Convection in a rotating annulus uniformly heated from below

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Abstract
We present a linear stability analysis, to second order in initial amplitude, of Bénard convection of a Boussinesq fluid in a thin rotating annulus for modest Taylor numbers T ([les ] 104). The work is motivated in part by the desire to study further a mechanism for maintaining, through horizontal Reynolds stresses induced in the convection, the sun's ‘equatorial acceleration’, which has been demonstrated for a rotating convecting spherical shell by Busse & Durney. The annulus is assumed to have stress free, perfectly conducting top and bottom (which allows separation of the equations) and non-conducting non-slip sides. A laboratory experiment which fulfills these conditions (except perhaps the free bottom) is being developed with H. Snyder.We study primarily annuli with gap-width to depth ratios a of order unity. The close, non-slip side-walls produce a number of effects not present in the infinite plane case, including overstability at high Prandtl numbers P, and multiple minima in Rayleigh number R on the stability boundary. The latter may give rise to vacillation. The eigenfunctions for stationary convection for a = 2, T [lsim ] 2000 clearly show momentum of the same sense as the rotation is transported from the inner to the outer half of the annulus, corresponding to transport toward equatorial latitudes on the sphere. The complete second-order solutions for the induced circulations indeed give faster rotation in the outer half, except for large P (> 102), in which case thermal stresses dominate. At all P, this differential rotation is qualitatively a thermal wind. Overstable convective cells, and stationary cells at higher T, induce more complicated differential rotations.