Nonlinear Force-free Research Articles

Multipoint study of the rapid filament evolution during a confined C2 flare on 28 March 2022, leading to eruption

This study focuses on the rapid evolution of the solar filament in active region 12975 during a confined C2 flare on 28 March 2022, which finally led to an eruptive M4 flare 1.5 h later. The event is characterized by the apparent breakup of the filament, the disappearance of its southern half, and the flow of the remaining filament plasma into a new, longer channel with a topology very similar to an extreme ultraviolet (EUV) hot channel observed during the flare. In addition, we outline the emergence of the original filament from a sheared arcade and discuss possible drivers for its rise and eruption. We took advantage of Solar Orbiter's favorable position, 0.33 AU from the Sun, and $83. 5^ west of the Sun-Earth line, to perform a multi-point study using the Spectrometer Telescope for Imaging X-rays (STIX) and the Extreme Ultraviolet Imager (EUI) in combination with the Atmospheric Imaging Assembly (AIA) and the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO) and Halpha images from the Earth-based Kanzelhöhe Observatory for Solar and Environmental Research (KSO) and the Global Oscillation Network Group (GONG). While STIX and EUI observed the flare and the filament's rise from close up and at the limb, AIA and HMI observations provided highly complementary on-disk observations from which we derived differential emission measure (DEM) maps and nonlinear force-free (NLFF) magnetic field extrapolations. According to our pre-flare NLFF extrapolation, field lines corresponding to both filament channels existed in close proximity before the flare. We propose a loop-loop reconnection scenario based on field structures associated with the AIA 1600 Å flare ribbons and kernels. It involves field lines surrounding and passing beneath the shorter filament channel, and field lines closely following the southern part of the longer channel. Reconnection occurs in an essentially vertical current sheet at a polarity inversion line (PIL) below the breakup region, which enables the formation of the flare loop arcade and EUV hot channel. This scenario is supported by concentrated currents and free magnetic energy built up by antiparallel flows along the PIL before the flare, and by non-thermal X-ray emission observed from the reconnection region. The reconnection probably propagated to involve the original filament itself, leading to its breakup and new geometry. This reconnection geometry also provides a general mechanism for the formation of the long filament channel and realizes the concept of tether cutting. It was probably active throughout the filament's continuous rise phase, which lasted from at least 30 min before the C2 flare until the filament eruption. The C2 flare represents a period of fast reconnection during this otherwise more steady period, during which most of the original filament was reconnected and joined the longer channel. These results demonstrate how rapid changes in solar filament topology can be driven by loop-loop reconnection with nearby field structures, and how this can be part of a long-lasting tether-cutting reconnection process. They also illustrate how a confined precursor flare due to loop-loop reconnection (Type I) can contribute to the evolution towards a full eruption, and that they can produce a flare loop arcade when the contact region between interacting loop systems has a sheet-like geometry similar to a flare current sheet.

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.

The effect of spatial sampling on magnetic field modeling and helicity computation

Context. Nonlinear force-free (NLFF) modeling is regularly used to indirectly infer the 3D geometry of the coronal magnetic field, which is not otherwise accessible on a regular basis by means of direct measurements. Aims. We study the effect of binning in time-series NLFF modeling of individual active regions (ARs) in order to quantify the effect of a different underlying spatial sampling on the quality of modeling as well as on the derived physical parameters. Methods. We apply an optimization method to sequences of Solar Dynamics Observatory (SDO) Helioseismic and Magnetic Imager (HMI) vector magnetogram data at three different plate scales for three solar active regions to obtain nine NLFF model time series. From the NLFF models, we deduce active-region magnetic fluxes, electric currents, magnetic energies, and relative helicities, and analyze those with respect to the underlying spatial sampling. We calculate various metrics to quantify the quality of the derived NLFF models and apply a Helmholtz decomposition to characterize solenoidal errors. Results. At a given spatial sampling, the quality of NLFF modeling is different for different ARs, and the quality varies along the individual model time series. For a given AR, modeling at a certain spatial sampling is not necessarily of superior quality compared to that performed with a different plate scale. Generally, the NLFF model quality tends to be higher for larger pixel sizes with the solenoidal quality being the ultimate cause for systematic variations in model-deduced physical quantities. Conclusions. Optimization-based modeling using SDO/HMI vector data binned to larger pixel sizes yields variations in magnetic energy and helicity estimates of ≲30% on overall, given that concise checks ensure the physical plausibility and high solenoidal quality of the tested model. Spatial-sampling-induced differences are relatively small compared to those arising from other sources of uncertainty, including the effects of applying different data calibration methods, those of using vector data from different instruments, or those arising from application of different NLFF methods to identical input data.

