Behavior Of Magnetic Field Lines Research Articles

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.

Local analysis of fast magnetic reconnection

Fast magnetic reconnection is defined by the topology of the magnetic field lines changing on a timescale that is approximately an order of magnitude longer than the topology-conserving ideal-evolution timescale. Fast reconnection is an intrinsic property of Faraday's law when the evolving magnetic field depends non-trivially on all three spatial coordinates and is commonly observed—even when the effects that allow topology breaking are arbitrarily small. The associated current density need only be enhanced by a factor of approximately ten and flows in thin but broad ribbons along the magnetic field. These results follow from the variation in the separation of neighboring pairs of magnetic field lines, which in an ideal evolution typically increases exponentially with time, and the existence of a spatial scale below which magnetic field lines freely change their identities due to non-ideal effects, such as resistivity. Traditional reconnection theory ignores exponentially large variations and relies on the current density reaching a magnitude that is exponentially larger than is actually required. Here, an analysis of the behavior of magnetic field lines in the neighborhood of an arbitrarily chosen line is used to obtain more precise and rigorous results on intrinsic reconnection. The maximum parallel kinetic energy of collisionless charged particles is shown to have an exponential increase in time during a generic magnetic evolution.

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.

Magnetic Energy Release, Plasma Dynamics, and Particle Acceleration in Relativistic Turbulent Magnetic Reconnection

Abstract In strongly magnetized astrophysical plasma systems, magnetic reconnection is believed to be the primary process during which explosive energy release and particle acceleration occur, leading to significant high-energy emission. Past years have witnessed active development of kinetic modeling of relativistic magnetic reconnection, supporting this magnetically dominated scenario. A much less explored issue in studies of relativistic reconnection is the consequence of three-dimensional dynamics, where turbulent structures are naturally generated as various types of instabilities develop. This paper presents a series of three-dimensional, fully kinetic simulations of relativistic turbulent magnetic reconnection (RTMR) in positron–electron plasmas with system domains much larger than kinetic scales. Our simulations start from a force-free current sheet with several different modes of long-wavelength magnetic field perturbations, which drive additional turbulence in the reconnection region. Because of this, the current layer breaks up and the reconnection region quickly evolves into a turbulent layer filled with coherent structures such as flux ropes and current sheets. We find that plasma dynamics in RTMR is vastly different from its 2D counterpart in many aspects. The flux ropes evolve rapidly after their generation, and can be completely disrupted by the secondary kink instability. This turbulent evolution leads to superdiffusive behavior of magnetic field lines as seen in MHD studies of turbulent reconnection. Meanwhile, nonthermal particle acceleration and the timescale for energy release can be very fast and do not depend strongly on the turbulence amplitude. The main acceleration mechanism is a Fermi-like acceleration process supported by the motional electric field, whereas the nonideal electric field acceleration plays a subdominant role. We also discuss possible observational implications of three-dimensional RTMR in high-energy astrophysics.

Helical magnetic self-organization of plasmas in toroidal pinches with transport barriers formation

Nonlinear MHD modeling of toroidal pinch configurations for hot plasma magnetic confinement describes several features of the helical self-organization process, which is observed in both reversed-field pinches and tokamaks. It can also give a hint on why transport barriers are formed, by far one of the more interesting observations in experiments. The work tackles these two topics, helical self-organization and transport barriers formation - adding further information and examples to the results already presented in [Veranda, et al, Nucl.Fus. 60 016007 (2020)]. Regarding the topic of helical self-organization, a synthesis of the results obtained by a 3D nonlinear viscoresistive magnetohydrodynamics model will be presented. Modelling predicts a technique to “channel” reversed-field pinches into a chosen macroscopic helical shape and also predicts that the features of such helical self-organization, studied in the RFX-mod experiment in Padova, depend on two parameters only: plasma dissipation coefficients and edge radial magnetic field. They can be exploited to calm the natural tendency of reversed-field pinches to a “sawtoothing” dynamics, i.e. by decreasing visco-resistive dissipation and using helical edge fields not resonating with the plasma safety factor. Regarding the MHD description of the process of formation of transport barriers by magnetic chaos healing, we will describe the computation of Lagrangian structures, hidden in the weakly stochastic behaviour of magnetic field lines, acting as barriers to the transport. The radial position of such structures is observed to correspond to higher gradients of magnetic field lines connection length to the edge: this provides a further indication of their possible role in the formation of electron temperature barriers.

Stickiness in double-curl Beltrami magnetic fields.

