## Methods of calculations

For this study Jones and Kang (2005a) have incorporated new, efficient 'Coarse Grained finite Momentum Volume' (CGMV) scheme for solving the CR diffusion-convection equation (Jones and Kang, 2005b) into 1D TVD MHD code. This MHD code has been used effectively by in the past to study diffusive shock acceleration using conventional finite difference methods to evolve the diffusion-convection equation (Frank et al., 1995; Kang and Jones, 1997). The results presented in Jones and Kang (2005a) are based on a prescribed spatial diffusion coefficient, although they are in the process of incorporating a CGMV-based routine to evolve the wave action equation for Alfvenic turbulence, so that a self-consistent treatment of the full system can be carried out. The CGMV scheme for evolving the CR distribution utilizes the first two momentum moments of f (t, x,p) over finite momentum bins; namely, n = APip2f (p)p and gj = ¡APip3f (p)p . (4.25.1)

Assuming a piecewise power law momentum subgrid model, the first moment of f (t, x, p) is, for example, n,i f3 )1 — df1 //(i — 3), (4.25.2)

where fi - f (pi MiWpi) fi+1, d = p+1/p, , (4.25.3)

and qj is the momentum index inside bin j. This leads to moments of the standard diffusion-convection equation (e.g., Skilling, 1975), such as dn, dn, ^ ^ du d ( „ dn, \ r, ,A „.

— + u—- = Fn — Fn , — n, — + — I Kn -1- I + Sn , (4.25.4)

where u = v + uw is the net velocity of CR scattering centers, including the gas motion, v, and the mean wave motion, uw (Jones and Kang, 2005b). In addition,

is a flux in momentum space, with p = — p(1/3)(du/dx/ Kn^ and Sn are the spatial diffusion coefficient, k(x, p), and a representative source term, S, averaged over the momentum interval. D(p) is the momentum diffusion coefficient. We henceforth express particle momentum in units of mc, where m is the particle mass. In the preliminary study Jones and Kang (2005a) assume the convenient spatial diffusion form, k(p) = Kopa, and ignore momentum diffusion (D = 0). Generally, spatial CR diffusion is not expected to be isotropic with respect to the direction of the local magnetic field. The degree of anisotropy has been shown to have important consequences, especially when the magnetic field is quasi-perpendicular to the shock normal (Baring et al., 1993; Frank et al., 1995; Jokipii, 1987). For these initial simulations, however, Jones and Kang (2005a) assume isotropic diffusion. They also assume a simple, common model for injection of low energy CR at the gas sub-shock; namely, that a fixed fraction, £inj, of the thermal proton flux through the sub-shock is able to escape upstream at momentum p,nj to join the diffusive CR population. This produces a source term in Eq. (4.25.4), for example,

where p is the upstream plasma density, p is the plasma molecular weight, vs is the plasma flow speed across the shock and <(x) is a normalized weighting function to distribute the injection across the numerical shock. These equations are coupled with the standard equations for compressible MHD by adding the pressure gradient of the CR, 3Pc/dx , into the MHD Euler equation and by including in the MHD energy conservation equation the term - udPc/dx to account for work done on the bulk fluid by the CR pressure gradient (this last contribution includes both the adiabatic compression of the plasma through the term vdPc/dx, as well as a term representing dissipation of scattering wave energy generated through CR streaming, according to McKenzie and Volk, 1982; Jones, 1993). In addition, to account properly for thermal energy taken from the bulk plasma during the CR injection process, an energy sink term,

must be, according to Jones and Kang (2005a), added to the MHD energy conservation equation.