Coronal Magnetic Field Models Research Articles

Total Solar Eclipse White-light Images as a Benchmark for Potential Field Source Surface Coronal Magnetic Field Models: An In-depth Analysis over a Solar Cycle

Potential field source surface (PFSS) models are widely used to simulate coronal magnetic fields. PFSS models use the observed photospheric magnetic field as the inner boundary condition and assume a perfectly radial field beyond a “source surface” (R ss). At present, total solar eclipse (TSE) white-light images are the only data that delineate the coronal magnetic field from the photosphere out to several solar radii (R ⊙). We utilize a complete solar cycle span of these images between 2008 and 2020 as a benchmark to assess the reliability of PFSS models. For a quantitative assessment, we apply the Rolling Hough Transform to the eclipse data and corresponding PFFS models to measure the difference, Δθ, between the data and model magnetic field lines throughout the corona. We find that the average Δθ, 〈Δθ〉, can be minimized for a given choice of R ss depending on the phase within a solar cycle. In particular, R ss ≈ 1.3 R ⊙ is found to be optimal for solar maximum, while R ss ≈ 3 R ⊙ yields a better match at solar minimum. Regardless, large (〈Δθ〉 > 10°) discrepancies between TSE data and PFSS-generated coronal field lines remain regardless of the choice of source surface. However, implementation of solar-cycle-dependent R ss optimal values does yield more reliable PFSS-generated coronal field lines for use in models and for tracing in situ measurements back to their sources at the Sun.

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

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

Field Line Universal relaXer (FLUX): A Fluxon Approach to Coronal Magnetic Field Modeling

We describe a novel method for modeling the global, steady solar wind using photospheric magnetic fields as a driving boundary condition. Prior wind models in this class include both rapid heuristic methods that use potential field extrapolation and variants thereof, trading rigor for computation speed, and detailed 3D magnetohydrodynamic (MHD) models that attempt to simulate the entire solar corona with a degree of physical rigor, but require large amounts of computation. The Field Line Universal relaXer, an open-source numerical code that implements the “fluxon” semi-Lagrangian approach to MHD modeling, provides an intermediate approach between these two general classes. In particular, the fluxon approach to MHD describes the magnetic field through discrete analogs of magnetic field lines, relaxing these structures to a stationary state of force balance. In this work we introduce a 1D solar wind solution along each field line, providing an ensemble of solutions that are interpolated back onto a uniform grid at an outer boundary surface. This provides advantages in physical rigor over heuristic semianalytic techniques, and in computational efficiency over full 3D MHD techniques. Here we describe the underlying methodology and the FLUXPipe modeling pipeline process.

The automatic identification and tracking of coronal flux ropes

Context. Constructing the relevant magnetic field lines from active region modelling data is crucial to understanding the underlying instability mechanisms that trigger the corresponding eruptions. Aims. We present a magnetic flux rope (FR) extraction tool for solar coronal magnetic field modelling data that builds upon a recent methodology. The newly developed method is then compared against its previous iteration. Furthermore, we apply the scheme to magnetic field simulations of active regions AR12473 (similar to our previous study) and AR11176. We compare the method to its predecessor and study the 3D movement of the newly extracted FRs up to heights of 200 and 300 Mm, respectively. Methods.The extraction method is based on the twist parameter Tw and a variety of mathematical morphology (MM) algorithms, including the opening transform and the morphological gradient. We highlight the differences between the methods by investigating the circularity of the FRs in the plane we extract from. The simulations for the active regions are carried out with a time-dependent data-driven magnetofrictional model (TMFM). We investigate the FR trajectories by tracking their apex throughout the full simulation time span. Results. Comparing the newly developed method to the previous extraction scheme, we demonstrate that this upgrade provides the user with more tools and less a priori assumptions about the FR shape, which in turn leads to a more accurate set of field lines. Despite some differences, both the newly extracted FR of AR12473 and the FR derived with the old iteration of the method show a similar general appearance, confirming that the two methods indeed extract the same structure. The methods differ the most in their emergence and formation stages, where the newly extracted FR deviates significantly from a perfectly circular cross-section (which was the basic assumption of the initial method). The propagation analysis yields that the erupting FR from AR12473 indeed shows stronger dynamics than the AR11176 FR and a significant deflection during its ascent through the domain. The modelling results are also verified with observations: AR12473 is dynamic and eruptive, while AR11176 only features an eruption outside of our simulation time window. Conclusions. We implemented a FR extraction method, incorporating mathematical morphology algorithms for 3D solar magnetic field simulations of active region FRs. This scheme was applied to AR12473 and AR11176. We find that the clearly eruptive FR of AR12473 experiences significant deflection during its rise. The AR11176 FR appears more stable, though there is still a notable deflection. This confirms that at these low coronal heights, FRs undergo significant changes in the direction of their propagation even for less dynamic cases.

