Field Line Equations Research Articles

Simulations of field line random walk in noisy slab turbulence

Magnetic field lines in magnetohydrodynamic turbulence are stochastic curves and, therefore, understanding their structure is a difficult problem of theoretical physics. One way to study the properties of such field lines is to perform simulations in which field line equations are solved numerically. In most previous articles studies have been performed in the context of the two-component turbulence model where one assumes a superposition of slab and two-dimensional modes. In the current paper we perform simulations for so-called noisy slab turbulence which is an example for three-dimensional turbulence with small Kubo numbers. We discuss in detail the challenges of the numerical magnetic field creation for this type of turbulence, perform the simulations, and compare our findings with analytical theory. It is demonstrated that a large number of wave modes is needed in order to obtain accurate results. Furthermore, we also show that analytical theories are very accurate for this type of turbulence model. The obtained results will also be useful for the analytical description of energetic particle transport in the considered turbulence model.

Electric field lines of an arbitrarily moving charged particle

Electromagnetic fields of relativistic charged particles have a broad frequency spectrum and a sophisticated spatial structure. Field lines offer a visual representation of this spatial structure. In this article, we derive a general set of equations for the field lines of any moving charged particle. The electric field lines are completely determined by the unit vector from the retarding point to the observation point. After proper transformations, the field line equations describe the rotation of this vector with an angular velocity coinciding with Thomas precession. In some cases, including all planar trajectories, the field line equations reduce to linear differential equations with constant coefficients. We present a detailed derivation of these equations and their general analytical solution. We then illustrate this method by constructing field lines for the “figure eight” motion of an electric charge moving under the influence of a plane wave, including complex field lines in three dimensions.

A General Curvilinear Magnetic Field‐Line‐Following Coordinate System for Ionosphere‐Plasmasphere Modeling

AbstractWe propose a simple way to define a field‐line‐following, general curvilinear coordinate system for a general magnetic field. This definition of field‐line‐following coordinate system reduces to a usual definition of dipole coordinate system when the magnetic field is approximated by an axisymmetric dipole. In this way, it can facilitate the numerical implementation by enabling validation of various metric terms computed numerically against those defined analytically in the case of the dipole field. Steps involved in grid generation are also sketched. Highly accurate results are obtained using the high‐order ordinary differential equation (ODE) solver to solve the general magnetic field line equations. The accuracy and consistency of the numerical implementation are validated against analytical results in the case of a dipole field. Numerical results show that this field‐line‐following coordinate system for the general magnetic field, like the coordinates for the dipole field, is also an Euler potential or Clebsch type coordinate system.

Magnetic dipole interaction with multipole magnetic field lines of neutron stars

Conservation of magnetic flux is associated with regions of the powerful magnetic fields (B ∽ 1013 G) near neutron stars' surface. The vector potential generated by moving electric charge Q is uniformly distributed within a Neutron star's surface (radius R). The evolution of the magnetic field of isolated neutron stars is studied and based on magnetic flux conservation; the multipolar magnetic fields for (l = 1; l = 2; l = 3; l = 4) have calculated. We developed the field line equations and simulated the magnetic field line geometry for the interaction between neutron stars’ dipole–multipolar magnetic fields using the MATLAB software program.

A geometrical and physical interpretation of quaternionic generalized magnetic flux tubes

In the present paper, we give a generalization for the magnetic flux tubes. Firstly, we define the flux tube via quaternions. We obtain the magnetic flux canal surfaces and flux tubes by quaternion product of a unit quaternion and the unit normal vector of the magnetic field line. Besides, we generate these surfaces by the homothetic motion. Then, we determine the magnetic field components by using the quaternionic representation of the flux tube. Secondly, the flux tube is dictated by the magnetic vector field defined at each point on a generating curve. A local orthonormal basis is attached to every point of the generating curve then the magnetic field is given in terms of the local coordinate directions, leading to a geometric viewpoint of the flux tubes. These provide direct access to the magnetic field line and kinematics equations related to the flux tube. Moreover, the magnetic energy function and magnetic diffusion of the flux tube that has an arbitrary cross-section curve are computed. The impacts of the stretch factor of the flux tube on the magnetic trajectories are investigated. Finally, the analytical solutions of the kinematics equations and several examples of the theory are provided.

