Force-free Magneitic Field Extrapolation for the Complex Sunspot Group of August 1972

Quantitative data on the magnetic field structure at all levels of the solar atmosphere are of basic importance for our understanding of physical processes in active regions in general and in flares in particular. Because photospheric longitudinal magnetograms are the most reliable data available, one has to look for a method of theoretical extrapolation of such data to higher levels. Such a method has been developed for force-free magnetic fields, i.e., ∇ X B = αB , with α = constant, satisfying more realistic boundary conditions as compared with earlier papers; e.g., the net magneitic flux through the magnetogram area is not required to be zero. The method has been used to calculate the magnetic field vector and lines of force in the flare-active region of August 1972. Calculated fields are compared with other observations such as structures in H α . Results of August 3rd show that the loop prominence systems observed during the flares of August 2nd and 4th are represented by a force-free field with positive α rather than by a current-free field ( α = 0) The extractable energy supply of this force-free field is of the order of magnitude of maximum flare demand (10³² erg); the height dependence of the magnetic field strengths agrees with that from radio and X-ray estimates. Similar results are obtained for the August 7th magnetic field structure.     

An electric-current description of solar flares

The presently prevailing theories of solar flares rely on the hypothetical presence of magnetic flux tubes beneath the photosphere and the two subsequent hypotheses, their emergence above the photosphere and explosive magnetic reconnection, converting magnetic energy carried by the flux tubes to solar flare energy. In this paper, we discuss solar flares from an entirely different point of view, namely in terms of power supply by a dynamo process in the photosphere. By this process, electric currents flowing along the magnetic field lines are generated and the familiar ‘force-free’ fields or the ‘sheared’ magnetic fields are produced. Upward field-aligned currents thus generated are carried by downward streaming electrons; these electrons can excite hydrogen atoms in the chromosphere, causing the optical Hα flares or ‘low temperature flares’. It is thus argued that as the ‘force-free’ fields are being built up for the magnetic energy storage, a flare must already be in progress.     

RECONSTRUCTION OF THE THREE-DIMENSIONAL CORONAL MAGNETIC FIELD

Studies of solar flares indicate that the mechanism of flares is magnetic in character and that the coronal magnetic field is a key to understanding solar high-energy phenomena. In our ongoing research we are conducting a systematic study of a large database of observations which includes both coronal structure (from the Soft X-ray Telescope on the Yohkoh spacecraft) and photospheric vector magnetic fields (from the Haleakala Stokes Polarimeter at Mees Solar Observatory). We compare the three-dimensional nonlinear force-free coronal magnetic field, computed from photospheric boundary data, to images of coronal structure. In this paper we outline our techniques and present results for active region AR 7220/7222. We show that the computed force-free coronal magnetic field agrees well with Yohkoh X-ray coronal loops, and we discuss the properties of the coronal magnetic field and the soft X-ray loops.     

A Comparison Between Nonlinear Force-Free Field and Potential Field Models Using Full-Disk SDO/HMI Magnetogram

Measurements of magnetic fields and electric currents in the pre-eruptive corona are crucial to the study of solar eruptive phenomena, like flares and coronal mass ejections (CMEs). However, spectro-polarimetric measurements of certain photospheric lines permit a determination of the vector magnetic field only at the photosphere. Therefore, there is considerable interest in accurate modeling of the solar coronal magnetic field using photospheric vector magnetograms as boundary data. In this work, we model the coronal magnetic field above multiple active regions with the help of a potential field and a nonlinear force-free field (NLFFF) extrapolation code over the full solar disk using Helioseismic and Magnetic Imager (SDO/HMI) data as boundary conditions. We compare projections of the resulting magnetic field lines with full-disk coronal images from the Atmospheric Imaging Assembly (SDO/AIA) for both models. This study has found that the NLFFF model reconstructs the magnetic configuration closer to observation than the potential field model for full-disk magnetic field extrapolation. We conclude that many of the trans-equatorial loops connecting the two solar hemispheres are current-free.     

