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Failed filament eruption remains mysterious on its initiation, magnetic environment, and erupting and failing mechanisms. We present multi-wavelength observations of a failed filament eruption and its associated hovered coronal mass ejection (hovered-CME) from limb observations of the Ahead of Solar Terrestrial Relations Observatory. On-disk observations from Solar Dynamics Observatory show the expansion of the anchored leg of an S-shaped filament during the pre-eruption phase. The main eruption starts as a sudden ejection of the erupted leg, which is followed by the appearance of EUV brightening in the S-shaped magnetic field. The brightening is spatio-temporal accompanied with hard X-ray emission enhancement, and cancellation of opposite magnetic polarities, which imply possible reconnection. After reaching the maximum displacement, the erupted material drains back to the Sun along the remaining anchored leg. The non-linear force free magnetic field extrapolation shows an S-shaped magnetic field, formed by two magnetic structures, with a strong enveloped magnetic field. The decay index at the possible apex of the filament is 0.8–1.2. Observations indicate that the failed filament eruption is triggered by tether cutting reconnection and is possibly confined by the upper magnetic field. The hovered-CME, resulting from the failed filament eruption and recording as a coronal mass ejection (CME), may cause the overestimation of the CME count.

We present and discuss properties of the following magnetic field models in a magnetic cloud: Miller and Turner solution, modified Miller–Turner solution, Romashets–Vandas toroidal and integral models, and Krittinatham–Ruffolo model. Helicity of the magnetic field in all the models is the main feature of magnetic clouds. The first three models describe the magnetic field inside an ideal torus. In the integral model, parameters of a generating torus ambiguously determine the volume and form of the magnetic field region. In the Krittinatham–Ruffolo model, the cross-section radius of the torus is variable, thereby it corresponds more closely to the real form of magnetic clouds in the inner heliosphere. These models can be used to interpret in-situ observations of the magnetic flux rope, to study a Forbush decrease in magnetic clouds and transport effects of solar energetic particles injected into a coronal mass ejection.

Recent advances in the field of neural networks have made convolutional neural networks (CNNs) a conventional algorithm for many computer vision tasks including image recognition and object detection. Because modeling the coronal magnetic field of the Sun is an important objective in heliophysics, this study extends the use of CNNs to the application of coronal magnetic field modeling. We employ a simple one-parameter model of linear force-free magnetic fields (LFFFs) to model active regions of multiple dipolar configurations including the Active Region (AR) 11117. We use state-of-the-art architectures such as ResNet and Inception networks, and develop our customized network “SolarNet” to determine the associated LFFF parameter alpha from a set of pseudo-coronal loop images, which are generated using the modeled active regions. Our results show very high accuracy of determining the LFFF parameter alpha, thereby demonstrating the effectiveness of the generic and customized deep CNN architectures to understand the coronal magnetic field.

The configuration and stability of an important kinetic plasma structure, the current sheet, determine the efficiency of magnetic energy storage, release, and transport in surrounding plasmas. These properties depend on β(the ratio of plasma pressures to magnetic field pressures) and Mach number M (the ratio of bulk velocities to magnetosonic velocities). For the most investigated current sheet, the near-Earth magnetotail current sheet, these parameters fall within a relatively narrow range of values (high β, low M). To investigate current sheet behavior for a wider range of parameters, we explore current sheets in the magnetotail of Mars using MAVEN observations. We find that low-β, high-M current sheets are abundant in Mars' magnetotail, but high-β, low-M current sheets can also be found there. Low-β current sheets are nearly force-free, whereas high M current sheets are balanced by a plasma flow gradient along the tail. We compare current sheet distributions in a (β,M) space for the Martian magnetotail, the near-Earth magnetotail (using THEMIS), and the distant magnetotail (using ARTEMIS). We also find that the pressure balance in the Martian magnetotail current sheet can occur by contributions from a wide range of ion species, or, in low beta cases, from field-aligned currents generation a force-free magnetic field configuration. The Martian magnetotail is a natural laboratory where current sheet of various types can be found and investigated.

