On the instability of toroidal magnetic fields and differential rotation in stars

Abstract

An electrically conducting spherical body of gas rotates in the presence of an azimuthal (toroidal) magnetic field B and its own gravitational field. Instabilities of the system due to either differential rotation or meridional gradients of B are examined by means of a local analysis. Account is taken of viscous, ohmic and thermal diffusion, the diffusivities being denoted by ν , η and κ respectively. Attention is mainly focused on the ‘rapidly rotating’ case in which the magnetic energy of the system is only a small fraction ( ε ) of the rotational energy. A discussion is given of some overlooked aspects of Goldreich-Schubert instability, which is usually said to occur if the angular momentum (per unit mass) decreases with distance r from the rotation axis or varies with distance z parallel to that axis. It is then shown that a toroidal magnetic field is not only less capable of suppressing the instability than has hitherto been supposed (when v ≪ n ) but actually acts as a catalyst for another quite different differential rotation instability if η is sufficiently small. This one is non-axisymmetric and substantially precedes that of Goldreich & Schubert by developing rapidly and with large azimuthal wavenumber if the angular velocity decreases more than a very small amount (0( ε )) with r . When the gas is strongly thermally stratified this instability still occurs if η is sufficiently small compared with κ . When the rotation is uniform, instability may still occur owing to the ( r, z ) distribution of the toroidal magnetic field itself. Its nature depends crucially on whether the region of interest is inside or outside a certain ‘critical radius’, the latter case being typically the more important astrophysically. Other geometrical effects of this kind complicate the issue, and though summarized at the end of the paper are difficult to report concisely here. The following results apply to a considerably simpler plane layer model previously investigated by Gilman (1970) and Roberts & Stewartson (1977). When the temperature gradient is almost adiabatic (as in a stellar convection zone) and rotation is absent, instability occurs (on the Alfvenic time scale) by Parker’s mechanism of magnetic buoyancy if B decreases with height. Rapid uniform rotation, such that, ε ≪ 1, stabilizes some field distributions, but those which decrease with height faster than the density ρ remain unstable (albeit with growth rates reduced by a factor of order ε 1 2 ) provided η is sufficiently small. When the gas is strongly thermally stratified (as in a stellar radiative interior) these results still apply if the thermal diffusivity κ is large enough to annul the effects of buoyancy, and this is the case if D * ≡ κV 2 /ηN 2 H 2 is large. Here V denotes the Alfvén speed, H the scale height and N the (conventional) buoyancy frequency. In the rapidly rotating case the stability of the system behaves in a curious way as D * is steadily decreased from an infinite value. The first significant effect of decreasing κ , or equivalently of increasing the stratification (!), is a destabilizing one, and only when D * drops below about unity does the stratification exert a significant stabilizing influence. The magnetic buoyancy instabilities above are all non-axisymmetric, but the possibility of axisymmetric instability, despite strong uniform rotation and stable stratification, is examined in an appendix. A somewhat novel instability, involving the simultaneous operation of two conceptually quite different doubly diffusive mechanisms, arises if ν/η is sufficiently small and κ/η is sufficiently large.