2024A&A...685A..68D

Recent observations have revealed detailed structures of radio relics across a wide range of frequencies. In this work, we performed three-dimensional magnetohydrodynamical (3D MHD) simulations of merger shocks propagating through a turbulent magnetized intracluster medium. We then employed on-the-fly Lagrangian particles to explore the physical processes behind the origination of radio substructures and their appearance in high and low-frequency observations. We employed two cosmic-ray (CR) electron acceleration models, with a fresh injection of electrons from the thermal pool and the re-acceleration of mildly relativistic electrons. We used the relative surface brightness fluctuations, δS_ν, to define a “degree of patchiness.” First, we found that patchiness is produced if the shock's surface has a distribution of Mach numbers, rather than a single Mach number. Second, radio relics appear patchier if the Mach number distribution consists of a large percentage of low Mach numbers (ℳ ≲ 2.5). Furthermore, as the frequency increases, the patchiness also becomes larger. Nevertheless, if radio relics are patchy at high frequencies (e.g., 18.6 GHz), they necessarily will also be patchy at low frequencies (e.g., 150 MHz). Then, to produce noticeable differences in the patchiness at low and high frequencies, the shock front should have a Mach number spread of σ_ℳ ≳ 0.3 - 0.4. Finally, the extent of the patchiness depends on the Mach number distribution as well as the CR acceleration model. We propose δS_ν as a potential tool for extracting merger shock properties and information on particle acceleration processes at shocks in radio observations.