Nonlinear particle reacceleration by successive shocks

Multiple shock acceleration is thought to take place in various environments, including chaotic winds, accretion disks and superbubbles. In the latter, multiple supernovae may explode within small timescales in compact regions and successively reaccelerate the distribution of cosmic rays which do not have time to escape far away from the star cluster. Reacceleration by successive shock waves is therefore expected to take place. We develop a model of nonlinear shock reacceleration and show that the downstream distribution of particles rapidly reach an asymptotic concave solution, such that the cosmic ray pressure saturates at the level of a few percent of the hydrodynamic pressure.