Preferred Horizontal Scale for Thermal Convection

Abstract

Linear stability theory for Rayleigh-Benard convection shows that for a specified Rayleigh number, greater than some critical value, only a finite range of horizontal wave numbers support convective instability in a horizontal layer of fluid heated from below. However, it is not possible to predict the preferred horizontal scale of established motions from this approach although it is clear from observations, particularly of the solar surface, that a preferred cell size does prevail. In an endeavour to establish a preferred horizontal scale appropriate non-linear modal equations have been integrated forward in time, initially incorporating a discrete band of wave numbers equally spaced across the range that supports convection, for a specific Rayleigh number. The horizontal resolution was improved in subsequent integrations by first deleting modes that had substantially decayed and then introducing new modes on a finer horizontal mesh in the vicinity of what appeared to be the evolutionary dominant mode. Finally, the multimode integrations were continued in time until the evolution of a dominant horizontal mode from within the restricted range was evident. Both the model characteristics and numerical scheme adopted placed limits on the degree of horizontal refinement that could be undertaken with confidence.