The onset of convection of a Boussinesq fluid in a horizontal plane layer is studied. The system rotates with constant angular velocity Omega, which is inclined at an angle upsilon to the vertical. The layer is sheared by keeping the upper boundary fixed, while the lower boundary moves parallel to itself with constant velocity U(0) normal to the plane containing the rotation vector and gravity g (i.e. U(0) parallel to g x Omega). The system is characterized by five dimensionless parameters: the Rayleigh number Ra, the Taylor number tau(2), the Reynolds number Re (based on U(0)), the Prandtl number Pr and the angle upsilon. The basic equilibrium state consists of a linear temperature profile and an Ekman-Couette flow, both dependent only on the vertical coordinate z. Our linear stability study involves determining the critical Rayleigh number Ra(c) as a function of tau and Re for representative values of upsilon and Pr. Our main results relate to the case of large Reynolds number, for which there is the possibility of hydrodynamic instability. When the rotation is vertical upsilon = 0 and tau much greater than1, so-called type I and type II Ekman layer instabilities are possible. With the inclusion of buoyancy Ra not equal 0 mode competition occurs. On increasing tau from zero, with fixed large Re, we identify four types of mode: a convective mode stabilized by the strong shear for moderate tau, hydrodynamic type I and 11 modes either assisted (Ra > 0) or suppressed (Ra < 0) by buoyancy forces at numerically large tau, and a convective mode for very large tau that is largely uninfluenced by the thin Ekman shear layer, except in that it provides a selection mechanism for roll orientation which would otherwise be arbitrary. Significantly, in the case of oblique rotation upsilon not equal 0, the symmetry associated with U(0) <----> -U(0) for the vertical rotation is broken and so the cases of positive and negative Re exhibit distinct stability characteristics, which we consider separately. Detailed numerical results were obtained for the representative case upsilon = pi/4. Though the overall features of the stability results are broadly similar to the case of vertical rotation, their detailed structure possesses a surprising variety of subtle differences.
