Nonlinear Force-free Field Model Research Articles

Stability of the coronal magnetic field around large confined and eruptive solar flares

Context. The coronal magnetic field, which overlies the current-carrying field of solar active regions, straps the magnetic configuration below. The characteristics of this overlying field are crucial in determining if a flare will be eruptive and accompanied by a coronal mass ejection (CME), or if it will remain confined without a CME. Aims. In order to improve our understanding of the pre-requisites of eruptive solar flares, we study and compare different measures that characterize the eruptive potential of solar active regions – the critical height of the torus instability (TI) as a local measure and the helicity ratio as a global measure – with the structural properties of the underlying magnetic field, namely the altitude of the center of the current-carrying magnetic structure. Methods. Using time series of 3D optimization-based nonlinear force-free magnetic field models of ten different active regions (ARs) around the time of large solar flares, we determined the altitudes of the current-weighted centers of the non-potential model structures. Based on the potential magnetic field, we inspected the decay index, n, in multiple vertical planes oriented alongside or perpendicular to the flare-relevant polarity inversion line, and estimated the critical height (hcrit) of TI using different thresholds of n. The critical heights were interpreted with respect to the altitudes of the current-weighted centers of the associated non-potential structures, as well as the eruptive character of the associated flares, and the eruptive potential of the host AR, as characterized by the helicity ratio. Results. Our most important findings are that (i) hcrit is more segregated in terms of the flare type than the helicity ratio, and (ii) coronal field configurations with a higher eruptive potential (in terms of the helicity ratio) also appear to be more prone to TI. Furthermore, we find no pronounced differences in the altitudes of the non-potential structures prior to confined and eruptive flares. An aspect that requires further investigation is that, generally, the modeled non-potential structures do not really reside in a torus-instable regime, so the applicability of the chosen nonlinear force-free modeling approach when targeting the structural properties of the coronal magnetic field is unclear.

Onset mechanism of an inverted U-shaped solar filament eruption revealed by NVST, SDO, and STEREO-A observations

Context. Solar filaments, also called solar prominences when appearing on the solar limb, consist of dense, cool plasma suspended in the hot and tenuous corona. They are the main potential sources of solar storms. Aims. To understand the onset mechanism of solar filaments, we investigated the eruption process of an inverted U-shaped solar filament and two precursory jet-like activities. Methods. Using observations from the New Vacuum Solar Telescope (NVST), the Solar Dynamics Observatory (SDO), and the Solar Terrestrial Relations Observatory-Ahead (STEREO-A), we investigated the event from two distinct observational perspectives: on the solar disk using NVST and SDO, and on the solar limb using STEREO-A. We employed both a nonlinear force-free field model and a potential field model to reconstruct the coronal magnetic field with the aim to understand its magnetic properties. Results. Two precursor jet-like activities were observed before the eruption, displaying an untwisted rotation. The second activity released an estimated twist of over two turns. During these two jet-like activities, Y-shaped brightenings, which are newly emerging magnetic flux accompanied by magnetic cancellation, and the formation of newly moving fibrils were identified. When these observational features are combined, it can be inferred that these two precursor jet-like activities released the magnetic field that constrained the filament and were caused by newly emerging magnetic flux. Before the filament eruption, it was observed that some moving flows had been ejected from the site as the onset of two jet-like activities, indicating the same physical process as in the two jet-like activities. Extrapolations revealed that the filament was below the height of the decay index of 1.0 and had a strong magnetic field (540 Gauss) and a high twist number (2.4 turns) before the eruption. An apparent rotational motion was observed during the filament eruption. Conclusions. We deduce that the solar filament, exhibiting an inverted U-shape, is a significantly twisted flux rope. The eruption of the filament was initiated by the release of constraining magnetic fields through continuous magnetic reconnection. This reconnection process was caused by the emergence of new magnetic flux.

