Magneto-frictional Method Research Articles

The effect of data-driving and relaxation models on magnetic flux rope evolution and stability

Understanding the flux rope eruptivity and effects of data driving in modelling solar eruptions is crucial for correctly applying different models and interpreting their results. We aim to investigate these by analysing the fully data-driven modelled eruption of the active regions (ARs) 12473 and AR11176, as well as preforming relaxation runs for AR12473 (found to be eruptive) where the driving is switched off systematically at different time steps. We intend to analyse the behaviour and evolution of fundamental quantities that are essential for understanding the eruptivity of magnetic flux ropes (MFRs). The data-driven simulations were carried out with the time-dependent magnetofrictional model (TMFM) for AR12473 and AR11176. For the relaxation runs, we employed the magnetofrictional method (MFM) and a zero-beta magnetohydrodynamic (MHD) model to investigate how significant the differences between the two relaxation procedures are when started from the same initial conditions. In total, 22 simulations were studied. To determine the eruptivity of the MFRs, we calculated and analysed characteristic geometric properties such as the cross-section, MFR height, and physical stability parameters such as MFR twist and the decay index. Furthermore, for the eruptive cases, we investigated the effect of sustained driving beyond the point of eruptivity on the MFR properties and evolution. We find that the fully driven AR12473 MFR is eruptive, while the AR11176 MFR is not. For the relaxation runs, we find that the MFM MFRs are eruptive when the driving is stopped around the flare time or later, while the MHD MFRs show eruptive behaviour even if the driving is switched off one and a half days before the flare occurs. We also find that characteristic MFR properties can vary greatly even for the eruptive cases of different relaxation simulations. The results suggest that data driving can significantly influence the evolution of the eruption, with differences appearing even when the relaxation time is set to later stages of the simulation when the MFRs have already entered an eruptive phase. Moreover, the relaxation model affects the results significantly, as highlighted by the differences between the MFM and MHD MFRs, showing that eruptivity in MHD does not directly translate to eruptivity in the MFM, despite the same initial conditions. Finally, if the exact critical values of instability parameters are unknown, tracking the evolution of typical MFR properties can be a powerful tool for determining MFR eruptivity.

Rotation and Confined Eruption of a Double Flux-rope System

We perform a data-constrained simulation with the zero-β assumption to study the mechanisms of strong rotation and failed eruption of a filament in active region 11474 on 2012 May 5 observed by Solar Dynamics Observatory and Solar Terrestrial Relations Observatory. The initial magnetic field is provided by nonlinear force-free field extrapolation, which is reconstructed by the regularized Biot–Savart laws and magnetofrictional method. Our simulation reproduces most observational features very well, e.g., the filament large-angle rotation of about 130°, the confined eruption, and the flare ribbons, allowing us to analyze the underlying physical processes behind observations. We discover two flux ropes in the sigmoid system, an upper flux rope (MFR1) and a lower flux rope (MFR2), which correspond to the filament and hot channel in observations, respectively. Both flux ropes undergo confined eruptions. MFR2 grows by tether-cutting reconnection during the eruption. The rotation of MFR1 is related to the shear-field component along the axis. The toroidal field tension force and the nonaxisymmetry forces confine the eruption of MFR1. We also suggest that the mutual interaction between MFR1 and MFR2 contributes to the large-angle rotation and the eruption failure. In addition, we calculate the temporal evolution of the twist and writhe of MFR1, which is a hint of probable reversal rotation.

Data-constrained Magnetohydrodynamic Simulation of an Intermediate Solar Filament Eruption

Solar eruptive activities could occur in weak magnetic field environments and over large spatial scales, which are especially relevant to eruptions involving intermediate or quiescent solar filaments. To handle the large scales, we implement and apply a flux rope embedding method using regularized Biot–Savart laws in the spherical coordinate system. Combined with a potential field source surface model and a magneto-frictional method, a nonlinear force-free field comprising a flux rope embedded in a potential field is constructed. Using the combined nonlinear force-free field as the initial condition, we then perform a zero-β data-constrained magnetohydrodynamic (MHD) simulation for an M8.7 flare at 03:38 UT on 2012 January 23. The MHD model reproduces the eruption process, flare ribbon evolution (represented by the quasi-separatrix layer evolution), and kinematics of the flux rope. This approach could potentially model global-scale eruptions from weak field regions.

