Scripted Finite Element Methods Applied to Global Geomagnetic Induction

Magnetic field observations from CHAMP, Ø rsted and SAC-C and improved techniques for comprehensive geomagnetic field modeling have generated renewed interest in using satellite and observatory data to study global scale electromagnetic induction in Earth's crust and mantle. The primary external source field derives from variations in the magnetospheric ring current, and recent studies show that over-simplified assumptions about its spatial structure lead to biased estimates of the frequency-dependent electromagnetic response functions generally used in inversions for mantle conductivity. The bias takes the form of local time dependence in the C- response estimates and highlights the need for flexible forward modeling tools for the global induction problem to accommodate 3D time-varying structure in both primary and induced fields. We are developing such tools using FlexPDE, a commercially available script-based finite element method (FEM) package for partial differential equations. Our strategy is to model the vector potential \mathbf{A}, where \mathbf{B} = \nabla × \mathbf{A}, thereby ensuring that \mathbf{B} is divergenceless. We compare our results with existing 2D and 3D analytical solutions to validate the solution techniques. We are able to mitigate time and memory constraints by adaptive refinement of a mesh that is initialized for high resolution in areas of interest. Solutions can be developed in either frequency or time domain, but we use time domain modeling because interpretation of satellite data is complicated by the motion of the satellites through magnetic fields that vary both temporally and spatially. We conduct 3D simulations of electromagnetic induction in a spherically symmetric planet with an asymmetric primary external field. This allows us to document expected differences in response estimates for ground versus satellite instruments. Suitable extensions of these tools are anticipated for use in induction studies of any planet or spherical body, allowing arbitrary spatial and temporal behavior in both primary and induced fields.