The onset of thermal convection in Ekman–Couette shear flow with oblique rotation
Gilbert, Andrew D.
Soward, Andrew M.
Journal of Fluid Mechanics
Cambridge University Press
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 ϑ to the vertical. The layer is sheared by keeping the upper boundary fixed, while the lower boundary moves parallel to itself with constant velocity U0 normal to the plane containing the rotation vector and gravity g (i.e. U0 || g × Ω). The system is characterized by five dimensionless parameters: the Rayleigh number Ra, the Taylor number τ2, the Reynolds number Re (based on U0), the Prandtl number Pr and the angle ϑ. 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 Rac as a function of τ and Re for representative values of ϑ 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 ϑ = 0 and τ >> 1, so-called type I and type II Ekman layer instabilities are possible. With the inclusion of buoyancy Ra ≠ 0 mode competition occurs. On increasing τ from zero, with fixed large Re, we identify four types of mode: a convective mode stabilized by the strong shear for moderate τ, hydrodynamic type I and II modes either assisted (Ra > 0) or suppressed (Ra < 0) by buoyancy forces at numerically large τ, and a convective mode for very large τ 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 ϑ _= 0, the symmetry associated with U0 ↔ −U0 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 ϑ = π/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.
Copyright © 2003 Cambridge University Press. Published version reproduced with the permission of the publisher.
Vol. 487, pp. 91-123