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 particle picture in the hydrodynamic approach. The model includes cosmic rays and two oppositely propagating Alfvén waves. It is a nonlinear model because the interactions between the cosmic rays, waves and background plasma depend on the waves themselves. The background is a simple fast shock. Besides the parameters characterising the nonlinear test particle picture, two more parameters are required for the shock, namely, the compression ratio and the location of the shock. In a uniform super-Alfvénic background cosmic rays can be accelerated. When the downstream of the background undergoes a fast shock transition, the cosmic ray may be accelerated further. We find that the cosmic ray energy flux far downstream remains unchanged when a shock is introduced. We define the efficiency of shock acceleration in terms of the difference in far downstream cosmic ray pressure between a uniform background and a shocked background with the same far upstream pressure. Depending on the parameters, the efficiency can take positive and negative values. It is possible that the cosmic ray pressure far downstream of the shock may be less than the pressure without the shock. We also comment on the role of the shock in accelerating particles.