Onset of oscillations in Rayleigh–Bénard convection: Horizontally unbounded slab

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Abstract
Rayleigh–Bénard convection in a laterally unbounded classical fluid layer with low Prandtl number P (ratio of kinematic viscosity to thermal diffusivity) is reexamined. An amplitude expansion with only a few normal modes yields lateral oscillations of the convective rolls, which are therefore only weakly nonlinear. For free boundary conditions, additional modes (absent for rigid boundaries) lead to long wavelength (‘‘hydrodynamic’’) oscillations, with explicit nonlinear distortions in the velocity and temperature fields. For oscillations with rigid boundaries, the finite critical wavenumbers are approximately independent of P for small P, and the calculated Rayleigh number, frequency, and wavenumber at onset agree well with observations in air. Discrepancies with experiments in dilute superfluid 3He–4He systems with small aspect ratios (ratio of horizontal to vertical dimensions) suggest that lateral boundaries or two‐fluid effects play an important role in these systems.