Rayleigh-Benard Convection at Finite Rayleigh Number in Large Aspect Ratio Boxes

Abstract

Our goal in these studies is to derive the phase-amplitude-mean drift equations for the Oberbeck-Boussinesq equations at finite Prandtl number. As a step along the way, we have derived the phase-amplitude equation in the infinite Prandtl number limit. Because the Oberbeck-Boussinesq equations are rather complicated, we illustrate the ideas and numerical methods using the real Swift-Hohenberg equation, which is a reasonable approximation to the full equations in the inertia-less limit. We consider finite Rayleigh numbers, and large aspect ratio boxes. The phase equation will come from a solvability condition in a multi-scale expansion. The method is analogous to the nonlinear WKB technique developped by Whitham (1973). Unlike the earlier case treated by Cross and Newell (1984), the equation considered here has no analytical solution for the nonlinear steady case so that this problem will contain all the ingredients used in the real convection problem. A phenomenological mean drift is introduced in the phase equation; this will be well justified with Oberbeck-Boussinesq equations. Numerical results for the long wavelength instabilities of the phase are presented.