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Generalized extended self-similarity in turbulence and its scaling hypothesis

Hirokazu Fujisaka

Department of Applied Analysis and Complex Dynamical Systems,
Graduate School of Informatics,
Kyoto University, Japan

One of eminent statistical characteristics of homogeneous, isotropic developed turbulence in the three dimensional system is the self-similar energy cascade in the wavenumber space. This results in the power law behavior $S_q(r)\sim r^{\zeta (q)}$ for the velocity structure function $S_q(r)$, the $q$-th moment of the velocity difference at two positions separated by $r$. $\zeta (q)$ is a universal function of $q$. Several years ago, it was found that even when the turbulence is not fully developed, i.e., the Reynolds number is not extremely high and the the scaling range where the above scaling law holds is not wide enough, scaling behaviors of $S_q(r)$ which has an extended form of that in developed turbulence holds. They are called the extended self-similarity (ESS) and the generalized extended self-similarity (GESS). I will talk about a phenomenological derivation of ESS and GESS, proposing a new scaling hypothesis on the basis of the large deviation theory of probability theory. Furthermore, using numerical and experimental data, I will examine the validity of the present approach.

References
Kolmogorov A N 1941, Dokl. Akad. Nauk SSSR 30 9
Kolmogorov A N 1962, J. Fluid Mech. 13 82
Obukhov A M 1962, J. Fluid Mech. 13 77.
Parisi G and Frisch U 1985, in Turbulence and Predictability in Geophysical Fluid Dynamics, Proceed. Intern. School of Physics `Enrico Fermi', 1983, Varenna, Italy, p.84, eds. M. Ghil, R. Benzi and G. Parisi, (Amsterdam: North-Holland)
Frisch U 1995, Turbulence: The legacy of A. N.Kolmogorov (Cambridge: Cambridge Univ. Press)
Benzi R, Ciliberto S, Tripiccione R, Baudet C and Succi S 1993, Phys. Rev. E 48 R29
Benzi R, Ciliberto S, Baudet C and Chavarria G R 1995, Physica D 80 385
Benzi R, Biferale L, Ciliberto S, Struglia M V and Tripiccione R 1996, Phys. Rev. E 53 R3025;
Benzi R, Biferale L, Ciliberto S, Struglia M V, Tripiccione R 1996, Physica D 96 162.
Fujisaka H and Inoue M 1987, Prog. Theor. Phys. 77 1334
Watanabe T and Fujisaka H 2000, J. of Phys. Soc. Japan 69 1672
Fujisaka H and Grossmann S 2001, Phys. Rev. E 63 026305
Fujisaka H, Nakayama Y, Watanabe T and Grossmann S 2002, Phys. Rev. E 65 046307.

Noise-induced pattern dynamics and intermittency

Hirokazu Fujisaka

Department of Applied Analysis and Complex Dynamical Systems,
Graduate School of Informatics,
Kyoto University, Japan

Intermittency is a quite ubiquitous phenomenon in nonlinear dynamics. The intermittency observed when a particular dynamical state undergoes the instability is called the modulational (often called the on-off) intermittency. Recently, an experimental confirmation of the on-off intermittency in the electrohydrodynamic convection in nematics under dichotomous noise was reported by John et al.. An eminent statistics of the observation is the intermittent generation of convective pattern.
In my talk, in order to elucidate the experiment I will first propose a phenomenological nonlinear stochastic model which has the structure of the Swift-Hohenberg equation for local convection variable with fluctuating threshold. Then, I will discuss results of numerical integration of the model equation associated with the intermittent emergence of convective pattern. Detailed analysis on the statistics of the intermittent pattern dynamics will be addressed.

References
Behn U, Lange A and John Th 1998, Phys. Rev. E 58 2047
John Th, Stannarius R and Behn U 1999, Phys. Rev. Lett. 83 749
Fujisaka H, Ouchi K and Ohara H 2001, Phys. Rev. E 64 036201
John Th, Behn U and R Stannarius 2002, Phys. Rev. E 65 046229
For the modulational intermittency (on-off intermittency), see the following references:
Fujisaka H and Yamada T 1985, Prog. Theor. Phys. 74 918
Fujisaka H and Yamada T 1986, Prog. Theor. Phys. 75 1087
Yamada T and Fujisaka H 1986, Prog. Theor. Phys. 76 582
Fujisaka H and Yamada T 1987, Prog. Theor. Phys. 77 1045
Platt N, Spiegel E A and Tresser C 1993, Phys. Rev. Lett. 70 279
Heagy J F, Platt N and Hammel S M 1994, Phys. Rev. E 49 1140
Yamada T, Fukushima K and Yazaki T 1989, Prog. Theor. Phys. Suppl. No.99, 120
Ott E and Sommerer J C 1994, Phys. Lett. A 188 39
Lai Y C and Grebogi C 1995, Phys. Rev. E 52 R3313
Cenys A, Namajunas A, Tamserius A and Schneider T 1996, Phys. Lett. A 213 259
Venkataramani S C, Antonsen Jr. T M, Ott E and Sommerer J C 1996, Physica D 96 66
Lai Y C 1996, Phys. Rev. E 54 321
Rödelsperger F, Cenys A and Benner H 1995, Phys. Rev. Lett. 75 2594
Becker J, Rödelsperger F, Weyrauch Th, Benner H, Just W and Cenys A 1999, Phys. Rev. E 59 1622
Fujisaka H, Ouchi K, Hata H, Masaoka B and Miyazaki S 1998, Physica D 114 237
Pikovsky A, Rosenblum M and Kurths J 2001, Synchronization: A universal concept in nonlinear sciences (Cambridge: Cambridge Univ. Press)


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Next: Gaspard Up: Abstracts Previous: Frank