Diffusion of relativistic charged particles and field lines in isotropic turbulence: II. Analytical models

TL;DR

This work addresses the perpendicular diffusion of high-energy charged particles in isotropic turbulence, where existing analytical theories fail at moderate rigidities. The authors develop an analytical framework that stratifies transport into three stages—along field lines, field-line diffusion, and particle-field-line decorrelation—with a subdiffusive phase in field-line transport, and derive the central relation d_perp(t) ≈ [d_FL(√⟨(Δz)^2⟩)/√⟨(Δz)^2⟩] · d_parallel(t) and ⟨(Δz)^2⟩ = 2∫_0^t d t' d_parallel(t'). They parameterize d_parallel(t) and d_FL(z) and compute an asymptotic diffusion coefficient κ_perp by solving for the transition time τ_c, enabling quantitative reproduction of the observed non-standard rigidity-dependence, including a ∝ (r_g/L_c)^{1/2} scaling at intermediate rigidities. The study demonstrates that the non-standard scaling arises from the subdiffusive phase in field-line transport and provides practical fitting forms to deploy in astrophysical applications, validating the broader physical picture of perpendicular cosmic-ray transport.

Abstract

The transport of high-energy particles in the presence of small-scale, turbulent magnetic fields is a long-standing issue in astrophysics. Analytical theories on transport perpendicular to the large-scale magnetic field disagree with numerical simulations at rigidities where the particles' gyroradii are slightly smaller than the correlation length of turbulence. At the same time, extending the numerical simulations to lower rigidities has proven computationally prohibitive. We present an analytical model for the perpendicular transport, based on (1) initial particle transport along field lines, (2) the transport of field lines and (3) the eventual decorrelation of particles from field lines. Transport parallel to the large-scale field is governed by pitch-angle scattering and so for times larger than the inverse pitch-angle diffusion coefficient, particles spatially diffuse in the parallel direction. Our results suggest that perpendicular diffusion occurs when particles have displaced in the perpendicular direction by a few correlation lengths of turbulence. We have tested the analytical theory by running a large suite of test particle simulations at unprecedentedly low rigidities, making extensive use of graphical processing units (GPUs). Our numerical results exhibit a non-standard rigidity-dependence for the perpendicular diffusion coefficient at intermediate rigidities. At the lowest rigidities, the standard rigidity-dependence is recovered. The simulated diffusion coefficients are nicely reproduced by our analytical model. We have traced the non-standard rigidity-dependence to a subdiffusive phase in the field line transport. Our study confirms our understanding of the escape of cosmic rays from the Galactic halo and its rigidity-dependence. [abridged]

Figures (5)

• Figure 1: Running field line diffusion coefficient $d_{\text{FL}}$. For turbulence levels of $\eta = 0.2$ and $0.5$ we compare the result of the numerically integration of eqs. \ref{['eqn:ODE_system_1']} and \ref{['eqn:ODE_system_2']} with the results of the numerical simulations Kuhlen:2022tov.
• Figure 2: Running field line diffusion coefficients $d_{\text{FL}}$ for the different turbulence levels $\eta$. The solid lines show the running field-line diffusion coefficient $d_{\text{FL}}$ (eq. \ref{['eqn:dFL_parametrisation']}) including a subdiffusive phase with parameters fitted to the asymptotic perpendicular mean free path $\lambda_{\perp}$. For the dotted line, we have set $\gamma = 0$ in eq. \ref{['eqn:dFL_parametrisation']} and thus there is no subdiffusive phase. The grey dashed line and shaded band indicate the results of simulations for field line transport at the respective turbulence level $\eta$.
• Figure 3: Running parallel and perpendicular diffusion coefficients $d_{\parallel}$ and $d_{\perp}$ for different reduced rigidities $r_{\text{g}}/L_{\text{c}}$. The grey lines and bands indicate the means and standard mean errors from the test particle simulations. The dashed (solid) line shows the running parallel (perpendicular) diffusion coefficient from the simulations. The asymptotic values are indicated by the dashed and dot-dashed lines, respectively. Finally, the dotted grey line indicates $L_{\text{c}}^2 / t$, that is the asymptotic perpendicular diffusion coefficient $\kappa_{\perp}$ is attained once the running perpendicular diffusion coefficient $d_{\perp}$ intersects this line.
• Figure 4: 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 $\eta$. The dots and squares are the results from test particle simulations. The dot-dashed and solid lines show the predictions from our model that includes a subdiffusive phase in the running field-line diffusion coefficient $d_{\text{FL}}$. The perpendicular mean-free path without a subdiffusive phase in $d_{\text{FL}}$ is indicated by the dotted lines.
• Figure 5: Ratio $\lambda_{\perp} / \lambda{\parallel}$ of the perpendicular and parallel mean free paths as a function of reduced rigidity $r_{\text{g}}/L_{\text{c}}$ for different turbulence levels $\eta$. The dots show the results of test particle simulations. Our model predictions including a subdiffusive phase in the running field-line diffusion coefficient $d_{\text{FL}}$ is shown by the solid lines. A model prediction without a subdiffusive phase in $d_{\text{FL}}$ is indicated by the dotted lines.