Rapidly Rotating Wall-Mode Convection

Abstract

In the rapidly rotating limit, we derive a balanced set of reduced equations governing the strongly nonlinear development of the convective wall-mode instability in the interior of a general container. The model illustrates that wall-mode convection is a multiscale phenomenon where the dynamics of the bulk interior diagnostically determine the small-scale dynamics within Stewartson boundary layers at the sidewalls. The sidewall boundary layers feedback on the interior via a nonlinear lateral heat-flux boundary condition, providing a closed system. Outside the asymptotically thin boundary layer, the convective modes connect to a dynamical interior that maintains scales set by the domain geometry. In many ways, the final system of equations resembles boundary-forced planetary geostrophic baroclinic dynamics coupled with barotropic quasi-geostrophic vorticity. The reduced system contains the results from previous linear instability theory but captured in an elementary fashion, providing a new avenue for investigating wall-mode convection in the strongly nonlinear regime. We also derive the dominant Ekman-flux correction to the onset Rayleigh number for large Taylor number, $\text{Ra}\approx 31.8{\text{Ta}}^{1/2}-4.43{\text{Ta}}^{5/12}+\mathcal{O}({\text{Ta}}^{1/3})$ for no-slip boundaries. However, we find that the linear onset in a finite cylinder differs noticeably compared to a Cartesian channel. We demonstrate some of the reduced model's nonlinear dynamics with numerical simulations in a cylindrical container.

44 pages, 14 figures. Accepted, J. Fluid Mech