Nonlinear Force-free Extrapolation Research Articles

Source Region and Launch Characteristics of Magnetic-arch-blowout Solar Coronal Mass Ejections Driven by Homologous Compact-flare Blowout Jets

We study the formation of four coronal mass ejections (CMEs) originating from homologous blowout jets. All of the blowout jets originated from NOAA Active Region (AR) 11515 on 2012 July 2, within a time interval of ≈14 hr. All of the CMEs were wide (angular widths ≈ 95°–150°), and propagated with speeds ranging between ≈300 and 500 km s−1 in LASCO coronagraph images. Observations at various EUV wavelengths in Solar Dynamics Observatory/Atmospheric Imaging Assembly images reveal that in all the cases, the source region of the jets lies at the boundary of the leading part of AR 11515 that hosts a small filament before each event. Coronal magnetic field modeling based on nonlinear force-free extrapolations indicates that in each case, the filament is contained inside of a magnetic flux rope that remains constrained by overlying compact loops. The southern footpoint of each filament is rooted in the negative polarity region where the eruption onsets occur. This negative polarity region undergoes continuous flux changes, including emergence and cancellation with opposite polarity in the vicinity of the flux rope, and the EUV images reveal brightening episodes near the filament’s southeastern footpoint before each eruption. Therefore, these flux changes are likely the cause of the subsequent eruptions. These four homologous eruptions originate near adjacent feet of two large-scale loop systems connecting from that positive polarity part of the AR to two remote negative polarity regions, and result in large-scale consequences in the solar corona.

Advancing Solar Magnetic Field Extrapolations through Multiheight Magnetic Field Measurements

Nonlinear force-free extrapolations are a common approach to estimate the 3D topology of coronal magnetic fields based on photospheric vector magnetograms. The force-free assumption is a valid approximation at coronal heights, but for the dense plasma conditions in the lower atmosphere, this assumption is not satisfied. In this study, we utilize multiheight magnetic field measurements in combination with physics-informed neural networks to advance solar magnetic field extrapolations. We include a flexible height-mapping, which allows us to account for the different formation heights of the observed magnetic field measurements. The comparison to analytical and simulated magnetic fields demonstrates that including chromospheric magnetic field measurements leads to a significant improvement of our magnetic field extrapolations. We also apply our method to chromospheric line-of-sight magnetograms from the Vector Spectromagnetograph (VSM) on the Synoptic Optical Long-term Investigations of the Sun (SOLIS) observatory, in combination with photospheric vector magnetograms from the Helioseismic Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO). The comparison to observations in extreme-ultraviolet wavelengths shows that the additional chromospheric information leads to a better agreement with the observed coronal structures. In addition, our method intrinsically provides an estimate of the corrugation of the observed magnetograms. With this new approach, we make efficient use of multiheight magnetic field measurements and advance the realism of coronal magnetic field simulations.

A New Global Nonlinear Force-Free Coronal Magnetic-Field Extrapolation Code Implemented on a Yin–Yang Grid

The solar magnetic field dominates and structures the solar coronal plasma. Detailed insights into the coronal magnetic field are important to understand most physical phenomena there. While direct, routine measurements of the coronal magnetic field are not available, field extrapolation of the photospheric vector-field measurements into the corona is the only way to study the structure and dynamics of the coronal field. Here we focus on global coronal structures traditionally modeled using spherical grids and synoptic vector magnetograms as boundary conditions. We developed a new code that performs nonlinear force-free magnetic-field extrapolations in spherical geometry. Our new implementation is based on a well-established optimization principle on a Cartesian grid and a single spherical finite-difference grid. In the present work, for the first time, the algorithm is able to reconstruct the magnetic field in the entire corona, including the polar regions. The finite-difference numerical scheme that was employed in previous spherical-code versions suffered from numerical inefficiencies because of the convergence of those grids on the poles. In our new code, we implement the so-called Yin–Yang overhead grid, the structure of which addresses this difficulty. Consequently, both the speed and accuracy of the optimization algorithm are improved compared to the previous implementations. We tested our new code using the well known semi-analytical model (Low and Lou solution). This is a commonly used benchmark for nonlinear force-free extrapolation codes.

