Nonlinear force-free field modeling of a solar active region using SDO/HMI and SOLIS/VSM data

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.Comment: 9 pages, 5 figure

Can We Improve the Preprocessing of Photospheric Vector Magnetograms by the Inclusion of Chromospheric Observations?

The solar magnetic field is key to understanding the physical processes in the solar atmosphere. Nonlinear force-free codes have been shown to be useful in extrapolating the coronal field upward from underlying vector boundary data. However, we can only measure the magnetic field vector routinely with high accuracy in the photosphere, and unfortunately these data do not fulfill the force-free condition. We must therefore apply some transformations to these data before nonlinear force-free extrapolation codes can be self-consistently applied. To this end, we have developed a minimization procedure that yields a more chromosphere-like field, using the measured photospheric field vectors as input. The procedure includes force-free consistency integrals, spatial smoothing, and -- newly included in the version presented here -- an improved match to the field direction as inferred from fibrils as can be observed in, e.g., chromospheric H$\alpha$ images. We test the procedure using a model active-region field that included buoyancy forces at the photospheric level. The proposed preprocessing method allows us to approximate the chromospheric vector field to within a few degrees and the free energy in the coronal field to within one percent.Comment: 22 pages, 6 Figur

How to optimize nonlinear force-free coronal magnetic field extrapolations from SDO/HMI vector magnetograms?

The SDO/HMI instruments provide photospheric vector magnetograms with a high spatial and temporal resolution. Our intention is to model the coronal magnetic field above active regions with the help of a nonlinear force-free extrapolation code. Our code is based on an optimization principle and has been tested extensively with semi-analytic and numeric equilibria and been applied before to vector magnetograms from Hinode and ground based observations. Recently we implemented a new version which takes measurement errors in photospheric vector magnetograms into account. Photospheric field measurements are often due to measurement errors and finite nonmagnetic forces inconsistent as a boundary for a force-free field in the corona. In order to deal with these uncertainties, we developed two improvements: 1.) Preprocessing of the surface measurements in order to make them compatible with a force-free field 2.) The new code keeps a balance between the force-free constraint and deviation from the photospheric field measurements. Both methods contain free parameters, which have to be optimized for use with data from SDO/HMI. Within this work we describe the corresponding analysis method and evaluate the force-free equilibria by means of how well force-freeness and solenoidal conditions are fulfilled, the angle between magnetic field and electric current and by comparing projections of magnetic field lines with coronal images from SDO/AIA. We also compute the available free magnetic energy and discuss the potential influence of control parameters.Comment: 17 Pages, 6 Figures, Sol. Phys., accepte

Nonlinear force-free magnetic field extrapolations: comparison of the Grad-Rubin and Wheatland-Sturrock-Roumeliotis algorithm

We compare the performance of two alternative algorithms which aim to construct a force-free magnetic field given suitable boundary conditions. For this comparison, we have implemented both algorithms on the same finite element grid which uses Whitney forms to describe the fields within the grid cells. The additional use of conjugate gradient and multigrid iterations result in quite effective codes. The Grad-Rubin and Wheatland-Sturrock-Roumeliotis algorithms both perform well for the reconstruction of a known analytic force-free field. For more arbitrary boundary conditions the Wheatland-Sturrock-Roumeliotis approach has some difficulties because it requires overdetermined boundary information which may include inconsistencies. The Grad-Rubin code on the other hand loses convergence for strong current densities. For the example we have investigated, however, the maximum possible current density seems to be not far from the limit beyond which a force free field cannot exist anymore for a given normal magnetic field intensity on the boundary.Comment: 21 pages, 13 figure

A Statistical Study on Force-Freeness of Solar Magnetic Fields in the Photosphere

It is an indisputable fact that solar magnetic fields are force-free in the corona, where force free fields means that current and magnetic fields are parallel and there is no Lorentz force in the fields. While the force-free extent of photospheric magnetic fields remains open. In this paper, the statistical results about it is given. The vector magnetograms (namely, $B_{x}$, $B_{y}$ and $B_{z}$ in heliocentric coordinates) are employed, which are deduced and calibrated from Stokes spectra, observed by Solar Magnetic Field Telescope (SMFT) at Huairou Solar Observing Station (HSOS) are used. We study and calibrated 925 magnetograms calibrated by two sets of calibration coefficients, that indicate the relations between magnetic fields and the strength of Stokes spectrum and can be calculated either theoretically or empirically. The statistical results show that the majority of active region magnetic fields are not consistent with the force-free model.Comment: 10 pages, 5 figures, has been accepted by PAS

Nonlinear force-free modeling of the solar coronal magnetic field

The coronal magnetic field is an important quantity because the magnetic field dominates the structure of the solar corona. Unfortunately direct measurements of coronal magnetic fields are usually not available. The photospheric magnetic field is measured routinely with vector magnetographs. These photospheric measurements are extrapolated into the solar corona. The extrapolated coronal magnetic field depends on assumptions regarding the coronal plasma, e.g. force-freeness. Force-free means that all non-magnetic forces like pressure gradients and gravity are neglected. This approach is well justified in the solar corona due to the low plasma beta. One has to take care, however, about ambiguities, noise and non-magnetic forces in the photosphere, where the magnetic field vector is measured. Here we review different numerical methods for a nonlinear force-free coronal magnetic field extrapolation: Grad-Rubin codes, upward integration method, MHD-relaxation, optimization and the boundary element approach. We briefly discuss the main features of the different methods and concentrate mainly on recently developed new codes.Comment: 33 pages, 3 figures, Review articl

