The effect of adiabatic focusing upon charged-particle propagation in random magnetic fields

Earl, J. A.

doi:10.1086/154346

Cited by 152 publications

(86 citation statements)

References 15 publications

(16 reference statements)

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“…An important physical effect due to spatial variations of the mean magnetic field is adiabatic focusing that causes coherent particle streaming along the mean field (Earl 1976;Kunstmann 1979). Schlickeiser & Shalchi (2008) gave a systematic derivation of the diffusion approximation in a weak focusing limit, when the adiabatic focusing length is much greater than the turbulent scattering length of cosmic-ray particles.…”

Section: Discussionmentioning

confidence: 99%

“…The purpose of this paper is to generalise the calculations of Earl (1974Earl ( , 1976 and Beeck & Wibberenz (1986) and derive the telegraph equation for the particle density in the weak focusing limit of Schlickeiser & Shalchi (2008). The resulting equation is valid for an anisotropic scattering rate, complementing A&A 554, A59 (2013) the derivation given by Litvinenko & Noble (2013).…”

Section: Introductionmentioning

confidence: 98%

“…Roelof (1969), Earl (1976), and Kunstmann (1979) identified the effect of adiabatic focusing on particle transport in spatially varying magnetic fields. Focused transport equations have been used extensively to model the propagation of energetic particles in interplanetary space following large solar flares (e.g., Bieber et al 2002;Sáiz et al 2008;Dröge et al 2010).…”

Section: Introductionmentioning

confidence: 99%

“…Alternative derivations of the telegraph equation are available in the limit of a uniform magnetic field (Earl 1974;Gombosi et al 1993; for a detailed discussion of the alternative derivations, see Schwadron & Gombosi 1994). In the case of focused transport, however, the complication arises that the telegraph equation describes the coefficient of an eigenfunction expansion rather than the particle density that is the physical quantity of interest (Earl 1976). Litvinenko & Noble (2013) used a new technique to obtain the telegraph equation for the particle density in the case of an arbitrary focusing strength, but their derivation is only valid under the assumption of isotropic pitch-angle scattering.…”

Section: Introductionmentioning

confidence: 99%

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The telegraph equation for cosmic-ray transport with weak adiabatic focusing

Litvinenko

Schlickeiser

2013

A&A

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Time-dependent solutions of a spatial diffusion equation are often used to describe the transport of solar energetic particles, accelerated in large solar flares. Approximate analytical solutions of the diffusion approximation can complement and guide detailed numerical solutions of the Fokker-Planck equation for the particle distribution function. The accuracy of the diffusion approximation is limited, however, because the signal propagation speed is infinite in the diffusion limit. An improved description of cosmic-ray transport is provided by the telegraph equation, characterised by a finite signal propagation speed. We derive the telegraph equation for the particle density, taking into account adiabatic focusing in a large-scale interplanetary magnetic field in a weak focusing limit. As an illustration, we calculate a propagating pulse solution of the telegraph equation, determine the rise time when the maximum particle intensity is reached at a given distance from the Sun, and compare the results with those obtained in the diffusion approximation. In comparison with the diffusion equation, the telegraph equation predicts an asymmetrical shape of the pulse and a shorter rise time. These potentially significant differences suggest that the more accurate telegraph equation should be used in analysis of the solar energetic particle data, at least to quantify the accuracy of the focused diffusion model.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The telegraph equation for cosmic-ray transport with weak adiabatic focusing

Litvinenko

Schlickeiser

2013

A&A

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“…The time-dependent one-dimensional diffusion equation including the effect of adiabatic focusing reads as (Earl 1976…”

Section: The Focused Diffusion Equationmentioning

confidence: 99%

A diffusive description of the focused transport of solar energetic particles

Artmann

Schlickeiser

Agueda

et al. 2011

A&A

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The transport of solar energetic charged particles along the interplanetary magnetic field in the ecliptic plane of the sun can be described roughly by a one-dimensional diffusion equation. Large-scale spatial variations of the guide magnetic field can be taken into account by adding an additional term to the diffusion equation that includes the effect of adiabatic focusing. We solve this equation analytically by assuming a point-like particle injection in time and space and a spatial power-law dependence for the focusing length and the spatial diffusion coefficient. We infer the intensity-and anisotropy-time profiles of solar energetic particles from this solution. Through these the influence of different assumptions for the diffusion parameters can be seen in a mathematically closed form. The comparison of calculated and measured intensity-and anisotropy-time profiles, which are a powerful diagnostic tool for interplanetary particle transport, gives information about the large-scale spatial dependence of the focusing length and the diffusion coefficient. For an exceptionally large solar energetic particle event, which did occur on 2001 April 15, we fit the 27−512 keV electron intensities and anisotropies observed by the Wind spacecraft using the theoretically derived profiles. We find a linear spatial dependence of the mean free path along the guiding magnetic field. We also find the mean free path to be energy independent, which supports the theory of "velocity-dependent diffusion". This means that the intensity profiles for the discussed energies exhibit the same shape if they are plotted against the traveled distance and not against the time. In this case the profiles differ only in their maximum values and we can determine the energy spectra of the solar flare electrons out of the scaling factor we need to fit the data. The derived spectra exhibits a power-law dependence ∝E −3 kin in an energy range from ∼50 keV to ∼500 keV.

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Wide longitudinal distribution of interplanetary electrons following the 7 February 2010 solar event: Observations and transport modeling

et al. 2014

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We analyze 65–105 keV electrons in the 7 February 2010 solar electron event observed simultaneously by STEREO‐A, STEREO‐B, and ACE. A method to reconstruct the full‐electron pitch angle distributions from the four Solar Electron and Proton Telescope sensors on STEREO‐A/B and the Solar Electron and Proton Telescope instrument on ACE in the energy range of approximately 60–300 keV for periods of incomplete angular coverage is presented. A transport modeling based on numerical solutions of a three‐dimensional particle propagation model which includes pitch angle scattering and focused transport is applied to the intensity and anisotropy profiles measured on all three spacecraft. Based on an analysis of intensity gradients observed between the three spacecraft, we find that the lateral transport of the electrons occurs partially close to the Sun, due to effects of nonradial divergence of magnetic field lines or particle diffusion, and partially in the interplanetary medium. For the mean free paths characterizing the electron diffusion parallel and perpendicular to the interplanetary magnetic field, we derive values of λ∥∼ 0.1 AU and λ⟂∼ 0.01 AU. In comparison with results from other particle events which we had previously analyzed in a similar manner we discuss whether the diffusion mean free paths parallel and perpendicular to the average magnetic field might be related with each other, and whether the particle transport perpendicular to the average magnetic field is more likely due to particles following meandering magnetic field lines, or due to particles being scattered off individual field lines.

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The effect of adiabatic focusing upon charged-particle propagation in random magnetic fields

Cited by 152 publications

References 15 publications

The telegraph equation for cosmic-ray transport with weak adiabatic focusing

The telegraph equation for cosmic-ray transport with weak adiabatic focusing

A diffusive description of the focused transport of solar energetic particles

Wide longitudinal distribution of interplanetary electrons following the 7 February 2010 solar event: Observations and transport modeling

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