In this thesis we investigate aspects of the Earth's magnetic field with a view to further understanding the physical processes at work in the Earth's core which act to maintain the geomagnetic field against ohmic decay. In particular, we consider problems relating to the stability of the Earth's magnetic field. The basic model of the Earth's fluid outer core we adopt throughout, chosen to simplify the problem whilst retaining the important features, is that of a cylindrical annulus of electrically conducting, incompressible fluid rotating rapidly about its axis. In this annulus we impose a toroidal magnetic field and velocity field which both depend only on the distance from the axis of rotation. This basic state is then perturbed and its stability investigated. In the first problem we consider in Chapter 2 the annulus is unbounded in the direction of the axis of rotation. We extend the problem solved by Fearn (1988) to include an inner core of finite conductivity and allow the mantle to be a perfect insulator or a perfect conductor. The effect of varying the inner core conductivity on the magnetic field strength required for the onset of instability is investigated and the result in the limit of an insulating inner core compared with that of Fearn (1988) to serve as a check on the results. In Chapter 3 the same model is extended to include a finitely conducting layer at the base of the mantle as a model for the D" layer. The inner core conductivity is in most cases assumed to be equal to that of the outer core and the influence of the D" layer on instability is investigated. The results are compared with those of the previous Chapter in the appropriate limits. In the final problem considered we extend the linear stability analysis into the weakly nonlinear regime. The annulus is bounded in the direction of the axis of rotation by perfectly conducting plates and the cylindrical walls are assumed to be perfectly insulating. To simplify the analysis the magnetostrophic approximation is made. Using a multiple scales expansion technique an amplitude equation is derived and the coefficients evaluated to determine if the instabilities are of sub- or supercritical type. As a check on the results the geostrophic flow that arises in the nonlinear regime is used as input into the linear code and its effect on the field strength required for instability investigated. The calculations in this thesis were carried out on the University of Glasgow's IBM 3090-150E/VF mainframe computer. The graphics have been produced using the GHOST80 graphics package. The results of Chapters 2 and 3 can also be found in Lamb (1994a,b).
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