Steepening of inertial Alfvén waves

Inertial Alfvén waves are thought to accelerate electrons to auroral energies via their parallel electric field in the Earth's magnetosphere. During active geomagnetic times, it is estimated that a significant percentage of electron precipitation energy into the Earth's ionosphere can be attributed to these waves. However, self-consistent wave/particle interactions of inertial Alfvén waves with the accelerated electron population are not well understood. We show that recent self-consistent models have a strong nonlinearity in them. A reduced set of equations which describe this nonlinear steepening is derived and shown to agree with drift-kinetic simulations and other published studies. From this reduced set of equations, many properties of the nonlinearity are derived and shown to agree with simulations. This includes the time and length scales and connecting the speed of the wave to the perturbation maximum value.