A nonlinear force-free field (NLFFF) extrapolation is widely used to reconstruct the three-dimensional magnetic field in the solar corona from the observed photospheric magnetic field. However, the pressure gradient and gravitational forces are ignored in the NLFFF model, even though the photospheric and chromospheric magnetic fields are not in general force-free. Here we develop a magnetohydrodynamic (MHD) relaxation method that reconstructs the solar atmospheric (chromospheric and coronal) magnetic field as a non-force-free magnetic field (NFFF) in magnetohydrostatic equilibrium where the Lorentz, pressure gradient, and gravitational forces are balanced. The system of basic equations for the MHD relaxation method is derived, and mathematical properties of the system are investigated. A robust numerical solver for the system is constructed based on the modern high-order shock capturing scheme. Two-dimensional numerical experiments that include the pressure gradient and gravitational forces are also demonstrated.

非线性无力场是天体物理中的重要数学模型，它是一套非线性偏微分方程组，经常用于太阳及恒星磁场的理论研究。在轴对称情形下，该方程组归化为满足特定边界条件的带有未知参数的二阶非线性常微分方程，此即所谓的Low-Lou解法。本文提出一种参数打靶法，作为对Low-Lou解法的探讨和补遗，并给出更多可选的数值无力场。 Nonlinear force-free magnetic field is an important mathematical model in astrophysics, which is a set of nonlinear partial differential equations, often used in the theoretical studies of solar and stellar magnetic fields. In the axisymmetric case, this set of partial differential equations is reduced into a nonlinear ordinary differential equation of second order with an unknown parameter, satisfying certain boundary condition. This is the so-called Low-Lou approach of the problem. In this paper, we propose a parametric shooting method as a technical supplement for the Low- Lou approach, offering more optional numerical force-free magnetic fields.

The solar active region (AR) 12192 was one of the most flare productive region of solar cycle 24, which produced many X-class flares; the most energetic being an X3.1 flare on October 24, 2014 at 21:10 UT. Customarily, such events are believed to be triggered by magnetic reconnection in coronal magnetic fields. Here we use the vector magnetograms from solar photosphere, obtained from Heliospheric Magnetic Imager (HMI) to investigate the magnetic field topology prior to the X3.1 event, and ascertain the conditions that might have caused the flare. To infer the coronal magnetic field, a novel non-force-free field (NFFF) extrapolation technique of the photospheric field is used, which suitably mimics the Lorentz forces present in the photospheric plasma. We also highlight the presence of magnetic null points and quasi-separatrix layers (QSLs) in the magnetic field topology, which are preferred sites for magnetic reconnections and discuss the probable reconnection scenarios., Comment: Proc. IAU Symp. 340. 2 pages, 4 figures, accepted for publication

Modeling the coronal magnetic field of the Sun is an important objective in heliophysics. In this study, we illustrate how to use deep learning to estimate the parameter for a magnetic field model. A linear force-free magnetic field configuration is employed to model, as an initial illustration, an active region by using two magnetic dipoles and to determine the associated linear force-free field (LFFF) alpha parameter from a set of pseudo-coronal loop images, which serve as training and validation sets to existing deep learning algorithms. Our results show very high accuracy of determining the LFFF parameter alpha from pseudo-coronal loop images.

The properties of four toroidal models of the magnetic flux rope are shown. The results of the analysis can be used to interpret in-situ observations of the magnetic flux rope and study Forbush-decrease in magnetic clouds.

In this paper three nonlinear force-free magnetic field equations such as the Liouville equation, Sine-Poisson equation and Sinh-Poisson equation are studied by (G′/G)-expansion method and exact periodic solutions are extracted. In all these cases the ratio of the current density and the magnetic field is not constant e.g. as happens in the solar atmosphere.

We study diffusion of charged particles in stationary stochastic magnetic field ${\bf B}$ with zero mean, $\langle {\bf B} \rangle = 0 $. In the case when electric current is carried by electrons, the field is force-free, $\mathrm{curl} \,{\bf B} = \alpha{\bf B} $, where $\alpha({\bf r})$ is an arbitrary scalar function. In a small region where the function $\alpha $ and the field magnitude $|{\bf B}|$ are approximately constant, the equations of motion of charged particles are integrated and reduced to the equation of mathematical pendulum. The transition from trapped to untrapped particles is continuously traced. Averaging over the magnetic field spectrum gives the spatial diffusion coefficient $D$ of particles as a function of the Larmor radius $r_L$ in the large-scale magnetic fields ($B_{LS}$) and magnetic field correlation length $L_0$. The diffusion coefficient turns out to be proportional to the Larmor radius, $D\propto r_L $, for $r_L L_0 /2\pi $. We apply obtained results to the diffusion of cosmic rays in the Galaxy, which contains a large number of independent regions with parameters $L_0$ and $B_{LS}$ varying in wide range. We average over $B_{LS}$ with the Kolmogorov spectrum and over $L_0$ with the distribution function $f(L_0)\propto L_0^{- 1+ \sigma}$. For the practically flat spectrum $\sigma = 1/15$, we have $ D\propto r_m^{0.7}$, which is consistent with observations., Comment: 10 pages, 5 figures

