Scaling and dissipation in the GOY shell model

Abstract

Abstract:This is a paper about multi-fractal scaling and dissipation in a shell model of turbulence, called the Gledzer-Ohkitani-Yamada model or GOY model. This set of equations describes a one dimensional cascade of energy towards higher wave vectors. When the model is chaotic, the high-wave-vector velocity is a product of roughly independent multipliers, one for each logarithmic momentum shell. The appropriate tool for studying the multifractal properties of this model is shown to be the energy ux on each shell rather than the velocity on each shell. Using this quantity, one can obtain better measurements of the deviations from Kolmogorov scaling (in the GOY dynamics) than were available up to now. These deviations are seen to depend upon the details of inertial-range structure of the model and hence are not universal. However, once the conserved quantities of the model are xed to have the same scaling structure as energy and helicity, these deviations seem to depend only weakly upon the scale parameter of the model. We analyze the connection between multifractality in the velocity distribution and multifractality in the dissipation. Our arguments suggest that the connection is universal for models of this character, but the model has a di erent behavior from that of real turbulence. We also predict the scaling behavior of time correlations of shell-velocities, of the dissipation, and of Lyapunov indices. These scaling arguments can be carried over, with little change, to multifractal models of real turbulence.

Multi-time, multi-scale correlation functions in turbulence and in turbulent models

“…The proposal can be seen as a merging of the proposal made in [3] -valid only for cases when the disconnected part is vanishing -and the proposal made in [8] -valid only in the asymptotic regime of large time-delays and large-scale separation. Our proposal is phenomenologically realistic and consistent with the dynamical constraints imposed by the equation of motion.…”

“…Let us notice that this proposal has already been presented in [8] and considered to express the leading term in the limit of large time-delays τ m → ∞; here, we want to refine the proposal made in [8] showing that by adding the proper time-dependencies it is possible to obtain a coherent description of the correlation functions for all time-delays. Expression (10) summarizes the idea that for time-delay larger than τ m but smaller than τ m−1 , velocity components with support on scales r > l m−1 did not have enough time to relax and therefore the local exponent, h, which describes fluctuations on those scales must be the same for both fields.…”

“…Some subclasses of the multi-scale, multi-time correlation functions (1) have lately attracted the attention of many scientists [2][3][4]6,8,10]. By evaluating (1) with r = R, at changing R, and at zero-time delay, t = 0, we have the celebrated structure functions of order p + q.…”

Critical “dimension” in shell model turbulence

“…In this respect, we apply the GOY model [6,7] which has been successful in giving results for intermittency corrections in agreement with experiments [8] (for other results on the GOY model, see [10][11][12][13]). The starting point is a set of wave numbers k n = k 0 2 n and an associated complex amplitude u n of the velocity field.…”

“…For δ < 1 this relation requires complex values of α, with ℑ(α) = π/ ln 2. In 3d turbulence helicity H = (∇ × u(x))·u(x)dx is conserved, which in terms of shell variables takes the form [11] …”