Magnetic topologies due to two bipolar regions

The Sun's atmosphere contains many diverse phenomena that are dominated by the coronal magnetic field. To understand these phenomena it is helpful to determine first the structure of the magnetic field, i.e., the magnetic topology. We study here the topological structure of the coronal magnetic field arising from the interaction of two bipolar regions, for which we find that four distinct, topologically stable states are possible. A bifurcation diagram is produced, showing how the magnetic configuration can change from one topology to another as the relative orientation and sizes of the bipolar regions are varied. The changes are produced either by a global separator bifurcation, a local double-separator bifurcation, a new, global separatrix quasi-bifurcation, or a new, global spine quasi-bifurcation.     

Separators in 3D Quiet-Sun Magnetic Fields

At the confluence of four regions of different magnetic connectivity lies a distinct topological candidate for coronal heating, namely the magnetic separator. In this study, a method for tracing separator curves is developed and the statistical properties of separators in coronal fields are subsequently explored by analysing a model field with an exponential source distribution, similar to that studied by Schrijver and Title (2002). Magnetic fields based on data from an observed sequence of MDI magnetograms are also considered as a case study. The picture that emerges is one in which there are many more magnetic separators than previously thought, since many separators arise from each null point. For an exponential source distribution, an average of 10.1±0.13 separators per null are found, of which 1.04±0.04 multiply link pairs of nulls (i.e., there is more than one separator linking such pairs of nulls). For the observed sequence of magnetograms, these figures are 7.63±0.2 and 0.99± 0.059, respectively. The results obtained here show that separators have a tendency to group together into trunks about a null. In the case of prone nulls, these trunks lie either normal to the photospheric surface or on it. It is also established that pairs of coronal nulls are frequently interconnected, suggesting that they may have been created by purely coronal bifurcations.     

Coronal Magnetic Topologies in a Spherical Geometry I. Two Bipolar Flux Sources

The evolution of the solar corona is dominated to a large extent by the hugely complicated magnetic field which threads it. Magnetic topology provides a tool to decipher the structure of this field and thus help to understand its behaviour. Usually, the magnetic topology of a potential field is calculated due to flux sources on a locally planar photospheric surface. We use a Green's function method to extend this theory to sources on a global spherical surface. The case of two bipolar flux-balanced source regions is studied in detail, with an emphasis on how the distribution and relative strengths of the source regions affect the resulting topological states. A new state with two spatially distinct separators connecting the same two magnetic null points, called the “dual intersecting“ state, is discovered. An erratum to this article is available at .     

A New Method for Finding Topological Separators in a Magnetic Field

Magnetic topology is a powerful tool for constraining certain physical properties of a given magnetic configuration, including the strengths and locations of current sheets, relative helicity and the magnetic free energy available for reconnection. A critical feature of magnetic topology is the separator, a field line bordering several different regions of connectivity. With existing methods, these field lines are at best computationally expensive and at worst impossible to find. A new method is presented for finding the Minimal Separator Set, all of the separators that necessarily exist in a configuration, and to use this information in combination with the optical analogy and a simulated annealing method to ‘cool’ an initial guess for each separator into a good approximation.     

A Framework for Understanding the Topology of Complex Coronal Structures

The Sun's coronal magnetic field is highly complex and provides the driving force for many dynamical processes. The topology of this complex field is made up mainly of discrete topological building blocks produced by small numbers of magnetic fragments. In this work we develop a method for predicting the possible topologies due to a potential field produced by three photospheric sources, and describe how this model accurately predicts the results of Brown and Priest (1999). We then sketch how this idea may be extended to more general non-symmetric configurations. It is found that, for the case of positive total flux, a local separator bifurcation may take place with three positive sources or with one positive and two negative sources, but not for two positive sources and one negative.     

