Turbulence and Radio Mini-halos in the Sloshing Cores of Galaxy Clusters

A number of relaxed, cool-core galaxy clusters exhibit diffuse, steep-spectrum radio sources in their central regions, known as radio mini-halos. It has been proposed that the relativistic electrons responsible for the emission have been reaccelerated by turbulence generated by the sloshing of the cool core gas. We present a high-resolution MHD simulation of gas sloshing in a galaxy cluster coupled with subgrid simulations of relativistic electron acceleration to test this hypothesis. Our simulation shows that the sloshing motions generate turbulence on the order of $\delta{v} \sim$ 50-200 km s$^{-1}$ on spatial scales of $\sim$50-100 kpc and below in the cool core region within the envelope of the sloshing cold fronts, whereas outside the cold fronts, there is negligible turbulence. This turbulence is potentially strong enough to reaccelerate relativistic electron seeds (with initial $\gamma \sim 100-500$) to $\gamma \sim 10^4$ via damping of magnetosonic waves and non-resonant compression. The seed electrons could remain in the cluster from, e.g., past AGN activity. In combination with the magnetic field amplification in the core, these electrons then produce diffuse radio synchrotron emission that is coincident with the region bounded by the sloshing cold fronts, as indeed observed in X-rays and the radio. The result holds for different initial spatial distributions of preexisting relativistic electrons. The power and the steep spectral index ($\alpha \approx 1-2$) of the resulting radio emission are consistent with observations of minihalos, though the theoretical uncertainties of the acceleration mechanisms are high. We also produce simulated maps of inverse-Compton hard X-ray emission from the same population of relativistic electrons.