Growth of a filament channel by intermittent small-scale magnetic reconnection

Context. A filament channel (FC), a plasma volume where the magnetic field is primarily aligned with the polarity inversion line, is believed to be the pre-eruptive configuration of coronal mass ejections. Nevertheless, evidence for how the FC is formed is still elusive. Aims. In this paper, we present a detailed study of the build-up of a FC in order to understand its formation mechanism. Methods. The New Vacuum Solar Telescope (NVST) of the Yunnan Observatory and the Optical and Near-infrared Solar Eruption Tracer (ONSET) of Nanjing University, as well as the Atmospheric Imaging Assembly (AIA) and Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO), are used to study the growth process of the FC. Furthermore, we reconstruct the nonlinear force-free field (NLFFF) of the active region using the regularized Biot-Savart laws (RBSL) and the magnetofrictional method to reveal the three-dimensional (3D) magnetic field properties of the FC. Results. We find that partial filament materials are quickly transferred to longer magnetic field lines formed by small-scale magnetic reconnection, as evidenced by dot-like Hα and extreme ultraviolet (EUV) brightenings and subsequent bidirectional outflow jets, as well as untwisting motions. The Hα and EUV bursts appear repeatedly at the same location and are closely associated with flux cancelation, which occurs between two small-scale opposite polarities and is driven by shearing and converging motions. The 3D NLFFF model reveals that the reconnection takes place in a hyperbolic flux tube that is located above the flux-cancelation site and below the FC. Conclusions. The FC is gradually built up toward a twisted flux rope via a series of small-scale reconnection events that occur intermittently prior to the eruption.

On the limitations of magneto-frictional relaxation

The magneto-frictional method is used in solar physics to compute both static and quasi-static models of the Sun's coronal magnetic field. Here, we examine how accurately magneto-friction (without fluid pressure) is able to predict the relaxed state in a one-dimensional test case containing two magnetic null points. Firstly, we show that relaxation under the full ideal magnetohydrodynamic equations in the presence of nulls leads necessarily to a non-force-free state, which could not be reached exactly by magneto-friction. Secondly, the magneto-frictional solutions are shown to lead to breakdown of magnetic flux conservation, whether or not the friction coefficient is scaled with magnetic field strength. When this coefficient is constant, flux is initially conserved, but only until discontinuous current sheets form at the null points. In the ensuing weak solution, we show that magnetic flux is dissipated at these current sheets. The breakdown of flux conservation does not occur for an alternative viscous relaxation scheme.

Estimating the Magnetic Structure of an Erupting CME Flux Rope From AR12158 Using Data-Driven Modeling

We investigate here the magnetic properties of a large-scale magnetic flux rope related to a coronal mass ejection (CME) that erupted from the Sun on September 12, 2014 and produced a well-defined flux rope in interplanetary space on September 14–15, 2014. We apply a fully data-driven and time-dependent magnetofrictional method (TMFM) using Solar Dynamics Observatory (SDO) magnetograms as the lower boundary condition. The simulation self-consistently produces a coherent flux rope and its ejection from the simulation domain. This paper describes the identification of the flux rope from the simulation data and defining its key parameters (e.g., twist and magnetic flux). We define the axial magnetic flux of the flux rope and the magnetic field time series from at the apex and at different distances from the apex of the flux rope. Our analysis shows that TMFM yields axial magnetic flux values that are in agreement with several observational proxies. The extracted magnetic field time series do not match well with in-situ components in direct comparison presumably due to interplanetary evolution and northward propagation of the CME. The study emphasizes also that magnetic field time-series are strongly dependent on how the flux rope is intercepted which presents a challenge for space weather forecasting.

