Magnetic field configurations associated with polarity intrusion in a solar active region

This paper presents a new class of exact solutions describing the non-linear force-free field above a spatially localized photospheric bipolar magnetic region. An essential feature is the variation in all three Cartesian directions and this could not be modelled adequately with previously known symmetric force-free fields. Sequences of force-free fields are constructed and analyzed to simulate the slow growth of a pair of spots on the photosphere. The axis connecting the spots executes rotational motion, distorting the photospheric neutral line separating fluxes of opposite signs. We show directly from the analytic solutions that the resulting reversal of the positions of the spots relative to the background field is associated with (i) the creation of magnetic free energy, (ii) the severe shearing of localized low-lying loops in the vicinity where the photospheric transverse field aligns with the photospheric neutral line, and (iii) the emergence and disappearance of flux from the photosphere at these highly stressed regions. The model relates theoretically for the first time these different magnetic field features that have been suggested by observation and theoretical considerations to be flare precursors. A general formula, based on the virial theorem, is also given for the free energy of a force-free field, strictly in terms of the field value at the photosphere. This formula has obvious practical application.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

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.     

Optimization code with weighting function for the reconstruction of coronal magnetic fields

We developed a code for the reconstruction of nonlinear force-free and non-force-free coronal magnetic fields. The 3D magnetic field is computed numerically with the help of an optimization principle. The force-free and non-force-free codes are compiled in one program. The force-free approach needs photospheric vector magnetograms as input. The non-force-free code additionally requires the line-of-sight integrated coronal density distribution in combination with a tomographic inversion code. Previously the optimization approach has been used to compute magnetic fields using all six boundaries of a computational box. Here we extend this method and show how the coronal magnetic field can be reconstructed only from the bottom boundary, where the boundary conditions are measured with vector magnetographs. The program is planed for use within the Stereo mission.     

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.     

Differential rotation and the structure and energy content of coronal magnetic fields

It is argued that differential rotation of the photospheric magnetic fields will induce currents in the corona. The work done against surface magnetic stresses will increase the energy content of the coronal magnetic field. The electrical conductivities are high and the foot points of field lines move with the differential rotation. The force-free field equations are solved with this constraint to obtain a minimum estimate of the energy increase for a quadrupole field. During a solar rotation the magnetic energy increases by 25%. Local release of this energy in the corona would have a significant effect. The expansion of field lines as a result of the differential rotation should increase the amount of flux and the field strength in the solar wind region.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Observational evidence for quantization in photospheric magnetic flux

Observations are presented which suggest that away from sunspots photospheric magnetic flux is quantized. Assuming the elemental area of a magnetic region to be 1 (arc-sec)² the elemental field strength is 525 G.Kitt Peak National Observatory Contribution No. 501.Operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.     

Including stereoscopic information in the reconstruction of coronal magnetic fields

We present a method to include stereoscopic information about the three-dimensional structure of flux tubes into the reconstruction of the coronal magnetic field. Due to the low plasma beta in the corona we can assume a force-free magnetic field, with the current density parallel to the magnetic field lines. Here we use linear force-free fields for simplicity. The method uses the line-of-sight magnetic field on the photosphere as observational input. The value of is determined iteratively by comparing the reconstructed magnetic field with the observed structures. The final configuration is the optimal linear force-free solution constrained by both the photospheric magnetogram and the observed plasma structures. As an example we apply our method to SOHO MDI/EIT data of an active region. In the future it is planned to apply the method to analyse data from the SECCHI instrument aboard the STEREO mission.     

The effects of viewing angle on the inference of magnetic shear in preflare active regions

The magnetic shear at a point within an active region field configuration can be defined (Hagyard et al., 1984b) as the difference in angle between the observed photospheric transverse field and that of a reference potential field calculated using the observed line-of-sight field as a boundary condition. Using analytic models for non-potential (but force-free) fields representative of preflaring active regions, we calculate the degree of magnetic shear along the magnetic neutral line that such fields would exhibit, as a function of the location and orientation of the active region on the solar disk. We find that, except for regions close to disk center, the position of the inferred neutral line (zero line-of-sight field) is significantly different from the actual neutral line (zero radial field), and that the calculated reference potential field also varies significantly with the position of the region. Thus the inferred degree of shear can vary significantly with the position and orientation of the region, due to (a) straightforward geometric projection effects, (b) the shift of the inferred neutral line relative to its true position, and (c) variations in the reference potential field. The significance of these results for flare prediction is considered.Presidential Young Investigator.     