Homologous Flaring Activity over a Sunspot Light Bridge in an Emerging Active Region

Abstract Sunspot light bridges are known to exhibit a variety of dynamic and persistent phenomena such as surges, small-scale jets, etc., in the chromosphere and transition region. While it has generally been proposed that magnetic reconnection is responsible for this small-scale dynamism, persistent flaring activity lasting several hours from the same spatial location on a sunspot light bridge has rarely been reported. We combine observations from the Atmospheric Imaging Assembly and the Helioseismic Magnetic Imager on board the Solar Dynamics Observatory to investigate homologous flaring activity over a small sunspot light bridge in an emerging flux region. The homologous flares all produced broad, collimated jets including a B6.4 class flare. The jets rise at a speed of about 200 km s−1, reach projected heights of about 98 Mm, and emerge from the same spatial location for nearly 14 hrs, after which they cease completely. A nonlinear force-free extrapolation of the photospheric magnetic field shows a low-lying flux rope connecting the light bridge to a remote opposite-polarity network. The persistent flares occur as a result of the rapid horizontal motion of the leading sunspot that causes the relatively vertical magnetic fields in the adjacent umbra to reconnect with the low-lying flux rope in the light bridge. Our results indicate that the flaring ceases once the flux rope has lost sufficient twist through repeated reconnections.

Observation and Modeling of High-temperature Solar Active Region Emission during the High-resolution Coronal Imager Flight of 2018 May 29

Abstract Excellent coordinated observations of NOAA active region 12712 were obtained during the flight of the High-resolution Coronal Imager (Hi-C) sounding rocket on 2018 May 29. This region displayed a typical active region core structure with relatively short, high-temperature loops crossing the polarity inversion line and bright “moss” located at the footpoints of these loops. The differential emission measure (DEM) in the active region core is very sharply peaked at about 4 MK. Further, there is little evidence for impulsive heating events in the moss, even at the high spatial resolution and cadence of Hi-C. This suggests that active region core heating is occurring at a high frequency and keeping the loops close to equilibrium. To create a time-dependent simulation of the active region core, we combine nonlinear force-free extrapolations of the measured magnetic field with a heating rate that is dependent on the field strength and loop length and has a Poisson waiting time distribution. We use the approximate solutions to the hydrodynamic loop equations to simulate the full ensemble of active region core loops for a range of heating parameters. In all cases, we find that high-frequency heating provides the best match to the observed DEM. For selected field lines, we solve the full hydrodynamic loop equations, including radiative transfer in the chromosphere, to simulate transition region and chromospheric emission. We find that for heating scenarios consistent with the DEM, classical signatures of energy release, such as transition region brightenings and chromospheric evaporation, are weak, suggesting that they would be difficult to detect.

The state of nonlinear force-free magnetic field extrapolation

Magnetic field extrapolation is the construction of a model solution for the coronal magnetic field in active regions from magnetic boundary data originating close to the Sun's surface. The nonlinear force-free model (in which the electric current density is parallel to the magnetic field) is often adopted to describe the coronal field. The solution of the nonlinear force-free equations is a challenging computational task, and the application of codes to available boundary data has revealed a number of significant problems with nonlinear force-free extrapolation. This paper summarises the present status of coronal field extrapolation, and describes some recent developments.

Full-disk nonlinear force-free field extrapolation of SDO/HMI and SOLIS/VSM magnetograms

