Monte Carlo simulations of intensity profiles for energetic particle propagation

Abstract

Aims. Numerical test-particle simulations are a reliable and frequently used tool for testing analytical transport theories and predicting mean-free paths. The comparison between solutions of the diffusion equation and the particle flux is used to critically judge the applicability of diffusion to the stochastic transport of energetic particles in magnetized turbulence. Methods. A Monte Carlo simulation code is extended to allow for the generation of intensity profiles and anisotropy-time profiles. Because of the relatively low number density of computational particles, a kernel function has to be used to describe the spatial extent of each particle. Results. The obtained intensity profiles are interpreted as solutions of the diffusion equation by inserting the diffusion coefficients that have been directly determined from the mean-square displacements. The comparison shows that the time dependence of the diffusion coefficients needs to be considered, in particular the initial ballistic phase and the often subdiffusive perpendicular coefficient. Conclusions. It is argued that the perpendicular component of the distribution function is essential if agreement between the diffusion solution and the simulated flux is to be obtained. In addition, time-dependent diffusion can provide a better description than the classic diffusion equation only after the initial ballistic phase.

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

“…For comparison purposes, either measurements by spacecraft such as STEREO and ACE or numerical simulation can be used. For instance, test-particle simulations calculate in detail the scattering of an ensemble of particles by a turbulent magnetic field, which allows one to register the number of particles arriving at a given position (Tautz et al 2016). A detailed description of this Monte-Carlo simulation technique can be found elsewhere (e.g., Tautz 2010;Tautz & Dosch 2013, and references therein).…”

Ergodicity of perpendicular cosmic ray transport

“…which can be obtained by recording the number (density) of particles in the form of an intensity profile 2 (Tautz et al, 2016).…”

“…This problem is of interest both for theoretical investigations as well as for the analysis of data for instance taken in situ by spacecraft. A non-diffusive behavior results in time-dependent ("running") diffusion coefficients, in which case the solution of the diffusion equation becomes considerably more involved and may even be replaced by a fractional differential equation (e. g., Metzler & Klafter, 2004;Tautz et al, 2016, and references therein), In addition, non-ergodic behavior emphasizes the individuality of the particles, thus requiring more care when conclusions are drawn based on a limited data set.…”

“…Recently, the role of subdiffusive perpendicular diffusion (e. g., Tautz & Shalchi 2010, but cf. Qin et al 2002Xu & Yan 2013) was emphasized in the extraction of diffusion coefficients from intensity profiles (Tautz et al, 2016). Generally, the necessity of anisotropic diffusion is increasingly recognized in the community (e. g., Kissmann, 2014;Girichidis et al, 2014).…”

“…Note that, unlike in previous work(Tautz et al, 2016;Ablaßmayer et al, 2016), here no kernel function is used. The reasons are that: (i) near the z axis, a sufficiently large number of particles will be registered at all times; (ii) potential inaccuracies due to the specific choice of a kernel will be avoided altogether.…”