Application of the three-dimensional telegraph equation to cosmic-ray transport

Tautz, R. C.; Lerche, I.

doi:10.1088/1674-4527/16/10/162

Cited by 22 publications

(18 citation statements)

References 20 publications

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“…Moreover, if a physical system has an intrinsic anisotropy, the telegraph equation preserves these anisotropic properties. On the contrary, those features are smeared out in the diffusive solution (Litvinenko & Noble 2016;Tautz & Lerche 2016).…”

Section: Appendix A: Cr Diffusionmentioning

confidence: 98%

Cosmic-ray hydrodynamics: Alfvén-wave regulated transport of cosmic rays

Thomas

Pfrommer

2019

Monthly Notices of the Royal Astronomical Society

108

128

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Star formation in galaxies appears to be self-regulated by energetic feedback processes. Among the most promising agents of feedback are cosmic rays (CRs), the relativistic ion population of interstellar and intergalactic plasmas. In these environments, energetic CRs are virtually collisionless and interact via collective phenomena mediated by kinetic-scale plasma waves and large-scale magnetic fields. The enormous separation of kinetic and global astrophysical scales requires a hydrodynamic description. Here, we develop a new macroscopic theory for CR transport in the self-confinement picture, which includes CR diffusion and streaming. The interaction between CRs and electromagnetic fields of Alfvénic turbulence provides the main source of CR scattering, and causes CRs to stream along the magnetic field with the Alfvén velocity if resonant waves are sufficiently energetic. However, numerical simulations struggle to capture this effect with current transport formalisms and adopt regularization schemes to ensure numerical stability. We extent the theory by deriving an equation for the CR momentum density along the mean magnetic field and include a transport equation for the Alfvén-wave energy. We account for energy exchange of CRs and Alfvén waves via the gyroresonant instability and include other wave damping mechanisms. Using numerical simulations we demonstrate that our new theory enables stable, self-regulated CR transport. The theory is coupled to magneto-hydrodynamics, conserves the total energy and momentum, and correctly recovers previous macroscopic CR transport formalisms in the steady-state flux limit. Because it is free of tunable parameters, it holds the promise to provide predictable simulations of CR feedback in galaxy formation.1 The Legendre polynomials are eigenfunctions of the pitch-angle Laplace operator ∂ t f | scatt = ∂ µ [ν(1 − µ 2 )/2 ∂ µ f ]. This operator describes pitch-angle diffusion and ν denotes the scattering frequency. Note that this simple Laplacian resembles the actual scattering operator as discussed in equation (51).

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Section: Appendix A: Cr Diffusionmentioning

confidence: 98%

Cosmic-ray hydrodynamics: Alfvén-wave regulated transport of cosmic rays

Thomas

Pfrommer

2019

Monthly Notices of the Royal Astronomical Society

108

128

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“…For example, in the case of the transport of energetic charged particle in turbulent magnetic fields such as low-energy cosmic rays in the solar wind, the diffusion equation can not be used to describe the transport for early times because it leads to a non-zero probability density everywhere, which would correspond to an infinite propagation speed. Using the telegraph equation in this case we get a more realistic model for the early phase transport because it combines diffusion with a finite propagation speed (see [18]). Telegraph equations have also an extraordinary importance in electrodynamics (the scalar Maxwell equations are of this type), in the theory of damped vibrations, and in probability because they are connected with finite velocity random motions (see [14,16]).…”

Section: Introductionmentioning

confidence: 99%

First and Second Fundamental Solutions of the Time-Fractional Telegraph Equation with Laplace or Dirac Operators

Ferreira

Rodrigues

Vieira

2018

Adv. Appl. Clifford Algebras

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In this work we obtain the first and second fundamental solutions (FS) of the multidimensional timefractional equation with Laplace or Dirac operators, where the two time-fractional derivatives of orders α ∈]0, 1] and β ∈]1, 2] are in the Caputo sense. We obtain representations of the FS in terms of Hankel transform, double Mellin-Barnes integrals, and H-functions of two variables. As an application, the FS are used to solve Cauchy problems of Laplace and Dirac type.

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“…One difference from the Fickian case is that the maximum signal propagation speed (parallel to magnetic field lines) is reduced to (κ /τ) 1/2 . It has been claimed that the telegraph equation is not suitable for modelling CR propagation as it does not conserve the total number of particles or, equivalently, their total energy (Malkov & Sagdeev 2015); see also Tautz & Lerche (2016). However, this assertion seems to be based on misunderstanding.…”

Section: Cr Propagation and The Telegraph Equationmentioning

confidence: 99%

Fickian and non-Fickian diffusion of cosmic rays

Rodrigues

Snodin

Sarson

et al. 2019

Monthly Notices of the Royal Astronomical Society

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Fluid approximations to cosmic ray (CR) transport are often preferred to kinetic descriptions in studies of the dynamics of the interstellar medium (ISM) of galaxies, because they allow simpler analytical and numerical treatments. Magnetohydrodynamic simulations of the ISM usually incorporate CR dynamics as an advection–diffusion equation for CR energy density, with anisotropic, magnetic-field-aligned diffusion with the diffusive flux assumed to obey Fick’s law. We compare test particle and fluid simulations of CRs in a random magnetic field. We demonstrate that a non-Fickian prescription of CR diffusion, which corresponds to the telegraph equation for the CR energy density, can be easily calibrated to match the test particle simulations with great accuracy. In particular, we consider a random magnetic field in the fluid simulation that has a lower spatial resolution than that used in the particle simulation to demonstrate that an appropriate choice of the diffusion tensor can account effectively for the unresolved (subgrid) scales of the magnetic field. We show that the characteristic time that appears in the telegraph equation can be physically interpreted as the time required for the particles to reach a diffusive regime and we stress that the Fickian description of the CR fluid is unable to describe complex boundary or initial conditions for the CR energy flux.

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Application of the three-dimensional telegraph equation to cosmic-ray transport

Cited by 22 publications

References 20 publications

Cosmic-ray hydrodynamics: Alfvén-wave regulated transport of cosmic rays

Cosmic-ray hydrodynamics: Alfvén-wave regulated transport of cosmic rays

First and Second Fundamental Solutions of the Time-Fractional Telegraph Equation with Laplace or Dirac Operators

Fickian and non-Fickian diffusion of cosmic rays

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