Moment Closures for the Equilibrium Statistics of Randomly-Forced Triads

Abstract

This paper is concerned with calculating the nonlinear transfer and partition of energy in a triad of interacting modes of a two-dimensional viscous flow, driven by random sources and sinks of vorticity. Our approach to the problem lies in deriving a moment expansion from the Fokker-Planck equation for the equilibrium probability distribution of a large ensemble of such flows. For sufficiently small values of a set of dimensionless ?ordering-parameters,? depending only on given external conditions, the moment expansion is convergent. A remarkably simple relation between the fourth and fifth moments leads to a tractable ?sixth- moment discard? closure, which is in some respects similar to the ?eddy-damped, quasi-normal? approximation, but has a clearer theoretical basis. The accuracy of the ?sixth-moment discard? closure is judged by comparing the theoretically-derived kinetic energy and energy-transfer spectra with those constructed from a large ensemble of numerical integrations of the original evolution equations. For nonhomogencous forcing corresponding to ?ordering parameters? only slightly less than unity, the theoretical and numerical results agree to within about 5%.