Nonlinear Force-Free Extrapolation of Emerging Flux with a Global Twist and Serpentine Fine Structures

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.     

CAN WE EXTRAPOLATE A MAGNETIC FIELD WHEN ITS TOPOLOGY IS COMPLEX?

In order to understand various solar phenomena controlled by the magnetic field, such as X-ray bright points, flares and prominence eruptions, the structure of the coronal magnetic field must be known. This requires a precise extrapolation of the photospheric magnetic field. Presently, only potential or linear force-free field approximations can be used easily. A more realistic modelling of the field is still an active research area because of well-known difficulties related to the nonlinear mixed elliptic-hyperbolic nature of the equations. An additional difficulty arises due to the complexity of the magnetic field structure which is caused by a discrete partition of the photospheric magnetic field. This complexity is not limited to magnetic regions having magnetic nulls (and so separatrices) but also occurs in those containing thin elongated volumes (called Quasi-Separatrix Layers) where the photospheric field-line linkage changes rapidly. There is a wide range for the thickness of such layers, which is determined by the character (bipolar or quadrupolar) of the magnetic region, by the sizes of the photospheric field concentrations and by the intensity of the electric currents. The aim of this paper is to analyse the recent nonlinear force-free field extrapolation techniques for complex coronal magnetic fields.     

Force-free magnetic fields in the solar atmosphere

Reliable measurements of the solar magnetic field are restricted to the level of the photosphere. For about half a century attempts have been made to calculate the field in the layers above the photosphere, i.e. in the chromosphere and in the corona, from the measured photospheric field. The procedure is known as magnetic field extrapolation. In the superphotospheric parts of active regions the magnetic field is approximately force-free, i.e. electric currents are aligned with the magnetic field. The practical application to solar active regions has been largely confined to constant-α or linear force-free fields, with a spatially constant ratio, α, between the electric current and the magnetic field. We review results obtained from extrapolations with constant-α force-free fields, in particular on magnetic topologies favourable for flares and on magnetic and current helicities. Presently, different methods are being developed to calculate non-constant-α or nonlinear force-free fields from photospheric vector magnetograms. We also briefly discuss these methods and present a comparison of a linear and a nonlinear force-free magnetic field extrapolation applied to the same photospheric boundary data. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Force-free magnetic arcades relevant to two-ribbon solar flares

Simple analytic models for the passive evolution of arcade-like magnetic fields through a series of force-free equilibria are presented. At the photospheric boundary, the normal magnetic field component is prescribed together with either the longitudinal field component or the photospheric shear. Analytic progress is made by considering either cylindrically symmetric solutions or using the separation of variables technique. Two distinct cylindrically symmetric force-free fields are obtained that possess the same normal field component and photospheric shear. The scond field contains a magnetic bubble. As the shear increases beyond a critical value, so the magnetic energy of the first configuration exceeds that of the second. The possibility is therefore suggested of an eruption of the first field outwards towards the second. Such an eruptive instability is proposed as the origin of a two-ribbon solar flare.A new analytic solution to the force-free field equations, of separable form, is discovered and it is pointed out that the existence of shear in a magnetic field does not preclude it from being potential.Now at AWRE, Aldermaston, Reading, Berkshire.     

Vertical current and a 3B/X12 flare in a highly sheared active region (NOAA 6659) on June 9, 1991

NOAA active region 6659, during its June 1991 transit across the solar disk, showed highly sheared vector magnetic field structures and produced numerous powerful flares, including five white-light flares. Photospheric vector magnetograms of this active region were obtained at the Huairou Solar Observing Station of the Beijing Astronomical Observatory. After the resolution of the 180° ambiguity of the transverse magnetic field and transformation of off-center vector magnetograms to the heliographic plane, we have determined the photospheric vertical current density and discussed the relationship with powerful flares. The following results were obtained: (a) The powerful 3B/X12 flare on June 9, 1991 was triggered by the interaction between the large-scale electric current system and magnetic flux of opposite polarity. (b) The kernels of the powerful Hβ flare (sites of the white-light flare) were close to the peaks of the vertical electric current density. (c) Some small-scale structures of the vertical current relative to the magnetic islands of opposite polarity have not been found. This probably implies that the electric current is not always parallel to the magnetic field in solar active regions.     

