Three-dimensional Coronal Magnetic Field Research Articles

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.

A Double-decker Flux Rope Model for the Solar Eruption on 2012 March 10

We present a magnetic configuration of a compound solar eruption observed on 2012 March 10, from NOAA AR 11429 near the disk center, which displayed a soft X-ray sigmoid before the eruption. We constructed a series of magnetic field models, including double-decker flux rope configurations, using the flux rope insertion method. This produces three-dimensional coronal magnetic field models constrained by the photospheric magnetogram and observed EUV coronal structures. We used different combinations of flux rope paths. We found that two flux ropes sharing the same path at different heights quickly experience a partial merging in the initial iteration of the magnetofrictional relaxation process. Different paths with less than 30% overlap allowed us to construct stable double-decker structures. The high spatial and temporal resolution of the Solar Dynamics Observatory/Atmospheric Imaging Assembly facilitated the selection of a best-fit model that matches the observations best. Moreover, by varying fluxes in this validated nonlinear force-free field double-decker configuration, we successfully reproduce all three scenarios of eruptions of double-decker configurations: (i) eruption due to the instability of higher flux rope; (ii) eruption due to rising lower flux rope and merging with higher flux rope; and (iii) eruption due to the instability of both flux ropes. This demonstrates that magnetofrictional simulation can capture the large-scale magnetic structure of eruptions for a realistic field configuration at eruption onset.

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.

An Electric-field-driven Global Coronal Magnetohydrodynamics Simulation Model Using Helioseismic and Magnetic Imager Vector-magnetic-field Synoptic Map Data

We present the simulation methodology and results of our new data-driven global coronal magnetohydrodynamics (MHD) simulation model. In this model, the solar-surface electric field is first calculated such that the curl will satisfy both the induction equation and the given temporal variations of the solar-surface magnetic field. We use the synoptic maps of the Helioseismic and Magnetic Imager three-component vector-magnetic-field data to specify the solar-surface magnetic-field vector for a period from Carrington Rotations (CRs) 2106 to 2110. A set of whole-Sun three-component electric-field maps are obtained for each CR transition interval of about 27.3 days. Using the inverted electric field as the driving variable, our new global coronal MHD model, with the angular resolution of π/64, can trace the evolution of the three-dimensional coronal magnetic field that matches the specified time-dependent solar-surface magnetic-field maps and simultaneously satisfies the divergence-free condition. A set of additional boundary treatments are introduced to control the contribution of the horizontal components of the magnetic field at the weak-field regions. The strength of the solar-surface magnetic field is limited to 20 Gauss for the sake of computational stability in this study. With these numerical treatments, the nonpotential coronal features, such as twisted loop structures, and their eruptive outward motions are obtained. This present model, capable of introducing three-component solar-surface magnetic-field observation data to coronal MHD simulations, is our first step toward a better model framework for the solar corona and hence solar wind.

Coronal magnetic field evolution over cycle 24

Context. The photospheric magnetic field vector is continuously derived from measurements, while reconstruction of the three-dimensional (3D) coronal magnetic field requires modelling with photospheric measurements as a boundary condition. For decades, the cycle variation of the magnetic field in the photosphere has been investigated. Until now, there has been no study of the evolution of the coronal magnetic flux in the corona or of the evolution of solar cycle magnetic free energy. Aims. The aim of this paper is to analyze the temporal variation of the magnetic field and free magnetic energy in the solar corona for solar cycle 24 and the behavior of the magnetic field in the two hemispheres. We want to investigate whether or not we can obtain better estimates of the magnetic field at Earth using the nonlinear force-free field extrapolation method. Methods. To model the magnetic field over cycle 24 we apply the nonlinear force-free field (NLFFF) optimization method to the entire set of the synoptic vector magnetic maps derived from observations made using the Heliospheric and Magnetic Imager (HMI) on board Solar Dynamic Observatory (SDO). Results. From our results, we find that during solar cycle 24, the maximum of the Sun’s dynamics is different than the sunspot number (SSN) maximum peak. The major contribution to the total unsigned flux is provided by the flux coming from the magnetic field structures other than sunspots (MSOS) within latitudes of −30° and +30°. The magnetic flux variation during solar cycle 24 shows a different evolution in the corona than in the photosphere. We find a correlation value of 0.8 between the derived magnetic energy from our model and the flare energy index derived from observations. On average, cycle 24 had a higher number of sunspots in the northern hemisphere (NH) but stronger flux in the southern hemisphere (SH) which could more effectively reach the higher layers of the atmosphere. The coupling between the hemispheres increases with height. The strongest asymmetries in the unsigned magnetic flux are between the two SSN peaks.

