The Efficiency of Magnetic Field Amplification at Shocks by Turbulence
  Turbulent dynamo field amplification has often been invoked to explain the
strong field strengths in thin rims in supernova shocks ($\sim 100 \, \mu$G)
and in radio relics in galaxy clusters ($\sim \mu$G). We present high
resolution MHD simulations of the interaction between pre-shock turbulence,
clumping and shocks, to quantify the conditions under which turbulent dynamo
amplification can be significant. We demonstrate numerically converged field
amplification which scales with Alfv\'en Mach number, $B/B_0 \propto {\mathcal
M}_{\rm A}$, up to ${\mathcal M}_{\rm A} \sim 150$. This implies that the
post-shock field strength is relatively independent of the seed field.
Amplification is dominated by compression at low ${\mathcal M}_{\rm A}$, and
stretching (turbulent amplification) at high ${\mathcal M}_{\rm A}$. For high
$\mathcal{M}_{\rm A}$, the $B$-field grows exponentially and saturates at
equipartition with turbulence, while the vorticity jumps sharply at the shock
and subsequently decays; the resulting field is orientated predominately along
the shock normal (an effect only apparent in 3D and not 2D). This agrees with
the radial field bias seen in supernova remnants. By contrast, for low
$\mathcal{M}_{\rm A}$, field amplification is mostly compressional, relatively
modest, and results in a predominantly perpendicular field. The latter is
consistent with the polarization seen in radio relics. Our results are
relatively robust to the assumed level of gas clumping. Our results imply that
the turbulent dynamo may be important for supernovae, but is only consistent
with the field strength, and not geometry, for cluster radio relics. For the
latter, this implies strong pre-existing $B$-fields in the ambient cluster
outskirts.
