Magnetic Stability Analysis for the Geodynamo

Kinematic dynamo modelling addresses the growth of magnetic fields under the action of fluid flow, effected in the geophysical case by the action of the convecting electrically conducting outer core and magnetic diffusion. In this study, we prescribe the flow to be stationary and geophysically motivated by some large scale process, for example, differential rotation or convection. Historically, workers have applied eigenvalue stability analysis and although in
some cases Earth-like solutions have been found, the results hinge critically on the precise choice of flow. We therefore cannot attribute physical mechanisms in such cases since it is rather case dependent.

Following the success of more generalised stability techniques applied to the transition to turbulence in various non-magnetic fluid dynamical problems, we investigate both the onset of and subsequent maximised transient growth of magnetic energy. These approaches differ from the eigenvalue methodology due to the non-normality of the underlying operator, which means that superposition of non-orthogonal decaying eigenmodes can result in sub-critical growth. The onset or instantaneous instability problem can be formulated using variational techniques which result in an equation amenable to a numerical Galerkin method. For the suite of flows studied, we find robust results that can be physically explained by field line stretching. Convectively
driven flows exhibit the greatest instability, the field structures giving this maximal instability being axisymmetric. All flows indicate an apparent asymptotic dependence on the magnetic Reynolds number Rm, which is reached when Rm = O(1000). For the flows studied, we find
improved lower bounds on Rm for energetic instability of between 5 and 14 times, compared to that resulting from the analytic analysis of Proctor (1977a).

In all the flows studied, without exception the geophysically dominant axisymmetric dipole field symmetry is preferentially transiently amplified. The associated physical mechanisms are either shearing of poloidal field into toroidal field by differential rotation, or advection into locations of radial upwelling followed by field line stretching in the convective case. Transient
energy growth of O(1000), which can be obtained when Rm = 1000 is robust and may explain the recovery of field intensity after a magnetic reversal. Assuming the flow to be a stationary solution of the geostrophic balance equation where buoyancy, pressure and the Coriolis forces are in equilibrium, we computed the geophysically scaled ratio of the Lorentz to the Coriolis
forces and found it to be O(1) for flows with a large convective component. This indicates that transient growth, in particular of axisymmetric fields that are ostensibly precluded by the theorem of Cowling (1933), can explain the entry into the non-linear regime without the need for eigenmode analysis.