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Register a new userThe Influence of Spatial Resolution on Nonlinear Force-Free Modeling
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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.
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Context: Solar magnetic fields are regularly extrapolated into the corona starting from photospheric magnetic measurements that can suffer from significant uncertainties. Aims: Here we study how inaccuracies introduced into the maps of the photospheric magnetic vector from the inversion of ideal and noisy Stokes parameters influence the extrapolation of nonlinear force-free magnetic fields. Methods: We compute nonlinear force-free magnetic fields based on simulated vector magnetograms, which have been produced by the inversion of Stokes profiles, computed froma 3-D radiation MHD simulation snapshot. These extrapolations are compared with extrapolations starting directly from the field in the MHD simulations, which is our reference. We investigate how line formation and instrumental effects such as noise, limited spatial resolution and the effect of employing a filter instrument influence the resulting magnetic field structure. The comparison is done qualitatively by visual inspection of the magnetic field distribution and quantitatively by different metrics. Results: The reconstructed field is most accurate if ideal Stokes data are inverted and becomes less accurate if instrumental effects and noise are included. The results demonstrate that the non-linear force-free field extrapolation method tested here is relatively insensitive to the effects of noise in measured polarization spectra at levels consistent with present-day instruments. Conclusions heading: Our results show that we can reconstruct the coronal magnetic field as a nonlinear force-free field from realistic photospheric measurements with an accuracy of a few percent, at least in the absence of sunspots.
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.
We use our semi-analytic solution of the nonlinear force-free field equation to construct three-dimensional magnetic fields that are applicable to the solar corona and study their statistical properties for estimating the degree of braiding exhibited by these fields. We present a new formula for calculating the winding number and compare it with the formula for the crossing number. The comparison is shown for a toy model of two helices and for realistic cases of nonlinear force-free fields; conceptually the formulae are nearly the same but the resulting distributions calculated for a given topology can be different. We also calculate linkages, which are useful topological quantities that are independent measures of the contribution of magnetic braiding to the total free energy and relative helicity of the field. Finally, we derive new analytical bounds for the free energy and relative helicity for the field configurations in terms of the linking number. These bounds will be of utility in estimating the braided energy available for nano-flares or for eruptions.
We use SDO/HMI and SOLIS/VSM photospheric magnetic field measurements to model the force-free coronal field above a solar active region, assuming magnetic forces to dominate. We take measurement uncertainties caused by, e.g., noise and the particular inversion technique into account. After searching for the optimum modeling parameters for the particular data sets, we compare the resulting nonlinear force-free model fields. We show the degree of agreement of the coronal field reconstructions from the different data sources by comparing the relative free energy content, the vertical distribution of the magnetic pressure and the vertically integrated current density. Though the longitudinal and transverse magnetic flux measured by the VSM and HMI is clearly different, we find considerable similarities in the modeled fields. This indicates the robustness of the algorithm we use to calculate the nonlinear force-free fields against differences and deficiencies of the photospheric vector maps used as an input. We also depict how much the absolute values of the total force-free, virial and the free magnetic energy differ and how the orientation of the longitudinal and transverse components of the HMI- and VSM-based model volumes compares to each other.
Presently, many models of the coronal magnetic field rely on photospheric vector magnetograms but these data have been shown to be problematic as the sole boundary information for nonlinear force-free field (NLFFF) extrapolations. Magnetic fields in the corona manifest themselves in high-energy images (X-rays and EUV) in the shapes of coronal loops, providing an additional constraint that at present is not used due to the mathematical complications of incorporating such input into numerical models. Projection effects and the limited number of usable loops further complicate the use of coronal information. We develop and test an algorithm to use images showing coronal loops in the modeling of the solar coronal magnetic field. We first fit projected field lines with field lines of constant-als force-free fields to approximate the three-dimensional distribution of currents in the corona along a sparse set of trajectories. We then apply a Grad-Rubin-like iterative technique to obtain a volume-filling nonlinear force-free model of the magnetic field, modifying method presented in citet{Wheatland2007}. We thoroughly test the technique on known analytical and solar-like model magnetic fields previously used for comparing different extrapolation techniques citep{Schrijver2006, Schrijver2008} and compare the results with those obtained by presently available methods that rely only on the photospheric data. We conclude that we have developed a functioning method of modeling the coronal magnetic field by combining the line-of-sight component of photospheric magnetic field with information from coronal images. Vector magnetograms over the full or partial photospheric boundary of the numerical domain could optionally be used.
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