A set of model equations is presented that allows for the self-consistent description of dynamic magnetosphere-ionosphere coupling phenomena with finite ionosphere. The model is very similar to magnetohydrodynamics (MHD), but the plasma-neutral gas interaction in the lower ionosphere is taken into account by a frictional force between ions and neutrals and an ionization-recombination term in the plasma transport equation. Further, the Hall term and the electron pressure term are retained in Ohm's law. It is shown that for the collision-dominated E layer, the familiar Pedersen and Hall conductivities can be derived directly from the basic equations. The model is used to numerically simulate the dynamic formation of a magnetospheric-ionospheric current system as the response to prescribed localized magnetospheric convection. Apart from a pair of Birkeland current sheets that are closed by Pedersen currents, associated Hall currents are generated. In addition, density irregularities in the E layer form as a direct consequence of the current closure. They are, however, not related to electron precipitation. For small length scales (≈10 km), these density perturbations result in considerably enhanced conductivities below the upward Birkeland currents which in turn lead to a spatial concentration of the latter compared to the downward currents. The timescale for the relaxation of the current system and the ionospheric convection toward a stationary state after the onset of magnetospheric convection is longer than the Alfv¿n travel time and depends on the height-integrated Pedersen conductivity. This is in good agreement with earlier theoretical predictions [Southwood and Kivelson, 1991>.¿ 1997 American Geophysical Union