The Magnetic Flux Rope Structure of a Triangulated Solar Filament

Abstract Solar magnetic flux ropes are core structures driving solar activities. We construct a magnetic flux rope for a filament/prominence observed at 01:11 UT on 2011 June 21 in AR 11236 with a combination of state-of-the-art methods, including triangulation from multiperspective observations, the flux rope embedding method, the regularized Biot–Savart laws, and the magnetofrictional method. First, the path of the filament is reconstructed via the triangulation with 304 Å images observed by the Atmospheric Imaging Assembly on board Solar Dynamics Observatory (SDO) and by the Extreme Ultraviolet Imager on board the twin Solar Terrestrial Relations Observatory. Then, a flux rope is constructed with the regularized Biot–Savart laws using the information of its axis. Next, it is embedded into a potential magnetic field computed from the photospheric radial magnetic field observed by the Helioseismic and Magnetic Imager on board SDO. The combined magnetic field is finally relaxed by the magnetofrictional method to reach a nonlinear force-free state. It is found that both models constructed by the regularized Biot–Savart laws and after the magnetofrictional relaxation coincide with the 304 Å images. The distribution of magnetic dips coincides with part of the filament/prominence material, and the quasi-separatrix layers wrap the magnetic flux ropes, displaying hyperbolic flux tube structures. These models have the advantages of constructing magnetic flux ropes in the higher atmosphere and weak magnetic field regions, which could be used as initial conditions for magnetohydrodynamic simulations of coronal mass ejections.

Solar Magnetic Flux Rope Eruption Simulated by a Data-driven Magnetohydrodynamic Model

Abstract The combination of magnetohydrodynamic (MHD) simulation and multi-wavelength observations is an effective way to study the mechanisms of magnetic flux rope eruption. We develop a data-driven MHD model using the zero-β approximation. The initial condition is provided by a nonlinear force-free field derived from the magneto-frictional method based on vector magnetic field observed by the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory. The bottom boundary uses observed time series of the vector magnetic field and the vector velocity derived by the Differential Affine Velocity Estimator for Vector Magnetograms. We apply the data-driven model to active region 11123 observed from 06:00 UT on 2010 November 11 to about 2 hr later. The evolution of the magnetic field topology coincides with the flare ribbons observed in the 304 and 1600 Å wavebands by the Atmospheric Imaging Assembly. The morphology, propagation path, and propagation range of the flux rope are comparable with the observations in 304 Å. We also find that a data-constrained boundary condition, where the bottom boundary is fixed to the initial values, reproduces a similar simulation result. This model can reproduce the evolution of a magnetic flux rope in its dynamic eruptive phase.

Effects of inner Alfvén surface location on black hole energy extraction in the limit of a force-free magnetosphere

An energy extracting black hole magnetosphere can be defined by the location of its inner Alfv\'{e}n surface, which determines the rate of black hole energy extraction along a given magnetic field line. We study how the location of the inner Alfv\'{e}n surface can modify the structure of energy extracting black hole magnetospheres in the force-free limit. Hundreds of magnetospheres are numerically computed via a general relativistic extension of the Newtonian magnetofrictional method for a full range of black hole spins and flow parameters. We find that jet-like structures naturally form very close to the horizon for Alfv\'{e}n surfaces near the boundary of the ergosphere and that energy is extracted towards the equatorial plane for Alfv\'{e}n surfaces close to the horizon. This suggests that two broad classes of energy extracting black hole magnetospheres might exist; those that transmit extracted energy directly to distant observers, and those that transmit extracted energy to nearby accreting matter. Applied to transient high energy phenomena, we find that they might also differ in timescale by a factor of 20 or more.

MAGNETO-FRICTIONAL MODELING OF CORONAL NONLINEAR FORCE-FREE FIELDS. I. TESTING WITH ANALYTIC SOLUTIONS

ABSTRACT We report our implementation of the magneto-frictional method in the Message Passing Interface Adaptive Mesh Refinement Versatile Advection Code (MPI-AMRVAC). The method aims at applications where local adaptive mesh refinement (AMR) is essential to make follow-up dynamical modeling affordable. We quantify its performance in both domain-decomposed uniform grids and block-adaptive AMR computations, using all frequently employed force-free, divergence-free, and other vector comparison metrics. As test cases, we revisit the semi-analytic solution of Low and Lou in both Cartesian and spherical geometries, along with the topologically challenging Titov–Démoulin model. We compare different combinations of spatial and temporal discretizations, and find that the fourth-order central difference with a local Lax–Friedrichs dissipation term in a single-step marching scheme is an optimal combination. The initial condition is provided by the potential field, which is the potential field source surface model in spherical geometry. Various boundary conditions are adopted, ranging from fully prescribed cases where all boundaries are assigned with the semi-analytic models, to solar-like cases where only the magnetic field at the bottom is known. Our results demonstrate that all the metrics compare favorably to previous works in both Cartesian and spherical coordinates. Cases with several AMR levels perform in accordance with their effective resolutions. The magneto-frictional method in MPI-AMRVAC allows us to model a region of interest with high spatial resolution and large field of view simultaneously, as required by observation-constrained extrapolations using vector data provided with modern instruments. The applications of the magneto-frictional method to observations are shown in an accompanying paper.

