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Fig. B.1. (Color online) Illustration of the rotation axis (blue), by which the z axis is rotated to be aligned with the background magnetic field vector B (red) as given by Eq. (B.1). This is necessary because the turbulent slab magnetic field obeys δB = δB(s) and δB ⊥ B, where s is the direction along the magnetic background field B.
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Context. Magnetized space plasmas such as the heliosphere contain a
spatially variable guide magnetic field that is superposed by
turbulence. Understanding the transport of energetic particles in such
an environment is important for many applications. Aims: The
modifications of the scattering mean free path caused by the effect of
adiabatic focusin...
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Observations from various interplanetary and other spacecraft missions evince that superthermal distributions are omnipresent in the solar wind and near Earth's plasma environment. These observations confirm the presence of coherent bipolar electric field pulses. In phase space, these electric field structures are observed as electron holes (EHs) o...
Citations
... Pitch-angle diffusion is caused by the magnetic fluctuations, and it is related to the parallel mean free path (Qin et al., 2004;Zhang et al., 2009). Magnetic focusing effect is important in the SEP transport (Tautz et al., 2012;Wang and Qin, 2016) and it is related with L B . Previous works often assumed a constant magnetic focusing length L B , which is defined as Roelof, 1969;Tautz et al., 2012). ...
... Magnetic focusing effect is important in the SEP transport (Tautz et al., 2012;Wang and Qin, 2016) and it is related with L B . Previous works often assumed a constant magnetic focusing length L B , which is defined as Roelof, 1969;Tautz et al., 2012). In a Parker spiral magnetic field, L B increases with heliocentric distance. ...
... We note, however, that thermodynamic arguments and the assumption of compressional turbulence may be sufficient to derive a similar spectral index (Fisk & Gloeckler, 2007b). Likewise, adiabatic cooling is a more efficient process than the interaction with turbulent electric fields (Zhang & Lee, 2011;Litvinenko & Schlickeiser, 2011;Tautz et al., 2012). In anisotropic helical turbulence, largescale electric fields may also give rise to particle acceleration (Fedorov & Stehlik, 2008). ...
Context: Typical space plasmas contain spatially and temporally variable
turbulent electromagnetic fields. Understanding the transport of energetic
particles and the acceleration mechanisms for charged particles is an important
goal of today's astroparticle physics. Aims: To understand the acceleration
mechanisms at the particle source, subsequent effects have to be known.
Therefore, the modification of a particle energy distribution, due to
stochastic acceleration, needs to be investigated. Methods: The diffusion in
momentum space was investigated by using both a Monte-Carlo simulation code and
by analytically solving the momentum-diffusion equation. For simplicity, the
turbulence was assumed to consist of one-dimensional Alfven waves. Results:
Using both methods, it is shown that, on average, all particles with velocities
comparable to the Alfven speeds are accelerated. This influences the energy
distribution by significantly increasing the energy spectral index.
Conclusions: Because of electromagnetic turbulence, a particle energy spectrum
measured at Earth can drastically deviate from its initial spectrum. However,
for particles with velocities significantly above the Alfven speed, the effect
becomes negligible.
... Notably, Artmann et al. (2011) employed a focused diffusion model to interpret the flare electron spectra obtained with the Wind spacecraft. Tautz et al. (2012) list several other applications of focused particle transport in astrophysics. derived the diffusion approximation taking into account adiabatic focusing (see also Earl 1981). ...
... Recently Tautz et al. (2012) simulated cosmic-ray transport with adiabatic focusing. They used a three-dimensional mean magnetic field and computed test-particle trajectories in a turbulent magnetic field. ...
... The computed mean free paths turned out to be much greater than the values predicted by . Tautz et al. (2012), however, did not describe the effects of coherent streaming and diffusion separately. By contrast, we used the diffusion approximation to interpret our numerical results, which enabled us to identify the separate effects of adiabatic focusing on both the parallel diffusion coefficient and the coherent speed. ...
Focused particle transport in a nonuniform large-scale magnetic field is investigated numerically in the case of isotropic pitch-angle scattering. Evolving particle density profiles and distribution moments are computed from solutions of a system of stochastic differential equations, equivalent to the original Fokker-Planck equation for the particle distribution. Conflicting analytical predictions for the transport coefficients in the diffusion limit, independently calculated by Beeck & Wibberenz and Shalchi, are compared with the numerical results. The reasons for the discrepancies among the analytical and numerical treatments, as well as the general limitations of the diffusion model, are discussed. The telegraph equation, derived in a higher-order expansion of the particle distribution function, is shown to describe the particle transport much more accurately than the diffusion model, especially ahead of a moving density pulse.
... whereê B0 andê δB are unit vectors denoting the directions of the mean and turbulent magnetic fields, respectively. While the background field is usually homogeneous (sometimes, special geometries are assumed such as the Parker spiral 19,20 or adiabatically focused fields 21,22 ), the spatial (and temporal, for example due to plasma waves 23 ) variation of the turbulent fields are considerably more complicated. Two basic onsets have been used over the years: First, the grid approach discretizes both the spatial and the wavenumber coordinates to enable a fast Fourier transform. ...
A modified method is presented to generate artificial magnetic turbulence
that is used for test-particle simulations. Such turbulent fields are obtained
from the superposition of a set of wave modes with random polarizations and
random directions of propagation. First, it is shown that the new method
simultaneously fulfils requirements of isotropy, equal mean amplitude and
variance for all field components, and vanishing divergence. Second, the number
of wave modes required for a stochastic particle behavior is investigated by
using a Lyapunov approach. For the special case of slab turbulence, it is shown
that already for 16 wave modes the particle behavior agrees with that shown for
considerably larger numbers of wave modes.
Understanding turbulent transport of charged particles in magnetized plasmas
often requires a model for the description of random variations in the
particle's pitch angle. The Fokker-Planck coefficient of pitch-angle
scattering, which is used to describe scattering parallel to the mean magnetic
field, is therefore of central importance. Whereas quasi-linear theory assumes
a homogeneous mean magnetic field, such a condition is often not fulfilled,
especially for high-energy particles. Here, a new derivation of the
quasi-linear approach is given that is based on the unperturbed orbit found for
an adiabatically focused mean magnetic field. The results show that, depending
on the ratio of the focusing length and the particle's Larmor radius, the
Fokker-Planck coefficient is significantly modified but agrees with the
classical expression in the limit of a homogeneous mean magnetic field.
The diffusion approximation to the Fokker-Planck equation is commonly used to model the transport of solar energetic particles in interplanetary space. In this study, we present exact analytical predictions of a higher order telegraph approximation for particle transport and compare them with the corresponding predictions of the diffusion approximation and numerical solutions of the full Fokker-Planck equation. We specifically investigate the role of the adiabatic focusing effect of a spatially varying magnetic field on an evolving particle distribution. Comparison of the analytical and numerical results shows that the telegraph approximation reproduces the particle intensity profiles much more accurately than does the diffusion approximation, especially when the focusing is strong. However, the telegraph approximation appears to offer no significant advantage over the diffusion approximation for calculating the particle anisotropy. The telegraph approximation can be a useful tool for describing both diffusive and wave-like aspects of the cosmic-ray transport.
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.
Appendix A is available in electronic form at http://www.aanda.org