Observational study of intermittent solar jets:p-mode modulation

Aims.Recurring jets are observed in the solar atmosphere. They can erupt intermittently over a long period of time. By the observation of intermittent jets, we wish to understand what causes the characteristics of the periodic eruptions.Methods.We report intermittent jets observed by the Goode Solar Telescope (GST) with the TiO Broadband Filter Imager (BFI), the Visible Imaging Spectrometer (VIS) in Hα, and the Near-InfraRed Imaging Spectropolarimeter (NIRIS). The analysis was aided and complemented by 1400 Å and 2796 Å data from the Interface Region Imaging Spectrograph (IRIS). These observational instruments allowed us to analyze the temporal characteristics of the jet events. By constructing the Hαdopplergrams, we found that the plasma first moves upward, but during the second phase of the jet, the plasma flows back. Working with time slice diagrams, we investigated the characteristics of the jet dynamics.Results.The jet continued for up to 4 h. The time-distance diagram shows that the peak of the jet has clear periodic-eruption characteristics (5 min) during 18:00 UT–18:50 UT. We also found a periodic brightening phenomenon (5 min) during the jet bursts in the observed bands in the transition region (1400 Å and 2796 Å), which may be a response to intermittent jets in the upper solar atmosphere. The time lag is 3 min. Evolutionary images in the TiO band revealed a horizontal movement of the granulation at the location of the jet. By comparison to the quiet region of the Sun, we found that the footpoint of the jet is enhanced at the center of the Hαspectral line profile, without significant changes in the line wings. This suggests prolonged heating at the footpoint of the jet. In the mixed-polarity magnetic field region of the jet, we observed the emergence of magnetic flux, its cancellation, and shear, indicating possible intermittent magnetic reconnection. This is confirmed by the nonlinear force-free field model, which was reconstructed using the magneto-friction method.Conclusions.The multiwavelength analysis indicates that the events we studied were triggered by magnetic reconnection that was caused by mixed-polarity magnetic fields. We suggest that the horizontal motion of the granulation in the photosphere drives the magnetic reconnection, which is modulated byp-mode oscillations.

First Insights into the Applicability and Importance of Different 3D Magnetic Field Extrapolation Approaches for Studying the Preeruptive Conditions of Solar Active Regions

The three-dimensional (3D) coronal magnetic field has not yet been directly observed. However, for a better understanding and prediction of magnetically driven solar eruptions, 3D models of solar active regions are required. This work aims to provide insight into the significance of different extrapolation models for analyzing the preeruptive conditions of active regions with morphological parameters in 3D. Here, we employed potential field (PF), linear force-free field (LFFF), and nonlinear force-free field (NLFFF) models and a neural network-based method integrating observational data and NLFFF physics (NF2). The 3D coronal magnetic field structure of a “flaring” (AR11166) and “flare-quiet” (AR12645) active region, in terms of their flare productivity, is constructed via the four extrapolation methods. To analyze the evolution of the field, six prediction parameters were employed throughout, from the photosphere up to the base of the lower corona. First, we find that the evolution of the adopted morphological parameters exhibits similarity across the investigated time period when considering the four types of extrapolations. Second, all the parameters exhibited preeruptive conditions not only at the photosphere but also at higher altitudes in the case of active region (AR) 11166, while three out of the six proxies also exhibited preeruptive conditions in the case of AR12645. We conclude that: (i) the combined application of several different precursor parameters is important in the lower solar atmosphere to improve eruption predictions, and (ii) to gain a quick yet reliable insight into the preflare evolution of active regions in 3D, the PF and LFFF are acceptable; however, the NF2 method is likely the more suitable option.

Characteristics of the Accelerated Electrons Moving along the Loop Derived from Cyclical Microwave Brightenings at the Footpoints

Many particles are accelerated during solar flares. To understand the acceleration and propagation processes of electrons, we require the pitch-angle distributions of the particles. The pitch angle of accelerated electrons has been estimated from the propagation velocity of a nonthermal microwave source archived in Nobeyama Radioheliograph data. We analyzed a flare event (an M-class flare on 2014 October 22) showing cyclical microwave brightenings at the two footpoint regions. Assuming that the brightenings were caused by the accelerated electrons, we approximated the velocity parallel to the magnetic field of the accelerated electrons as ∼7.7 × 104 and 9.0 × 104 km s −1. The estimated pitch angle of the accelerated electrons is 69°–80° and the size of the loss cone at the footpoint (estimated from the magnetic field strength in the nonlinear force-free field model) is approximately 43°. Most of the accelerated electrons could be reflected at the footpoint region. This feature can be interpreted as brightenings produced by bouncing motion of the accelerated electrons.

