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Phase-winding solutions in a finite container above the convective threshold

An analysis is presented of the steady states of two-dimensional convection near threshold in a laterally finite container with aspect ratio 2L [Gt ] 1. It is shown that the allowed wavevectors which can occur in the bulk of the container are reduced from a band |q| ∼ [(R − R0)/R0]½ in the laterally infinite system to a band |q| ∼ (R − R0)/R0 in a system with sidewalls (R is the Rayleigh number and R0 its critical value in the infinite system). The analysis involves an expansion of the hydrodynamic equations in the small parameter [(R − R0)/R0]½, and leads to amplitude equations with boundary conditions, which generalize to higher order those previously obtained by Newell & Whitehead and Segel. The precise values of the allowed wavevectors depend on the Prandtl number of the fluid and the thermal properties of the sidewalls. For certain values of these parameters all the allowed wavevectors are less than the critical value q0. The applicability of the results to convection in a rectangular container is briefly discussed.