Journal of Low Temperature Physics

, Volume 149, Issue 1–2, pp 65–77 | Cite as

Two-Particle Separation in the Point Vortex Gas Model of Superfluid Turbulence

  • S. Wang
  • Y. A. Sergeev
  • C. F. Barenghi
  • M. A. Harrison
Article

Abstract

We study numerically statistical properties of the random mean square separation, Δ(t)=〈Δ21/2 of fluid particle pairs in the Onsager’s point vortex gas, i.e. in a random, inviscid, two-dimensional flow generated by the motion of quantized point vortices of random polarities. This model is perhaps the simplest, two-dimensional model of superfluid turbulence in helium II at low temperatures when the normal fluid is absent. The existence of the inverse energy cascade through scales in such a system was discovered in the pioneering work by Onsager (Nuovo Cimento Suppl. 6:279, 1949). We found that the mean diffusivity of particle pairs, d〈Δ2〉/dt obeys, to a good degree of accuracy, the Richardson’s diffusion ‘four-third’ law, d〈Δ2〉/dt∼Δ4/3. We also analyzed the details of time evolution of the rms particle separation. At small time the particle separation is ballistic, while at large time, when the motion of individual particles in the considered pair becomes uncorrelated, the mean square separation follows the diffusion law, 〈Δ2〉∼t. For intermediate time we found that the root mean square separation, Δ=〈Δ21/2 is very sensitive to the initial separation, due to the memory of initial separation kept by the particle pair during the initial period of exponential ballistic separation, so that the rms separation never follows the Richardson’s t3/2 scaling law. We characterized the time evolution of the rms particle separation in the vicinity of the inflexion point of the curve Δ(t) by the power law, \(\Delta\sim\gamma t^{\alpha(\Delta_{0})}\) , and found the dependence of α and γ on the initial separation Δ0, thus generalizing the Richardson’s t3/2-law for the considered two-dimensional point vortex gas.

Keywords

Superfluid turbulence Point vortex gas Particle separation Diffusivity of particle pairs 

PACS

47.37.+q 67.40.Vs 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Onsager, Nuovo Cimento Suppl. 6, 279 (1949) MathSciNetGoogle Scholar
  2. 2.
    W.F. Vinen, J.J. Niemela, J. Low Temp. Phys. 128, 167 (2002) CrossRefGoogle Scholar
  3. 3.
    J. Maurer, P. Tabeling, Europhys. Lett. 43, 29 (1998) CrossRefADSGoogle Scholar
  4. 4.
    S.R. Stalp, L. Skrbek, R.J. Donnelly, Phys. Rev. Lett. 24, 4831 (1999) CrossRefADSGoogle Scholar
  5. 5.
    D.R. Poole, C.F. Barenghi, J. Low Temp. Phys. 134, 483 (2004) CrossRefADSGoogle Scholar
  6. 6.
    D. Montgomery, Phys. Lett. A39, 7 (1972) ADSGoogle Scholar
  7. 7.
    D. Montgomery, G. Joyce, Phys. Fluids 17, 1139 (1974) CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    G.S. Deem, N.J. Zabusky, Phys. Rev. Lett. 27, 396 (1971) CrossRefADSGoogle Scholar
  9. 9.
    E.D. Siggia, H. Aref, Phys. Fluids 24, 171 (1981) CrossRefADSGoogle Scholar
  10. 10.
    R.H. Kraichnan, D. Montgomery, Rep. Prog. Phys. 43, 547 (1980) CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    S.V. Nazarenko, V.E. Zakharov, Physica D 56, 381 (1992) MATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    L.F. Richardson, Proc. Roy. Soc. A 10, 523 (1926) Google Scholar
  13. 13.
    A.S. Monin, A.M. Yaglom, Statistical Fluid Mechanics; Mechanics of Turbulence, vol. 2 (MIT Press, Cambridge, 1975), Chapter 8, pp. 555–556 Google Scholar
  14. 14.
    P. Morel, M. Larchevêque, J. Atmos. Sci. 31, 2189 (1974) CrossRefADSGoogle Scholar
  15. 15.
    J.C.H. Fung, J.C. Vassilicos, Phys. Rev. E 57, 1677 (1998) CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    G.K. Batchelor, Proc. Camb. Philos. Soc. 48, 345 (1952) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    V.I. Tatarski, Izv. Vysš. Učebn. Zaved., Radiofiz. 4, 551 (1960) Google Scholar
  18. 18.
    A.M. Obukhov, Bull. Acad. Sci. U.S.S.R., Géog. & Géophys., Mosc. 5, 453 (1941) Google Scholar
  19. 19.
    J.C.H. Fung, J.C.R. Hunt, N.A. Malik, R.J. Perkins, J. Fluid Mech. 236, 281 (1992) MATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    F. Nicolleau, J.C. Vassilicos, Phys. Rev. Lett. 90, 024503 (2003) CrossRefADSGoogle Scholar
  21. 21.
    F. Nicolleau, G. Yu, Phys. Fluids 16, 2309 (2004) CrossRefADSGoogle Scholar
  22. 22.
    P.K. Yeung, Phys. Fluids 6, 3416 (1994) MATHCrossRefADSGoogle Scholar
  23. 23.
    D.J. Thomson, B.J. Devenish, J. Fluid Mech. 526, 277 (2005) MATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    L. Biferale, G. Boffetta, A. Celani, B.J. Devenish, A. Lanotte, F. Toschi, Phys. Fluids 17, 115101 (2005) CrossRefADSMathSciNetGoogle Scholar
  25. 25.
    M.S. Borgas, P.K. Yeung, J. Fluid Mech. 502, 125 (2004) CrossRefADSMathSciNetGoogle Scholar
  26. 26.
    M. Bourgoin, N.T. Ouellette, H. Xu, J. Berg, E. Bodenschatz, Science 311, 835 (2006) CrossRefADSGoogle Scholar
  27. 27.
    M.-C. Jullien, J. Paret, P. Tabeling, Phys. Rev. Lett. 82, 2872 (1999) CrossRefADSGoogle Scholar
  28. 28.
    H. Aref, Ann. Rev. Fluid Mech. 15, 345 (1983) CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • S. Wang
    • 1
  • Y. A. Sergeev
    • 1
  • C. F. Barenghi
    • 2
  • M. A. Harrison
    • 2
  1. 1.School of Mechanical and Systems EngineeringNewcastle UniversityNewcastle upon TyneUK
  2. 2.School of Mathematics and StatisticsNewcastle UniversityNewcastle upon TyneUK

Personalised recommendations