Circular ribbon flare triggered from an incomplete fan-spine configuration

Context. Circular ribbon flares are characterised by circular, semi-circular, or elliptical ribbon brightenings. As the physics of such solar events involves a true 3D magnetic topology, they have been extensively studied in contemporary solar research. Aims. In order to understand the triggering processes and the complex magnetic topology involved in circular ribbon flares, we carried out a thorough investigation of an M-class circular ribbon flare that originated within close proximity of a quasi-separatrix layer (QSL). Methods. We combined multi-wavelength Atmospheric Imaging Assembly (AIA) and Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) observations with photospheric Helioseismic and Magnetic Imager (HMI) observations and coronal magnetic field modelling analysis using the non-linear force free field (NLFFF) model. Results. The circular ribbon flare occurred from a complex magnetic configuration characterised by negative magnetic patches surrounded by positive-polarity regions on three sides. As the negative polarity patches were not surrounded by positive-polarity regions on all four sides, the corresponding coronal field was devoid of any null points. This led to the formation of an incomplete fan-spine-like configuration that deviated from classical fan-spine configurations in null-point topology. Further, an observationally identified QSL structure was situated within the active region, very close to the flaring region. The presence of the QSL was verified by the NLFFF modelling. The far end of the spine-like lines terminated very close to one footpoint location of the QSL lines. Our analysis suggests that activities at this location led to the activation of a flux rope situated within the fan-like lines and triggering of the circular ribbon flare via perturbation of the overall fan-spine-like structure. Further, we identified RHESSI X-ray sources from the footpoints of the QSL structure, which suggests that slipping reconnections can also lead to discernible signatures of particle acceleration.

The Initiation and Back-reaction of the X5.4 Flare on 2012 March 7

In this paper, we study the evolution of the X5.4 flare (SOL2012-03-07T00:02) in NOAA Active Region 11429, focusing on its initiation mechanisms and back-reaction effects. To help our study, three-dimensional (3D) coronal magnetic field models are extrapolated from the photospheric magnetograms of the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory under the assumptions of nonlinear force-free field (NLFFF) and non-force-free field (non-FFF). We investigate the 3D magnetic structure and MHD kink instability, torus instability, and double-arc instability (DAI), and find that this flare is most likely triggered by the tether-cutting reconnection and the subsequent DAI. For the back-reactions of the flare, both NLFFF and non-FFF models clearly show an increase in horizontal magnetic field (B h ) and a decrease in inclination angle (ϕ) of the magnetic field near the polarity inversion line, from the photosphere up to a certain height (5 Mm and 8 Mm for non-FFF and NLFFF, respectively). In addition, the non-FFF model shows an enhancement of the downward Lorentz force acting on the photosphere, and the location of the enhancement spatially coincides with the location of the flare onset. The observed back-reaction is likely a consequence of magnetic reconnection.

