Convective instabilities of a self-gravitating, rapidly rotating fluid spherical shell are investigated in the presence of an imposed azimuthal axisymmetric magnetic field in the form of the toroidal decay mode that satisfies electrically insulating boundary conditions and has dipole symmetry. Concentration is on two major questions: how purely thermal convection of the different forms (Zhang 1992, 1994) is affected by the Lorentz force, the strength of which is measured by the Elsasser number ∧, and in what manner purely magnetic instabilities in a spherical shell (Zhang & Fearn 1993, 1994) are associated with magnetic convection. It is found that the two-dimensionality of purely thermal convection (Busse 1970) survives under the influence of a strong Lorentz force. Convective motions always attempt to satisfy the Proudman–Taylor constraint and remain predominantly two-dimensional in the whole range of ∧, 0 ≤ ∧ ≤ ∧
c
, where ∧
c
═
O
(10) is the critical Elsasser number for purely magnetic instabilities. Though the optimum azimuthal wave number
m
of convection rolls decreases drastically, from
m
~
O
(
T
1/6
) at ∧ ═ 0 to
m
═
O
(5) at ∧ ═
O
(1). We show that there exist no optimum values of ∧ that can give rise to an overall minimum of the (modified) Rayleigh number
R
*; the optimum value of
R
* is a monotonically, smoothly decreasing function of ∧, from
R
* ═
O
(
T
1/6
) at ∧ <
O
(
T
-1/6
) to
R
* ═ O (–10) at ∧ ═ 20. We also show that the influence of the magnetic field on thermal convection is crucially dependent on the size of the Prandtl number. At sufficiently small Prandtl number, the Poincaré convection mode (Zhang 1994) is preferred in the region 0 ≤ ∧ < ∧
c
, and is only slightly affected by the presence of the toroidal magnetic field. Analytical solutions of the magnetic convection problem are then obtained based on a perturbation analysis, showing a good agreement with the numerical solution.
On a relationship between magnetohydrodynamic planetary eigenmodes and second-class inertial elgenmodes
