Galaxy clusters are permeated by a hot, tenuous plasma known as the intra-cluster medium (ICM). While the ICM can often be approximated as a thermalized plasma in hydrostatic equilibrium, it also contains significant non-thermal components such as magnetic fields, relativistic particles, and turbulence. Little is known about how magnetic fields and relativistic particles are energized in the cluster environment, or about the microinstabilities that limit the particle mean free path and thereby affect the viscosity and turbulence of the ICM.

We address this problem by theoretically and numerically modeling diffuse synchrotron emission observed at radio frequencies. In this work, we study the acceleration of relativistic particles by turbulence in the inter-cluster region between two merging clusters. The evolution of the merging system is simulated with the cosmological MHD code Enzo.

We improve a run-time Lagrangian tracer method implemented in Enzo and follow the trajectories of baryonic matter using tracer particles. In post-processing, we solve the Fokker–Planck equation for all tracers, with cooling and reacceleration efficiencies evaluated from the local quantities recorded along each tracer trajectory.

The simulated bridge reproduces several properties of the radio bridge observed between Abell 399 and Abell 401, including its spectral shape, intensity profile, and pixel-by-pixel correlation between radio and X-ray intensities. This work demonstrates that our tracer-particle method is suitable for exploring the interplay among magnetic fields, turbulence, and cosmic rays in the cluster environment.