Tracking magnetic flux and helicity from the Sun to Earth

Aims.We analyze the complete chain of effects – from the Sun to Earth – caused by a solar eruptive event in order to better understand the dynamic evolution of magnetic-field-related quantities in interplanetary space, in particular that of magnetic flux and helicity.Methods.We study a series of connected events – a confined C4.5 flare, a flare-less filament eruption, and a double-peak M-class flare – that originated in NOAA active region (AR) 12891 on late 2021 November 1 and early 2021 November 2. We deduce the magnetic structure of AR 12891 using stereoscopy and nonlinear force-free (NLFF) magnetic field modeling, allowing us to identify a coronal flux rope and to estimate its axial flux and helicity. Additionally, we compute reconnection fluxes based on flare ribbon and coronal dimming signatures from remote sensing imagery. Comparison to corresponding quantities for the associated magnetic cloud (MC) deduced from in situ measurements from Solar Orbiter and near-Earth spacecraft allows us to draw conclusions on the evolution of the associated interplanetary coronal mass ejection (CME). The latter analysis is aided by the application of geometric fitting techniques (graduated cylindrical shell modeling; GCS) and interplanetary propagation models (drag-based ensemble modeling; DBEM) to the interplanetary CME.Results.NLFF modeling suggests the magnetic structure of the host AR was in the form of a left-handed (negative-helicity) flux rope reaching altitudes of 8−10 Mm above photospheric levels, which is in close agreement with the corresponding stereoscopic estimate. GCS and DBEM modeling suggest that the ejected flux rope propagated in a self-similar expanding manner through interplanetary space. Comparison of magnetic fluxes and helicities processed by magnetic reconnection in the solar source region and the respective budgets of the MC indicate a considerable contribution from the eruptive process, though the pre-eruptive budgets also appear to be relevant.

Formation and Eruption of Hot Channels during an M6.5 Class Solar Flare

We investigate the formation and eruption of hot channels associated with the M6.5 class flare (SOL 2015-06-22T18:23) that occurred in NOAA AR 12371 on 2015 June 22. Two flare precursors are observed before the flare’s main phase. Observations in 94 and 131 Å by SDO/AIA revealed the early morphology of the first hot channel as a group of hot loops, termed the seed hot channel. A few seed hot channels are formed above the polarity inversion line (PIL) and the formation is associated with the parallel motion of the footpoint brightenings along the PIL, which proceeds into the early stage of the flare main phase. During this process, seed hot channels build up and rise slowly, accelerating at the peak of the second precursor. They merge during the process of acceleration, forming a larger hot channel, which then forms an inverted-γ-shape kinking structure. Before the flare peak, the second kinking hot channel with negative crossing appears near the first kinking hot channel that erupted. The eruption of these two hot channels produces two peaks on the main flare's GOES light curve. The propagation of the footpoint brightenings along the PIL indicates that the first kinking hot channel may be formed due to zipper reconnection. The occurrence of merging between seed hot channels observed by AIA is supported by the extrapolated nonlinear force-free field models. The observed writhing motion of the first kinking hot channel may be driven by the Lorentz force.

Topological Evolution of an Unwinding Blowout Jet

We investigate the topological evolution of coronal jet containing a sigmoid-like flux rope using a nonlinear force-free field model obtained with the flux-rope insertion method and magnetofrictional relaxation. We examine the topological evolution predicted by the unstable model with respect to observations from Solar Dynamic Observatory’s Atmospheric Imaging Array. We also calculate the squeezing factor, an approximation for sharp discontinuities in the magnetic field, and the coiling rate, an approximation for the amount of twist in the field. We identify at least two topological features where magnetic reconnection is likely taking place: an internal anemone-like region, near the filament, and an external region between the closed dome of the coronal jet and the ambient field. We also find evidence of reconnection below the filament, but it is not clear if the two inner regions are the same. We find that the internal region inflates the jet dome into the external region, which in turn initiates the fast eruption, allowing the inner region to unwind and the filament to escape. Finally, we examine the thermal evolution of the jet and trace the regions of enhanced emission-measure-weighted temperature (T EM) to the location of the expected reconnection regions. We find that magnetic field lines associated with the internal reconnection region are tied to increased T EM and emission in extreme-UV observations, indicative of heating. We identify this eruption as an untwisting jet, where unwinding magnetic field lines impart energy along the magnetic field forming the observed features of the coronal jet.

