Numerical analysis of the coupled hydromagnetic wave equations in the magnetosphere

Inside the magnetosphere two basic hydromagnetic modes, the poloidal and toroidal, are always coupled. The problem of coupled waves is of fundamental importance for nontrivial geometries and general field configurations. As a first step the box model is investigated here, using the ideal MHD equations for the cold plasma. The spectrum and eigenfunctions are calculated for several cases by applying the finite element method. Both the improper continuum eigenmodes as well as the discrete solutions are computed. Existing analytical results are confirmed, and it is found that the discrete modes situated inside the Alfvén continuum are the best candidates for explaining observed hydromagnetic resonances. Other properties of these global modes are derived and compared with experimental data, and satisfactory agreement is obtained despite the simple geometric model employed. Generally, damping of the discrete modes is found to be considerably lower than that of the continuum. The results indicate that the weakly damped global modes lose their energy directly through ionospheric Joule dissipation, and not via mode conversion to field line resonances.