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.

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.

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.

The Magnetic Properties of Heating Events on High-temperature Active-region Loops

Abstract Understanding the relationship between the magnetic field and coronal heating is one of the central problems of solar physics. However, studies of the magnetic properties of impulsively heated loops have been rare. We present results from a study of 34 evolving coronal loops observed in the Fe xviii line component of 94 Å filter images obtained by the Atmospheric Imaging Assembly (AIA)/Solar Dynamics Observatory (SDO) from three active regions with different magnetic conditions. We show that the peak intensity per unit cross section of the loops depends on their individual magnetic and geometric properties. The intensity scales proportionally to the average field strength along the loop (B avg) and inversely with the loop length (L) for a combined dependence of . These loop properties are inferred from magnetic extrapolations of the photospheric Helioseismic and Magnetic Imager (HMI)/SDO line-of-sight and vector magnetic field in three approximations: potential and two nonlinear force-free (NLFF) methods. Through hydrodynamic modeling (enthalpy-based thermal evolution loop (EBTEL) model) we show that this behavior is compatible with impulsively heated loops with a volumetric heating rate that scales as .

On Flare-CME Characteristics from Sun to Earth Combining Remote-Sensing Image Data with In Situ Measurements Supported by Modeling

We analyze the well-observed flare and coronal mass ejection (CME) from 1 October 2011 (SOL2011-10-01T09:18) covering the complete chain of effects – from Sun to Earth – to better understand the dynamic evolution of the CME and its embedded magnetic field. We study in detail the solar surface and atmosphere associated with the flare and CME using the Solar Dynamics Observatory (SDO) and ground-based instruments. We also track the CME signature off-limb with combined extreme ultraviolet (EUV) and white-light data from the Solar Terrestrial Relations Observatory (STEREO). By applying the graduated cylindrical shell (GCS) reconstruction method and total mass to stereoscopic STEREO-SOHO (Solar and Heliospheric Observatory) coronagraph data, we track the temporal and spatial evolution of the CME in the interplanetary space and derive its geometry and 3D mass. We combine the GCS and Lundquist model results to derive the axial flux and helicity of the magnetic cloud (MC) from in situ measurements from Wind. This is compared to nonlinear force-free (NLFF) model results, as well as to the reconnected magnetic flux derived from the flare ribbons (flare reconnection flux) and the magnetic flux encompassed by the associated dimming (dimming flux). We find that magnetic reconnection processes were already ongoing before the start of the impulsive flare phase, adding magnetic flux to the flux rope before its final eruption. The dimming flux increases by more than 25% after the end of the flare, indicating that magnetic flux is still added to the flux rope after eruption. Hence, the derived flare reconnection flux is most probably a lower limit for estimating the magnetic flux within the flux rope. We find that the magnetic helicity and axial magnetic flux are lower in the interplanetary space by ∼ 50% and 75%, respectively, possibly indicating an erosion process. A CME mass increase of 10% is observed over a range of {sim},4,mbox{--},20~mathrm{R}_{odot }. The temporal evolution of the CME-associated core-dimming regions supports the scenario that fast outflows might supply additional mass to the rear part of the CME.

Distribution characteristics of coronal electric current density as an indicator for the occurrence of a solar flare

Abstract In this paper we investigate the distribution characteristics of the coronal electric current density in a flare-producing active region (AR12158; SOL2014-09-10) by reconstructing nonlinear force-free (NLFF) fields from photospheric magnetic field data. A time series of NLFF fields shows the spatial distribution and its temporal development of coronal current density in this active region. A fractal dimensional analysis shows that a concentrated coronal current forms a structure of fractal spatiality. Furthermore, the distribution function of coronal current density is featured with a double power-law profile, and the value of electric current density at the breaking point of a double power-law fitting function shows a noticeable time variation toward the onset of an X-class flare. We discuss that this quantity will be a useful indicator for the occurrence of a flare.