Preprocessing of Vector Magnetograph Data for a Nonlinear Force-Free Magnetic Field Reconstruction

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 nonlinear force-free optimization code. Unfortunately, the measurements of the photospheric magnetic field contain inconsistencies and noise. In particular, the transversal components (say B _x and B _y) of current vector magnetographs have their uncertainties. Furthermore, the magnetic field in the photosphere is not necessarily force free and often not consistent with the assumption of a force-free field above the magnetogram. 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.     

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.     

Global Solar Free Magnetic Energy and Electric Current Density Distribution of Carrington Rotation 2124

Solar eruptive phenomena, like flares and coronal mass ejections (CMEs), are governed by magnetic fields. To describe the structure of these phenomena one needs information on the magnetic flux density and the electric current density vector components in three dimensions throughout the atmosphere. However, current spectro-polarimetric measurements typically limit the determination of the vector magnetic field to only 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 for global solar atmosphere using nonlinear force-free field (NLFFF) extrapolation codes implemented to a synoptic maps of photospheric vector magnetic field synthesized from the Vector Spectromagnetograph (VSM) on Synoptic Optical Long-term Investigations of the Sun (SOLIS) as boundary condition. Using the resulting three-dimensional magnetic field, we calculate the three-dimensional electric current density and magnetic energy throughout the solar atmosphere for Carrington rotation 2124 using our global extrapolation code. We found that spatially, the low-lying, current-carrying core field demonstrates a strong concentration of free energy in the active-region core, from the photosphere to the lower corona (about 70 Mm). The free energy density appears largely co-spatial with the electric current distribution.     

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.     

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)     

The three-dimensional magnetic field structure of Hale region 16747 (1980 April) and its evolution

Using the force-free model and the observed photospheric fields as boundary conditions, we calculated the three-dimensional magnetic field of the active region Hale 16747 and discussed its structure and evolution. The results show that 1) the magnetic flux tubes twisted continuously during their rise; 2) the twisting was mainly due to the counter-clock wise rotation of the preceding sunspots and the development of the δ structure in the middle of the region; 3) the thin dark filaments seen on April 5 east of the preceding sunspots ran in the direction of the transverse field, hence were probably the trajectories of mass motion; 4) the time rate of solar flare occurrence in this region can be explained by the evolution of the magnetic field; 5) the two homologons flares on April 7 and 8 resulted from two sets of magnetic arches located in series. These results support to some extent speculations given in Refs. [1] and [2].     

Computation of solar magnetic fields from photospheric observations

The observational difficulties of obtaining the magnetic field distribution in the chromosphere and corona of the Sun has led to methods of extending photospheric magnetic measurements into the solar atmosphere by mathematical procedures. A new approach to this problem presented here is that a constant alpha force-free field can be uniquely determined from the tangential components of the measured photospheric flux alone. The vector magnetographs now provide measurements of both the solar photospheric tangential and the longitudinal magnetic field. This paper presents derivations for the computation of the solar magnetic field from these type of measurements. The fields considered are assumed to be a constant alpha force-free fields or equivalent, producing vanishing Lorentz forces. Consequently, magnetic field lines and currents are related by a constant and hence show an identical distribution. The magnetic field above simple solar regions are described from the solution of the field equations.     

An essay on sunspots and solar flares

The presently prevailing theories of sunspots and 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 for solar flare energy.In this paper, we pay attention to the fact that there are large-scale magnetic fields which divide the photosphere into positive and negative (line-of-sight) polarity regions and that they are likely to be more fundamental than sunspot fields, as emphasized most recently by McIntosh (1981). A new phenomenological model of the sunspot pair formation is then constructed by considering an amplification process of these largescale fields near their boundaries by shear flows, including localized vortex motions. The amplification results from a dynamo process associated with such vortex flows and the associated convergence flow in the largescale fields.This dynamo process generates also some of the familiar “force-free” fields or the “sheared” magnetic fields in which the magnetic field-aligned currents are essential. Upward field-aligned currents generated by the dynamo process are carried by downward streaming electrons which are expected to be accelerated by an electric potential structure; a similar structure is responsible for accelerating auroral electrons in the magnetosphere. Depending on the magnetic field configuration and the shear flows, the current-carrying electrons precipitate into different geometrical patterns, causing circular flares, umbral flares, two-ribbon flares, etc. Thus, it is suggested that “low temperature flares” are directly driven by the photospheric dynamo process.     

