Magnetic Field Amplification and Saturation in Turbulence Behind a Relativistic Shock

We have investigated via two-dimensional relativistic MHD simulations the long-term evolution of turbulence created by a relativistic shock propagating through an inhomogeneous medium. In the postshock region, magnetic field is strongly amplified by turbulent motions triggered by preshock density inhomogeneities. Using a long-simulation box we have followed the magnetic-field amplification until it is fully developed and saturated. The turbulent velocity is sub-relativistic even for a strong shock. Magnetic-field amplification is controled by the turbulent motion and saturation occurs when the magnetic energy is comparable to the turbulent kinetic energy. Magnetic-field amplification and saturation depend on the initial strength and direction of the magnetic field in the preshock medium, and on the shock strength. If the initial magnetic field is perpendicular to the shock normal, the magnetic field is first compressed at the shock and then can be amplified by turbulent motion in the postshock region. Saturation occurs when the magnetic energy becomes comparable to the turbulent kinetic energy in the postshock region. If the initial magnetic field in the preshock medium is strong, the postshock region becomes turbulent but significant field amplification does not occur. If the magnetic energy after shock compression is larger than the turbulent kinetic energy in the postshock region, significant field amplification does not occur. We discuss possible applications of our results to gamma-ray bursts and active galactic nuclei.