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

The linear stability of the sheet pinch is examined for a rapidly rotating fluid. The sheet pinch is a horizontal layer of uniform, incompressible, inviscid fluid of density ρ, electrical conductivity σ, magnetic permeability μ, confined between two perfectly conducting planes z=0, d, where z is height. The prevailing magnetic field, B0(z), is horizontal; at each height it is unidirectional, but that direction turns continuously (in one sense only) as z increases; the system is in magnetostatic equilibrium. The layer rotates about the vertical with an angular velocity, Ω, that is large: Ω-VA/d, where [formula omitted] is the Alfvén velocity and μ is the magnetic permeability. The dimensionless measure of σ is the Elsasser number [formula omitted]. The principal example studied here is force-free (J0×B0 = 0) since the chosen electric current density, [formula omitted], is a multiple of B0. Both are of constant strength, but turn uniformly in direction with height, completing a total of q radians between z = 0 and z = d. The growth rate, s, of small perturbations of horizontal wave vector k, is determined in a number of cases. It is found that instabilities do not occur if B0 makes less than one half turn between the boundaries. For q>n, stability is lost when A exceeds a critical value, Ac, the instability being direct (i.e. the largests is real, and this s is zero for Λ=Λc); as q increases, Λc decreases and s increases for any A. As A increases beyond Λc, s attains a maximum and then decreases monotonically to zero as Λ→∞. The asymptotic form of the eigenmodes in the limit Λ→∞ is analysed in detail for general B0, especially their structure within the critical layers [of thickness [formula omitted] surrounding the critical levels, at which k is orthogonal to B0. The equilibrium is found to be more unstable when J0 × B0/J0. B0 is antiparallel to Ω than when it is parallel. It is shown that, provided this critical level is not asymptotically within one of the boundary layers [thickness [formula omitted], at a wall, s = [formula omitted], i.e. the instability develops more rapidly than the rate [formula omitted] at which B0 evolves through ohmic diffusion. Numerical evidence is presented, however, which indicates that, for the mode of maximum instability, the critical level moves into a boundary layer as Λ→∞. © 1990, Taylor & Francis Group, LLC. All rights reserved.