Nonlinear force-free field (NLFFF) models are thought to be viable tools for investigating the structure, dynamics and evolution of the coronae of solar active regions. In a series of NLFFF modeling studies, we have found that NLFFF models are successful in application to analytic test cases, and relatively successful when applied to numerically constructed Sun-like test cases, but they are less successful in application to real solar data. Different NLFFF models have been found to have markedly different field line configurations and to provide widely varying estimates of the magnetic free energy in the coronal volume, when applied to solar data. NLFFF models require consistent, forcefree vector magnetic boundary data. However, vector magnetogram observations sampling the photosphere, which is dynamic and contains significant Lorentz and buoyancy forces, do not satisfy this requirement, thus creating several major problems for force-free coronal modeling efforts. In this article, we discuss NLFFF modeling of NOAA Active Region 10953 using Hinode/SOT-SP, Hinode/XRT, STEREO/SECCHI-EUVI, and SOHO/MDI observations, and in the process illustrate the three such issues we judge to be critical to the success of NLFFF modeling: (1) vector magnetic field data covering larger areas are needed so that more electric currents associated with the full active regions of interest are measured, (2) the modeling algorithms need a way to accommodate the various uncertainties in the boundary data, and (3) a more realistic physical model is needed to approximate the photosphere-to-corona interface in order to better transform the forced photospheric magnetograms into adequate approximations of nearly force-free fields at the base of the corona. We make recommendations for future modeling efforts to overcome these as yet unsolved problems.

We compare six algorithms for the computation of nonlinear force-free (NLFF) magnetic fields (including optimization, magnetofrictional, Grad-Rubin based, and Green's function-based methods) by evaluating their performance in blind tests on analytical force-free-field models for which boundary conditions are specified either for the entire surface area of a cubic volume or for an extended lower boundary only. Figures of merit are used to compare the input vector field to the resulting model fields. Based on these merit functions, we argue that all algorithms yield NLFF fields that agree best with the input field in the lower central region of the volume, where the field and electrical currents are strongest and the effects of boundary conditions weakest. The NLFF vector fields in the outer domains of the volume depend sensitively on the details of the specified boundary conditions; best agreement is found if the field outside of the model volume is incorporated as part of the model boundary, either as potential field boundaries on the side and top surfaces, or as a potential field in a skirt around the main volume of interest. For input field (B) and modeled field (b), the best method included in our study yields an average relative vector error E n = |B − b| / |B| of only 0.02 when all sides are specified and 0.14 for the case where only the lower boundary is specified, while 162 CAROLUS J. SCHRIJVER ET AL. the total energy in the magnetic field is approximated to within 2%. The models converge towards the central, strong input field at speeds that differ by a factor of one million per iteration step. The fastest-converging, best-performing model for these analytical test cases is the Wheatland, Sturrock, and Roumeliotis (2000) optimization algorithm as implemented by Wiegelmann (2004).

Solar flares and coronal mass ejections are associated with rapid changes in field connectivity and are powered by the partial dissipation of electrical currents in the solar atmosphere. A critical unanswered question is whether the currents involved are induced by the motion of preexisting atmospheric magnetic flux subject to surface plasma flows or whether these currents are associated with the emergence of flux from within the solar convective zone. We address this problem by applying state-of-the-art nonlinear force-free field (NLFFF) modeling to the highest resolution and quality vector-magnetographic data observed by the recently launched Hinode satellite on NOAA AR 10930 around the time of a powerful X3.4 flare. We compute 14 NLFFF models with four different codes and a variety of boundary conditions. We find that the model fields differ markedly in geometry, energy content, and force-freeness. We discuss the relative merits of these models in a general critique of present abilities to model the coronal magnetic field based on surface vector field measurements. For our application in particular, we find a fair agreement of the best-fit model field with the observed coronal configuration, and argue (1) that strong electrical currents emerge together with magnetic flux preceding the flare, (2) that these currents are carried in an ensemble of thin strands, (3) that the global pattern of these currents and of field lines are compatible with a large-scale twisted flux rope topology, and (4) that the $10 32 erg change in energy associated with the coronal electrical currents suffices to power the flare and its associated coronal mass ejection.