Sympathetic Filament Eruptions within a Fan-spine Magnetic System

Abstract It is unclear whether successive filament eruptions at different sites within a short time interval are physically connected or not. Here, we present the observations of successive eruptions of a small and a large filament in a tripolar magnetic field region whose coronal magnetic field showed as a fan-spine magnetic system. By analyzing the multiwavelength observations taken by the Solar Dynamic Observatory and the extrapolated three-dimensional coronal magnetic field, we find that the two filaments resided respectively in the two lobes that make up the inner fan structure of the fan-spine magnetic system. In addition, a small fan-spine system was also revealed by the squashing factor Q map, which located in the east lobe of the fan structure of the large fan-spine system. The eruption of the small filament was a failed filament eruption, which did not cause any coronal mass ejection (CME) except for three flare ribbons and two post-flare-loop systems connecting the three magnetic polarities. The eruption of the large filament not only caused similar post-flare-loop systems and flare ribbons, as observed in the small filament eruption, but also a large-scale CME. Based on our analysis results, we conclude that the two successive filament eruptions were physically connected, in which the topology change caused by the small filament eruption is thought to be the physical linkage. In addition, the eruption of the small fan-spine structure further accelerated the instability and violent eruption of the large filament.

Data-driven MHD Simulation of the Formation and Initiation of a Large-scale Preflare Magnetic Flux Rope in AR 12371

Abstract Solar eruptions are the most powerful drivers of space weather. To understand their cause and nature, it is crucial to know how the coronal magnetic field evolves before eruption. Here we study the formation process of a relatively large-scale magnetic flux rope (MFR) in active region NOAA 12371 that erupts with a major flare and coronal mass ejection on 2015 June 21. A data-driven numerical magnetohydrodynamic model is employed to simulate three-dimensional coronal magnetic field evolution of 1 day duration before the eruption. Comparison between the observed features and our modeled magnetic field discloses how the pre-eruption MFR forms. Initially, the magnetic field lines were weakly twisted as being simple sheared arcades. Then a long MFR was formed along the polarity inversion line due to the complex photospheric motion, which is mainly shearing rather than twisting. The presence of the MFR is evidenced by a coherent set of magnetic field lines with twist number above unity. Below the MFR a current sheet is shown in the model, suggesting that tether-cutting reconnection plays a key role in the MFR formation. The MFR’s flux grows as more and more field lines are twisted due to continuous injection of magnetic helicity by the photospheric motions. Meanwhile, the height of the MFR’s axis increases monotonely from its formation. By an analysis of the decay index of its overlying field, we suggest that this is because the MFR runs into the torus instability regime and becomes unstable, which finally triggers the eruption.

The Formation of CME from Coupling Fan-spine Magnetic System: A Difficult Journey

Abstract We present the eruption of a mini-filament that caused a large-scale complicated coronal mass ejection (CME) on 2014 March 28, using the high-resolution observations taken by the Solar Dynamic Observatory. Three-dimensional coronal magnetic field extrapolated from the nonlinear force-free field code reveals that the magnetic environment of the eruption source region was a large fan-spine magnetic system that hosted another small fan-spine system under its fan, and the mini-filament was located under the fan structure of the small fan-spine system. Our analysis results suggest that the eruption of the mini-filament underwent three reconnection processes before the formation of the CME. First, the erupting filament triggered the null point reconnection in the small fan-spine system. During this stage, the sudden expansion of the spine field lines also excited a large-scale extreme-ultraviolet wave. Second, the spine field lines of the small fan-spine system were pushed up by the erupting filament and therefore further triggered the null point reconnection in the large fan-spine magnetic system. Third, during the second stage, magnetic reconnection also occurred between the two legs of the stretched confining field lines of the mini-filament. The present study suggests that the formation of the observed CME from the coupling fan-spine magnetic system was more complicated than previously thought, needing to undergo multistage magnetic reconnection processes in the low corona.

