Modelling cosmic-ray transport: magnetised versus unmagnetised motion in astrophysical magnetic turbulence

Abstract

Cosmic-ray transport in turbulent astrophysical environments remains a multifaceted problem and, despite decades of study, the impact of complex magnetic field geometry -- evident in simulations and observations -- has only recently received more focussed attention. To understand how ensemble-averaged transport behaviour emerges from the intricate interactions between cosmic rays and structured magnetic turbulence, we run test-particle experiments in snapshots of a strongly turbulent magnetohydrodynamics simulation. We characterise particle-turbulence interactions via the gyro radii of particles and their experienced field-line curvatures, which reveals two distinct transport modes: magnetised motion, where particles are tightly bound to strong coherent flux tubes and undergo large-scale mirroring; and unmagnetised motion, characterised by chaotic scattering through weak and highly tangled regions of the magnetic field. We formulate an effective stochastic process for each mode: compound subdiffusion with long mean free paths for magnetised motion, and a Langevin process with short mean free paths for unmagnetised motion. A combined stochastic walker that alternates between these two modes accurately reproduces the mean squared displacements observed in the test-particle data. Our results emphasise the critical role of coherent magnetic structures in comprehensively understanding cosmic-ray transport and lay a foundation for developing a theory of geometry-mediated transport.

Figures (13)

• Figure 1: Radially averaged power spectra of flow and magnetic field in the statistically saturated state of the simulation with $h=2$. Indicated are the wavenumbers of the integral scales $k_u=2.139$ and $k_B=8.983$, the effective Kolmogorov dissipation scales $k_\nu=409.148$ and $k_\eta=511.787$, as well as the r.m.s. gyro wavenumbers ${\left(\upi/2\right)^{-1}}{\hat{\omega}_g}$ of the considered gyro frequencies $\hat{\omega}_g=64; 90.51; 128; 181.019; 256$.
• Figure 2: Isosurfaces of the magnetic field strength $B$ ( blue) and the current density magnitude $j=\|\nabla\times\boldsymbol{B}\|$ ( red). (left) Whole box with $B_\mathrm{iso}/B_\mathrm{max}=0.7$ and $j_\mathrm{iso}/j_\mathrm{max}=0.418$. (middle) Cutout with $B_\mathrm{iso}/B_\mathrm{max}=0.489$ and $j_\mathrm{iso}/j_\mathrm{max}=0.303$. (right) Cutout with $B_\mathrm{iso}/B_\mathrm{max}=0.245$ and $j_\mathrm{iso}/j_\mathrm{max}=0.115$. The subscript iso denotes the value at which the isosurfaces are drawn. The structures of the magnetic isosurfaces correspond to flux tubes, as indicated in figure \ref{['fig:trajectories']}, which are amplified by the fluctuating dynamo action. Most of the magnetic energy is concentrated on large scales in a few intense flux tubes, while small scales reveal less intense and tightly folded flux tubes. Current sheets appear in close proximity to intense flux tubes and are embedded between folds.
• Figure 3: Slices through a magnetic flux tube, surrounded by tight folds, current sheets and plasmoids. (top) Field-line curvature divided by the field strength $\kappa/B$ for comparison with the magnetisation criterion $\kappa r_g\sim 1$ with $r_g\propto B^{-1}$. (centre) Magnitude of the current density $j$ indicating intense current sheets. (bottom) Alignment between the magnetic field and current density $\sigma_{j,B}=\hat{\boldsymbol{j}\,}\!\bcdot\hat{\boldsymbol{B}\,}\!$ indicating cellularisation into approximately force-free patches. Further indicated are the correlation scale of the magnetic field and the locations of the flux tube and example plasmoids.
• Figure 4: Magnetic field lines coloured by field strength $B$ and test-particle trajectories coloured by pitch angle cosine $\mu$ in the flux tube (top) and one of the plasmoids (bottom) from figure \ref{['fig:slice']}. In the coherent flux tube, particles are closely bound to field lines with occasional large-scale mirroring. The plasmoid exhibits highly tangled field lines and effectively confines particles with a mixture of small-scale mirroring and unmagnetised scattering. The magnetic correlation scale is indicated for reference.
• Figure 5: (a) Average of the relative magnetic moment variation $\,\overline{\!{\delta M}}/\,\overline{\!{M}}$ conditional on particle gyro radius $\bar{r}_g$ and field-line curvature $\bar{\kappa}$. All recorded quantities are gyro-averaged. The colour scale is centred at $\,\overline{\!{\delta M}}/\,\overline{\!{M}}=1$, where particles can be considered magnetised for smaller variations and unmagnetised for larger variations. Further, the colour scale is capped to $\log_{10}\,\overline{\!{\delta M}}/\,\overline{\!{M}}\in(-0.6, 0.6)$ to highlight the transition region. This transition region is compared with the magnetisation criterion $\bar{\kappa}\bar{r}_g\sim1$ expected from the field-line curvature picture. The joint density $p(\bar{\kappa}, \bar{r}_g)$ is indicated for reference. (b) The conditional average $\langle\,\overline{\!{\delta M}}/\,\overline{\!{M}}|\bar{\kappa}\bar{r}_g\rangle$ also shows the transition from predominantly magnetised and to predominantly unmagnetised motion as $\bar{\kappa}\bar{r}_g$ increases, although the joint density $p(\,\overline{\!{\delta M}}/\,\overline{\!{M}},\bar{\kappa}\bar{r}_g)$ reveals some uncertainty of this criterion.