Self-similar evolution of cosmic-ray-modified quasi-parallel plane shocks

Using an improved version of the previously introduced Cosmic Ray Acceleration SHock (CRASH) code, we have calculated the time evolution of cosmic-ray (CR) modified quasi-parallel plane shocks for Bohm-like diffusion, including self-consistent models of Alfvén wave drift and dissipation, along with “thermal leakage injection” of CRs. The new simulations follow evolution of the CR distribution to much higher energies than our previous study, providing a better examination of evolutionary and asymptotic behaviors. The postshock CR pressure becomes constant after quick initial adjustment, since the evolution of the CR partial pressure expressed in terms of a momentum similarity variable is self-similar. The shock precursor, which scales as the diffusion length of the highest energy CRs, subsequently broadens approximately linearly with time, independent of diffusion model, so long as CRs continue to be accelerated to ever-higher energies. This means the nonlinear shock structure can be described approximately in terms of the similarity variable, x/(ust)x/(ust), where us is the shock speed once the postshock pressure reaches an approximate time asymptotic state. As before, the shock Mach number is the key parameter determining the evolution and the CR acceleration efficiency, although finite Alfvén wave drift and wave energy dissipation in the shock precursor reduce the effective velocity change experienced by CRs. This reduces acceleration efficiency noticeably, thus, providing a second important parameter at low and moderate Mach numbers. For low Mach numbers (M0≲5M0≲5) the CR acceleration efficiency depends on the thermal leakage injection rate, the Alfvénic Mach number, and any preexisting CR population. However, these dependences become weak for high shock Mach numbers of M0>30M0>30. To evaluate CR acceleration efficiencies in the simulated shocks we present for a wide range of shock parameters a “CR energy ratio”, Φ(M0)Φ(M0), comparing the time asymptotic volume-integrated energy in CRs to the time-integrated kinetic energy flux through the shock. This ratio asymptotes to roughly 0.5 for sufficiently strong shocks. The postshock CR pressure is also approximately 1/2 the momentum flux through the shock for very high Mach numbers.