A NEW CODE FOR NONLINEAR FORCE-FREE FIELD EXTRAPOLATION OF THE GLOBAL CORONA

Reliable measurements of the solar magnetic field are still restricted to the photosphere, and our present knowledge of the three-dimensional coronal magnetic field is largely based on extrapolation from photospheric magnetogram using physical models, e.g., the nonlinear force-free field (NLFFF) model as usually adopted. Most of the currently available NLFFF codes have been developed with computational volume like Cartesian box or spherical wedge while a global full-sphere extrapolation is still under developing. A high-performance global extrapolation code is in particular urgently needed considering that Solar Dynamics Observatory (SDO) can provide full-disk magnetogram with resolution up to $4096\times 4096$. In this work, we present a new parallelized code for global NLFFF extrapolation with the photosphere magnetogram as input. The method is based on magnetohydrodynamics relaxation approach, the CESE-MHD numerical scheme and a Yin-Yang spherical grid that is used to overcome the polar problems of the standard spherical grid. The code is validated by two full-sphere force-free solutions from Low & Lou's semi-analytic force-free field model. The code shows high accuracy and fast convergence, and can be ready for future practical application if combined with an adaptive mesh refinement technique.

RAPID CHANGES OF PHOTOSPHERIC MAGNETIC FIELD AFTER TETHER-CUTTING RECONNECTION AND MAGNETIC IMPLOSION

The rapid, irreversible change of the photospheric magnetic field has been recognized as an important element of the solar flare process. This Letter reports such a rapid change of magnetic fields during the 2011 February 13 M6.6 flare in NOAA AR 11158 that we found from the vector magnetograms of the Helioseismic and Magnetic Imager with 12-min cadence. High-resolution magnetograms of Hinode that are available at ~-5.5, -1.5, 1.5, and 4 hrs relative to the flare maximum are used to reconstruct three-dimensional coronal magnetic field under the nonlinear force-free field (NLFFF) assumption. UV and hard X-ray images are also used to illuminate the magnetic field evolution and energy release. The rapid change is mainly detected by HMI in a compact region lying in the center of the magnetic sigmoid, where the mean horizontal field strength exhibited a significant increase by 28%. The region lies between the initial strong UV and hard X-ray sources in the chromosphere, which are cospatial with the central feet of the sigmoid according to the NLFFF model. The NLFFF model further shows that strong coronal currents are concentrated immediately above the region, and that more intriguingly, the coronal current system underwent an apparent downward collapse after the sigmoid eruption. These results are discussed in favor of both the tether-cutting reconnection producing the flare and the ensuing implosion of the coronal field resulting from the energy release.

FIELD TOPOLOGY ANALYSIS OF A LONG-LASTING CORONAL SIGMOID

We present the first field topology analysis based on nonlinear force-free field (NLFFF) models of a long-lasting coronal sigmoid observed in 2007 February with the X-Ray Telescope on Hinode. The NLFFF models are built with the flux rope insertion method and give the three-dimensional coronal magnetic field as constrained by observed coronal loop structures and photospheric magnetograms. Based on these models, we have computed horizontal maps of the current and the squashing factor Q for 25 different heights in the corona for all six days of the evolution of the region. We use the squashing factor to quantify the degree of change of the field line linkage and to identify prominent quasi-separatrix layers (QSLs). We discuss the major properties of these QSL maps and devise a way to pick out important QSLs since our calculation cannot reach high values of Q. The complexity in the QSL maps reflects the high degree of fragmentation of the photospheric field. We find main QSLs and current concentrations that outline the flux rope cavity and that become characteristically S-shaped during the evolution of the sigmoid. We note that, although intermittent bald patches exist along the length of the sigmoid during its whole evolution, themore » flux rope remains stable for several days. However, shortly after the topology of the field exhibits hyperbolic flux tubes (HFT) on February 7 and February 12 the sigmoid loses equilibrium and produces two B-class flares and associated coronal mass ejections (CMEs). The location of the most elevated part of the HFT in our model coincides with the inferred locations of the two flares. Therefore, we suggest that the presence of an HFT in a coronal magnetic configuration may be an indication that the system is ready to erupt. We offer a scenario in which magnetic reconnection at the HFT drives the system toward the marginally stable state. Once this state is reached, loss of equilibrium occurs via the torus instability, producing a CME.« less

