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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, force-free 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., Comment: Accepted to ApJ

The space mission STEREO will provide images from two viewpoints. An important aim of the STEREO mission is to get a 3D view of the solar corona. We develop a program for the stereoscopic reconstruction of 3D coronal loops from images taken with the two STEREO spacecraft. A pure geometric triangulation of coronal features leads to ambiguities because the dilute plasma emissions complicates the association of features in image 1 with features in image 2. As a consequence of these problems the stereoscopic reconstruction is not unique and multiple solutions occur. We demonstrate how these ambiguities can be resolved with the help of different coronal magnetic field models (potential, linear and non-linear force-free fields). The idea is that, due to the high conductivity in the coronal plasma, the emitting plasma outlines the magnetic field lines. Consequently the 3D coronal magnetic field provides a proxy for the stereoscopy which allows to eliminate inconsistent configurations. The combination of stereoscopy and magnetic modelling is more powerful than one of these tools alone. We test our method with the help of a model active region and plan to apply it to the solar case as soon as STEREO data become available., Comment: 18 pages, 5 figures

AIMS: We develop an optimization principle for computing stationary MHD equilibria. METHODS: Our code for the self-consistent computation of the coronal magnetic fields and the coronal plasma uses non-force-free MHD equilibria. Previous versions of the code have been used to compute non-linear force-free coronal magnetic fields from photospheric measurements. The program uses photospheric vector magnetograms and coronal EUV images as input. We tested our reconstruction code with the help of a semi-analytic MHD-equilibrium. The quality of the reconstruction was judged by comparing the exact and reconstructed solution qualitatively by magnetic field-line plots and EUV-images and quantitatively by several different numerical criteria. RESULTS: Our code is able to reconstruct the semi-analytic test equilibrium with high accuracy. The stationary MHD optimization code developed here has about the same accuracy as its predecessor, a non-linear force-free optimization code. The computing time for MHD-equilibria is, however, longer than for force-free magnetic fields. We also extended a well-known class of nonlinear force-free equilibria to the non-force-free regime for purposes of testing the code. CONCLUSIONS: We demonstrate that the code works in principle using tests with analytical equilibria, but it still needs to be applied to real data., Comment: 6 pages, 3 figures

CONTEXT: As the coronal magnetic field can usually not be measured directly, it has to be extrapolated from photospheric measurements into the corona. AIMS: We test the quality of a non-linear force-free coronal magnetic field extrapolation code with the help of a known analytical solution. METHODS: The non-linear force-free equations are numerically solved with the help of an optimization principle. The method minimizes an integral over the force-free and solenoidal condition. As boundary condition we use either the magnetic field components on all six sides of the computational box in Case I or only on the bottom boundary in Case II. We check the quality of the reconstruction by computing how well force-freeness and divergence-freeness are fulfilled and by comparing the numerical solution with the analytical solution. The comparison is done with magnetic field line plots and several quantitative measures, like the vector correlation, Cauchy Schwarz, normalized vector error, mean vector error and magnetic energy. RESULTS: For Case I the reconstructed magnetic field shows good agreement with the original magnetic field topology, whereas in Case II there are considerable deviations from the exact solution. This is corroborated by the quantitative measures, which are significantly better for Case I. CONCLUSIONS: Despite the strong nonlinearity of the considered force-free equilibrium, the optimization method of extrapolation is able to reconstruct it; however, the quality of reconstruction depends significantly on the consistency of the input data, which is given only if the known solution is provided also at the lateral and top boundaries, and on the presence or absence of flux concentrations near the boundaries of the magnetogram., Comment: 6 pages, 2 figures, Research Note

Knowledge regarding the coronal magnetic field is important for the understanding of many phenomena, like flares and coronal mass ejections. Because of the low plasma beta in the solar corona the coronal magnetic field is often assumed to be force-free and we use photospheric vector magnetograph data to extrapolate the magnetic field into the corona with the help of a non-linear force-free optimization code. Unfortunately the measurements of the photospheric magnetic field contain inconsistencies and noise. In particular the transversal components (say Bx and By) of current vector magnetographs have their uncertainties. Furthermore the magnetic field in the photosphere is not necessary force-free and often not consistent with the assumption of a force-free field above. We develop a preprocessing procedure to drive the observed non force-free data towards suitable boundary conditions for a force-free extrapolation. As a result we get a data set which is as close as possible to the measured data and consistent with the force-free assumption., Comment: 20 pages, 5 figures

We describe a newly developed code for the extrapolation of nonlinear force-free coronal magnetic fields in spherical coordinates. The program uses measured vector magnetograms on the solar photosphere as input and solves the force-free equations in the solar corona. The method is based on an optimization principle and the heritage of the newly developed code is a corresponding method in Cartesian geometry. We test the newly developed code with the help of a semi-analytic solution and rate the quality of our reconstruction qualitatively by magnetic field line plots and quantitatively with a number of comparison metrics. We find that we can reconstruct the original test field with high accuracy. The method is fast if the computation is limited to low co-latitudes (say $30^\circ \leq \theta \leq 150^\circ$), but becomes significantly slower if the polar regions are included., Comment: 16 pages, 3 figures, Solar Physics, accepted