Extrapolation codes in Cartesian geometry for modelling the magnetic field in the corona do not take the curvature of the Sun's surface into account and can only be applied to relatively small areas, e.g., a single active region. We compare the analysis of the photospheric magnetic field and subsequent force-free modeling based on full-disk vector maps from Helioseismic and Magnetic Imager (HMI) on board solar dynamics observatory (SDO) and Vector Spectromagnetograph (VSM) of the Synoptic Optical Long-term Investigations of the Sun (SOLIS). We use Helioseismic and Magnetic Imager and Vector Spectromagnetograph photospheric magnetic field measurements to model the force-free coronal field above multiple solar active regions, assuming magnetic forces to dominate. We solve the nonlinear force-free field equations by minimizing a functional in spherical coordinates over a full disk excluding the poles. After searching for the optimum modeling parameters for the particular data sets, we compare the resulting nonlinear force-free model fields. We compare quantities like the total magnetic energy content and free magnetic energy, the longitudinal distribution of the magnetic pressure and surface electric current density using our spherical geometry extrapolation code. The magnetic field lines obtained from nonlinear force-free extrapolation based on Helioseismic and Magnetic Imager and Vector Spectromagnetograph data have good agreement. However, the nonlinear force-free extrapolation based on Helioseismic and Magnetic Imager data have more contents of total magnetic energy, free magnetic energy, the longitudinal distribution of the magnetic pressure and surface electric current density compared to the one from Vector Spectromagnetograph data.

The Force-Free Electrodynamics Method for the Extrapolation of Coronal Magnetic Fields from Vector Magnetograms

We present a new improved version of our force-free electrodynamics (FFE) numerical code in spherical coordinates that extrapolates the magnetic field in the inner solar corona from a photospheric vector magnetogram. The code satisfies the photospheric boundary condition and the condition divB=0 to machine accuracy. The performance of our method is evaluated with standard convergence parameters, and is found to be comparable to that of other nonlinear force-free extrapolations.

MAGNETIC ENERGY AND HELICITY BUDGETS IN THE ACTIVE-REGION SOLAR CORONA. II. NONLINEAR FORCE-FREE APPROXIMATION

Expanding on an earlier work that relied on linear force-free magnetic fields, we self-consistently derive the instantaneous free magnetic energy and relative magnetic helicity budgets of an unknown three-dimensional nonlinear force-free magnetic structure extending above a single, known lower-boundary magnetic field vector. The proposed method does not rely on the detailed knowledge of the three-dimensional field configuration but is general enough to employ only a magnetic connectivity matrix on the lower boundary. The calculation yields a minimum free magnetic energy and a relative magnetic helicity consistent with this free magnetic energy. The method is directly applicable to photospheric or chromospheric vector magnetograms of solar active regions. Upon validation, it basically reproduces magnetic energies and helicities obtained by well-known, but computationally more intensive and non-unique, methods relying on the extrapolated three-dimensional magnetic field vector. We apply the method to three active regions, calculating the photospheric connectivity matrices by means of simulated annealing, rather than a model-dependent nonlinear force-free extrapolation. For two of these regions we correct for the inherent linear force-free overestimation in free energy and relative helicity that is larger for larger, more eruptive, active regions. In the third studied region, our calculation can lead to a physical interpretation of observed eruptive manifestations. We conclude that the proposed method, including the proposed inference of the magnetic connectivity matrix, is practical enough to contribute to a physical interpretation of the dynamical evolution of solar active regions.

How Should One Optimize Nonlinear Force-Free Coronal Magnetic Field Extrapolations from SDO/HMI Vector Magnetograms?

The Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO) provides photospheric vector magnetograms with a high spatial and temporal resolution. Our intention is to model the coronal magnetic field above active regions with the help of a nonlinear force-free extrapolation code. Our code is based on an optimization principle and has been tested extensively with semianalytic and numeric equilibria and applied to vector magnetograms from Hinode and ground-based observations. Recently we implemented a new version which takes into account measurement errors in photospheric vector magnetograms. Photospheric field measurements are often affected by measurement errors and finite nonmagnetic forces inconsistent for use as a boundary for a force-free field in the corona. To deal with these uncertainties, we developed two improvements: i) preprocessing of the surface measurements to make them compatible with a force-free field, and ii) new code which keeps a balance between the force-free constraint and deviation from the photospheric field measurements. Both methods contain free parameters, which must be optimized for use with data from SDO/HMI. In this work we describe the corresponding analysis method and evaluate the force-free equilibria by how well force-freeness and solenoidal conditions are fulfilled, by the angle between magnetic field and electric current, and by comparing projections of magnetic field lines with coronal images from the Atmospheric Imaging Assembly (SDO/AIA). We also compute the available free magnetic energy and discuss the potential influence of control parameters.

