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.
Existence and Homogenization of the Rayleigh-Bénard Problem