Induction Effects in Terrestrial Magnetism Part I. Theory

Abstract

The paper deals with the electromagnetic effect of motions in the earth's core, considered as a fluid metallic sphere. On the basis of simple estimates the electric conductivity of the core is assumed of the same order of magnitude as that of common metals. The mathematical treatment follows Hansen and Stratton: three independent vector solutions of the vectorial wave equation are introduced; two of these have vanishing divergence, and they are designated as toroidal and poloidal vector fields. The vector potential and electric field are toroidal, whereas the magnetic field is poloidal. These vectors, expressed in terms of spherical harmonics and Bessel functions, possess some notable properties of orthogonality which are briefly discussed. The theory of the free, exponentially decaying current modes is then given, leading to decay periods of the order of some tens of thousands of years. Next, the field equations in the presence of mechanical motions of the conducting fluid are set up. The field is developed in a series of the fundamental, orthogonal vectors, and the field equations are transformed into a system of ordinary differential equations for the coefficients of this development. The behavior of the solutions depends on the symmetry of the "coupling matrix" that arises from the term of the field equations expressing the induction effects. In order to evaluate this matrix the velocity field is developed into a series of the fundamental vectors similar to the series for the electromagnetic field. It is then shown that when the velocity is a toroidal vector field the coupling matrix is antisymmetrical. When the velocity field is poloidal, the coupling matrix is neither purely symmetrical nor purely antisymmetrical. For stationary fluid motion the linear differential equations can be integrated in closed form by a transformation to new normal modes, whenever the matrix of the system is either symmetrical or antisymmetrical. In the latter case the eigenvalues are purely imaginary and the coefficients of the new normal modes are harmonic functions of time, representing oscillatory changes in amplitude of the field components. For a symmetrical matrix the eigenvalues are real and the time factors of the new normal modes are real exponentials representing amplification or de-amplification as the case may be, depending on the sign of the velocity. For a matrix without specific symmetry, normal modes do not, as a rule, exist but similar, somewhat less stringent results can be derived in special cases. In the case of toroidal flow, in particular, the oscillatory changes of the field components are superposed upon the slow exponential decay characteristic of the free modes.