Coronal Magnetic Field Structure and Evolution for Flaring AR 11117 and Its Surroundings

In this study, photospheric vector magnetograms obtained with the Synoptic Optical Long-term Investigations of the Sun survey (SOLIS), are used as boundary conditions to model the three-dimensional nonlinear force-free (NLFF) coronal magnetic fields as a sequence of nonlinear force-free equilibria in spherical geometry. We study the coronal magnetic field structure inside active regions and its temporal evolution. We compare the magnetic field configuration obtained from NLFF extrapolation before and after flaring event in active region (AR) 11117 and its surroundings observed on 27 October 2010. We compare the magnetic field topologies and the magnetic energy densities and study the connectivities between AR 11117 and its surroundings. During the investigated time period, we estimate the change in free magnetic energy from before to after the flare to be 1.74x10^{32}erg which represents about 13.5% of nonlinear force-free magnetic energy before the flare. In this study, we find that electric currents from AR 11117 to its surroundings were disrupted after the flare.

The Free Energy of NOAA Solar Active Region AR 11029

The NOAA active region (AR) 11029 was a small but highly active sunspot region which produced 73 GOES soft X-ray flares during its transit of the disk in late October 2009. The flares appear to show a departure from the well-known power law frequency-size distribution. Specifically, too few GOES C-class and no M-class flares were observed by comparison with a power law distribution (Wheatland, Astrophys. J. 710, 1324, 2010). This was conjectured to be due to the region having insufficient magnetic energy to power the missing large events. We construct nonlinear force-free extrapolations of the coronal magnetic field of AR 11029 using data taken on 24 October by the SOLIS Vector SpectroMagnetograph (SOLIS/VSM) and data taken on 27 October by the Hinode Solar Optical Telescope SpectroPolarimeter (Hinode/SP). Force-free modeling with photospheric magnetogram data encounters problems, because the magnetogram data are inconsistent with a force-free model. We employ a recently developed “self-consistency” procedure which addresses this problem and accommodates uncertainties in the boundary data (Wheatland and Régnier, Astrophys. J. 700, L88, 2009). We calculate the total energy and free energy of the self-consistent solution, which provides a model for the coronal magnetic field of the active region. The free energy of the region was found to be ≈ 4×1029 erg on 24 October and ≈ 7×1031 erg on 27 October. An order of magnitude scaling between RHESSI non-thermal energy and GOES peak X-ray flux is established from a sample of flares from the literature and is used to estimate flare energies from the observed GOES peak X-ray flux. Based on the scaling, we conclude that the estimated free energy of AR 11029 on 27 October when the flaring rate peaked was sufficient to power M-class or X-class flares; hence, the modeling does not appear to support the hypothesis that the absence of large flares is due to the region having limited energy.

Nonlinear Force-Free Extrapolation of Emerging Flux with a Global Twist and Serpentine Fine Structures

We study the flux emergence process in NOAA active region 11024, between 29 June and 7 July 2009, by means of multi-wavelength observations and nonlinear force-free extrapolation. The main aim is to extend previous investigations by combining, as much as possible, high spatial resolution observations to test our present understanding of small-scale (undulatory) flux emergence, whilst putting these small-scale events in the context of the global evolution of the active region. The combination of these techniques allows us to follow the whole process, from the first appearance of the bipolar axial field on the east limb, until the buoyancy instability could set in and raise the main body of the twisted flux tube through the photosphere, forming magnetic tongues and signatures of serpentine field, until the simplification of the magnetic structure into a main bipole by the time the active region reaches the west limb. At the crucial time of the main emergence phase high spatial resolution spectropolarimetric measurements of the photospheric field are employed to reconstruct the three-dimensional structure of the nonlinear force-free coronal field, which is then used to test the current understanding of flux emergence processes. In particular, knowledge of the coronal connectivity confirms the identity of the magnetic tongues as seen in their photospheric signatures, and it exemplifies how the twisted flux, which is emerging on small scales in the form of a sea-serpent, is subsequently rearranged by reconnection into the large-scale field of the active region. In this way, the multi-wavelength observations combined with a nonlinear force-free extrapolation provide a coherent picture of the emergence process of small-scale magnetic bipoles, which subsequently reconnect to form a large-scale structure in the corona.

