Relations between Electrical Conductivity of a Mantle and Fluctuating Magnetic Fields

TLDR: In this paper, it is shown that the electrical conductivity within an object can often be related to simple properties of fluctuating magnetic fields observed and generated outside of it, and the specific question to which we are addressing ourselves is the determination of the electric conductivity as a function of depth into a geological body by observing the properties of fluctuations magnetic fields outside that body.

Abstract: Summary It is shown that the ratio of the vertical field to the horizontal field gradient outside a geological body can often be used to measure the electrical conductivity at depth within it. The in-phase part of this ratio, at a given frequency, gives the penetration depth of the magnetic field, the out-of-phase component yields approximately the conductivity at the penetration depth. In this paper it will be shown that the electrical conductivity within an object can often be related to simple properties of fluctuating magnetic fields observed and generated outside of it. The specific question to which we address ourselves is the determination of the electrical conductivity as a function of depth into a geological body by observing the properties of fluctuating magnetic fields outside that body. In the case of the Earth, we are concerned with the fluctuating magnetic fields which are excited by electric currents flowing in the ionosphere, in the case of the Moon by magnetic fields which are convected by the solar wind. We shall mathematically analyse the interaction of an oscillating magnetic field which is slightly non-uniform in the horizontal plane with a laterally uniform body whose electrical conductivity increases with depth. It will be shown that, above and outside the body, the magnetic field has two simply interpretable properties whose importance for interpreting magnetic deep sounding data has not been realized heretofore. The ratio of the ' in phase ' part of the vertical field to the ' gradient ' of the oscillating horizontal field component is equal to an appropriately defined field penetration depth. The ratio of the out-of-phase part of the vertical field and the same horizontal field gradient is approximately equal to the field attenuation length associated with the electrical conductivity of the body evaluated at the field penetration depth. Chapman (1919) pointed out that the fluctuating magnetic fields observed at the surface of the Earth could be resolved into normal modes and that the part of each normal mode which is due to eddy currents flowing inside the body could be separated from that due to currents flowing externally. By noting that the internal currents were coupled to and excited by the external currents, he obtained an approximate value for the Earth's conductivity. Various combinations of the ' complex ' ratio of the internal current and the external current have subsequently been denoted as magnetic response functions. By studying these response functions as a function frequency means have been devised to generate conductivity profiles of the Earth and Moon from experimental data (Rikitake 1966). Most analyses have seriously utilized only the real part of this response function (e.g. Banks 1969), i.e. they have considered only the magnitude of the horizontal