Hydromagnetic waves in rapidly rotating spherical shells generated by poloidal decay modes

Abstract

This paper presents the first attempt to examine the stability of a poloidal magnetic field in a rapidly rotating spherical shell of electrically conducting fluid. We find that a steady axisymmetric poloidal magnetic field loses its stability to a non-axisymmetric perturbation when the Elsasser number A based on the maximum strength of the field exceeds a value about 20. Comparing this with observed fields, we find that, for any reasonable estimates of the appropriate parameters in planetary interiors, our theory predicts that all planetary poloidal fields are stable, with the possible exception of Jupiter. The present study therefore provides strong support for the physical relevance of magnetic stability analysis to planetary dynamos. We find that the fluid motions driven by magnetic instabilities are characterized by a nearly two-dimensional columnar structure attempting to satisfy the Proudman-Taylor theorm. This suggests that the most rapidly growing perturbation arranges itself in such a way that the geostrophic condition is satisfied to leading order. A particularly interesting feature is that, for the most unstable mode, contours of the non-axisymmetric azimuthal flow are closely aligned with the basic axisymmetric poloidal magnetic field lines. As a result, the amplitude of the azimuthal component of the instability is smaller than or comparable with that of the poloidal component, in contrast with the instabilities generated by toroidal decay modes (Zhang and Fearn, 1994). It is shown, by examining the same system with and without fluid inertia, that fluid inertia plays a secondary role when the magnetic Taylor number T , 2 lo5. We find that the direction of propagation of hydromagnetic waves driven by the instability is influenced strongly by the size of the inner core.

Fast Quasi‐Geostrophic Magneto‐Coriolis Modes in the Earth's Core

“…In this section, we present a set of basis vectors for the 3‐D magnetic field, satisfying insulating boundary conditions at the CMB. Unlike in classical geodynamo simulations, where the boundary condition is enforced at each time step of the forward iteration, the boundary condition is included in the basis elements (Chen et al., 2018; Li et al., 2010; Zhang & Fearn, 1995). The detailed derivation of such a basis is given in Appendix .…”

Is the geodynamo process intrinsically unstable?

“…scaled so that | B P | max =1 and | B T | max =1. We use the same functional form of B T and B P as in a previous study of magnetic field instability (Zhang & Fearn 1994; Zhang & Fearn 1995). Let us look at the form of the poloidal magnetic field B P as an example.…”

Fast Quasi-Geostrophic Magneto-Coriolis Modes in the Earth's core

“…In this section we present a set of basis vectors for the 3-D magnetic field, satisfying insulating boundary conditions at the CMB. Unlike in classical geodynamo simulations, where the boundary condition is enforced at each time step of the forward iteration, the boundary condition is included in the basis elements (Zhang & Fearn, 1995;Li et al, 2010;Chen et al, 2018). The detailed derivation of such a basis is given in Appendix A.…”