Nonlinear Force-free Magnetic Field 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.

Are Nonthermal Velocities in Active Region Coronal Loops Anisotropic?

We have measured line widths in active region coronal loops in order to determine whether the nonthermal broadening is anisotropic with respect to the magnetic field direction. These nonthermal velocities are caused by unresolved fluid motions. Our analysis method combines spectroscopic data and a magnetic field extrapolation. We analyzed spectra from the Extreme Ultraviolet Imaging Spectrometer on Hinode. A differential emission measure analysis showed that many spectral lines that are commonly considered to be formed in the active region have a substantial contribution from the background quiet Sun. From these spectra we identified lines whose emission was dominated by the active region loops rather than background sources. Using these lines, we constructed maps of the nonthermal velocity. With data from the Helioseismic Magnetic Imager on the Solar Dynamics Observatory and the Coronal Modeling System nonlinear force-free magnetic field reconstruction code, we traced several of the magnetic field lines through the active region. Comparing the spectroscopic and magnetic data, we looked for correlations of the nonthermal velocity with the viewing angle between the line of sight and the magnetic field. We found that nonthermal velocities show a weak anticorrelation with the viewing angle. That is, the tendency is for the nonthermal velocity to be slightly larger in the parallel direction. This parallel broadening may be due to acoustic waves or unresolved parallel flows.

A Large-Scale Dataset of Three-Dimensional Solar Magnetic Fields Extrapolated by Nonlinear Force-Free Method

It has been widely accepted that solar magnetic field manipulates all solar activities, especially violent solar bursts in solar corona. Thus, it is extremely important to reconstruct three-dimentional (3D) magnetic field of solar corona from really observed photospheric magnetogram. In this paper, a large-scale dataset of 3D solar magnetic fields of active regions is built by using the nonlinear force-free magnetic field (NLFFF) extrapolation from vector magnetograms of Helioseismic and Magnetic Imager (HMI) on Solar Dynamics Observatory (SDO). In this dataset, all space-weather HMI active region patches (SHARPs) with the corresponding serial numbers of national oceanic and atmospheric administration (NOAA) are included. They are downloaded from the SHARP 720 s series of JSOC every 96 minutes. In addition, each sample is labelled with a finer grained label for solar flare forecast. This paper is with the purpose of open availability of data resource and source code to the peers for refraining from repeated labor of data preparation. Meanwhile, with such a large-scale, high spatio-temporal resolution and high quality scientific data, we anticipate a wide attention and interest from artificial intelligence (AI) and computer vision communities, for exploring AI for astronomy over such a large-scale dataset.

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.

Multi-Wavelength Observations of a Failed Filament Eruption and Associated Hovered Coronal Mass Ejection

Failed filament eruption remains mysterious on its initiation, magnetic environment, and erupting and failing mechanisms. We present multi-wavelength observations of a failed filament eruption and its associated hovered coronal mass ejection (hovered-CME) from limb observations of the Ahead of Solar Terrestrial Relations Observatory. On-disk observations from Solar Dynamics Observatory show the expansion of the anchored leg of an S-shaped filament during the pre-eruption phase. The main eruption starts as a sudden ejection of the erupted leg, which is followed by the appearance of EUV brightening in the S-shaped magnetic field. The brightening is spatio-temporal accompanied with hard X-ray emission enhancement, and cancellation of opposite magnetic polarities, which imply possible reconnection. After reaching the maximum displacement, the erupted material drains back to the Sun along the remaining anchored leg. The non-linear force free magnetic field extrapolation shows an S-shaped magnetic field, formed by two magnetic structures, with a strong enveloped magnetic field. The decay index at the possible apex of the filament is 0.8–1.2. Observations indicate that the failed filament eruption is triggered by tether cutting reconnection and is possibly confined by the upper magnetic field. The hovered-CME, resulting from the failed filament eruption and recording as a coronal mass ejection (CME), may cause the overestimation of the CME count.

