Title:

Differential rotation and the geodynamo

This thesis investigates the effect of differential rotation on convection in a rapidly rotating, electrically conducting fluid, motivated by the problem of the generation of the Earth's magnetic field. First we consider how an imposed differential rotation affects the thermal stability of the fluid in the presence of an imposed toroidal magnetic field. We choose a magnetic field profile that is stable. The familiar role of differential rotation is a stabilising one. We wish to examine the less well known destabilising effect that it can have. In a plane layer model (for which we are restricted to Roberts number q = 0 to allow separation of variables) with differential rotation, U = sO(z)1&phis;, no choice of O(z) led to a destabilising effect. However, in a cylindrical geometry (for which our model permits all values of q) we found that differential rotations U = sO(z)1&phis; which include a substantial proportion of negative gradient (dO/ds < 0) give a destabilising effect which is largest when the magnetic Reynolds number Rm = O(10); the critical Rayleigh number, Rac, is about 7% smaller at minimum than at Rm = 0 for q = 106. We also found that as q is reduced, the destabilising effect is diminished and at q = 106, which may be more appropriate to the Earth's core, the effect causes a dip in the critical Rayleigh number of only about 0.001%. In the above results, the Elsasser number Lambda = 1 but the effect of differential rotation is also dependent on Lambda. Earlier work has shown a smooth transition from thermal to differential rotation driven instability at high Lambda [Lambda = 0(100)]. We find, at intermediate Lambda [Lambda = 0(10)], a dip in the Rac vs. Rm curve similar to the Lambda = 1 case. However, it has Rac < 0 at its minimum and unlike the results for high Lambda, larger values of Rm result in a restabilisation. We move on to take a brief look at how the above effect may influence magnetic instability. Working in the cylindrical geometry we again find that certain differential rotations have a destabilising effect, this time with a minimum in the critical value of the Elsasser number, Lambdac, at Rm = O(100) about 30% of its value at Rm = 0. Further, the forms of O(s) which cause this stability dip are the same ones as in the thermal stability investigation. The above forms a preliminary to pursuing some nonlinear calculations. In the magnetostrophic approximation, which we employ, the most important nonlinear effect is the geostrophicflow, UG(s)1&phis; which is dependent on the magnetic field through a modified Taylor's condition. Proceeding in the cylindrical geometry we study the nonlinear influence of this dynamically determined differential rotation on convection. The expected effect is for the geostrophic flow to equilibrate a growing solution in what is known as an Ekman state. This breaks down at greater forcing of the system and a Taylor state is reached. We approach this problem from the standpoint of trying to find Ekman states for various parameters and conditions. This we do successfully. However, we discover that our numerical solution sometimes breaks down when Ra is increased beyond a certain point (Ra = RaT). Here, we find good evidence to indicate that we are approaching a Taylor state. We also discover that o → 0 as Ra → RaT. We find that the most important factor affecting the numerical breakdown is the orientation of gravity. No numerical breakdown occurs (i.e. we can reach very high values of Ra) with radially directed gravity. However, this is not the case for axially directed gravity. The range of Ra over which an Ekman state can be found is of 0(1) or less. Interestingly, there is also a correspondence between the gravity direction and the shape of the geostrophic flow. Radial gravity seems to promote a form of O which wouldn't have a stability dip (see above) whereas axial gravity favours a form which would.
