Cosmic-ray transport in inhomogeneous media

TL;DR

This paper develops a mean-field theory for cosmic-ray transport along magnetic-field lines in a patchy, multi-phase medium where the local diffusion coefficient $\kappa(x)$ fluctuates across scales. It proves that long-time diffusion is diffusive with $\bar{\kappa}(t) \to \kappa_H$, the harmonic mean of the local diffusivity, while short-time diffusion tends toward $\kappa_A$, the arithmetic mean, with a transient sub-diffusive regime in between; for a two-phase, multi-scale model the authors introduce a progressive harmonic mean and validate it against Monte Carlo simulations. The strength and duration of the sub-diffusive transient depend on the patch-size distribution exponent $\alpha$ of the high- and low-diffusion patches, linking microstructure to transport. Allowing even modest cross-field diffusion $\kappa_{\perp}$ can let CRs circumnavigate low-diffusion zones, imprinting or modifying the energy dependence of transport depending on $\alpha$ and the energy scalings of the parallel and perpendicular diffusivities, with implications for CR confinement and escape in astrophysical environments.

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.

Figures (8)

• Figure 1: Cartoon of a patchy two-phase medium. In the left panel, the two-phase medium is depicted with regions of high parallel-diffusion coefficient in white and regions of low parallel-diffusion coefficient shaded in blue. CRs are sourced in localised areas of space, shown in red, across multiple field lines. The magnetic-field lines threading the media, to which CRs would be tied, are shown as black lines. The patchy stochastic structure is multi-scale, meaning that the distributions of patch lengths along magnetic-field lines, shown in the upper right panel, will typically be power laws. From these distributions, $P_{\mathrm{high}}(l)$ for high-diffusion patches and $P_{\mathrm{low}}(l)$ for low-diffusion patches, a single realisation of the spatially varying diffusion coefficient $\kappa(x)$ along a magnetic-field line can be constructed and is shown in the lower right panel.
• Figure 2: The solution of the diffusion model (\ref{['eqn:1:1']}) in a patchy environment. The upper panel shows the density of an initial point source in an inhomogeneous diffusive medium at five, logarithmically spaced, times. The spatially varying diffusion coefficient, shown in the lower panel, alternates between high and low values. The dashed lines demarcate the transitions between high- and low-diffusion regions, which create a characteristic 'staircase' structure---the high-diffusion regions have flatter density profiles, separated by regions with larger gradients enabled in the low-diffusion regions. At late times, the overall density envelope spreads with a diffusion coefficient equal to the harmonic mean of the local diffusion coefficient (\ref{['eqn:1:3']})---this envelope is shown by the dash--dotted line. For this example, the distributions of patch sizes in high- and low-diffusion regions are $P_{\mathrm{high}}(l)= P_{\mathrm{low}}(l) \propto l^{-2.5}$.
• Figure 3: The running diffusion coefficient $\bar{\kappa}(t)$ of CRs propagating through inhomogeneous media with different $\alpha \equiv -\mathrm{d}\ln P_{\mathrm{high}}/\mathrm{d}\ln l$, where $P_{\mathrm{high}}(l)$ is the probability of patch size $l \in [l_{\mathrm{min}},l_{\mathrm{max}}]$ for high-diffusion patches (but for these simulations, we chose $P_{\mathrm{low}} = P_{\mathrm{high}}$). Numerical Monte-Carlo solutions are shown (in yellow for random walks in position and red for random walks in pitch angle; see Appendix \ref{['Section:C']} for details). All diffusion coefficients begin at the arithmetic mean (\ref{['eqn:1:2p5']}) and then reach the harmonic mean (\ref{['eqn:1:3']}) after a period of sub-diffusion. The decay law of $\bar{\kappa}(t)$ with time in the sub-diffusive regime can be computed (up to order-unity factors) easily from the scalings (\ref{['eqn:2:2']}), shown as the black dotted lines. A more quantitative theoretical prediction, derived in Appendix \ref{['Section:B']}, is shown by the the blue curves. The inset shows the $\alpha = 2.0$ case on a semi-log scale, as the diffusion coefficient in this case is expected to decay logarithmically.
• Figure 4: The escape rate (inverse time for a CR to diffuse past a distance $L_{\mathrm{esc}}$, normalised to $t_0 = l_{\mathrm{esc}}^{2}/2\kappa_{\mathrm{A}}(E_{0})$) as a function of the CR energy $E$ relative to a reference energy $E_{0}$. In this numerical set-up (chosen entirely for illustrative purpose), we considered the CRs to have an energy-dependent diffusion coefficient $\propto E^{1/2}$ in the low-diffusion regions and $\propto E^{2}$ in the high-diffusion regions (cf. Reichherzera_2024). We chose ${l_{\mathrm{esc}} = 10\,l_{\mathrm{min}} = l_{\mathrm{max}}/10}$. As a result, for low $\alpha$ (where large patches are frequent), the escape rate is dominated by CRs' diffusion in long high-diffusion patches, while for $\alpha > 2$ long patches become rarer, so slow diffusion dominates the CR escape rate.
• Figure 5: Effective diffusion coefficient in a 2D diffusive medium. The upper panel shows a small sample of our model diffusive medium through which CRs propagate. The lower-left panel shows the running parallel diffusion coefficient $\bar{\kappa}_{\mathrm{2D}}(t)$; the black line is the case with $\kappa_{\perp}=0$ (the 1D running diffusion coefficient $\bar{\kappa}(t)$). The lower-right panel shows the converged effective parallel diffusion $\bar{\kappa}_{\mathrm{2D}}(t\to \infty)$, which interpolates between the harmonic mean $\kappa_{\mathrm{H}}$ (CRs must pass through every patch on the field line) and the arithmetic mean $\kappa_{\mathrm{A}}$ (CRs diffuse so rapidly perpendicularly that they uniformly sample the diffusion coefficient). This curve is in reasonable agreement with the simple heuristic (grey line) given by (\ref{['eqn:4:4']}) from (\ref{['eqn:4:3p5']}). For this simulation, $\kappa_{\mathrm{high}}/\kappa_{\mathrm{low}} = 10^{3}$, the pdf of high-diffusion patches was chosen to be $P_{\mathrm{high}}(l) \propto l^{-3}$ with $l_{\mathrm{max}}/l_{\mathrm{low}} = 400$, whereas the low-diffusion patches were chosen to be single-scale: $P_{\mathrm{low}}(l) = \delta(l-l_{0})$ with $l_{0}=2l_{\mathrm{min}}$ (giving $f=0.5$). In the perpendicular direction, the system had a single correlation length $l_{\perp} = l_{\mathrm{min}}/5$.