An analytical approximation of a nonlinear force-free magnetic field (NLFFF) solution was developed in Paper I, while a numerical code that performs fast forward-fitting of this NLFFF approximation to a line-of-sight magnetogram and coronal 3D loops has been described and tested in Paper II. Here we calculate the free magnetic energy $E_{\rm free}=E_{\rm N}-E_{\rm P}$, i.e. the difference of the magnetic energies between the nonpotential field and the potential field. A second method to estimate the free energy is obtained from the mean misalignment angle change $\Delta\mu=\mu_{\rm P}-\mu_{\rm N}$ between the potential and nonpotential field, which scales as $E_{\rm free}/E_{\rm P} \approx \tan^2{(\Delta\mu)}$. For four active regions observed with STEREO in 2007 we find free energies in the range of $q_{\rm free}=(E_{\rm free}/E_{\rm P}) \approx 1%-10%$, with an uncertainty of less than $\pm 2%$ between the two methods, while the free energies obtained from 11 other NLFFF codes exhibit a larger scatter of order $\approx\pm10%$. We find also a correlation between the free magnetic energy and the GOES flux of the largest flare that occurred during the observing period, which can be quantified by an exponential relationship, $F_{\rm GOES} \propto \exp{(q_{\rm free}/0.015)}$, implying an exponentiation of the dissipated currents., Comment: 23 pages, 8 Figures, 1 Table

Based on a second-order approximation of nonlinear force-free magnetic field solutions in terms of uniformly twisted field lines derived in Paper I, we develop here a numeric code that is capable to forward-fit such analytical solutions to arbitrary magnetogram (or vector magnetograph) data combined with (stereoscopically triangulated) coronal loop 3D coordinates. We test the code here by forward-fitting to six potential field and six nonpotential field cases simulated with our analytical model, as well as by forward-fitting to an exactly force-free solution of the Low and Lou (1990) model. The forward-fitting tests demonstrate: (i) a satisfactory convergence behavior (with typical misalignment angles of $\mu \approx 1^\circ-10^\circ$), (ii) relatively fast computation times (from seconds to a few minutes), and (iii) the high fidelity of retrieved force-free $\alpha$-parameters ($\alpha_{\rm fit}/\alpha_{\rm model} \approx 0.9-1.0$ for simulations and $\alpha_{\rm fit}/\alpha_{\rm model} \approx 0.7\pm0.3$ for the Low and Lou model). The salient feature of this numeric code is the relatively fast computation of a quasi-forcefree magnetic field, which closely matches the geometry of coronal loops in active regions, and complements the existing {\sl nonlinear force-free field (NLFFF)} codes based on photospheric magnetograms without coronal constraints., Comment: Solar PHysics, (in press), 25 pages, 11 figures

Recent general relativistic magnetohydrodynamic (MHD) simulations have suggested that relativistic jets from active galactic nuclei are powered by the rotational energy of central black holes through magnetic fields. Some mechanisms for the extraction of rotational energy of black holes have been proposed, such as the Penrose process, the Blandford–Znajek mechanism, the MHD/magnetic Penrose process, and superradiance. Among them, the Blandford–Znajek mechanism is the most promising as an important process of the central engines of the high-energy phenomena. However, a distinctive dynamic process of this mechanism has not yet been demonstrated intuitively, whereas extraction via the Penrose process has a simple interpretation. In this study, we performed numerical simulations of 1D force-free magnetodynamics (FFMD) with respect to the emergence of the Blandford–Znajek mechanism at the equatorial plane of the rapidly spinning black hole. A radial monopole-like magnetic field is used as an initial condition. We found a numerical solution of the steady-state region spreading from the ergosphere to the infinity point like a "tsunami." We can intuitively understand that this tsunami is caused and sustained by the degeneracy of the electromagnetic field of the perfect conducting plasma in the ergosphere. However, this degeneracy is not mathematically guaranteed in the time-dependent equations of FFMD. The degeneracy is given by an appropriate initial condition and is sustained by the condition at the "stretched horizon" of the black hole.