Nonlinear force-free field extrapolation in spherical geometry: improved boundary data treatment applied to a SOLIS/VSM vector magnetogram

Routine measurements of the solar magnetic field are mainly carried out in the photosphere. Therefore, one has to infer the field strength in the higher layers of the solar atmosphere from the measured photospheric field based on the assumption that the corona is force-free. Meanwhile, those measured data are inconsistent with the above force-free assumption. Therefore, one has to apply some transformations to these data before nonlinear force-free extrapolation codes can be applied. Extrapolation codes in cartesian geometry for modelling the magnetic field in the corona do not take the curvature of the Sun's surface into account and can only be applied to relatively small areas, e.g., a single active region. Here we apply a method for nonlinear force-free coronal magnetic field modelling and preprocessing of photospheric vector magnetograms in spherical geometry using the optimization procedure.We solve the nonlinear force-free field equations by minimizing a functional in spherical coordinates over a restricted area of the Sun. We extend the functional by an additional term, which allows to incorporate measurement error and treat regions with lacking observational data. We use vector magnetograph data from the Synoptic Optical Long-term Investigations of the Sun survey (SOLIS) to model the coronal magnetic field. We study two neighbouring magnetically connected active regions observed on May 15 2009. For vector magnetograms with variable measurement precision and randomly scattered data gaps (e.g., SOLIS/VSM) the new code yields field models which satisfy the solenoidal and force-free condition significantly better as it allows deviations between the extrapolated boundary field and observed boundary data within measurement errors. Data gaps are assigned to an infinite error. We extend this new scheme to spherical geometry and apply it for the first time to real data.

Study on Two Methods for Nonlinear Force-Free Extrapolation Based on Semi-Analytical Field

In this paper, two semi-analytical solutions of force free fields (\citeauthor{low90}, \citeyear{low90}) have been used to test two nonlinear force-free extrapolation methods. One is the boundary integral equation (BIE) method developed by \citeauthor{yan00} (\citeyear{yan00}), and another is the approximate vertical integration (AVI) method developed by \citeauthor{son06} (\citeyear{son06}). Some improvements for the AVI method have been taken to avoid the singular points in the process of calculation. It is found that the correlation coefficients between the first semi-analytical field and extrapolated field by BIE, and also that by improved AVI, are greater than $90\%$ below a height 10 of the $64 \times 64$ lower boundary. While for the second semi-analytical field, these correlation coefficients are greater than $80\%$ below the same relative height. Although the differences between the semi-analytical solutions and the extrapolated fields exist for both BIE and AVI methods, these two methods can give reliable results for the height of about 15$\%$ of the extent of the lower boundary.

NONLINEAR FORCE-FREE MODELING OF MAGNETIC FIELDS IN A SOLAR FILAMENT

We present a striking filament pattern in the nonlinear force-free (NLFF) chromospheric magnetic field of the active region NOAA 10956. The NLFF chromospheric field is extrapolated from the Hinode high-resolution photospheric vector magnetogram using the weighted optimization method. The modeled structure is characterized by a highly sheared field with strong horizontal magnetic components and has a virtually identical shape and location as the filament seen in Hα. The modeled field strength agrees with the recent He I 10830 A observations by Kuckein et al.. The unequivocal resemblance between the NLFF extrapolation and the Hα observation not only demonstrates the ability of the NLFF field to reproduce chromospheric features, but also provides a valuable diagnostic tool for the filament magnetic fields.

