Abstract
Fermi acceleration can take place at ultra-relativistic shock waves if theupstream or downstream magnetic field has been remodeled so that most of themagnetic power lies on short spatial scales. The relevant conditions underwhich Fermi acceleration become efficient in the presence of both a coherentand a short scale turbulent magnetic field are addressed. Within the MHDapproximation, this paper then studies the amplification of a pre-existingmagnetic field through the streaming of cosmic rays upstream of a relativisticshock wave. The magnetic field is assumed to be perpendicular in the shockfront frame, as generally expected in the limit of large shock Lorentz factor.In the MHD regime, compressive instabilities seeded by the net cosmic-raycharge in the shock precursor (as seen in the shock front frame) develop on theshortest spatial scales but saturate at a moderate level deltaB/B ~ 1, which isnot sufficient for Fermi acceleration. As we argue, it is possible that otherinstabilities outside the MHD range provide enough amplification to allowsuccessful Fermi acceleration.