Non-linear particle reacceleration by multiple shocks

When the pressure of particles accelerated at shock waves is no longer negligible compared to the kinetic pressure of the gas, the linear theory of diffusive shock acceleration breaks down. This is expected in particular when the shock sweeps up pre-existing cosmic rays, or when multiple shocks reaccelerate successively the same particles. To describe these systems, one has to account for the non-linear backreaction of the particles on the magnetohydrodynamic flow. Using an up-to-date semi-analytical model of particle reacceleration at non-linear shocks, we show that the presence of pre-existing energetic particles strongly affects the shock profile, in such a way that the reacceleration of non-thermal particles or the acceleration of particles from the thermal bath becomes less efficient. We further describe the evolution of the distribution of particles after several shocks and study the properties of the asymptotic solution. We detail the case of identical shocks as well as more realistic scenarios, including the heating of the medium or superbubble environments. When the particles are efficiently confined in the acceleration region, it is generally found that the spectrum converges toward a concave solution after a few tens of shocks, with a spectral index around 3.5 at the highest energy. The post-shock cosmic ray pressure reaches an asymptotic value of about 4-5 per cent of the ram pressure of one shock. Most of the shock pressure is transferred to escaping particles.