Physica lOD (1984) 249-276 North-Holland, Amsterdam SYMMETRIES AND PATTERN SELECTION IN RAYLEIGH-B]~NARD CONVECTION M. GOLUBITSKY Department of Mathematics, University of Houston, Houston, Texas 77004, USA and J.W. SWIFT and E. KNOBLOCH Department of Physics, University of California, Berkeley, California 94720, USA Received 10 May 1983 Revised 29 June 1983 This paper describes the process of pattern selection between rolls and hexagons in Rayleigh-l~nard convection with reflectional symmetry in the horizontal midplane. This symmetry is a consequence of the Boussinesq approximation, provided the boundary conditions are the same on the top and bottom plates. All possible local bifurcation diagrams (assuming certain non-degeneracy conditions) are found using only group theory. The results are therefore applicable to other systems with the same symmetries. Rolls, hexagons, or a new solution, regular triangles, can be stable depending on the physical system. Rolls are stable in ordinary Rayleigh-B~nard convection. The results are compared to those of Buzano and Golubitsky [1] without the midplane reflection symmetry. The bifurcation behavior of the two cases is quite different, and a connection between them is established by considering the effects of breaking the reflectional symmetry. Finally, the relevant experimental results are described. 1. Introduction Rayleigh-B~nard convection provides perhaps the best studied example of nonlinear pattern selection. In the simplest version of the problem a layer of fluid confined between infinite, stress-free, horizontal boundaries is heated uniformly from below. For small temperature differences, mea- sured by the Rayleigh number R, energy is trans- ported by molecular conduction. As R is increased, the conduction state loses stability. At R =/~, the point of neutral stability, the linear stability prob- lem admits several qualitatively different plan- forms: rolls, squares, hexagons, and in fact any linear combination of rolls with the critical wave- length. For supercritical values of R the amplitude of each planform grows exponentially until the nonlinear effects become important. The non- linear terms are responsible for selecting one of the patterns admitted by the linearized problem. In the laboratory, this process will be affected by random initial conditions and imperfections in the appara- tus, as well as by the presence of sidewalls, all of which will have an effect on pattern selection. Much theoretical work on convection assumes the Boussinesq approximation, in which all mate- rial properties are independent of temperature, with the exception of the density entering in the driving buoyancy term. If, in addition, the bound- ary conditions are the same on the top and bottom plates, and the mean temperature in the layer is time-independent [2], then the resulting problem is symmetric under a reflection in the horizontal midplane, together with a temperature reversal. Under such conditions it has been predicted (Sch- liiter et al. [3]) that in a large aspect ratio container rolls will be observed at the onset of convection. On the other hand, in systems lacking the reflectional symmetry, i.e., non-Boussinesq fluids, or systems with asymmetrical boundary conditions or time-dependent heating, hexagons are usually observed. This tendency has been explained for a 0167-2789/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)