Exact solutions of dispersion equation for MHD waves with short-wavelength modification

Dispersive magnetohydrodynamic (MHD) waves with short-wavelength modification have an important role in transforming energy from waves into particles. In this paper, based on the two-fluid mode, a dispersion equation, including the short-wavelength effect, and its exact solution are presented. The outcome is responsible for the short-wavelength modification versions of the three ideal MHD modes (i.e. the fast, slow and Alfvén). The results show that the fast and Alfvén modes are modified considerably by the shortwavelength effect mainly in the quasi-parallel and quasi-perpendicular propagation directions, respectively, while the slow mode can be affected by the short-wavelength effect in all propagation directions. On the other hand, the dispersive modification occurs primarily in the finite-β regime of 0:001 < β < 1 for the fast mode and in the high-β regime of 0:1 < β < 10 for the slow mode. For the Alfvén mode, the dispersive modification occurs from the low-β regime of β < 0:001 through the high-β regime of β > 1.