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.

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)

Two Episodes of a Filament Eruption from a Fan-spine Magnetic Configuration

Abstract In this paper, we present detailed observations of a filament eruption associated with a B6.0 flare, a jet-like coronal mass ejection (CME), and a type-III radio burst on 2013 March 2. The filament, which is located at the northwest edge of active region (AR) 11183, experienced a partial and then a full eruption. Each episode of the filament eruption produced a circular flare ribbon and a blowout jet, which is a mixture of hot and cool plasma that can be observed by the Solar Dynamics Observatory. Extrapolated coronal magnetic configuration using both the potential field source surface and the nonlinear force-free magnetic field models shows that the filament is embedded in a fan-spine magnetic topology. Considering the photospheric magnetic evolution, we suggest that the first episode of the filament eruption (partial eruption) is caused by the continuous photospheric magnetic flux cancellation below its western segment, while the eruption that follows results from the reduction of magnetic constraint above the filament in the aftermath of the first eruption episode and reconnection. Combining the observations with the extrapolated coronal magnetic configuration, we find that the event is an example that fits the null-point reconnection scenario. Our event is a result of null-point reconnection (interchange reconnection) between a closed filament magnetic field and the ambient open field at the edge of an AR, thus it has implications for the source of the slow speed solar wind and the source of the cold and dense plasma detected within.

Magnetic Helicity Budget of Solar Active Regions Prolific of Eruptive and Confined Flares

Abstract We compare the coronal magnetic energy and helicity of two solar active regions (ARs), prolific in major eruptive (AR 11158) and confined (AR 12192) flaring, and analyze the potential of deduced proxies to forecast upcoming flares. Based on nonlinear force-free (NLFF) coronal magnetic field models with a high degree of solenoidality, and applying three different computational methods to investigate the coronal magnetic helicity, we are able to draw conclusions with a high level of confidence. Based on real observations of two solar ARs we checked trends regarding the potential eruptivity of the active-region corona, as suggested earlier in works that were based on numerical simulations, or solar observations. Our results support that the ratio of current-carrying to total helicity, , shows a strong ability to indicate the eruptive potential of a solar AR. However, does not seem to be indicative for the magnitude or type of an upcoming flare (confined or eruptive). Interpreted in the context of earlier observational studies, our findings furthermore support that the total relative helicity normalized to the magnetic flux at the NLFF model’s lower boundary, , represents no indicator for the eruptivity.

Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

The Godbillon–Vey invariant occurs in homology theory, and algebraic topology, when conditions for a co-dimension 1, foliation of a three-dimensional manifold are satisfied. The magnetic Godbillon–Vey helicity invariant in magnetohydrodynamics (MHD) is a higher-order helicity invariant that occurs for flows in which the magnetic helicity density$h_{m}=\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B}=\boldsymbol{A}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{A})=0$, where$\boldsymbol{A}$is the magnetic vector potential and$\boldsymbol{B}$is the magnetic induction. This paper obtains evolution equations for the magnetic Godbillon–Vey field$\unicode[STIX]{x1D6C8}=\boldsymbol{A}\times \boldsymbol{B}/|\boldsymbol{A}|^{2}$and the Godbillon–Vey helicity density$h_{\text{gv}}=\unicode[STIX]{x1D6C8}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D6C8})$in general MHD flows in which either$h_{m}=0$or$h_{m}\neq 0$. A conservation law for$h_{\text{gv}}$occurs in flows for which$h_{m}=0$. For$h_{m}\neq 0$the evolution equation for$h_{\text{gv}}$contains a source term in which$h_{m}$is coupled to$h_{\text{gv}}$via the shear tensor of the background flow. The transport equation for$h_{\text{gv}}$also depends on the electric field potential$\unicode[STIX]{x1D713}$, which is related to the gauge for$\boldsymbol{A}$, which takes its simplest form for the advected$\boldsymbol{A}$gauge in which$\unicode[STIX]{x1D713}=\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{u}$where$\boldsymbol{u}$is the fluid velocity. An application of the Godbillon–Vey magnetic helicity to nonlinear force-free magnetic fields used in solar physics is investigated. The possible uses of the Godbillon–Vey helicity in zero helicity flows in ideal fluid mechanics, and in zero helicity Lagrangian kinematics of three-dimensional advection, are discussed.

