Waves in the Earth’s core. III. A perturbative approach to quasi-free-decay modes

Abstract

We consider the canonical problem of magnetic field decay in an electrically conducting fluid ball. The problem is closely allied to the problem of the decay modes of a rigid ball, and the spatial form of the eigenmodes survives largely intact. The decaying but oscillatory behaviour of the new fluid eigenmodes first discovered by Schmitt a decade ago (and named quasi-free-decay (QFD) modes) is deduced by application of perturbation methods to the case of rapid rotation and a static applied background magnetic field that is uniform and axial. Some, but not all, of the rigid-case poloidal eigenmodes share decay rates with other toroidal modes, necessitating the use of both degenerate and non-degenerate perturbation theory within this paper. The perturbation theory is developed in terms of the Elsasser number Λ (measuring the competition between Coriolis and Lorentz forces), and the analytic results are in striking accord with numerical calculations even when Λ is of O ( 1 ) . We find linear scaling of the QFD eigenfrequency with Λ and small changes in the decay rate that scale with Λ 2 . Although the modes are overdamped (quality factor Q < 1 ), they are not strongly overdamped when the applied field is strong Λ ∼ 1 .