Magnetic stability and non-linear evolution of a selection of mean field dynamos

The work presented herein is a continuation and extension of previous magnetic stability and non-linear evolution analyses by Hutcheson and Fearn [J. Fluid Mech. 291 (1995) 343; Phys. Earth Planet. Int. 97 (1996) 43; Phys. Earth Planet. Int. 99 (1997) 19]. The earlier works are based on imposed magnetic fields in cylindrical geometries which inevitably suffer from a certain degree of artificiality. We have therefore taken two important steps towards geophysical realism; firstly, our basic state field is now dynamically generated as part of a mean field dynamo mechanism, and secondly, we perform the analysis in a spherical shell with a finitely conducting inner core. alysis is undertaken numerically using a pseudo spectral time stepping procedure. This allows us to examine Ekman numbers as low as 2.5×10−4 whilst assuming that the Rossby number is identically zero. Our results indicate that magnetic instabilities, in the form of symmetry breaking bifurcations from the axisymmetric basic states supplied by the mean field dynamos, occur at Elsasser numbers of around O(10) or greater. Also, in each model, the symmetry breaking bifurcation was supercritical in nature. These results, although similar to the ones obtained in the previous cylindrical geometry work, hold when the basic state is oscillatory, a feature which was not explored in the studies mentioned above.