Advertisement

JETP Letters

, Volume 83, Issue 12, pp 541–545 | Cite as

Differential model for 2D turbulence

  • V. S. L’vov
  • S. Nazarenko
Article

Abstract

We present a phenomenological model for 2D turbulence in which the energy spectrum obeys a nonlinear fourth-order differential equation. This equation respects the scaling properties of the original Navier-Stokes equations, and it has both the −5/3 inverse-cascade and the −3 direct-cascade spectra. In addition, our model has Raleigh-Jeans thermodynamic distributions as exact steady state solutions. We use the model to derive a relation between the direct-cascade and the inverse-cascade Kolmogorov constants, which is in good qualitative agreement with the laboratory and numerical experiments. We discuss a steady state solution where both the enstrophy and the energy cascades are present simultaneously, and we discuss it in the context of the Nastrom-Gage spectrum observed in atmospheric turbulence. We also consider the effect of the bottom friction on the cascade solutions and show that it leads to an additional decrease and finite-wavenumber cutoffs of the respective cascade spectra, which agrees with the existing experimental and numerical results.

PACS numbers

47.27.-i 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. Leith, Phys. Fluids 10, 1409 (1967); Phys. Fluids 11, 1612 (1968).CrossRefGoogle Scholar
  2. 2.
    R. Kraichnan and D. Montgomery, Rep. Prog. Phys. 43, 547 (1980).MathSciNetCrossRefADSGoogle Scholar
  3. 3.
    C. Connaughton and S. Nazarenko, Phys. Rev. Lett. 92, 044501 (2004).Google Scholar
  4. 4.
    V. S. Lvov, S. V. Nazarenko, and G. Volovik, JETP Lett. 80, 535 (2004).CrossRefGoogle Scholar
  5. 5.
    V. S. L’vov, G. Ooms, and A. Pomyalov, Phys. Rev. E 67, 046314 (2003).Google Scholar
  6. 6.
    S. Hasselmann and K. Hasselmann, J. Phys. Oceanogr. 15, 1369 (1985).CrossRefGoogle Scholar
  7. 7.
    R. S. Iroshnikov, Sov. Phys. Dokl. 30, 126 (1985).zbMATHGoogle Scholar
  8. 8.
    S. V. Nazarenko, J. Stat. Mech. L02002 doi: 10.1088/1742-5468/2006/02/L02002 (2006), nlin.CD/0510054.Google Scholar
  9. 9.
    S. V. Nazarenko, JETP Lett. 83, 198 (2005), condmat/0511136.CrossRefGoogle Scholar
  10. 10.
    V. S. L’vov, A. Pomyalov, and I. Procaccia, Phys. Rev. Lett. 89, 064501 (2002).Google Scholar
  11. 11.
    J. Paret, M.-C. Jullien, and P. Tabeling, physics/9904044 v. 1 (1999).Google Scholar
  12. 12.
    J. Paret and P. Tabeling, Phys. Rev. Lett. 79, 4162 (1997).CrossRefADSGoogle Scholar
  13. 13.
    V. Borue, Phys. Rev. Lett. 71, 3967 (1993).CrossRefADSGoogle Scholar
  14. 14.
    T. Ishihara and Y. Kaneda, Phys. Fluids 13, 544 (2001).CrossRefADSGoogle Scholar
  15. 15.
    S. Danilov and D. Gurarie, Phys. Rev. E 63, 020203(R) (2001).Google Scholar
  16. 16.
    L. M. Smith and V. Yakhot, Phys. Rev. Lett. 71, 352 (1993).CrossRefADSGoogle Scholar
  17. 17.
    G. D. Nastrom and K. S. Gage, J. Atmos. Sci. 42, 950 (1984).CrossRefADSGoogle Scholar
  18. 18.
    G. D. Nastrom, K. S. Gage, and W. H. Jasperson, Nature 310, 36 (1984).CrossRefADSGoogle Scholar
  19. 19.
    D. K. Lilly, Atmos. Sci. 46, 2026 (1989).CrossRefADSGoogle Scholar
  20. 20.
    M. Maltrud and G. K. Vallis, J. Fluid Mech. 228, 321 (1991).ADSGoogle Scholar
  21. 21.
    S. Danilov and D. Gurarie, Phys. Usp. 43, 863 (2000).CrossRefGoogle Scholar
  22. 22.
    D. K. Lilly, Geophys. Fluid Dyn. 3, 289 (1972).Google Scholar
  23. 23.
    L. M. Smith and V. Yakhot, J. Fluid Mech. 274, 115 (1994).zbMATHMathSciNetCrossRefADSGoogle Scholar
  24. 24.
    A. Colin de Verdiere, Geophys. Astrophys. Fluid Dyn. 15, 213 (1980).zbMATHGoogle Scholar
  25. 25.
    J. Sommeria, J. Fluid Mech. 170, 139 (1986).CrossRefADSGoogle Scholar
  26. 26.
    F. V. Dolzhanskii, V. A. Krymov, and D. Yu. Manin, Sov. Phys. Usp. 33, 495 (1990).CrossRefGoogle Scholar
  27. 27.
    D. Yu. Manin, Atmos. Ocean Phys. 26, 426 (1990).Google Scholar
  28. 28.
    G. Boffetta, A. Cenedese, S. Espa, and S. Musacchio, Europhys. Lett. 71, 590 (2005).CrossRefADSGoogle Scholar
  29. 29.
    L. Biferale, M. Cencini, A. Lanotte, and D. Vergni, Phys. Fluids 15, 1012 (2003).MathSciNetCrossRefADSGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2006

Authors and Affiliations

  • V. S. L’vov
    • 1
    • 3
  • S. Nazarenko
    • 2
  1. 1.Department of Chemical Physicsthe Weizmann Institute of ScienceRehovotIsrael
  2. 2.Mathematics InstituteThe University of WarwickCoventryUK
  3. 3.Low Temperature LaboratoryHelsinki University of TechnologyFinland

Personalised recommendations