A cold, nonlinear, quasi-neutral, two-fluid model of field-aligned current (FAC) sheets with a uniform tangential electric field describes stationary, spatially periodic electromagnetic perturbations (stationary inertial Alfv¿n (SIA) waves) which carry enhanced current, which can accelerate electrons parallel to a strong background magnetic field B0, and which cause large variations in plasma density. The waves propagate along B0 at speeds less than the Alfv¿n speed VA. Waves can be divided into two branches according to the sign of their field-aligned phase velocity with respect to the direction of field-aligned electron drift. Waves propagating parallel to electron drift can accelerate low-energy electrons up to VA within steady state parallel sheets with widths of the order of the electron inertial length &lgr;e=c/&ohgr;pe, where &ohgr;pe is the electron plasma frequency. Waves propagating antiparallel to electron drift can accelerate electrons to velocities much greater than VA in steady state sheets which are orders of magnitude wider than &lgr;e and which are depleted in density. SIA waves also form when only an energetic subpopulation of electrons carries FAC through an ambient population initially at rest and can accelerate the ambient electrons to approach the energy of the subpopulation. These results extend the phenomenon of steady state electron acceleration by inertial Alfv¿n waves to include much higher energies and wider spatial scales than previously recognized. Although the relatively simple model described here omits certain effects which are known to be important in the Earth's auroral regions, several properties of SIA wave solutions suggest they may play a role in auroral arc formation, including steady state field-aligned electron acceleration over a broad range of spatial scales and energies, density variation, and the formation of multiple parallel structures. ¿ American Geophysical Union 1996