DUAL-STAGE RECONNECTION DURING SOLAR FLARES OBSERVED IN HARD X-RAY

In this Letter, we present hard X-ray (HXR) observation by the Reuven Ramaty High Energy Solar Spectroscopic Imager of the 2003 October 29 X10 flare. Two pairs of HXR conjugate footpoints have been identified during the early impulsive phase. This geometric configuration is very much in the manner predicted by the tether-cutting scenario first proposed by Moore & Roumeliotis. The HXR light curves show that the outer pair of footpoints disappeared much faster than the other pair. This temporal behavior further confirms that this event is a good example of the tether-cutting model. In addition, we reconstructed a three-dimensional magnetic field based on the nonlinear force-free extrapolation and found that each pair of HXR footpoints were indeed linked by corresponding magnetic field lines.

The correlation of fractal structures in the photospheric and the coronal magnetic field

This work examines the relation between the fractal properties of the photospheric magnetic patterns and those of the coronal magnetic fields in solar active regions.Our results show no correlation between the fractal dimensions of the observed 2d photospheric structures and the extrapolated unstable volumes in the corona, when nonlinear force-free extrapolation is used. This result is independent of efforts to (1) bring the photospheric magnetic fields closer to a nonlinear force-free equilibrium and (2) omit the lower part of the modeled magnetic field volume that is almost completely filled by unstable volumes. A significant correlation between the fractal dimensions of the photospheric and coronal magnetic features is only observed at the zero level (lower limit) of approximation of a current-free (potential) magnetic field extrapolation. We conclude that the complicated transition from photospheric non-force-free fields to coronal force-free ones hampers any direct correlation between the fractal dimensions of the 2d photospheric patterns and their 3d counterparts in the corona at the nonlinear force-free limit.

Evolution of the flaring active region NOAA 10540 as a sequence of nonlinear force-free field extrapolations

Context. The solar corona is structured by magnetic fields. As direct measurements of the coronal magnetic field are not routinely available, it is extrapolated from photospheric vector magnetograms. When magnetic flux emerges from below the solar surface and expands into the corona, the coronal magnetic field is destabilized, leading to explosive phenomena like flares or coronal mass ejections. Aims. We study the temporal evolution of the flaring active region NOAA 10540 and are in particular interested in the free magnetic energy available to power the flares associated with it. Methods. We extrapolated photospheric vector magnetograms measured with the Solar Flare Telescope, located in Tokyo, into the corona with the help of a nonlinear force-free field model. This coronal magnetic field model is based on a well-tested multigrid-like optimization code with which we were able to estimate the energy content of the 3D coronal field, as well as an upper limit for its free magnetic energy. Furthermore, the evolution of the energy density with height and time was studied. Results. The coronal magnetic field energy in active region 10540 increases slowly during the three days before an M6.1 flare and drops significantly after it. We estimated the energy that was set free during this event as ∝10 25 J. A sequence of nonlinear force-free extrapolations of the coronal magnetic field shows a build up of magnetic energy before a flare and release of energy during the flare. The drop in magnetic energy of the active region is sufficient to power an M6.1 flare.

Can We Improve the Preprocessing of Photospheric Vector Magnetograms by the Inclusion of Chromospheric Observations?

The solar magnetic field is key to understanding the physical processes in the solar atmosphere. Nonlinear force-free codes have been shown to be useful in extrapolating the coronal field upward from underlying vector boundary data. However, we can only measure the magnetic field vector routinely with high accuracy in the photosphere, and unfortunately these data do not fulfill the force-free condition. We must therefore apply some transformations to these data before nonlinear force-free extrapolation codes can be self-consistently applied. To this end, we have developed a minimization procedure that yields a more chromosphere-like field, using the measured photospheric field vectors as input. The procedure includes force-free consistency integrals, spatial smoothing, and – newly included in the version presented here – an improved match to the field direction as inferred from fibrils as can be observed in, for example, chromospheric Hα images. We test the procedure using a model active-region field that included buoyancy forces at the photospheric level. The proposed preprocessing method allows us to approximate the chromospheric vector field to within a few degrees and the free energy in the coronal field to within one percent.