Some aspects of finite element calculations applied to ULF magnetospheric cavity waves

Abstract

The finite element method is applied to describe hydromagnetic pulsations of the cavity formed by the ionosphere and the magnetopause. The variation of the density and magnetic field in the cavity leads to the existence of resonant field lines (shells), which form the Alfvén continuum. More or less perfectly reflecting boundary conditions imply discrete solutions of the linearized MHD wave equations in the cavity. The presence of part of the discrete spectrum within the the Alfvén continuum frequency range poses both analytical as well as numerical challenges. Several numerical aspects of current problems pertaining to cavity modes are here illuminated from the viewpoint of the finite element approach. The presence of a resonance in the eigensolutions and a coefficient function with large gradient require high spatial resolution. Still, the finite element method turns out to be very successful in calculating the spectrum and eigensolutions. Generally, lowest-order base functions in conjunction with an adaptive mesh size give good results for a reasonable number of nodal points. However, it is argued that when kinetic effects are considered, a proper discretization can only be obtained with higher-order base functions.