Magnetic Diffusion in a Spherically-Symmetric Conducting Mantle

Abstract

Summary The diffusion of changes in the Earth’s main magnetic field through a spherically-symmetric, conducting mantle is studied. It is shown that the theoretical development is independent of the assumed radial conductivity distribution. Two separate problems are considered ; the diffusion of a change in the radial component of the poloidal field at the core-mantle boundary through to the surface of the Earth, which is of relevance to the determination of an effective conductivity for the lower mantle from the secular variation observations; and the diffusion into the mantle of the toroidal field induced at the core-mantle boundary by the shearing of the poloidal field in the boundary-layer, which is of relevance in the study of fluctuations in the core-mantle electromagnetic coupling. Numerical results are obtained for an assumed uniform conductivity in the lower 2 000 km of the mantle. The presence of the mantle inhibits the study of the structure of the Earth’s magnetic field at the core-mantle boundary in two opposing ways. The conductivity in the lower mantle is sufficiently high to screen considerably the poloidal part of the field we observe at the surface of the Earth, while the conductivity of the upper mantle is small enough compared to that of the lower mantle effectively to block the toroidal field from observation at the surface. In studies where a knowledge of the field at the core-mantle boundary is important, such as in the study of electromagnetic core-mantle coupling, where a complete knowledge of the field at the core-mantle boundary would determine the coupling completely (Rochester 1962), the question of how the magnetic field diffuses through the conducting mantle is a primary one. Up to the present, studies of magnetic diffusion in spherical geometry (Lahiri & Price 1939, McDonald 1957, Rochester 1960) have proceeded on some assumption concerning the radial distribution of electrical conductivity. Either the distribution was assumed uniform or to depend inversely on a power of the radius. The theory of the diffusion of changes in the field through the mantle, however, can be developed independently of any assumption about the form of the distribution of conductivity, apart from the usual one of spherical symmetry. The related numerical computations involve the integration of ordinary differential equations subject to initial conditions.