An analytical study of the MHD clamshell instability on a sphere

TL;DR

This paper analytically investigates the MHD clamshell instability on a sphere, revealing how weak shear and magnetic field behavior influence the formation of the characteristic tilting pattern and instability.

Contribution

It provides an analytical framework for understanding the clamshell instability using asymptotic methods and semicircle rules, highlighting the role of local magnetic field behavior.

Findings

Weak shear excites the critical layer leading to instability.

Magnetic field influences the instability mainly through the critical layer.

Existence of a neutral tilting mode in solid body rotation.

Abstract

This paper studies the instability of two-dimensional magnetohydrodynamic (MHD) systems on a sphere using analytical methods. The underlying flow consists of a zonal differential rotation and a toroidal magnetic field is present. Semicircle rules that prescribe the possible domain of the wave velocity in the complex plane for general flow and field profiles are derived. The paper then sets out an analytical study of the `clamshell instability', which features field lines on the two hemispheres tilting in opposite directions (Cally 2001, Sol. Phys. vol. 199, pp. 231--249). An asymptotic solution for the instability problem is derived for the limit of weak shear of the zonal flow, via the method of matched asymptotic expansions. It is shown that when the zonal flow is solid body rotation, there exists a neutral mode that tilts the magnetic field lines, referred to as the `tilting mode'. A weak shear of the zonal flow excites the critical layer of the tilting mode, which reverses the tilting direction to form the clamshell pattern and induces the instability. The asymptotic solution provides insights into properties of the instability for a range of flow and field profiles. A remarkable feature is that the magnetic field affects the instability only through its local behaviour in the critical layer.