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Title: Comparison of the telegraph and hyperdiffusion approximations in cosmic-ray transport

Abstract

The telegraph equation and its generalizations have been repeatedly considered in the models of diffusive cosmic-ray transport. Yet the telegraph model has well-known limitations, and analytical arguments suggest that a hyperdiffusion model should serve as a more accurate alternative to the telegraph model, especially on the timescale of a few scattering times. We present a detailed side-by-side comparison of an evolving particle density profile, predicted by the telegraph and hyperdiffusion models in the context of a simple but physically meaningful initial-value problem, compare the predictions with the solution based on the Fokker–Planck equation, and discuss the applicability of the telegraph and hyperdiffusion approximations to the description of strongly anisotropic particle distributions.

Authors:
;  [1]
  1. Department of Mathematics, University of Waikato, P. B. 3105, Hamilton (New Zealand)
Publication Date:
OSTI Identifier:
22600126
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physics of Plasmas; Journal Volume: 23; Journal Issue: 6; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
70 PLASMA PHYSICS AND FUSION TECHNOLOGY; 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ANISOTROPY; APPROXIMATIONS; COMPARATIVE EVALUATIONS; COSMIC RADIATION; DENSITY; DISTRIBUTION; FOKKER-PLANCK EQUATION; PARTICLES; SCATTERING; TRANSPORT THEORY

Citation Formats

Litvinenko, Yuri E., and Noble, P. L.. Comparison of the telegraph and hyperdiffusion approximations in cosmic-ray transport. United States: N. p., 2016. Web. doi:10.1063/1.4953564.
Litvinenko, Yuri E., & Noble, P. L.. Comparison of the telegraph and hyperdiffusion approximations in cosmic-ray transport. United States. doi:10.1063/1.4953564.
Litvinenko, Yuri E., and Noble, P. L.. 2016. "Comparison of the telegraph and hyperdiffusion approximations in cosmic-ray transport". United States. doi:10.1063/1.4953564.
@article{osti_22600126,
title = {Comparison of the telegraph and hyperdiffusion approximations in cosmic-ray transport},
author = {Litvinenko, Yuri E. and Noble, P. L.},
abstractNote = {The telegraph equation and its generalizations have been repeatedly considered in the models of diffusive cosmic-ray transport. Yet the telegraph model has well-known limitations, and analytical arguments suggest that a hyperdiffusion model should serve as a more accurate alternative to the telegraph model, especially on the timescale of a few scattering times. We present a detailed side-by-side comparison of an evolving particle density profile, predicted by the telegraph and hyperdiffusion models in the context of a simple but physically meaningful initial-value problem, compare the predictions with the solution based on the Fokker–Planck equation, and discuss the applicability of the telegraph and hyperdiffusion approximations to the description of strongly anisotropic particle distributions.},
doi = {10.1063/1.4953564},
journal = {Physics of Plasmas},
number = 6,
volume = 23,
place = {United States},
year = 2016,
month = 6
}
  • Diffusive cosmic-ray transport in nonuniform large-scale magnetic fields in the presence of boundaries is considered. Reflecting and absorbing boundary conditions are derived for a modified telegraph equation with a convective term. Analytical and numerical solutions of illustrative boundary problems are presented. The applicability and accuracy of the telegraph approximation for focused cosmic-ray transport in the presence of boundaries are discussed, and potential applications to modeling cosmic-ray transport are noted.
  • Cosmic rays (CR), constrained by scattering on magnetic irregularities, are believed to propagate diffusively. However, a well-known defect of diffusive approximation, whereby some of the particles propagate unrealistically fast, has directed interest toward an alternative CR transport model based on the “telegraph” equation. Though, 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 timescales. Nevertheless, this term may formally be converted from the fourthmore » order hyper-diffusive term of the expansion. However, 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. Because 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.« less
  • From joint meeting of the American Nuclear Society and the Atomic Industrial Forum and Nuclear Energy Exhibition; San Francisco, California, USA (11 Nov 1973). See CONF-731101-.
  • Two previously derived approximations to the linear-linear nodal transport method, the linear-nodal (LN) and the linear-linear (LL) methods, are reexamined, together with a new approximation, the bilinear (BL) method, that takes into account the bilinear nodal flux moment. The three methods differ in the degree of analyticity retained in the final discrete variable equations; however, they all possess the very high accuracy characteristic of nodal methods. Unlike previous work, the final equations are manipulated and cast in the form of the classical weighted diamond-difference (WDD) equations (not just a WDD algorithm). This makes them simple to implement in a computermore » code, especially for those users who have experience with WDD algorithms. Other algorithms, such as the nodal algorithm, also can be used to solve the WDD-form equations. A computer program that solves two-dimensional transport problems using the LN, LL, or the BL method is used to solve three test problems. The results are used to confirm our algebraic manipulations of the nodal equations and also to compare the performance of the three methods from the computational, as well as the theoretical, point of view. The three methods are found to have comparable accuracies for the problems studies, especially on meshes that are sufficiently fine.« less