On Fermi acceleration and magnetohydrodynamic instabilities at ultra-relativistic magnetized shock waves

Fermi acceleration can take place at ultra-relativistic shock waves if the upstream or downstream magnetic field has been remodelled so that most of the magnetic power lies on short spatial scales. The relevant conditions under which Fermi acceleration becomes efficient in the presence of both coherent and short-scale turbulent magnetic fields are addressed. Within the magnetohydrodynamic (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 delta B/B similar to 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.