Complexity methods applied to turbulence in plasma astrophysics

In this review many of the well known tools for the analysis of Complex systems are used in order to study the global coupling of the turbulent convection zone with the solar atmosphere where the magnetic energy is dissipated explosively. Several well documented observations are not easy to interpret with the use of Magnetohydrodynamic (MHD) and/or Kinetic numerical codes. Such observations are: (1) The size distribution of the Active Regions (AR) on the solar surface, (2) The fractal and multi fractal characteristics of the observed magnetograms, (3) The Self-Organised characteristics of the explosive magnetic energy release and (4) the very efficient acceleration of particles during the flaring periods in the solar corona. We review briefly the work published the last twenty five years on the above issues and propose solutions by using methods borrowed from the analysis of complex systems. The scenario which emerged is as follows: (a) The fully developed turbulence in the convection zone generates and transports magnetic flux tubes to the solar surface. Using probabilistic percolation models we were able to reproduce the size distribution and the fractal properties of the emerged and randomly moving magnetic flux tubes. (b) Using a Non Linear Force Free (NLFF) magnetic extrapolation numerical code we can explore how the emerged magnetic flux tubes interact nonlinearly and form thin and Unstable Current Sheets (UCS) inside the coronal part of the AR. (c) The fragmentation of the UCS and the redistribution of the magnetic field locally, when the local current exceeds a Critical threshold, is a key process which drives avalanches and forms coherent structures. This local reorganization of the magnetic field enhances the energy dissipation and influences the global evolution of the complex magnetic topology. Using a Cellular Automaton and following the simple rules of Self Organized Criticality (SOC), we were able to reproduce the statistical characteristics of the observed time series of the explosive events, (d) finally, when the AR reaches the turbulently reconnecting state (in the language of the SOC theory this is called SOC state) it is densely populated by UCS which can act as local scatterers (replacing the magnetic clouds in the Fermi scenario) and enhance dramatically the heating and acceleration of charged particles.

FORMATION OF MAGNETIC FLUX ROPES DURING CONFINED FLARING WELL BEFORE THE ONSET OF A PAIR OF MAJOR CORONAL MASS EJECTIONS

NOAA Active Region (AR) 11429 was the source of twin super-fast Coronal Mass Ejections (CMEs). The CMEs took place within a hour from each other, with the onset of the first taking place in the beginning of March 7, 2012. This AR fulfills all the requirements for a "super active region"; namely, Hale's law incompatibility and a $\delta$-spot magnetic configuration. One of the biggest storms of Solar Cycle 24 to date ($D_{st}=-143$ nT) was associated with one of these events. Magnetic Flux Ropes (MFRs) are twisted magnetic structures in the corona, best seen in $\sim$10 MK hot plasma emission and are often considered the core of erupting structures. However, their "dormant" existence in the solar atmosphere (i.e. prior to eruptions), is an open question. Aided by multi-wavelength observations (SDO/HMI/AIA and STEREO EUVI B) and a Non-Linear Force-Free (NLFFF) model for the coronal magnetic field, our work uncovers two separate, weakly-twisted magnetic flux systems which suggest the existence of pre-eruption MFRs that eventually became the seeds of the two CMEs. The MFRs could have been formed during confined (i.e. not leading to major CMEs) flaring and sub-flaring events which took place the day before the two CMEs in the host AR 11429.

Validation of the magnetic energy vs. helicity scaling in solar magnetic structures

We assess the validity of the free magnetic energy - relative magnetic helicity diagram for solar magnetic structures. We used two different methods of calculating the free magnetic energy and the relative magnetic helicity budgets: a classical, volume-calculation nonlinear force-free (NLFF) method applied to finite coronal magnetic structures and a surface-calculation NLFF derivation that relies on a single photospheric or chromospheric vector magnetogram. Both methods were applied to two different data sets, namely synthetic active-region cases obtained by three-dimensional magneto-hydrodynamic (MHD) simulations and observed active-region cases, which include both eruptive and noneruptive magnetic structures. The derived energy--helicity diagram shows a consistent monotonic scaling between relative helicity and free energy with a scaling index 0.84$\pm$0.05 for both data sets and calculation methods. It also confirms the segregation between noneruptive and eruptive active regions and the existence of thresholds in both free energy and relative helicity for active regions to enter eruptive territory. We consider the previously reported energy-helicity diagram of solar magnetic structures as adequately validated and envision a significant role of the uncovered scaling in future studies of solar magnetism.

