Ion Alfv\'en velocity fluctuations and implications for the diffusion of streaming cosmic rays

The interstellar medium (ISM) of star-forming galaxies is magnetized and turbulent. Cosmic rays (CRs) propagate through it, and those with energies from $\sim\,\rm{GeV} - \rm{TeV}$ are likely subject to the streaming instability, whereby the wave damping processes balances excitation of resonant ionic Alfv\'en waves by the CRs, reaching an equilibrium in which the propagation speed of the CRs is very close to the local ion Alfv\'en velocity. The transport of streaming CRs is therefore sensitive to ionic Alfv\'en velocity fluctuations. In this paper we systematically study these fluctuations using a large ensemble of compressible MHD turbulence simulations. We show that for sub-Alfv\'enic turbulence, as applies for a strongly magnetized ISM, the ionic Alfv\'en velocity probability density function (PDF) is determined solely by the density fluctuations from shocked gas forming parallel to the magnetic field, and we develop analytical models for the ionic Alfv\'en velocity PDF up to second moments. For super-Alfv\'enic turbulence, magnetic and density fluctuations are correlated in complex ways, and these correlations as well as contributions from the magnetic fluctuations sets the ionic Alfv\'en velocity PDF. We discuss the implications of these findings for underlying "macroscopic" diffusion mechanisms in CRs undergoing the streaming instability, including modeling the macroscopic diffusion coefficient for the parallel transport in sub-Alfv\'enic plasmas. We also describe how, for highly-magnetized turbulent gas, the gas density PDF, and hence column density PDF, can be used to access information about ionic Alfv\'en velocity structure from observations of the magnetized ISM.