Three-dimensional convection in a horizontal fluid layer subjected to a constant shear

Rayleigh-Bénard convection in the presence of a plane Couette flow is investigated by numerical computations. From earlier work it is well known that longitudinal rolls are preferred at the onset of convection and that at Prandtl numbers of the order unity or less these rolls become unstable with respect to the wavy instability which introduces wavy distortions perpendicular to the axis of the rolls. In the present analysis the three-dimensional flows arising from these distortions are studied and their stability is considered. A main result is the subcritical existence of three-dimensional flows at Rayleigh numbers far below the critical value for onset of convection.