Topology of Magnetic Field and Coronal Heating in Solar Active Regions

Force-free magnetic fields can be computed by making use of a new numerical technique, in which the fields are represented by a boundary integral equation based on a specific Green's function. Vector magnetic fields observed on the photospheric surface can be taken as the boundary conditions of this equation. In this numerical computation, the following two points are emphasized: (1) A new method for data reduction is proposed, for removing uncertainties in boundary data and determining the parameter in this Green's function, which is important for solving the boundary integral equation. In this method, the transverse components of the observed boundary field are calibrated with a linear force-free field model without changing their azimuth. (2) The computed 3-D fields satisfy the divergence-free and force-free conditions with high precision. The alignment of these field lines is mostly in agreement with structures in Hα and Yohkoh soft X-ray images. Since the boundary data are calibrated with a linear force-free field model, the computed 3-D magnetic field can be regarded as a quasi-linear force-free field approximation. The reconstruction of 3-D magnetic field in active region NOAA 7321 was taken as an example to quantitatively exhibit the capability of our new numerical technique.     

Magnetic diffusion and flare energy buildup

Photospheric motion shears or twists solar magnetic fields to increase magnetic energy in the corona, because this process may change a current-free state of a coronal field to force-free states which carry electric current. This paper analyzes both linear and nonlinear two-dimensional force-free magnetic field models and derives relations of magnetic energy buildup with photospheric velocity field. When realistic data of solar magnetic field (B ₀ 10³ G) and photospheric velocity field (v _max 1 km s^–1) are used, it is found that 3–4 hours are needed to create an amount of free magnetic energy which is of the order of the current-free field energy. Furthermore, the paper studies situations in which finite magnetic diffusivities in photospheric plasma are introduced. The shearing motion increases coronal magnetic energy, while the photospheric diffusion reduces the energy. The variation of magnetic energy in the coronal region, then, depends on which process dominates.     

On practical representation of magnetic field

Various manners of determination of a magnetic field are reviewed briefly from the standpoint of practicality and uniqueness. Then a practical representation of magnetic fields in terms of a class of force-free magnetic field is described. The proposed scheme is based on the physical consideration that in the chromosphere and lower corona a quasistatic magnetic field must be nearly force-free and that for the class of force-free magnetic field, i.e., ×B=B with = constant, the magnetic field can be determined uniquely from the observed distribution of the vertical component of a magnetic field. The applicability of the representation is demonstrated by examples and the limitations are discussed.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Development of a topological model for solar flares

The main theoretical studies of the process involved in solar flares have been made in the two-dimensional approximation. However, the preliminary studies made with three field components suggest that reconnection could take place in the separatrices, the separator (intersection of separatrices) being a privileged location for this process. As a consequence the sites of flare kernels must be located on the intersections of the separatrices with the photosphere. Therefore, in order to understand the role of interacting large-scale structures in solar flares, we have analysed the topology of three-dimensional potential and linear force-free fields. The magnetic field has been modelled by a distribution of charges or dipoles located below the photosphere. This modelling permits us to define the field connectivity by the charges or the dipoles at both ends of every field line.We found that the appearance of a separator above the photosphere is more likely when a parasitic bipole emerges outside the axis that joins the main polarities and when the field lines are characteristic of a field created by dipoles. The separatrices derived in the potential and force-free hypothesis have different shapes. However, in the strong field regions where flares usually occur, the separatrices of the potential and force-free field models become closer. This property makes possible the use of the potential field, as a first estimate, for computing the location in the photosphere of the separatrices and for comparing this location with the position of observed H kernels. Displacements of the separatrices of a force-free field result from modifications of the free energy of the field. Then force-free fields have the further capability of predicting the kernel displacement. In all cases a configuration suitable for prominence support is found above the separator.     

