Cosmic ray momentum diffusion in the presence of nonlinear Alfvén waves

Abstract. The relation between the spatial diffusion coefficient along the magnetic field, kII, and the momentum diffusion coefficient, Dp, for relativistic cosmic ray particles is modelled using Monte Carlo simulations. Wave fields with vanishing wave helicity and cross-helicity, constructed by superposing 'Alfvén-like' waves are considered. As the result, particle trajectories in high amplitude wave fields and then - by averaging over these trajectories - the values of transport coefficients are derived. The modelling is performed at various wave amplitudes, from δ B/B₀ = 0.15 to 2.0, and for a number of wave field types. At our small amplitudes approximately the quasi-linear theory (QLT) estimates for k_II and D_p are reproduced. However, with growing wave amplitude the simulated results show a small divergence from the QLT ones, with k_II decreasing slower than theoretical prediction and the opposite being true for D_p. The wave field form gives only a slight influence on the wave-particle interactions at large wave amplitudes δ B/B₀ ~ 1. The parameter characterizing the relative efficiency of the second-order to the first-order acceleration at shock waves, D_p κ_II is given in the QLT approximation by the Skilling formula V²_A p² / 9. In simulations together with increasing δ B it increases above this scale in all the cases under our study. Consequences of the present results for the second-order Fermi acceleration at shock waves are briefly addressed.