General principles of describing second- and higher-order null points of a potential magnetic field in 3D

General principles of describing secondand higher-order null points of a potential magnetic field are formulated. The potential near a second-order null of the general form can be specified by a linear combination of four basic functions, the list of which is presented. Near secondand higher-order null points, field line equations often cannot be integrated analytically; however, in some cases, it is possible to present a qualitative description of the geometry of null vicinities with consideration of the behavior of field lines near rays outgoing from null, at which the field is radial or equals zero.

A complete topological invariant for braided magnetic fields

A topological flux function is introduced to quantify the topology of magnetic braids: non-zero line-tied magnetic fields whose field lines all connect between two boundaries. This scalar function is an ideal invariant defined on a cross-section of the magnetic field, whose integral over the cross-section yields the relative magnetic helicity. Recognising that the topological flux function is an action in the Hamiltonian formulation of the field line equations, a simple formula for its differential is obtained. We use this to prove that the topological flux function uniquely characterises the field line mapping and hence the magnetic topology. A simple example is presented.

Heat flux modeling using ion drift effects in DIII-D H-mode plasmas with resonant magnetic perturbations

The heat flux patterns measured in low-collisionality DIII-D H-mode plasmas strongly deviate from simultaneously measured CII emission patterns, used as indicator of particle flux, during applied resonant magnetic perturbations. While the CII emission clearly shows typical striations, which are similar to magnetic footprint patterns obtained from vacuum field line tracing, the heat flux is usually dominated by one large peak at the strike point position. The vacuum approximation, which only considers applied magnetic fields and neglects plasma response and plasma effects, cannot explain the shape of the observed heat flux pattern. One possible explanation is the effect of particle drifts. This is included in the field line equations and the results are discussed with reference to the measurement. Electrons and ions show different drift motions at thermal energy levels in a guiding center approximation. While electrons hardly deviate from the field lines, ions can drift several centimetres away from field line flux surfaces. A model is presented in which an ion heat flux, based on the ion drift motion from various kinetic energies as they contribute to a thermal Maxwellian distribution, is calculated. The simulated heat flux is directly compared to measurements with a varying edge safety factor q95. This analysis provides evidence for the dominate effect of high-energy ions in carrying heat from the plasma inside the separatrix to the target. High-energy ions are deposited close to the unperturbed strike line, while low-energy ions can travel into the striated magnetic topology.

Unique topological characterization of braided magnetic fields

We introduce a topological flux function to quantify the topology of magnetic braids: non-zero, line-tied magnetic fields whose field lines all connect between two boundaries. This scalar function is an ideal invariant defined on a cross-section of the magnetic field, and measures the average poloidal magnetic flux around any given field line, or the average pairwise crossing number between a given field line and all others. Moreover, its integral over the cross-section yields the relative magnetic helicity. Using the fact that the flux function is also an action in the Hamiltonian formulation of the field line equations, we prove that it uniquely characterizes the field line mapping and hence the magnetic topology.

Structure of magnetic field lines

In this paper the Hamiltonian structure of magnetic lines is studied in many ways. First it is used vector analysis for defining the Poisson bracket and Casimir variable for this system. Second it is derived Pfaffian equations for magnetic field lines. Third, Lie derivative and derivative of Poisson bracket is used to show structure of this system. Finally, it is shown Nambu structure of the magnetic field lines.

Calculation of the magnetic surface function gradient in stellarators with broken stellarator symmetry

The computation of the gradient of the magnetic surface function, ∇ψ, plays an essential role in plasma physics, e.g., for investigations of plasma equilibrium currents or transport fluxes in stellarators. The evaluation of ∇ψ becomes more complicated if the magnetic field B does not exhibit stellarator symmetry. Here, a scheme for computation of ∇ψ for magnetic configurations which do not show stellarator symmetry is presented. The proposed method is based on computations of gradients of integrals of magnetic field line equations. This new technique for ∇ψ calculations is applied to Uragan-2M [O. S. Pavlichenko for the U-2M group, Plasma Phys. Controlled Fusion 35, B223 (1993)]. Taking into account the influence of current feeds and detachable joints of the helical winding the magnetic configuration does not exhibit stellarator symmetry. Computations of ∇ψ, the effective ripple ϵeff, and the geometrical factor λb for the bootstrap current in the 1/ν transport regime are performed.

