Cosmic-ray transport in inhomogeneous media

TL;DR

This paper develops a theory for cosmic-ray transport in inhomogeneous media with spatially fluctuating diffusion coefficients, revealing diffusive behavior at long times, transient sub-diffusion at intermediate times, and the influence of medium structure on confinement and energy dependence.

Contribution

It introduces a comprehensive model for cosmic-ray transport in multi-phase media with large spatial diffusion fluctuations, including effects on confinement and energy dependence.

Findings

Transport is diffusive at long times with harmonic mean diffusion coefficient.

Transient sub-diffusion occurs due to low-diffusion regions.

Medium structure influences confinement times and energy dependence.

Abstract

A theory of cosmic-ray transport in multi-phase diffusive media is developed, with the specific application to cases in which the cosmic-ray diffusion coefficient has large spatial fluctuations that may be inherently multi-scale. We demonstrate that the resulting transport of cosmic rays is diffusive in the long-time limit, with an average diffusion coefficient equal to the harmonic mean of the spatially varying diffusion coefficient. Thus, cosmic-ray transport is dominated by areas of low diffusion even if these areas occupy a relatively small, but not infinitesimal, fraction of the volume. On intermediate time scales, the cosmic rays experience transient effective sub-diffusion, as a result of low-diffusion regions interrupting long flights through high-diffusion regions. In the simplified case of a two-phase medium, we show that the extent and extremity of the sub-diffusivity of cosmic-ray transport is controlled by the spectral exponent of the distribution of patch sizes of each of the phases. We finally show that, despite strongly influencing the confinement times, the multi-phase medium is only capable of altering the energy dependence of cosmic-ray transport when there is a moderate (but not excessive) level of perpendicular diffusion across magnetic-field lines.