A critique of frozen-flux inverse modelling of a nearly steady geodynamo

Abstract

Most inversions of geomagnetic secular variation for fluid flow at the core's surface utilize the so-called frozen-flux approximation where diffusion of the magnetic field is neglected. Here we note that the frozen-flux approximation can fail miserably for the geophysically relevant case of a nearly steady dynamo. We draw this conclusion from an inspection of the induction equation, after expanding the velocity and magnetic fields in terms of temporal mean and fluctuating parts (as is standard in mean-field electrodynamics). The resulting pair of equations, one describing steady dynamo action and the other describing secular variation, are coupled, and since steady dynamo action relies on diffusion, secular variation cannot be approximately described by the (diffusive-free) frozen-flux approximation. In support of this conclusion we present two new kinematic dynamo models. The first dynamo exhibits no secular variation even though it has a non-zero surface flow; for this dynamo no purely poloidal surface flow model can be constructed by using the frozen-flux approximation, but the actual surface flow is purely poloidal. The second dynamo exhibits westward drift of the magnetic field but has surface flow that is eastwards. Both of these models have magnetic Reynolds numbers much greater than unity, and so satisfy the criterion usually invoked for justifying the use of the frozen-flux approximation, and yet both of these models are contradictory to the frozen-flux approximation. Thus, inversions for core flow should consider the effects of diffusion. Core flow models deduced assuming that the frozen-flux approximation is valid, and any results dependent on such flow models, should be treated with a great deal of caution.