Generic magnetic field model in poloidal divertor tokamaks in the presence of resonant magnetic perturbations

A generic analytical model for the description of magnetic field lines in poloidal divertor tokamaks in the presence of external resonant magnetic perturbations is proposed. It is based on the Hamiltonian description of magnetic field lines in tokamaks. The safety factor and the spectra of magnetic perturbations are chosen by the requirement to satisfy their generic behaviour near the magnetic separatrix and at the magnetic axis. The field line equations are integrated by the construction of two symplectic and computationally efficient mappings of field lines. The model for internal MHD modes is also proposed. The mapping procedure for field lines which includes the MHD modes is described. It is shown that the numerically calculated diffusion and convection coefficients of field lines are in close agreement with the quasilinear ones. It is found that in the presence of internal MHD modes at the plasma edge the convectional outward transport of field lines may reverse its direction to inward convectional transport.

Integrable maps with non-trivial topology: application to divertor configurations

We explore a method for constructing two-dimensional area-preserving, integrable maps associated with Hamiltonian systems, with a given set of fixed points and given invariant curves. The method is used to find an integrable Poincaré map for the field lines in a large aspect ratio tokamak with a poloidal single-null divertor. The divertor field is a superposition of a magnetohydrodynamic equilibrium with an arbitrarily chosen safety factor profile, with a wire carrying an electric current to create an X-point. This integrable map is perturbed by an impulsive perturbation that describes non-axisymmetric magnetic resonances at the plasma edge. The non-integrable perturbed map is applied to study the structure of the open field lines in the scrape-off layer, reproducing the main transport features obtained by integrating numerically the magnetic field line equations, such as the connection lengths and magnetic footprints on the divertor plate.

On the quasi-periodic oscillations in magnetars

We study torsional Alfv\'en oscillations of magnetars, i.e., neutron stars with a strong magnetic field. We consider the poloidal and toroidal components of the magnetic field and a wide range of equilibrium stellar models. We use a new coordinate system (X,Y), where $X=\sqrt{a_1} \sin \theta$, $Y=\sqrt{a_1}\cos \theta$ and $a_1$ is the radial component of the magnetic field. In this coordinate system, the 1+2-dimensional evolution equation describing the quasi-periodic oscillations, QPOs, see Sotani et al. (2007), is reduced to a 1+1-dimensional equation, where the perturbations propagate only along the Y-axis. We solve the 1+1-dimensional equation for different boundary conditions and open magnetic field lines, i.e., magnetic field lines that reach the surface and there match up with the exterior dipole magnetic field, as well as closed magnetic lines, i.e., magnetic lines that never reach the stellar surface. For the open field lines, we find two families of QPOs frequencies; a family of "lower" QPOs frequencies which is located near the X-axis and a family of "upper" frequencies located near the Y-axis. According to Levin (2007), the fundamental frequencies of these two families can be interpreted as the turning points of a continuous spectrum. We find that the upper frequencies are constant multiples of the lower frequencies with a constant equaling 2n+1. For the closed lines, the corresponding factor is n+1 . By these relations, we can explain both the lower and the higher observed frequencies in SGR 1806-20 and SGR 1900+14.

Model of a magnetic field in poloidal divertor tokamaks affected by resonant magnetic perturbations

A generic analytical model for the description of magnetic field lines in poloidal divertor tokamaks in the presence of external resonant magnetic perturbations is proposed. It is based on the Hamiltonian description of magnetic field lines in tokamaks. The safety factor and the spectra of magnetic perturbations are chosen by the requirement to satisfy their generic behavior near the magnetic separatrix and at the magnetic axis. The field line equations of the model are integrated using symplectic efficient mappings of field lines. The analytical formulas for the quasilinear diffusion and convection coefficients of field lines are obtained. The latter describes the outwardly directed transport of field lines at the plasma edge. It was shown that they are in a good agreement with the corresponding numerically calculated coefficients.

Tokamak magnetic field lines described by simple maps

The magnetic field line structure in a tokamak can be obtained by direct numerical integration of the field line equations. However, this is a lengthy procedure and the analysis of the solution may be very time-consuming. Otherwise we can use simple two-dimensional, area-preserving maps, obtained either by approximations of the magnetic field line equations, or from dynamical considerations. These maps can be quickly iterated, furnishing solutions that mirror the ones obtained from direct numerical integration, and which are useful when long-term studies of field line behavior are necessary (e.g. in diffusion calculations). In this work we focus on a set of simple tokamak maps for which these advantages are specially pronounced.

