Waves in the Earth’s core. I. Mildly diffusive torsional oscillations

Abstract

Axisymmetric oscillations of fluid in a rapidly rotating whole sphere immersed in a magnetic field can be supported by the elastic tension of the magnetic field lines. This special class of Alfvén waves is largely geostrophic (invariant along the rotation axis) and describes a set of normal modes that has been extensively studied in the ideal, lossless case, a limit in which regular solutions do not exist when the background magnetic field is axisymmetric. We study the geophysically relevant limit with parameters such that magnetic diffusion plays a realistic role appropriate to the Earth’s core, by choosing a Lundquist numberLuappropriately. We demonstrate for the first time the existence of normal modes in the presence of an axisymmetric background field, and obtain eigenfrequencies and decay rates that lead us to deduce quality factorsQfor these modes for two simple background fields of dipole and quadrupole parity. Two scaling behaviours ofQare seen depending on the background field and normal modes’ frequency, one scaling asLu1/2and another asLu, so that likelyQ>10in the core of the Earth. A one-dimensional theory is presented that is able to capture the frequencies of oscillation quite accurately.