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The interstellar medium (ISM) of star-forming galaxies is magnetized andturbulent. 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 ionicAlfv\'en waves by the CRs, reaching an equilibrium in which the propagationspeed of the CRs is very close to the local ion Alfv\'en velocity. Thetransport of streaming CRs is therefore sensitive to ionic Alfv\'en velocityfluctuations. In this paper we systematically study these fluctuations using alarge ensemble of compressible MHD turbulence simulations. We show that forsub-Alfv\'enic turbulence, as applies for a strongly magnetized ISM, the ionicAlfv\'en velocity probability density function (PDF) is determined solely bythe density fluctuations from shocked gas forming parallel to the magneticfield, and we develop analytical models for the ionic Alfv\'en velocity PDF upto second moments. For super-Alfv\'enic turbulence, magnetic and densityfluctuations are correlated in complex ways, and these correlations as well ascontributions from the magnetic fluctuations sets the ionic Alfv\'en velocityPDF. We discuss the implications of these findings for underlying "macroscopic"diffusion mechanisms in CRs undergoing the streaming instability, includingmodeling the macroscopic diffusion coefficient for the parallel transport insub-Alfv\'enic plasmas. We also describe how, for highly-magnetized turbulentgas, the gas density PDF, and hence column density PDF, can be used to accessinformation about ionic Alfv\'en velocity structure from observations of themagnetized ISM.

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