In many geophysical and astrophysical contexts, thermal convection is influenced by bothrotation and an underlying shear flow. The linear theory for thermal convection ispresented, with attention restricted to a layer of fluid rotating about a horizontal axis,and plane Couette flow driven by differential motion of the horizontal boundaries.

The eigenvalue problem to determine the critical Rayleigh number is solved numericallyassuming rigid, fixed-temperature boundaries. The preferred orientation of the convectionrolls is found, for different orientations of the rotation vector with respect to the shearflow. For moderate rates of shear and rotation, the preferred roll orientation depends onlyon their ratio, the Rossby number.

It is well known that rotation alone acts to favour rolls aligned with the rotationvector, and to suppress rolls of other orientations. Similarly, in a shear flow, rollsparallel to the shear flow are preferred. However, it is found that when the rotation vectorand shear flow are parallel, the two effects lead counter-intuitively (as in other,analogous convection problems) to a preference for oblique rolls, and a critical Rayleighnumber below that for Rayleigh–Bénard convection.

When the boundaries are poorly conducting, the eigenvalue problem is solved analyticallyby means of an asymptotic expansion in the aspect ratio of the rolls. The behaviour of thestability problem is found to be qualitatively similar to that for fixed-temperatureboundaries.

Fully nonlinear numerical simulations of the convection are also carried out. These aregenerally consistent with the linear stability theory, showing convection in the form ofrolls near the onset of motion, with the appropriate orientation. More complicated statesare found further from critical.