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Relative magnetic helicity, as a conserved quantity of ideal magnetohydrodynamics, has been highlighted as an important quantity to study in plasma physics. Due to its nonlocal nature, its estimation is not straightforward in both observational and numerical data. In this study we derive expressions for the practical computation of the gauge-independent relative magnetic helicity in three-dimensional finite domains. The derived expressions are easy to implement and rapid to compute. They are derived in Cartesian coordinates, but can be easily written in other coordinate systems. We apply our method to a numerical model of a force-free equilibrium containing a flux rope, and compare the results with those obtained employing known half-space equations. We find that our method requires a much smaller volume than half-space expressions to derive the full helicity content. We also prove that values of relative magnetic helicity of different magnetic fields can be compared with each other in the same sense as free-energy values can. Therefore, relative magnetic helicity can be meaningfully and directly compared between different datasets, such as those from different active regions, but also within the same dataset at different times. Typical applications of our formulae include the helicity computation in three-dimensional models of the solar atmosphere, e.g., coronal-field reconstructions by force-free extrapolation and discretized magnetic fields of numerical simulations.

We study the flux emergence process in NOAA active region 11024, between 29 June and 7 July 2009, by means of multi-wavelength observations and nonlinear force-free extrapolation. The main aim is to extend previous investigations by combining, as much as possible, high spatial resolution observations to test our present understanding of small-scale (undulatory) flux emergence, whilst putting these small-scale events in the context of the global evolution of the active region. The combination of these techniques allows us to follow the whole process, from the first appearance of the bipolar axial field on the east limb, until the buoyancy instability could set in and raise the main body of the twisted flux tube through the photosphere, forming magnetic tongues and signatures of serpentine field, until the simplification of the magnetic structure into a main bipole by the time the active region reaches the west limb. At the crucial time of the main emergence phase high spatial resolution spectropolarimetric measurements of the photospheric field are employed to reconstruct the three-dimensional structure of the nonlinear force-free coronal field, which is then used to test the current understanding of flux emergence processes. In particular, knowledge of the coronal connectivity confirms the identity of the magnetic tongues as seen in their photospheric signatures, and it exemplifies how the twisted flux, which is emerging on small scales in the form of a sea-serpent, is subsequently rearranged by reconnection into the large-scale field of the active region. In this way, the multi-wavelength observations combined with a nonlinear force-free extrapolation provide a coherent picture of the emergence process of small-scale magnetic bipoles, which subsequently reconnect to form a large-scale structure in the corona.

Context. Extrapolations of solar photospheric vector magnetograms into three-dimensional magnetic fields in the chromosphere and corona are usually done under the assumption that the fields are force-free. This condition is violated in the photosphere itself and a thin layer in the lower atmosphere above. The field calculations can be improved by preprocessing the photospheric magnetograms. The intention here is to remove a non-force-free component from the data.
Aims. We compare two preprocessing methods presently in use, namely the methods of Wiegelmann et al. (2006, Sol. Phys., 233, 215) and Fuhrmann et al. (2007, A&A, 476, 349).
Methods. The two preprocessing methods were applied to a vector magnetogram of the recently observed active region NOAA AR 10 953. We examine the changes in the magnetogram effected by the two preprocessing algorithms. Furthermore, the original magnetogram and the two preprocessed magnetograms were each used as input data for nonlinear force-free field extrapolations by means of two different methods, and we analyze the resulting fields.
Results. Both preprocessing methods managed to significantly decrease the magnetic forces and magnetic torques that act through the magnetogram area and that can cause incompatibilities with the assumption of force-freeness in the solution domain. The force and torque decrease is stronger for the Fuhrmann et al. method. Both methods also reduced the amount of small-scale irregularities in the observed photospheric field, which can sharply worsen the quality of the solutions. For the chosen parameter set, the Wiegelmann et al. method led to greater changes in strong-field areas, leaving weak-field areas mostly unchanged, and thus providing an approximation of the magnetic field vector in the chromosphere, while the Fuhrmann et al. method weakly changed the whole magnetogram, thereby better preserving patterns present in the original magnetogram. Both preprocessing methods raised the magnetic energy content of the extrapolated fields to values above the minimum energy, corresponding to the potential field. Also, the fields calculated from the preprocessed magnetograms fulfill the solenoidal condition better than those calculated without preprocessing.

Context. Reliable measurements of the solar magnetic field are restricted to the phoptosphere. As an alternative to measurements, the field in the higher layers of the atmosphere is calculated from the measured photospheric field, mostly under the assumption that it is force-free. However, the magnetic field in the photosphere is not force-free. Moreover, most methods for the extrapolation of the photospheric magnetic field into the higher layers prescribe the magnetic vector on the whole boundary of the considered volume, which overdetermines the force-free field. Finally, the extrapolation methods are very sensitive to small-scale noise in the magnetograph data, which, however, if sufficienly resolved numerically, should affect the solution only in a thin boundary layer close to the photosphere. Aims. A new method for the preprocessing of solar photospheric vector magnetograms has been developed that, by improving their compatibility with the condition of force- freeness and removing small-scale noise, makes them more suitable for extrapolations into three- dimensional nonlinear force-free magnetic fields in the chromosphere and corona. Methods. A functional of the photospheric field values is minimized whereby the total magnetic force and the total magnetic torque on the considered volume above the photosphere, as well as a quantity measuring the degree of small-scale noise in the photospheric boundary data, are simultaneously made small. For the minimization, the method of simulated annealing is used and the smoothing of noisy magnetograph data is attained by windowed median averaging. Results. The method was applied to a magnetogram derived from a known nonlinear force-free test field to which an artificial noise had been added. The algorithm recovered all main structures of the magnetogram and removed small- scale noise. The main test was to extrapolate from the noisy photospheric vector magnetogram before and after the preprocessing. The preprocessing was found to significantly improve the agreement of the extrapolated with the exact field.

Testing magnetofrictional extrapolation with the Titov-Demoulin model of solar active regions
(2010)

We examine the nonlinear magnetofrictional extrapolation scheme using the solar active region model by Titov and Demoulin as test field. This model consists of an arched, line-tied current channel held in force-free equilibrium by the potential field of a bipolar flux distribution in the bottom boundary. A modified version with a parabolic current density profile is employed here. We find that the equilibrium is reconstructed with very high accuracy in a representative range of parameter space, using only the vector field in the bottom boundary as input. Structural features formed in the interface between the flux rope and the surrounding arcade - "hyperbolic flux tube" and "bald patch separatrix surface" - are reliably reproduced, as are the flux rope twist and the energy and helicity of the configuration. This demonstrates that force-free fields containing these basic structural elements of solar active regions can be obtained by extrapolation. The influence of the chosen initial condition on the accuracy of reconstruction is also addressed, confirming that the initial field that best matches the external potential field of the model quite naturally leads to the best reconstruction. Extrapolating the magnetogram of a Titov-Demoulin equilibrium in the unstable range of parameter space yields a sequence of two opposing evolutionary phases, which clearly indicate the unstable nature of the configuration: a partial buildup of the flux rope with rising free energy is followed by destruction of the rope, losing most of the free energy.

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 paper, 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 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.