Ray Diffusion Approach

• Ray Diffusion Approach is a comprehensive framework defining cosmic-ray transport via gyroresonant scattering and magnetic mirror effects in turbulent plasmas.
• It utilizes high-resolution MHD simulations and mode decomposition to quantify the distinct impacts of Alfvénic and magnetosonic turbulence on parallel and perpendicular diffusion.
• The approach explains the observed slow cosmic-ray diffusion near sources like supernova remnants by integrating both resonant scattering and mirror trapping mechanisms.

The ray diffusion approach provides a comprehensive framework for describing the transport of cosmic-ray (CR) particles in turbulent astrophysical environments. This approach integrates gyroresonant scattering and magnetic mirror effects, accounts for the distinct roles of magnetohydrodynamic (MHD) turbulence modes, and quantifies their impact through detailed numerical simulations, with particular application to explaining anomalously slow diffusion near CR sources such as supernova remnants and pulsar halos (2506.15031).

1. Diffusion Mechanisms: Scattering and Mirror Diffusion

Two principal mechanisms govern CR diffusion in MHD turbulence:

• Scattering Diffusion involves gyroresonant interactions between CRs and turbulent MHD waves. Under quasi-linear theory (QLT), the resonance condition is

$$R_n = \pi\,\delta(k_\parallel v_\parallel - \omega \pm n\Omega),$$

where $k_\parallel$ and $v_\parallel$ are the wavevector and particle velocity components parallel to the local magnetic field, $\omega$ is the wave frequency, $\Omega$ the CR gyrofrequency, and $n$ the harmonic number. This interaction leads to pitch angle scattering and randomizes the direction of CRs along the field lines.

• Mirror Diffusion is a nonresonant process whereby CRs interact with large-scale turbulent compressions—magnetic mirrors—causing particles (with sufficiently large pitch angles and $R_g < l_{\mathrm{mir}}$) to reflect between high-field regions while conserving the first adiabatic invariant,

$M = \frac{\gamma m u_\perp^2}{2B} = \text{const}.$

Here, $u_\perp$ is the particle perpendicular velocity and $B$ the local field strength. Mirror trapping reduces the effective parallel transport, confining CRs more efficiently than scattering alone.

These mechanisms are not mutually exclusive; the overall diffusion is a complex superposition where the relative contribution depends on turbulence properties and CR pitch angle distributions.

2. Influence of MHD Turbulence Regimes

The properties of the underlying MHD turbulence—mode composition and magnetization regime—crucially determine the diffusion behavior:

• Alfvén Modes (incompressible, anisotropic with $L_\parallel \propto L_\perp^{2/3}$) drive perpendicular diffusion through field-line wandering. Their contribution dominates transversal CR transport.
• Magnetosonic Modes (compressible fast and slow modes) are the primary source of parallel scattering via gyroresonance, efficiently randomizing the parallel component of CR motion.

Turbulence regimes are further categorized:

• Super-Alfvénic ($M_A > 1$): Turbulent energies exceed magnetic energies, yielding nearly isotropic diffusion ($D_\perp \approx D_\parallel$).
• Sub-Alfvénic ($M_A < 1$): Magnetic field effects dominate, with anisotropy scaling as $D_\perp \approx D_\parallel M_A^4$. Below specific transition scales ($L_A, L_\text{tr}$), anisotropy increases.

The net result is that fast/slow magnetosonic modes chiefly determine parallel CR scattering, while Alfvénic modes set the rate of perpendicular (transverse) diffusion, especially in the small $M_A$ regime.

3. Numerical Simulation Approach

The paper utilizes high-resolution 3D ideal MHD simulations with the following methodology:

• Magnetized Turbulence Generation: The ideal MHD equations (mass, momentum, induction) are solved with uniform mean fields and solenoidal Fourier-space forcing to maintain statistical stationarity.
• MHD Mode Decomposition: Wavelet and Fourier-based strategies partition the resulting turbulence into Alfvén, slow, and fast modes, allowing isolation of each mode's diffusion impact.
• Test-Particle Method: Large ensembles of CRs are propagated in the simulated fields, with equations of motion

$$\frac{d}{dt}\left(\frac{m\mathbf{u}}{\sqrt{1-u^2/c^2}}\right) = q\,(\mathbf{u} \times \mathbf{B}),$$

solved using adaptive Bulirsch–Stoer integration. The displacements parallel ($\Delta \ell_\parallel$) and perpendicular ($\Delta \ell_\perp$) to the mean field are tracked.

