Collisionless shocks form ubiquitously in tenuous cosmic plasmas via collective, electromagnetic viscosities. The formation process of such shocks inevitably produces supra-thermal particles, which can be further accelerated to very high energies through the interactions with resonantly scattering Alfvén waves in the converging flows across a shock (Drury, 1983; Malkov and Drury, 2001). In the kinetic approach to study numerically the CR acceleration at shocks, the diffusion-convection equation for the particle momentum distribution, f (p), is solved with suitably modified gas-dynamic equations. This numerical task is challenging, because the full CR shock transition includes a very wide range of length scales associated with the particle diffusion lengths, k(p)/us , from CR injection scales near the shock to outer diffusion scales for the highest energy particles. To follow the acceleration of highly relativistic CR from supra-thermal energies, Kang and Jones (2005) have developed the CRASH (Cosmic-Ray Amr sHock) code in one dimensional (1D) plane-parallel geometry by combining a powerful Adaptive Mesh Refinement (AMR) technique and a shock tracking technique (Kang et al., 2001). Time-dependent nonlinear simulations of diffusive shock acceleration found that 10-4 -10-3 of incoming thermal particles can be injected into the CR population via thermal leakage at quasi-parallel shocks, and that up to 50-60 % of the shock kinetic energy can be converted into CR at strong shocks with Ms > 10 (Kang et al., 2002; Kang and Jones, 2005b). The presence of a preexisting CR population is equivalent to having efficient thermal leakage injection at the shock.

It is believed that the CR pressure is important in the evolution of the interstellar medium of our Galaxy and most of galactic CR protons with energies up to = 1014 eV are accelerated by supernova remnant shocks (Blandford and Eichler, 1987). Simulations of diffusive shock acceleration in spherical supernova remnants also indicate that CR can absorb up to 50% of the initial blast energies (Berezhko and Volk, 1997, 2000). In paper of Kang and Jones (2005a) is described a new CRASH code in 1D spherical geometry. Kang and Jones (2005a) solved the flow equations in a frame comoving with the spherical shock, so the shock and refined region around it stay at the same grid locations. They present the numerical simulation results for a typical supernova remnant expanding into the uniform hot interstellar medium.

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