Reversals of the earth's magnetic field: An attempt at a relaxation model

A mathematical model that might be relevant to the dynamic properties of the earth's dynamo is constructed. The model involves one mechanical variable (convection velocity at constant geometry) and two electromagnetic degrees of freedom. It represents a generalization of the Rikitake coupled dynamos. We first show that the problem involves two very different time scales, a fast MHD scale and a slow inductive scale. Using the ratio of these scales as an expansion parameter, we solve analytically the non-linear differential system which describes the dynamics of our model. We first isolate a slow motion, and we study its stability against fast MHD ocillations. For large amplitudes, we show how to construct an expansion in the drift of the fast orbit. We survey in some detail the various possible regimes. Among them, we find a typical “ratchet like” relaxation behaviour, which may perhaps explain the very sudden reversals of the earth's field. Orders of magnitude and possible generalizations are briefly sketched.