M. L. DeRosa1, M. S. Wheatland2, K. D. Leka3, G. Barnes3, T. Amari4, A. Canou4, S. A. Gilchrist5,2, J. K. Thalmann6, G. Valori7, T. Wiegelmann8, C. J. Schrijver1, A. Malanushenko9, X. Sun10, S. Régnier11
1 Lockheed Martin Solar and Astrophysics Laboratory, Palo Alto, CA, USA
2 School of Physics, University of Sydney, Australia
3 NorthWest Research Associates, Boulder, CO, USA
4 Centre de Physique Théorique de l’École Polytechnique, Palaiseau, France
5 Observatoire de Paris, Meudon, France
6 Institute of Physics/IGAM, University of Graz, Austria
7 Mullard Space Science Laboratory, University College London, UK
8 Max-Planck-Institut für Sonnensystemforschung, Göttingen, Germany
9 Montana State University, Bozeman, MT, USA
10 Stanford University, Stanford, CA, USA
11 Northumbria University, Newcastle-Upon-Tyne, UK
Calculating nonlinear force-free fields (NLFFFs) is an increasingly popular way to estimate magnetic field geometries and connectivities, free energies, helicities, and other important physical quantities of interest associated with solar active regions. This popularity stems in part from the wider availability of vector magnetic field data from recent instrumentation such as Hinode/SOT, SOLIS, and SDO/HMI. Tests of the NLFFF model on boundary data with known solution fields confirm that the extrapolations can reproduce key features of the test fields1,2, however the application to solar cases has proved more problematic and less reliable3.
NLFFFs are distinguished from potential fields by the presence of currents within the computational domain that are parallel to magnetic field lines, and as a result boundary conditions on the currents are required. Vertical currents determined from photospheric vector magnetograms of solar active regions are typically used. However, such currents often show structure on fine scales, and thus the spatial resolution of the vector magnetogram data may be an important factor for successful extrapolation. NLFFF extrapolations using lower-resolution magnetograms may not always capture all of the currents important for structuring the magnetic fields overlying solar active regions.
Figure 1 | Comparison of the vertical magnetic field Bz and vertical magnetic current Jz for AR 10978 at two resolution levels. The images of panels (a) and (b), which show Bz and Jz, have a pixel size of approximately 0.1″, or close to the native resolution of Hinode/SOT-SP. In panels (c) and (d), the Hinode data have been rebinned down by a factor of eight, and many small-size features lost in the process. The pixel size of SDO/HMI vector magnetogram data of approximately 0.5″ lies in between these resolution levels. (This figure originally appeared as Fig. 2 in ref. 4 and is reproduced by permission of the AAS.)
To explore the effects of the spatial resolution on the NLFFF technique, we use rebinned polarization spectra of AR 10978 as observed by the Hinode/SOT-SP to generate vector magnetograms at various spatial resolution levels4. The resulting magnetograms have pixel sizes ranging from about 0.1 Mm to 1.7 Mm. These vector magnetograms were used as input for multiple NLFFF method solvers, and the solutions were intercompared. Maps of the vertical magnetic field Bz and vertical current Jz sampled at two of the resolution levels used in the study are shown in Fig. 1. The figure illustrates that many small-scale features present in the higher-resolution images of the active region become less distinct in the lower-resolution versions. Fig. 4 of ref. 4 provides a more quantitative evaluation of these effects.
Figure 2 | Energy versus pixel size for the full set of NLFFF calculations. The colored lines in each panel represent values of (a) total magnetic energy E, (b) energy of the associated potential field E0, and (c) free energy Ef = E – E0 for NLFFF calculations using boundary data having the specified pixel sizes, with each color representing the results from a different NLFFF solver. Four of five solvers show greater Ef as the pixel sizes become smaller, i.e., as more highly-resolved boundary data are used. The solvers are optimization (purple), magnetofrictional (blue), CFIT (orange), XTRAPOL (red), and FEMQ (black). (This figure is adapted from Fig. 6 of ref. 4 and is reproduced by permission of the AAS.)
This study indicates that, in general, NLFFFs constructed using more highly resolved boundary data have higher free energies and are more force- and divergence-free. As an example of these trends, Fig. 2 shows the total, potential, and free energies as a function of pixel size for the full set of NLFFF solutions considered in ref. 4. Four out of the five solvers yielded solutions with higher free energies at the higher resolution levels than at the lower levels, sometimes by as much as a factor of two. (Note that the plots in the figure are presented as a function of pixel sizes, and thus higher resolutions are located leftward in each plot.)
This study also shows that differences between methods are as significant as the spatial resolution of the boundary data. The different solvers (optimization, magnetofrictional, CFIT, XTRAPOL, and FEMQ) are color-coded in Fig. 2, and the range of energies across solvers is at least as large as the variation with pixel size. Additionally, the construction of NLFFFs is affected by other issues, the most prominent of which is that data from photospheric sources are known not to be force-free5, in direct contradiction to the model assumptions. As a result, changes are made by the solvers to the boundary data in order to be consistent with the model, and different solvers resolve the inconsistency in different ways. Given all of these factors, it is therefore recommended that users of NLFFF solutions check agreement with observed coronal loops, and evaluate the force-free and divergence-free properties before using these solutions in a scientific setting.
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