Stability of a system with cosmic rays and waves

Aims. We investigate the the stability of a cosmic-ray plasma system using the hydrodynamic approach. Methods. The system we are considering comprises cosmic rays and two oppositely propagating Alfven waves. The thermal plasma flow is treated as a background, such that the feedback of the cosmic rays and waves is neglected. Steady-state solutions exist for the system, and when two waves are present, the steady state is non-uniform. Linear stability analysis on the steady-state solutions shows that for short wavelengths, large cosmic ray energy density, and small wave energy density the system is stable. Results. We present in detail the case of uniform background flow. The system is most unstable in the upstream region and becomes stable in the downstream region in the linear regime for short-enough wavelengths. We also speculate on the implication of the instability.