Random Walk Of Magnetic Field Lines Research Articles

Magnetic field line random walk and solar energetic particle path lengths

Context. In 2020 May-June, six solar energetic ion events were observed by the Parker Solar Probe/IS⊙IS instrument suite at ≈0.35 AU from the Sun. From standard velocity-dispersion analysis, the apparent ion path length is ≈0.625 AU at the onset of each event. Aims. We develop a formalism for estimating the path length of random-walking magnetic field lines to explain why the apparent ion path length at an event onset greatly exceeds the radial distance from the Sun for these events. Methods. We developed analytical estimates of the average increase in path length of random-walking magnetic field lines, relative to the unperturbed mean field. Monte Carlo simulations of field line and particle trajectories in a model of solar wind turbulence were used to validate the formalism and study the path lengths of particle guiding-center and full-orbital trajectories. The formalism was implemented in a global solar wind model, and the results are compared with ion path lengths inferred from IS⊙IS observations. Results. Both a simple estimate and a rigorous theoretical formulation are obtained for field-lines’ path length increase as a function of path length along the large-scale field. From simulated field line and particle trajectories, we find that particle guiding centers can have path lengths somewhat shorter than the average field line path length, while particle orbits can have substantially longer path lengths due to their gyromotion with a nonzero effective pitch angle. Conclusions. The long apparent path length during these solar energetic ion events can be explained by (1) a magnetic field line path length increase due to the field line random walk and (2) particle transport about the guiding center with a nonzero effective pitch angle due to pitch angle scattering. Our formalism for computing the magnetic field line path length, accounting for turbulent fluctuations, may be useful for application to solar particle transport in general.

Magnetic Field Line Random Walk and Solar Energetic Particle Path Lengths: Stochastic Theory and PSP/ISoIS Observation

We thank Junxiang Hu for useful discussions. This research is partially supported by the Parker Solar Probe mission and the IS IS project (contract NNN06AA01C) and a subcontract to University of Delaware from Princeton University (SUB0000165). Additional support is acknowledged from the NASA Living With a Star (LWS) program (NNX17AB79G) and HSR program (80NSSC18K1210 & 80NSSC18K1648) and Thailand Science Research and Innovation (RTA6280002). The IS IS data and visualization tools are available to the community at https://spacephysics.princeton.edu/missions-instruments/isois; data are also available via the NASA Space Physics Data Facility. PSP was designed, built, and is now operated by the Johns Hopkins Applied Physics Laboratory as part of NASA’s LWS program (contract NNN06AA01C). Support from the LWS management and technical team has played a critical role in the success of the PSP mission.

Effects of the Mean Field Gradients on Magnetic Field Line Random Walk

Abstract The 2D model of the field line random walk (FLRW) is developed by considering a space-dependent mean magnetic field with perpendicular and parallel gradients, and a component in the perpendicular plane The impact of the configuration of the mean field on FLRW is explored. We have found that both the diffusion (random walk) and the convection (ordered walk) are significantly modified. The diffusion is strongly influenced by the parallel gradient and by , while the perpendicular gradient generates a flow of the field lines along its direction. A synergistic effect between the three elements of the configuration of the mean field is found.

Magnetic field line random walk in two-dimensional dynamical turbulence

The field line random walk (FLRW) of magnetic turbulence is one of the important topics in plasma physics and astrophysics. In this article, by using the field line tracing method, the mean square displacement (MSD) of FLRW is calculated on all possible length scales for pure two-dimensional turbulence with the damping dynamical model. We demonstrate that in order to describe FLRW with the damping dynamical model, a new dimensionless quantity R is needed to be introduced. On different length scales, dimensionless MSD shows different relationships with the dimensionless quantity R. Although the temporal effect affects the MSD of FLRW and even changes regimes of FLRW, it does not affect the relationship between the dimensionless MSD and dimensionless quantity R on all possible length scales.

Coronal Heating Topology: The Interplay of Current Sheets and Magnetic Field Lines

Abstract The magnetic topology and field line random walk (FLRW) properties of a nanoflare-heated and magnetically confined corona are investigated in the reduced magnetohydrodynamic regime. Field lines originating from current sheets form coherent structures, called current sheet connected (CSC) regions, which extend around them. CSC FLRW is strongly anisotropic, with preferential diffusion along the current sheets’ in-plane length. CSC FLRW properties remain similar to those of the entire ensemble but exhibit enhanced mean square displacements and separations due to the stronger magnetic field intensities in CSC regions. The implications for particle acceleration and heat transport in the solar corona and wind, and for solar moss formation are discussed.