A comparison of coronal X-ray structures of active regions with magnetic fields computed from photospheric observations

The appearances of several X-ray active regions observed on March 7, 1970 and June 15, 1973 are compared with the corresponding coronal magnetic field topology. Coronal fields have been computed from measurements of the longitudinal component of the underlying magnetic fields, under the current-free hypothesis. An overall correspondence between X-ray structures and calculated field lines is established, and the magnetic counterpart of different X-ray features is also examined. A correspondence between enhanced X-ray emission and the location of compact closed field lines is suggested by this study. Representative magnetic field values calculated under the assumption of current-free fields are given for heights up to 200″.     

Magnetic diffusion and flare energy buildup

Photospheric motion shears or twists solar magnetic fields to increase magnetic energy in the corona, because this process may change a current-free state of a coronal field to force-free states which carry electric current. This paper analyzes both linear and nonlinear two-dimensional force-free magnetic field models and derives relations of magnetic energy buildup with photospheric velocity field. When realistic data of solar magnetic field (B ₀ 10³ G) and photospheric velocity field (v _max 1 km s^–1) are used, it is found that 3–4 hours are needed to create an amount of free magnetic energy which is of the order of the current-free field energy. Furthermore, the paper studies situations in which finite magnetic diffusivities in photospheric plasma are introduced. The shearing motion increases coronal magnetic energy, while the photospheric diffusion reduces the energy. The variation of magnetic energy in the coronal region, then, depends on which process dominates.     

Theory of quadrupolar sunspots and the active region of August, 1972

Energy is stored when the force-free magnetic field in an active region departs from a potential field, the departure showing up as a shear in the field. As soon as the field untwists, energy will be released to produce flares. Based on this idea, we derived an analytical solution of the equation of force-free field under the assumption of a constant force-free factor, and found expressions for seven important quantities for quadrupolar sunspots: the magnetic energy of the twisted field, that of potential field, the extractable free energy ΔM, the magnetic flux, the total current, the force-free factor and the field decay factor, in terms of three observables: the field intensity, the twist angle and the distance between two spots of the same polarity. The expression for ΔM can be useful in solar prediction work. For the active region of August, 1972, we found ΔM up to 6 × 10³² erg, sufficient to supply the energy of the observed flare activity. Observations of this active region are in good general agreement with our theoretical expectations: in the entire twisting of the quadrupolar sunspot group, each spot assumes the form of a complete spiral in the clockwise direction for each of the four spots.     

Magnetic flux of colinear bipolar spot pairs

It has been widely conjectured that solar flares are energized by the magnetic energy stored in complex active regions. Paradoxically, however, in attempting to show that magnetic changes cause or characterize flares, solar magnetic observations have produced equivocal results.In previous attempts at resolving the paradox, it has been contended that magnetic measurements are simply imprecise or that magnetic theories of flares are incorrect. We present an alternative explanation: the present use of magnetograms to examine active region structure through numerical integration of miscellaneous field lines (under various force-free assumptions) provides qualitative information only and does not utilize the quantitative information available. Therefore, we propose a new approach to the analysis of magnetograms which is illustrated with a highly symmetrized example that permits integration in closed form. The proposed approach exploits the cellular structure of the flux of field lines present in a complex active region. The various topological connectivities distinguish parent and daughter flux cells. A function F is developed expressing the flux partitioned into the daughter cell of interconnected field lines in a potential field. This F is a function of the location, strength, and relative motions of the photospheric sources. Then dF/dt is used as an EMF in the direct calculation of the stored magnetic energy available for flare production. In carrying out this program the flux partitioning surface (separatrix) is calculated along with its line of self-intersection (separator). The separator is the location of the principal energy release site.     