Sigmoid Formation through Slippage of a Single J-shaped Coronal Loop

A well-known precursor of an imminent solar eruption is the appearance of a hot S-shaped loop, also known as a sigmoid, in an active region (AR). Classically, the formation of such an S-shaped loop is envisaged to be implemented by magnetic reconnection of two oppositely oriented J-shaped loops. However, the details of reconnection are elusive due to weak emission and subtle evolution during the preeruptive phase. In this paper, we investigate how a single J-shaped loop transforms into an S-shaped one through the slippage of one of its footpoints in NOAA AR 11719 on 2013 April 11. During an interval of about 16 minutes, the J-shaped loop slips through a low-corona region of strong electric current density in a bursty fashion, reaching a peak apparent speed of the slipping footpoint as fast as 1000 km s−1 and over. The enhancement of electric current density, as suggested by nonlinear force-free field modeling, indicates that the “nonidealness” of coronal plasma becomes locally important, which may facilitate magnetic reconnection. The loop segment undergoing slipping motions is heated; meanwhile, above the fixed footpoint coronal emission dims due to a combination effect of the lengthening and heating of the loop; the latter of which is manifested in the temporal variation of dimming slope and of emission measure. These features together support an asymmetric scenario of sigmoid formation through slipping reconnection of a single J-shaped loop, which differs from the standard tether-cutting scenario involving a double J-shaped loop system.

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

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

Eruptions from coronal bright points: A spectroscopic view by IRIS of a mini-filament eruption, QSL reconnection, and reconnection-driven outflows

Context. Our study investigates a mini-filament eruption associated with cancelling magnetic fluxes. The eruption originates from a small-scale loop complex commonly known as a coronal bright point (CBP). The event is uniquely recorded in both the imaging and spectroscopic data taken with the Interface Region Imaging Spectrograph (IRIS). Aims. The investigation aims to gain a better understanding of the physical processes driving these ubiquitous small-scale eruptions. Methods. We analysed IRIS spectroscopic and slit-jaw imaging observations as well as images taken in the extreme-ultraviolet channels of the Atmospheric Imaging Assembly (AIA) and line-of-sight magnetic-field data from the Helioseismic Magnetic Imager (HMI) on board the Solar Dynamics Observatory. As the observations can only indicate the possible physical processes at play, we also employed a non-linear force-free field (NLFFF) relaxation approach based on the HMI magnetogram time series. This allowed us to further investigate the evolution of the magnetic-field structures involved in the eruption process. Results. We identified a strong small-scale brightening as a micro-flare in a CBP, recorded in emission from chromospheric to flaring plasmas. The mini-eruption is manifested via the ejection of hot (CBP loops) and cool (mini-filament) plasma recorded in both the imaging and spectroscopic data. The micro-flare is preceded by the appearance of an elongated bright feature in the IRIS slit-jaw 1400 Å images, located above the polarity inversion line. The micro-flare starts with an IRIS pixel size brightening and propagates bi-directionally along the elongated feature. We detected, in both the spectral and imaging IRIS data and AIA data, strong flows along and at the edges of the elongated feature; we believe that these represent reconnection outflows. Both edges of the elongated feature that wrap around the edges of the erupting MF evolve into a J-type shape, creating a sigmoid appearance. A quasi-separatrix layer (QSL) is identified in the vicinity of the polarity inversion line by computing the squashing factor, Q, in different horizontal planes of the NLFFF model. Conclusions. This CBP spectro-imaging study provides further evidence that CBPs represent downscaled active regions and, as such, they may make a significant contribution to the mass and energy balance of the solar atmosphere. They are the sources of all range of typical active-region features, including magnetic reconnection along QSLs, (mini-)filament eruptions, (micro-)flaring, reconnection outflows, etc. The QSL reconnection site has the same spectral appearance as the so-called explosive events identified by strong blue- and red-shifted emission, thus providing an answer to an outstanding question regarding the true nature of this spectral phenomenon.

Magnetic Structure in Successively Erupting Active Regions: Comparison of Flare-Ribbons With Quasi-Separatrix Layers

