Self-similar fluctuation and large deviation statistics in the shell model of turbulence.

Both static and dynamic multiscalings of fluctuations of energy flux and energy dissipation rate in the Gledzer-Ohkitani-Yamada (GOY) shell model of turbulence are numerically investigated. We compute the large deviation rate function of energy flux not only in the inertial range (IR) but also around the crossover between the inertial range and the dissipation range (DR). The rate function in IR exists to be concave, which assures the applicability of the Legendre transformation with the anomalous scaling exponents that have been investigated so far, and turns out to be independent of the Reynolds number. On the contrary, near the crossover scale, an intermediate dissipation range (IMDR) scaling is observed with the rate function in IMDR, which is accounted with the argument on dissipation scale fluctuation dominated by the energy flux fluctuation in the inertial range. Furthermore, to study the difference between IR intermittency and DR intermittency, we compute finite time-averaged quantities of energy flux and energy dissipation rate and investigate their multiscaling behavior. The difference observed in terms of their dynamic multiscaling is discussed.

On-off convection: Noise-induced intermittency near the convection threshold.

A phenomenological nonlinear stochastic model of intermittency experimentally observed by Behn, Lange, and John [Phys. Rev. E 58, 2047 (1998)] in the electrohydrodynamic convection in nematics under dichotomous noise is proposed. This has the structure of the two-dimensional Swift-Hohenberg equation for local convection variable with fluctuating threshold. Numerical integration of the model equation shows intermittent emergence of convective pattern. Its statistics are found to obey those known, so far, for on-off intermittency. In the course of time, although the pattern intensity changes intermittently, no evident pattern change is observed. Adding additive noise, we observe an intermittent change of convective pattern.

Transport and entropy production due to chaos or turbulence.

A projection-operator method is developed for the statistical-mechanical formulation of chaotic or turbulent transport, such as chaos-induced friction in a forced damped pendulum and turbulent viscosity in a turbulent fluid. Then the nonlinear deterministic equations of motion for these dynamical systems are transformed into linear stochastic equations with chaotic or turbulent fluctuating forces. This leads to a fluctuation-dissipation formula which relates the chaotic or turbulent transport coefficients to the time correlation of the fluctuating forces. Applying this theory to the forced damped pendulum, we explore the chaos-induced friction and the power spectra of chaotic orbits. Applying it to the fluid turbulence governed by the Navier-Stokes equation, we find that the turbulent viscosity in the inertial subrange depends on wave number k as k(-beta) with beta=4 / 3+1 / 2/micro(2/3)/, micro (q) being the intermittency exponent of order q.

Scaling hypothesis leading to extended self-similarity in turbulence.

A scaling hypothesis leading to extended self-similarity (ESS) for structure functions (the qth order moments of the magnitude of the longitudinal component velocity differences) in isotropic, homogeneous turbulence is proposed. This is done by generalizing the scale variable r to rg(r/L), with a crossover function g. By extending the refined self-similarity, it is shown that the presented scaling also leads to ESS for structure functions of the energy dissipation rate fluctuations, and to ESS bridging relations between velocity and dissipation rate moments. Extended self-similarity on the basis of a universal crossover function g strictly holds toward the outer scale (L) range only. Yet we find at least approximate ESS toward the viscous, inner scale (l) range. Furthermore, the probability densities for the velocity differences and the energy dissipation rate fluctuations which are compatible with this ESS are offered.

Dynamic phase transition in a time-dependent Ginzburg-Landau model in an oscillating field.

The Ginzburg-Landau model below its critical temperature in a temporally oscillating external field is studied both theoretically and numerically. As the frequency or the amplitude of the external field is changed, a nonequilibrium phase transition is observed. This transition separates spatially uniform, symmetry-restoring oscillations from symmetry-breaking oscillations. Near the transition a perturbation theory is developed, and a switching phenomenon is found in the symmetry-broken phase. Our results confirm the equivalence of the present transition to that found in Monte Carlo simulations of kinetic Ising systems in oscillating fields, demonstrating that the nonequilibrium phase transition in both cases belongs to the universality class of the equilibrium Ising model in zero field. This conclusion is in agreement with symmetry arguments [G. Grinstein, C. Jayaprakash, and Y. He, Phys. Rev. Lett. 55, 2527 (1985)] and recent numerical results [G. Korniss, C. J. White, P. A. Rikvold, and M. A. Novotny, Phys. Rev. E 63, 016120 (2001)]. Furthermore, a theoretical result for the structure function of the local magnetization with thermal noise, based on the Ornstein-Zernike approximation, agrees well with numerical results in one dimension.

Large deviation statistics of the energy-flux fluctuation in the shell model of turbulence

The energy-flux fluctuation in the shell model of turbulence is numerically analyzed from the large deviation statistical point of view. We first observe that the rate function defined in the inertial range is independent of the Reynolds number. The rate function derived by the cascade model of the log-Poisson statistics turns out to be in good agreement with the present numerical result in the region where strong singularity of fluctuation exits. This fact may imply the universality as well as the robustness of the large deviation statistical quantities in turbulence.