Cosmic ray momentum diffusion in the presence of non-linear Alfven waves

The relation between the spatial diffusion coefficient along the magnetic field, kappa(parallel to), and the momentum diffusion coefficient, D-p, for relativistic cosmic ray particles is modeled using Monte Carlo simulations. Wave fields with vanishing wave helicity and cross-helicity, constructed by superposing 'Alfven-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 modeling is performed at various wave amplitudes, from delta B/B-0 = 0.15 to 2.0, and for a number of wave field types. At OUT small amplitudes approximately the quasi-linear theory (QLT) estimates for kappa(parallel to) and D-p are reproduced. However, with growing wave amplitude the simulated results show a small divergence from the QLT ones, with kappa(parallel to) 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 delta B/B-0 similar to 1. The parameter characterizing the relative efficiency of the second-order to the first-order acceleration at shock waves, D-p-kappa(parallel to), is given in the QLT approximation by the Skilling formula V(A)(2)p(2)/9. In simulations together with increasing delta 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.