Validation and Benchmarking of a Practical Free Magnetic Energy and Relative Magnetic Helicity Budget Calculation in Solar Magnetic Structures

In earlier works we introduced and tested a nonlinear force-free (NLFF) method designed to self-consistently calculate the coronal free magnetic energy and the relative magnetic helicity budgets of observed solar magnetic structures. In principle, the method requires only a single, photospheric or low-chromospheric, vector magnetogram of a quiet-Sun patch or an active region and performs calculations without three-dimensional magnetic and velocity-field information. In this work we strictly validate this method using three-dimensional coronal magnetic fields. Benchmarking employs both synthetic, three-dimensional magnetohydrodynamic simulations and nonlinear force-free field extrapolations of the active-region solar corona. Our time-efficient NLFF method provides budgets that differ from those of more demanding semi-analytical methods by a factor of approximately three, at most. This difference is expected to come from the physical concept and the construction of the method. Temporal correlations show more discrepancies that are, however, soundly improved for more complex, massive active regions, reaching correlation coefficients on the order of, or exceeding, 0.9. In conclusion, we argue that our NLFF method can be reliably used for a routine and fast calculation of the free magnetic energy and relative magnetic helicity budgets in targeted parts of the solar magnetized corona. As explained in this article and in previous works, this is an asset that can lead to valuable insight into the physics and triggering of solar eruptions.

SEPARABLE SOLUTIONS OF FORCE-FREE SPHERES AND APPLICATIONS TO SOLAR ACTIVE REGIONS

In this paper, we present a systematic study of the force-free field equation for simple axisymmetric configurations in spherical geometry and apply it to the solar active regions. The condition of separability of solutions in the radial and angular variables leads to two classes of solutions: linear and nonlinear force-free fields. We have studied these linear solutions and extended the nonlinear solutions for the radial power law index to the irreducible rational form $n= p/q$, which is allowed for all cases of odd $p$ and cases of $q>p$ for even $p$, where the poloidal flux $\psi\propto1/r^n$ and field $\mathbf{B}\propto 1/r^{n+2}$). We apply these solutions to simulate photospheric vector magnetograms obtained using the spectropolarimeter on board Hinode. The effectiveness of our search strategy is first demonstrated on test inputs of dipolar, axisymmetric,and non axisymmetric linear force-free fields. Using the best-fit to these magnetograms, we build three-dimensional axisymmetric field configurations and calculate the energy and relative helicity with two independent methods, which are in agreement. We have analyzed five magnetograms for AR 10930 spanning a period of three days during which two X-class flares occurred which allowed us to find the free energy and relative helicity of the active region before and after the flare; our analysis indicates a peak in these quantities before the flare events which is consistent with the results mentioned in literature. We also analyzed single-polarity regions AR 10923 and 10933, which showed very good fits with potential fields. This method can provide useful reconstruction of the nonlinear force-free (NLFF) fields as well as reasonably good input fields for other numerical techniques.

Accuracy analysis and application of extrapolation of force-free fields in solar active and quiet regions

AbstractIn this paper, the availability, applicability and deviation of nonlinear force-free (NLFF) fields extrapolated by Approximate Vertical Integration (AVI), Boundary Integral Equation (BIE) and Optimization (Opt.) methods are studied based on the comparison with two semi-analytical fields (Low & Lou 1990). These NLFF extrapolations based on the observational vector magnetograms are used to study the spatial magnetic field in the quiet Sun.

Coronal Magnetic Field Structure and Evolution for Flaring AR 11117 and Its Surroundings

In this study, photospheric vector magnetograms obtained with the Synoptic Optical Long-term Investigations of the Sun survey (SOLIS), are used as boundary conditions to model the three-dimensional nonlinear force-free (NLFF) coronal magnetic fields as a sequence of nonlinear force-free equilibria in spherical geometry. We study the coronal magnetic field structure inside active regions and its temporal evolution. We compare the magnetic field configuration obtained from NLFF extrapolation before and after flaring event in active region (AR) 11117 and its surroundings observed on 27 October 2010. We compare the magnetic field topologies and the magnetic energy densities and study the connectivities between AR 11117 and its surroundings. During the investigated time period, we estimate the change in free magnetic energy from before to after the flare to be 1.74x10^{32}erg which represents about 13.5% of nonlinear force-free magnetic energy before the flare. In this study, we find that electric currents from AR 11117 to its surroundings were disrupted after the flare.