Evolution of helically twisted prominence structures of March 11, 1979

Helical structures are generally associated with many eruptive solar prominences. Thus, study of their evolution in the solar atmosphere assumes importance. We present a study of a flare-associated erupting prominence of March 11, 1979, with conspicuous helically twisted structure, observed in H line center. We have attempted to understand the role played by twisted force-free magnetic fields in this event. In the analysis, we have assumed that the helical structures visible in H outline the field lines in which prominence tubes are embedded. Untwisting of observed prominence tubes and later, formation of open prominence structures provide evidence of restructuring of the magnetic field configuration over the active region during the course of prominence eruption. Temporal evolution of the force-free parameter is obtained for two main prominence tubes observed to be intertwined in a rope-like structure. Axial electric currents associated with the prominence tubes are estimated to be of the order of 10¹¹ A which decreased with time. Correspondingly, it is estimated that the rate of energy release was 10²⁸ erg s^–1 during the prominence eruption.     

A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. I. Theory

We derive an analytical approximation of nonlinear force-free magnetic field solutions (NLFFF) that can efficiently be used for fast forward-fitting to solar magnetic data, constrained either by observed line-of-sight magnetograms and stereoscopically triangulated coronal loops, or by 3D vector-magnetograph data. The derived NLFFF solutions provide the magnetic field components B _x (x), B _y (x), B _z (x), the force-free parameter α(x), the electric current density j(x), and are accurate to second-order (of the nonlinear force-free α-parameter). The explicit expressions of a force-free field can easily be applied to modeling or forward-fitting of many coronal phenomena.     

On the topology of filaments and chromospheric fibrils near sunspots

The similarity between the spiral topology of chromospheric fibrils and filaments observed in H near sunspots and the configuration of an axisymmetric force-free magnetic field is examined. It is suggested that some of the observed features could be interpreted in terms of the configuration of lines of force of an axisymmetric force-free chromospheric magnetic field. Implications of the results of analysis to the possible interpretations of other observed topological features near a sunspot are discussed.Visiting scientist at the High Altitude Observatory.The National Center for Atmospheric Research is sponsored by the National Science Foundation.Sponsored by the Office of Naval Research, Contract No. N 00014-67-C-0290.     

Probing Twisted Magnetic Field Using Microwave Observations in an M Class Solar Flare on 11 February, 2014

This work demonstrates the possibility of magnetic-field topology investigations using microwave polarimetric observations. We study a solar flare of GOES M1.7 class that occurred on 11 February, 2014. This flare revealed a clear signature of spatial inversion of the radio-emission polarization sign. We show that the observed polarization pattern can be explained by nonthermal gyrosynchrotron emission from the twisted magnetic structure. Using observations of the Reuven Ramaty High Energy Solar Spectroscopic Imager, Nobeyama Radio Observatory, Radio Solar Telescope Network, and Solar Dynamics Observatory, we have determined the parameters of nonthermal electrons and thermal plasma and identified the magnetic structure where the flare energy release occurred. To reconstruct the coronal magnetic field, we use nonlinear force-free field (NLFFF) and potential magnetic-field approaches. Radio emission of nonthermal electrons is simulated by the GX Simulator code using the extrapolated magnetic field and the parameters of nonthermal electrons and thermal plasma inferred from the observations; the model radio maps and spectra are compared with observations. We have found that the potential-magnetic-field approach fails to explain the observed circular polarization pattern; on the other hand, the Stokes-\(V\) map is successfully explained by assuming nonthermal electrons to be distributed along the twisted magnetic structure determined by the NLFFF extrapolation approach. Thus, we show that the radio-polarization maps can be used for diagnosing the topology of the flare magnetic structures where nonthermal electrons are injected.     

A two-dimensional model for a solar prominence: Effect of an external magnetic field

Using analytical approximations we study the effects of different external magnetic configurations on the half-width, mass, and internal magnetic structure of a quiescent solar prominence, modelled as a thin vertical sheet of cool plasma. Firstly, we build up a zeroth-order model and analyse the effects produced by a potential coronal field or a constant- force-free field. This model allows us to obtain the half-width and mass of the prominence for different values of the external field, pressure and shear angle. Secondly, the effects of these external magnetic configurations on a two-dimensional model proposed by Ballester and Priest (1987) are studied. The main effects are a change of the half-width with height, an increase of the mass, a decrease of the magnetic field strength with height and a change in the shape of the magnetic field lines.     