Context. In recent years the accuracy of magnetic field observations in the solar atmosphere has made considerable progress. Similar progress is being made in computer modeling of complex plasma systems and computer capabilities. Aims. To consider observed solar magnetic field structure in numerical simulations a new extrapolation method for solar magnetic fields is used to incorporate such fields into three-dimensional MHD simulations Methods. The simulation employs a new modified linear magnetic field extrapolation which is specifically designed to satisfy symmetry conditions which are generic to MHD models. The model domain includes photosphere, chromosphere, and corona. Results. The new model is applied to several solar field configurations and results are compared to three-dimensional field structure from observations and another extrapolation method. The new model provides a simple and efficient method for the simulation of observed solar magnetic field structures by constructing a three-dimensional initial field that is consistent with symmetry boundary conditions of MHD simulations.

Force-free extrapolations are widely used to study the magnetic field in the solar corona based on surface measurements. The extrapolations assume that the ratio of internal energy of the plasma to magnetic energy, the plasma-beta is negligible. Despite the widespread use of this assumption observations, models, and theoretical considerations show that beta is of the order of a few percent to more than 10%, and thus not small. We investigate what consequences this has for the reliability of extrapolation results. We use basic concepts starting with the force and the energy balance to infer relations between plasma-beta and free magnetic energy, to study the direction of currents in the corona with respect to the magnetic field, and to estimate the errors in the free magnetic energy by neglecting effects of the plasma (beta<, Accepted for publication in A&A (Oct.14,2015), 8 pages, 4 figures

Some new approaches for nonlinear force-free magnetic field are presented and new exact solutions are found analytically. Examples are given and some implications of results to astrophysical solar plasmas as well as tokamak or/and spheromak plasmas are discussed.

We compare six algorithms for the computation of nonlinear force-free (NLFF) magnetic fields (including optimization, magnetofrictional, Grad-Rubin based, and Green's function-based methods) by evaluating their performance in blind tests on analytical force-free-field models for which boundary conditions are specified either for the entire surface area of a cubic volume or for an extended lower boundary only. Figures of merit are used to compare the input vector field to the resulting model fields. Based on these merit functions, we argue that all algorithms yield NLFF fields that agree best with the input field in the lower central region of the volume, where the field and electrical currents are strongest and the effects of boundary conditions weakest. The NLFF vector fields in the outer domains of the volume depend sensitively on the details of the specified boundary conditions; best agreement is found if the field outside of the model volume is incorporated as part of the model boundary, either as potential field boundaries on the side and top surfaces, or as a potential field in a skirt around the main volume of interest. For input field (B) and modeled field (b), the best method included in our study yields an average relative vector error En =� |B− b|� /�| B|� of only 0.02 when all sides are specified and 0.14 for the case where only the lower boundary is specified, while

We compare the performance of two alternative algorithms which aim to construct a force-free magnetic field given suitable boundary conditions. For this comparison, we have implemented both algorithms on the same finite element grid which uses Whitney forms to describe the fields within the grid cells. The additional use of conjugate gradient and multigrid iterations result in quite effective codes. The Grad-Rubin and Wheatland-Sturrock-Roumeliotis algorithms both perform well for the reconstruction of a known analytic force-free field. For more arbitrary boundary conditions the Wheatland-Sturrock-Roumeliotis approach has some difficulties because it requires overdetermined boundary information which may include inconsistencies. The Grad-Rubin code on the other hand loses convergence for strong current densities. For the example we have investigated, however, the maximum possible current density seems to be not far from the limit beyond which a force free field cannot exist anymore for a given normal magnetic field intensity on the boundary., 21 pages, 13 figures

Knowledge regarding the coronal magnetic field is important for the understanding of many phenomena, like flares and coronal mass ejections. Because of the low plasma beta in the solar corona the coronal magnetic field is often assumed to be force-free and we use photospheric vector magnetograph data to extrapolate the magnetic field into the corona with the help of a non-linear force-free optimization code. Unfortunately the measurements of the photospheric magnetic field contain inconsistencies and noise. In particular the transversal components (say Bx and By) of current vector magnetographs have their uncertainties. Furthermore the magnetic field in the photosphere is not necessary force-free and often not consistent with the assumption of a force-free field above. We develop a preprocessing procedure to drive the observed non force-free data towards suitable boundary conditions for a force-free extrapolation. As a result we get a data set which is as close as possible to the measured data and consistent with the force-free assumption., Comment: 20 pages, 5 figures

It is generally assumed that magnetic fields inside interplanetary magnetic clouds and flux ropes in the solar photosphere are force-free. In order to model such fields, the solution of rot B = α B is commonly used where α = const. But comparisons of this solutions with observations show significant difference. To treat this problem,we examine the solutions with α = α 0 / ( 1 + r / r 0 ) 3 / 2 .

