When the perpendicular wavelength of the shear Alfvén wave becomes comparable to or smaller than the electron inertial length, the electric field associated with the wave has a component parallel to the magnetic field. An inertial Alfvén wave can therefore accelerate electrons parallel to the magnetic field. This acceleration mechanism demands the fine-scale structure so characteristic of the discrete aurora. Moreover, thicknesses of the discrete auroral forms turn out to be comparable to electron inertial lengths at altitudes where auroral electrons are accelerated. Numerical simulations of acceleration by inertial Alfvén waves show both upward and downward acceleration and that a large fraction of accelerated electrons are accelerated into trapped orbits. Some of the aurora we see can be due to upward accelerated electrons originating from the southern hemisphere. The acceleration process can also serve as a mechanism for replenishing the electron radiation belts following a substorm. The presentation will conclude with a digression on a powerful, substorm-associated source of shear Alfvén waves.