Preprocessing of Vector Magnetograph Data for a Nonlinear Force-Free Magnetic Field Reconstruction

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 nonlinear force-free optimization code. Unfortunately, the measurements of the photospheric magnetic field contain inconsistencies and noise. In particular, the transversal components (say B _x and B _y) of current vector magnetographs have their uncertainties. Furthermore, the magnetic field in the photosphere is not necessarily force free and often not consistent with the assumption of a force-free field above the magnetogram. 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.     

A Magnetostatic Grad�CRubin Code for Coronal Magnetic Field Extrapolations

The coronal magnetic field cannot be directly observed, but, in principle, it can be reconstructed from the comparatively well observed photospheric magnetic field. A?popular approach uses a nonlinear force-free model. Non-magnetic forces at the photosphere are significant, meaning the photospheric data are inconsistent with the force-free model, and this causes problems with the modeling (De Rosa et?al., Astrophys. J. 696, 1780, 2009). In this paper we present a numerical implementation of the Grad?CRubin method for reconstructing the coronal magnetic field using a magnetostatic model. This model includes a pressure force and a non-zero magnetic Lorentz force. We demonstrate our implementation on a simple analytic test case and obtain the speed and numerical error scaling as a function of the grid size.     

Validation and Benchmarking of a Practical Free Magnetic Energy and Relative Magnetic Helicity Budget Calculation in Solar Magnetic Structures

In earlier works we introduced and tested a nonlinear force-free (NLFF) method designed to self-consistently calculate the coronal free magnetic energy and the relative magnetic helicity budgets of observed solar magnetic structures. In principle, the method requires only a single, photospheric or low-chromospheric, vector magnetogram of a quiet-Sun patch or an active region and performs calculations without three-dimensional magnetic and velocity-field information. In this work we strictly validate this method using three-dimensional coronal magnetic fields. Benchmarking employs both synthetic, three-dimensional magnetohydrodynamic simulations and nonlinear force-free field extrapolations of the active-region solar corona. Our time-efficient NLFF method provides budgets that differ from those of more demanding semi-analytical methods by a factor of approximately three, at most. This difference is expected to come from the physical concept and the construction of the method. Temporal correlations show more discrepancies that are, however, soundly improved for more complex, massive active regions, reaching correlation coefficients on the order of, or exceeding, 0.9. In conclusion, we argue that our NLFF method can be reliably used for a routine and fast calculation of the free magnetic energy and relative magnetic helicity budgets in targeted parts of the solar magnetized corona. As explained in this article and in previous works, this is an asset that can lead to valuable insight into the physics and triggering of solar eruptions.     

How To Use Magnetic Field Information For Coronal Loop Identification

The structure of the solar corona is dominated by the magnetic field because the magnetic pressure is about four orders of magnitude higher than the plasma pressure. Due to the high conductivity the emitting coronal plasma (visible, e.g., in SOHO/EIT) outlines the magnetic field lines. The gradient of the emitting plasma structures is significantly lower parallel to the magnetic field lines than in the perpendicular direction. Consequently information regarding the coronal magnetic field can be used for the interpretation of coronal plasma structures. We extrapolate the coronal magnetic field from photospheric magnetic field measurements into the corona. The extrapolation method depends on assumptions regarding coronal currents, e.g., potential fields (current-free) or force-free fields (current parallel to magnetic field). As a next step we project the reconstructed 3D magnetic field lines on an EIT-image and compare with the emitting plasma structures. Coronal loops are identified as closed magnetic field lines with a high emissivity in EIT and a small gradient of the emissivity along the magnetic field.     

A solar magnetic field model and its 3-D boundary element method solution

A force-free magnetic field model with constant is established, and a boundary element method is proposed to solve the problem. The procedure ensures a unique solution as well as a finite magnetic energy content. The proposed formulation is effective in solving magnetic fields above the solar surface, and the validity of our procedure is demonstrated by satisfactory agreement between calculated and observed magnetograms.     

