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 obtains 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 a
sub-Alfv\'enic plasma. 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.
