Turbulent Field Lines Research Articles

Variation in Path Lengths of Turbulent Magnetic Field Lines and Solar Energetic Particles

Modeling of time profiles of solar energetic particle (SEP) observations often considers transport along a large-scale magnetic field with a fixed path length from the source to the observer. Here, we point out that variability in the turbulent field line path length can affect the fits to SEP data and the inferred mean free path and injection profile. To explore such variability, we perform Monte Carlo simulations in representations of homogeneous 2D MHD + slab turbulence adapted to spherical geometry and trace trajectories of field lines and full particle orbits, considering proton injection from a narrow or wide angular region near the Sun, corresponding to an impulsive or gradual solar event, respectively. We analyze our simulation results in terms of field line and particle path length statistics for 1° × 1° pixels in heliolatitude and heliolongitude at 0.35 and 1 au from the Sun, for different values of the turbulence amplitude b/B 0 and turbulence geometry as expressed by the slab fraction f s . Maps of the most probable path lengths of field lines and particles at each pixel exhibit systematic patterns that reflect the fluctuation amplitudes experienced by the field lines, which in turn relate to the local topology of 2D turbulence. We describe the effects of such path length variations on SEP time profiles, both in terms of path length variability at specific locations and the motion of the observer with respect to turbulence topology during the course of the observations.

A detailed numerical study of field line random walk in magnetic turbulence

ABSTRACT A fundamental problem in space plasma physics and astrophysics is to understand the behaviour of magnetic field lines in turbulence. In the past it was controversial what aspects of turbulence are most important in field line random walk theory. In the current paper we employ numerical tools, commonly referred to as simulations, to gain more insight. In particular, we explore the importance of the energy range of the turbulence spectrum, spectral anisotropy, as well the existence of a component of the turbulent magnetic field parallel with respect to the mean magnetic field. The latter point is directly related to the question whether field line random walk in compressible turbulence behaves differently compared to random walk in incompressible turbulence.

Heuristic construction of field line random walk diffusion coefficients

To understand the statistics of magnetic field lines in turbulence is complicated. Previous descriptions have been based on either pure numerical treatments, often called simulations, or analytical theory. In the latter case one operates with nonlinear integral or differential equations which are difficult to solve. In order to achieve a complete analytical description, one needs to employ a set of approximations which are often difficult to justify. Although, previously developed numerical and analytical treatments of field line random walk agree with each other, they do not provide simple arguments for the analytical form of field line diffusion coefficients. It is the purpose of the current article to derive field line random walk coefficients entirely based on simple heuristic arguments. Special cases such as slab, two-dimensional, three-dimensional, and two-component turbulence are considered. Heuristically constructed diffusion parameters are compared with previously obtained analytical results. It is shown that the heuristic arguments provide the same scaling behavior as the formulas derived by employing systematic analytical theories.

Field line random walk in magnetic turbulence

The stochastic behavior of magnetic field lines in turbulence is explored analytically and numerically. This problem is a fundamental aspect of turbulence research but also highly relevant in the theory of energetic particles. In the current paper, previous approaches are reviewed and some simple heuristic arguments are provided helping the reader to understand the reason for the form of analytical results. The importance of the so-called Kubo number in field line random walk theory is also discussed. Furthermore, analytical results for a position-dependent field line diffusion coefficient are provided. For more realistic turbulence configurations, the field line diffusion coefficients are computed numerically. This includes quasi-slab, quasi-2D, two-component, and three-dimensional turbulence. Specific aspects of the field line random walk in each model are also discussed. Results based on a diffusion approximation are compared with numerical results obtained without employing this approximation with the aim to explore its validity and accuracy. Numerical results based on simulations for incompressible and compressible turbulence are also discussed.

