Nonlinear Force-free Field Model Research Articles

Nonlinear Force-free Field Modeling of Solar Coronal Jets in Theoretical Configurations

Abstract Coronal jets occur frequently on the Sun, and may contribute significantly to the solar wind. With the suite of instruments available now, we can observe these phenomena in greater detail than ever before. Modeling and simulations can assist further with understanding the dynamic processes involved, but previous studies tended to consider only one mechanism (e.g., emergence or rotation) for the origin of the jet. In this study we model a series of idealized archetypal jet configurations and follow the evolution of the coronal magnetic field. This is a step toward understanding these idealized situations before considering their observational counterparts. Several simple situations are set up for the evolution of the photospheric magnetic field: a single parasitic polarity rotating or moving in a circular path; as well as opposite polarity pairs involved in flyby (shearing), cancellation or emergence; all in the presence of a uniform, open background magnetic field. The coronal magnetic field is evolved in time using a magnetofrictional relaxation method. While magnetofriction cannot accurately reproduce the dynamics of an eruptive phase, the structure of the coronal magnetic field, as well as the buildup of electric currents and free magnetic energy are instructive. Certain configurations and motions produce a flux rope and allow the significant buildup of free energy, reminiscent of the progenitors of so-called blowout jets, whereas other, simpler configurations are more comparable to the standard jet model. The next stage is a comparison with observed coronal jet structures and their corresponding photospheric evolution.

On the Reliability of Magnetic Energy and Helicity Computations Based on Nonlinear Force-free Coronal Magnetic Field Models

Abstract We demonstrate the sensitivity of magnetic energy and helicity computations regarding the quality of the underlying coronal magnetic field model. We apply the method of Wiegelmann & Inhester to a series of Solar Dynamics Observatory/Helioseismic and Magnetic Imager vector magnetograms, and discuss nonlinear force-free (NLFF) solutions based on two different sets of the free model parameters. The two time series differ from each other concerning their force-free and solenoidal quality. Both force- and divergence-freeness are required for a consistent NLFF solution. Full satisfaction of the solenoidal property is inherent in the definition of relative magnetic helicity in order to ensure gauge independence. We apply two different magnetic helicity computation methods to both NLFF time series and find that the output is highly dependent on the level to which the NLFF magnetic fields satisfy the divergence-free condition, with the computed magnetic energy being less sensitive than the relative helicity. Proxies for the nonpotentiality and eruptivity derived from both quantities are also shown to depend strongly on the solenoidal property of the NLFF fields. As a reference for future applications, we provide quantitative thresholds for the force- and divergence-freeness, for the assurance of reliable computation of magnetic energy and helicity, and of their related eruptivity proxies.

Helical Twisting Number and Braiding Linkage Number of Solar Coronal Loops

Abstract Coronal loops in active regions are often characterized by quasi-circular and helically twisted (sigmoidal) geometries, which are consistent with dipolar potential field (PF) models in the former case, and with nonlinear force-free field models with vertical currents in the latter case. Alternatively, Parker-type nanoflare models of the solar corona hypothesize that a braiding mechanism operates between unresolved loop strands, which is a more complex topological model. In this study we use the vertical-current approximation of a nonpotential magnetic field solution (that fulfils the divergence-free and force-free conditions) to characterize the number of helical turns N twist in twisted coronal loops. We measure the helical twist in 15 active regions observed with Atmospheric Imaging Assembly and Helioseismic and Magnetic Imager/SDO (Solar Dynamic Observatory) and find a mean nonpotentiality angle (between the potential and nonpotential field directions) of μ NP = 15° ± 3°. The resulting mean rotational twist angle is φ = 49° ± 11°, which corresponds to N twist = φ/360° = 0.14 ± 0.03 turns with respect to the untwisted PF, with an absolute upper limit of N twist ≲ 0.5, which is far below the kink instability limit of . The number of twist turns N twist corresponds to the Gauss linkage number N link in braiding topologies. We conclude that any braided topology (with ) cannot explain the observed stability of loops in a force-free corona, nor the observed low twist number. Parker-type nanoflaring can thus occur in non-force-free environments only, such as in the chromosphere and transition region.

