Abstract

The linearised magnetohydrodynamic stability of an electrically-conducting liquid sphere is considered. Such studies complement non-linear dynamically-consistent dynamo calculations. The magnetic field, the velocity, the pressure and the temperature in the sphere are governed by the magnetic induction equation, the heat equation and the Navier-Stokes equation in the Boussinesq approximation with uniform diffusivities. The momentum equation may include inertial, buoyancy, viscous and magnetic Lorentz forces. The gravitational field may be asymmetric. If the reference frame is rotating, there is also a Coriolis force. The magnetic field and the velocity are solenoidal. The governing equations are linearised about a steady axisymmetric basic-state.

The linearised equations are discretised in angle using vector spherical harmonic expansions of all vector fields and spherical harmonic expansions of all scalar fields, for both the basic and perturbation states. Toroidal-poloidal representations are also used, but only for the perturbation solenoidal vector fields, not the basic state fields. Thus there are five independent perturbation scalar fields: the temperature, and the toroidal and poloidal potentials of the magnetic field and the velocity. The resulting hybrid angular spectral equations are more compact and less error prone to code, than if the toroidal-poloidal spectral interactions of products are fully expanded. Moreover, although there are eleven products, only three subroutines are required to evaluate their angular dependence. The subroutines calculate the three angular coupling integrals, which occur, in terms of 3j-, 6j- and 9j-symbols. Further, only six combinations of radial functions occur, which greatly simplifies radial discretisation. The code uses second-order finite-differences on a uniform radial grid. The resulting large-scale complex non-hermitian generalised eigen- and critical-value problems are solved using inverse and Newton-Raphson iteration methods, respectively. Results are presented for several models: free-decay, rotating viscous free-decay, rotating thermal convection, Dudley-James kinematic roll-dynamos and magneto-convection.

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