On the influence of supra-thermal particle acceleration on the
  morphology of low-Mach, high-$\beta$ shocks
  When two galaxy clusters encounter each other, the interaction results in a
collisionless shock that is characterized by a low (1-4) sonic Mach number, and
a high Alfv{\'e}nic Mach number. Our goal is to determine if, and to what
extent, such shocks can accelerate particles to sufficient velocities that they
can contribute to the cosmic ray spectrum. We combine two different
computational methods, magnetohydrodynamics (MHD) and particle-in-cell (PIC)
into a single code that allows us to take advantage of the high computational
efficiency of MHD while maintaining the ability to model the behaviour of
individual non-thermal particles. Using this method, we perform a series of
simulations covering the expected parameter space of galaxy cluster collision
shocks. Our results show that for shocks with a sonic Mach number below 2.25 no
diffusive shock acceleration can take place because of a lack of instabilities
in the magnetic field, whereas for shocks with a sonic Mach number $\geq\,3$
the acceleration is efficient and can accelerate particles to relativistic
speeds. In the regime between these two extremes, diffusive shock acceleration
can occur but is relatively inefficient because of the time- and
space-dependent nature of the instabilities. For those shocks that show
efficient acceleration, the instabilities in the upstream gas increase to the
point where they change the nature of the shock, which, in turn, will influence
the particle injection process.
