Production of Magnetic Energy by Macroscopic Turbulence in GRB Afterglows

Afterglows of gamma-ray bursts are believed to require magnetic fields much stronger than that of the compressed preshock medium. As an alternative to microscopic plasma instabilities, we propose amplification of the field by macroscopic turbulence excited by the interaction of the shock with a clumpy preshock medium, for example, a stellar wind. Using a recently developed formalism for localized perturbations to an ultrarelativistic shock, we derive constraints on the length scale, amplitude, and volume filling factor of density clumps required to produce a given magnetic energy fraction within the expansion time of the shock, assuming that the energy in the field achieves equipartion with the turbulence. Stronger and smaller scale inhomogeneities are required for larger shock Lorentz factors. Hence, it is likely that the magnetic energy fraction evolves as the shock slows. This could be detected by monitoring the synchrotron cooling frequency if the radial density profile ahead of the shock, smoothed over clumps, is known.