MAGNETO-FRICTIONAL MODELING OF CORONAL NONLINEAR FORCE-FREE FIELDS. II. APPLICATION TO OBSERVATIONS

ABSTRACT A magneto-frictional module has been implemented and tested in the Message Passing Interface Adaptive Mesh Refinement Versatile Advection Code (MPI-AMRVAC) in the first paper of this series. Here, we apply the magneto-frictional method to observations to demonstrate its applicability in both Cartesian and spherical coordinates, and in uniform and block-adaptive octree grids. We first reconstruct a nonlinear force-free field (NLFFF) on a uniform grid of 1803 cells in Cartesian coordinates, with boundary conditions provided by the vector magnetic field observed by the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO) at 06:00 UT on 2010 November 11 in active region NOAA 11123. The reconstructed NLFFF successfully reproduces the sheared and twisted field lines and magnetic null points. Next, we adopt a three-level block-adaptive grid to model the same active region with a higher spatial resolution on the bottom boundary and a coarser treatment of regions higher up. The force-free and divergence-free metrics obtained are comparable to the run with a uniform grid, and the reconstructed field topology is also very similar. Finally, a group of active regions, including NOAA 11401, 11402, 11405, and 11407, observed at 03:00 UT on 2012 January 23 by SDO/HMI is modeled with a five-level block-adaptive grid in spherical coordinates, where we reach a local resolution of pixel−1 in an area of 790 Mm × 604 Mm. Local high spatial resolution and a large field of view in NLFFF modeling can be achieved simultaneously in parallel and block-adaptive magneto-frictional relaxations.

Evolution of flare ribbons, electric currents, and quasi-separatrix layers during an X-class flare

The standard model for eruptive flares has in the past few years been extended to 3D. It predicts typical J-shaped photospheric footprints of the coronal current layer, forming at similar locations as the Quasi-Separatrix Layers (QSLs). Such a morphology is also found for flare ribbons observed in the EUV band, as well as in non-linear force-free field (NLFFF) magnetic field extrapolations and models. We study the evolution of the photospheric traces of the current density and flare ribbons, both obtained with the SDO instruments. We investigate the photospheric current evolution during the 6 September 2011 X-class flare (SOL2011-09-06T22:20) from observational data of the magnetic field obtained with HMI. This evolution is compared with that of the flare ribbons observed in the EUV filters of the AIA. We also compare the observed electric current density and the flare ribbon morphology with that of the QSLs computed from the flux rope insertion method/NLFFF model. The NLFFF model shows the presence of a fan-spine configuration of overlying field lines, due to the presence of a parasitic polarity, embedding an elongated flux rope that appears in the observations as two parts of a filament. The QSLs, evolved via a magnetofrictional method, also show similar morphology and evolution as both the current ribbons and the EUV flare ribbons obtained at several times during the flare. For the first time, we propose a combined analysis of the photospheric traces of an eruptive flare, in a complex topology, with direct measurements of electric currents and QSLs from observational data and a magnetic field model. The results, obtained by two different and independent approaches, 1) confirm previous results of current increase during the impulsive phase of the flare, 2) show how NLFFF models can capture the essential physical signatures of flares even in a complex magnetic field topology.

Force-free twisted magnetospheres of neutron stars

The X-ray spectra observed in the persistent emission of magnetars are evidence for the existence of a magnetosphere. The high-energy part of the spectra is explained by resonant cyclotron upscattering of soft thermal photons in a twisted magnetosphere, which has motivated an increasing number of efforts to improve and generalize existing magnetosphere models. We want to build more general configurations of twisted, force-free magnetospheres as a first step to understanding the role played by the magnetic field geometry in the observed spectra. First we reviewed and extended previous analytical works to assess the viability and limitations of semi-analytical approaches. Second, we built a numerical code able to relax an initial configuration of a nonrotating magnetosphere to a force-free geometry, provided any arbitrary form of the magnetic field at the star surface. The numerical code is based on a finite-difference time-domain, divergence-free, and conservative scheme, based of the magneto-frictional method used in other scenarios. We obtain new numerical configurations of twisted magnetospheres, with distributions of twist and currents that differ from previous analytical solutions. The range of global twist of the new family of solutions is similar to the existing semi-analytical models (up to some radians), but the achieved geometry may be quite different. The geometry of twisted, force-free magnetospheres shows a wider variety of possibilities than previously considered. This has implications for the observed spectra and opens the possibility of implementing alternative models in simulations of radiative transfer aiming at providing spectra to be compared with observations.

