Effects of Electron inertia on the Waves in Hall Magnetohydrodynamics

Electromagnetic waves in Hall Magnetohydrodynamics (HMHD) exhibit a variety of complex and interesting features. Earlier works have shown the mode conversion of incompressible MHD shear Alfv\'en wave into a compressible, the fast wave becoming mostly electromagnetic and the slow wave becomes almost a fluid-dynamical wave. These features have been supported partially by numerical simulations of Shaikh \& Zank (ApJ,640,L195,2006). Also, recent numerical simulations on solar wind turbulence show some interesting features for the power spectra, like the predominance of Whistler cascades in the range kde $<$ 1 , where k is the wave number and de is the electron inertial length. In this work, we first consider the linear dispersion relation for a complete two-fluid plasma with a finite electron mass. We analyze the different branches of the dispersion relation for different ranges of kde, i.e., kde $>$ 1 and kde $<$ 1, in addition to the ranges kdi $>$ 1 and kdi $<$ 1. The wave characteristics,revealing many significant features, are discussed.