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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The depth of penetration of Jupiter's zonal winds into the planet's interior is unknown. A possible way to determine the depth is to measure the effects of the winds on the planet's high-order zonal gravitational coefficients, a task to be undertaken by the Juno spacecraft. It is shown here that the equatorial winds alone largely determine these coefficients which are nearly independent of the depth of the non-equatorial winds.

This paper presents the first attempt to examine the stability of a poloidal magnetic field in a rapidly rotating spherical shell of electrically conducting fluid. We find that a steady axisymmetric poloidal magnetic field loses its stability to a non-axisymmetric perturbation when the Elsasser number A based on the maximum strength of the field exceeds a value about 20. Comparing this with observed fields, we find that, for any reasonable estimates of the appropriate parameters in planetary interiors, our theory predicts that all planetary poloidal fields are stable, with the possible exception of Jupiter. The present study therefore provides strong support for the physical relevance of magnetic stability analysis to planetary dynamos. We find that the fluid motions driven by magnetic instabilities are characterized by a nearly two-dimensional columnar structure attempting to satisfy the Proudman-Taylor theorm. This suggests that the most rapidly growing perturbation arranges itself in such a way that the geostrophic condition is satisfied to leading order. A particularly interesting feature is that, for the most unstable mode, contours of the non-axisymmetric azimuthal flow are closely aligned with the basic axisymmetric poloidal magnetic field lines. As a result, the amplitude of the azimuthal component of the instability is smaller than or comparable with that of the poloidal component, in contrast with the instabilities generated by toroidal decay modes (Zhang and Fearn, 1994). It is shown, by examining the same system with and without fluid inertia, that fluid inertia plays a secondary role when the magnetic Taylor number T , 2 lo5. We find that the direction of propagation of hydromagnetic waves driven by the instability is influenced strongly by the size of the inner core.