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Magneto-static modelling from SUNRISE/IMaX: Application to an active region observed with SUNRISE II  

Thomas Wiegelmann   Submitted: 2017-01-13 10:08

Magneto-static models may overcome some of the issues facing force-free magnetic field extrapolations. So far they have seen limited use and have faced problems when applied to quiet-Sun data. Here we present a first application to an active region. We use solar vector magnetic field measurements gathered by the IMaX polarimeter during the flight of the Sunrise balloon-borne solar observatory in June 2013 as boundary condition for a magneto-static model of the higher solar atmosphere above an active region. The IMaX data are embedded in active region vector magnetograms observed with SDO/HMI. This work continues our magneto-static extrapolation approach, which has been applied earlier (Paper I) to a quiet Sun region observed with Sunrise I. In an active region the signal-to-noise-ratio in the measured Stokes parameters is considerably higher than in the quiet Sun and consequently the IMaX measurements of the horizontal photospheric magnetic field allow us to specify the free parameters of the model in a special class of linear magneto-static equilibria. The high spatial resolution of IMaX (110-130 km, pixel size 40 km) enables us to model the non-force-free layer between the photosphere and the mid chromosphere vertically by about 50 grid points. In our approach we can incorporate some aspects of the mixed beta layer of photosphere and chromosphere, e.g., taking a finite Lorentz force into account, which was not possible with lower resolution photospheric measurements in the past. The linear model does not, however, permit to model intrinsic nonlinear structures like strongly localized electric currents.

Authors: T. Wiegelmann, T. Neukirch, D.H. Nickeler, S.K. Solanki, P. Barthol, A. Gandorfer, L. Gizon, J. Hirzberger, T.L. Riethmüller, M. van Noort, J. Blanco Rodríguez, J. C. Del Toro Iniesta, D. Orozco Suárez, W. Schmidt, V. Martínez Pillet, M. Knölker
Projects: Other

Publication Status: ApJS, accepted
Last Modified: 2017-01-13 13:22
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Magneto-static modelling of the mixed plasma Beta solar atmosphere based on SUNRISE/IMaX data  

Thomas Wiegelmann   Submitted: 2015-11-19 07:12

Our aim is to model the 3D magnetic field structure of the upper solar atmosphere, including regions of non-negligible plasma beta. We use high-resolution photospheric magnetic field measurements from SUNRISE/IMaX as boundary condition for a magneto-static magnetic field model. The high resolution of IMaX allows us to resolve the interface region between photosphere and corona, but modelling this region is challenging for the following reasons. While the coronal magnetic field is thought to be force-free (the Lorentz-force vanishes), this is not the case in the mixed plasma β environment in the photosphere and lower chromosphere. In our model, pressure gradients and gravity forces are taken self-consistently into account and compensate the non-vanishing Lorentz-force. Above a certain height (about 2 Mm) the non-magnetic forces become very weak and consequently the magnetic field becomes almost force-free. Here we apply a linear approach, where the electric current density consists of a superposition of a field-line parallel current and a current perpendicular to the Sun's gravity field. We illustrate the prospects and limitations of this approach and give an outlook for an extension towards a non-linear model.

Authors: T. Wiegelmann, T. Neukirch, D.H. Nickeler, S.K. Solanki, V. Martinez Pillet, J.M. Borrero
Projects: None

Publication Status: ApJ, accepted
Last Modified: 2015-11-20 15:38
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The Magnetic Field in the Solar Atmosphere  

Thomas Wiegelmann   Submitted: 2014-10-17 02:08

This publication provides an overview of magnetic fields in the solar atmosphere with the focus lying on the corona. The solar magnetic field couples the solar interior with the visible surface of the Sun and with its atmosphere. It is also responsible for all solar activity in its numerous manifestations. Thus, dynamic phenomena such as coronal mass ejections and flares are magnetically driven. In addition, the field also plays a crucial role in heating the solar chromosphere and corona as well as in accelerating the solar wind. Our main emphasis is the magnetic field in the upper solar atmosphere so that photospheric and chromospheric magnetic structures are mainly discussed where relevant for higher solar layers. Also, the discussion of the solar atmosphere and activity is limited to those topics of direct relevance to the magnetic field. After giving a brief overview about the solar magnetic field in general and its global structure, we discuss in more detail the magnetic field in active regions, the quiet Sun and coronal holes.