A proof is given of the following statement: if B is a smooth force-free magnetic field contained in a cylindrical domain of axis parallel toand of star-shaped cross-section, and if B is topologically equivalent to the uniform field B0 = B0 ˆ, then B = B0. In addition to being a very first step in the general study of the uniqueness of a magnetostatic equilibrium subject to a topological constraint, this result has a direct relevance to the approach recently developed by Ng & Bhattacharjee (1998, Phys. Plasma, 5, 4028) to support Parker's theory of current sheets formation in the solar corona.

Inspired by the analogy between the magnetic field and velocity field of incompressible fluid flow, we propose a fluid dynamics approach for computing nonlinear force-free magnetic fields. This method has the advantage that the divergence-free condition is automatically satisfied, which is a sticky issue for many other algorithms, and we can take advantage of modern high resolution algorithms to process the force-free magnetic field. Several tests have been made based on the well-known analytic solution proposed by Low & Lou. The numerical results are in satisfactory agreement with the analytic ones. It is suggested that the newly proposed method is promising in extrapolating the active region or the whole sun magnetic fields in the solar atmosphere based on the observed vector magnetic field on the photosphere.

Starting from Bernstein's principle of magnetohydrodynamic energy, a general analysis is presented for the stability of a kind of 1-D force-free magnetic fields with singular current density surfaces and a single parameter in cylindrical coordinates. It is found that in the parameter space of this kind of force-free magnetic fields there simultaneously exist stable and unstable regions. Their stability is solely determined by the radial distribution of the magnetic pitch in the neighborhood of the cylinder axis, and is independent of the presence of singular current density surface at the boundary of the field.

This paper considers single-particle trajectories in a planar sheared force-free magnetic field. A specific feature of this magnetic configuration is the absence of both gradient and curvature magnetic drifts, as well as a diamagnetic force along field lines. Therefore, in the framework of the drift approximation, the motion of the particle guiding centre does not feel the magnetic field's non-uniformity. Here we discuss how the latter affects actual particle trajectories, making them quite different from simple circular gyromotion even when the Larmor radius is small. It is also shown how magnetic confinement ceases to work when the Larmor radius becomes comparable to the spatial scale of the field variation.

We present a method to include stereoscopic information about the three dimensional structure of flux tubes into the reconstruction of the coronal magnetic field. Due to the low plasma beta in the corona we can assume a force free magnetic field, with the current density parallel to the magnetic field lines. Here we use linear force free fields for simplicity. The method uses the line of sight magnetic field on the photosphere as observational input. The value of $\alpha$ is determined iteratively by comparing the reconstructed magnetic field with the observed structures. The final configuration is the optimal linear force solution constrained by both the photospheric magnetogram and the observed plasma structures. As an example we apply our method to SOHO MDI/EIT data of an active region. In the future it is planned to apply the method to analyse data from the SECCHI instrument aboard the STEREO mission.

The magnetic fields in the solar corona are generally neither force-free nor axisymmetric and have complex dynamics that are difficult to characterize. Here we simulate the topological evolution of solar coronal magnetic field lines (MFLs) using a magnetohydrodynamic model. The simulation is initialized with a non-axisymmetric non-force-free magnetic field that best correlates with the observed vector magnetograms of solar active regions (ARs). To focus on these ideas, simulations are performed for the flaring AR 11283 noted for its complexity and well-documented dynamics. The simulated dynamics develops as the initial Lorentz force pushes the plasma and facilitates successive magnetic reconnections at the two X-type null lines present in the initial field. Importantly, the simulation allows for the spontaneous development of mass flow, unique among contemporary works, that preferentially reconnects field lines at one of the X-type null lines. Consequently, a flux rope consisting of low-lying twisted MFLs, which approximately traces the major polarity inversion line, undergoes an asymmetric monotonic rise. The rise is attributed to a reduction in the magnetic tension force at the region overlying the rope, resulting from the reconnection. A monotonic rise of the rope is in conformity with the standard scenario of flares. Importantly, the simulated dynamics leads to bifurcations of the flux rope, which, being akin to the observed filament bifurcation in AR 11283, establishes the appropriateness of the initial field in describing ARs., 25 pages, 11 figures, published in The Astrophysical Journal

