Alfvén wave transport effects in the time evolution of parallel cosmic-ray-modified shocks

This paper presents a numerical study of the time evolution of plane, cosmic-ray modified shocks with magnetic field parallel to the shock normal, based on the diffusive shock acceleration formalism and including the effects from the finite propagation speed and energy of Alfvén waves responsible for controlling the transport of the cosmic rays. The simulations discussed are based on a three-fluid model for the dynamics, but a more complete formalism is laid out for future work. The results of the simulations confirm earlier, steady state analyses that found these Alfvén transport effects to be potentially important when the upstream Alfvén speed and the gas sound speed are comparable, i.e., when the plasma and magnetic pressures are similar. It is also clear however, that the impact of Alfvén transport effects, which tend to slow shock evolution and reduce the time asymptotic cosmic-ray pressure in the shock, is strongly dependent upon uncertain details in the transport models. Both cosmic-ray advection tied to streaming Alfvén waves and dissipation of wave energy are important to include in the models. Further, Alfvén transport properties on both sides of the shock are also influential.