Magnetohydrodynamic-Particle-in-Cell Method for Coupling Cosmic Rays with a Thermal Plasma: Application to Non-relativistic Shocks

摘要: We formulate a magnetohydrodynamic-particle-in-cell (MHD-PIC) method for describing the interaction between collisionless cosmic ray (CR) particles and thermal plasma. The plasma is treated as fluid, obeying equations of ideal MHD, while CRs are relativistic Lagrangian subject to Lorentz force. Backreaction from gas included in form momentum energy feedback. In addition, we include electromagnetic feedback due CR-induced Hall effect that becomes important when electron-ion drift velocity background induced by approaches Alfv\'en velocity. Our applicable on scales much larger than ion inertial length, bypassing microscopic must be resolved conventional PIC methods, retaining full kinetic nature CRs. have implemented tested this Athena MHD code, where overall scheme second-order accurate fully conservative. As first application, describe numerical experiment study particle acceleration non-relativistic shocks. Using simplified prescription injection, reproduce shock structure CR spectra obtained with more self-consistent hybrid-PIC simulations, but at substantially reduced computational cost. also show reduces growth rate Bell instability affects dynamics vicinity front. step forward, able capture transition non regimes, spectrum $f(p)\sim p^{-4}$ connecting smoothly through transition, expected theory Fermi acceleration.