Fluid simulations of cosmic ray-modified shocks

We consider cosmic ray (CR) modified shocks with both streaming and diffusion in the two-fluid description. Previously, numerical codes were unable to incorporate streaming in this demanding regime, and have never been compared against analytic solutions. First, we find a new analytic solution highly discrepant in acceleration efficiency from the standard solution. It arises from bi-directional streaming of CRs away from the subshock, similar to a Zeldovich spike in radiative shocks. Since fewer CRs diffuse back upstream, this results in a much lower acceleration efficiency, typically $\sim 10\%$ as opposed to $\sim 50\%$ found in previous analytic work. At Mach number $\gtrsim 10$, the new solution bifurcates into 3 branches, with efficient, intermediate and inefficient CR acceleration. Our two-moment code (Jiang & Oh 2018) accurately recovers these solutions across the entire parameter space probed, with no ad hoc closure relations. For generic initial conditions, the inefficient branch is the most robust and preferred solution. The intermediate branch is unstable, while the efficient branch appears only when the inefficient branch is not allowed (for CR dominated or high plasma $\beta$ shocks). CR modified shocks have very long equilibration times ($\sim 1000$ diffusion time) required to develop the precursor, which must be resolved by $\gtrsim 10$ cells for convergence. Non-equilibrium effects, poor resolution and obliquity of the magnetic field all reduce CR acceleration efficiency. Shocks in galaxy scale simulations will generally contribute little to CR acceleration without a subgrid prescription.