can be described by classical fluid shocks, but in others may not be. Like classical shocks a region develops where most of the particles entering from upstream interact strongly and have much of their directed energy “dissipated”; that is they are “thermalized”. In this context we will identify CRs as those particles that are not thermalized because they interact too weakly with the plasma turbulence to become trapped inside the primary dissipative layer. The dissipative layer is usually called the shock, although it is only part of the shock structure, so I will use the more appropriate “subshock” label. The thickness of the subshock is not easily characterized, and it depends on the orientation of the local mean magnetic field. Often it is described in terms of the characteristic gyroradius of thermalized particles.

If we can ignore the mass, energy and momentum carried by the CRs then the usual hydrodynamic or MHD shock jump conditions apply across the subshock. CRs then can be treated as test particles interacting with this flow. The central idea behind DSA is that these CRs still are sufficiently strongly scattered by local plasma turbulence that their motions become randomized (but not thermalized) close to the 