Understanding coronal heating remains a central problem in solar physics. Many mechanisms have been proposed to explain how energy is transferred to and deposited in the corona. We summarise past observational studies that attempted to identify the heating mechanism and point out the difficulties in reproducing the observations of the solar corona from the heating models. The aim of this paper is to study whether the observed EUV emission in individual coronal loops in solar active regions can provide constraints on the volumetric heating function, and to develop a diagnostics for the heating function for a subset of loops that are found close to static thermal equilibrium. We reconstruct the coronal magnetic field from SDO/HMI data using a non-linear force free magnetic field model. We model selected loops using a 1D stationary model, with a heating rate dependent locally on the magnetic field strength along the loop, and calculate the emission from these loops in various EUV wavelengths for different heating rates. We present a method to measure a power index β defining the dependence of the volumetric heating rate EH on the magnetic field, EH ∝ Bβ, and controlling also the shape of the heating function: concentrated near the loop top, uniform and concentrated near the footpoints. The diagnostics is based on the dependence of the electron density on the index β. This method is free from the assumptions of the loop filling factor but requires spectroscopic measurements of the density-sensitive lines. The range of applicability for loops of different length and heating distributions is discussed, and the steps to solving the coronal heating problem are outlined.

In this paper, we present a model of the plasma beta above an active region and discuss its consequences in terms of coronal magnetic field modeling. The β-plasma model is representative and derived from a collection of sources. The resulting β variation with height in the solar atmosphere is used to emphasize that the assumption that the magnetic pressure dominates over the plasma pressure must be carefully employed when extrapolating the magnetic field. This paper points out (1) that the paradigm that the coronal magnetic field can be constructed from a force-free magnetic field must be used in the correct context, since the force-free region is sandwiched between two regions which have β>1, (2) that the chromospheric Mg ii–C iv magnetic measurements occur near the β-minimum, and (3) that, moving from the photosphere upwards, β can return to ∼1 at relatively low coronal heights, e.g., R∼1.2 R s.

A new method for reconstructing force-free magnetic fields from their boundary values, based on minimizing the global departure of an initial field from a force-free and solenoidal state, is presented. The method is tested by application to a known nonlinear solution. We discuss the obstacles to be overcome in the application of this method to the solar case: the reconstruction of force-free fields in the corona from measurements of the vector magnetic field in the low atmosphere.

Based on the energy integral and stability theorems given by Newcomb (1960) for linear pinch plasma, this paper proves that the one-dimensional force-free magnetic field with singular current density surface obtained by Low (1993) in cylindrical coordinates is definitely stable.

The description of linear force-free magnetic fields in terms of the Moses eigenfunctions of the curl operator begun previously is here completed by the derivation of a general expression for the field’s spherical curl transform. This enables the transform space representation of a given field to be determined and compared with that of other fields, assisting the analysis and classification of this type of magnetic field as well as providing a basis for generalization. The result obtained gives the spherical curl transform as a weighted projection of the vector Radon transform of the field on the appropriate curl eigenvector. The process is exemplified by the determination of the transforms of three fields: the simplest force-free magnetic field, and the Lundquist and classical spheromak fields. The latter two are both of interest as models of the magnetic fields of solar magnetic clouds, while the classical spheromak field is relevant to the design of nuclear fusion reactors as well. The use of the transform in generalizing the Lundquist field is briefly discussed. As before, all results apply equally well to the description of the Trkalian subset of Beltrami fields in fluid dynamics.

The nonaxisymmetric magnetodynamic instability of a streaming liquid jet endowed with surface tension and ambient with force-free magnetic field has been developed. The boundary value problem is formulated, simplified and solved. Upon applying the pressure balance across the perturbed interface, a general dispersion relation is derived and discussed. Some stability criteria are obtained as limiting cases. The capillary force is destabilizing only in a small domain of axisymmetric disturbance while it is stabilizing otherwise. The streaming has a strong destabilizing effect in all modes of perturbation and that influence is independent of the perturbed wave-length. The uniform magnetic field interior the liquid jet is stabilizing in all modes of perturbation. Each of the longitudinal and azimuthal Lundquist force-free fields separately is stabilizing but their combined effect is extensively destabilizing. The principle of the exchange of stability is allowed for the present model due to the non-uniform molity behaviour of the force-free fields. In spite of all that the electro-magnetic force stresses are able to endow the fluid with a sort of rigidity that improves the stability states.