EVOLUTION OF THE MAGNETIC FIELD LINE DIFFUSION COEFFICIENT AND NON-GAUSSIAN STATISTICS

ABSTRACT The magnetic field line random walk (FLRW) plays an important role in the transport of energy and particles in turbulent plasmas. For magnetic fluctuations that are transverse or almost transverse to a large-scale mean magnetic field, theories describing the FLRW usually predict asymptotic diffusion of magnetic field lines perpendicular to the mean field. Such theories often depend on the assumption that one can relate the Lagrangian and Eulerian statistics of the magnetic field via Corrsin’s hypothesis, and additionally take the distribution of magnetic field line displacements to be Gaussian. Here we take an ordinary differential equation (ODE) model with these underlying assumptions and test how well it describes the evolution of the magnetic field line diffusion coefficient in 2D+slab magnetic turbulence, by comparisons to computer simulations that do not involve such assumptions. In addition, we directly test the accuracy of the Corrsin approximation to the Lagrangian correlation. Over much of the studied parameter space we find that the ODE model is in fairly good agreement with computer simulations, in terms of both the evolution and asymptotic values of the diffusion coefficient. When there is poor agreement, we show that this can be largely attributed to the failure of Corrsin’s hypothesis rather than the assumption of Gaussian statistics of field line displacements. The degree of non-Gaussianity, which we measure in terms of the kurtosis, appears to be an indicator of how well Corrsin’s approximation works.

MAGNETIC FIELD LINE RANDOM WALK IN ISOTROPIC TURBULENCE WITH VARYING MEAN FIELD

ABSTRACT In astrophysical plasmas, the magnetic field line random walk (FLRW) plays an important role in guiding particle transport. The FLRW behavior is scaled by the Kubo number for rms magnetic fluctuation b, large-scale mean field , and coherence scales parallel ( ) and perpendicular ( ) to . Here we use a nonperturbative analytic framework based on Corrsin’s hypothesis, together with direct computer simulations, to examine the R-scaling of the FLRW for varying B 0 with finite b and isotropic fluctuations with , instead of the well-studied route of varying for . The FLRW for isotropic magnetic fluctuations is also of astrophysical interest regarding transport processes in the interstellar medium. With a mean field, fluctuations may have variance anisotropy, so we consider limiting cases of isotropic variance and transverse variance (with b z = 0). We obtain analytic theories, and closed-form solutions for extreme cases. Padé approximants are provided to interpolate all versions of theory and simulations to any B 0. We demonstrate that, for isotropic turbulence, Corrsin-based theories generally work well, and with increasing R there is a transition from quasilinear to Bohm diffusion. This holds even with b z = 0, when different routes to are mathematically equivalent; in contrast with previous studies, we find that a Corrsin-based theory with random ballistic decorrelation works well even up to R = 400, where the effects of trapping are barely perceptible in simulation results.

Cross-field transport in Goldreich-Sridhar magnetohydrodynamic turbulence.

I derive analytically the temporal dependence of the perpendicular transport coefficient of a charged particle in the three-dimensional anisotropic turbulence conjectured by Goldreich-Sridhar by implementing multispacecraft constraints on the turbulence power spectrum. The particle motion away from the turbulent local field line is assessed as gradient-curvature drift of the guiding center and compared with the magnetic field line random walk. At inertial scales much smaller than the turbulence outer scale, particles decorrelate from field lines in a free-streaming motion, with no diffusion. In the solar wind at 1 AU, for energy sufficiently small (<1 keV protons), the perpendicular average displacement due to field line tangling generally dominates over two decades of turbulent scales. However, for higher energies (≃25 MeV protons) within the range of multispacecraft measurements, the longitudinal spread originating from transport due to gradient-curvature drift reaches up to ≃10^{∘}-20^{∘}. This result highlights the role of perpendicular transport in the interpretation of interplanetary and interstellar data.

NONLINEAR EFFECTS IN PARTICLE TRANSPORT IN STOCHASTIC MAGNETIC FIELDS

Collisional particle transport in stochastic magnetic fields is studied using a semi-analytical method. The aim is to determine the influence of the nonlinear effects that occur in the magnetic field line random walk on particle transport. We show that particle transport coefficients can be strongly influenced by the magnetic line trapping. The conditions that correspond to these nonlinear regimes are determined. We also analyze the effects produced by the space variation of the large-scale magnetic field. We show that an average drift is generated by the gradient of the magnetic field, which strongly increases and reverses its orientation in the nonlinear regime.

