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.