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 source of unusual coronal upflows with photospheric abundance in a solar active region

Context. Upflows in the corona are of importance, as they may contribute to the solar wind. There has been considerable interest in upflows from active regions (ARs). The coronal upflows that are seen at the edges of active regions have coronal elemental composition and can contribute to the slow solar wind. The sources of the upflows have been challenging to determine because they may be multiple, and the spatial resolution of previous observations is not yet high enough. Aims. In this article, we analyse coronal upflows in AR 12960 that are unusually close to the sunspot umbra. We analyse their properties, and we attempt to determine if it is possible that they feed into the slow solar wind. Methods. We analysed the activity in the upflow region in detail using a combination of Solar Orbiter EUV images at high spatial and temporal resolution, Hinode/EUV Imaging Spectrometer data, and observations from instruments on board the Solar Dynamics Observatory. This combined dataset was acquired during the first Solar Orbiter perihelion of the science phase, which provided a spatial resolution of 356 km for two pixels. Doppler velocity, density, and plasma composition determinations, as well as coronal magnetic field modelling, were carried out to understand the source of the upflows. Results. We observed small magnetic fragments, called moving magnetic features (MMFs), moving away from the sunspot in the active region. Specifically, they moved towards the sunspot from the edge of the penumbra where a small positive polarity connects to the umbra via small-scale and very dynamic coronal loops. At this location, small dark grains are evident and flow along penumbral filaments in continuum images. The magnetic field modelling showed small low-lying loops anchored close to the umbral magnetic field. The high-resolution data of the Solar Orbiter EUV Imagers showed the dynamics of these small loops, which last on time scales of only minutes. The edges of these small loops are the location of the coronal upflow that has photospheric abundance.

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.

Evolution of Magnetic Fields and Energy Release Processes during Homologous Eruptive Flares

We explore the processes of the repetitive buildup and the explosive release of magnetic energy, together with the formation of magnetic flux ropes, which eventually resulted in three homologous eruptive flares of successively increasing intensities (i.e., M2.0, M2.6, and X1.0). The flares originated from NOAA active region 12017 between 2014 March 28 and 29. EUV observations and magnetogram measurements, together with coronal magnetic field modeling, suggest that the flares were triggered by the eruption of flux ropes that were embedded in a densely packed system of loops within a small part of the active region. In X-rays, the first and second events show similar evolutions, with single compact sources, while the third event exhibits multiple emission centroids, with a set of strong nonthermal conjugate sources at 50–100 keV during the hard X-ray peak. Over an interval of ≈ 44 hr, the photospheric magnetic field encompassing the three flares undergoes important phases of emergence and cancellation, together with significant changes near the polarity inversion lines within the flaring region. Our observations point toward the tether-cutting mechanism being the plausible triggering process of the eruptions. Between the second and third events, we observe a prominent phase of flux emergence that temporally correlates with the buildup phase of free magnetic energy in the active region corona. In conclusion, our analysis reveals efficient coupling between the rapidly evolving photospheric and coronal magnetic fields in the active region, leading to a continued phase of the buildup of free energy, which results in the homologous flares of successively increasing intensities.

Coronal Field Geometry and Solar Wind Speed

The Wang–Sheeley–Arge (WSA) solar wind (SW) model is based on the idea that weakly expanding coronal magnetic field tubes are associated with sources of fast SWs and vice versa. A parameter called the “flux tube expansion” (FTE) is used to determine the degree of expansion of magnetic tubes. The FTE is calculated based on the coronal magnetic field model, usually in the potential approximation. The second input parameter for the WSA model is the great circle distance from the base of the open magnetic field line in the photosphere to the boundary of the corresponding coronal hole (DCHB). These two coronal magnetic field parameters are related by an empirical relationship with the solar wind velocity near the Sun. The WSA model has shortcomings and does not fully explain the solar wind formation mechanisms. In the present work, we model various coronal magnetic field parameters in the potential-field source-surface (PFSS) approximation from a long series of magnetographic observations: the Solar Telescope-magnetograph for Operative Prognoses (STOP) (Kislovodsk Mountain Astronomical Station), the Helioseismic and magnetic imager (SDO/HMI), and data from the Wilcox Solar Observatory (WSO). Our main goal is to identify correlations between the coronal magnetic field parameters and the observed SW velocity in order to use them for modeling SW. We found that the SW velocity correlates relatively well with some geometric properties of the magnetic tubes, including the force line length, the latitude of the force line footpoints, and the DCHB. We propose a formula for calculating the SW velocity based on these parameters. The presented relationship does not use FTE and showed a better correlation with observations compared to the WSA model.

