Recent observations of particle distributions that are narrow in pitch angle and broad in energy have suggested that kinetic Alfvén waves are a significant contributor to auroral particle acceleration. Oscillations at frequencies near 1 Hz are a natural consequence of the propagation of Alfvén waves in the strongly varying Alfvén speed profile above the auroral ionosphere, the so-called ionospheric Alfvén resonator. These waves often propagate in the presence of perpendicular density gradients at various spatial scales. Simulations have been performed to study the evolution of these fields including both parallel and perpendicular inhomogeneity. Phase mixing at the boundaries of the density cavity leads to small-scale Alfvén waves, which can develop the parallel electric fields needed to accelerate the Alfvénic aurora. These simulations verify the kinetic Alfvén wave dispersion relation in the electron inertial limit, which predicts that the perpendicular phase and group velocity of these waves are in the opposite direction. In addition, the results show that narrow spatial scales are favored by high ionospheric conductance.