Computation of the dimension of a model of fully developed turbulence

Abstract

We compute numerically the attractor dimension of a model of fully developed MHD turbulence both using Lyapounov exponents, and via a counting algorithm. The Lyapounov dimension is found to be essentially given by the total number of modes which lie in the inertial range; the maximal exponent, i.e. the divergence rate of nearby trajectories, is given by the inverse of the shortest time scale available in the inertial range. The correlation dimension, on the other hand, is found to be given by roughly one third of the modes in the inertial range. Both evaluations of dimension coincide only at very low Reynolds, i.e. at low dimension (equal to about 3). If extrapolated to real 3-dimensional Navier-Stokes turbulence, our results are consistent with the two scaling laws : 1) the maximal Lyapounov exponent scales as Re1/2, 2) the Lyapounov dimension and Kolmogorov entropy scale as Re9/4 (no saturation of the dimension of the turbulent attractor).