  Using an improved version of the previously introduced CRASH (Cosmic Ray
Acceleration SHock) 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 Alfven 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/(u_s t), where u_s 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 Alfven wave drift
and wave energy dissipation in the shock precursor reduce the effective
velocity change experienced by CRs, so reduce acceleration efficiency
noticeably, thus, providing a second important parameter at low and moderate
Mach numbers.