This paper studies the magnetic topology of successively erupting active regions (ARs) 11,429 and 12,371. Employing vector magnetic field observations from Helioseismic and Magnetic Imager, the pre-eruptive magnetic structure is reconstructed by a model of non-linear force-free field (NLFFF). For all the five CMEs from these ARs, the pre-eruptive magnetic structure identifies an inverse-S sigmoid consistent with the coronal plasma tracers in EUV observations. In all the eruption cases, the quasi-separatrix layers (QSLs) of large Q values are continuously enclosing core field bipolar regions in which inverse-S shaped flare ribbons are observed. These QSLs essentially represent the large connectivity gradients between the domains of twisted core flux within the inner bipolar region and the surrounding potential like arcade. It is consistent with the observed field structure largely with the sheared arcade. The QSL maps in the chromosphere are compared with the flare-ribbons observed at the peak time of the flares. The flare ribbons are largely inverse-S shape morphology with their continuity of visibility is missing in the observations. For the CMEs in the AR 12371, the QSLs outline the flare ribbons as a combination of two inverse J-shape sections with their straight parts being separated. These QSLs are typical with the weakly twisted flux rope. Similarly, for the CMEs in the AR 11429, the QSLs are co-spatial with the flare ribbons both in the middle of the PIL and in the hook sections. In the frame work of standard model of eruptions, the observed flare ribbons are the characteristic of the pre-eruptive magnetic structure being sigmoid which is reproduced by the NLFFF model with a weakly twisted flux rope at the core.

Magnetic helicity and energy budget around large confined and eruptive solar flares

Context. In order to better understand the underlying processes and prerequisites for solar activity, it is essential to study the time evolution of the coronal magnetic field of solar active regions (ARs) associated with flare activity. Aims. We investigate the coronal magnetic energy and helicity budgets of ten solar ARs around the times of large flares. In particular, we are interested in a possible relation of the derived quantities to the particular type of the flares that the AR produces, namely, whether they are associated with a CME or whether they are confined (i.e., not accompanied by a CME). Methods. Using an optimization approach, we employed time series of 3D nonlinear force-free magnetic field models of ten ARs, covering a time span of several hours around the time of occurrence of large solar flares (GOES class M1.0 and larger). We subsequently computed the 3D magnetic vector potentials associated to the model 3D coronal magnetic field using a finite-volume method. This allows us to correspondingly compute the coronal magnetic energy and helicity budgets, as well as related (intensive) quantities such as the relative contribution of free magnetic energy, EF/E (energy ratio), the fraction of non-potential (current-carrying) helicity, |HJ|/|HV| (helicity ratio), and the normalized current-carrying helicity, |HJ|/ϕ′2. Results. The total energy and helicity budgets of flare-productive ARs (extensive parameters) cover a broad range of magnitudes, with no obvious relation to the eruptive potential of the individual ARs, that is, whether or not a CME is produced in association with the flare. The intensive eruptivity proxies, EF/E and |HJ|/|HV|, and |HJ|/ϕ′2, however, seem to be distinctly different for ARs that produce CME-associated large flares compared to those which produce confined flares. For the majority of ARs in our sample, we are able to identify characteristic pre-flare magnitudes of the intensive quantities that are clearly associated with subsequent CME-productivity. Conclusions. If the corona of an AR exhibits characteristic values of ⟨|HJ|/|HV|⟩ > 0.1, ⟨EF/E⟩ > 0.2, and ⟨|HJ|/ϕ′2⟩ > 0.005, then the AR is likely to produce large CME-associated flares. Conversely, confined large flares tend to originate from ARs that exhibit coronal values of ⟨|HJ|/|HV|⟩ ≲ 0.1, ⟨EF/E⟩ ≲ 0.1, and ⟨|HJ|/ϕ′2⟩ ≲ 0.002.

Determining the source and eruption dynamics of a stealth CME using NLFFF modelling and MHD simulations

