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We explore how wave-particle interactions affect diffusive shock acceleration(DSA) at astrophysical shocks by performing time-dependent kinetic simulations,in which phenomenological models for magnetic field amplification (MFA),Alfvenic drift, thermal leakage injection, Bohm-like diffusion, and a freeescape boundary are implemented. If the injection fraction of cosmic-ray (CR)particles is greater than 2x10^{-4}, for the shock parameters relevant foryoung supernova remnants, DSA is efficient enough to develop a significantshock precursor due to CR feedback, and magnetic field can be amplified up to afactor of 20 via CR streaming instability in the upstream region. If scatteringcenters drift with Alfven speed in the amplified magnetic field, the CR energyspectrum can be steepened significantly and the acceleration efficiency isreduced. Nonlinear DSA with self-consistent MFA and Alfvenic drift predictsthat the postshock CR pressure saturates roughly at 10 % of the shock rampressure for strong shocks with a sonic Mach number ranging 20< M_s< 100. Sincethe amplified magnetic field follows the flow modification in the precursor,the low energy end of the particle spectrum is softened much more than the highenergy end. As a result, the concave curvature in the energy spectra does notdisappear entirely even with the help of Alfvenic drift. For shocks with amoderate Alfven Mach number (M_A<10), the accelerated CR spectrum can become assteep as E^{-2.1}-E^{-2.3}, which is more consistent with the observed CRspectrum and gamma-ray photon spectrum of several young supernova remnants.

Diffusive shock acceleration with magnetic field amplification and Alfvenic drift