Ion Alfvén velocity fluctuations and implications for the diffusion of streaming cosmic rays

Abstract

The interstellar medium (ISM) of star-forming galaxies is magnetized and turbulent. Cosmic rays (CRs) propagate through it, and those with energies from ∼ GeV − TeV are likely subject to the streaming instability, whereby the wave damping processes balances excitation of resonant ionic Alfv én waves by the CRs, reaching an equilibrium in which the propagation speed of the CRs is very close to the local ion Alfv én velocity. The transport of streaming CRs is therefore sensitive to ionic Alfv én 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 énic turbulence, as obtains for a strongly magnetized ISM, the ionic Alfv én 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 én velocity PDF up to second moments. For super-Alfv énic 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 én 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 énic 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 én velocity structure from observations of the magnetized ISM.

Turbulent diffusion of streaming cosmic rays in compressible, partially ionised plasma

“…However, there is less studies, and less agreement in the literature regarding the behaviour of 𝜅 with M A0 in the M A0 1 regime. Here a number of authors report 𝜅 ∼ M 4 𝐴 (Yan & Lazarian 2008;Xu & Yan 2013;Cohet & Marcowith 2016) 9 , i.e., the diffusion scaling exactly with the magnetic field variance (Beattie et al 2022). While the direction of the scaling of 𝜅 for sub-Alfvénic turbulence found in these simulations is qualitatively consistent with our findings, we find a vastly different scaling, 𝜅 ∼ M −1 A0 , which is closer to results from Casse et al (2001).…”

“…Because 𝜎 2 𝑠 ∼ ln(1 + M 2 ) (Federrath et al 2008;Federrath et al 2010b;Molina et al 2012;Beattie et al 2021b), this explains the scatter in M that we see in the top left panel of Figure 7. We postpone a more detailed comparison between Beattie et al (2022) theory and our results for a future study.…”

“…Gas density fluctuations in compressible turbulence are extremely inhomogenous, and are not isotropic when the large-scale field is strong 7 . Since 𝑣 str ∝ 𝐵/ √ 𝜌, inhomogeneities in the gas density can produce macroscopic diffusion in the positions of SCRs, regardless of the value of M A0 , as temporally and/or spatially separated SCRs populations experience different density fluctuations along their trajectories (Beattie et al 2022). For M A0 < 1 turbulence, when the magnetic field fluctuations are negligible, Beattie et al (2022) showed that, in the absence of strong advection, density fluctuations formed from gas flows along the magnetic fields solely determine the macroscopic diffusion in ℓ .…”

“…Since 𝑣 str ∝ 𝐵/ √ 𝜌, inhomogeneities in the gas density can produce macroscopic diffusion in the positions of SCRs, regardless of the value of M A0 , as temporally and/or spatially separated SCRs populations experience different density fluctuations along their trajectories (Beattie et al 2022). For M A0 < 1 turbulence, when the magnetic field fluctuations are negligible, Beattie et al (2022) showed that, in the absence of strong advection, density fluctuations formed from gas flows along the magnetic fields solely determine the macroscopic diffusion in ℓ . For M A0 > 1, both the 𝐵-field fluctuations and correlations between the 𝐵 and 𝜌 grow and contribute to setting the fluctuations in 𝑣 str .…”

“…As 𝜅 ⊥ is independent of 𝜒 in this regime, with a systematic scatter set by M (see top left panel of Figure 7) this shows that 𝜅 ∝ 𝑣 str,0 /2. In this regime Beattie et al (2022) finds that 𝜅 ∝ (1/4)𝜎 2 𝑠 𝑣 str,0 ℓ cor,𝜌/𝜌 0 , where 𝜎 2 𝑠 is the logarithmic gas density variance and ℓ cor,𝜌/𝜌 0 is the gas density correlation scale. By substituting this model into equation 19,…”