  Fermi acceleration can take place at ultra-relativistic shock waves if the
upstream or downstream magnetic field has been remodeled so that most of the
magnetic power lies on short spatial scales. The relevant conditions under
which Fermi acceleration become efficient in the presence of both a coherent
and a short scale turbulent magnetic field are addressed. Within the MHD
approximation, this paper then studies the amplification of a pre-existing
magnetic field through the streaming of cosmic rays upstream of a relativistic
shock wave. The magnetic field is assumed to be perpendicular in the shock
front frame, as generally expected in the limit of large shock Lorentz factor.
In the MHD regime, compressive instabilities seeded by the net cosmic-ray
charge in the shock precursor (as seen in the shock front frame) develop on the
shortest spatial scales but saturate at a moderate level deltaB/B ~ 1, which is
not sufficient for Fermi acceleration. As we argue, it is possible that other
instabilities outside the MHD range provide enough amplification to allow
successful Fermi acceleration.