The double-curl Beltrami magnetic field in the presence of a uniform mean field is considered for investigating the nonlinear dynamical behavior of magnetic field lines. The solutions of the double-curl Beltrami equation being non-force-free in nature belong to a large class of physically interesting magnetic fields. A particular choice of solution for the double-curl equation in three dimensions leads to a wholly chaotic phase space. In the presence of a strong mean field, the phase space is a combination of closed magnetic surfaces and weakly chaotic regions that slowly tends to global randomness with a decreasing mean field. Stickiness is an important feature of the mixed phase space that describes the dynamical trapping of a chaotic trajectory at the border of regular regions. The global behavior of such trajectories is understood by computing the recurrence length statistics showing a long-tail distribution in contrast to a wholly chaotic phase space that supports a distribution which decays rapidly. Also, the transport characteristics of the field lines are analyzed in connection with their nonlinear dynamical properties.

Recurrence-based analysis of barrier breakup in the standard nontwist map.

We study the standard nontwist map that describes the dynamic behaviour of magnetic field lines near a local minimum or maximum of frequency. The standard nontwist map has a shearless invariant curve that acts like a barrier in phase space. Critical parameters for the breakup of the shearless curve have been determined by procedures based on the indicator points and bifurcations of periodical orbits, a methodology that demands high computational cost. To determine the breakup critical parameters, we propose a new simpler and general procedure based on the determinism analysis performed on the recurrence plot of orbits near the critical transition. We also show that the coexistence of islands and chaotic sea in phase space can be analysed by using the recurrence plot. In particular, the measurement of determinism from the recurrence plot provides us with a simple procedure to distinguish periodic from chaotic structures in the parameter space. We identify an invariant shearless breakup scenario, and we also show that recurrence plots are useful tools to determine the presence of periodic orbit collisions and bifurcation curves.

Chaotic behavior of magnetic field lines near simplest current systems

In the context of studying the problem of simulation of magnetic fields on the Sun, the structure of the field in the vicinity of two circular current loops with different mutual arrangement in space is considered. When the symmetry in the arrangement is sufficient, a system of magnetic surfaces created by the closed field lines arises. With a reduction in symmetry, isolated closed lines may exist. For the case of two identical current loops coupled perpendicularly, it is shown that the subsystems of these lines may be ordered in space in a complex manner. At large distances, a system of loops is equivalent to a dipole with a high degree of accuracy, while an approximate winding of the lines on the deformed toroids, encircling each of the loops, occurs at small distances. At intermediate distances, there are regions of both ordered and chaotic behavior of field lines. Results were obtained with the use of the numerical simulation method.

An innovative experiment on superconductivity, based on video analysis and non-expensive data acquisition

In this paper we present a new experiment on superconductivity, designed for university undergraduate students, based on the high-speed video analysis of a magnet falling through a ceramic superconducting cylinder (Tc = 110 K). The use of an Atwood’s machine allows us to vary the magnet’s speed and acceleration during its interaction with the superconductor. In this way, we highlight the existence of two interaction regimes: for low crossing energy, the magnet is levitated by the superconductor after a transient oscillatory damping; for higher crossing energy, the magnet passes through the superconducting cylinder. The use of a commercial-grade high speed imaging system, together with video analysis performed using the Tracker software, allows us to attain a good precision in space and time measurements. Four sensing coils, mounted inside and outside the superconducting cylinder, allow us to study the magnetic flux variations in connection with the magnet’s passage through the superconductor, permitting us to shed light on a didactically relevant topic as the behaviour of magnetic field lines in the presence of a superconductor. The critical discussion of experimental data allows undergraduate university students to grasp useful insights on the basic phenomenology of superconductivity as well as on relevant conceptual topics such as the difference between the Meissner effect and the Faraday-like ‘perfect’ induction.

Hamiltonian description of closed configurations of the vacuum magnetic field

Methods of obtaining and using the Hamiltonians of closed vacuum magnetic configurations of fusion research systems are reviewed. Various approaches to calculate the flux functions determining the Hamiltonian are discussed. It is shown that the Hamiltonian description allows one not only to reproduce all traditional results, but also to study the behavior of magnetic field lines by using the theory of dynamic systems. The potentialities of the Hamiltonian formalism and its close relation to traditional methods are demonstrated using a large number of classical examples adopted from the fundamental works by A.I. Morozov, L.S. Solov’ev, and V.D. Shafranov.

Subsurface structure of the Evershed flows in sunspots

The radial outflows in sunspot penumbrae, known as the Evershed effect, are of significant interest for understanding the dynamics of sunspots. Local helioseismology has not been able to determine the depth of these flows nor their relationship to mass circulation in sunspots. Recent radiative MHD simulations have provided a convincing explanation of the Evershed flow as a natural consequence of magnetoconvection in the strongly inclined magnetic field region of the penumbra. The simulations reproduce many observational features of penumbra dynamics, including the filamentary structure, the high-speed non-stationary "Evershed clouds", and the "sea-serpent" behavior of magnetic field lines. We present the subsurface structure of the Evershed effect, obtained from numerical simulations, and determine the depth of the radial outflows for various magnetic field strengths and inclinations. The simulations predict that Evershed flows are rather shallow and concentrated in the top 0.5 – 1 Mm layer of the convection zone. This prediction can be tested by local helioseismology methods.

