Convection in rotating annulus: An asymptotic theory and numerical solutions

Abstract

Thermal instabilities in the form of two-dimensional convection rolls in a rotating annulus with the flat rigid ends and a moderate gap are investigated by both asymptotic and numerical methods. It is shown that the thin Ekman boundary layers at the ends of the annulus, to which the convection rolls attach, play an active controlling role. An asymptotic theory for an asymptotically large Taylor number T is developed to obtain complete analytical solutions of the convection rolls, indicating at leading order Rc = R0+C(T1/4/λ), ω = 0, where λ is the aspect ratio, ω is the frequency, Rc is the critical Rayleigh number with the presence of the Ekman boundary layers and R0 is the critical Rayleigh number without the influence of the boundary layers. While R0 can be determined exactly by using Bessel functions as the eigenfunction, constant C is obtained by matching the interior convection rolls to explicit solutions of the Ekman boundary layers. In the corresponding numerical analysis, convection solutions in a rotating annulus are calculated up to T = 108 with λ = 1. The analytical and numerical convection solutions are then compared to show a remarkable quantitative agreement when T>106. © 1998 American Institute of Physics.