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Two-fluid or Hall MHD description of weakly-ionized stratified atmosphere polarization effect on MHD waves

We use two-fluid or Hall effect MHD description of weakly ionized stratified atmosphere to describe several polarization features of the MHD wave penetration. We employ a pair of functions for the electric and magnetic field components ratio which can be treated analytically. As examples, we derive an approximation to the case of the MHD waves in the Earth's Hall ionosphere and demonstrate its different responses (ellipticity and rotation) for Alfvén and magnetosonic modes in the long-wavelength limit, k⊥lz < 1, where k⊥ is the horizontal wavelength and lz the Hall ionosphere thickness. In the short wavelength limit, k⊥lz > 1, one gets a relevant estimate of the Alfvén and magnetosonic wave electric field distribution in height.