skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Numerical Test of Analytical Theories for Perpendicular Diffusion in Small Kubo Number Turbulence

Abstract

In the literature, one can find various analytical theories for perpendicular diffusion of energetic particles interacting with magnetic turbulence. Besides quasi-linear theory, there are different versions of the nonlinear guiding center (NLGC) theory and the unified nonlinear transport (UNLT) theory. For turbulence with high Kubo numbers, such as two-dimensional turbulence or noisy reduced magnetohydrodynamic turbulence, the aforementioned nonlinear theories provide similar results. For slab and small Kubo number turbulence, however, this is not the case. In the current paper, we compare different linear and nonlinear theories with each other and test-particle simulations for a noisy slab model corresponding to small Kubo number turbulence. We show that UNLT theory agrees very well with all performed test-particle simulations. In the limit of long parallel mean free paths, the perpendicular mean free path approaches asymptotically the quasi-linear limit as predicted by the UNLT theory. For short parallel mean free paths we find a Rechester and Rosenbluth type of scaling as predicted by UNLT theory as well. The original NLGC theory disagrees with all performed simulations regardless what the parallel mean free path is. The random ballistic interpretation of the NLGC theory agrees much better with the simulations, but compared to UNLT theory themore » agreement is inferior. We conclude that for this type of small Kubo number turbulence, only the latter theory allows for an accurate description of perpendicular diffusion.« less

Authors:
;  [1]
  1. Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB R3T 2N2 (Canada)
Publication Date:
OSTI Identifier:
22663676
Resource Type:
Journal Article
Resource Relation:
Journal Name: Astrophysical Journal; Journal Volume: 839; Journal Issue: 2; Other Information: Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; COMPARATIVE EVALUATIONS; COSMIC RADIATION; DIFFUSION; MAGNETIC FIELDS; MAGNETOHYDRODYNAMICS; MEAN FREE PATH; NONLINEAR PROBLEMS; RANDOMNESS; SIMULATION; SLABS; TURBULENCE; TWO-DIMENSIONAL CALCULATIONS

