Resonant micro-instabilities at quasi-parallel collisionless shocks: cause or consequence of shock (re)formation

A case of two interpenetrating, cold and quasi-neutral ion-electron plasmas is investigated with the multi-fluid approach. We consider that one plasma flows quasi-parallel to the lines of a background magnetic field embedded in another static plasma. If the flow turns super-Alfv\'enic, we show that parallel R/L-modes and perpendicular X/O-modes become unstable and grow in amplitude. Within the linear theory, we find that the growth rate curve of an unstable mode has a maximum at some wavenumber specific to each mode. If we consider a shock-like plasma configuration, we find that the fastest growing mode is the resonant one (with $k \sim r_{gi}^{-1}$) which strongly interacts with ions. In Particle-In-Cell (PIC) simulations, we observe that a resonant wave with the same properties is excited during the early phases of shock formation. Once the wave becomes non-linear, it efficiently scatters ions and triggers the initial shock formation. This implies that the typical compression ratio of $\sim 4$ could naturally arise as a consequence of a highly resonant micro-physical process. We model the interaction of ions reflected from the reforming shock barrier in a weak-beam case, and we show that the upstream wave now matches the instability we expect from the equations. By using PIC simulations, we explain how the strong-beam resonant instability triggers shock formation in the non-linear stage, and how the weak-beam instability reforms and transmits the shock afterwards. The micro-instabilities that we study here could largely contribute to shock triggering as well as to the reformation and transmission of the shock itself.