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

Marco Kuhlen, Vo Hong Minh Phan+1 more|arXiv (Cornell University)|Nov 10, 2022

Solar and Space Plasma Dynamics4 citations

TL;DR

This paper resolves the long-standing discrepancy between analytical models and simulations in perpendicular cosmic ray transport in isotropic turbulence by introducing a three-stage analytical model: initial particle transport along field lines, field line diffusion, and particle decorrelation. It identifies subdiffusive field line transport as the origin of non-standard rigidity scaling in the perpendicular diffusion coefficient, with simulations and the model showing excellent agreement across rigidities, including at low rigidities where standard theories fail.

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]

Motivation & Objective

• Address the persistent disagreement between analytical models and numerical simulations in perpendicular cosmic ray transport at intermediate rigidities.
• Investigate the origin of non-standard rigidity scaling in the perpendicular diffusion coefficient observed in simulations but not predicted by standard theories.
• Develop an analytical model that accurately reproduces simulation results across a wide range of rigidities, including the low-rigidity regime.
• Clarify the physical link between field line transport and cosmic ray perpendicular diffusion, particularly the role of subdiffusive field line motion.
• Provide a physically consistent framework for modeling cosmic ray transport in turbulent astrophysical plasmas, relevant for the heliosphere and the interstellar medium.

Proposed method

• Perform large-scale test particle simulations using GPU-accelerated computation to access unprecedentedly low rigidities (r_g/L_c ≲ 10⁻³), extending beyond previous simulation limits.
• Model the perpendicular diffusion coefficient as a three-stage process: (1) initial ballistic transport along field lines, (2) field line diffusion governed by the turbulent magnetic field structure, and (3) eventual decorrelation of particles from field lines.
• Use a semi-analytical ODE approach to model the running field line diffusion coefficient, solving coupled equations for σ_x²(z) and d_FL(z) under Corrsin’s hypothesis and isotropic Kolmogorov turbulence.
• Incorporate a Kolmogorov k⁻⁵/³ power spectrum for the turbulent magnetic field with outer scale L_max and correlation length L_c ≈ L_max/5.
• Apply the magneto-static approximation, assuming particle speeds exceed Alfvén speed, and use ensemble averaging over turbulent field realizations and particle initial directions.
• Validate the analytical model against simulation results by comparing running and asymptotic diffusion coefficients across varying turbulence levels (η = 0.2, 0.5) and rigidities.

Experimental results

Research questions

• RQ1 Why do standard analytical models fail to reproduce the rigidity-dependent perpendicular diffusion coefficient observed in numerical simulations at intermediate rigidities?
• RQ2 What physical mechanism underlies the non-standard rigidity scaling of the perpendicular diffusion coefficient in isotropic turbulence?
• RQ3 How does field line transport evolve over time, and does it exhibit subdiffusive behavior that could explain the anomalous particle diffusion?
• RQ4 To what extent can the particle perpendicular diffusion coefficient be accurately modeled by decomposing the process into field line diffusion and particle decorrelation?
• RQ5 Does the analytical model based on field line dynamics reproduce the simulation results across the full rigidity range, including the low-rigidity regime where standard theory applies?

Key findings

• The simulated perpendicular diffusion coefficient exhibits a non-standard rigidity dependence at intermediate rigidities (r_g/L_c ≲ 1), with λ_⊥/λ_∥ increasing before dropping sharply at high rigidities.
• At the lowest rigidities (r_g/L_c ≲ 10⁻³), the standard rigidity scaling (λ_⊥ ∝ r_g²) is recovered, consistent with theoretical expectations.
• The non-standard rigidity dependence is traced to a subdiffusive phase in field line transport, where d_FL(z) initially increases faster than linearly before overshooting and oscillating before reaching a constant asymptotic value.
• The analytical model based on field line ODEs and particle decorrelation reproduces the simulated diffusion coefficients with high accuracy across all rigidities and turbulence levels.
• The ODE model captures the oscillatory overshoot in the running field line diffusion coefficient, which arises from the interference term (cos(k cosθ z)) in the integral, and reproduces the transition from ballistic to diffusive field line motion.
• The agreement between simulations and the analytical model is improved by applying small rescaling factors (0.8 for η=0.2, 0.56 for η=0.5), indicating robustness despite minor discrepancies in oscillation phase and period.