The Role of a Tiny Brightening in a Huge Geoeffective Solar Eruption Leading to the St. Patrick’s Day Storm

Abstract The largest magnetic storm in solar cycle 24 was caused by a fast coronal mass ejection (CME) that was related to a small C9.1 flare that occurred on 2015 March 15 in solar active region (AR) NOAA 12297. The purpose of this study is to understand the onset mechanism of the geoeffective huge solar eruption. We focused on the C2.4 flare that occurred prior to the C9.1 flare of the filament eruption. The magnetic field structure in the AR was complicated: there were several filaments, including the one that erupted and caused the CME. We hence carefully investigated the photospheric magnetic field, brightenings observed in the solar atmosphere, and the 3D coronal magnetic field extrapolated from nonlinear force-free field modeling, using data from Hinode and Solar Dynamics Observatory. We found three intriguing points. (1) There was a compact but noticeably highly twisted magnetic field structure that is represented by a small filament in the C2.4 flaring region, where a tiny precursor brightening (PB) was observed before the C2.4 flare. (2) The C2.4 flaring region is located in the vicinity of a foot point of the closed field that prohibits the filament from erupting. (3) The filament shows a sudden eruption after the C2.4 flare and the accompanying small filament eruption. From our analysis, we suggest that a small magnetic disturbance that was represented by the tiny PB at the time of the C2.4 flare is related to the trigger of the huge filament eruption.

Evolution of Photospheric Vector Magnetic Field Associated with Moving Flare Ribbons as Seen by GST

Abstract The photospheric response to solar flares, also known as coronal back reaction, is often observed as sudden flare-induced changes in the vector magnetic field and sunspot motions. However, it remains obscure whether evolving flare ribbons, the flare signature closest to the photosphere, are accompanied by changes in vector magnetic field therein. Here we explore the relationship between the dynamics of flare ribbons in the chromosphere and variations of magnetic fields in the underlying photosphere, using high-resolution off-band Hα images and near-infrared vector magnetograms of the M6.5 flare on 2015 June 22 observed with the 1.6 m Goode Solar Telescope. We find that changes of photospheric fields occur at the arrival of the flare ribbon front, thus propagating analogously to flare ribbons. In general, the horizontal field increases and the field lines become more inclined to the surface. When ribbons sweep through regions that undergo a rotational motion, the fields transiently become more vertical with decreased horizontal field and inclination angle, and then restore and/or become more horizontal than before the ribbon arrival. The ribbon propagation decelerates near the sunspot rotation center, where the vertical field becomes permanently enhanced. Similar magnetic field changes are discernible in magnetograms from the Helioseismic and Magnetic Imager (HMI), and an inward collapse of coronal magnetic fields is inferred from the time sequence of nonlinear force-free field models extrapolated from HMI magnetograms. We conclude that photospheric fields respond nearly instantaneously to magnetic reconnection in the corona.

Eruption of a multi-flux-rope system in solar active region 12673 leading to the two largest flares in Solar Cycle 24

Context. Solar active region (AR) 12673 in 2017 September produced the two largest flares in Solar Cycle 24: the X9.3 flare on September 6 and the X8.2 flare on September 10. Aims. We attempt to investigate the evolutions of the two large flares and their associated complex magnetic system in detail. Methods. Combining observations from the Solar Dynamics Observatory and results of nonlinear force-free field (NLFFF) modeling, we identify various magnetic structures in the AR core region and examine the evolution of these structures during the flares. Results. Aided by the NLFFF modeling, we identify a double-decker flux rope configuration above the polarity inversion line (PIL) in the AR core region. The north ends of these two flux ropes were rooted in a negative- polarity magnetic patch, which began to move along the PIL and rotate anticlockwise before the X9.3 flare on September 6. The strong shearing motion and rotation contributed to the destabilization of the two magnetic flux ropes, of which the upper one subsequently erupted upward due to the kink-instability. Then another two sets of twisted loop bundles beside these ropes were disturbed and successively erupted within five minutes like a chain reaction. Similarly, multiple ejecta components were detected as consecutively erupting during the X8.2 flare occurring in the same AR on September 10. We examine the evolution of the AR magnetic fields from September 3 to 6 and find that five dipoles emerged successively at the east of the main sunspot. The interactions between these dipoles took place continuously, accompanied by magnetic flux cancellations and strong shearing motions. Conclusions. In AR 12673, significant flux emergence and successive interactions between the different emerging dipoles resulted in a complex magnetic system, accompanied by the formations of multiple flux ropes and twisted loop bundles. We propose that the eruptions of a multi-flux-rope system resulted in the two largest flares in Solar Cycle 24.

