Computation of a dimension for a model of fury developed turbulence

Abstract

One computes the Lyapounov dimension of a model of fully developed turbulence studied in ref.(1,2). Non-linear interactions act between nearest neighbours in a discretized wavenumbers space and conservation properties are verified. Equations are of the form: dX_n/d_t= k_n Σ A_ij X_iX_j νk_n ² X_n δ_n1 where A_ij are coup] ing constants and i, j =n-1,n, or n+1, and n goes from 1 to N. The wavenumbers k_n are discretized in a geometric way: k_n+1/k_n=const; X_n are scalar velocity or magnetic field amplitudes, ν is the diffusivity (kinetic or magnetic) and _{δ n1} is a forcing term acting only on the first kinetic mode. This model exhibits time fluctuations at all scales, and time-averaged power-law spectra, as well as intermittency at small (dissipative) scales. We have studied here the case where the maximum over minimum wavenumber is 256, varying the dissipation. We found that the maximal Lyapounov exponent is scaled by the inverse of the turn-over time of the smallest scale in the inertial range. The Lyapounov dimension is approximatively given by the total number of modes which lie in the inertial range. (We found in particular no indication of saturation of dimension with the Reynolds number).

Figure 1 below (left) shows the kinetic and magnetic energy spectra averaged over about 5 10⁵ time steps (T=1024). One sees the growth of the inertial range when viscosity is reduced. Figure 2 (right) gives the Lyapounov exponents; Table below gives resulting Lyapounov dimensions.

$$dX_n /dt = k_n \Sigma A_{ij} X_i X_j - vk_n^2 X_n + \delta _{n1}$$

We conjecture that these results hold also for real fully developed 3-dimensional Navier-Stokes turbulence:

• 1) the maximum Lyapounov number scales as Re^1/2

• 2) the Lyapounov dimension scales as Re^9/4