Authors: Thomas Wiegelmann, Julia K. Thalmann and Sami K. Solanki
Projects: None

Publication Status: 109 pages, 30 Figures, to be published in A&ARv
Last Modified: 2014-10-22 12:37
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Solar Force-free Magnetic Fields  

Thomas Wiegelmann   Submitted: 2012-08-28 05:13

The structure and dynamics of the solar corona is dominated by the magnetic field. In most areas in the corona magnetic forces are so dominant that all non-magnetic forces like plasma pressure gradient and gravity can be neglected in the lowest order. This model assumption is called the force-free field assumption, as the Lorentz force vanishes. This can be obtained by either vanishing electric currents (leading to potential fields) or the currents are co-aligned with the magnetic field lines. First we discuss a mathematically simpler approach that the magnetic field and currents are proportional with one global constant, the so-called linear force-free field approximation. In the generic case, however, the relation between magnetic fields and electric currents is nonlinear and analytic solutions have been only found for special cases, like 1D or 2D configurations. For constructing realistic nonlinear force-free coronal magnetic field models in 3D, sophisticated numerical computations are required and boundary conditions must be obtained from measurements of the magnetic field vector in the solar photosphere. This approach is currently of large interests, as accurate measurements of the photospheric field become available from ground-based (for example SOLIS) and space-born (for example Hinode and SDO) instruments. If we can obtain accurate force-free coronal magnetic field models we can calculate the free magnetic energy in the corona, a quantity which is important for the prediction of flares and coronal mass ejections. Knowledge of the 3D structure of magnetic field lines also help us to interpret other coronal observations, e.g., EUV-images of the radiating coronal plasma.

Authors: Thomas Wiegelmann, Takashi Sakurai
Projects: None

Publication Status: Accepted for publication in Living Reviews in Solar Physics
Last Modified: 2012-08-28 12:30
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How to optimize nonlinear force-free coronal magnetic field extrapolations from SDO/HMI vector magnetograms?  

Thomas Wiegelmann   Submitted: 2012-02-17 01:02

The SDO/HMI instruments provide photospheric vector magnetograms with a highspatial and temporal resolution. Our intention is to model the coronal magneticfield above active regions with the help of a nonlinear force-freeextrapolation code. Our code is based on an optimization principle and has beentested extensively with semi-analytic and numeric equilibria and been appliedbefore to vector magnetograms from Hinode and ground based observations.Recently we implemented a new version which takes measurement errors inphotospheric vector magnetograms into account. Photospheric field measurementsare often due to measurement errors and finite nonmagnetic forces inconsistentas a boundary for a force-free field in the corona. In order to deal with theseuncertainties, we developed two improvements: 1.) Preprocessing of the surfacemeasurements in order to make them compatible with a force-free field 2.) Thenew code keeps a balance between the force-free constraint and deviation fromthe photospheric field measurements. Both methods contain free parameters,which have to be optimized for use with data from SDO/HMI. Within this work wedescribe the corresponding analysis method and evaluate the force-freeequilibria by means of how well force-freeness and solenoidal conditions arefulfilled, the angle between magnetic field and electric current and bycomparing projections of magnetic field lines with coronal images from SDO/AIA.We also compute the available free magnetic energy and discuss the potentialinfluence of control parameters.

Authors: T. Wiegelmann, J.K. Thalmann, B. Inhester, T. Tadesse, X. Sun, J.T. Hoeksema
Projects: SDO-HMI

Publication Status: Sol. Phys., accepted
Last Modified: 2012-02-17 08:42
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Magnetic loops in the quiet Sun  

Thomas Wiegelmann   Submitted: 2010-10-04 03:12

We investigate the fine structure of magnetic fields in the atmosphere of the quiet Sun. We use photospheric magnetic field measurements from {sc Sunrise}/IMaX with unprecedented spatial resolution to extrapolate the photospheric magnetic field into higher layers of the solar atmosphere with the help of potential and force-free extrapolation techniques. We find that most magnetic loops which reach into the chromosphere or higher have one foot point in relatively strong magnetic field regions in the photosphere. 91% of the magnetic energy in the mid chromosphere (at a height of 1 Mm) is in field lines, whose stronger foot point has a strength of more than 300 G, i.e. above the equipartition field strength with convection. The loops reaching into the chromosphere and corona are also found to be asymmetric in the sense that the weaker foot point has a strength B < 300 G and is located in the internetwork. Such loops are expected to be strongly dynamic and have short lifetimes, as dictated by the properties of the internetwork fields.

