Magnetic Field Line Diffusion at the Onset of Stochasticity

1987 ◽  
Vol 42 (10) ◽  
pp. 1181-1192
Author(s):  
K. Elsässer ◽  
P. Deeskow
Keyword(s):  
Magnetic Field ◽  
Diffusion Coefficient ◽  
Field Line ◽  
Magnetic Field Line ◽  
Time Integration ◽  
Kinetic Equations ◽  
Theoretical Formula ◽  
Stochasticity Threshold ◽  
Field Lines ◽  
Line Diffusion

The Hamiltonian equations of a particle in a random set of waves just above the stochasticity threshold are considered both theoretically and numerically. First we derive the diffusion coefficient and the autocorrelation time perturbatively without using the thermodynamic limit, and we discuss the relevance of the H am iltonian problem for particle acceleration and magnetic field line flow. Then we integrate the equations for an ensemble of magnetic field lines numerically for a model problem [15] and show the time evolution of moments and correlations. Twice above the threshold we observe diffusive behaviour from the beginning, but the diffusion coefficient includes also the non-resonant modes. Just at threshold we find first a short phase o f free acceleration, later a diffusion which is slower than predicted by the theoretical formula. The best way to analyze the problem is in terms of cumulants, but a reliable com parison with any theory requires also a time integration of the corresponding kinetic equations.

scholarly journals A Monte Carlo simulation of magnetic field line tracing in the solar wind

2001 ◽  
Vol 8 (3) ◽  
pp. 151-158 ◽  
Author(s):  
P. Pommois ◽  
G. Zimbardo ◽  
P. Veltri
Keyword(s):  
Magnetic Field ◽  
Monte Carlo Simulation ◽  
Solar Wind ◽  
Monte Carlo ◽  
Diffusion Coefficient ◽  
Field Line ◽  
Magnetic Field Line ◽  
Magnetic Turbulence ◽  
Field Lines ◽  
The Magnetic Field

Abstract. It is well known that the structure of magnetic field lines in solar wind can be influenced by the presence of the magnetohydrodynamic turbulence. We have developed a Monte Carlo simulation which traces the magnetic field lines in the heliosphere, including the effects of magnetic turbulence. These effects are modelled by random operators which are proportional to the square root of the magnetic field line diffusion coefficient. The modelling of the random terms is explained, in detail, in the case of numerical integration by discrete steps. Furthermore, a proper evaluation of the diffusion coefficient is obtained by a numerical simulation of transport in anisotropic magnetic turbulence. The scaling of the fluctuation level and of the correlation lengths with the distance from the Sun are also taken into account. As a consequence, plasma transport across the average magnetic field direction is obtained. An application to the propagation of energetic particles from corotating interacting regions to high heliographic latitudes is considered.

scholarly journals The development of magnetic field line wander by plasma turbulence

2017 ◽  
Vol 83 (4) ◽  
Author(s):  
Gregory G. Howes ◽  
Sofiane Bourouaine
Keyword(s):  
Magnetic Field ◽  
Field Line ◽  
Magnetic Field Line ◽  
Large Scale ◽  
Magnetic Energy ◽  
Plasma Turbulence ◽  
Alfven Wave ◽  
Astrophysical Plasmas ◽  
Field Lines ◽  
The Magnetic Field

Plasma turbulence occurs ubiquitously in space and astrophysical plasmas, mediating the nonlinear transfer of energy from large-scale electromagnetic fields and plasma flows to small scales at which the energy may be ultimately converted to plasma heat. But plasma turbulence also generically leads to a tangling of the magnetic field that threads through the plasma. The resulting wander of the magnetic field lines may significantly impact a number of important physical processes, including the propagation of cosmic rays and energetic particles, confinement in magnetic fusion devices and the fundamental processes of turbulence, magnetic reconnection and particle acceleration. The various potential impacts of magnetic field line wander are reviewed in detail, and a number of important theoretical considerations are identified that may influence the development and saturation of magnetic field line wander in astrophysical plasma turbulence. The results of nonlinear gyrokinetic simulations of kinetic Alfvén wave turbulence of sub-ion length scales are evaluated to understand the development and saturation of the turbulent magnetic energy spectrum and of the magnetic field line wander. It is found that turbulent space and astrophysical plasmas are generally expected to contain a stochastic magnetic field due to the tangling of the field by strong plasma turbulence. Future work will explore how the saturated magnetic field line wander varies as a function of the amplitude of the plasma turbulence and the ratio of the thermal to magnetic pressure, known as the plasma beta.

Fast magnetic field-line reconnexion in a compressible fluid. Part 1. Coplanar field lines

1982 ◽  
Vol 28 (2) ◽  
pp. 335-367 ◽  
Author(s):  
A. M. Soward ◽  
E. R. Priest
Keyword(s):  
Magnetic Field ◽  
Field Line ◽  
Magnetic Field Line ◽  
Compressible Fluid ◽  
Similarity Solution ◽  
Gas Pressure ◽  
Local Similarity ◽  
Mathematical Foundation ◽  
Previous Suggestion ◽  
Field Lines

The Petschek model for incompressible reconnexion has been put on a firm mathematical foundation in an earlier paper by Soward & Priest, who discovered a ‘local’ similarity solution for the process. The present paper extends that analysis to compressible reconnexion, in which the previous Alfvén waves are replaced by slow magneto-acoustic shocks of switch-off type. By contrast with a previous suggestion, it is found unnecessary to include intermediate waves standing ahead of the slow shocks. The maximum reconnexion rate is typically half of Petschek's stated value, though faster rates are achieved when the external gas pressure is sufficiently low.

