A nonlinear energy balance model of particle acceleration by collisionless parallel shock waves

We describe in this Letter a new way to model processes of particle acceleration in quasi-parallel shocks and report some promising preliminary results of numerical analysis. The treatment of plasma and waves is self-consistent and time-dependent but nevertheless relatively simple from a physical point of view. The model assumes that resonant wave-particle interaction is the most important mechanism for both shock formation and particle acceleration but does not use the diffusion-convection approach for the interaction. Instead it uses conservation laws and resonance conditions to find where waves will be generated or dumped and hence particles pitch-angle scattered. Because the distribution function for bulk plasma and not just the high-energy tail is included in the model, no special bootstrap or termination assumptions are required ( neither the introduction of a separate population of seed particles nor some ad hoc escape rate of accelerated particles is needed). In spite of all the simplicity, the preliminary results not only show remarkable agreement with diffusive shock acceleration models in the prediction of power spectra for accelerated particles in the upstream region but also reveal the presence of a spectral break in the high-energy part of the spectra. The results also confirm that acceleration can start from the thermal particles and confirm the importance of second-order Fermi acceleration.