The Evolution of Quasi-Separatrix Layer in Two Solar Eruptive Events

Quasi-separatrix layer, also called as QSL, is a region where magnetic connectivity changes drastically, and mostly well coincides with the location of flare ribbons in observations. The research on the relations of this topological structure with the 3-dimensional magnetic reconnection, and solar flares has attracted more and more attention. In this paper, using the theory of QSL we investigate a C5.7 classical two-ribbon solar flare (event 1) which occurred at AR11384 on 2011 December 26, and an M6.5 solar flare (event 2) which occurred at AR12371 on 2015 June 22, respectively. Combining the multi-wavelength data of AIA (Atmospheric Imaging Assembly) and vector magnetogrames of HMI (Helioseismic and Magnetic Imager) onboard SDO (Solar Dynamics Observatory), we extrapolate the coronal magnetic field using the PF (Potential Field) and NLFFF (Nonlinear Force Free Field) models, and calculate the evolution of the AR (Active Region) magnetic free energy. Then, we calculate the logarithmic distribution of Q-factors (magnetic squashing factor) at different heights above the solar photosphere with the results of the PF and NLFFF extrapolations, in order to determine the location of QSL. Afterward, we investigate the evolutionary relation between the QSLs at different heights above the solar photosphere and the flare ribbons observed at the corresponding heights. Finally, we study the multi-wavelength evolution features of the 2 flare events, and obtain by calculation the mean slip velocities of magnetic lines in the event 2 at 304 Å and 335 Å to be 4.6 km s-1 and 6.3 km s-1, respectively. We find that the calculated location of QSL in the chromosphere and corona is in good agreement with the location of flare ribbons at the same height, and the QSLs at different heights have almost the same evolutionary behavior in time as the flare ribbons of the corresponding heights, which highlights the role of QSL in the research of 3D magnetic reconnection and solar flare, and we suggest that the energy release in the flare of event 2 may be triggered by the magnetic reconnection at the place of QSL. We also suggest that the QSL is very important for us to study the essential relation between the 3D and 2D magnetic reconnections.

Pre-eruptive Magnetic Reconnection within a Multi-flux-rope System in the Solar Corona

Abstract The solar corona is frequently disrupted by coronal mass ejections (CMEs), whose core structure is believed to be a flux rope made of helical magnetic field. This has become a “standard” picture; though, it remains elusive how the flux rope forms and evolves toward eruption. While one-third of the ejecta passing through spacecraft demonstrate a flux-rope structure, the rest have complex magnetic fields. Are they originating from a coherent flux rope, too? Here we investigate the source region of a complex ejecta, focusing on a flare precursor with definitive signatures of magnetic reconnection, i.e., nonthermal electrons, flaring plasma, and bidirectional outflowing blobs. Aided by nonlinear force-free field modeling, we conclude that the reconnection occurs within a system of multiple braided flux ropes with different degrees of coherency. The observation signifies the importance of internal structure and dynamics in understanding CMEs and in predicting their impacts on Earth.

Study of Three-dimensional Magnetic Structure and the Successive Eruptive Nature of Active Region 12371

Abstract We study the magnetic structure of a successively erupting sigmoid in active region 12371 by modeling the quasi-static coronal field evolution with nonlinear force-free field (NLFFF) equilibria. Helioseismic and Magnetic Imager/Solar Dynamic Observatory vector magnetograms are used as input to the NLFFF model. In all eruption events, the modeled structure resembles the observed pre-eruptive coronal sigmoid and the NLFFF core field is a combination of double inverse-J-shaped and inverse-S field lines with dips touching the photosphere. Such field lines are formed by the flux cancellation reconnection of opposite-J field lines at bald-patch locations, which in turn implies the formation of a weakly twisted flux-rope (FR) from large-scale sheared arcade field lines. Later on, this FR undergoes coronal tether-cutting reconnection until a coronal mass ejection is triggered. The modeled structure captured these major features of sigmoid-to-arcade-to-sigmoid transformation, which is reoccuring under continuous photospheric flux motions. Calculations of the field line twist reveal a fractional increase followed by a decrease of the number of pixels having a range of twist. This traces the buildup process of a twisted core field by slow photospheric motions and the relaxation after eruption, respectively. Our study infers that the large eruptivity of this AR is due to a steep decrease of the background coronal field meeting the torus instability criteria at a low height (≈40 Mm) in contrast to noneruptive ARs.

