Cosmic Ray Hydrodynamics at Shock Fronts

When a magnetised, thermal plasma containing a shock wave eeciently accelerates Cosmic Rays (CR) a reaction is exerted on the uid structure. The simplest description of this process involves three conservation equations for the background uid, with the CR pressure gradient included as an extra force, plus a hydrodynamic equation for the CR energy density. Since the CR spectrum is not determined in this approach two quantities are introduced, the ratio of CR pressure to energy density and a mean diiusion coeecient, that must be speciied to close the system of equations. Physical arguments are presented for how these quantities should evolve and then compared with time dependent numerical solutions of acceleration at a plane parallel, piston driven shock.