Dispersion and Spatial Structure of Coupled Alfvn and Slow Magnetosonic Modes in the Dipole Magnetosphere

Numerical analysis of the coupling between the Alfven and slow azimuthally small scale modes in the dipolar model of the magnetosphere is performed. The field line curvature, inhomogeneity both across the magnetic shells and along the field lines, and finite plasma pressure are taken into account. The field line curvature causes coupling Alfven and slow modes, while the contribution from the fast mode is neglected due to the assumed small scale of the oscillations in the azimuthal direction. The plasma pressure decreases with distance from the Earth, which is a characteristic for the ring current region. The wave's transverse dispersion, that is, dependence of the radial wave vector kr on the frequency omega, was studied. It was found that the wave can exist in two frequency ranges. Both ranges are bounded by resonant frequency on one side and the cutoff frequency on the other. The higher frequency range corresponds to the Alfven mode. However, its properties are modified due to the coupling with the slow mode. For example, the divergence of the plasma displacement and the magnetic field compressional component appear. In the slow magnetosonic region, in contrast, the cutoff frequency is always smaller than the resonant one. If the pressure gradient is strong and negative, the slow mode cutoff frequency can disappear, that is, the radial wave vector squared even for the zero frequency. It means that the kr value goes to zero at imaginary frequency. Mode structure along the field line for different plasma pressure values and its pressure gradients was calculated.