Dynamical Chaos: Problems in Turbulence

  • R. Z. Sagdeev
  • G. M. Zaslavsky
Conference paper


Chaos of streamlines of steady Beltrami-type flows leads to formation of liquid stochastic web. Structural properties of the web are inherited by fluid particles. Their dynamics is intermittent. There are two origins of intermittency: trappings and flights. These effects result in anomalous diffusion. We consider random walks on multifractals of the Levy flight type and discuss differences in scaling properties of spatial and temporal averages. This reflects different fractal properties of the time evolution, and the space on which it occurs.


Fractal Property Random Walk Phase Portrait Fluid Particle Chaotic Region 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L.D. Landau and E.M. Lifshitz. Fluid Mechanics (Pergamon, L. 1959).Google Scholar
  2. 2.
    A.N. Kolmogorov. J. Fluid Mech. 13, 82 (1962).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    B. Mandelbrot. In Turbulence and Navier-Stokes Equations. Ed. R. Temam, Lecture Notes and Mathematics. Vol. 565 (Berlin, 1976), p. 121.Google Scholar
  4. B.
    Mandelbrot. J. Fluid Mech. 62, 331 (1974).zbMATHCrossRefGoogle Scholar
  5. 4.
    G. Parisi and U. Frisch. In Turbulence and Predictability in Geophysical fluid Dynamics and Climate Dynamics, Ed. M. Ghil, R. Benzi and G. Parisi, North-Holland, Amsterdam (1985), p. 71.Google Scholar
  6. 5.
    Paladin and Vulpiani, Physics Reports, 156, 149 (1987).MathSciNetCrossRefGoogle Scholar
  7. 6.
    V. I. Arnold. Mathematical Methods in Classical Mechanics. (Springer New York, 1980).Google Scholar
  8. 7.
    H. Aref. J. Fluid Mech. 143, 1 (1984) ibid.Google Scholar
  9. 7a.
    J.M. Ottino, C.W. Leong, H. Rising and P.D. Swanson. Nature, 333, 419 (1988).Google Scholar
  10. 8.
    T. Dombre, U. Frisch, J.M. Green, M. Henon, A. Mehr and A.M. Soward. J. Fluid Mech. 167, 353 (1986).MathSciNetzbMATHCrossRefGoogle Scholar
  11. 9.
    G.M. Zaslavsky, R.Z. Sagdeev, and A.A. Chernikov. Sov. Phys. JETP, 67, 270 (1988).MathSciNetGoogle Scholar
  12. 10.
    H.K. Moffatt. J. Fluid Mech. 159, 359 (1985).MathSciNetzbMATHCrossRefGoogle Scholar
  13. 11.
    V.V. Beloshapkin, A.A. Chernikov, M.Ya. Natenzon, B.A. Petrovichev, R.Z. Sagdeev and G.M. Zaslavsky, Nature 337, 133 (1989).CrossRefGoogle Scholar
  14. 12.
    D. Ruelle and F. Takens. Commun. Math. Phys. 20, 167 (1971).MathSciNetzbMATHCrossRefGoogle Scholar
  15. 13.
    J.P. Eckmann, Rev. Mod. Phys. 53, 655 (1981).MathSciNetCrossRefGoogle Scholar
  16. 14.
    R.Z. Sagdeev, D.A. Usikov and G.M. Zaslavsky. Nonlinear Physics (Harwood Acad. Publ. N.Y. 1988).Google Scholar
  17. 15.
    Cellular structures in instabilities. Eds. J.E. Wesfreid and S. Zaleski (Springer, Berlin, 1984).Google Scholar
  18. 16.