Context. Coronal mass ejections (CMEs) that exhibit weak or no eruption signatures in the low corona, known as stealth CMEs, are problematic as upon arrival at Earth they can lead to geomagnetic disturbances that were not predicted by space weather forecasters. Aims. We investigate the origin and eruption of a stealth event that occurred on 2015 January 3 that was responsible for a strong geomagnetic storm upon its arrival at Earth. Methods. To simulate the coronal magnetic field and plasma parameters of the eruption we use a coupled approach. This approach combines an evolutionary nonlinear force-free field model of the global corona with a MHD simulation. Results. The combined simulation approach accurately reproduces the stealth event and suggests that sympathetic eruptions occur. In the combined simulation we found that three flux ropes form and then erupt. The first two flux ropes, which are connected to a large AR complex behind the east limb, erupt first producing two near-simultaneous CMEs. These CMEs are closely followed by a third, weaker flux rope eruption in the simulation that originated between the periphery of AR 12252 and the southern polar coronal hole. The third eruption coincides with a faint coronal dimming, which appears in the SDO/AIA 211 Å observations, that is attributed as the source responsible for the stealth event and later the geomagnetic disturbance at 1 AU. The incorrect interpretation of the stealth event being linked to the occurrence of a single partial halo CME observed by LASCO/C2 is mainly due to the lack of STEREO observations being available at the time of the CMEs. The simulation also shows that the LASCO CME is not a single event but rather two near-simultaneous CMEs. Conclusions. These results show the significance of the coupled data-driven simulation approach in interpreting the eruption and that an operational L5 mission is crucial for space weather forecasting.

Energy partition in a confined flare with an extreme-ultraviolet late phase

Aims. In this paper, we reanalyze the M1.2 confined flare with a large extreme-ultraviolet (EUV) late phase on 2011 September 9, with a focus on its energy partition. Methods. The flare was observed by the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamics Observatory (SDO). The three-dimensional (3D) magnetic fields of the active region 11283 prior to the flare were obtained using nonlinear force free field modeling and the vector magnetograms observed by the Helioseismic and Magnetic Imager (HMI) on board the SDO. Properties of the nonthermal electrons injected into the chromosphere were obtained from the hard X-ray observations of the Ramaty Hight Energy Solar Spectroscopic Imager (RHESSI). Soft X-ray fluxes of the flare were recorded by the GOES spacecraft. Irradiance in 1−70 Å and 70−370 Å were measured by the EUV Variability Experiment (EVE) on board the SDO. We calculated various energy components of the flare. Results. The radiation (∼5.4 × 1030 erg) in 1−70 Å is nearly eleven times larger than the radiation in 70−370 Å, and is nearly 180 times larger than the radiation in 1−8 Å. The peak thermal energy of the post-flare loops is estimated to be (1.7−1.8) × 1030 erg based on a simplified schematic cartoon. Based on previous results of the enthalpy-based thermal evolution of loops (EBTEL) simulation, the energy inputs in the main flaring loops and late-phase loops are (1.5−3.8) × 1029 erg and 7.7 × 1029 erg, respectively. The nonthermal energy ((1.7−2.2) × 1030 erg) of the flare-accelerated electrons is comparable to the peak thermal energy and is sufficient to provide the energy input of the main flaring loops and late-phase loops. The magnetic free energy (9.1 × 1031 erg) before flare is large enough to provide the heating requirement and radiation, indicating that the magnetic free energy is sufficient to power the flare.

Transverse oscillation of a coronal loop induced by a flare-related jet

Context. Kink oscillations in coronal loops are ubiquitous, and we apply the observed parameters of oscillations to estimate the magnetic field strength of the loops. Aims. In this work, we report our multiwavelength observations of the transverse oscillation of a large-scale coronal loop with a length of ≥350 Mm. The oscillation was induced by a blowout coronal jet, which was related to a C4.2 circular-ribbon flare (CRF) in active region 12434 on 2015 October 16. We aim to determine the physical parameters in the coronal loop, including the Alfvén speed and the magnetic field strength. Methods. The jet-induced kink oscillation was observed in extreme ultraviolet (EUV) wavelengths by the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamics Observatory (SDO). Line-of-sight magnetograms were observed by the Helioseismic and Magnetic Imager (HMI) on board the SDO. We took several slices along the loop to assemble time-distance diagrams and used an exponentially decaying sine function to fit the decaying oscillation. The initial amplitude, period, and damping time of kink oscillations were obtained. Coronal seismology of the kink mode was applied to estimate the Alfvén speed and the magnetic field strength in the oscillating loop. In addition, we measured the magnetic field of the loop through nonlinear force-free field (NLFFF) modeling using the flux rope insertion method. Results. The oscillation is most pronounced in AIA 171 and 131 Å. The oscillation is almost in phase along the loop with a peak initial amplitude of ∼13.6 Mm, meaning that the oscillation belongs to the fast standing kink mode. The oscillation lasts for ∼3.5cycles with an average period of ∼462 s and an average damping time of ∼976 s. The values of τ/P lie in the range of 1.5–2.5. Based on coronal seismology, the Alfvén speed in the oscillating loop is estimated to be ∼1210 km s−1. Two independent methods are applied to calculate the magnetic field strength of the loop, resulting in 30–43 G using coronal seismology and 21–23 G using NLFFF modeling. Conclusions. The magnetic field strength estimated using two different approaches are on the same order of magnitude, which confirms the reliability of coronal seismology by comparing with NLFFF modeling.

