The onset of thermal convection in Ekman–Couette 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 $\Omega$, which is inclined at an angle $\vartheta$ to the vertical. The layer is sheared by keeping the upper boundary fixed, while the lower boundary moves parallel to itself with constant velocity ${\bm U}_0$ normal to the plane containing the rotation vector and gravity ${\bm g}$ (i.e. ${\bm U}_0\,\parallel\,{\bm g}\,{\times}\,\Omega)$. The system is characterized by five dimensionless parameters: the Rayleigh number $Ra$, the Taylor number $\tau^2$, the Reynolds number $Re$ (based on ${\bm U}_0$), the Prandtl number $Pr$ and the angle $\vartheta$. 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 $\vartheta$ 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 $\vartheta=0$ and $\tau\gg 1$, so-called type I and type II Ekman layer instabilities are possible. With the inclusion of buoyancy $Ra\not=0$ 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 II 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 $\vartheta\not=0$, the symmetry associated with ${\bm U}_0\leftrightarrow -{\bm U}_0$ 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 $\vartheta=\pi/4$. 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.