Cosmic Ray Transport with Magnetic Focusing and the "Telegraph" model

TL;DR

This paper rigorously analyzes cosmic ray transport models, demonstrating that the telegraph equation does not naturally arise from the Chapman-Enskog expansion and is unphysical during the initial relaxation period.

Contribution

The study clarifies the mathematical origin of the telegraph term, showing it is not inherent to cosmic ray transport and only appears as an unphysical artifact during early relaxation.

Findings

No second order time derivative (telegraph term) emerges in proper asymptotic expansion.

The telegraph term is only relevant during a short initial relaxation period.

The telegraph solution introduces unphysical singularities during this period.

Abstract

Cosmic rays (CR), constrained by scattering on magnetic irregularities, are believed to propagate diffusively. But a well-known defect of diffusive approximation whereby some of the particles propagate unrealistically fast has directed interest towards an alternative CR transport model based on the "telegraph" equation. However, its derivations often lack rigor and transparency leading to inconsistent results. We apply the classic Chapman-Enskog method to the CR transport problem. We show that no "telegraph" (second order time derivative) term emerges in any order of a proper asymptotic expansion with systematically eliminated short time scales. Nevertheless, this term may formally be\emph{ converted} from the \emph{fourth} order hyper-diffusive term of the expansion. But, both the telegraph and hyperdiffusive terms may only be important for a short relaxation period associated with either strong pitch-angle anisotropy or spatial inhomogeneity of the initial CR distribution. Beyond this period the system evolves diffusively in both cases. The term conversion, that makes the telegraph and Chapman-Enskog approaches reasonably equivalent, is possible only after this relaxation period. During this period, the telegraph solution is argued to be unphysical. Unlike the hyperdiffusion correction, it is not uniformly valid and introduces implausible singular components to the solution. These dominate the solution during the relaxation period. As they are shown not to be inherent in the underlying scattering problem, we argue that the telegraph term is involuntarily acquired in an asymptotic reduction of the problem.