Resistive instability and the magnetostrophic approximation

Abstract

We investigate resistive instability of the toroidal magnetic field B*0 = B*0 (s*)1ø, [where (s*, ø, z*) are cylindrical polar coordinates] permeating a conducting fluid confined in an infinite cylindrical annulus. With application to planetary cores in mind, the system is rapidly rotating with uniform angular velocity ω0= ω0lz. Resistive instability is most often associated with critical levels k . B*0 = 0 (where k is the wave vector). For our choice of field, critical levels are located at zeros of B*0. In this paper, the main emphasis is on studying resistive instability when no critical levels are present and we find instabilities for certain choices of B*0 when the cylindrical container is electrically insulating. Asymptotic results are obtained in the limit of high conductivity and in the limit of small axial wavenumber. A very careful approach is necessary if the conductivity of the fluid is large. The Elsasser number Λ is a non-dimensional measure of the conductivity. When a critical level is present, instability is concentrated in a critical layer of width O(Λ-1/3) in the limit Λ → ∞ For the cylindrical geometry, when no critical level is present, the magnetic boundary layer that develops in the limit of large A has width O(Λ-1). This has two immediate consequences. Numerically, it means that the boundary layer can only be resolved for modest values of Λ. Physically, as the lengthscale in the boundary layer decreases, it means that viscous effects (which are normally neglected along with inertial effects in the magnetostrophic approximation) must become important. Reinstating viscosity modifes the boundary-layer scalings and permits following instability to much higher values of A. Clearly, if boundary layers are not passive, care must be taken when using the magnetostrophic approximation when resistive effects are weak in order to obtain physically correct results.

Key words:

• Earth's core
• hydromagnetic waves
• magnetic instabilities
• magnetostrophic approximation
• resistive instability