UTILIZATION OF MULTIPLE MEASUREMENTS FOR GLOBAL THREE-DIMENSIONAL MAGNETOHYDRODYNAMIC SIMULATIONS

Magnetic field measurements, line of sight (LOS) and/or vector magnetograms, have been used in a variety of solar physics studies. Currently, the global transverse velocity measurements near the photosphere from the Global Oscillation Network Group (GONG) are available. We have utilized these multiple observational data, for the first time, to present a data-driven global three-dimensional and resistive magnetohydrodynamic (MHD) simulation, and to investigate the energy transport across the photosphere to the corona. The measurements of the LOS magnetic field and transverse velocity reflect the effects of convective zone dynamics and provide information from the sub-photosphere to the corona. In order to self-consistently include the observables on the lower boundary as the inputs to drive the model, a set of time-dependent boundary conditions is derived by using the method of characteristics. We selected GONG's global transverse velocity measurements of synoptic chart CR2009 near the photosphere and SOLIS full-resolution LOS magnetic field maps of synoptic chart CR2009 on the photosphere to simulate the equilibrium state and compute the energy transport across the photosphere. To show the advantage of using both observed magnetic field and transverse velocity data, we have studied two cases: (1) with the inputs of the LOS magnetic field and transverse velocity measurements, and (2) with the input of the LOS magnetic field and without the input of transverse velocity measurements. For these two cases, the simulation results presented here are a three-dimensional coronal magnetic field configuration, density distributions on the photosphere and at 1.5 solar radii, and the solar wind in the corona. The deduced physical characteristics are the total current helicity and the synthetic emission. By comparing all the physical parameters of case 1 and case 2 and their synthetic emission images with the EIT image, we find that using both the measured magnetic field and the velocity distribution would give more cohesive results.

Three-Dimensional Modeling of the Solar Active Region

In this study we developed an extrapolation code based on the MHD relaxation method to reconstruct a three-dimensional (3D) coronal magnetic field. A 3D magnetic field based on a nonlinear force-free field (NLFFF) was extrapolated from a photospheric vector-field map. A benchmark test using the semi-analytical solution introduced by Low & Lou (1990) found that the extrapolated solution were able to well reproduce an original Low and Lou solution. We then applied the NLFFF extrapolation to solar active region (AR) NOAA 10930. The energy accumulation region formed by the strong magnetic shear above the polarity inversion line was resultantly reproduced and this structure is greatly different from the potential field characterized by the minimum energy state. In this paper we discuss the reliability of our extrapolated field and a relationship between quasi-separatrix layers (QSLs) and a flare ribbon from the CaII image observed by Hinode/SOT.

SUNSPOT ROTATION, FLARE ENERGETICS, AND FLUX ROPE HELICITY: THE ERUPTIVE FLARE ON 2005 MAY 13

We use the Michelson Doppler Imager and TRACE observations of photospheric magnetic and velocity fields in NOAA 10759 to build a three-dimensional coronal magnetic field model. The most dramatic feature of this active region is the 34 ◦ rotation of its leading polarity sunspot over 40 hr. We describe a method for including such rotation in the framework of the Minimum Current Corona model. We apply this method to the buildup of energy and helicity associated with the eruptive flare of 2005 May 13. We find that including the sunspot rotation almost triples the modeled flare energy (1.0 × 10 31 erg) and flux rope self-helicity (−7.1 × 10 42 Mx 2 ). This makes the results consistent with observations: the energy derived from GOES is 1.0 × 10 31 erg, the magnetic cloud helicity from WIND is −5 × 10 42 Mx 2 . Our combined analysis yields the first quantitative picture of the helicity and energy content processed through a flare in an active region with an obviously rotating sunspot and shows that rotation dominates the energy and helicity budget of this event.

Coronal Mass Ejections as Loss of Confinement of Kinked Magnetic Flux Ropes

We perform MHD simulations in a spherical geometry of the evolution of the three-dimensional coronal magnetic field as a twisted magnetic flux tube emerges slowly into the low-β corona previously occupied by a potential arcade. We study the evolution of the emerged flux rope in the corona as its twist and magnetic energy increases due to flux emergence. We find two distinct stages of the evolution. The earlier evolution is nearly quasi-static, with the flux rope being able to settle into an neighboring equilibrium when the emergence is stopped. When the twist in the emerged flux rope reaches some critical level, the flux rope is found to undergo kink motion and erupt through the arcade field at a localized region, with most of the arcade field remaining closed. The nonlinear evolution of the kink instability facilitates the loss of confinement of the flux rope by changing its orientation at the apex such that it becomes easier for the flux rope to part and erupt through the arcade. The eruption of the writhing flux rope produces prominence field with Λ-shaped and cross-legged morphologies that have been seen in the cores of some CMEs.

