Thermally Driven Hydromagnetic Dynamos

There are still many challenges to be overcome before we can claim to have a full understanding of the generation of the Earth's magnetic field. Prom a mathematical point of view, the governing equations are nonlinear and must be solved in fully three dimensions, meaning that a numerical method must be employed, although this would probably also be the case for a two-dimensional problem. However, it is only relatively recently that the computer technology has become available to make this possible. Obtaining these solutions remains a highly computationally intensive task, making it difficult to find solutions for a range of parameter values. This is extremely important as a great deal of uncertainty still surrounds the present (and past) geophysical values of the main parameters in the governing equations. Our aim is to try and further understanding of the effect of varying some of these key parameters in simplified, but fully self-consistent hydromagnetic dynamo models. These models will allow us to examine the effect of including the full inertial term to the equations, which has in the past been neglected due to the small geophysical value of the parameter which controls its effect. Further physical insight into the magnetic field generation mechanism will be provided, and we will examine some key issues in numerical dynamo modelling. A broad introduction to the Earth's magnetic field, the properties of the core, the possible energy sources and the current state of successful numerical dynamo models, is given in Chapter 1. In Chapter 2 we will describe in detail the governing equations and associated theory of magnetohydrodynamic (MHD) flows in a rapidly rotating spherical shell, as is appropriate for the Earth. Chapter 3 presents the results of varying the Rayleigh number, Ra, and the azimuthal wavenumber, m, for a 21/2D dynamo model, and also examines the effect (if any) of different forms of thermal driving. We show that the dynamo can exhibit very different types of behaviour for small changes in Ra, and in one particular case the magnetic field can be shut off, leaving only a convective solution. This type of behaviour is not observed for a different value of m. Our model is therefore too severely truncated in azimuth, but also suggests that care should be exercised when interpreting the results from a single run of a numerical dynamo model with a fixed value of Ra. The different forms of thermal driving produce qualitatively very similar dynamos, with the case of internal heating seeming to give the most efficient dynamo at any given value of Ra. However our definition of Ra is most suitable for internal heating, and this probably accounts for the difference in efficiency. In Chapter 4, the same 21/2D model is used to examine the effect of varying the inner core radius. This is the first detailed study to be performed in a fully self- consistent dynamo model, and will aid understanding of the long term behaviour of the geodynamo, because the inner core is slowly growing as it freezes out of the outer core fluid. We find that the critical Rayleigh number for the onset of convection is dependent on the inner core radius. This plays a crucial role in determining the behaviour of the solution, along with the geometry and the diffusion time of the inner core. We show that not only does a large inner core stabilise the magnetic field, due to the diffusion time of the inner core, but that a small inner core also stabilises the magnetic field, due to the simpler geometry. The inertial term has not been included in a 21/2D model before, although it is included in some form in most SD models. In Chapter 5, we use a different 21/2D model to examine the effect of including the inertial term, and choosing different values of the Rossby number, Ro, while keeping the Ekman number, E, fixed. In addition the imposed equatorial symmetry constraint has been removed in this new model. We find a rather complicated pattern of behaviour, with the inertia of the fluid strongly affecting the time dependence of the solution obtained, but having less effect on the structure on the flow. There are two possible solutions, one which is chaotic and one which is periodic. As the value of Ro is increased we find that it becomes increasingly difficult to maintain a magnetic field, and above a certain value of Ro no solutions could be obtained. A solution obtained with m = 4 intermittently changes between chaotic and periodic states