Hydrodynamic approach to cosmic ray propagation - II. Nonlinear test particle picture in a shocked background

In this article, cosmic ray acceleration is studied in a nonlinear test par ticle picture in the hydrodynamic approach. The model includes cosmic rays and two oppositely propagating Alfven waves. It is a nonlinear model becaus e the interactions between the cosmic rays, waves and background plasma dep end on the waves themselves. The background is a simple fast shock. Besides the parameters characterising the nonlinear test particle picture, two mor e parameters are required for the shock, namely, the compression ratio and the location of the shock. In a uniform super-Alfvenic background cosmic ra ys can be accelerated. When the downstream of the background undergoes a fa st shock transition, the cosmic ray may be accelerated further. We find tha t the cosmic ray energy flux far downstream remains unchanged when a shock is introduced. We define the efficiency of shock acceleration in terms of t he difference in far downstream cosmic ray pressure between a uniform backg round and a shocked background with the same far upstream pressure. Dependi ng on the parameters, the efficiency can take positive and negative values. It is possible that the cosmic ray pressure far downstream of the shock ma y be less than the pressure without the shock. We also comment on the role of the shock in accelerating particles.