Citation Formats

Heusen, M., and Shalchi, A., E-mail: husseinm@myumanitoba.ca, E-mail: andreasm4@yahoo.com. Numerical Test of Analytical Theories for Perpendicular Diffusion in Small Kubo Number Turbulence. United States: N. p., 2017. Web. doi:10.3847/1538-4357/AA6A55.
Heusen, M., & Shalchi, A., E-mail: husseinm@myumanitoba.ca, E-mail: andreasm4@yahoo.com. Numerical Test of Analytical Theories for Perpendicular Diffusion in Small Kubo Number Turbulence. United States. doi:10.3847/1538-4357/AA6A55.
Heusen, M., and Shalchi, A., E-mail: husseinm@myumanitoba.ca, E-mail: andreasm4@yahoo.com. Thu . "Numerical Test of Analytical Theories for Perpendicular Diffusion in Small Kubo Number Turbulence". United States. doi:10.3847/1538-4357/AA6A55.
@article{osti_22663676,
title = {Numerical Test of Analytical Theories for Perpendicular Diffusion in Small Kubo Number Turbulence},
author = {Heusen, M. and Shalchi, A., E-mail: husseinm@myumanitoba.ca, E-mail: andreasm4@yahoo.com},
abstractNote = {In the literature, one can find various analytical theories for perpendicular diffusion of energetic particles interacting with magnetic turbulence. Besides quasi-linear theory, there are different versions of the nonlinear guiding center (NLGC) theory and the unified nonlinear transport (UNLT) theory. For turbulence with high Kubo numbers, such as two-dimensional turbulence or noisy reduced magnetohydrodynamic turbulence, the aforementioned nonlinear theories provide similar results. For slab and small Kubo number turbulence, however, this is not the case. In the current paper, we compare different linear and nonlinear theories with each other and test-particle simulations for a noisy slab model corresponding to small Kubo number turbulence. We show that UNLT theory agrees very well with all performed test-particle simulations. In the limit of long parallel mean free paths, the perpendicular mean free path approaches asymptotically the quasi-linear limit as predicted by the UNLT theory. For short parallel mean free paths we find a Rechester and Rosenbluth type of scaling as predicted by UNLT theory as well. The original NLGC theory disagrees with all performed simulations regardless what the parallel mean free path is. The random ballistic interpretation of the NLGC theory agrees much better with the simulations, but compared to UNLT theory the agreement is inferior. We conclude that for this type of small Kubo number turbulence, only the latter theory allows for an accurate description of perpendicular diffusion.},
doi = {10.3847/1538-4357/AA6A55},
journal = {Astrophysical Journal},
number = 2,
volume = 839,
place = {United States},
year = {Thu Apr 20 00:00:00 EDT 2017},
month = {Thu Apr 20 00:00:00 EDT 2017}
}
  • The magnetic field line diffusion coefficients D{sub x} and D{sub y} are obtained by numerical simulations in the case that all the magnetic turbulence correlation lengths l{sub x}, l{sub y}, and l{sub z} are different. We find that the variety of numerical results can be organized in terms of the Kubo number, the definition of which is extended from R=({delta}B/B{sub 0})(l{sub {parallel}}/l{sub {perpendicular}}) to R=({delta}B/B{sub 0})(l{sub z}/l{sub x}), for l{sub x}{ge}l{sub y}. Here, l{sub {parallel}} (l{sub {perpendicular}}) is the correlation length along (perpendicular to) the average field B{sub 0}=B{sub 0}{cflx e}{sub z}. We have anomalous, non-Gaussian transport for R{approx_lt}0.1, inmore » which case the mean square deviation scales nonlinearly with time. For R{approx_gt}1 we have several Gaussian regimes: an almost quasilinear regime for 0.1{approx_lt}R{approx_lt}1, an intermediate, transition regime for 1{approx_lt}R{approx_lt}10, and a percolative regime for R{approx_gt}10. An analytical form of the diffusion coefficient is proposed, D{sub i}=D({delta}Bl{sub z}/B{sub 0}l{sub x}){sup {mu}}(l{sub i}/l{sub x}){sup {nu}}l{sub x}{sup 2}/l{sub z}, which well describes the numerical simulation results in the quasilinear, intermediate, and percolative regimes.« less
  • Motion of charged particles in a collisional plasma with stochastic magnetic field lines is investigated on the basis of the so-called A-Langevin equation. Compared to the previously used V-Langevin model, here finite Larmor radius effects are taken into account. The A-Langevin equation is solved under the assumption that the Lagrangian correlation function for the magnetic field fluctuations is related to the Eulerian correlation function (in Gaussian form) via the Corrsin approximation. The latter is justified for small Kubo numbers. The velocity correlation function, being averaged with respect to the stochastic variables including collisions, leads to an implicit differential equation formore » the mean square displacement. From the latter, different transport regimes, including the well-known Rechester-Rosenbluth diffusion coefficient, are derived. Finite Larmor radius contributions show a decrease of the diffusion coefficient compared to the guiding center limit. The case of small (or vanishing) mean fields is also discussed.« less
  • We explore perpendicular diffusion based on the unified nonlinear transport theory. We derive simple analytical forms for the perpendicular mean free path and investigate the influence of different model spectra. We show that for cases where the field line random walk is normal diffusive, the perpendicular diffusion coefficient consists of only two transport regimes. Details of the spectral shape are less important, especially those of the inertial range. Only the macroscopic properties of the turbulence spectrum control the perpendicular diffusion coefficient. Simple formulae for the perpendicular diffusion coefficient are derived which can easily be implemented in solar modulation or shockmore » acceleration codes.« less
  • We explore perpendicular diffusion based on the unified nonlinear transport theory. In Paper I, we focused on magnetostatic turbulence, whereas in the present article we include dynamical turbulence effects. For simplicity, we assume a constant correlation time. We show that there is now a nonvanishing contribution of the slab modes. We explore the parameter regimes in which the turbulence dynamics becomes important for perpendicular diffusion. Analytical forms for the perpendicular diffusion coefficient are derived, which can be implemented easily in solar modulation or shock acceleration codes.
  • In several astrophysical applications one needs analytical forms of cosmic-ray diffusion parameters. Some examples are studies of diffusive shock acceleration and solar modulation. In the current article we explore perpendicular diffusion based on the unified nonlinear transport theory. While we focused on magnetostatic turbulence in Paper I, we included the effect of dynamical turbulence in Paper II of the series. In the latter paper we assumed that the temporal correlation time does not depend on the wavenumber. More realistic models have been proposed in the past, such as the so-called damping model of dynamical turbulence. In the present paper wemore » derive analytical forms for the perpendicular diffusion coefficient of energetic particles in two-component turbulence for this type of time-dependent turbulence. We present new formulas for the perpendicular diffusion coefficient and we derive a condition for which the magnetostatic result is recovered.« less