Parametric Instability of Arc-Polarized Alfvén Waves and Wave Packets in 1D Periodicand Open Systems*

Although Alfvén waves are an exact solution to the nonlinear magnetohydrodynamic (MHD) equations, they are subject to the parametric decay instability (PDI) at large amplitudes. Though PDI has been widely studied, few investigations have examined wave packets of finite size and the effect of different boundary conditions on the growth rate. From a linear analysis of circular and arc-polarized wave packets in periodic and open boundary systems, we find that both types of wave are 4-5 times more stable in open boundary conditions compared to periodic. Additionally, once the wave packet width $\ell$ becomes smaller than the system size L, the growth rate decreases with a power law $\ell$/L. We demonstrate that the growth rate of daughter waves depends on the conditions upstream and downstream the pump wave and on the fraction of volume it fills. Implications of our results for the interpretation of simulations, experiment, and solar wind observations are discussed.