Magnetic Connectivity between Active Regions 10987, 10988, and 10989 by Means of Nonlinear Force-Free Field Extrapolation

Extrapolation codes for modelling the magnetic field in the corona in cartesian geometry do not take the curvature of the Sun's surface into account and can only be applied to relatively small areas, \textit{e.g.}, a single active region. We apply a method for nonlinear force-free coronal magnetic field modelling of photospheric vector magnetograms in spherical geometry which allows us to study the connectivity between multi-active regions. We use vector magnetograph data from the Synoptic Optical Long-term Investigations of the Sun survey (SOLIS)/Vector Spectromagnetograph(VSM) to model the coronal magnetic field, where we study three neighbouring magnetically connected active regions (ARs: 10987, 10988, 10989) observed on 28, 29, and 30 March 2008, respectively. We compare the magnetic field topologies and the magnetic energy densities and study the connectivities between the active regions(ARs). We have studied the time evolution of magnetic field over the period of three days and found no major changes in topologies as there was no major eruption event. From this study we have concluded that active regions are much more connected magnetically than the electric current.Comment: Solar Physic

Analysis and interpretation of a fast limb CME with eruptive prominence, C-flare and EUV dimming

Coronal Mass ejections or CMEs are large dynamical solar-corona events. The mass balance and kinematics of a fast limb CME, including its prominence progenitor and the associated flare, will be compared with computed magnetic structures to look for their origin and effect. Multi-wavelength ground-based and space-borne observations are used to study a fast W-limb CME event of December 2, 2003, taking into account both on and off disk observations. Its erupting prominence is measured at high cadence with the Pic du Midi full H-alpha line-flux imaging coronagraph. EUV images from space instruments are processed including difference imaging. SOHO/LASCO images are used to study the mass excess and motions. A fast bright expanding coronal loop is identified in the region recorded slightly later by GOES as a C7.2 flare, followed by a brightening and an acceleration phase of the erupting material with both cool and hot components. The total coronal radiative flux dropped by 5 percent in the EUV channels, revealing a large dimming effect at and above the limb. The typical 3-part structure observed 1 hour later shows a core shaped similarly to the eruptive filament/prominence. The total measured mass of the escaping CME (1.5x10to16 g from C2 LASCO observations) definitely exceeds the estimated mass of the escaping cool prominence material although assumptions made to analyse the Ha erupting prominence, as well as the corresponding EUV darkening of the filament observed several days before, made this evaluation uncertain by a factor of 2. From the current free extrapolation we discuss the shape of the magnetic neutral surface and a possible scenario leading to an instability, including the small scale dynamics inside and around the filament.Comment: 11 pages, 9 figure

On the Force-Freeness of the Photospheric Sunspot Magnetic Fields as Observed from Hinode (SOT/SP)

A magnetic field is force-free if there is no interaction between the magnetic field and plasma in surrounding atmosphere i.e., electric currents are aligned with the magnetic field, giving rise to zero Lorentz force. Computation of various magnetic parameters such as magnetic energy, gradient of twist of sunspot fields and any kind of extrapolations, heavily hinge on the force-free approximation of the photospheric sunspot magnetic fields. Thus it is important to inspect the force-freeness of sunspot fields. The force-freeness of sunspot magnetic fields has been examined earlier by some researchers ending with incoherent results. Accurate photospheric vector field measurements with high spatial resolution are required to inspect the force-free nature of sunspots. We use several such vector magnetograms obtained from the Solar Optical Telescope/Spectro-Polarimeter aboard the Hinode. Both necessary and sufficient conditions for force-freeness are examined by checking global and local nature of magnetic forces over sunspots. We find that the sunspot magnetic fields are not much away from force-free configuration, although they are not completely force-free on the photosphere. The umbral and inner penumbral fields are more force-free than the middle and the outer penumbral fields. During their evolution, sunspot magnetic fields are found to maintain their proximity to force-free behaviour. Although a dependence of net Lorentz force components is seen on the evolutionary stages of the sunspots, we don't find a systematic relationship between the nature of sunspot fields and associated flare activity. Further, we examine whether the fields at photosphere follow linear or non-linear force free conditions. After examining this in various complex and simple sunspots we conclude that,in either case,the photospheric sunspot fields are closer to satisfy non linear force-free field approximation.Comment: 34 pages, 4 figures, 2 tables; accepted for publication in the Astrophysical Journa

Using models of the ocean's mean dynamic topography to identify errors in coastal geodetic levelling

Identifying errors (blunders and systematic errors) in coastal geodetic levelling networks has often been problematic. This is because (1) mean sea level (MSL) at tide gauges cannot be directly compared to height differences from levelling because the geoid/quasigeoid and MSL are not parallel, being separated by the ocean’s mean dynamic topography (MDT) and (2) the lack of redundancy at the edge of the levelling network. This paper sets out a methodology to independently identify blunders and/or systematic errors (over long distances) in geodetic levelling using MDT models to account for the separation between the geoid/quasigeoid and MSL at tide gauges. This method is then tested in a case study using an oceanographic MDT model, MSL observations, GNSS data and a quasigeoid model. The results are significant because the errors found could not be detected by standard levelling misclosure checks alone, with supplementary data from an MDT model, with cross-validation from GNSS-quasigeoid allowing their detection. In addition, it appears that an oceanographic-only MDT is as effective as GNSS and a quasigeoid model for detecting levelling errors, which could be particularly useful for countries with coastal levelling errors in their levelling networks that cannot be identified by conventional levelling closure checks