Temporal Intermittency in the Energy Cascade Process and Local Lyapunov Analysis in Fully-Developed Model Turbulence

Abstract

Energy cascade process is investigated numerically on a scalar model of fully-developed threedimensional turbulence. It is found that energy propagates through the inertial range intermittently like bursts which are separated by quiescent periods (Siggia's view revisited). During the activated phase the first local Lyapunov exponent oscillates violently, and the support of the first Lyapunov vector spreads over the inertial subrange. 1. Introduction Various turbulent phenomena observed in diverse fields of nonlinear physics have their origin in strange attractors in the phase space_ One of the most important examples of such phenomena is fully-developed three-dimensional (3D) turbulence governed by the N avier-Stokes equation, where the phase space is an infinite dimensional function space and the strange attract or has a huge dimension_ Since its discovery, the concept of the strange attractor has been expected to give a unified approach to the turbulent phenomena.!) However, up to now, the notion of the strange attractor is useful mainly for lower-dimensional turbulent motions, and has not yet given a satisfactory understanding of fully-developed turbulence. Particularly, no interpretation is found for the fundamental scaling law of Kolmogorov in the theory2) of chaotic dynamical systems. This is firstly because the fully-developed turbulence has much more degrees of freedom than the present-day supercomputers can reasonably deal with, and secondly because most of the charatterizations of strange attractors devised so far are practically applicable to systems with small degrees of freedom. At present, at least from a numerical point of view, the fullydeveloped turbulence seems to be still beyond the approach of the methods in the dynamical system theory.