Energy-Dependent Transport of Cosmic Rays in the Multiphase, Dynamic Interstellar Medium

We investigate the transport of spectrally resolved cosmic ray (CR) protons with kinetic energies between $1-100$ GeV within the dynamic, multiphase interstellar medium (ISM), using a two-moment CR fluid solver applied to a TIGRESS MHD simulation with conditions similar to the solar neighborhood. Our CR transport prescription incorporates space- and momentum-dependent CR scattering coefficients $σ=κ^{-1}$, computed from the local balance between streaming-driven Alfvèn wave growth and damping processes. We find that advection combines with momentum-dependent diffusion to produce a CR distribution function $f(p)\propto~p^{-γ}$ with $γ\approx4.6$ that agrees with observations, steepened from an injected power law slope $γ_\mathrm{inj}=4.3$. The CR pressure is uniform in the highly diffusive, mostly neutral midplane region, but decreases exponentially in the ionized extraplanar region where scattering is efficient. To interpret these numerical results, we develop a two-zone analytic model that captures and links the two (physically and spatially) distinct regimes of CR transport in the multiphase, dynamic ISM. At low momenta, CR transport is dominated by gas advection, while at high momenta, both advection and diffusion contribute. At high momentum, the analytic prediction for the spectral slope approaches $γ=(4/3)γ_\mathrm{inj}-1$, and the predicted scaling of grammage with momentum is $X\propto p^{1-γ_\mathrm{inj}/3}$, consistent with the simulations. These results support a physical picture in which CRs are confined within the neutral midplane by the surrounding ionized gas, with their escape regulated by both the CR scattering rate in the ionized extraplanar gas and the velocity and Alfvén speed of that gas, at effective speed $v_\mathrm{c,eff}\approx(1/2)[κ_\parallel~d(v+v_\mathrm{A,i})/dz]^{1/2}$.