    H. Chate and P. Manneville. Phys. Rev. Lett. 58, 112 (1987).CrossRefGoogle Scholar
  19. 17.
    F. Heslot, B. Castaing and A. Libchaber. Phys. Rev. A, 36, 5870 (1987).CrossRefGoogle Scholar
  20. 18.
    B. Nicolaenko. Los Alamos National Laboratory. 1988.Google Scholar
  21. 19.
    L.D. Meshalkin, Ya.G. Sinai, Prikl. Matern. i Mekhanika, 25, 1140 (1961).Google Scholar
  22. 20.
    V.V. Beloshapkin, A.A. Chernikov, R.Z. Sagdeev and G.M. Zaslavsky. Phys. Lett. A, 133, 395 (1988).MathSciNetCrossRefGoogle Scholar
  23. 21.
    V. Yakhot, J.B. Bayly, and S.A. Orszag. Phys. Fluids 29, 2025 (1986).CrossRefGoogle Scholar
  24. 22.
    A. Tsinober and E. Levich. Phys. Lett. A, 99, 321 (1983).CrossRefGoogle Scholar
  25. 23.
    R.B. Pelz, V. Yakhot, S.A. Orszag, L. Shtilman, and E. Levich. Phys. Rev. Lett. 54, 2505 (1985).CrossRefGoogle Scholar
  26. 23a.
    R.H. Kraichnan and R. Panda. Phys. Fluids 31, 2395 (1988).zbMATHCrossRefGoogle Scholar
  27. 24.
    B.J. Bayly, and V. Yakhot. Phys. Rev. A, 34, 381 (1986).CrossRefGoogle Scholar
  28. 25.
    N.N. Filonenko, R.Z. Sagdeev, and G.M. Zaslavsky. Nucl. Fusion 7, 253 (1967).CrossRefGoogle Scholar
  29. 26.
    B.A. Dubrovin, S.P. Novikov and A.T. Fomenko. Contemporary Geometry (in Russian). Nauka, Moscow, 1979, chap. 7.Google Scholar
  30. 27.
    G.M. Zaslavsky. Chaos in Dynamic Systems. Harwood Acad. Publ. N.Y., 1985.Google Scholar
  31. 28.
    A.A. Chernikov, R.Z. Sagdeev, and G.M. Zaslavsky (to be published).Google Scholar
  32. 29.
    G.M. Zaslavsky, M.Yu. Zakharov, R.Z. Sagdeev, D.A. Usikov and A.A. Chernikov. Sov. Phys. JETP 64, 294 (1986).MathSciNetGoogle Scholar
  33. 30.
    G.M. Zaslavsky, R.Z. Sagdeev, A.A. Chernikov. Uspekhi Fiz. Nauk, 156, 193 (1988).MathSciNetCrossRefGoogle Scholar
  34. 31.
    Y. Pomeau and P. Manneville. Comm. Math. Phys. 74, 149 (1980).MathSciNetCrossRefGoogle Scholar
  35. 32.
    G.M. Zaslavsky, R.Z. Sagdeev, D.K. Tchaikovsky, and A.A. Chernikov. Zhurn. Eksp. i Teor. Fiz. (1989).Google Scholar
  36. 33.
    P. Levy, Theorie de l’addition des variables aleatoires (Gauthier-Villars, Paris, 1937).Google Scholar
  37. 34.
    E.W. Montroll and M.F. Shlesinger. In Studies in statistical mechanics. Eds. J. Lebowitz and E.W. Montroll, v. 11, p. 5 (North Holland, Amsterdam, 1984).Google Scholar
  38. 35.
    R.S. Mac Kay, J.D. Meiss, and I.C. Percival. Physical D, 13, 55 (1984).CrossRefGoogle Scholar
  39. 36.
    W. Feller. An Introduction to Probability Theory and Its Applications. Vol. 2 (Wiley, New York, 1966).zbMATHGoogle Scholar
  40. 37.
    A.A. Chernikov, B.A. Petrovichev, A. Rogalskii, R.Z. Sagdeev and G.M. Zaslavsky (to be published).Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1991

Authors and Affiliations

  • R. Z. Sagdeev
  • G. M. Zaslavsky

There are no affiliations available

Personalised recommendations