Particle reacceleration by compressible turbulence in galaxy clusters: effects of a reduced mean free path

Direct evidence for in situ particle acceleration mechanisms in the intergalactic medium (IGM) is provided by the diffuse Mpc-scale synchrotron emissions observed from galaxy clusters. It has been proposed that magnetohydrodynamic turbulence, generated during cluster-cluster mergers, may be a source of particle reacceleration in the IGM. Calculations of turbulent acceleration must self-consistently account for the complex non-linear coupling between turbulent waves and particles. This has been calculated in some detail with the assumption that turbulence interacts in a collisionless way with the IGM. In this paper, we explore a different picture of acceleration by compressible turbulence in galaxy clusters, where the interaction between turbulence and the IGM is mediated by plasma instabilities and maintained collisional at scales much smaller than the Coulomb mean free path. In this regime, most of the energy of fast modes is channelled into the reacceleration of relativistic particles and the acceleration process approaches a universal behaviour, being self-regulated by the back-reaction of the accelerated particles on the turbulence itself. Assuming that relativistic protons contribute to several per cent (or less) of the cluster energy, consistent with the Fermi observations of nearby clusters, we find that compressible turbulence at the level of a few per cent of the thermal energy can reaccelerate relativistic electrons at GeV energies, which are necessary to explain the observed diffuse radio emission in the form of giant radio haloes.