The theory of diffusive particle acceleration explains the spectral properties of the cosmic rays below energies of approx. 10^6 GeV as produced at strong shocks in supernova remnants (SNRs). To supply the observed flux of cosmic rays, a significant fraction of the energy released by a supernova has to be transfered to cosmic rays. The key to the question of the efficiency of SNRs in producing cosmic rays is the injection process from thermal energies. A self-consistent model has to take into account the interaction of the accelerated particles with magneto-hydrodynamic waves, which generate the particle diffusion, a requisite for the acceleration process. Such a nonlinear model of the turbulent background plasma has been developed recently (Malkov, 1998, Phys. Rev. E 58, 4911). We use this model for the first numerical treatment of the gas dynamics and the diffusion-convection equation at a quasi-parallel strong shock, which incorporates a plasma-physical injection model to investigate the cosmic ray production.
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