The Alfvén speed is given by

and B is constant in the one dimensional problem.

Ko (1992) gave a simple model of the coupling between plasma, CR and waves. The diffusion coefficient k, the stochastic acceleration rate 1/t and e± (which are related to streaming instability) are given by k =

1 16aV2P+P-

1 xt

where c is the speed of light, and «indicates the strength of coupling.

3.4.3. Steady state profiles of the CR-plasma system

According to Ko (1999), in steady state there are six integration constants: 1) magnetic flux

2) mass flux

3) entropy constant

4) total energy flux

5) total momentum and 6) wave-action

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