• Quantitative Diffusion Metrics:

$$D_\parallel = \frac{\langle \Delta\ell_\parallel^2 \rangle}{2\Delta t}\,,\qquad D_\perp = \frac{\langle \Delta\ell_\perp^2 \rangle}{2\Delta t}$$

and mean free paths

$$\lambda_\parallel = \frac{3 D_\parallel}{u}\,,\qquad \lambda_\perp = \frac{3 D_\perp}{u}$$

This enables detailed exploration of how transport regimes, turbulence spectra, and CR energies interact.

4. Key Numerical and Physical Findings

The simulations reveal several major outcomes:

• Transition from Superdiffusion to Normal Diffusion: At early times, perpendicular CR transport is superdiffusive (mean square displacement $\propto t^{3/2}$) due to field-line wandering; at late times (thousands of gyroperiods), normal diffusion is established ($\propto t$).
• Mirror Diffusion Enhances Confinement: Particles with large pitch angles (smaller $|\mu|$) are preferentially trapped by magnetic mirrors, resulting in reduced parallel displacement; this effect is more pronounced in high compressibility (strong magnetosonic mode dominance).
• Energy and Scaling Laws: Power-law scalings are observed: In strong turbulence, $\lambda_\perp \propto R_g^{2/3}$, $\lambda_\parallel \propto R_g^{1/3}$ (sub-Alfvénic case); in super-Alfvénic, both scale $\propto R_g$. At higher energies ($R_g >$ transition scale), parallel MFP can steepen ($\propto R_g^2$) and perpendicular MFP may saturate.
• Mode-Decomposed Contributions: Magnetosonic modes dominate parallel scattering (setting $D_\parallel$), while Alfvénic fluctuations set the behavior of $D_\perp$ through spatial field wander. This division maps directly onto observed transport anisotropies in the ISM and near CR sources.

5. Astrophysical Implications: Slow Diffusion near CR Sources

These simulation results have critical astrophysical consequences:

• Explaining Slow Diffusion Zones: The dominance of mirror diffusion in confining CRs, especially with strong compressible turbulence, accounts for the observed suppressed diffusion coefficients in TeV $\gamma$-ray halos around pulsars and SNRs—often up to two orders of magnitude below standard ISM values.
• CR Trapping and Escape: The interplay between mirror trapping and resonant scattering defines escape times, energy-dependent leakage rates, and spatial extents of CR halos—key for interpreting nonthermal emission profiles and for constraining CR acceleration and feedback models.
• Dependence on Turbulence Regimes: The diversity of ISM environments (from super- to sub-Alfvénic turbulence, variable mode mixture) implies substantial spatial variation in CR propagation properties. This affects CR energy deposition, ionization rates, and the resulting secondary emission signatures.
• CR Propagation Modeling: Realistic propagation models must combine both processes—mirror and scattering diffusion—in a manner dependent on local MHD properties, rather than relying solely on quasi-linear or Bohm-like prescriptions.

6. Broader Significance for Diffusion Theory and Observations

The ray diffusion approach, as evidenced by these simulations, redefines the paradigm of cosmic-ray transport by:

• Integrating distinct microphysical processes (scattering and mirroring) tied to turbulence mode content.
• Providing quantitative predictions for the crossover from superdiffusive to diffusive regimes, with energy-dependence modulated by local turbulence.
• Contextualizing gamma-ray and synchrotron observations around CR sources within the framework of spatially and mode-dependent diffusion.
• Demonstrating the necessity of high-fidelity turbulence characterization in accurate CR phenomenology.

The approach directly links fundamental plasma processes to observable features in the high-energy sky and highlights the need for next-generation observational programs and simulation efforts focused on turbulence diagnostics and CR feedback effects.