Simulating the Coronal Evolution of Bipolar Active Regions to Investigate the Formation of Flux Ropes

The coronal magnetic field evolution of 20 bipolar active regions (ARs) is simulated from their emergence to decay using the time-dependent nonlinear force-free field method of Mackay, Green, and van Ballegooijen (Astrophys. J. 729, 97, 2011). A time sequence of cleaned photospheric line-of-sight magnetograms, which covers the entire evolution of each AR, is used to drive the simulation. A comparison of the simulated coronal magnetic field with the 171 and 193 Å observations obtained by the Solar Dynamics Observatory (SDO)/Atmospheric Imaging Assembly (AIA), is made for each AR by manual inspection. The results show that it is possible to reproduce the evolution of the main coronal features such as small- and large-scale coronal loops, filaments and sheared structures for 80% of the ARs. Varying the boundary and initial conditions, along with the addition of physical effects such as Ohmic diffusion, hyperdiffusion and a horizontal magnetic field injection at the photosphere, improves the match between the observations and simulated coronal evolution by 20%. The simulations were able to reproduce the build-up to eruption for 50% of the observed eruptions associated with the ARs. The mean unsigned time difference between the eruptions occurring in the observations compared to the time of eruption onset in the simulations was found to be ≈5 hrs. The simulations were particularly successful in capturing the build-up to eruption for all four eruptions that originated from the internal polarity inversion line of the ARs. The technique was less successful in reproducing the onset of eruptions that originated from the periphery of ARs and large-scale coronal structures. For these cases global, rather than local, nonlinear force-free field models must be used. While the technique has shown some success, eruptions that occur in quick succession are difficult to reproduce by this method and future iterations of the model need to address this.

Deducing the reliability of relative helicities from nonlinear force-free coronal models

Aims. We study the relative helicity of active region (AR) NOAA 12673 during a ten-hour time interval centered around a preceding X2.2 flare (SOL2017-09-06T08:57) and also including an eruptive X9.3 flare that occurred three hours later (SOL2017-09-06T11:53). In particular, we aim for a reliable estimate of the normalized self-helicity of the current-carrying magnetic field, the so-called helicity ratio, |HJ|/|H𝒱|, a promising candidate to quantity the eruptive potential of solar ARs. Methods. Using Solar Dynamics Observatory Helioseismic and Magnetic Imager vector magnetic field data as an input, we employ nonlinear force-free (NLFF) coronal magnetic field models using an optimization approach. The corresponding relative helicity, and related quantities, are computed using a finite-volume method. From multiple time series of NLFF models based on different choices of free model parameters, we are able to assess the spread of |HJ|/|H𝒱|, and to estimate its uncertainty. Results. In comparison to earlier works, which identified the non-solenoidal contribution to the total magnetic energy, Ediv/E, as selection criterion regarding the required solenoidal quality of magnetic field models for subsequent relative helicity analysis, we propose to use in addition the non-solenoidal contribution to the free magnetic energy, |Emix|/EJ, s. As a recipe for a reliable estimate of the relative magnetic helicity (and related quantities), we recommend to employ multiple NLFF models based on different combinations of free model parameters, to retain only those that exhibit smallest values of both Ediv/E and |Emix|/EJ, s at a certain time instant, to subsequently compute mean estimates, and to use the spread of the individually contributing values as an indication for the uncertainty.