According to Jones and Kang (2005a), CR-modified shocks are known to differ significantly from ordinary gas-dynamical or MHD shocks. CR diffusion upstream produces a pressure gradient, adding adiabatic compression to the gas. This preheats the gas and substantially increases compression through the transition. It also weakens the gas sub-shock. Typically, in fact, the compression through a strong CR shock precursor dominates the total shock compression, so that most of the CR acceleration actually takes place in the precursor, rather than in the thin, weakened sub-shock. Furthermore, energy extraction from the thermal plasma by the CR cools the bulk gas with respect to adiabatic gas shocks. Particularly if the CR escape upstream this allows the shock transition to resemble a radiatively cooled shock transition, further amplifying the total compression through the transition, while reducing the down stream temperature and pressure compared to gas-dynamical behaviors. Previous theoretical studies of CR-MHD shocks have emphasized major differences with respect to gas dynamical models for CR-modified shocks. For weak to moderate strength oblique shocks, magnetic field compression restricts the enhanced compression mentioned above. That becomes a relatively small effect in very strong MHD shocks, however, once the total downstream pressure is dominated by the CR (Webb et al., 1986; Frank et al., 1995). A much more significant influence in strong MHD shocks can be the difference between the motion of the bulk plasma and the motion of the CR scattering centers (Jones,

1993); i.e., the drift of the CR with respect to the fluid. If the upstream scattering is due primarily to Alfven waves amplified by CR streaming away from the shock, then one expects the mean motion of the scattering centers to propagate upstream along the mean magnetic field at approximately the local Alfven velocity with respect to the fluid flow. What matters is the component of this velocity along the shock normal, since it parallels the flow and CR gradients. This Alfven velocity component will scale as vax ^ Bxj-Jp , where Bx is the mean magnetic field component along the shock normal. Since Bx does not change through the shock, vax x 1/VP, while the bulk flow speed varies as 1/p, through mass conservation. Consequently, the effective rate of mirror convergence responsible for diffusive shock acceleration, du/ dx, is reduced compared to a pure gas flow, reducing the rate of CR acceleration. Moreover, the magnitude of u is increased by the upstream-facing drift, which increases the rate of precursor heating for a given CR pressure gradient. That reduces the strength of the shock transition, which also reduces the efficiency of diffusive shock acceleration. Jones and Kang (2005a) illustrate these effects by comparing in Fig. 4.25.1 three simulations involving a Mach 40 piston-generated shock.

In Fig. 4.25.1 each flow has an upstream magnetic field inclined 45 degrees from the shock normal with a magnetic pressure there equal to the gas pressure. The simple diffusion model assumed Ko = 0.1 and a = 0.51. CR injection was included with £inj = 0.001. Fourteen momentum bins were used in solving the diffusion-convection equation. The black, solid curves indicate behaviors when uw = 0. This simulated shock is very similar to a gas-dynamical shock discussed in Jones and Kang (2005b) and illustrated in Fig. 4.25.2. The shock quickly becomes CR dominated, but only approaches dynamical equilibrium at the last time shown. The CR momentum distribution at the sub-shock shows the strongly concave form typical of such simulated CR shocks. By comparison the other plotted results include effects of CR drift. The red, dotted curves represent the behavior when uw = vax, while the blue, dashed curves come from a simulation in which uw was included in the MHD energy equation, but not the momentum equation. That approach has been used by some authors e.g., Berezhko and Volk (2000) in non-MHD models to approximate the influence of MHD by allowing for dissipation of wave energy. The differences between the gas-dynamical solutions and the MHD solutions are obvious. The magnetic, Maxwell stresses have had little impact on the MHD solutions. However, as anticipated from the above discussion, compression through the shock is much reduced through CR drift, while the efficiency of CR acceleration is reduced by about one third. These two effects have also mostly eliminated the strong concavity in the CR momentum distribution, since the shock precursor is a much less important contributor to diffusive shock acceleration. On the other hand, there is relatively little difference between the two CR-drift models.

In this one strong shock case, at least, modeling the MHD shock by estimating Alfven wave heating of the bulk plasma would provide a reasonable approximation to the more complete model.

Figure 4.25.1. Three simulated MHD Mach 40 CR-modified shocks formed off a piston on the left boundary. The gas density p, gas pressure Pg, and CR pressure Pc spatial distributions are shown along with the CR momentum distribution f (p) at the sub-shock. From Jones and Kang (2005a).

Figure 4.25.1. Three simulated MHD Mach 40 CR-modified shocks formed off a piston on the left boundary. The gas density p, gas pressure Pg, and CR pressure Pc spatial distributions are shown along with the CR momentum distribution f (p) at the sub-shock. From Jones and Kang (2005a).

Fig 4.25.2. The same as in Fig. 4.25.1, but for gas-dynamical shock. Acording to Jones and Kang (2005b).

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