Cosmic ray transport and acceleration with magnetic mirroring

We analyse the transport of cosmic rays (CR) in magnetic fields that are structured on scales greater than the CR Larmor radius. We solve the Vlasov-Fokker-Planck (VFP) equation for various mixes of mirroring and small-angle scattering and show that relatively small deviations from a uniform magnetic field can induce mirroring and inhibit CR transport to levels that mimic Bohm diffusion in which the CR mean free path is comparable with the CR Larmor radius. Our calculations suggest that shocks may accelerate CR to the Hillas (1984) energy without the need for magnetic field amplification on the Larmor scale. This re-opens the possibility, subject to more comprehensive simulations, that young supernova remnants may be accelerating CR to PeV energies, and maybe even to higher energies beyond the knee in the energy spectrum. We limit our discussion of CR acceleration to shocks that are non-relativistic.