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

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

Preprocessing photospheric vector magnetograms for nonlinear force-free field extrapolation of the global corona

Nonlinear force-free magnetic field (NLFFF) extrapolation based on the observed photospheric magnetic field is the most important method to obtain the coronal magnetic field nowadays. However, raw photospheric magnetograms contain magnetic forces and small-scale noises, and fail to be consistent with the force-free assumption of NLFFF models. The procedure for removing the forces and noises in observed data is called preprocessing. In this paper, we extend the preprocessing code of Jiang & Feng to spherical coordinates for a full sphere. We first smooth the observed data with Gaussian smoothing, and then split the smoothed magnetic field into a potential field and a non-potential field. The potential part is computed by a numerical potential field model, and the non-potential part is preprocessed using an optimization method to minimize the magnetic forces and magnetic torques. Applying the code to synoptic charts of the vector magnetic field from SDO/HMI, we find it can effectively reduce the noises and forces, and improve the quality of data for a better input which will be used for NLFFF extrapolations applied to the global corona.

A Comparative Study of the Eruptive and Non-eruptive Flares Produced by the Largest Active Region of Solar Cycle 24

We investigate the morphological and magnetic characteristics of solar active region (AR) NOAA 12192. AR 12192 was the largest region of Solar Cycle 24; it underwent noticeable growth and produced 6 X-class flares, 22 M-class flares, and 53 C-class flares in the course of its disc passage. However, the most peculiar fact of this AR is that it was associated with only one CME in spite of producing several X-class flares. In this work, we carry out a comparative study between the eruptive and non-eruptive flares produced by AR 12192. We find that the magnitude of abrupt and permanent changes in the horizontal magnetic field and Lorentz force are significantly smaller in the case of the confined flares compared to the eruptive one. We present the areal evolution of AR 12192 during its disc passage. We find the flare-related morphological changes to be weaker during the confined flares, whereas the eruptive flare exhibits a rapid and permanent disappearance of penumbral area away from the magnetic neutral line after the flare. Furthermore, from the extrapolated nonlinear force-free magnetic field, we examine the overlying coronal magnetic environment over the eruptive and non-eruptive zones of the AR. We find that the critical decay index for the onset of torus instability was achieved at a lower height over the eruptive flaring region, than for the non-eruptive core area. These results suggest that the decay rate of the gradient of overlying magnetic field strength may play a decisive role to determine the CME productivity of the AR. In addition, the magnitude of changes in the flare-related magnetic characteristics are found to be well correlated with the nature of solar eruptions.

The Causes of Quasi-homologous CMEs

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

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

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

DIAGNOSTICS OF CORONAL HEATING IN ACTIVE-REGION LOOPS

ABSTRACT Understanding coronal heating remains a central problem in solar physics. Many mechanisms have been proposed to explain how energy is transferred to and deposited in the corona. We summarize past observational studies that attempted to identify the heating mechanism and point out the difficulties in reproducing the observations of the solar corona from the heating models. The aim of this paper is to study whether the observed extreme ultraviolet (EUV) emission in individual coronal loops in solar active regions can provide constraints on the volumetric heating function, and to develop a diagnostic for the heating function for a subset of loops that are found close to static thermal equilibrium. We reconstruct the coronal magnetic field from Solar Dynamics Observatory/HMI data using a nonlinear force-free magnetic field model. We model selected loops using a one-dimensional stationary model, with a heating rate dependent locally on the magnetic field strength along the loop, and we calculate the emission from these loops in various EUV wavelengths for different heating rates. We present a method to measure a power index β defining the dependence of the volumetric heating rate E H on the magnetic field, , and controlling also the shape of the heating function: concentrated near the loop top, uniform and concentrated near the footpoints. The diagnostic is based on the dependence of the electron density on the index β. This method is free from the assumptions of the loop filling factor but requires spectroscopic measurements of the density-sensitive lines. The range of applicability for loops of different length and heating distributions is discussed, and the steps to solving the coronal heating problem are outlined.

