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

Abstract

SUMMARY Most inversions of geomagnetic secular variation for £uid £ow at the core’s surface utilize the so-called frozen-£ux approximation where diiusion of the magnetic ¢eld is neglected. Here we note that the frozen-£ux 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 ¢elds in terms of temporal mean and £uctuating parts (as is standard in mean-¢eld 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 diiusion, secular variation cannot be approximately described by the (diiusive-free) frozen-£ux approximation. In support of this conclusion we present two new kinematic dynamo models. The ¢rst dynamo exhibits no secular variation even though it has a non-zero surface £ow; for this dynamo no purely poloidal surface £ow model can be constructed by using the frozen-£ux approximation, but the actual surface £ow is purely poloidal.The second dynamo exhibits westward drift of the magnetic ¢eld but has surface £ow 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-£ux approximation, and yet both of these models are contradictory to the frozen-£ux approximation. Thus, inversions for core £ow should consider the eiects of diiusion. Core £ow models deduced assuming that the frozen-£ux approximation is valid, and any results dependent on such £ow models, should be treated with a great deal of caution.