Resistive instabilities in rapidly rotating fluids

Details Resistive instabilities in rapidly rotating fluids Resistive instabilities in rapidly rotating fluids

Nov 1991

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1991gapfd..60..295k&link_type=abstract

Geophysical and Astrophysical Fluid Dynamics, vol. 60, Issue 1, pp.295-333

Physics

Fluid Dynamics

11

Resistive Instability, Tearing Mode, Rotating Magnetohydrodynamics

Scientific paper

The linear stability is examined of a stratified sheet pinch in a rapidly rotating fluid lying hydrostatically in a gravitational field, g, perpendicular to the sheet. The sheet pinch is a horizontal layer of inviscid, Boussinesq fluid of electrical conductivity , magnetic permeabilitym, and almost uniform density 0, confined between two perfectly conducting planes z = 0, d, where z is height. The prevailing magnetic field, B0(z), is horizontal; it is unidirectional at each z level, but that direction depends on z. The layer rotates about the vertical with a large angular velocity, : VA/d, where VA = B0/(mρ0) is the Alfvén velocity. The Elsasser number, = s B20/2 Ωρ0, measures s. A (modified) Rayleigh number, R=gβd2/ρ0V2A, measures the buoyancy force, where β is the imposed density gradient, antiparallel to gravity, g. Diffusion of density differences is ignored. The gravitationless case, R = 0, was studied in Part 1 of this series. It was shown that "resistive instabilities", known as "tearing modes", exist when A is large enough, the horizontal wavenumber, k, of the instability is small enough, and when at least one "critical level" exists within the layer, i.e. a value of z at which B is perpendicular to the horizontal wavevector, k. When R is sufficiently large, instability occurs for any k; critical layers need not exist; k need not be small. For each k, a critical value, Rc(k), exists such that if R>RC the layer is ideally unstable (i.e. unstable for =); when a critical level exists, Rc is independent of k. When, the growth rate, s, of instability approaches from above that of the ideal mode as, but for R < Rc only resistive instabilities exist, i.e. those in which s+0 as. These "g-modes" are the main topic of this study. When critical levels are absent, they grow no more rapidly (except near R = Rc) than the rate at which B itself evolves ohmically; when a critical level exists, and especially when k is large (the so-called "fast g-modes"), they grow more rapidly. When l and one or more critical levels exist, s is determined entirely by the structure of the "critical layer" surrounding a critical level. When R>0, instability is always "direct" (s = 0), but when R <0, overstability may occur (so, s>0). Since such a mode bifurcates from a tearing instability, it exists only if is large enough, k is small enough, and a critical level exists. The main example studied here is a sheet pinch in which B is of constant strength but turns uniformly in direction with height; it is force-free.

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