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Nonlinear convection in a rigid channel uniformly heated from below

A weakly nonlinear theory is developed for convection in an infinite rigid horizontal rectangular channel uniformly heated from below. A combination of analytical and numerical techniques along and in the cross-section of the channel leads to the derivation of an amplitude equation governing the spatial and temporal evolution of the flow above the critical Rayleigh number. Results are obtained for general Prandtl numbers and a wide range of aspect ratios. Overall trends are confirmed by comparison with results for an idealized model with stress-free horizontal boundaries. For wide channels, where the aspect ratio is large, the limiting form of the amplitude equation is predicted by reference to the two-dimensional equation describing roll patterns in infinite layers. The connection with this well-developed theory is established for both rigid and stress-free horizontal boundary conditions.