Quantitative magnetospheric magnetic field modelling with toroidal and poloidal vector fields

Toroidal and poloidal vector fields allow divergence free magnetic field representations in regions where currents flow. We derive general magnetospheric magnetic fields using combinations of spherical harmonic expansions of the toroidal and poloidal fields. Adding restrictive conditions like the field line topology symmetry or the magnetic field measurements, more specific magnetospheric magnetic field models can be derived. Two examples of this technique are given : an axisymmetric model with a ring current in the equatorial region and a time-dependent model of the Earth's magnetosphere. Our results are compared with the Olson-Pfitzer model.