Magnetohydrodynamic-Particle-in-Cell Simulations of the Cosmic-Ray Streaming Instability: Linear Growth and Quasi-linear Evolution

The gyro-resonant cosmic-ray (CR) streaming instability is believed to play a crucial role in CR transport, leading to growth of Alfv\'en waves at small scales that scatter CRs, and impacts the interaction of CRs with the ISM on large scales. However, extreme scale separation ($\lambda \ll \rm pc$), low cosmic ray number density ($n_{\rm CR}/n_{\rm ISM} \sim 10^{-9}$), and weak CR anisotropy ($\sim v_A/c$) pose strong challenges for proper numerical studies of this instability on a microphysical level. Employing the recently developed magnetohydrodynamic-particle-in-cell (MHD-PIC) method, which has unique advantages to alleviate these issues, we conduct one-dimensional simulations that quantitatively demonstrate the growth and saturation of the instability in the parameter regime consistent with realistic CR streaming in the large-scale ISM. Our implementation of the $\delta f$ method dramatically reduces Poisson noise and enables us to accurately capture wave growth over a broad spectrum, equally shared between left and right handed Alfv\'en modes. We are also able to accurately follow the quasi-linear diffusion of CRs subsequent to wave growth, which is achieved by employing phase randomization across periodic boundaries. Full isotropization of the CRs in the wave frame requires pitch angles of most CRs to efficiently cross $90^\circ$, and can be captured in simulations with relatively high wave amplitude and/or high spatial resolution. We attribute this crossing to non-linear wave-particle interaction (rather than mirror reflection) by investigating individual CR trajectories. We anticipate our methodology will open up opportunities for future investigations that incorporate additional physics.