Automated Driving for Global Nonpotential Simulations of the Solar Corona

We describe a new automated technique for active region emergence in coronal magnetic field models, based on the inversion of the electric field locally from a single line-of-sight magnetogram for each region. The technique preserves the arbitrary shapes of magnetic field distribution associated with individual active regions and incorporates emerging magnetic helicity (twist) in a parametrized manner through a noninductive electric field component. We test the technique with global magnetofrictional simulations of the coronal magnetic field during Solar Cycle 24 Maximum from 2011 June 1 to 2011 December 31. The active regions are determined in a fully automated and objective way using Spaceweather HMI Active Region Patch (SHARP) data. Our primary aim is to constrain two free parameters in the emergence algorithm: the duration of emergence and the twist parameter for each individual active region. While the duration has a limited effect on the resulting coronal magnetic field, changing the sign and amplitude of the twist parameters profoundly influences the amount of nonpotentiality generated in the global coronal magnetic field. We explore the possibility of constraining both the magnitude and sign of the twist parameter using estimates of the current helicity derived from vector magnetograms and supplied in the SHARP metadata for each region. Using the observed sign of twist for each region reduces the overall nonpotentiality in the corona, highlighting the importance of scatter in the emerging active region helicities.

Origin of extreme solar eruptive activity from the active region NOAA 12673 and the largest flare of solar cycle 24

AbstractDuring 2017, when the Sun was moving toward the minimum phase of solar cycle 24, an exceptionally eruptive active region (AR) NOAA 12673 emerged on the Sun during August 28-September 10. During the highest activity level, the AR turned into a δ-type sunspot region, which manifests the most complex configuration of magnetic fields from the photosphere to the coronal heights. The AR 12673 produced four X-class and 27 M-class flares, along with numerous C-class flares, making it one of the most powerful ARs of solar cycle 24. Notably, it produced the largest flare of solar cycle 24, namely, the X9.3 event on 2017 September 6. In this work, we highlight the results of our comprehensive analysis involving multi-wavelength imaging and coronal magnetic field modeling to understand the evolution and eruptivity from AR 12673. We especially focus on the morphological, spectral and kinematical evolution of the two X-class flares on 6 September 2017. We explore various large- and small-scale magnetic field structures of the active region which are associated with the triggering and subsequent outbursts during the powerful solar transients.

Homologous Compact Major Blowout-eruption Solar Flares and their Production of Broad CMEs

We analyze the formation mechanism of three homologous broad coronal mass ejections (CMEs) resulting from a series of solar blowout-eruption flares with successively increasing intensities (M2.0, M2.6, and X1.0). The flares originated from NOAA Active Region 12017 during 2014 March 28–29 within an interval of ≈24 hr. Coronal magnetic field modeling based on nonlinear force-free field extrapolation helps to identify low-lying closed bipolar loops within the flaring region enclosing magnetic flux ropes. We obtain a double flux rope system under closed bipolar fields for all the events. The sequential eruption of the flux ropes led to homologous flares, each followed by a CME. Each of the three CMEs formed from the eruptions gradually attained a large angular width, after expanding from the compact eruption-source site. We find these eruptions and CMEs to be consistent with the “magnetic-arch-blowout” scenario: each compact-flare blowout eruption was seated in one foot of a far-reaching magnetic arch, exploded up the encasing leg of the arch, and blew out the arch to make a broad CME.

Quantitative Evaluation of Coronal Magnetic Field Models Using Tomographic Reconstructions of Electron Density

Abstract We introduce a new quantitative approach for assessing the quality of coronal magnetic field models. The method compares the location of the magnetic neutral line at a specified height in the magnetic field model with the locations of localized density peaks in the coronal electron density, as measured using coronal rotational tomography. This approach is flexible to the presence of pseudostreamers in the coronal magnetic field, as well as folds in the streamer belt. We present an example application during mid-2010 when the white-light streamer-belt structure is complex and the emergence of a large active region on the far side of the Sun presents a challenge for modeling the coronal magnetic structure.

Validation scheme for solar coronal models: Constraints from multi-perspective observations in EUV and white light

Context. In this paper, we present a validation scheme to investigate the quality of coronal magnetic field models, which is based on comparisons with observational data from multiple sources. Aims. Many of these coronal models may use a range of initial parameters that produce a large number of physically reasonable field configurations. However, that does not mean that these results are reliable and comply with the observations. With an appropriate validation scheme, which is the aim of this work, the quality of a coronal model can be assessed. Methods. The validation scheme was developed with the example of the EUropean Heliospheric FORecasting Information Asset (EUHFORIA) coronal model. For observational comparison, we used extreme ultraviolet and white-light data to detect coronal features on the surface (open magnetic field areas) and off-limb (streamer and loop) structures from multiple perspectives (Earth view and the Solar Terrestrial Relations Observatory – STEREO). The validation scheme can be applied to any coronal model that produces magnetic field line topology. Results. We show its applicability by using the validation scheme on a large set of model configurations, which can be efficiently reduced to an ideal set of parameters that matches best with observational data. Conclusions. We conclude that by using a combined empirical visual classification with a mathematical scheme of topology metrics, a very efficient and objective quality assessment for coronal models can be performed.

