Strategies for improved global representation of magnetospheric electric potential structure on a polar-capped ionosphere

In some simple models of magnetospheric electrodynamics [e.g., Volland, Ann. Géophys., 31, 154-173, 1975], the normal component of the convection electric field is discontinuous across the boundary between closed and open magnetic field lines, and this discontinuity facilitates the formation of auroral arcs there. The requisite discontinuity in E is achieved by making the scalar potential proportional to a positive power (typically 1 or 2) of L on closed field lines and to a negative power (typically −1/2) of L on open (i.e., polar-cap) field lines. This suggests it may be advantageous to construct more realistic (and thus more complicated) empirical magnetospheric and ionospheric electric-field models from superpositions of mutually orthogonal (or not) vector basis functions having this same analytical property (i.e., discontinuity at L = L*, the boundary surface between closed and open magnetic field lines). The present work offers a few examples of ways to make such constructions. A major challenge in this project has been to devise a coordinate system that simplifies the required analytical expansions of electric scalar potentials and accommodates the anti-sunward offset of each polar cap's centroid relative to the corresponding magnetic pole. For circular northern and southern polar caps containing equal amounts of magnetic flux, one can imagine a geometrical construction of coordinate contours such that arcs of great circles connect points of equal quasi-longitude (analogous to MLT) on the northern and southern polar-cap boundaries. For more general polar-cap shapes and (in any case) to assure mutual orthogonality of respective coordinate surfaces on the ionosphere, a formulation based on harmonic coordinate functions (expanded in solutions of the two-dimensional Laplace equation) may be preferable.