Diffusion of stochastic magnetic field lines with average poloidal magnetic component

We study the diffusion of stochastic isotropic and anisotropic magnetic field lines in turbulence with average poloidal magnetic field component. In the current work we employ the so-called DeCorrelation Trajectory method in order to compute the running and asymptotic diffusion coefficients of stochastic magnetic field lines with average poloidal magnetic field component. A modification of the diffusion coefficients is caused by the average magnetic field component parameter in the presence of trajectory trapping. We calculate the field line diffusion coefficients Dxx and Dyy for different values of the magnetic Kubo number K, the average magnetic field component parameter Kav and the anisotropy parameter Λ. Furthermore we compare our new findings with those already obtained in papers dealing with the calculus of the diffusion coefficients for a stochastic sheared magnetic field.

Stochastic field-line wandering in magnetic turbulence with shear. II. Decorrelation trajectory method

We study the diffusion of magnetic field lines in turbulence with magnetic shear. In the first part of the series, we developed a quasi-linear theory for this type of scenario. In this article, we employ the so-called DeCorrelation Trajectory method in order to compute the diffusion coefficients of stochastic magnetic field lines. The magnetic field configuration used here contains fluctuating terms which are described by the dimensionless functions bi(X, Y, Z), i = (x, y) and they are assumed to be Gaussian processes and are perpendicular with respect to the main magnetic field B0. Furthermore, there is also a z-component of the magnetic field depending on radial coordinate x (representing the gradient of the magnetic field) and a poloidal average component. We calculate the diffusion coefficients for magnetic field lines for different values of the magnetic Kubo number K, the dimensionless inhomogeneous magnetic parallel and perpendicular Kubo numbers KB∥, KB⊥, as well as Kav=byavKB∥/KB⊥.

THREE-DIMENSIONAL SPONTANEOUS MAGNETIC RECONNECTION

ABSTRACT Magnetic reconnection is best known from observations of the Sun where it causes solar flares. Observations estimate the reconnection rate as a small, but non-negligible fraction of the Alfvén speed, so-called fast reconnection. Until recently, the prevailing pictures of reconnection were either of resistivity or plasma microscopic effects, which was contradictory to the observed rates. Alternative pictures were either of reconnection due to the stochasticity of magnetic field lines in turbulence or the tearing instability of the thin current sheet. In this paper we simulate long-term three-dimensional nonlinear evolution of a thin, planar current sheet subject to a fast oblique tearing instability using direct numerical simulations of resistive-viscous magnetohydrodynamics. The late-time evolution resembles generic turbulence with a −5/3 power spectrum and scale-dependent anisotropy, so we conclude that the tearing-driven reconnection becomes turbulent reconnection. The turbulence is local in scale, so microscopic diffusivity should not affect large-scale quantities. This is confirmed by convergence of the reconnection rate toward with increasing Lundquist number. In this spontaneous reconnection, with mean field and without driving, the dissipation rate per unit area also converges to , and the dimensionless constants 0.015 and 0.006 are governed only by self-driven nonlinear dynamics of the sheared magnetic field. Remarkably, this also means that a thin current sheet has a universal fluid resistance depending only on its length to width ratio and to .

NONLINEAR MULTISCALE SIMULATION OF TURBULENT FLUX TUBES

A new method for the full nonlinear computation of sets of turbulent field lines has recently been introduced that allows inclusion of the equivalent of more than four decades of turbulent scales with a fully three-dimensional distribution of wavevectors. The integration scheme is here detailed, which, through transformation of the set of differential equations into mappings, compounds the efficiency and accuracy of the method. The potential of the method is then demonstrated with multiscale simulations of magnetic flux tubes ranging over nearly four decades of length scales both along and across the background field. Magnetic flux tubes of various sizes are computed for a turbulence spectrum typical of the quiet slow solar wind near 1 AU. Implications of the simulation results for the transport of energetic particles, and in particular, for the interpretation of impulsive solar-energetic-particle and upstream ion-event observations are discussed.

STATISTICS OF FIELD-LINE DISPERSAL: RANDOM-WALK CHARACTERIZATION AND SUPRADIFFUSIVE REGIME

A study of the statistics of field-line dispersal recently showed that beyond its expected ballistic and diffusive regimes, the mean separation logarithm of simulated turbulent field lines displays supradiffusive behavior. The reason for this unexpected supradiffusion is investigated. The approach taken in this new study is to try to better characterize the statistics of the field-line separations in terms of random walk.

