Hall magnetohydrodynamics of partially ionized plasmas

The Hall effect arises in a plasma when electrons are able to drift with the magnetic field but ions cannot. In a fully-ionized plasma this occurs for frequencies between the ion and electron cyclotron frequencies because of the larger ion inertia. Typically this frequency range lies well above the frequencies of interest (such as the dynamical frequency of the system under consideration) and can be ignored. In a weakly-ionized medium, however, the Hall effect arises through a different mechanism -- neutral collisions preferentially decouple ions from the magnetic field. This typically occurs at much lower frequencies and the Hall effect may play an important role in the dynamics of weakly-ionised systems such as the Earth's ionosphere and protoplanetary discs. To clarify the relationship between these mechanisms we develop an approximate single-fluid description of a partially ionized plasma that becomes exact in the fully-ionized and weakly-ionized limits. Our treatment includes the effects of ohmic, ambipolar, and Hall diffusion. We show that the Hall effect is relevant to the dynamics of a partially ionized medium when the dynamical frequency exceeds the ratio of ion to bulk mass density times the ion-cyclotron frequency, i.e. the Hall frequency. The corresponding length scale is inversely proportional to the ion to bulk mass density ratio as well as to the ion-Hall beta parameter.