Ionospheric Electrodynamics Using Magnetic Apex Coordinates.

The equations of ionospheric electrodynamics are developed for a geomagnetic field of general configuration, with specific application to coordinate systems based on Magnetic Apex Coordinates. Two related coordinate systems are proposed: Modified Apex Coordinates, appropriate for calculations involving electric fields and magnetic-field-aligned currents; and Quasi-Dipole Coordinates, appropriate for calculations involving height-integrated ionospheric currents. Distortions of the geomagnetic field from a dipole cause modifications to the equations of electrodynamics, with distortion factors exceeding 50% at some geographical locations. Under the assumption of equipotential geomagnetic-field lines, it is shown how the field-line-integrated electrodynamic equations can be expressed in two dimensions in magnetic latitude and longitude, and how the height-integrated and field-aligned current densities can be calculated. Expressions are derived for the simplified calculation of magnetic perturbations above and below the ionosphere associated with the three-dimensional current system. It is shown how the base vectors for the Modified Apex coordinate system can be applied to map electric fields, plasma-drift velocities, magnetic perturbations, and Poynting fluxes along the geomagnetic field to other altitudes, automatically taking into account changes in magnitude and direction of these vector quantities along the field line. Similarly, it is shown how Quasi-Dipole coordinates are useful for expressing horizontal ionospheric currents, equivalent currents, and ground-level magnetic perturbations. A computer code is made available for efficient calculation of the various coordinates, base vectors, and related quantities described in this article.