A computationally compact representation of Magnetic-Apex and Quasi-Dipole coordinates with smooth base vectors

Many structural and dynamical features of the ionized and neutral upper atmosphere are strongly organized by the geomagnetic field, and several magnetic coordinate systems have been developed to exploit this organization. Quasi-Dipole coordinates are appropriate for calculations involving horizontally stratified phenomena like height-integrated currents, electron densities, and thermospheric winds; Modified Apex coordinates are appropriate for calculations involving electric fields and magnetic field-aligned currents. The calculation of these coordinates requires computationally expensive tracing of magnetic field lines to their apexes. Interpolation on a precomputed grid provides faster coordinate conversions, but requires the overhead of a sufficiently fine grid, as well as finite differencing to obtain coordinate base vectors. In this paper, we develop a compact and robust representation of the transformation from geodetic to Quasi-Dipole (QD), Apex, and Modified Apex coordinates, by fitting the QD coordinates to spherical harmonics in geodetic longitude and latitude. With this representation, base vectors may be calculated directly from the expansion coefficients. For an expansion truncated at order 6, the fitted coordinates deviate from the actual coordinates by a maximum of 0.4 degrees, and typically by 0.1 degrees. The largest errors occur in the equatorial Atlantic region. Compared to interpolation on a pre-computed grid, the spherical harmonic representation is much more compact and produces smooth base vectors. An algorithm for efficiently and concurrently computing scalar and vector spherical harmonic functions is provided in the appendix. Computer code for producing the expansion coefficients and evaluating the fitted coordinates and base vectors is included in the auxiliary material.