Simulation of amplitude‐modulated circularly polarized Alfvén waves for beta less than one

We study the nonlinear properties of the amplitude‐modulated circulary polarized Alfvén wave for β < 1. The temporal behavior of the wave packet of the electromagnetic hybrid simulation is compared with a numerical solution of the derivative nonlinear Schrödinger (DNLS) equation. It is shown that the left‐hand‐polarized mode evolves into a shocklike structure due to the modulational instability. However, both cyclotron damping and a snowplow effect near the steepened wave packet suppress its further steepening, contrary to the predictions of the DNLS equation. For the right‐hand mode, formation of the shock does not take place, and the initial time development is well described by the DNLS equation. The daughter Alfvén wave and ion acoustic waves are excited due to the decay instability at a later time. Heating or acceleration of the particles takes place for both left‐ and right‐hand waves. Energy transfer from the wave to the particles occurs effectively when substantial modulation in the wave amplitude is present. The significance of this result is briefly discussed.