Reflection of shear Alfvén waves at the ionosphere and the divergent Hall current

A general expression for the reflection and mode conversion of MHD waves at the ionosphere is derived. On the basis of the expression, the effect of ionospheric divergent Hall current (Hall part of the induction current associated with the inductive electric field of fast magnetosonic wave) on localized toroidal oscillation is examined. When the horizontal scale of localized oscillation is of the order of several times of the height of ionosphere, the inductive electric field can play a significant role in the reflection of shear Alfvén waves with longer periods (e.f., - 100 s) in the high-latitude region, especially, in the auroral zone. Then, the contribution of the divergent Hall current to the field-aligned one can be no longer neglected and so the eigenfrequency of localized toroidal oscillation is effectively controlled by the height-integrated Hall conductivity in the ionosphere.