Enstrophy Cascade in a Model of 2D Turbulence: Comparison with 3D Energy Cascade

The average property of cascade process in fully-developed turbulence can basically be described universally by the Kolmogorov theoryl) (K41) in the 3D case and by the Batchelor-Kraichnan-Leith)-4) (BKL) theory in the 2D case. However the fluctuations around the average state, whether temporal or spatial, are poorly understood. In order to obtain an insight into such a problem, temporal intermittency of the energy cascade has been studied in a chaotic cascade model of 3D turbulence and a: close correlation was observed between instantaneous behavior of energy flux and that of local Lyapunbv exponent.) In this paper we treat a chaotic model of 2D turbulence whose average property has been found to be consistent with the BKL enstrophy cascade theory.6) Here we investigate fluctuations around the BKL scaling law mainly by examining instantaneous enstrophy flux and local Lyapunov exponent. The purpose of this paper is to compare the results· with the previously reported 3D case and to point out differences in the dynamics between the enstrophy and the energy cascade processes~ The model, first proposed by Gledzer,7) is constructed in the waveulJ.mber space discretized as kn=2n-lO(n=1 ~ N). The dependent variable Un represents the collective velocity of the shell at k=kn, and 1/2· Un2 is the energy associated with the n-th shell. The total energy and total enstrophy are E='!E.nun2/2 and Q='!E.nkn2un2/2, respectively. The evolution equation for Un is taken to be,