Magnetized Accreting Stars: Studies of the Inner Accretion‐Disk Edge by a Magnetohydrodynamic Approach. II. Ideal Instabilities of the Inner Disk Edge

The interaction between an accretion disk and the magnetosphere of a rotating compact star is studied in two- and three-dimensional simulations to address the role of ideal (nonresistive) magnetohydrodynamic instabilities of the differentially rotating plasma. Two-dimensional simulations show the dominance of the gravitationally driven Rayleigh-Taylor instability over the shear-flow-driven Kelvin-Helmholtz instability. Three-dimensional simulations starting from the full equilibria obtained by a numerical magnetofrictional method rather than a radial cut at z = 0 generally show smaller growth rates and larger mode wavelengths due to magnetic line-tying. Thus growth rates of instability that cannot be calculated analytically because of the complicated magnetic field and plasma model can now be computed using numerical magnetohydrodynamic simulations. Variations of the accretion rate of a pulsar can be explained in terms of fast gravitational instability of the inner accretion disk edge and slow reloading of disk plasma.

Evolution of Solar Magnetic Arcades. I. Ideal MHD Evolution under Footpoint Shearing

The ideal MHD evolution of a single magnetic arcade undergoing footpoint motions in a two-dimensional Cartesian geometry is investigated using numerical simulation. Also, force-free states of the same arcade are constructed with the use of a magnetofrictional method, which is formulated differently from those used in previous studies. In MHD simulations, no instability or nonequilibrium is found to the value of shear 100 times as large as the footpoint separation in the potential field. The evolutionary sequence is composed of three distinct phases. The first phase is characterized by the increase of the toroidal field strength and the second phase by a sort of self-similar expansion. In the third phase, the formation and growth of a central current layer are conspicuous. With increasing shear, the maximum current density increases, the width of the current layer decreases, and the feet of the current layer, which bifurcates above the bottom boundary, get closer to each other. The field lines in the current layer tend to thread the bottom boundary nearly horizontally for a large shear. From our results, it is inductively inferred that the magnetic arcade in a two-dimensional Cartesian geometry approaches an open field as the shear increases indefinitely.

Cylindrically symmetric force-free magnetic fields

view Abstract Citations (16) References (21) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Cylindrically Symmetric Force-free Magnetic Fields Porter, Lisa J. ; Klimchuk, James A. ; Sturrock, Peter A. Abstract The magnetofrictional method was used to study the energy buildup in stressed coronal fields possessing cylindrical symmetry. Four different nonlinear, force-free magnetic-field configurations were examined. It was determined that, in all cases, a reasonable amount of twist in the field lines can produce enough free magnetic energy to power a typical flare. Furthermore, it was found that the rate of energy buildup is enhanced if the greatest twist and/or the magnetic flux is concentrated closer to the neutral line. It is thought that the open-field configuration (a configuration for which the field lines extend to infinity and the current is confined to a current sheet separating outgoing and incoming field lines) is the limiting state as one imposes infinite shear. The results of this work do not contradict this theory once numerical errors are taken into account. Publication: The Astrophysical Journal Pub Date: February 1992 DOI: 10.1086/170980 Bibcode: 1992ApJ...385..738P Keywords: Force-Free Magnetic Fields; Magnetic Field Configurations; Solar Corona; Solar Magnetic Field; Current Sheets; Solar Flares; Solar Physics; MAGNETOHYDRODYNAMICS: MHD; SUN: CORONA; SUN: FLARES; SUN: MAGNETIC FIELDS full text sources ADS |

Coronal magnetic fields produced by photospheric shear

The magneto-frictional method is used for computing force free fields to examine the evolution of the magnetic field of a line dipole, when there is relative shearing motion between the two polarities. It found that the energy of the sheared field can be arbitrarily large compared with the potential field. It is also found that it is possible to fit the magnetic energy, as a function of shear, by a simple functional form.