NONLINEAR SIMULATION OF TURBULENT FIELD LINES: DISPERSAL STATISTICS

A new method for the full nonlinear computation of sets of turbulent field lines is introduced that extends the sums of random numbers distribution method previously applied to the computation of individual field lines. With a multiscale variation of the phases consistent with in situ observations of intermittent solar wind (SW) turbulence, the new method allows inclusion of the equivalent of more than four decades of turbulent scales with a fully three-dimensional distribution of wavevectors. As a first application, pairs of magnetic field lines are computed in independent realizations of the turbulence, for spectra typical of the quiet slow SW near 1 AU. The statistics of field-line dispersal are then studied from the simulated pairs of magnetic field lines and compared to earlier theoretical predictions. It appears that while the earlier theoretical picture remains relatively accurate as long as the mean variation of separation logarithm {Lambda} is less than one, the qualitative picture is quickly altered as {Lambda} grows past one.

NONLINEAR SIMULATION OF TURBULENT FIELD LINES AND COMPARISON OF THEIR MEAN CROSS-FIELD DISPLACEMENT WITH THEORETICAL PREDICTION

A new method for the full nonlinear computation of turbulent field lines is presented that extends the Sums of Random Numbers Distribution method previously applied to the computation of generalized quasilinear (GQL) field lines. A study of the mean cross-field displacement confirms the theoretical prediction of a nonlinear diffusion regime, on the large scales where the mean cross-field displacement exceeds twice the perpendicular correlation length. The simulation results duplicate at all length scales the variations of the mean cross-field displacement predicted by the theoretical calculations, including the transitions from GQL supradiffusion to nonlinear diffusion. Both GQL and full nonlinear computations also reproduce, accurately, the predicted variations with turbulence level of the large-scale GQL and nonlinear diffusion coefficients, respectively.

Lengths of Wandering Magnetic Field Lines in the Turbulent Solar Wind

Charged energetic and suprathermal particles propagating in the solar wind (SW), as in most astrophysical environments, follow magnetic field-line irregularities down to very short scales of turbulence. Their minimal path length, or path length in the absence of scattering, is determined by the length of the traced magnetic field line resolved to those scales. The lengths of turbulent field lines are investigated here as functions of resolution scale and turbulence condition in the SW. In situ measurements on board Helios 2, turbulent field-line simulations, and theory all give excellent agreement for the dependence of the turbulent length on the resolution scale, as well as for the average field-line lengthening. Close to 1 AU, the length of a turbulent field segment is increased on average by close to 50%, with even longer field lines in some slow SW streams. While in fast SW most of the field-line lengthening is due to short-scale irregularities, wandering on a 0.1 AU scale can produce in slow SW as much lengthening as irregularities on shorter scales, explaining a strong stream-to-stream variability. In solar impulsive particle events, noticeable travel delays of the first arriving particles should result from the turbulent lengthening of the magnetic field lines, with significant variations of these travel delays from one event to another.

Distributions of Magnetic Field Orientations in the Turbulent Solar Wind

The Parker spiral model for interplanetary magnetic fields gives a first-order approximation to the magnetic field geometry in the solar wind (SW). How much real magnetic fields deviate from a Parker spiral in the turbulent SW, however, remains a broadly open question from the theory-modeling viewpoint. By generalizing to three-dimensional space a method, recently introduced, that allows for the simulation of realistic turbulent field lines with as many as 1015 turbulent modes, the distributions of the magnetic field orientations relative to the Parker spiral fields or mean fields are computed in quiet slow and fast SW at 0.3 and 1 AU. A comparison of these distributions with distributions determined from Helios 2 in situ data shows excellent agreement, further confirming the validity of the method. The angle distributions in three-dimensional space are relatively broad and peak away from 0°. The locations of these peaks, anywhere between 5° and 65° on timescales of 1.5 minutes to 10 hr in quiet wind within 1 AU from the Sun, are studied as a function of the SW conditions, length of the field segments, and realization of the turbulent field. Analytical predictions for the statistical distributions of field-line orientations are also made and compared to the numerical results.