Classification of Harmonic Functions in the Exterior of a Unit Sphere and Its Application to the Description of the External Magnetic Field of the Earth

Abstract

The magnetic fields of a subterranean source observed on the Earth’s surface or from the orbit of a satellite are studied. We analyze the space of all harmonic functions g in the external spherical domain, which are defined as series in a standard basis. The functions decrease at infinity; their gradients are normal to the field of directions specified at a boundary sphere, and the field of directions is determined by the gradient of a basis function f. The results can be used in the solution of the general problem in which the functions f and g are assumed to be independent of two spherical coordinates on the boundary sphere. Specific calculations were made only under the assumption of the independence of both g and f of the longitude coordinate. The known results are refined for the case in which f is the dipole or quadrupole harmonic, and a new result is obtained for the case in which f is the octupole harmonic.