Preprocessing photospheric vector magnetograms for nonlinear force-free field extrapolation of the global corona

Nonlinear force-free magnetic field(NLFFF) extrapolation based on the observed photospheric magnetic field is the most important method to obtain the coronal magnetic field nowadays.However, raw photospheric magnetograms contain magnetic forces and small-scale noises, and fail to be consistent with the force-free assumption of NLFFF models. The procedure for removing the forces and noises in observed data is called preprocessing. In this paper, we extend the preprocessing code of Jiang Feng to spherical coordinates for a full sphere. We first smooth the observed data with Gaussian smoothing, and then split the smoothed magnetic field into a potential field and a non-potential field.The potential part is computed by a numerical potential field model, and the non-potential part is preprocessed using an optimization method to minimize the magnetic forces and magnetic torques. Applying the code to synoptic charts of the vector magnetic field from SDO/HMI, we find it can effectively reduce the noises and forces, and improve the quality of data for a better input which will be used for NLFFF extrapolations applied to the global corona.     

Flare build-up in preflare magnetic loops and nonlinear force-free magnetic fields

In this paper, we extend B. C. Low's study on nonlinear force-free magnetic fields. Based on Low's mathematical method, a revised boundary-value problem of the two-dimensional nonlinear force-free magnetic field is solved analytically. The solution shows that higher magnetic loops evolve towards preflare loops when the gradient of longitudinal magnetic field at the photospheric level and the angle (shear) included between the magnetic field line and magnetic neutral line increase with time. The density, temperature and the current density are higher in the preflare loops than in the high-lying magnetic loops. We believe it is the loops that provide conditions for the eruption of the flare.The original has been published in the Acta Astronomica Sinica 21 (1980), 152, in Chinese. The present paper completes the discussion and revises some of the preliminary results.     

An analysis of photospheric vector magnetograms, Hα images and soft X-ray images in a superactive region NOAA 7321

Based on the photospheric vector magnetograms taken at Huairou Solar Observing Station (HSOS), H images taken at Hiraiso Solar Terrestrial Research Center of Communication Research Laboratory, soft X-ray images taken by Yohkoh and an extrapolation method, the magnetic field structures and some active phenomena of the active region AR 7321 around 04:12 UT on 27 October 1992 are analyzed in this paper. A divergence of the transverse magnetic field, located at a newly formed spot, was found. At least four highly sheared magnetic field systems separately spread from this divergence to four other sites around this divergence. Vertical current concentrations are upward in this region and downward in the other four sites, and the corresponding H bright patches and soft X-ray bright loops coincide with these structures, confirming the existence of these four systems. The extrapolated magnetic force lines reconstructed by the Boundary Element Method (BEM), force-free field assumption, and boundary condition of observed photospheric vector magnetic field, coincide in space with the H bright patches and soft X-ray bright loops, showing that this extrapolation method is very effective and suggesting that the H bright patches and soft X-ray bright loops in this case represent the magnetic field structures in the upper atmosphere of the Sun.The bright structures in the H images and the soft X-ray images have a close correlation with the non-potential characteristics of the magnetic fields.     

Relationship between electric currents,photospheric motions,chromospheric activity,and magnetic field topology

Through coordinated observations made during the Max'91 campaign in June 1989 in Potsdam (magnetograms), Debrecen (white light and H), and Meudon (MSDP), we follow the evolution of the sunspot group in active region NOAA 5555 for 6 days. The topology of the coronal magnetic field is investigated by using a method based on the concept of separatrices - applied previously (Mandriniet al., 1991) to a magnetic region slightly distorted by field-aligned currents. The present active region differs by having significant magnetic shear. We find that the H flare kernels and the main photospheric electric current cells are located close to the intersection of the separatrices with the chromosphere, in a linear force-free field configuration adapted to the observed shear. Sunspot motions, strong currents, isolated polarities, or intersecting separatrices are not in themselves sufficient to produce a flare. A combination of them all is required. This supports the idea that flares are due to magnetic reconnection, when flux tubes with field-aligned currents move towards the separatrix locations.