Authors: Thomas Wiegelmann, Sami K Solanki, Juan Borrero, Valentin Martinez Pillet, J. C. del Toro Iniesta, Vicente Domingo, J. A. Bonet Navarro, Peter Barthol, Achim Gandorfer, Michael Knoelker, Wolfgang Schmidt, Alan M. Title
Projects: None

Publication Status: accepted for ApJL Sunrise special issue
Last Modified: 2010-10-04 18:07
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Thin current sheets caused by plasma flow gradients in space and astrophysical plasma  

Thomas Wiegelmann   Submitted: 2010-08-18 04:22

Strong gradients in plasma flows play a major role in space and astrophysical plasmas. A typical situation is that a static plasma equilibrium is surrounded by a plasma flow, which can lead to strong plasma flow gradients at the separatrices between field lines with different magnetic topologies, e.g., planetary magnetospheres, helmet streamers in the solar corona, or at the boundary between the heliosphere and interstellar medium. Within this work we make a first step to understand the influence of these flows towards the occurrence of current sheets in a stationary state situation. We concentrate here on incompressible plasma flows and 2D equilibria, which allow us to find analytic solutions of the stationary magnetohydrodynamics equations (SMHD). First we solve the magnetohydrostatic (MHS) equations with the help of a Grad-Shafranov equation and then we transform these static equilibria into a stationary state with plasma flow. We are in particular interested to study SMHD-equilibria with strong plasma flow gradients perpendicular to separatrices. We find that induced thin current sheets occur naturally in such situations. The strength of the induced currents depend on the Alfvén Mach number and its gradient, and on the magnetic field.

Authors: Dieter H. Nickeler, Thomas Wiegelmann
Projects: None

Publication Status: Ann. Geophys., 28, 1523-1532, 2010, DOI: 10.5194/angeo-28-1523-2010
Last Modified: 2010-08-18 10:20
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How to deal with measurement errors and lacking data in nonlinear force-free coronal magnetic field modelling?  

Thomas Wiegelmann   Submitted: 2010-06-01 09:52

Context: The measured solar photospheric magnetic field vector is extrapolated into the solar corona under the assumption of a force-free plasma. In the generic case this problem is nonlinear. Aims: We aim to improve an algorithm for computing the nonlinear force-free coronal magnetic field. We are in particular interested to incorporate measurement errors and to handle lacking data in the boundary conditions. Methods: We solve the nonlinear force-free field equations by minimizing a functional. Within this work we extend the functional by an additional term, which allows us to incorporate measurement errors and treat regions with lacking observational data. We test the new code with the help of a well known semi-analytic test case. We compare coronal magnetic field extrapolations from ideal boundary conditions and boundary conditions where the transversal magnetic field information is lacking or has a poor signal-to-noise ratio in weak field regions. Results: For ideal boundary conditions the new code gives the same result as the old code. The advantage of the new approach, which includes an error matrix, is visible only for non-ideal boundary conditions. The force-free and solenoidal conditions are fulfilled significantly better and the solutions agrees somewhat better with the exact solution. The new approach also relaxes the boundary and allows a deviation from the boundary data in poor signal-to-noise ratio areas. Conclusions: The incorporation of measurement errors in the updated extrapolation code significantly improves the quality of nonlinear force-free field extrapolation from imperfect boundary conditions.

Authors: T. Wiegelmann and B. Inhester
Projects: None

Publication Status: A&A, in press, DOI: 10.1051/0004-6361/201014391
Last Modified: 2010-06-01 14:47
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Nonlinear force-free modelling: influence of inaccuracies in the measured magnetic vector  

Thomas Wiegelmann   Submitted: 2009-12-15 04:45

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.

Authors: T. Wiegelmann, L. Yelles Chaouche, S. K. Solanki, A. Lagg
Projects: Hinode/SOT

Publication Status: Astronomy and Astrophysics, accepted,
Last Modified: 2009-12-15 12:20
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Solar stereoscopy ? where are we and what developments do we require to progress?  

Thomas Wiegelmann   Submitted: 2009-08-25 05:20

Observations from the two STEREO-spacecraft give us for the first time the possibility to use stereoscopic methods to reconstruct the 3-D solar corona. Classical stereoscopy works best for solid objects with clear edges. Consequently an application of classical stereoscopic methods to the faint structures visible in the optically thin coronal plasma is by no means straight forward and several problems have to be treated adequately: 1) First there is the problem of identifying one-dimensional structures ? e.g. active region coronal loops or polar plumes- from the two individual EUVimages observed with STEREO/EUVI. 2) As a next step one has the association problem to find corresponding structures in both images. This becomes more difficult as the angle between STEREO-A and B increases. 3) Within the reconstruction problem stereoscopic methods are used to compute the 3-D-geometry of the identified structures. Without any prior assumptions, e.g., regarding the footpoints of coronal loops, the reconstruction problem has not one unique solution. 4) One has to estimate the reconstruction error or accuracy of the reconstructed 3-D-structure, which depends on the accuracy of the identified structures in 2-D, the separation angle between the spacecraft, but also on the location, e.g., for east-west directed coronal loops the reconstruction error is highest close to the loop top. 5) Eventually we are not only interested in the 3-D-geometry of loops or plumes, but also in physical parameters like density, temperature, plasma flow, magnetic field strength etc. Helpful for treating some of these problems are coronal magnetic field models extrapolated from photospheric measurements, because observed EUV-loops outline the magnetic field. This feature has been used for a new method dubbed "magnetic stereoscopy". As examples we show recent application to active region loops.