Error analysis regarding the calculation of nonlinear force-free field

Magnetic field extrapolation is an alternative method to study chromospheric and coronal magnetic fields. In this paper, two semi-analytical solutions of force-free fields (Low and Lou in Astrophys. J. 352:343, 1990) have been used to study the errors of nonlinear force-free (NLFF) fields based on force-free factor α. Three NLFF fields are extrapolated by approximate vertical integration (AVI) Song et al. (Astrophys. J. 649:1084, 2006), boundary integral equation (BIE) Yan and Sakurai (Sol. Phys. 195:89, 2000) and optimization (Opt.) Wiegelmann (Sol. Phys. 219:87, 2004) methods. Compared with the first semi-analytical field, it is found that the mean values of absolute relative standard deviations (RSD) of α along field lines are about 0.96–1.19, 0.63–1.07 and 0.43–0.72 for AVI, BIE and Opt. fields, respectively. While for the second semi-analytical field, they are about 0.80–1.02, 0.67–1.34 and 0.33–0.55 for AVI, BIE and Opt. fields, respectively. As for the analytical field, the calculation error of 〈|RSD|〉 is about 0.1∼0.2. It is also found that RSD does not apparently depend on the length of field line. These provide the basic estimation on the deviation of extrapolated field obtained by proposed methods from the real force-free field.

Comparison of Nonlinear Force-Free Field and Potential Field in the Quiet Sun

In this paper, a potential field extrapolation and three nonlinear force-free (NLFF) field extrapolations (optimization, direct boundary integral (DBIE) and approximate vertical integration (AVI) methods) are used to study the spatial configuration of magnetic field in the quiet Sun. It is found that the strength differences between the three NLFF and potential fields exist in the low layers. However, they tend to disappear as the height increases, which are of the order of 0.1 G when the height exceeds $\sim$ 2000 km above the photosphere. The absolute azimuth difference between one NLFF field and the potential field is as follows: for the optimization field, it decreases evidently as the height increases; for the DBIE field, it almost keeps constant and shows no significant change as the height increases; for the AVI field, it increases slowly as the height increases. The analysis shows that the reconstructed NLFF fields deviate significantly from the potential field in the quiet Sun.

Nonlinear Force-Free and Potential-Field Models of Active-Region and Global Coronal Fields during the Whole Heliosphere Interval

Between 2008/3/24 and 2008/4/2, the three active regions NOAA active regions 10987, 10988 and 10989 were observed daily by the Synoptic Optical Long-term Investigations of the Sun (SOLIS) Vector Spectro-Magnetograph (VSM) while they traversed the solar disk. We use these measurements and the nonlinear force-free magnetic field code XTRAPOL to reconstruct the coronal magnetic field for each active region and compare model field lines with images from the Solar Terrestrial RElations Observatory (STEREO) and Hinode X-ray Telescope (XRT) telescopes. Synoptic maps made from continuous, round-the-clock Global Oscillations Network Group (GONG) magnetograms provide information on the global photospheric field and potential-field source-surface models based on these maps describe the global coronal field during the Whole Heliospheric Interval (WHI) and its neighboring rotations. Features of the modeled global field, such as the coronal holes and streamer belt locations, are discussed in comparison with extreme ultra-violet and coronagraph observations from STEREO. The global field is found to be far from a minimum, dipolar state. From the nonlinear models we compute physical quantities for the active regions such as the photospheric magnetic and electric current fluxes, the free magnetic energy and the relative helicity for each region each day where observations permit. The interconnectivity of the three regions is addressed in the context of the potential-field source-surface model. Using local and global quantities derived from the models, we briefly discuss the different observed activity levels of the regions.

NONLINEAR FORCE-FREE MODELING OF MAGNETIC FIELDS IN A SOLAR FILAMENT

We present a striking filament pattern in the nonlinear force-free (NLFF) chromospheric magnetic field of the active region NOAA 10956. The NLFF chromospheric field is extrapolated from the Hinode high-resolution photospheric vector magnetogram using the weighted optimization method. The modeled structure is characterized by a highly sheared field with strong horizontal magnetic components and has a virtually identical shape and location as the filament seen in Hα. The modeled field strength agrees with the recent He I 10830 A observations by Kuckein et al.. The unequivocal resemblance between the NLFF extrapolation and the Hα observation not only demonstrates the ability of the NLFF field to reproduce chromospheric features, but also provides a valuable diagnostic tool for the filament magnetic fields.