Numerical modeling of collapse in ideal incompressible hydrodynamics

  • V. A. Zheligovsky
  • E. A. Kuznetsov
  • O. M. Podvigina
Plasma, Gases


The appearance of a singularity in the velocity-field vorticity ω at an isolated point irrespective of the symmetry of initial distribution is demonstrated numerically. The behavior of maximal vorticity |ω| near the collapse point is well approximated by the dependence (t0t)−1, where t0 is the collapse time. This is consistent with the interpretation of collapse as the breaking of vortex lines.

PACS numbers

47.15.Ki 47.32.Cc 


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  1. 1.
    A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (Gidrometeoizdat, St. Petersburg, 1996, 2nd ed.; MIT Press, Cambridge, 1975), Vol. 2; V. S. L’vov, Phys. Rep. 257, 1 (1991).Google Scholar
  2. 2.
    A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 9 (1941).Google Scholar
  3. 3.
    U. Frisch, Turbulence. The Legacy of A. N. Kolmogorov (Cambridge Univ. Press, Cambridge, 1995).Google Scholar
  4. 4.
    R. M. Kerr, Phys. Fluids A 4, 2845 (1993).MathSciNetGoogle Scholar
  5. 5.
    R. B. Pelz, Phys. Rev. E 55, 1617 (1997); O. N. Boratav and R. B. Pelz, Phys. Fluids 6, 2757 (1994).CrossRefADSGoogle Scholar
  6. 6.
    R. Grauer, C. Marliani, and K. Germaschewski, Phys. Rev. Lett. 80, 4177 (1998).CrossRefADSGoogle Scholar
  7. 7.
    E. A. Kuznetsov and V. P. Ruban, Pis’ma Zh. Éksp. Teor. Fiz. 67, 1015 (1998) [JETP Lett. 67, 1076 (1998)].Google Scholar
  8. 8.
    E. A. Kuznetsov and V. P. Ruban, Zh. Éksp. Teor. Fiz. 118, 853 (2000) [JETP 91, 775 (2000)].Google Scholar
  9. 9.
    V. I. Arnold, Catastrophe Theory (Znanie, Moscow, 1981; Springer-Verlag, Berlin, 1986).Google Scholar
  10. 10.
    V. E. Zakharov and E. A. Kuznetsov, Zh. Éksp. Teor. Fiz. 91, 1310 (1986) [Sov. Phys. JETP 64, 773 (1986)].ADSGoogle Scholar
  11. 11.
    J. T. Beals, T. Kato, and A. J. Majda, Commun. Math. Phys. 94, 61 (1984).Google Scholar
  12. 12.
    P. Constantin, Ch. Feferman, and A. J. Majda, Commun. Partial Diff. Eqns. 21, 559 (1996).Google Scholar

Copyright information

© MAIK "Nauka/Interperiodica" 2001

Authors and Affiliations

  • V. A. Zheligovsky
    • 1
  • E. A. Kuznetsov
    • 1
  • O. M. Podvigina
    • 1
  1. 1.International Institute of the Theory of Earthquake Forecasting and Mathematical GeophysicsRussian Academy of SciencesMoscowRussia

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