Long-time evolution of magnetic fields in relativistic gamma-ray burst shocks

We investigate the long-time evolution of magnetic fields generated by the two-stream instability at ultra- and subrelativistic astrophysical collisionless shocks. Based on three-dimensional particle-in-cell (PIC) simulation results, we introduce a two-dimensional toy model of interacting current filaments. Within the framework of this model, we demonstrate that the field correlation scale in the region far downstream of the shock grows nearly as the light crossing time, lambda(t) similar to ct, thus making the diffusive field dissipation inefficient. The obtained theoretical scaling is tested using numerical PIC simulations. This result extends our understanding of the structure of collisionless shocks in gamma-ray bursts and other astrophysical objects.