Numerical Simulations of Three‐dimensional Coronal Magnetic Fields Resulting from the Emergence of Twisted Magnetic Flux Tubes

We present the results of MHD simulations in the low-β regime of the evolution of the three-dimensional coronal magnetic field as an arched, twisted magnetic flux tube emerges into a preexisting coronal potential magnetic arcade. We find that the line-tied emerging flux tube becomes kink-unstable when a sufficient amount of twist is transported into the corona. For an emerging flux tube with a left-handed twist (which is the preferred sense of twist for active region flux tubes in the northern hemisphere), the kink motion of the tube and its interaction with the ambient coronal magnetic field lead to the formation of an intense current layer that displays an inverse-S shape, consistent with the X-ray sigmoid morphology preferentially seen in the northern hemisphere. The position of the current layer in relation to the lower boundary magnetic field of the emerging flux tube is also in good agreement with the observed spatial relations between the X-ray sigmoids and their associated photospheric bipolar magnetic regions. We argue that the inverse-S-shaped current layer formed is consistent with being a magnetic tangential discontinuity limited by numerical resolution and thus may result in the magnetic reconnection and significant heating that causes X-ray sigmoid brightenings.

Measurements of Three‐dimensional Coronal Magnetic Fields from Coordinated Extreme‐Ultraviolet and Radio Observations of a Solar Active Region Sunspot

We observed NOAA Active Region 8108 around 1940 UT on 1997 November 18 with the Very Large Array and with three instruments aboard the NASA/ESA Solar and Heliospheric Observatory satellite, including the Coronal Diagnostic Spectrometer, the EUV Imaging Telescope, and the Michelson Doppler Imager. We used the right-hand and left-hand circularly polarized components of the radio observing frequencies, along with the coordinated EUV observations, to derive the three-dimensional coronal magnetic field above the region's sunspot and its immediate surroundings. This was done by placing the largest possible harmonic (which corresponds to the smallest possible magnetic field strength) for each component of each radio frequency into appropriate atmospheric temperature intervals such that the calculated radio brightness temperatures at each spatial location match the corresponding observed values. The temperature dependence of the derived coronal magnetic field, B(x,y,T), is insensitive to uncertainties on the observed parameters and yields field strengths in excess of 580 G at 2 × 106 K and in excess of 1500 G at 1 × 106 K. The height dependence of the derived coronal magnetic field, B(x,y,h), varies significantly with our choice of magnetic scale height LB. Based on LB = 3.8 × 109 cm derived from the relative displacements of the observed radio centroids, we find magnetic field strengths in excess of 1500 G at heights of 15,000 km and as great as 1000 G at 25,000 km. By observing a given target region on several successive days, we would obtain observations at a variety of projection angles, thus enabling a better determination of LB and, ultimately, B(x,y,h). We compare coronal magnetic fields derived from our method with those derived from a potential extrapolation and find that the magnitudes of the potential field strengths are factors of 2 or more smaller than those derived from our method. This indicates that the sunspot field is not potential and that currents must be present in the corona. Alfven speeds between 25,000 and 57,000 km s-1 are derived for the 1 × 106 K plasma at the centroids of the radio observing frequencies. Filling factors between 0.003 and 0.1 are derived for the 1 × 106 K plasma at the centroids of the radio observing frequencies.

The Three‐dimensional Coronal Magnetic Field during Whole Sun Month

Combining models and observations, we study the three-dimensional coronal magnetic field during a period of extensive coordinated solar observations and analysis known as the Whole Sun Month (WSM) campaign (1996 August 10-September 8). The two main goals of the WSM campaign are addressed in this paper, namely, (1) to use the field configuration to link coronal features observed by coronagraphs and imaging telescopes to solar wind speed variations observed in situ and (2) to study the role of the three-dimensional coronal magnetic field in coronal force balance. Specifically, we consider how the magnetic field connects the two fastest wind streams to the two regions that have been the main foci of the WSM analysis: the equatorial extension of the north coronal hole (known as the Elephant's Trunk) and the axisymmetric streamer belt region on the opposite side of the Sun. We then quantitatively compare the different model predictions of coronal plasma and solar wind properties with observations and consider the implications for coronal force balance and solar wind acceleration.

Reconstruction of the Three-Dimensional Coronal Magnetic Field

Studies of solar flares indicate that the mechanism of flares is magnetic in character and that the coronal magnetic field is a key to understanding solar high-energy phenomena. In our ongoing research we are conducting a systematic study of a large database of observations which includes both coronal structure (from the Soft X-ray Telescope on the Yohkoh spacecraft) and photospheric vector magnetic fields (from the Haleakala Stokes Polarimeter at Mees Solar Observatory). We compare the three-dimensional nonlinear force-free coronal magnetic field, computed from photospheric boundary data, to images of coronal structure. In this paper we outline our techniques and present results for active region AR 7220/7222. We show that the computed force-free coronal magnetic field agrees well with Yohkoh X-ray coronal loops, and we discuss the properties of the coronal magnetic field and the soft X-ray loops.