The onset of thermal convection in EkmanCouette shear flow with oblique rotation

Abstract

The onset of convection of a Boussinesq fluid in a horizontal plane layer is studied. The system rotates with constant angular velocity Ω, which is inclined at an angle V 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 ∥ g × Ω). The system is characterized by five dimensionless parameters: the Rayleigh number Ra, the Taylor number τ 2 , the Reynolds number Re (based on U 0 ), the Prandtl number Pr and the angle V. 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 r and Re for representative values of V and Pr. Our main results relate to the case of large Reynolds number, for which there is the possibility of hydrodynamic instability