Joint Action and Competition between Centrifugal, Magnetorotational, and Magnetic Buoyancy Instabilities

Abstract

Instabilities driven by some combination of rotation, velocity shear, and magnetic field in a stratified fluid under gravity play an important role in many astrophysical settings. Of particular note are the centrifugal instability, the magnetorotational instability, and magnetic buoyancy instability. Here, we consider a Cartesian model of an equatorial region incorporating all the physical ingredients necessary to study their competition. We investigate the linear instability to interchange (“axisymmetric”) modes of an inviscid, perfectly conducting, isothermal gas, including the effects of rotation, velocity shear, and poloidal and toroidal magnetic fields. The stability problem can be reduced to a second-order boundary value problem, with the growth rate as the eigenvalue. We can make analytic progress through consideration of the physically relevant regime in which the transverse horizontal wavenumber k ≫ 1. Via a perturbation analysis, with 1/ k as the small parameter, we can derive the growth rate and the spatial dependence of the eigenfunctions: the unstable modes are strongly localized in the vertical direction, being either wall modes (localized near a boundary of the domain) or body modes (localized in the interior). We describe the conditions under which the joint action of the separate instability mechanisms leads to enhancement or suppression of the instability. Our analytical results are supplemented by numerical solutions of the stability problem. The most unstable mode found analytically is typically in excellent agreement with that found numerically through consideration of a wide range of wavenumbers. Finally, we discuss how our results relate to the solar tachocline.

Keywords

• Magnetohydrodynamics
• Solar magnetic fields
• Perturbation methods