无力场模型黑子上空的纵向磁场梯度

本文按常α无力场模型计算了1980年10月23日Boulder 2744活动区前导黑子的纵向磁场随高度的变化,并与用CIV 1548谱线观测得到的色球一日冕过渡区的磁场资料相结合,求得CIV 1548发射区的有效高度。这些结果与文献[4]中对同一黑子用势场模型推求的结果有很大差别。从而表明,势场和无力场在某些方面导致的结果是极不相同的。鉴于观测已表明活动区上空存在电流的事实,在活动区磁场的模拟中,特别是在强扭曲活动区磁场的计算中,应当避免采用势场,而尽可能采用无力场模型。     

High energetic solar proton flares on the declining phase of solar cycle 22

The year 1991 is a part of the declining phase of the solar cycle 22, during which high energetic flares have been produced by active regions NOAA/USAF 6659 in June. The associated solar proton events have affected the Earth environment and their proton fluxes have been measured by GOES space craft. The evaluation of solar activity during the first half of June 1991, have been carried out by applying a method for high energetic solar flares prediction on the flares of June 1991. The method depends on cumulative summation curves according to observed H-alpha flares, X-ray bursts, in the active region 6659 during one rotation when the energetic solar flares of June 1991 have occurred. It has been found that the steep trend of increased activity sets on several tens of hours prior to the occurrence of the energetic flare, which can be used, together with other methods, for forecasts of major flares. All the used data at the present work are received from NOAA, Boulder, Colorado, USA.     

The Free Energy of NOAA Solar Active Region AR 11029

The NOAA active region (AR) 11029 was a small but highly active sunspot region which produced 73 GOES soft X-ray flares during its transit of the disk in late October 2009. The flares appear to show a departure from the well-known power law frequency-size distribution. Specifically, too few GOES C-class and no M-class flares were observed by comparison with a power law distribution (Wheatland, Astrophys. J. 710, 1324, 2010). This was conjectured to be due to the region having insufficient magnetic energy to power the missing large events. We construct nonlinear force-free extrapolations of the coronal magnetic field of AR 11029 using data taken on 24 October by the SOLIS Vector SpectroMagnetograph (SOLIS/VSM) and data taken on 27 October by the Hinode Solar Optical Telescope SpectroPolarimeter (Hinode/SP). Force-free modeling with photospheric magnetogram data encounters problems, because the magnetogram data are inconsistent with a force-free model. We employ a recently developed “self-consistency” procedure which addresses this problem and accommodates uncertainties in the boundary data (Wheatland and Régnier, Astrophys. J. 700, L88, 2009). We calculate the total energy and free energy of the self-consistent solution, which provides a model for the coronal magnetic field of the active region. The free energy of the region was found to be ≈?4×10²⁹?erg on 24 October and ≈?7×10³¹?erg on 27 October. An order of magnitude scaling between RHESSI non-thermal energy and GOES peak X-ray flux is established from a sample of flares from the literature and is used to estimate flare energies from the observed GOES peak X-ray flux. Based on the scaling, we conclude that the estimated free energy of AR 11029 on 27 October when the flaring rate peaked was sufficient to power M-class or X-class flares; hence, the modeling does not appear to support the hypothesis that the absence of large flares is due to the region having limited energy.     

Gradients of the line-of-sight magnetic fields in active region NOAA 6659

The gradients of line-of-sight magnetic fields in active region NOAA 6659 on 1991 June 8 have been calculated based on the photospheric and chromospheric magnetograms taken at Huairou Solar Observing Station. We found that high gradients coincided with high strengths of the transverse magnetic fields, implying a complicated configuration of the magnetic field in the lower atmosphere.For this extraordinarily flare-prolific region, a possible relationship between the gradients and the flares was inferred.