Cosmic ray acceleration at parallel relativistic shocks in the presence of finite-amplitude magnetic field perturbations

We study the first-order Fermi acceleration process at parallel shock waves by means of Monte Carlo simulations. A 'realistic' model of the magnetic field turbulence is applied involving sinusoidal waves imposed on the background mean field parallel to the shock normal. The finite-amplitude magnetic field perturbations lead to locally oblique field configurations at the shock and the respective magnetic field compression. It results in modification of the particle acceleration process introducing some features observed in oblique shocks, e.g. particle reflections from the shock. For the parallel mildly relativistic shocks we demonstrate for the first time a (non-monotonic) variation of the accelerated particle spectral index with the turbulence amplitude.