Magnetic Field Amplification and Particle Acceleration in Weakly Magnetized Trans-relativistic Electron-ion Shocks

We investigate the physics of quasi-parallel trans-relativistic shocks propagating in weakly magnetized plasmas by means of long-duration two-dimensional particle-in-cell simulations. The structure of the shock precursor is shaped by a competition between the Bell instability and the Weibel (filamentation) instability. The Bell instability is dominant at relatively high magnetizations $(\sigma\gtrsim10^{-3})$, whereas the Weibel instability prevails at lower magnetizations $(\sigma\lesssim10^{-4})$. Shocks with precursors shaped by Bell modes efficiently accelerate ions, converting a fraction $\varepsilon_{\mathrm{i}}\sim0.2$ of the upstream flow energy into downstream nonthermal ion energy. The maximum energy of nonthermal ions exhibits a Bohm scaling in time, as $E_{\max}\propto t$. A much smaller fraction $\varepsilon_{\mathrm{e}}\ll0.1$ of the upstream flow energy goes into downstream nonthermal electrons in the Bell regime. On the other hand, when the precursor is dominated by Weibel modes, the shock efficiently generates both nonthermal ions and electrons with $\varepsilon_{\mathrm{i}}\sim\varepsilon_{\mathrm{e}}\sim0.1$, albeit with a slower scaling for the maximum energy, $E_{\mathrm{max}}\propto t^{1/2}$. Our results are applicable to a wide range of trans-relativistic shocks, including the termination shocks of extragalactic jets, the late stages of gamma-ray burst afterglows, and shocks in fast blue optical transients.