Magnetic Energy Storage and Current Density Distributions for Different Force-Free Models

In the last decades, force-free-field modelling has been used extensively to describe the coronal magnetic field and to better understand the physics of solar eruptions at different scales. Especially the evolution of active regions has been studied by successive equilibria in which each computed magnetic configuration is subject to an evolving photospheric distribution of magnetic field and/or electric-current density. This technique of successive equilibria has been successful in describing the rate of change of the energetics for observed active regions. Nevertheless the change in magnetic configuration due to the increase/decrease of electric current for different force-free models (potential, linear and nonlinear force-free fields) has never been studied in detail before. Here we focus especially on the evolution of the free magnetic energy, the location of the excess of energy, and the distribution of electric currents in the corona. For this purpose, we use an idealised active region characterised by four main polarities and a satellite polarity, allowing us to specify a complex topology and sheared arcades to the coronal magnetic field but no twisted flux bundles. We investigate the changes in the geometry and connectivity of field lines, the magnetic energy and current-density content as well as the evolution of null points. Increasing the photospheric current density in the magnetic configuration does not dramatically change the energy-storage processes within the active region even if the magnetic topology is slightly modified. We conclude that for reasonable values of the photospheric current density (the force-free parameter α<0.25 Mm⁻¹), the magnetic configurations studied do change but not dramatically: i) the original null point stays nearly at the same location, ii) the field-line geometry and connectivity are slightly modified, iii) even if the free magnetic energy is significantly increased, the energy storage happens at the same location. This extensive study of different force-free models for a simple magnetic configuration shows that some topological elements of an observed active region, such as null points, can be reproduced with confidence only by considering the potential-field approximation. This study is a preliminary work aiming at understanding the effects of electric currents generated by characteristic photospheric motions on the structure and evolution of the coronal magnetic field.     

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.     

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.     

Photospheric and heliospheric magnetic fields

The magnetic field in the heliosphere evolves in response to the photospheric field at its base. This evolution, together with the rotation of the Sun, drives space weather through the continually changing conditions of the solar wind and the magnetic field embedded within it. We combine observations and simulations to investigate the sources of the heliospheric field from 1996 to 2001. Our algorithms assimilate SOHO/MDI magnetograms into a flux-dispersal model, showing the evolving field on the full sphere with an unprecedented duration of 5.5 yr and temporal resolution of 6 hr. We demonstrate that acoustic far-side imaging can be successfully used to estimate the location and magnitude of large active regions well before they become visible on the solar disk. The results from our assimilation model, complemented with a potential-field source-surface model for the coronal and inner-heliospheric magnetic fields, match Yohkoh/SXT and KPNO/He?10830 Å coronal hole boundaries quite well. Even subject to the simplification of a uniform, steady solar wind from the source surface outward, our model matches the polarity of the interplanetary magnetic field (IMF) at Earth ～3% of the time during the period 1997–2001 (independent of whether far-side acoustic data are incorporated into the simulation). We find that around cycle maximum, the IMF originates typically in a dozen disjoint regions. Whereas active regions are often ignored as a source for the IMF, the fraction of the IMF that connects to magnetic plage with absolute flux densities exceeding 50 Mx cm^?2 increases from ?10% at cycle minimum up to 30–50% at cycle maximum, with even direct connections between sunspots and the heliosphere. For the overall heliospheric field, these fractions are ?1% to 20–30%, respectively. Two case studies based on high-resolution TRACE observations support the direct connection of the IMF to magnetic plage, and even to sunspots. Parallel to the data assimilation, we run a pure simulation in which active regions are injected based on random selection from parent distribution functions derived from solar data. The global properties inferred for the photospheric and heliospheric fields for these two models are in remarkable agreement, confirming earlier studies that no subtle flux-emergence patterns or field-dispersal properties are required of the solar dynamo beyond those that are included in the model in order to understand the large-scale solar and heliospheric fields.