Variations in Finite-difference Potential Fields

Abstract The potential field (PF) solution of the solar corona is a vital modeling tool for a wide range of applications, including minimum energy estimates, coronal magnetic field modeling, and empirical solar wind solutions. Given its popularity, it is important to understand how choices made in computing a PF may influence key properties of the solution. Here we study PF solutions for the global coronal magnetic field on 2012 June 13, computed with our high-performance finite-difference code POT3D. Solutions are analyzed for their global properties and locally around NOAA AR 11504, using the net open flux, open-field boundaries, total magnetic energy, and magnetic structure as metrics. We explore how PF solutions depend on (1) the data source, type, and processing of the inner boundary conditions; (2) the choice of the outer boundary condition height and type; and (3) the numerical resolution and spatial scale of information at the lower boundary. We discuss the various qualitative and quantitative differences that naturally arise by using different maps as input, and we illustrate how coronal morphology and open flux depend most strongly on the outer boundary condition. We also show how large-scale morphologies and the open magnetic flux are remarkably insensitive to model resolution, while the surface mapping and embedded magnetic complexity vary considerably. This establishes important context for past, current, and future applications of the PF for coronal and solar wind modeling.

3D Solar Coronal Loop Reconstructions with Machine Learning

Abstract The magnetic field plays an essential role in the initiation and evolution of different solar phenomena in the corona. The structure and evolution of the 3D coronal magnetic field are still not very well known. A way to ascertain the 3D structure of the coronal magnetic field is by performing magnetic field extrapolations from the photosphere to the corona. In previous work, it was shown that by prescribing the 3D-reconstructed loops’ geometry, the magnetic field extrapolation produces a solution with a better agreement between the modeled field and the reconstructed loops. This also improves the quality of the field extrapolation. Stereoscopy, which uses at least two view directions, is the traditional method for performing 3D coronal loop reconstruction. When only one vantage point of the coronal loops is available, other 3D reconstruction methods must be applied. Within this work, we present a method for the 3D loop reconstruction based on machine learning. Our purpose for developing this method is to use as many observed coronal loops in space and time for the modeling of the coronal magnetic field. Our results show that we can build machine-learning models that can retrieve 3D loops based only on their projection information. Ultimately, the neural network model will be able to use only 2D information of the coronal loops, identified, traced, and extracted from the extreme-ultraviolet images, for the calculation of their 3D geometry.

Impact of Inner Heliospheric Boundary Conditions on Solar Wind Predictions at Earth

AbstractPredictions of the physical parameters of the solar wind at Earth are at the core of operational space weather forecasts. Such predictions typically use line‐of‐sight observations of the photospheric magnetic field to drive a heliospheric model. The models Wang‐Sheeley‐Arge (WSA) and ENLIL for the transport in the heliosphere are commonly used for these respective tasks. Here we analyze the impact of replacing the potential field coronal boundary conditions from WSA with two alternative approaches. The first approach uses a more realistic nonpotential rather than potential approach, based on the Durham Magneto Frictional Code (DUMFRIC) model. In the second approach the ENLIL inner boundary conditions are based on Inter Planetary Scintillation observations (IPS). We compare predicted solar wind speed, plasma density, and magnetic field magnitude with observations from the WIND spacecraft for two 6‐month intervals in 2014 and 2016. Results show that all models tested produce fairly similar output when compared to the observed time series. This is not only reflected in fairly low correlation coefficients (<0.3) but also large biases. For example, for solar wind speed some models have average biases of more than 150 km/s. On a positive note, the choice of coronal magnetic field model has a clear influence on the model results when compared to the other models in this study. Simulations driven by IPS data have a high success rate with regard to detection of the high speed solar wind. Our results also indicate that model forecasts do not degrade for longer forecast times.

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.

Erratum: “On the Reliability of Magnetic Energy and Helicity Computations Based on Nonlinear Force-free Coronal Magnetic Field Models” (2019, ApJL, 880, L6)

Erratum: “On the Reliability of Magnetic Energy and Helicity Computations Based on Nonlinear Force-free Coronal Magnetic Field Models” (2019, ApJL, 880, L6)