Ingredients of a theory of convective textures close to onset

Abstract

The patterns of convection to be expected in laterally large Rayleigh-B\'enard cells close to onset are analyzed using the amplitude equation approach of Newell and Whitehead and Segel. This equation allows the introduction of a Lyapunov functional, which may be used to order the stability of various patterns. Two competing effects are identified: a surface effect that favors rolls approaching the sidewalls normally, and so tends to produce $O(1)$ rotations of the roll orientation over the cell, and bulk effects that favor straight parallel rolls. The competition between these effects, and the role of defects in the structure, are studied in some simple examples.