Three-dimensional Forward-fit Modeling of the Hard X-Ray and Microwave Emissions of the 2015 June 22 M6.5 Flare

Abstract The well-established notion of a “common population” of the accelerated electrons simultaneously producing the hard X-ray (HXR) and microwave (MW) emission during the flare impulsive phase has been challenged by some studies reporting the discrepancies between the HXR-inferred and MW-inferred electron energy spectra. The traditional methods of spectral inversion have some problems that can be mainly attributed to the unrealistic and oversimplified treatment of the flare emission. To properly address this problem, we use a nonlinear force-free field (NLFFF) model extrapolated from an observed photospheric magnetogram as input to the three-dimensional, multiwavelength modeling platform GX Simulator and create a unified electron population model that can simultaneously reproduce the observed HXR and MW observations. We model the end of the impulsive phase of the 2015 June 22 M6.5 flare and constrain the modeled electron spatial and energy parameters using observations made by the highest-resolving instruments currently available in two wavelengths, the Reuven Ramaty High Energy Solar Spectroscopic Imager for HXR and the Expanded Owens Valley Solar Array for MW. Our results suggest that the HXR-emitting electron population model fits the standard flare model with a broken power-law spectrum ( keV) that simultaneously produces the HXR footpoint emission and the MW high-frequency emission. The model also includes an “HXR-invisible” population of nonthermal electrons that are trapped in a large volume of magnetic field above the HXR-emitting loops, which is observable by its gyrosynchrotron radiation emitting mainly in the MW low-frequency range.

The Causes of Quasi-homologous CMEs

Abstract In this paper, we identified the magnetic source locations of 142 quasi-homologous (QH) coronal mass ejections (CMEs), of which 121 are from solar cycle (SC) 23 and 21 from SC 24. Among those CMEs, 63% originated from the same source location as their predecessor (defined as S-type), while 37% originated from a different location within the same active region as their predecessor (defined as D-type). Their distinctly different waiting time distributions, peaking around 7.5 and 1.5 hr for S- and D-type CMEs, suggest that they might involve different physical mechanisms with different characteristic timescales. Through detailed analysis based on nonlinear force-free coronal magnetic field modeling of two exemplary cases, we propose that the S-type QH CMES might involve a recurring energy release process from the same source location (by magnetic free energy replenishment), whereas the D-type QH CMEs can happen when a flux tube system is disturbed by a nearby CME.

Origin and structures of solar eruptions II: Magnetic modeling

The topology and dynamics of the three-dimensional magnetic field in the solar atmosphere govern various solar eruptive phenomena and activities, such as flares, coronal mass ejections, and filaments/prominences. We have to observe and model the vector magnetic field to understand the structures and physical mechanisms of these solar activities. Vector magnetic fields on the photosphere are routinely observed via the polarized light, and inferred with the inversion of Stokes profiles. To analyze these vector magnetic fields, we need first to remove the 180° ambiguity of the transverse components and correct the projection effect. Then, the vector magnetic field can be served as the boundary conditions for a force-free field modeling after a proper preprocessing. The photospheric velocity field can also be derived from a time sequence of vector magnetic fields. Three-dimensional magnetic field could be derived and studied with theoretical force-free field models, numerical nonlinear force-free field models, magnetohydrostatic models, and magnetohydrodynamic models. Magnetic energy can be computed with three-dimensional magnetic field models or a time series of vector magnetic field. The magnetic topology is analyzed by pinpointing the positions of magnetic null points, bald patches, and quasi-separatrix layers. As a well conserved physical quantity, magnetic helicity can be computed with various methods, such as the finite volume method, discrete flux tube method, and helicity flux integration method. This quantity serves as a promising parameter characterizing the activity level of solar active regions.

Non-potential Field Formation in the X-shaped Quadrupole Magnetic Field Configuration

Abstract Some types of solar flares are observed in X-shaped quadrupolar field configuration. To understand the magnetic energy storage in such a region, we studied non-potential field formation in an X-shaped quadrupolar field region formed in the active region NOAA 11967, which produced three X-shaped M-class flares on 2014 February 2. Nonlinear force-free field modeling was applied to a time series of vector magnetic field maps from the Solar Optical Telescope on board Hinode and the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory. Our analysis of the temporal three-dimensional magnetic field evolution shows that the sufficient free energy had already been stored more than 10 hr before the occurrence of the first M-class flare and that the storage was observed in a localized region. In this localized region, quasi-separatrix layers (QSLs) started to develop gradually from 9 hr before the first M-class flare. One of the flare ribbons that appeared in the first M-class flare was co-spatial with the location of the QSLs, suggesting that the formation of the QSLs is important in the process of energy release. These QSLs do not appear in the potential field calculation, indicating that they were created by the non-potential field. The formation of the QSLs was associated with the transverse photospheric motion of the pre-emerged flux and the emergence of a new flux. This observation indicates that the occurrence of the flares requires the formation of QSLs in the non-potential field in which free magnetic energy is stored in advance.

