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Prog. Theor. Phys. Vol. 63 No. 6 (1980) pp. 1931-1944

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Statistical Dynamics of Chaotic Flows

Hazime Mori and Hirokazu Fujisaka

Department of Physics, Kyushu University, Fukuoka 812

(Received January 19, 1980)

Abstract:

A cascade model for chaotic flows proposed in a previous note is further developed to determine the Hausdorff dimension D of some strange attractors and to relate the turbulence statistics to local vortex dynamics. It thus turns out that the Lorenz attractor has D=2.06 at r=40, σ=16, b=4, and the Smale solenoid has D=1+[ln 2/ln (1/ε)], (ε<1/2)and the Hénon mapping has D=1.26 at a=1.4, b=0.3.
The statistics of fully-developed fluid turbulence is determined by the intermittency exponent µ which is related to the Hausdorff dimension D of the dissipative structure of vorticity by µ=3-D. It is shown that µ can be expressed in terms of a statistical average of the local expansion rates of an active region of vorticity over an orbit of the chaotic flow in fluid space and over an ensemble of extrinsic randomness, and represents how fast spatial spottiness is generated by the vortex stretching. Thus a generalized β-model is formulated in terms of local vortex dynamics without the assumption of self-similarity. It also turns out that the validity of the universality and the scaling laws depends on the speed of the mixing of flow due to the vortex stretching.


URL : http://ptp.ipap.jp/link?PTP/63/1931/
DOI : 10.1143/PTP.63.1931

[ Full Text PDF : FREE ACCESS (1121K) ] Citation:

References:

  1. A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge) (1971), vol. 1, (1975), vol. 2.
  2. P. C. Martin, in “Statistical Physics”, Proceedings of the 12th International Conference on Statistical Physics (Akdémiai Kiadó, Budapest, 1976).
  3. B. B. Mandelbrot, Fractals: Form, Chance and Dimension (Freeman, San Francisco, 1977); in Turbulence Seminar, Lecture Notes in Math., vol. 615, eds. A. Dold and B. Eckmann (Springer-Verlag, Berlin, 1977).
  4. U. Frisch, P. L. Sulem and M. Nelkin, J. Fluid Mech. 87 (1978), 719, [CrossRef]and references cited therein.
  5. H. Fujisaka and H. Mori, Prog. Theor. Phys. 62 (1979), 54[PTP]; Phys. Lett. A 72 (1979), 444[CrossRef].
  6. V. I. Oseledec, Trans. Moscow Math. Soc. 19 (1968), 197.
  7. T. Nagashima and I. Shimada, Prog. Theor. Phys. 58 (1977), 1318[PTP].
    I. Shimada and T. Nagashima, Prog. Theor. Phys. 59 (1978), 1033[PTP]; ibid. 61 (1979), 1605[PTP].
  8. S. D. Feit, Commun. Math. Phys. 61 (1978), 249[CrossRef].
  9. D. Ruelle, Prog. Theor. Phys. Suppl. No. 64 (1978), 339[PTP].
  10. H. Mori, Prog. Theor. Phys. 63 (1980), 1044[PTP]. This is referred to as I.
  11. E. N. Lorenz, J. Atmos, Sci. 20 (1963), 130.
  12. D. Ruelle, Statistical Mechanics and Dynamical Systems, Duke Univ. Math. Series III (Math. Dept., Duke Univ., Durham, U. S. A., 1977).
  13. S. Smale, in Turbulence Seminar, Lecture Notes in Math., vol. 615, eds. A. Dold and B. Eckmann (Springer-Verlag, Berlin, 1977).
  14. M. Hénon, Commun. Math. Phys. 50 (1976), 69[CrossRef].
  15. G. K. Batchelor and A. A. Townsend, Proc. R. Soc. A 199 (1949), 238.
    A. Y. S. Kuo and S. Corrsin, J. Fluid Mech. 56 (1972), 447[CrossRef].
  16. A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, 1975), vol. 2, §21, §25.
  17. R. H. Kraichnan, J. Fluid Mech. 62 (1974), 305, [CrossRef]and references cited therein.
  18. M. Nelkin and T. L. Bell, Phys. Rev. A 17 (1978), 363[APS].
    M. Nelkin, “Universality and Scaling in Fully Developed Turbulence,” Proceedings of the 13th International Conference on Statistical Physics (Haifa, Israel, 1977).
  19. H. Mori and H. Fujisaka, in “Pattern Formation and Pattern Recognition,” Proceedings of the International Symposium on Synergetics, ed. H. Haken (Springer-Verlag, Berlin, 1979).
  20. S. A. Orszag, in Fluid Dynamics, eds. R. Balian and J. -L. Peube (Gordon & Breach, London, 1977).
  21. A. Zippelius and M. Lucke, “The Effect of External Noise in the Lorenz Model,” preprint 1979.
  22. D. Ruelle, Phys. Lett. A 72 (1979), 81[CrossRef].
  23. B. B. Mandelbrot, “Geometric Facets of Statistical Physics,” Proceedings of the 13th International Conference on Statistical Physics (Haifa, Israel, 1977).

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 65 No. 3 (1981) pp. 1085-1087 :
    Anomalous Diffusion of Vorticity in Fully-Developed Turbulence
    Hazime Mori
  2. Progress of Theoretical Physics Vol. 67 No. 5 (1982) pp. 1630-1632 :
    Statistics of Active Regions in the β-Model of Turbulence
    Shigeo Kida
  3. Progress of Theoretical Physics Vol. 68 No. 1 (1982) pp. 64-84 :
    Global Aspects of the Dissipative Dynamical Systems. I
    Yoji Aizawa
  4. Progress of Theoretical Physics Vol. 68 No. 2 (1982) pp. 439-447 :
    Diffusion of Particles in Fully-Developed Turbulence
    Kiyofumi Takayoshi and Hazime Mori
  5. Progress of Theoretical Physics Vol. 68 No. 4 (1982) pp. 1105-1121 :
    Multiperiodic Flows, Chaos and Lyapunov Exponents
    Hirokazu Fujisaka
  6. Progress of Theoretical Physics Vol. 68 No. 6 (1982) pp. 2180-2183 :
    On the β-Model of Intermittent Fully-Developed Turbulence
    Hazime Mori and Hirokazu Fujisaka
  7. Progress of Theoretical Physics Vol. 69 No. 3 (1983) pp. 725-741 :
    Vortex Stretching and Relative Diffusion in Grid Turbulence
    Hazime Mori and Kiyofumi Takayoshi
  8. Progress of Theoretical Physics Vol. 69 No. 3 (1983) pp. 756-772 :
    Enstrophy Cascade and Relative Diffusion in Two-Dimensional Fully-Developed Turbulence
    Hazime Mori
  9. Progress of Theoretical Physics Supplement No.69 (1980) pp. 111-121 :
    Anomalous Diffusion of Vorticity in Fully-Developed Turbulence
    Hazime Mori

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