Convection in a spherical shell under rapid rotation is of interest not only in the geophysical and astrophysical fields but also from a fundamental point of view. This large-scale phenomenon in planets and other celestial bodies can lead to a self-sustained dynamo if the fluid is a conductor of electricity. The problem has also been the subject of bifurcation-theoretic studies in order to analyze the nature and stability of the solutions obtained for various parameter ranges.

Consequently, we consider Rayleigh-Benard convection in a conducting fluid in a rotating spherical shell. The fluid is subjected to a radial gravitational force, to centrifugal forces and to the Lorentz force. We use a pseudo-spectral code to solve the magnetohydrodynamic (MHD) equations, which couple the fluid velocity, governed by the Navier-Stokes equations, with an electromagnetic field, in a spherical fluid shell. We perform direct numerical simulations with explicit and implicit treatment of the Coriolis term to compare their stability and accuracy, in addition to establishing a range of convergence for both methods. We adapt the code to compute steady states and rotating waves via Newton’s method in order to access unstable states and trace bifurcation diagrams.