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 Ω 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 U 0 normal to the plane containing the rotation vector and gravity g (i.e. U 0 g × Ω Ω 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 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 Ra c 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 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 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.

Transient growth of Ekman-Couette flow

“…4, type I instability is characterized by a large wavenumber and a negative angle while type II has a smaller wavenumber and a positive angle. The results agree very well with the ones previously reported in [3,4].…”

Converging flow between coaxial cones

“…With convergence and rotation, a link may be made to classical stability problems involving Ekman instability and Couette-Poiseuille flows, not to mention Taylor vortices (Hoffman et al, 1998;Hoffman & Busse, 1999, 2001Ponty et al, 2003). Finally there is the question of the stability of the plug flows and boundary layers whose structure we have determined in our study.…”

Dynamo action in flows with cat's eyes

“…This flow is designed to be broadly similar to the flows seen for the Ekman instability (Hoffman et al 1998, Ponty et al 2001, 2003. However, note that in this flow the velocity w is constant on stream lines of constant .…”