Theory of magnetic field line random walk in noisy reduced magnetohydrodynamic turbulence

When a magnetic field consists of a mean part and fluctuations, the stochastic wandering of its field lines is often treated as a diffusive process. Under suitable conditions, a stable value is found for the mean square transverse displacement per unit parallel displacement relative to the mean field. Here, we compute the associated field line diffusion coefficient for a highly anisotropic “noisy” reduced magnetohydrodynamic model of the magnetic field, which is useful in describing low frequency turbulence in the presence of a strong applied DC mean magnetic field, as may be found, for example, in the solar corona, or in certain laboratory devices. Our approach is nonperturbative, based on Corrsin's independence hypothesis, and makes use of recent advances in understanding factors that control decorrelation over a range of parameters described by the Kubo number. Both Bohm and quasilinear regimes are identified.

MAGNETIC FIELD LINE RANDOM WALK FOR DISTURBED FLUX SURFACES: TRAPPING EFFECTS AND MULTIPLE ROUTES TO BOHM DIFFUSION

The magnetic field line random walk (FLRW) is important for the transport of energetic particles in many astrophysical situations. While all authors agree on the quasilinear diffusion of field lines for fluctuations that mainly vary parallel to a large-scale field, for the opposite case of fluctuations that mainly vary in the perpendicular directions, there has been an apparent conflict between concepts of Bohm diffusion and percolation/trapping effects. Here computer simulation and non-perturbative analytic techniques are used to re-examine the FLRW in magnetic turbulence with slab and two-dimensional (2D) components, in which 2D flux surfaces are disturbed by the slab fluctuations. Previous non-perturbative theories for D ⊥, based on Corrsin's hypothesis, have identified a slab contribution with quasilinear behavior and a 2D contribution due to Bohm diffusion with diffusive decorrelation (DD), combined in a quadratic formula. Here we present analytic theories for other routes to Bohm diffusion, with random ballistic decorrelation (RBD) either due to the 2D component itself (for a weak slab contribution) or the total fluctuation field (for a strong slab contribution), combined in a direct sum with the slab contribution. Computer simulations confirm the applicability of RBD routes for weak or strong slab contributions, while the DD route applies for a moderate slab contribution. For a very low slab contribution, interesting trapping effects are found, including a depressed diffusion coefficient and subdiffusive behavior. Thus quasilinear, Bohm, and trapping behaviors are all found in the same system, together with an overall viewpoint to explain these behaviors.

TIME-DEPENDENT PERPENDICULAR TRANSPORT OF FAST CHARGED PARTICLES IN A TURBULENT MAGNETIC FIELD

We present an analytic derivation of the temporal dependence of the perpendicular transport coefficient of charged particles in magnetostatic turbulence, for times smaller than the time needed for charged particles to travel the turbulence correlation length. This time window is left unexplored in most transport models. In our analysis all magnetic scales are taken to be much larger than the particle gyroradius, so that perpendicular transport is assumed to be dominated by the guiding center motion. Particle drift from the local magnetic field lines (MFLs) and magnetic field line random walk are evaluated separately for slab and three-dimensional (3D) isotropic turbulence. Contributions of wavelength scales shorter and longer than the turbulence coherence length are compared. In contrast to the slab case, particles in 3D isotropic turbulence unexpectedly diffuse from local MFLs; this result questions the common assumption that particle magnetization is independent of turbulence geometry. Extensions of this model will allow for a study of solar wind anisotropies.

Magnetic field line random walk in non-axisymmetric turbulence

Including a random component of a magnetic field parallel to an ambient field introduces a mean perpendicular motion to the average field line. This effect is normally not discussed because one customarily chooses at the outset to ignore such a field component in discussing random walk and diffusion of field lines. A discussion of the basic effect is given, indicating that any random magnetic field with a non-zero helicity will lead to such a non-zero perpendicular mean motion. Several exact analytic illustrations are given of the effect as well as a simple numerical illustration.

Random walk of magnetic field lines: analytical theory versus simulations

The analytical nonlinear theory of magnetic field line random walk predicts the existence of nondiffusive transport for certain forms of the turbulence spectrum. In the present article we use computer simulations to test these predictions made for well-established one- and two-dimensional models of magnetic field fluctuations. For the first time it is shown by using simulations, that for a whole family of spectra a non-diffusive behavior of field line wandering can be found.

Detailed analytical investigation of magnetic field line random walk in turbulent plasmas: I. Two-component slab/two-dimensional turbulence

AbstractThe random displacement of magnetic field lines in the presence of magnetic turbulence in plasmas is investigated from first principles. A two-component (slab/two-dimensional composite) model for the turbulence spectrum is employed. An analytical investigation of the asymptotic behavior of the field-line mean square displacement (FL-MSD) is carried out. It is shown that the magnetic field lines behave superdiffusively for very large values of the position variable z, since the FL-MSD σ varies as σ ~ z4/3. An intermediate diffusive regime may also possibly exist for finite values of z under conditions which are explicitly determined in terms of the intrinsic turbulent plasma parameters. The superdiffusive asymptotic result is confirmed numerically via an iterative algorithm. The relevance to previous results is discussed.

