Probability Distribution of Modal Amplitudes in Interacting Triads with Arbitrary Random Forcing

Abstract

This paper deals with a statistical-mechanical approach to the problem of calculating the statistics of randomly forced triads of modes in the two-dimensional flow of a viscous fluid. We first construct the probability distribution of modal amplitudes as an approximate solution of the Fokker-Planck equation. The net nonlinear energy-transfer and kinetic energy spectrum are then calculated directly from the probability distribution. These theoretical results agree very well with the statistics of a large ensemble of numerical integrations of the original evolution equations, particularly for only moderate departures from ?white-noise? forcing.