Diffusion of relativistic charged particles and field lines in isotropic turbulence: I. Numerical simulations

TL;DR

The paper tackles how relativistic charged particles diffuse across a large-scale magnetic field in isotropic Kolmogorov turbulence, addressing the rigidity dependence of parallel and perpendicular diffusion. It employs GPU-accelerated test-particle simulations in synthetic turbulence to reach unprecedentedly low reduced rigidities $r_g/L_c$, revealing that $\λ_{\parallel}$ follows the expected $ (r_g/L_c)^{1/3} $ scaling, while $\λ_{\perp}$ shows a more complex behavior with a faster-than-$(r_g/L_c)^{1/3}$ scaling at intermediate rigidities before returning to $(r_g/L_c)^{1/3}$ at very small $r_g/L_c$. The study also analyzes field-line diffusion, finding subdiffusive phases near the correlation scale and highlighting the role of turbulence strength $\eta$ in shaping transport. These results have direct implications for Galactic cosmic-ray propagation and acceleration at shocks, providing a refined understanding of perpendicular diffusion that informs large-scale CR transport models. $\lambda_{\parallel}$, $\lambda_{\perp}$, and field-line diffusion together elucidate how CRs escape, how turbulence governs diffusion, and how transport transitions across rigidity regimes.

Abstract

The transport of non-thermal particles across a large-scale magnetic field in the presence of magnetised turbulence has been a long-standing issue in high-energy astrophysics. Of particular interest is the dependence of the parallel and perpendicular mean free paths $λ_{\parallel}$ and $λ_{\perp}$ on rigidity $\mathcal{R}$. We have revisited this important issue with a view to applications from the transport of Galactic cosmic rays to acceleration at astrophysical shocks. We have run test particle simulations of cosmic ray transport in synthetic, isotropic Kolmogorov turbulence at unprecedentedly low reduced rigidities $\rg/\Lc \simeq 10^{-4}$, corresponding to $\mathcal{R} \simeq 10 \, \text{TV}$ for a turbulent magnetic field of $\Brms = 4 \, μ\text{G}$ and correlation length $\Lc = 30 \, \text{pc}$. Extracting the (asymptotic) parallel and perpendicular mean free paths $λ_{\parallel}$ and $λ_{\perp}$, we have found $λ_{\parallel} \propto (\rg/\Lc)^{1/3}$ as expected for a Kolmogorov turbulence spectrum. In contrast, $λ_{\perp}$ has a faster dependence on $\rg/\Lc$ for $10^{-2} \lesssim \rg/\Lc \lesssim 1$, but for $\rg/\Lc \ll 10^{-2}$, also $λ_{\perp} \propto (\rg/\Lc)^{1/3}$. Our results have important implications for the transport of Galactic cosmic rays.

Figures (12)

• Figure 1: Pitch-angle autocorrelation function $C(t)$ for $r_{\text{g}} / L_{\text{c}} = 8.9 \times 10^{-3}$ and different turbulence levels $\eta$. The dashed vertical line indicates $\Omega t = 2 \pi$. The dotted lines are fits of the form $A \exp[-t / \tau]$ to the different $C(t)$ for $\Omega t > 2 \pi$.
• Figure 3: Running parallel diffusion coefficient $d_{\parallel}(t)$ at $r_{\text{g}}/L_{\text{c}} = 8.9 \times 10^{-3}$ and for various turbulence levels $\eta$. The solid lines show the mean over the ensemble of turbulent magnetic fields, the shaded band indicate the standard mean error. The dashed lines indicate the suppressed ballistic growth. The dotted horizontal lines show the asymptotic diffusion coefficients $\kappa_{\parallel}$ and the vertical dot-dashed lines show the scattering time $\tau_{\text{s}}'$, increased due to the suppression of the ballistic growth.
• Figure 5: Running perpendicular diffusion coefficient $d_{\perp}(t)$ at $r_{\text{g}}/L_{\text{c}} = 8.9 \times 10^{-3}$ and for various turbulence levels $\eta$. The solid lines show the mean over the ensemble of turbulent magnetic fields, the shaded band indicate the standard mean error. The dashed lines indicate the suppressed ballistic growth. The dotted horizontal lines show the asymptotic diffusion coefficients $\kappa_{\perp}$ and the vertical dot-dashed lines show the scattering time $\tau_{\text{s}}'$, increased due to the suppression of the ballistic growth.
• Figure 7: Running parallel and perpendicular diffusion coefficients $d_{\parallel}(t)$ and $d_{\perp}(t)$, respectively, for $\eta = 0.5$ and for various reduced rigidities $r_{\text{g}}/L_{\text{c}}$. The dotted and dash-dotted lines indicate the asymptotic diffusion coefficients $\kappa_{\parallel}$ and $\kappa_{\perp}$, respectively.
• Figure 8: Asymptotic parallel and perpendicular mean free paths $\lambda_{\parallel}$ and $\lambda_{\perp}$ as a function of reduced rigidity $r_{\text{g}}/L_{\text{c}}$ for different turbulence levels $\eta=0.2, \, 0.5, \, 0.8$ and $1$. We also compare the results of our test particle simulations with the data of giacinti2017 (empty circles, squares, and pentagons) as well as Lopez-Coto:2017pbk (empty hexagons). We have indicated power laws $\propto (r_{\text{g}}/L_{\text{c}})^{1/3}$ for $r_{\text{g}}/L_{\text{c}} < 10^{-1}$ and $\propto (r_{\text{g}}/L_{\text{c}})^2$ for $r_{\text{g}}/L_{\text{c}} > 1$.