Context. The origin of cosmic rays (CRs) and how they propagate remain unclear. Studying the propagation of CRs in magnetohydrodynamic (MHD) turbulence can help to comprehend many open issues related to CR origin and the role of turbulent magnetic fields. Aims. To comprehend the phenomenon of slow diffusion in the near-source region, we study the interactions of CRs with the ambient turbulent magnetic field to reveal their universal laws. Methods. We numerically study the interactions of CRs with the ambient turbulent magnetic field, considering pulsar wind nebula as a general research case. Taking the magnetization parameter and turbulence spectral index as free parameters, together with radiative losses, we perform three group simulations to analyze the CR spectral, spatial distributions, and possible CR diffusion types. Results. Our studies demonstrate that (1) CR energy density decays with both its effective radius and kinetic energy in the form of power-law distributions; (2) the morphology of the CR spatial distribution strongly depends on the properties of magnetic turbulence and the viewing angle; (3) CRs suffer a slow diffusion near the source and a fast or normal diffusion away from the source; (4) the existence of a power-law relationship between the averaged CR energy density and the magnetization parameter is independent of both CR energy and radiative losses; and (5) radiative losses can suppress CR anisotropic diffusion and soften the power-law distribution of CR energy density. Conclusions. The distribution law established between turbulent magnetic fields and CRs presents an intrinsic property, providing a convenient way to understand complex astrophysical processes related to turbulence cascades.
Gao et al. (2025) studied this question.