In CR-plasma systems without waves, or in systems where the thermal plasma is dominant (the so called nonlinear test particle picture) physical solutions can be classified completely (Drury and Volk, 1982; Axford, Leer, and McKenzie, 1982; Jiang, Chan, and Ko, 1996; Ko, Chan, and Webb, 1997; Ko, 1998). Unfortunately the mathematics of the full system is too complicated to sort out every physical solution.

Ko (1999) works out several typical solutions numerically. A solution of structure is deemed physical if its pressures are non-negative, and it approaches uniform states both far upstream (x ^) and far downstream (x ). Moreover, owing to stochastic acceleration at least one of the three pressures

Pc, P-, P- must be zero as x ^ (see Eqs. 3.4.5 and 3.4.6).

Recall that in CR-plasma systems without waves there are two generic steady state structures. The flow profile is monotonically decreasing and it is either continuous or have one discontinuity (i.e., a sub-shock): Drury and Volk (1981), Axford, Leer and McKenzie (1982), Ko, Chan and Webb (1997). For systems with a unidirectional wave it is possible to consider only continuous flow, because a sub-shock generates both waves downstream. In this case the flow profile is also monotonically decreasing (McKenzie and Volk, 1982). It is necessary to point out that uniform states are physically allowable solutions in the simplified systems mentioned above but not in the full system. Ko (1999) concentrates only on the continuous flow profile of the full system (i.e., with both forward and backward waves). Furthermore, in paper of Ko (1999) has super-Alfvenic flows were considered only (i.e., u/Va > 1 everywhere). In all these calculations the magnetic field, velocity, density, pressures, and length are nominated as following: Bo,uo,po,Po,Lo, where bO//o = pouO = Po and Lo = c2(aPouo)-1. To integrate the set of equations, besides assigning values to Yg and Yc, eight constants are required, e.g., three integration constants O, Fm, Ftot, and five initial values of u, Pth, Pc, P-, P- at x = 0. Results of numerical calculations for Yg = 5/3, Yc = 4/3 are shown in Fig. 3.4.1-3.4.4.

10 x

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