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

Abstract

[1] 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 heightintegrated 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°, and typically by 0.1°. 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.

Seasonal variations and north–south asymmetries in polar wind outflow due to solar illumination

“…We do this by centring the circular polar caps around the invariant magnetic poles (e.g. Emmert et al, 2010;Förster and Cnossen, 2013) instead of the geomagnetic poles. These are located at 82…”

Comparison between IRI and preliminary Swarm Langmuir probe measurements during the St. Patrick storm period

“…In fact, most of processes occurring in the ionosphere have a marked magnetic latitudinal dependence (Davies 1990;Kelley 2009). So, we converted geographical coordinates into quasi-dipole coordinates (Emmert et al 2010) and considered the following magnetic latitude bands: between −90°S and −60°S (SP, south pole), between −60°S and −30°S (SM, south mid ), between −30°S and 30°N (EQ, equator), between 30°N and 60°N (NM, north mid), between 60°N and 90°N (NP, north pole). The limits of these bands have been chosen also on the base of the magnetic latitude distribution of Swarm N e measurements.…”

“…Partition into MLT is made to consider that, for each Swarm orbit, half measurements are taken in the morning sector (descending phase of satellite orbit) and half in the evening sector (ascending phase of satellite orbit). So, dividing data in this way we distinguish among the different dynamics characterizing morning and evening ionospheric sectors, especially at low and equatorial latitudes that are characterized by the fountain effect (Davies 1990;Kelley 2009). Since Swarm satellites move along near-polar orbits, MLTs are clustered around morning and evening sectors and partially spread over the entire 24 h MLT range at the poles.…”

Climatology of GPS signal loss observed by Swarm satellites

“…For example, MLT = 12 (noon) is defining the magnetic meridian facing the Sun at the magnetic equator. Here we used the Quasi-Dipole magnetic field model (Richmond, 1995;Emmert et al, 2010) to calculate MLAT and MLT. Similarly, the events have been first sorted into 2 • × 0.5 h bins (in MLAT and MLT, respectively), and then divided by the orbit number of Swarm C in each bin.…”