Topological analysis of a perturbed MHD equilibrium using magnetic field-line invariants

1997 ◽  
Vol 58 (3) ◽  
pp. 553-569 ◽  
Author(s):  
JASON W. BATES ◽  
H. RALPH LEWIS
Keyword(s):  
Magnetic Field ◽  
Field Line ◽  
Magnetic Field Line ◽  
Order Term ◽  
Topological Analysis ◽  
Small Perturbations ◽  
Level Curves ◽  
Mhd Equilibrium ◽  
Field Lines ◽  
Order Invariant

A procedure has previously been developed for the iterative construction of invariants associated with magnetic field-line Hamiltonians that consist of an axisymmetric zeroth-order term plus a non-axisymmetric perturbation. Approximate field-line invariants obtained with this procedure are used to examine the topological properties of magnetic field lines in a parabolic-current MHD equilibrium that was slightly perturbed from axisymmetry in the limit of a periodic cylindrical configuration. Excellent agreement between Poincaré maps and the level curves of the first-order invariant is found for small perturbations. A means of circumventing the ‘small-divisor problem’ in some cases is identified and implemented with outstanding results. These results indicate that this perturbation method can have valuable consequences for future investigations of magnetic field-line topology.

scholarly journals Uncertainties in field-line tracing in the magnetosphere. Part I: the axisymmetric part of the internal geomagnetic field

1997 ◽  
Vol 15 (2) ◽  
pp. 165-180 ◽  
Author(s):  
D. M. Willis ◽  
J. Robin Singh ◽  
J. Comer
Keyword(s):  
Magnetic Field ◽  
Field Line ◽  
Magnetic Field Line ◽  
Spherical Harmonic ◽  
Geomagnetic Field ◽  
Frame Of Reference ◽  
Standard Errors ◽  
Analytic Equation ◽  
Axisymmetric Part ◽  
Field Lines

Abstract. The technique of tracing along magnetic field lines is widely used in magnetospheric physics to provide a "magnetic frame of reference'' that facilitates both the planning of experiments and the interpretation of observations. The precision of any such magnetic frame of reference depends critically on the accurate representation of the various sources of magnetic field in the magnetosphere. In order to consider this important problem systematically, a study is initiated to estimate first the uncertainties in magnetic-field-line tracing in the magnetosphere that arise solely from the published (standard) errors in the specification of the geomagnetic field of internal origin. Because of the complexity in computing these uncertainties for the complete geomagnetic field of internal origin, attention is focused in this preliminary paper on the uncertainties in magnetic-field-line tracing that result from the standard errors in just the axisymmetric part of the internal geomagnetic field. An exact analytic equation exists for the magnetic field lines of an arbitrary linear combination of axisymmetric multipoles. This equation is used to derive numerical estimates of the uncertainties in magnetic-field-line tracing that are due to the published standard errors in the axisymmetric spherical harmonic coefficients (i.e. gn0 ± δgn0). Numerical results determined from the analytic equation are compared with computational results based on stepwise numerical integration along magnetic field lines. Excellent agreement is obtained between the analytical and computational methods in the axisymmetric case, which provides great confidence in the accuracy of the computer program used for stepwise numerical integration along magnetic field lines. This computer program is then used in the following paper to estimate the uncertainties in magnetic-field-line tracing in the magnetosphere that arise from the published standard errors in the full set of spherical harmonic coefficients, which define the complete (non-axisymmetric) geomagnetic field of internal origin. Numerical estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, calculated here for the axisymmetric part of the internal geomagnetic field, should be regarded as "first approximations'' in the sense that such estimates are only as accurate as the published standard errors in the set of axisymmetric spherical harmonic coefficients. However, all procedures developed in this preliminary paper can be applied to the derivation of more realistic estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, following further progress in the determination of more accurate standard errors in the spherical harmonic coefficients.

Diffusion coefficient and Kolmogorov entropy of magnetic field lines

1984 ◽  
Vol 32 (1) ◽  
pp. 141-158 ◽  
Author(s):  
Gaetano Zimbardo ◽  
Pierluigi Veltri ◽  
Francesco Malara
Keyword(s):  
Magnetic Field ◽  
Diffusion Coefficient ◽  
Random Walk ◽  
Diffusion Equation ◽  
Field Line ◽  
Statistical Techniques ◽  
Kolmogorov Entropy ◽  
Closed Set ◽  
Field Lines ◽  
Single Field

A diffusion equation for magnetic field lines of force in a turbulent magnetic field, which describes both the random walk of a single field line and how two nearby lines separate from each other, has been obtained using standard statistical techniques. Starting from such an equation, a closed set of equations for the moments may be obtained, in general, with suitable assumptions. From such a set of equations the Kolmogorov entropy may be explicitly calculated. The results have been applied to the most interesting examples of magnetic field geometries.

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