Rapid diffusion of the poloidal geomagnetic field through the weakly conducting mantle: a perturbation solution

A systematic regular perturbation procedure is developed to account for weak mantle conduction as unsteady electromagnetic fields are extrapolated downward from the Earth's surface to the core-mantle boundary. The mantle is treated as a radially symmetric conductor of highly variable conductivity. The unique poloidal-toroidal decomposition of a magnetic vector potential leads first to three-dimensional and then, after spherical harmonic analysis, to one-dimensional linear diffusion equations for the two defining scalar functions. Emphasis is placed on a regular perturbation solution to the inverse poloidal diffusion problem, for the case where diffusion through the mantle is rapid on the time-scale for changes in forcing at the core-mantle boundary. The perturbation theory is evaluated with reference to a range of proposed conductivity profiles and two geomagnetic field models. It is found that uncertainty in the conductivity is at present less important than errors in the field models, and that the first-order corrections to the main field and secular variation at the core surface are likely to be negligible and small, respectively.