RESISTIVE INSTABILITIES IN RAPIDLY ROTATING FLUIDS - LINEAR-THEORY OF THE G-MODE

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 σ, magnetic permeability α, and almost uniform density p0, 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, ω: ωVA/d, where [formula omitted] is the Alfvén velocity. The Elsasser number, Λ [formula omitted] measures σ. A (modified) Rayleigh number, [formula omitted] measures the buoyancy force, where βis 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 Λ= ∞); when a critical level exists, Rc is independent of k. When λ ∞, the growth rate, s, of instability approaches from above that of the ideal mode as A→∞ but for R<RC only resistive instabilities exist, i.e. those in which s→+0 as λ-→.∞ These “(/-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 Al 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”, but when R<0, overstability may occur [formula omitted]. Since such a mode bifurcates from a tearing instability, it exists only if A 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. © 1991 Taylor & Francis Group. All rights reserved.