let 99, 164502 (2007); What drives mesoscale atmospheric turbulence? arXiv:0805.0390

  15. Atmospheric spectrum Lab experiment, weak spectral condensate Nastrom, Gage, J. Atmosph. Sci. 1985 Shats et al, PRL2007

  16. Inverse cascades lead to emerging symmetries but eventually to condensates which break symmetries in a different way for different moments Mean shear flow (condensate) changes all velocity moments:

  17. Mean subtraction recovers isotropic turbulence 1.Compute time-average velocity field (N=400): 2. Subtract from N=400 instantaneous velocity fields Recover ~ k-5/3 spectrum in the energy range Kolmogorov law – linear S3 (r) dependence in the “turbulence range”; Kolmogorov constant C≈7 Skewness Sk ≈ 0 , flatness slightly higher, F ≈ 6

  18. Weak condensate Strong condensate

  19. Conclusion Inverse cascades seems to be scale invariant. Within experimental accuracy, isolines of advected quantities are conformal invariant (SLE) in turbulent inverse cascades. Condensation into a system-size coherent mode breaks symmetries of inverse cascades. Condensates can enhance and suppress fluctuations in different systems For Gross-Pitaevsky equation, condensate may make turbulence conformal invariant

  20. Case of weak condensate Weak condensate case shows small differences with isotropic 2D turbulence ~ k-5/3 spectrum in the energy range Kolmogorov law – linear S3 (r) dependence; Kolmogorov constant C≈5.6 Skewness and flatness are close to their Gaussian values (Sk=0, F=3)

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