Magnetohydrodynamic Instabilities in a Rapidly Rotating System

Magnetohydrodynamics and its use in understanding the Earth's magnetic field has enjoyed much attention in the last fifty years. This has much to do with the recent explosion in computer technology which has allowed the formulation and numerical solution of model problems which are not immediately analytically tractable. In this thesis, we approach the hydromagnetic dynamo problem from a stability point of view. We do not concern ourselves with the generation of the main (or basic) field, but consider its stability to small perturbations. Any instabilities found are important since they give constraints on the unknown field and sustaining motions in the core. After the introduction in Chapter 1, Chapter 2 formulates a linearised hydromagnetic stability problem as an eigenvalue problem. For a hydromagnetic system in the geometry of an infinite cylindrical annulus, we have revealed the presence of double eigenvalues at various locations in the parameter space. We show that tracking a particular eigenvalue around a closed path in parameter space need not necessarily return the original eigenvalue. This phenomena was first examined by Jones (1987), in the context of Poiseuille flow. In the hydromagnetic problem, we find that the most unstable mode (i.e. the mode we are most interested in) often behaves in this manner. We show that classifying magnetic instabilities as being either of the resistive or ideal class is not possible at geophysically relevant field strengths. In a nonlinear eigenvalue analysis, Fearn, Lamb, McLean & Ogden (1997) demonstrated qualitative differences between the viscid and the inviscid (magnetostrophic) approaches indicating that finite viscosity models cannot yet reach a parameter regime characteristic of the Earth's core. In Chapters 3 and 4 we present a nonlinear hydromagnetic stability analysis in a bounded annular model of the Earth's core. We adopt the magnetostrophic approximation in the fluid main body but incorporate viscous effects from the boundary layers in the form of the geostrophic flow. The nonlinear problem is then solved numerically using a time-stepping method. Chapter 3 corroborates and extends the work of Fearn et al (1997) and Chapter 4 considers the stability and nonlinear development of more geophysically relevant basic fields that depend not only on the radial coordinate, but also on the axial coordinate. This work is then compared with the viscous analyses of Hutcheson & Fearn (1995a,b, 1996, 1997)