Morphology of radio relics I: What causes the substructure of synchrotron emission?

High-resolution radio observations of cluster radio relics often show complex spatial and spectral features. However, it is not clear what these features reveal about the underlying magnetic field properties. We performed three-dimensional magneto-hydrodynamical simulations of merger shock waves propagating through a magnetised, turbulent intracluster medium. Our model includes the diffusive shock acceleration of cosmic-ray electrons, their spatial advection and energy losses at run-time. With this set-up we can investigate the relation between radio substructure and pre-shock plasma conditions in the host cluster. We find that upstream turbulence plays a major role in shaping the properties of radio relics produced downstream. Within the assumption of diffusive shock acceleration, we can reproduce the observed discrepancy between the X-ray derived Mach number of shocks, and the Mach number inferred from radio spectra. Our simulated spectral index maps and profiles across the radio relic also suggest that the standard deviation of the upstream magnetic field must be relatively small ($\sigma_B\leq 1 \, \mu$G) in order to reproduce observations and therefore, radio relics can potentially constrain the distribution of magnetic fields in galaxy clusters outskirts.