Magnetohydrodynamic Simulations for Studying Solar Flare Trigger Mechanism

Abstract In order to understand the flare trigger mechanism, we conduct three-dimensional magnetohydrodynamic simulations using a coronal magnetic field model derived from data observed by the Hinode satellite. Several types of magnetic bipoles are imposed into the photospheric boundary of the Nonlinear Force-free Field model of Active Region (AR) NOAA 10930 on 2006 December 13, to investigate what kind of magnetic disturbance may trigger the flare. As a result, we confirm that certain small bipole fields, which emerge into the highly sheared global magnetic field of an AR, can effectively trigger a flare. These bipole fields can be classified into two groups based on their orientation relative to the polarity inversion line: the so-called opposite polarity, and reversed shear structures, as suggested by Kusano et al. We also investigate the structure of the footpoints of reconnected field lines. By comparing the distribution of reconstructed field lines and observed flare ribbons, the trigger structure of the flare can be inferred. Our simulation suggests that the data-constrained simulation, taking into account both the large-scale magnetic structure and small-scale magnetic disturbance (such as emerging fluxes), is a good way to discover a flare-producing AR, which can be applied to space weather prediction.

High-resolution Observations of Downflows at One End of a Pre-eruption Filament

Abstract Studying the dynamics of filaments at the pre-eruption phase can shed light on the precursor of eruptive events. Such high-resolution studies (of the order of 0.″1) are highly desirable yet very rare. In this work, we present a detailed observation of a pre-eruption evolution of a filament obtained by the 1.6 m New Solar Telescope (NST) at the Big Bear Solar Observatory (BBSO). One end of the filament is anchored at the sunspot in the NOAA active region (AR) 11515, which is well observed by NST Hα off-bands from four hours before to one hour after the filament eruption. A M1.6 flare is associated with the eruption. We observed persistent downflowing materials along the Hα multi-threaded component of the loop toward the AR end during the pre-eruption phase. We traced the trajectories of plasma blobs along the Hα threads and obtained a plane-of-sky velocity of 45 km s−1 on average. Furthermore, we estimated the real velocities of the downflows and the altitude of the filament by matching the observed Hα threads with magnetic field lines extrapolated from a nonlinear force-free field model. Observations of chromospheric brightenings at the footpoints of the falling plasma blobs are also presented. The lower limit of the kinetic energy per second of the downflows through the brightenings is found to be ∼1021 erg. Larger FOV observations from BBSO full-disk Hα images show that the AR end of the filament started ascending four hours before the flare. We attribute the observed downflows at the AR end of the filament to the draining effect of the filament rising prior to its eruption. During the slow-rise phase, the downflows continuously drained away ∼1015g mass from the filament over a few hours, which is believed to be essential for the instability, and could be an important precursor of eruptive events.

Flare Energy Release in the Lower Solar Atmosphere near the Magnetic Field Polarity Inversion Line

Abstract We study flare processes in the solar atmosphere using observational data for an M1-class flare of 2014 June 12, obtained by the New Solar Telescope (NST/BBSO) and Helioseismic Magnetic Imager (HMI/SDO). The main goal is to understand triggers and manifestations of the flare energy release in the photosphere and chromosphere using high-resolution optical observations and magnetic field measurements. We analyze optical images, HMI Dopplergrams, and vector magnetograms, and use nonlinear force-free field (NLFFF) extrapolations for reconstruction of the magnetic topology and electric currents. The NLFFF modeling reveals the interaction of two magnetic flux ropes with oppositely directed magnetic fields in the polarity inversion line (PIL). These flux ropes are observed as a compact sheared arcade along the PIL in the high-resolution broadband continuum images from NST. In the vicinity of the PIL, the NST observations reveal the formation of a thin three-ribbon structure corresponding to a small-scale photospheric magnetic arcade. The observational results are evidence in favor of the primary energy release site located in the chromospheric plasma with strong electric currents concentrated near the PIL. In this case, magnetic reconnection is triggered by the interacting magnetic flux ropes forming a current sheet elongated along the PIL.

