Numerical Solution of the Time‐dependent Kinetic Equation for Anisotropic Pitch‐Angle Scattering

Lu, J. Y.; Zank, G. P.; Webb, G. M.

doi:10.1086/319722

Cited by 16 publications

(9 citation statements)

References 50 publications

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“…An effective numerical approach for solving is to separate the distribution function f to two parts, one being the unscattered part, F and the other describing the scattering effects f s and then solve using a polynomial expansion method [ Zank et al , 2000b]. Instead of entering the details of that paper, we are interested here in relating the results of Zank et al [2000b] [see also Lu et al , 2002] to single particle motion in the absence of scatterings. Equations (4.10) and (4.11) of Zank et al [2000b] describe the separation of f into F and f s with adiabatic cooling and focusing included.…”

Section: Particle Transportmentioning

confidence: 99%

Energetic particle acceleration and transport at coronal mass ejection–driven shocks

2003

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[1] Evidence now exists which suggests that in large solar energetic particle (SEP) events, particles are often accelerated to $ MeV energies (and perhaps up to GeV energies) at shock waves driven by coronal mass ejections (CMEs). These energetic particles are of considerable importance to space weather studies since they serve as a precursor signal for possible disruptive events at the Earth. As a CME-driven shock propagates, expands and weakens, particles accelerated diffusively at the shock can escape upstream and downstream into the interplanetary medium. The escaping energized particles propagate along the interplanetary magnetic field, experiencing only weak scattering from fluctuations in the interplanetary magnetic field (IMF). In this work, we study the timedependent transport of energetic particles accelerated at a propagating shock using a Monte-Carlo approach. This treatment, together with our previous work on particle acceleration at shocks, allows us to investigate the characteristics (intensity profiles, angular distribution, particle anisotropies) of high-energy particles arriving at various distances from the sun. Such an approach is both easy to implement and allows us to study the affect of interplanetary turbulence on particle transport in a systematic manner. These theoretical models form an excellent basis on which to interpret observations of highenergy particles made in situ at 1 AU by spacecraft such as ACE and WIND.

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Section: Particle Transportmentioning

confidence: 99%

Energetic particle acceleration and transport at coronal mass ejection–driven shocks

2003

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“…The influence of the solar wind is manifest through the presence of V sw in the second, third, and fourth terms. We have ignored in equation (5) the following: terms of OðvV sw =c 2 Þ and time-varying magnetic field (Kulsrud & Pearce 1969), the spiral Parker IMF and variable solar wind velocity (Skilling 1975;Ruffolo 1995;Isenberg 1997;Lu, Zank, & Webb 2001), and particle momentum transport associated with the tensor elements D lP and D PP (Schlickeiser 1989a(Schlickeiser , 1989b, which are smaller than the pitch-cosine diffusion coefficient D ll by OðV A =vÞ and O½ðV A =vÞ 2 , respectively (Appendix B). Equation (5) reduces to the focused transport equation (Kulsrud & Pearce 1969;Roelof 1969) for V sw ¼ 0.…”

Section: Energetic Particle Transportmentioning

confidence: 99%

Modeling Shock‐accelerated Solar Energetic Particles Coupled to Interplanetary Alfven Waves

Reames

Tylka

2003

ApJ

171

183

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We present an idealized model simulating the coupled evolution of the distributions of multispecies shockaccelerated energetic ions and interplanetary Alfvén waves in gradual solar energetic particle (SEP) events. Particle pitch-angle diffusion coefficients are expressed in terms of wave intensities, and wave growth rates in terms of momentum gradients of SEP distributions, by the same quasilinear theory augmented with resonance broadening. The model takes into consideration various physical processes: for SEPs, particle motion, magnetic focusing, scattering by Alfvén waves, solar wind convection, and adiabatic deceleration; for the waves, WKB transport and amplification by streaming SEPs. Shock acceleration is heuristically represented by continuous injection of prescribed spectra of SEPs at a moving shock front. We show the model predictions for two contrasting sets of SEP source spectra, fast weakening and softening in one case and long lasting and hard in the other. The results presented include concurrent time histories of multispecies SEP intensities and elemental abundance ratios, as well as sequential snapshots of the following: SEP intensity energy spectra, Alfvén wave spectra, particle mean free paths as functions of rigidity, and spatial profiles of SEP intensities and mean free paths. Wave growth plays a key role in both cases, although the magnitude of the wave growth differs greatly, and quite different SEP abundance variations are obtained. In these simulations, the maximum wave growth rate is large, but small relative to the wave frequency, and everywhere the total wave magnetic energy density remains small relative to that of the background magnetic field. The simulations show that, as the energetic protons stream outward, they rapidly amplify the ambient Alfvén waves, by several orders of magnitude in the inner heliosphere. Energetic minor ions find themselves traveling through resonant Alfvén waves previously amplified by higher velocity protons. The nonuniformly growing wave spectra alter the rigidity dependence of particle scattering, resulting in complex time variations of SEP abundances at large distances from the Sun. The greatly amplified waves travel outward in an expanding and weakening '' shell,'' creating an expanding and falling '' reservoir '' of SEPs with flat spatial intensity profiles behind, while in and beyond the shell the intensities drop steeply. The wave-particle resonance relation dynamically links the evolving characteristics of the SEP and Alfvén wave distributions in this new mode of SEP transport. We conclude that wave amplification, the counterpart to the scattering of streaming particles required by energy conservation, plays an essential role in the transport of SEPs in gradual SEP events. The steep proton-amplified wave spectra just upstream of the shock suggest that they may also be important in determining the elemental abundances of shock-accelerated SEP sources.

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“…The surprisingly effective quasinumerical approach has been used to model the transport of interstellar pickup ions. 42,43 …”

Section: Discussionmentioning

confidence: 99%

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

Litvinenko

Noble

2016

Physics of Plasmas

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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 sideby-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. Published by AIP Publishing. [http://dx

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Numerical Solution of the Time‐dependent Kinetic Equation for Anisotropic Pitch‐Angle Scattering

Cited by 16 publications

References 50 publications

Energetic particle acceleration and transport at coronal mass ejection–driven shocks

Energetic particle acceleration and transport at coronal mass ejection–driven shocks

Modeling Shock‐accelerated Solar Energetic Particles Coupled to Interplanetary Alfven Waves

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

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