A formalism for the inversion of geomagnetic data for core motions with diffusion

Fluid motion in the Earth's core may be mapped by inverting secular variation assuming the magnetic field lines are frozen to the fluid (zero diffusion). The inverse problem may be formulated by integrating the induction equation in time to give a relationship between geomagnetic field differences and fluid motion on the core surface. Numerical estimates of the time scales involved suggest diffusion will be important, if small, for secular variation, and some workers found diffusion to be essential in explaining some aspects of recent secular variation. Diffusion is here retained in the induction equation and integration with respect to time is replaced with inversion of the diffusion operator, yielding a relationship between field differences and a space-time average of the core motions through a thin surface layer at the top of the core whose depth is determined by the time interval between measurements. An asymptotic analysis in the limit (eta/p)(1/2) --> 0, where p is the Laplace transform variable, recovers the frozen flux approximation to first order and gives higher correction terms for the diffusion. The largest diffusive correction for the core is likely to come from expulsion of toroidal flux by upwelling, which changes the flux linkage through the core-mantle boundary. A radial gradient of toroidal field of about 20nT m(-1) throughout the top 60 km of the core is needed to account for the flux changes inferred on the core-mantle boundary beneath the south Atlantic between Epochs 1905.5 and 1969.5, giving a toroidal field of 1.2 mT 60 km below the core surface. The horizontal components of magnetic field (or secular variation) cannot be used as data: the formalism ensures the correct boundary conditions are satisfied at the surface and once the radial component of the induction equation is satisfied the other components are satisfied automatically.