Nonlinear evolution of parallel-propagating hydromagnetic waves.

The nonlinear evolution of plane hydromagnetic fluctuations propagating along the unperturbed magnetic field direction is considered. From an expansion of the ideal magnetohydrodynamic equations and the hydromagnetic shock jump conditions, it is shown that a wave in which the magnitude of the magnetic field is nonconstant steepens into a shock and subsequently evolves toward a purely Alfvenic fluctuation of lower mean energy density. Explicit expressions are derived for the asymptotic state and for the characteristic lines which describe the evolution toward that state. A class of fluctuations which includes linearly polarized waves is shown to evolve into rotational discontinuities. The results are applied to observations of hydromagnetic fluctuations in the solar wind.