Geomagnetic induction in a heterogenous sphere: Azimuthally symmetric test computations and the response of an undulating 660‐km discontinuity

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Abstract
A finite element numerical method is presented for computing electromagnetic induction in a heterogeneous conducting sphere by external source current excitation. The numerical model has applicability in the problem of determining the three‐dimensional electrical conductivity structure of Earth's mantle. The formulation is in terms of vector and scalar potentials. Boundary value problems are derived for fully three‐dimensional (3‐D) and azimuthally symmetric geometries. To validate the code, we check against a quasi‐analytic solution for an azimuthally symmetric configuration of eccentrically nested spheres and against an integral equation solution for an azimuthally symmetric, buried thin spherical shell model. To illustrate the capability of the code for modeling the electromagnetic response of realistic mantle structural variations, we compute solutions for an azimuthally symmetric electrical model whose lateral heterogeneity corresponds to zonal averages of the topographic relief on the 660‐km seismic discontinuity. The latter is derived from differential shear wave travel times. The anomalous responses are small but either spatially correlated or anticorrelated with the relief, depending on the period. The behavior can be reproduced by simple one‐dimensional (1‐D) modeling. The volumetric heterogeneity associated with recent seismic tomographic models should yield a greater surface anomaly, since electromagnetic data are inherently more sensitive to bulk electrical properties than sharp internal interfaces. The finite element method presented here is general and can account for galvanic, oceanic, and non‐P10 source effects. The present implementation does not include these complications, but the code is designed so that they can be added easily. The major obstacle to their inclusion at present is the scarcity of other 3‐D solutions to validate our calculated results.