Surface Integral Formulae for Geomagnetic Studies

Abstract

Many geomagnetic studies are concerned with time-varying magnetic fields at the Earth’s surface, and their relation to electric current systems in the upper atmosphere and in the Earth. There are, for example, many recent (as well as many earlier) studies of the Solar and Lunar diurnal variation fields, and of the ionospheric and earth current systems associated with them. These studies have led to the development of various methods and formulae for separating the surface field into parts of external and internal origin, and for determining the ionospheric currents that correspond to the external part. Further, the relationship found between the fields of internal and external origin is of the form that would be expected if the earth currents were due simply to electromagnetic induction by the moving and varying ionospheric current systems. A study of this relationship can lead to information about the conductivity of the Earth at various depths. This requires the solution of various mathematical problems on the electromagnetic induction of currents in concentric spheres and spherical shells of non-uniform conductivity, and involves the evaluation of self and mutual induction effects in these conductors. One of the important problems is to estimate the influence of currents induced in the relatively highly conducting oceans, and the screening effects of these currents on the conducting layers of the earth below. In the earlier studies the current systems and magnetic fields were expressed in terms of series of spherical harmonics and many valuable results were obtained in this way. When, however, it is desired to study detailed features of certain fields, such as the great enhancement of the Sq field near the dip equator, or, again, when one attempts to solve an induction problem which has, as one of the conductors, a thin shell with abrupt changes in conductivity, as at the surface of the Earth, then this method leads to difficulties owing to the slow convergence of the spherical harmonic expansions required. In the solution of such induction problems, the numerical work involved in solving the infinite sets of simultaneous equations, that are found for the coefficients of the required spherical harmonic expansions, is often prohibitive, and the solution obtained by truncating the series at a point that would make the numerical work possible very inaccurate. (See for example, the discussion by Ashour (1965a) of a problem considered by Rikitake & Yokoyama (1955)). Hence, alternative methods of treating these problems have been sought.