Pattern formation in spatially ramped Rayleigh-B'enard systems

Abstract

We study pattern formation and selection in Rayleigh-Bsystems confined between well conducting horizontal boundaries and subjected to a weak horizontal gradient of the Rayleigh number. The study is based on the numerical integration of the Swift-Hohenberg equation and addresses the questions of the preferred orientation of the patterns with respect to the gradient of the Rayleigh number, boundary effects observed at subcritial sidewalls, the characteristics of long-term evolution of the patterns with emphasis on the wa- velength selection, and the effect of non-potential modifications of the Swift-Hohenberg equation. It is shown that, contrary to common belief, the rolls do not align with the direction of the horizontal temperature gradient (Dewel, 1989; Malomed, 1993), due to the influence of the walls. Rather, rolls approach a sidewall perpendicularly when the lo- cal bifurcation parameter is sufficiently beyond the threshold but tend to be parallel to a subcritical or critical sidewall. Simulati- ons performed with non-potential modifications of the Swift-Hohenberg equation lead in most cases, to asymptotic time-dependent behaviours.