UNRAVELLING THE COMPONENTS OF A MULTI-THERMAL CORONAL LOOP USING MAGNETOHYDRODYNAMIC SEISMOLOGY

ABSTRACT Coronal loops, constituting the basic building blocks of the active Sun, serve as primary targets to help understand the mechanisms responsible for maintaining multi-million Kelvin temperatures in the solar and stellar coronae. Despite significant advances in observations and theory, our knowledge on the fundamental properties of these structures is limited. Here, we present unprecedented observations of accelerating slow magnetoacoustic waves along a coronal loop that show differential propagation speeds in two distinct temperature channels, revealing the multi-stranded and multithermal nature of the loop. Utilizing the observed speeds and employing nonlinear force-free magnetic field extrapolations, we derive the actual temperature variation along the loop in both channels, and thus are able to resolve two individual components of the multithermal loop for the first time. The obtained positive temperature gradients indicate uniform heating along the loop, rather than isolated footpoint heating.

TEMPORAL AND SPATIAL RELATIONSHIP OF FLARE SIGNATURES AND THE FORCE-FREE CORONAL MAGNETIC FIELD

ABSTRACT We investigate the plasma and magnetic environment of active region NOAA 11261 on 2011 August 2 around a GOES M1.4 flare/CME (SOL2011-08-02T06:19). We compare coronal emission at the (extreme) ultraviolet and X-ray wavelengths, using SDO AIA and RHESSI images, in order to identify the relative timing and locations of reconnection-related sources. We trace flare ribbon signatures at ultraviolet wavelengths in order to pin down the intersection of previously reconnected flaring loops in the lower solar atmosphere. These locations are used to calculate field lines from three-dimensional (3D) nonlinear force-free magnetic field models, established on the basis of SDO HMI photospheric vector magnetic field maps. Using this procedure, we analyze the quasi-static time evolution of the coronal model magnetic field previously involved in magnetic reconnection. This allows us, for the first time, to estimate the elevation speed of the current sheet’s lower tip during an on-disk observed flare as a few kilometers per second. A comparison to post-flare loops observed later above the limb in STEREO EUVI images supports this velocity estimate. Furthermore, we provide evidence for an implosion of parts of the flaring coronal model magnetic field, and identify the corresponding coronal sub-volumes associated with the loss of magnetic energy. Finally, we spatially relate the build up of magnetic energy in the 3D models to highly sheared fields, established due to the dynamic relative motions of polarity patches within the active region.

非线性无力场的Low-Lou解法探讨

非线性无力场是天体物理中的重要数学模型，它是一套非线性偏微分方程组，经常用于太阳及恒星磁场的理论研究。在轴对称情形下，该方程组归化为满足特定边界条件的带有未知参数的二阶非线性常微分方程，此即所谓的Low-Lou解法。本文提出一种参数打靶法，作为对Low-Lou解法的探讨和补遗，并给出更多可选的数值无力场。 Nonlinear force-free magnetic field is an important mathematical model in astrophysics, which is a set of nonlinear partial differential equations, often used in the theoretical studies of solar and stellar magnetic fields. In the axisymmetric case, this set of partial differential equations is reduced into a nonlinear ordinary differential equation of second order with an unknown parameter, satisfying certain boundary condition. This is the so-called Low-Lou approach of the problem. In this paper, we propose a parametric shooting method as a technical supplement for the Low- Lou approach, offering more optional numerical force-free magnetic fields.