Resistive Instabilities in Rapidly Rotating Fluids.
Abstract
Resistive instabilities of a forcefree magnetic field are analyzed in this dissertation. The main example of such a system studied here is a sheet pinch rotating rapidly about the vertical. The sheet pinch is a layer of incompressible, inviscid, fluid of almost uniform density rho_0, electrical conductivity sigma, magnetic permeability mu, confined between two horizontal perfectly conducting planes z = 0, d. The prevailing magnetic field, B_0(z), is horizontal, sheared and of constant strength. The angular velocity Omega of the layer is large: Omegagg V_{A}/d, where V_ {A}=B_0/sqrt{murho _0} is the Alfven velocity. The magnetic diffusivity eta=1/musigma is measured by the Elsasser number Lambda=V _sp{A}{2}/2Omegaeta . The layer is prone to magnetic "tearing" instabilities and, if the layer is stratified, also to "gmode" instabilities. They are analyzed by linear stability theory. Normal mode solutions are obtained by means of a compact finite difference method for finite Lambda and an asymptotic analysis for the limit Lambdatoinfty. . The linear analysis of the tearing modes shows that instabilities occur if the shearing of the field B_0 is strong enough, if Lambda exceeds a certain critical value, Lambda_{c}, and if the horizontal wave number k is small enough. When Lambdagg1, singular layers of thickness O(Lambda^{1/4}d) develop around singular levels z_{s } at which the prevailing field B_0 is perpendicular to the propagation direction of the perturbations. The (dimensionless) linear growth rate, s, of the perturbations decreases to zero with the power law s~Lambda^{3/4 } as Lambdatoinfty. . In the linear analysis of the gmodes, the density stratification is measured by a (nondimensional) modified Rayleigh number R and the diffusion of density differences is neglected. We find that the system is unstable in the ideal limit (Lambda=infty) if R exceeds a critical value R_{c }. When R<R_{c} and R = O(1), resistive gmodes exist with an O(k Lambda^{1/2}) growth rate. The nonlinear evolution of tearing modes is analyzed when Lambda is slightly above critical and when Lambdatoinfty. When the amplitude A of the perturbations is finite, "Taylor state" solutions exist due to the shear in B_0 . We find that the critical mode evolves to a nonlinear steady and stable state with the amplitude  A_{s}~sqrt {Lambda  Lambda_{c}} . In the limit Lambdatoinfty , there are no steady stable states. The singular levels are driven out of the singular layers by the nonlinear interaction. In other words, the singular layer structures of the linear theory are destroyed by the nonlinear interactions. A full nonlinear analysis is therefore necessary to study the nonlinear evolution in the limit Lambda toinfty.
 Publication:

Ph.D. Thesis
 Pub Date:
 January 1992
 Bibcode:
 1992PhDT........78K
 Keywords:

 Mathematics; Physics: Fluid and Plasma; Geophysics