Piezomagneti field associated with the Mogi model revisited: Analytic solution for finite spherical source.

Abstract

The piezomagnetic field associated with the Mogi model was reexamined for the point source and the finite spherical source problems. In the point source case the question was how to deal with divergent stresses near the pressure source. In SASAI's (1979) method, the magnetized crust was divided into two layers, i.e. the upper half shallower than the source (0<z<D) and the lower half deeper than the source (D<z<H). The solution (type I) was simply the sum of the two contributions as calculated by the Fourier transform method. An alternative calculation (type II) integrates the whole magnetized area except for an infinitesimally small sphere involving the pressure source. This was achieved with the aid of a theorem for spherical harmonics. These two procedures gave different results. In order to identify the cause of the discrepancy, the finite source problem was further investigated. The piezomagnetic field was represented in the form of a one-dimensional integral containing complete elliptic integrals. It can be evaluated numerically with the aid of the double exponential formula (DEF). We called it the type III solution. The present analytic approach verifies the results of numerical 3-D integrations by DAVIS (1976), and SUZUKI and OSHIMAN (1990). The solution as the spherical radius diminishes to zero coincides with the type II solution. The type II solution had a gap across H=D. The potential value at H=D was just the average of the two limiting values for H=D-e and H=D+e. In the type I solution, the magnetization along a thin layer at H=D was ignored, which produced the pressure source term in the type II solution. Accordingly the type II solution was appropriate for the point source problem: the previous result by SASAI (1979) should be rejected.