No abstract

The nonlinear force-free field (NLFFF) model is often used to describe the solar coronal magnetic field, however a series of earlier studies revealed difficulties in the numerical solution of the model in application to photospheric boundary data. We investigate the sensitivity of the modeling to the spatial resolution of the boundary data, by applying multiple codes that numerically solve the NLFFF model to a sequence of vector magnetogram data at different resolutions, prepared from a single Hinode/SOT-SP scan of NOAA Active Region 10978 on 2007 December 13. We analyze the resulting energies and relative magnetic helicities, employ a Helmholtz decomposition to characterize divergence errors, and quantify changes made by the codes to the vector magnetogram boundary data in order to be compatible with the force-free model. This study shows that NLFFF modeling results depend quantitatively on the spatial resolution of the input boundary data, and that using more highly resolved boundary data yields more self-consistent results. The free energies of the resulting solutions generally trend higher with increasing resolution, while relative magnetic helicity values vary significantly between resolutions for all methods. All methods require changing the horizontal components, and for some methods also the vertical components, of the vector magnetogram boundary field in excess of nominal uncertainties in the data. The solutions produced by the various methods are significantly different at each resolution level. We continue to recommend verifying agreement between the modeled field lines and corresponding coronal loop images before any NLFFF model is used in a scientific setting.standing the physics underlying magnetic energy release, and improving the ability to predict space weather storms caused by large events.A popular model for the coronal magnetic field B is the nonlinear force-free field (NLFFF) model (see the reviews by Wiegelmann & Sakurai 2012 and Régnier 2013), which assumes a static configuration with a zero Lorentz force,where J = µ −1 0 ∇×B is the electric current density, together with the solenoidal condition, arXiv:1508.05455v1 [astro-ph.SR]

To better understand eruptive events in the solar corona, we combine sequences of multi-wavelength observations and modelling of the coronal magnetic field of NOAA AR 8210, a highly flare-productive active region. From the photosphere to the corona, the observations give us information about the motion of magnetic elements (photospheric magnetograms), the location of flares (e.g., Hα, EUV or soft X-ray brightenings), and the type of events (Hα blueshift events). Assuming that the evolution of the coronal magnetic field above an active region can be described by successive equilibria, we follow in time the magnetic changes of the 3D nonlinear force-free (nlff) fields reconstructed from a time series of photospheric vector magnetograms. We apply this method to AR 8210 observed on May 1, 1998 between 17:00 UT and 21:40 UT. We identify two types of horizontal photospheric motions that can drive an eruption: a clockwise rotation of the sunspot, and a fast motion of an emerging polarity. The reconstructed nlff coronal fields give us a scenario of the confined flares observed in AR 8210: the slow sunspot rotation enables the occurence of flare by a reconnection process close to a separatrix surface whereas the fast motion is associated with small-scale reconnections but no detectable flaring activity. We also study the injection rates of magnetic energy, Poynting flux and relative magnetic helicity through the photosphere and into the corona. The injection of magnetic energy by transverse photospheric motions is found to be correlated with the storage of energy in the corona and then the release by flaring activity. The magnetic helicity derived from the magnetic field and the vector potential of the nlff configuration is computed in the coronal volume. The magnetic helicity evolution shows that AR 8210 is dominated by the mutual helicity between the closed and potential fields and not by the self helicity of the closed field which characterizes the twist of confined flux bundles. We conclude that for AR 8210 the complex topology is a more important factor than the twist in the eruption process.

Nonlinear force-free solutions for the magnetic field in the solar corona constructed using photospheric vector magnetic field boundary data suffer from a basic problem: the observed boundary data are inconsistent with the nonlinear force-free model. Specifically, there are two possible choices of boundary conditions on vertical current provided by the data, and the two choices lead to different force-free solutions. A novel solution to this problem is described. Bayesian probability is used to modify the boundary values on current density, using field-line connectivity information from the two force-free solutions and taking into account uncertainties, so that the boundary data are more consistent with the two nonlinear force-free solutions. This procedure may be iterated until a set of self-consistent boundary data (the solutions for the two choices of boundary conditions are the same) is achieved. The approach is demonstrated to work in application to Hinode/SOT observations of NOAA active region 10953.

Abstract. The Active Region 8151 (AR 8151) observed in February 1998 is the site of an eruptive event associated with a filament and a S-shaped structure, and producing a slow Coronal Mass Ejection (CME). In order to determine how the CME occurs, we compute the 3D coronal magnetic field and we derive some relevant parameters such as the free magnetic energy and the relative magnetic helicity. The 3D magnetic configuration is reconstructed from photospheric magnetic magnetograms (IVM, Mees Solar Observatory) in the case of a non-constant-α force-free (nlff) field model. The reconstruction method is divided into three main steps: the analysis of vector magnetograms (transverse fields, vertical density of electric current, ambiguity of 180• ), the numerical scheme for the nlff magnetic field, the interpretation of the computed magnetic field with respect to the observations. For AR 8151, the nlff field matches the coronal observations from EIT/SOHO and from SXT/Yohkoh. In particular, three characteristic flux tubes are shown: a highly twisted flux tube, a long twisted flux tube and a quasi-potential flux tube. The maximum energy budget is estimated to 2.6 × 10 31 erg and the relative magnetic helicity to 4.7 × 10 34 G 2 cm 4 . From the simple photospheric magnetic distribution and the evidence of highly twisted flux tubes, we argue that the flux rope model is the most likely to describe the initiation mechanism of the eruptive event associated with AR 8151.