Journal of Fluid Mechanics


The onset of thermal convection in Ekman–Couette shear flow with oblique rotation

Y. PONTY a1, A. D. GILBERT a2 and A. M. SOWARD a2
a1 Observatoire de la Cote d'Azur – Nice, Laboratoire Cassini – CNRS UMR 6529, BP 4229, 06304 Nice Cedex 4, France
a2 School of Mathematical Sciences, University of Exeter, Exeter EX4 4QE, UK


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.

(Received June 17 2002)
(Revised November 27 2002)