Quasi‐static magnetospheric MHD processes and the “ground state” of the magnetosphere

From the viewpoint of magnetic equilibrium and convection theories, we discuss whether there exists a theoretically well‐defined ground state of the Earth's magnetosphere. As a basis for this discussion, we review the quasi‐static MHD theory and show in a two‐dimensional (2‐D) model that the convecting magnetosphere can (mathematically) reach a steady state. The steady state configuration has fairly stretched magnetotail field lines and is, therefore, likely to be unstable to the tearing mode instability which ultimately leads to substorm onset in magnetotail regions with stretched field lines. We present an interpretation of the substorm phenomenon in terms of sequences of quasi‐static MHD equilibrium states and argue that, under the influence of convection, magnetic substorms occur periodically in Earth's magnetosphere, thus being an integral part of the entire convection cycle. After explosively releasing plasma and magnetic field energies in a substorm, the magnetosphere starts the convection process again from a low‐energy state. In addition to “smooth” tail field configurations, there exist oscillatory equilibrium tail structures with lower potential energy.We discuss the concept of the ground state of the magnetosphere within the context of 2‐D quasi‐static MHD models for the near‐Earth magnetosphere. However, we are unable to find a definition of a single magnetospheric ground state that is conceptually simple and computationally feasible. The term “ground state” denotes the lowest energy state of a system, which for a magnetosphere in quasi‐static force equilibrium is the empty vacuum magnetic field configuration with zero thermal plasma pressure. The real magnetosphere never approaches that state. Therefore we propose the term “average magnetosphere” for defining a baseline configuration that corresponds to average solar wind conditions. Earth's magnetosphere is convectively driven and powered by the solar wind. Thus all potential energy states W = ∫ [P/(γ − 1) + B²/2µ0] dV that belong to a particular convection time sequence between substorm events are transient in time. Therefore the newly defined “average magnetosphere” has a potential energy 〈W − Wvacuum〉 representing the time average of equilibrium states over the substorm cycle. Thus a “quiet” magnetosphere would have a net potential energy lower than the average, whereas an “active” magnetosphere would have a higher energy. It is an unresolved theoretical problem, however, to determine the lowest and highest possible energy states for the magnetosphere that are compatible with the thermodynamic condition of quasi‐static convection.Three‐dimensional (3‐D) magnetospheric equilibria have not been computed yet, but we describe our experience with Stern's stretch transformations and a magnetofrictional method for calculating 3‐D magnetospheric MHD equilibria.

Force-free magnetic fields - The magneto-frictional method

view Abstract Citations (251) References (9) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Force-free Magnetic Fields: The Magneto-frictional Method Yang, W. H. ; Sturrock, P. A. ; Antiochos, S. K. Abstract The problem under discussion is that of calculating magnetic field configurations in which the Lorentz force j x B is everywhere zero, subject to specified boundary conditions. We choose to represent the magnetic field in terms of Clebsch variables in the form B = grad alpha x grad beta. These variables are constant on any field line so that each field line is labeled by the corresponding values of alpha and beta. When the field is described in this way, the most appropriate choice of boundary conditions is to specify the values of alpha and beta on the bounding surface. We show that such field configurations may be calculated by a magneto-frictional method. We imagine that the field lines move through a stationary medium, and that each element of magnetic field is subject to a frictional force parallel to and opposing the velocity of the field line. This concept leads to an iteration procedure for modifying the variables alpha and beta, that tends asymptotically towards the force-free state. We apply the method first to a simple problem in two rectangular dimensions, and then to a problem of cylindrical symmetry that was previously discussed by Barnes and Sturrock (1972). In one important respect, our new results differ from the earlier results of Barnes and Sturrock, and we conclude that the earlier article was in error. Publication: The Astrophysical Journal Pub Date: October 1986 DOI: 10.1086/164610 Bibcode: 1986ApJ...309..383Y Keywords: Boundary Value Problems; Force-Free Magnetic Fields; Lorentz Force; Magnetic Field Configurations; Aerospace Environments; Boundary Conditions; Friction; Iteration; Velocity; Astrophysics; HYDROMAGNETICS full text sources ADS |