Magnetohydrodynamic drag on an oscillating sphere in a rotating cavity

TL;DR

This paper develops a comprehensive boundary-layer theory for magnetohydrodynamic drag on an oscillating sphere in a rotating cavity, relevant to planetary interiors and confined MHD systems.

Contribution

It introduces a unified framework for magnetic and viscous effects in bounded rotating MHD flows, extending previous simplified models and validating with simulations.

Findings

Derived boundary layers and magnetohydrodynamic drag from Alfvén-wave radiation, viscous effects, and Ohmic dissipation.

Extended the theory to non-axisymmetric modes and rotation perturbations, including boundary layer corrections.

Validated the theoretical model with magnetohydrodynamic simulations, providing a quantitative planetary interior framework.

Abstract

We analyse the magnetohydrodynamic drag on a sphere undergoing small-amplitude translational oscillations in a rotating spherical cavity. This provides a canonical model for oscillatory flows in confined rotating magnetohydrodynamic systems, where dissipation arises from the poorly constrained coupling between magnetic fields, rotation and viscosity. Such flows occur in planetary interiors, notably driven by the translational oscillations of the Earth's inner core along linear or circular trajectories (the polar and equatorial Slichter modes). They may also arise in the thin subsurface oceans of icy moons where strong confinement is expected. Previous theoretical studies considered only simplified limits, restricted to the polar mode: Stokes (1851) solved the viscous bounded problem without rotation or magnetic effects, revealing the importance of pressure, whereas Buffett and Goertz (1995) examined magnetic tension in a non-rotating inviscid unbounded fluid, neglecting magnetic pressure and confinement. We develop a unified boundary-layer framework for magnetic and viscous effects in a fluid shell bounded by two solid regions with possibly different electromagnetic properties. Large electromagnetic contrasts arise even in simple laboratory configurations, such as an iron sphere oscillating in a liquid-metal (e.g. Galinstan). We derive boundary layers and obtain the magnetohydrodynamic drag from Alfv\'en-wave radiation, viscous effects and Ohmic dissipation (accounting for pressure effects). The theory is extended to non-axisymmetric equatorial modes and to rotation perturbations of the polar mode, with stronger rotation treated through magnetohydrodynamic Stokes-Ekman boundary layers and a corrected inviscid solution of Busse (1974). Our magnetohydrodynamic simulations validate our theory, providing a quantitative framework for planetary interiors.