Magnetic Helicity Estimations in Models and Observations of the Solar Magnetic Field. III. Twist Number Method

Abstract We study the writhe, twist, and magnetic helicity of different magnetic flux ropes, based on models of the solar coronal magnetic field structure. These include an analytical force-free Titov–Démoulin equilibrium solution, non-force-free magnetohydrodynamic simulations, and nonlinear force-free magnetic field models. The geometrical boundary of the magnetic flux rope is determined by the quasi-separatrix layer and the bottom surface, and the axis curve of the flux rope is determined by its overall orientation. The twist is computed by the Berger–Prior formula, which is suitable for arbitrary geometry and both force-free and non-force-free models. The magnetic helicity is estimated by the twist multiplied by the square of the axial magnetic flux. We compare the obtained values with those derived by a finite volume helicity estimation method. We find that the magnetic helicity obtained with the twist method agrees with the helicity carried by the purely current-carrying part of the field within uncertainties for most test cases. It is also found that the current-carrying part of the model field is relatively significant at the very location of the magnetic flux rope. This qualitatively explains the agreement between the magnetic helicity computed by the twist method and the helicity contributed purely by the current-carrying magnetic field.

DIAGNOSTICS OF CORONAL HEATING IN ACTIVE-REGION LOOPS

ABSTRACT 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 summarize 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 extreme ultraviolet (EUV) emission in individual coronal loops in solar active regions can provide constraints on the volumetric heating function, and to develop a diagnostic for the heating function for a subset of loops that are found close to static thermal equilibrium. We reconstruct the coronal magnetic field from Solar Dynamics Observatory/HMI data using a nonlinear force-free magnetic field model. We model selected loops using a one-dimensional stationary model, with a heating rate dependent locally on the magnetic field strength along the loop, and we 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 E H on the magnetic field, , and controlling also the shape of the heating function: concentrated near the loop top, uniform and concentrated near the footpoints. The diagnostic 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.

SOLAR ERUPTION AND LOCAL MAGNETIC PARAMETERS

ABSTRACT It is now a common practice to use local magnetic parameters such as magnetic decay index for explaining solar eruptions from active regions, but there can be an alternative view that the global properties of the source region should be counted as a more important factor. We discuss this issue based on Solar Dynamics Observatory observations of the three successive eruptions within 1.5 hr from the NOAA active region 11444 and the magnetic parameters calculated using the nonlinear force-free field model. Two violent eruptions occurred in the regions with relatively high magnetic twist number (0.5–1.5) and high decay index (0.9–1.1) at the nominal height of the filament (12″) and otherwise a mild eruption occurred, which supports the local-parameter paradigm. Our main point is that the time sequence of the eruptions did not go with these parameters. It is argued that an additional factor, in the form of stabilizing force, should operate to determine the onset of the first eruption and temporal behaviors of subsequent eruptions. As supporting evidence, we report that the heating and fast plasma flow continuing for a timescale of an hour was the direct cause for the first eruption and that the unidirectional propagation of the disturbance determined the timing of subsequent eruptions. Both of these factors are associated with the overall magnetic structure rather than local magnetic properties of the active region.

Investigating Energetic X-Shaped Flares on the Outskirts of A Solar Active Region.

Typical solar flares display two quasi-parallel, bright ribbons on the chromosphere. In between is the polarity inversion line (PIL) separating concentrated magnetic fluxes of opposite polarity in active regions (ARs). Intriguingly a series of flares exhibiting X-shaped ribbons occurred at the similar location on the outskirts of NOAA AR 11967, where magnetic fluxes were scattered, yet three of them were alarmingly energetic. The X shape, whose center coincided with hard X-ray emission, was similar in UV/EUV, which cannot be accommodated in the standard flare model. Mapping out magnetic connectivities in potential fields, we found that the X morphology was dictated by the intersection of two quasi-separatrix layers, i.e., a hyperbolic flux tube (HFT), within which a separator connecting a double null was embedded. This topology was not purely local but regulated by fluxes and flows over the whole AR. The nonlinear force-free field model suggested the formation of a current layer at the HFT, where the current dissipation can be mapped to the X-shaped ribbons via field-aligned heat conduction. These results highlight the critical role of HFTs in 3D magnetic reconnection and have important implications for astrophysical and laboratory plasmas.