The Biermann Catastrophe in Numerical MHD

The Biermann Battery effect is frequently invoked in cosmic magnetogenesis and studied in High-Energy Density laboratory physics experiments. Generation of magnetic fields by the Biermann effect due to mis-aligned density and temperature gradients in smooth flow <i>behind</i> shocks is well known. We show that a Biermann-effect magnetic field is also generated <i>within</i> shocks. Direct implementation of the Biermann effect in MHD codes does not capture this physical process, and worse, produces unphysical magnetic fields at shocks whose value does not converge with resolution. We show that this convergence breakdown is due to naive discretization, which fails to account for the fact that discretized irrotational vector fields have spurious solenoidal components that grow without bound near a discontinuity. We show that careful consideration of the kinetics of ion viscous shocks leads to a formulation of the Biermann effect that gives rise to a convergent algorithm. We note two novel physical effects: a <i>resistive magnetic precursor</i> in which Biermann-generated field in the shock "leaks" resistively upstream; and a <i>thermal magnetic precursor</i>, in which field is generated by the Biermann effect ahead of the shock front due to gradients created by the shock's electron thermal conduction precursor. Both effects appear to be potentially observable in experiments at laser facilities. We re-examine published studies of magnetogenesis in galaxy cluster formation, and conclude that the simulations in question had inadequate resolution to reliably estimate the field generation rate. Corrected estimates suggest primordial field values in the range $B\sim 10^{-22}$G --- $10^{-19}$G by $z=3$.