MHD eigenmodes in the inner magnetosphere

Many ultra-low-frequency phenomena observed in the inner magnetosphere are attributed to magnetohydrodynamic (MHD) eigenmodes. We review theoretical investigations of the compressional and shear (Alfven) eigenmodes and the coupling between them. The properties of the eigenmodes are strongly controlled by the spatial variation of the Alfven speed, especially at the plasmapause, and by the ionospheric boundary conditions. We first illustrate the plasmapause effects by using numerical simulations and explain the spatial mode structure by using the concept of virtual resonance. We then discuss the ionospheric damping of the coupled shear and compressional modes. We find that the ionosphere with finite conductivity damps the shear wave component much faster than it does the compressional component. We also discuss some important but not fully understood eigenmode properties that appear when local time asymmetry is introduced into the background medium.