Authors: T. Wiegelmann, B. Inhester, L. Feng
Projects: STEREO

Publication Status: Annales Geophysicae, Volume 27, Issue 7, pp.2925-2936, 2009
Last Modified: 2009-08-25 09:00
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Subject will be restored when possible  

Thomas Wiegelmann   Submitted: 2008-01-08 08:10

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, {it e.g.}, chromospheric Hα 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.

Authors: Wiegelmann, T; Thalmann, J.K.; Schrijver, C.J.; DeRosa, M.L.; Metcalf, T.R.
Projects: Hinode/SOT

Publication Status: Solar Physics, accepted
Last Modified: 2008-01-08 09:53
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Nonlinear force-free modeling of the solar coronal magnetic field.(Review article for CSWM special issue)  

Thomas Wiegelmann   Submitted: 2007-06-06 09:22

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.

Authors: Thomas Wiegelmann
Projects: None

Publication Status: JGR, accepted
Last Modified: 2007-06-06 12:39
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Magnetic Stereoscopy  

Thomas Wiegelmann   Submitted: 2006-12-21 09:45

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.

Authors: T. Wiegelmann, B. Inhester
Projects: None

Publication Status: Solar Physics, Vol. 236, 25-40, 2006
Last Modified: 2006-12-21 09:54
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Preprocessing of vector magnetograph data for a nonlinear force-free magnetic field reconstruction  

Thomas Wiegelmann   Submitted: 2006-12-21 09:39

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.

Authors: T. Wiegelmann, B. Inhester, T.Sakurai
Projects: None

Publication Status: Solar Physics, Vol. 233, 215-232 (2006)
Last Modified: 2006-12-21 09:54
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Testing non-linear force-free coronal magnetic field extrapolations with the Titov-Demoulin equilibrium  

Thomas Wiegelmann   Submitted: 2006-12-21 09:32

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.

Authors: T. Wiegelmann, B. Inhester, B. Kliem, G. Valori and T. Neukirch
Projects: None

Publication Status: A&A, Vol. 453, 737-741 (2006)
Last Modified: 2006-12-21 09:54
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An optimization principle for the computation of MHD equilibria in the solar corona  

Thomas Wiegelmann   Submitted: 2006-12-20 09:37

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.

Authors: Thomas Wiegelmann and Thomas Neukirch
Projects: None

Publication Status: A&A, Vol. 457, 1053-1058, 2006
Last Modified: 2006-12-20 11:35
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COMPUTING NONLINEAR FORCE-FREE CORONAL MAGNETIC FIELDS IN SPHERICAL GEOMETRY  

Thomas Wiegelmann   Submitted: 2006-12-06 05:10

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° leq heta leq 150°), but becomes significantly slower if the polar regions are included.

Authors: Thomas Wiegelmann
Projects: None

Publication Status: Solar Physics, accepted
Last Modified: 2006-12-06 11:24
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Abstracts by Author
Magneto-static modelling from SUNRISE/IMaX: Application to an active region observed with SUNRISE II
Magneto-static modelling of the mixed plasma Beta solar atmosphere based on SUNRISE/IMaX data
The Magnetic Field in the Solar Atmosphere
Solar Force-free Magnetic Fields
How to optimize nonlinear force-free coronal magnetic field extrapolations from SDO/HMI vector magnetograms?
Magnetic loops in the quiet Sun
Thin current sheets caused by plasma flow gradients in space and astrophysical plasma
How to deal with measurement errors and lacking data in nonlinear force-free coronal magnetic field modelling?
Nonlinear force-free modelling: influence of inaccuracies in the measured magnetic vector
Solar stereoscopy ? where are we and what developments do we require to progress?
Subject will be restored when possible
Nonlinear force-free modeling of the solar coronal magnetic field.(Review article for CSWM special issue)
Magnetic Stereoscopy
Preprocessing of vector magnetograph data for a nonlinear force-free magnetic field reconstruction
Testing non-linear force-free coronal magnetic field extrapolations with the Titov-Demoulin equilibrium
An optimization principle for the computation of MHD equilibria in the solar corona
COMPUTING NONLINEAR FORCE-FREE CORONAL MAGNETIC FIELDS IN SPHERICAL GEOMETRY

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