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.