In this paper some soft X-ray loops observed by Yohkoh/SXT and extrapolated magnetic fields from the vector magnetogram data observed at the Mitaka Flare Telescope are compared and analyzed. The computed field lines generally agree with Yohkoh/SXT coronal loops. Typical examples of the magnetic field intensities, loop widths, and loop heights along field lines are presented.

A method for the reconstruction of the linear force-free magnetic field in a bounded domain (as a rectangular box, Ω) is presented here. The Dirichlet boundary-value problem for the Helmholtz equation is solved for the B z component specified at the Ω boundary. Chebyshev's iteration method with the optimal rearrangement of the iteration parameters sequence was used. The solution is obtained as for the positive-definite, and for the non-sign-definite difference analogue of the differential operator ▽2 u + α2 u. Specifying two scalar functions, B x and B y on the intersection of the lateral part of the Ω boundary with one selected plane z = constant, and using B z inside the Ω, we have found B x and B y throughout Ω. The algorithm was tested with the numerical procedure which gives the analytic solution B of the linear force-free field (LFFF) equations for the dipole in a half-space. The root-mean-square deviation of the analytic solution B from the calculated B′ does not exceed 1.0%. Boundary conditions for the B′ calculation were taken as given by the analytic LFFF solution B. Comparison of B′ with B″, which was calculated by the potential non-photospheric boundary conditions, show that they differ significantly. Thus, the specification of boundary conditions at non-photospheric boundaries of the volume under consideration is of particular importance when modeling the LFFF in a bounded volume. The algorithm proposed here allows one to use the information about magnetic fields in the corona for the modeling of LFFF in a limited domain above an active region on the Sun.

An expression is derived giving the curl transform of a linear force-free magnetic field in terms of the curl eigenfunction projection on the sum of two integrals over the field in physical space. The result is verified by direct substitution of the curl eigenfunction expansion of the field and exemplified by deriving the curl transform for the simplest force-free magnetic field. This provides the means to compare the curl transforms of previously-worked-out fields with those derived directly from the transform space representations, effectively completing the description of linear force-free magnetic fields in terms of the curl eigenfunction expansion begun previously. The direct and inverse curl transforms provide a new approach to the practical and theoretical study of linear force-free magnetic fields (as well as Trkalian-Beltrami flows) and their classification.

We developed a new nonlinear force-free magnetic field (NLFFF) forward-fitting algorithm based on an analytical approximation of force-free and divergence-free NLFFF solutions, which requires as input a line-of-sight magnetogram and traced 2D loop coordinates of coronal loops only, in contrast to stereoscopically triangulated 3D loop coordinates used in previous studies. Test results of simulated magnetic configurations and from four active regions observed with STEREO demonstrate that NLFFF solutions can be fitted with equal accuracy with or without stereoscopy, which relinquishes the necessity of STEREO data for magnetic modeling of active regions (on the solar disk). The 2D loop tracing method achieves a 2D misalignment of $\mu_2=2.7^\circ\pm 1.3^\circ$ between the model field lines and observed loops, and an accuracy of $\approx 1.0%$ for the magnetic energy or free magnetic energy ratio. The three times higher spatial resolution of TRACE or SDO/AIA (compared with STEREO) yields also a proportionally smaller misalignment angle between model fit and observations. Visual/manual loop tracings are found to produce more accurate magnetic model fits than automated tracing algorithms. The computation time of the new forward-fitting code amounts to a few minutes per active region., Comment: ApJ, Febr 2013, (in press), 11 Figures

Force-free magnetic fields are described as solutions of special nonlinear partial differential equations which are replaced frequently through linear equations. To record the diversity of the structures of these fields, a discussion of the nonlinear equations is necessary. For this purpose the method of similarity analysis is used. The Lie symmetry groups admitted by the nonlinear equations for force-free magnetic fields are presented. To record and classify the different types of group-invariant solutions, one-and two-dimensional optimal systems of subalgebras are listed. The reduced equations of the two-dimensional optimal system are systems of ordinary differential equations, and their solutions define similarity solutions which are force-free magnetic fields. Only in one case is it necessary to calculate similarity solutions numerically. The corresponding reduced equations are a nonautonomous dynamical system with the similarity variable in the place of time.

