The stability of toroidal magnetic fields with equatorial symmetry: Evolution of instabilities
 
Ken Hutcheson and David R Fearn

We examine the nonlinear evolution of instabilities of toroidal magnetic fields that are either antisymmetric or symmetric with respect to the equatorial plane. Strong antisymmetric toroidaI  fields will be present in planetary interiors because of strong differential rotation, and the stability of such fields will be an important consideration for the overall dynamo mechanism. An earlier detailed linear stability analysis has provided the  foundation for the calculations, and previous nonlinear  stability calculations of z-independent fields form the basis  for our numerical approach. The magnetic field strength is parameterised by the Elsasser number, L, such that Lc is the critical field strength, and we use an Ekman number, e = 10-4. We find that finite amplitude solutions exist for  L > Lc and that no such solutions exist for L < L (a weakly nonlinear analysis in the magnetostrophic regime had previously shown that subcritical finite amplitude solutions do exist). By investigating the dominant nonlinear effects, we conclude that our solutions are influenced by the viscosity of the model fluid, and that at e = 10-4 the solutions are not yet dominated by the geostrophic flow, as we might expect in the small e limit.