  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.