Extrapolations of solar photospheric vector magnetograms into three-dimensional magnetic fields in the chromosphere and corona are usually done under the assumption that the fields are force-free. The field calculations can be improved by preprocessing the photospheric magnetograms. We compare two preprocessing methods presently in use, namely the methods of Wiegelmann et al. (2006) and Fuhrmann et al. (2007). The two preprocessing methods were applied to a recently observed vector magnetogram. We examine the changes in the magnetogram effected by the two preprocessing algorithms. Furthermore, the original magnetogram and the two preprocessed magnetograms were each used as input data for nonlinear force-free field extrapolations by means of two different methods, and we analyze the resulting fields. Both preprocessing methods managed to significantly decrease the magnetic forces and magnetic torques that act through the magnetogram area and that can cause incompatibilities with the assumption of force-freeness in the solution domain. Both methods also reduced the amount of small-scale irregularities in the observed photospheric field, which can sharply worsen the quality of the solutions. For the chosen parameter set, the Wiegelmann et al. method led to greater changes in strong-field areas, leaving weak-field areas mostly unchanged, and thus providing an approximation of the magnetic field vector in the chromosphere, while the Fuhrmann et al. method weakly changed the whole magnetogram, thereby better preserving patterns present in the original magnetogram. Both preprocessing methods raised the magnetic energy content of the extrapolated fields to values above the minimum energy, corresponding to the potential field. Also, the fields calculated from the preprocessed magnetograms fulfill the solenoidal condition better than those calculated without preprocessing., Accepted for Astron. Astrophys

The helical force-free equation, [triangledown] [times] B = [alpha]B, has been solved analytically in a toroidal coordinates system for a torus of arbitrary aspect ratio without the approximation of a large aspect ratio. The three-dimensional force-free equation is reduced to a scalar Helmholtz equation. A set of analytical solutions for the Helmholtz equation in the torus is presented. With these solutions, the eigenvalues have been obtained for an aspect ration R/a [>=] 1.5 and toroidal mode number -5 [ =] 1.5 14 refs., 3 figs., 1 tab.

Models of the magnetic field in the solar chromosphere and corona are still mainly based on theoretical extrapolations of photospheric measurements. For the practical calculation of the global field, the so-called source-surface model has been introduced, in which the influence of the solar wind is described by the requirement that the field be radial at some exterior (source) surface. Then the assumption that the field is current-free in the volume between the photosphere and this surface allows for its determination from the photospheric measurement. In the present paper a generalization of the source-surface model to force-free fields is proposed. In the generalized model the parameterα( = ∇ ×B·B/B2)must be non-constant (or vanish identically) and currents are restricted to regions with closed field lines. A mathematical algorithm for computing the field from boundary data is devised.

Two dimensional, three components compressible MHD equations are solved with multistep implicit scheme. The spontaneous reconnection of current sheets with force free field background is simulated. In comparison with the case where the background is potential field, a great difference is found. Our conclusion is that the force free magnetic field can accelerate reconnection. It may have important scientific significance to solar physics, for force free field is widely present in solar atmosphere.

[1] A torus-type flux rope model with an arbitrary aspect ratio was applied to an interplanetary magnetic cloud observed by ACE and Nozomi on 16–18 April 1999, when Nozomi was 0.2 AU downstream of ACE in the solar wind within 3° of heliocentric longitude. The large and small radii of the torus, the direction of the symmetric axis, and the crossing points of the spacecraft were determined so that they would minimize the sum of the square of the difference between the model field and the hourly averages of the observed field. Self-similar expansion of the flux rope was assumed in proportion with the heliocentric distance. The best fit model had large and small radii of 0.16 and 0.09 AU, respectively. Both spacecraft passed through the northern part of the torus. Difference in the magnetic field observed by the two spacecraft was explained by the difference in their paths through the magnetic cloud. The model fit was consistent with the direction of the vector normal to the preceding planar magnetic structures. The chirality of the flux rope was positive (left handed), suggesting that the solar source was on the Northern Hemisphere. Assuming a probable association with the filament disappearance observed on 13 April 1999 at N16 E00, it is inferred that the filament had traveled in interplanetary space across the ecliptic plane. It was also found that nearly the same fitting result was obtained using a single-spacecraft observation in the case of a torus-shaped magnetic cloud with a small aspect ratio.