Quasi‐linear Magnetic Field Lines: Anomalous Transport and Self‐Similarity, from the Analytic to the Numerical

Charged energetic particles propagating in solar wind magnetic fields span field irregularities down to very short turbulent scales, not described by the original quasi-linear theory for weak magnetic turbulence. This theory only predicts a field line diffusion on the largest scales, well above the correlation length, inverse of the spectral flattening wavenumber. The quasi-linear prediction for the transport and behavior of magnetic field lines is generalized here to all scales and arbitrary three-dimensional turbulence spectra. New analytical expressions are derived for the field line mean square cross-field displacement Δx2, and analytical proof is presented for the anomalous transport of the field lines. We find Δx2 ∝ (Δz)β, where Δz is the elapsed distance along the average field and β, the transport exponent, can take any value between 0 and 2. A decreasing turbulence spectrum results in a field line supradiffusion (β > 1), while an inverted spectrum implies a subdiffusion (β < 1). Simple expressions are derived for the transport exponent and coefficient. A powerful new method is presented to compute magnetic field lines in the quasi-linear regime of turbulence that allows rapid computation of field lines generated from any three-dimensional turbulence spectrum, including some 1015 modes and more. Individual field lines computed with this method show how a spectral steepening results in a smoothing of the field lines and how harder spectra give increasingly more short-scale fluctuations. The field line self-similarity, characteristic of power-law spectra, is demonstrated visually, and the anomalous transport of the field lines is confirmed numerically.

String behaviour of magnetic field lines and their rotation in neutron star’s interior

It is found from Maxwell’s equations that the magnetic field lines are good analogues of relativistic strings. The Lorentz force per unit length of magnetic tube is interpretable as Magnus force acting on each individual magnetic tube. It is shown that the superconducting current in pulsar’s interior causes local rotation of magnetic flux tubes carrying quantized flux. Such local rotation remains operative as long as the induced magnetic field of normal electron fluid is above the lower critical field but below the upper. The conservation of magnetic flux leads to a geometrical condition in the form of the Weingarten identity which ensures the existence of family of “magnetic world sheetrdquo;. Each “magnetic world sheet” is a magnetic flux conserving surface. In the process of collapse, a compact spacelike cross-section of a magnetic tube terminates into a trapped surface if the magnetic energy grows faster along the fluid flow lines than that along the magnetic field lines.

Modelling of plasma and impurity behaviour in a tokamak with a stochastic layer

The behaviour of magnetic field lines and transport of charged particles and heat in a stochastic layer at the plasma edge is discussed. Results of numerical modelling by the one-dimensional code RITM of particle and energy balance in Tore Supra discharges without and with an ergodic divertor are presented and compared with the experimental data. The effects of stochastization on the temperature profile, on the radiation losses from impurities and on impurity concentration in the plasma core are analysed on the basis of simple analytical models.

Study of the behaviour of magnetic field lines after perturbation of the toroidal field on magnetic surfaces

The paper presents a study of the effect of a perturbing magnetic field on a field whose magnetic surfaces are tori nested around a closed central line. This perturbation effect creates magnetic islands around surfaces with rational rotational transform. These islands are investigated analytically, which makes it possible to evaluate their size. The resulting turbulence of the medium can then be studied by calculating the interaction of two neighbouring islands.

A class of exact solutions and physical behaviour of magnetic field lines in plasma column

Using hodograph and Legendre transformation functions the basic equations are recast in terms of this function, and the conditions which this function should satisfy are stated. Several applications of this method are considered, and the geometry of magnetic field lines is discussed.

Ergodic Behavior of Magnetic Field Lines in a Poloidal Divertor Tokamak

The ergodic behavior of magnetic field lines is investigated numerically in a Tokamak divertor system under the influence of magnetic perturbations. The entropylike quantity introduced into the magnetic system characterizes the property of the magnetic field lines and is related closely to the width of the ergodic layer. The metric entropy of the ergodic field lines is given in terms of the entropylike quantity.

Lie series and invariant functions for analytic symplectic maps

Symplectic maps (canonical transformations) are treated from the Lie algebraic point of view using Lie series and Lie algebraic techniques. It is shown that under very general conditions an analytic symplectic map can be written as a product of Lie transformations. Under certain conditions this product of Lie transformations can be combined to form a single Lie transformation by means of the Campbell–Baker–Hausdorff theorem. This result leads to invariant functions and generalizes to several variables a classic result of Birkhoff for the case of two variables. It also provides a new approach since the connection between symplectic maps, Lie algebras, invariant functions, and Birkhoff’s work has not been previously recognized and exploited. It is expected that the results obtained will be applicable to the normal form problem in Hamiltonian mechanics, the use of the Poincaré section map in stability analysis, and the behavior of magnetic field lines in a toroidal plasma device.