Subexponential Divergence and Diffusive Twist of Turbulent Magnetic Field Lines in the Limit of the Very Short Separations

Turbulent magnetic field lines have long been thought to be diverging from each other (or converging toward each other) at exponential rates known as Lyapunov exponents. It is argued here that in a turbulent magnetized plasma, subexponential divergence (convergence) and diffusive twist better characterize the dispersal of magnetic field lines (MFLs) in the limit of the very small separations ρ than do the usual Lyapunov exponents or exponentiation rates. In that limit of the very small separations, the field-line equations give a variation rate for ln ρ , not ρ, and the implied log-normality of the ρ distribution makes (ln ρ/ρ0)2 a much better probe of the exponential divergence of core MFLs. A fully nonlinear calculation shows that the separation logarithm, ln ρ , and twist or rotation angle, Δ θ , between pairs of MFLs diffuse with the distance Δ z elapsed along the main field, as soon as Δ z exceeds min (k−1II,ζII) , the minimum of the parallel correlation length k−1II ≡ L∥ ∇ of the turbulent field gradients and of the associated nonlinear scale, ζII ≡ ζ∇, defined as the field-aligned length scale for which the mean cross-field displacement (rζ − r0)21/2 reaches 21/2ξ k−1II ≡ 21/2L⊥ ∇, with kII the wavenumber where the turbulence spectrum becomes steeper than (k2∥ + ξ2k2⊥)−1 and ξ the anisotropy parameter of the turbulence. The average growth of the core field-line separation ρ0e (ln ρ/ρ0)21/2 = e∝ (Δ z)1/2 along the direction of fastest growth, being subexponential, is not compatible with the definition of Lyapunov exponents. The largest exponentiation rate of the core MFLs actually decreases with the distance Δ z. Application of the new nonlinear calculation to the solar wind shows a substantial MFL rotation in a plane transverse to the main field.

Visualization of the solar magnetic field using coefficients of harmonic expansion in the potential approximation

A method for visualization of the solar magnetic field is suggested and implemented. The harmonic expansion coefficients obtained in the potential approximation from magnetic-field observations at the photospheric level are used as an input. Three-dimensional equations for field lines are numerically integrated, and the results are presented in the form of projections onto the picture plane. The density of the line number is assumed to be proportional to the magnetic flux density. The lines of different topological connectedness are differently colored and show areas with open and closed configurations, thus mapping the information about the field direction and magnitude. This method is useful for research and applied studies related to investigation and forecasting of space weather.

Mappings of stochastic field lines in poloidal divertor tokamaks

Two mapping methods to study magnetic field lines near the separatrix of poloidal divertor tokamaks in the presence of external non-axisymmetric magnetic perturbations are proposed. The first mapping method is based on the Hamiltonian formulation of field line equations in the Boozer coordinates and solving it by the canonical transformation of variables (Abdullaev et al 1999 Phys. Plasmas 6 153). The second mapping is a canonical mapping near the separatrix which is constructed using the recently developed method (Abdullaev 2004 Phys. Rev. E 70 064202, Abdullaev 2005 Phys. Rev. E 72 064202). We construct the corresponding mappings for magnetic field lines in divertor tokamaks in the presence of non-axisymmetric magnetic perturbations. The mappings are applied to study the properties of open stochastic field lines near the separatrix for the wire model of the plasma. Poincaré sections, the so-called laminar and magnetic footprint plot (a contour plot of wall to wall connections lengths) in the plasma region and on the divertor plates are obtained. The quasilinear diffusion coefficients of field lines are also estimated.

Topology of toroidal helical fields in non-circular cross-sectional tokamaks

The ordinary differential magnetic field line equations are solved numerically; the tokamak magnetic structure is studied on Hefei Tokamak-7 Upgrade (HT-7U) when the equilibrium field with a monotonic q-profile is perturbed by a helical magnetic field. We find that a single mode (m, n) helical perturbation can cause the formation of islands on rational surfaces with q = m/n and q = (m +/- 1, +/- 2, +/- 3,...)/n due to the toroidicity and plasma shape (i.e. elongation and triangularity), while there are many undestroyed magnetic surfaces called Kolmogorov-Arnold-Moser (KAM) barriers on irrational surfaces. The islands on the same rational surface do not have the same size. When the ratio between the perturbing magnetic field B-r(r) and the toroidal magnetic field amplitude B(phi)0 is large enough, the magnetic island chains on different rational surfaces will overlap and chaotic orbits appear in the overlapping area, and the magnetic field becomes stochastic. It is remarkable that the stochastic layer appears first in the plasma edge region.