Continuous solutions of the hydrodynamic approach to cosmic-ray propagation

A hydrodynamic approach is employed to study a cosmic-ray-plasma system, which comprises thermal plasma, cosmic rays and two oppositely propagating Alfvén waves. The hydrodynamic approach is a good approximation in dealing with the structure or dynamics of the system. In this paper, we concentrate on the steady-state structures of the system, in particular, structures with continuous (or smooth) profiles. Three mechanisms are responsible for the energy exchange between different components. They are work done by plasma flow via pressure gradients, cosmic-ray streaming instability and stochastic acceleration. The inter-play between these mechanisms generates several morphologically different structures. They may be divided into two categories: one looks like the test-particle picture and the other looks like a modified shock. Very often the profiles are non-monotonic, which is in sharp contrast to systems with only thermal plasma and cosmic rays, whose flow velocity (and cosmic-ray pressure) profiles are alway menotonically decreasing (and increasing).