Atmospheric oscillations—V. The propagator matrix and the transmission of an electrostatic potential along the geomagnetic field lines

Abstract

An account is given of a new mathematical formulation of the process of transmission of an electrostatic field along magnetic field lines, assuming the latter to be equipotentials of the electrostatic field. The theory is presented in terms of a dipole magnetic field, but is capable of extension to higher multipoles.By means of vector and matrix representations in Hilbert space, corresponding to the description of the field variables in terms of their spherical harmonic components, a propagator matrix is constructed which, given the harmonic expansion of the potential over a sphere of radius r o, enables the corresponding expansion at a general radius to be determined. Proposed areas of application of the technique include three-dimensional ionospheric dynamo theory, and tidal motions in the thermosphere which are excited by electric fields transmitted upwards from the dynamo region; in both cases an analytical rather than a geometrical or numerical specification of the potential is essential.