In contrast to the well-known columnar convection mode in rapidly rotating spherical fluid systems, the viscous dissipation of the preferred convection mode at sufficiently small Prandtl numberPrtakes place only in the Ekman boundary layer. It follows that different types of velocity boundary condition lead to totally different forms of the asymptotic relationship between the Rayleigh numberRand the Ekman numberEfor the onset of convection. We extend both perturbation and numerical analyses with the stress-free boundary condition (Zhang 1994) in rapidly rotating spherical systems to those with the non-slip boundary condition. Complete analytical solutions – the critical parameters for the onset of convection and the corresponding flow and temperature structure – are obtained and a new asymptotic relation betweenRandEis derived. While an explicit solution of the Ekman boundary-layer problem can be avoided by constructing a proper surface integral in the case of the stress-free boundary problem, an explicit solution of the spherical Ekman boundary layer is required and then obtained to derive the solvability condition for the present problem. In the corresponding numerical analysis, velocity and temperature are expanded in terms of spherical harmonics and Chebychev functions. Accurate numerical solutions are obtained in the asymptotic regime of smallEandPr, and comparison between the analytical and numerical solutions is then made to demonstrate that a satisfactory quantitative agreement between the analytical and numerical analyses is reached.

[1] To first approximation the interiors of many planetary bodies consist of a core and mantle with significantly different densities. The shapes of the surface and interface between the core and the mantle are basic properties reflecting planetary structure and rotation. In addition, interface shape is an important parameter controlling the dynamics of a fluid core. We present a theory for the rotational distortion of a two-layer model of a planet (two-layer Maclaurin spheroid) that determines the shapes of both the interface and the outer free surface without treating departure from sphericity as a small perturbation. Since the interface and the outer free surface, in general, have different shapes, two different spheroidal coordinates are required in the mathematical analysis, and the transformation between them is at the heart of the complexity of the theory. Furthermore, two different cases have to be considered. In the first case, the core is sufficiently large, or the rate of rotation is sufficiently small, that the foci of the outer free surface are located within the core. In the second case, the core is sufficiently small, or the rate of rotation is sufficiently fast, that the foci of the free surface are located within the outer layer. In comparison to the classical Maclaurin solution which is explicitly analytical, the relevant multiple integrals for the equilibrium solution of a two-layer Maclaurin spheroid have to be evaluated numerically. The shape of a two-layer rotating planet is characterized by three dimensionless parameters that are explored systematically in the present study.

Solutions of the Poincaré equation describing equatorially trapped three-dimensional boundary travelling waves in rotating spherical systems are discussed. It is shown that the combined effects of Coriolis forces and spherical curvature enable the equatorial region to form an equatorial waveguide tube with characteristic latitudinal radius (2/m)1/2 and radial radius (1/m), where m is azimuthal wavenumber. Inertial waves with sufficiently simple structure along the axis of rotation and sufficiently small azimuthal wavelength must be trapped in the equatorial waveguide tube. The structure and frequency of the inertial waves are thus hardly affected by the presence of an inner sphere or by the condition of higher latitudes. Further calculations on rotating spherical fluid shells of finite internal viscosity and stressfree boundaries are also discussed.

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