Dispersion relation for ballooning modes and condition of their stability in the near-earth plasma

The ballooning disturbances in a finite-pressure plasma in a curvilinear magnetic field are described by a system of coupled equations for Alfv,n and slow magnetosonic modes. The local dispersion relation obtained in a WKB approximation is the simplest and most evident method that can be used to characterize the properties of these disturbances. This dispersion relation is widely used to predict the possible instabilities and spectral properties of LF oscillations in the nightside magnetosphere. The formal derivation of the dispersion relation from the initial system of coupled MHD modes and the transition to different limiting cases have been traced. The behavior of dispersion curves in different oscillation branches and the possible development of instabilities and formation of regions where waves cannot propagate have been studied in detail. This made it possible to specify the results of previous works and even indicate the incorrectness in some works. In particular, it has been indicated that a fast Alfv,n branch of oscillations is always stable and an aperiodic instability can originate on a slow magnetosonic oscillation branch.