Order parameter and amplitude equations for the Rayleigh-Bénard convection

(Received 20 August 1993, revised 18 November 1993, accepted 6 December 1993)

Abstract
The reduced description of the roll patterns and their stability near onset is investigated in detail for Rayleigh-Bénard convection. The starting point is a novel order parameter equation (OPE) in Fourier space that is rigorous up to cubic order in the amplitudes of the critical modes at threshold. Comparison with rigorous results from a Galerkin analysis exhibits the range of validity of this order parameter description. In particular in the case of gases (Prandtl number ~ 1), the reduced description is fairly satisfactory. In a next step the OPE are used to derive coupled amplitude equations in real space. In this way a complete description of the mean drift mode is achieved for the first time. It is shown that at least for intermediate Prandtl numbers a large number of derivative terms is necessary to get good agreement with the rigorous results. From the results one can also judge, under what conditions common model equations are likely to describe the real systems.