In many geophysical and astrophysical contexts, thermal convection is influenced by both
rotation and an underlying shear flow. The linear theory for thermal convection is
presented, 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 numerically
assuming rigid, fixed-temperature boundaries. The preferred orientation of the convection
rolls is found, for different orientations of the rotation vector with respect to the shear
flow. For moderate rates of shear and rotation, the preferred roll orientation depends only
on their ratio, the Rossby number.
It is well known that rotation alone acts to favour rolls aligned with the rotation
vector, and to suppress rolls of other orientations. Similarly, in a shear flow, rolls
parallel to the shear flow are preferred. However, it is found that when the rotation vector
and 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 Rayleigh
number below that for Rayleigh–Bénard convection.
When the boundaries are poorly conducting, the eigenvalue problem is solved analytically
by means of an asymptotic expansion in the aspect ratio of the rolls. The behaviour of the
stability problem is found to be qualitatively similar to that for fixed-temperature
boundaries.
Fully nonlinear numerical simulations of the convection are also carried out. These are
generally consistent with the linear stability theory, showing convection in the form of
rolls near the onset of motion, with the appropriate orientation. More complicated states
are found further from critical.