Linear Simulations of Boussinesq Convection in a Deep Rotating Spherical Shell

Abstract

We present extensive linear numerical simulations of Boussinesq convection in a rotating spherical shell of finite depth. The motivation for the study is the problem of general circulation of the solar convection zone. We solve the marching equations on a staggered grid in the meridian plane for the amplitudes of the most unstable Fourier mode of longitudinal wavenumber m between 0 and 24, for Taylor number T between 0 and 106, at a Prandtl number P=1, for a shell of depth 20% of the outer radius. Stress-free, fixed-temperature boundary conditions are used at the inner and outer bounding surfaces. Modes of two symmetries, symmetric and antisymmetric about the equator, are studied. The principal results are as follows: Increasing Taylor number T splits the most unstable solutions for each m into two classes: a broad band of high m solutions which peak at or near the equator, and a small number of low m solutions which peak at or near the poles. The equatorial modes are unstable at lower Rayleigh number R. The polar modes appear to be similar in many respects to plane-parallel convection with rotation parallel to gravity. Modes symmetric about the equator are unstable at lower R than those which are antisymmetric, by a percentage which increases with T in the range studied. Equatorial modes of both symmetries propagate prograde (frequency ?>0) in longitude at high T and retrograde (?<0) at low T, in agreement with earlier work. Polar modes propagate, too, but very slowly. Critical (first unstable) equatorial modes are shown to have or be closely approaching asymptotic dependence Rc?T?, mc?T?, ?c?T? with increasing T, in agreement with analytical analyses of Roberts and Busse. With increasing T, symmetric equatorial modes take on the form of rolls swirling about an axis parallel to the rotation axis and extending across both Northern and Southern Hemispheres in agreement with earlier results. Antisymmetric modes also assume a roll shape, but with swirl oppositely directed in the two hemispheres, together with fluid pumped across the equator parallel to the rotation axis. Polar modes become a ring of vortices more and more tightly arranged around the pole. Outward radial heat flux peaks at the equator for symmetric equatorial modes, and at a low latitude for antisymmetric modes. Both are suppressed near the equator near the outer boundary at high T. Symmetric modes also transport heat toward the equator, while antisymmetric modes transport heat poleward at the lowest latitudes, equatorward at somewhat higher latitudes. Symmetric equatorial modes transport angular momentum radially inward at low T, radially outward at high T. These modes transport angular momentum toward the equator from higher latitudes at all T.