Spherical magnetic instabilities in the Earth’s core and equatorial symmetries

Abstract

The Boussinesq rotating magnetohydrodynamic equations in a conducting fluid sphere, linearised about a steady basic state, are discretised using scalar and vector spherical harmonics. Vector spherical harmonic representations are used for the basic state vector fields as well as the interaction terms in the equations. The solenoidal property of the perturbation magnetic and velocity fields are imposed on the fields and equations using relations between vector spherical harmonics and toroidal–poloidal representations. The method has several distinct advantages over previous discretisation methods: it produces compact angular spectral equations which require only five core subroutines to generate all interactions and product terms, and only six core subroutines to radially discretise the equations using finite-differences; the core subroutines can be extensively tested using only simple models; and its generality permits very flexible code. We apply our approach to the magnetic instabilities of a steady axisymmetric azimuthal magnetic field. To isolate magnetic instabilities and suppress other instabilities the magnetostrophic approximation is made. The perturbation velocity and magnetic field satisfy the inviscid boundary and insulating exterior conditions, respectively. Results are presented for the simplest equatorially symmetric (quadrupole) and equatorially antisymmetric (dipole) magnetic field models which vanish on a single spherical surface. For each model the perturbation fields decouple into two subproblems, in which the perturbation magnetic field has the same or opposite symmetry to the basic state field. A strong preference is found for the equatorially symmetric perturbation magnetic field in the equatorially symmetric model. However, there is only a marginal preference for the equatorial antisymmetric perturbation magnetic field in the equatorially antisymmetric model. We also consider a class of equatorially unsymmetric basic state magnetic fields which are convex combinations of the first two models. Although the critical Elsasser number does not change appreciably, the most unstable mode changes totally from equatorially symmetric to equatorially antisymmetric. Finally, we consider the effect on the equatorially symmetric model of varying the spherical surface on which the field vanishes. The most unstable mode is always equatorially symmetric. When the dimensionless magnetic-zero radius is decreased below about 0.7 the most unstable mode switches from eastward to westward propagation. The magnetic field perturbation in the westward (eastward) case is strongly confined to the region outside (inside) the magnetic-zero radius.