The Cosmic‐Ray Diffusion Tensor in Nonaxisymmetric Turbulence

Previous investigations of magnetic field line random walk and charged particle transport in turbulence mostly employed an axisymmetric turbulence model. However, real turbulence is assumed to be nonaxisymmetric. In this article we employ a simple model for nonaxisymmetric turbulence based on a very recent article. By analytically describing the random walk of magnetic field lines and by combining these results with a general compound transport model, we derive the cosmic-ray diffusion tensor. As demonstrated, the nonaxisymmetry leads to different components of the cosmic-ray diffusion tensor in the perpendicular direction. These new results are essential for several applications such as solar modulation studies.

Field line wandering and perpendicular scattering of charged particles in Alfvénic slab turbulence

We describe analytically the magnetic field line random walk in Alfvenic slab turbulence. This result can easily be combined with a compound transport model to describe analytically the perpendicular scattering of charged cosmic rays. We demonstrate that the well known subdiffusive solution for magnetostatic turbulence is only valid for shorter time scales. For longer times, perpendicular diffusion of charged test-particles is recovered due to wave propagation effects. For low energy cosmic rays, this new result could be important for understanding heliospheric observations.

Suppressed Diffusive Escape of Topologically Trapped Magnetic Field Lines

Many processes in astrophysical plasmas are directly related to magnetic connection in the presence of turbulent fluctuations. Even statistically homogeneous turbulence can contain closed topological structures that inhibit otherwise random transport of field line trajectories, thus temporarily trapping certain trajectories. When a coherent random field perturbation is added, the trapped field lines can escape diffusively but at a suppressed rate that is much lower than what would be estimated based on the perturbation field alone. Here we demonstrate both trapping and escape, and show, using a novel quasi-linear theory, how to compute the suppressed diffusion that affects the escape from the trapping structure. The effect is relevant to understanding filamentary magnetic connection in interplanetary space and the observed dropouts in moderately energetic particles from impulsive solar flares. Expressed here in terms of a magnetic field line random walk, this phenomenon also has analogies in a broad range of dynamical systems that evolve as an incompressible flow in phase space with a coherent perturbation.

Magnetic connection from the Earth to the solar corona, flare positions and solar energetic particle observations

Solar energetic particle (SEP) events are often associated with solar flares. Such particles propagate through the interplanetary medium, where significant levels of magnetic turbulence are found. We study the magnetic connection from the Earth to the solar corona in three dimensional magnetic turbulence. In the numerical simulation, different turbulence levels and solar wind velocities can be used. Input to the simulation is from web-based data sets, and comparison is made with the solar flare observations contained in the Goes catalogue for the years 1996, 1997, 1998, following solar minimum. For this data set, we find that SEPs can reach the Earth when the difference in the heliographic longitudes of the flare and of the magnetic foot point of the Earth is 25°–30° at most. On the other hand the longitudinal angular semi-width of the magnetic field line random walk in the solar wind, when mapped to the solar corona, is found to be typically $6^\circ$–10°. The discrepancy between the two values can be explained either by the presence of a flare – associated coronal mass ejection shock, with a longitudinal semi-size of ~$20^\circ$, or by particle propagation, which could be enhanced over the field line random walk by, e.g., gyroresonant effects, or by the presence of magnetic shear between the fast and the slow streams which enhances the longitudinal spread of field lines.

Separation of Magnetic Field Lines in Two‐Component Turbulence

The problem of the separation of random magnetic field lines in collisionless astrophysical plasmas is closely related to the problem of the magnetic field line random walk and is highly relevant to the transport of charged particles in turbulent plasmas. In order to generalize treatments based on quasi-linear theory, here we examine the separation of nearby magnetic field lines by employing a nonperturbative technique based on the Corrsin independence hypothesis. Specifically, we consider the case of two-component turbulence in which the magnetic field fluctuations are a mixture of one-dimensional (slab) and two-dimensional ingredients, as a concrete example of anisotropic turbulence that provides a useful description of turbulence in the solar wind. We find that random field trajectories can separate in general through three regimes of the behavior of the running diffusion coefficient: slow diffusive separation, an intermediate regime of superdiffusion, and fast diffusive separation at large distances. These features are associated with the gradual, exponential divergence of field lines within islands of two-dimensional turbulence, followed by diffusive separation at long distances. The types of behavior are determined not by the Kubo number but rather a related ratio that takes the turbulence anisotropy into account. These results are confirmed by computer simulations. We discuss implications for space observations of energetic charged particles, including ‘‘dropouts’’ of solar energetic particles.