Interpretation of multiwavelength observations of November 5, 1980 solar flares by the magnetic topology of AR 2766

We present a detailed analysis of the magnetic topology of AR 2776 together with Hα UV, X-rays, and radio observations of the November 5, 1980 flares in order to understand the role of the active region large-scale topology on the flare process. As at present the coronal magnetic field is modeled by an ensemble of sub-photospheric sources whose positions and intensities are deduced from a least-square fit between the computed and observed longitudinal magnetic fields. Charges and dipole representations are shown to lead to similar modeling of the magnetic topology provided that the number of sources is great enough. However, for AR 2776, departure from a potential field has to be taken into account, therefore a linear force-free field extrapolation is used. The locations of the four bright off-band Hα kernels in quadrupolar active regions have been studied previously. In this new study the active region is bipolar and shows a two-ribbon structure. We show that these two ribbons are a consequence of the bipolar photospheric field (the four kernels of quadrupolar regions merge into two bipolar regions). The two ribbons are found to be located at the intersection of the separatrices with the chromosphere when the shear, deduced from the fibril direction, is taken into account. This study supports the hypothesis that magnetic energy is stored in field-aligned currents and released by magnetic reconnection at the location of the separator, before being transported along field lines to the chromospheric level. It is also possible that part of the magnetic energy could be stored and released on the separatrices. Our study shows that meeting just one of two conditions- the presence of intense coronal currents or of a separator in a magnetic field configuration - is not sufficient for flaring. In order to release the stored energy, the coronal currents need either to be formed along the separatrices or to be transported towards the separator or separatrices. The location of the observed photospheric current concentrations on the computed separatrices supports this view. Member of the Carrera del Investigador Científico, CONICET.     

Aspects of Three-Dimensional Magnetic Reconnection – (Invited Review)

In this review paper we discuss several aspects of magnetic reconnection theory, focusing on the field-line motions that are associated with reconnection. A new exact solution of the nonlinear MHD equations for reconnective annihilation is presented which represents a two-fold generalization of the previous solutions. Magnetic reconnection at null points by several mechanisms is summarized, including spine reconnection, fan reconnection and separator reconnection, where it is pointed out that two common features of separator reconnection are the rapid flipping of magnetic field lines and the collapse of the separator to a current sheet. In addition, a formula for the rate of reconnection between two flux tubes is derived. The magnetic field of the corona is highly complex, since the magnetic carpet consists of a multitude of sources in the photosphere. Progress in understanding this complexity may, however, be made by constructing the skeleton of the field and developing a theory for the local and global bifurcations between the different topologies. The eruption of flux from the Sun may even sometimes be due to a change of topology caused by emerging flux break-out. A CD-ROM attached to this paper presents the results of a toy model of vacuum reconnection, which suggests that rapid flipping of field lines in fan and separator reconnection is an essential ingredient also in real non-vacuum conditions. In addition, it gives an example of binary reconnection between a pair of unbalanced sources as they move around, which may contribute significantly to coronal heating. Finally, we present examples in TRACE movies of geometrical changes of the coronal magnetic field that are a likely result of large-scale magnetic reconnection. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1005248007615     

Evolution of the photospheric magnetic field and coronal null points before solar flares

Based on a topological model for the magnetic field of a solar active region (AR), we suggest a criterion for the existence of magnetic null points on the separators in the corona. With the problem of predicting solar flares in mind, we have revealed a model parameter whose decrease means that the AR evolves toward a major eruptive flare. We analyze the magnetic field evolution for AR 9077 within two days before the Bastille Day flare on July 14, 2000. The coronal conditions are shown to have become more favorable for magnetic reconnection, which led to a 3B/X5.7 eruptive flare.     

Three-dimensional Separator Reconnection – How Does It Occur?

In two dimensions magnetic energy release takes place at locations where the magnetic field strength becomes zero and has an x-point topology. The x-point topology can collapse into two y-points connected by a current sheet when the advection of magnetic flux into the x-point is larger than the dissipation of magnetic flux at the x-point. In three dimensions magnetic fields may also contain singularities in the form of three-dimensional null points. Three-dimensional nulls are created in pairs and are therefore, at least in the initial stages, always connected by at least one field line – the separator. The separator line is defined by the intersection of the fan planes of the two nulls. In the plane perpendicular to a single separator the field line topology locally has a two dimensional x-point structure. Using a numerical approach we find that the collapse of the separator can be initiated at the two nulls by a velocity shear across the fan plane. It is found that for a current concentration to connect the two nulls along the separator, the current sheet can only obtain two different orientations relative to the field line structure of the nulls. The sheet has to have an orientation midway between the fan plane and the spine axis of each null. As part of this process the spine axes are found to lose their identity by transforming into an integrated part of the separator surfaces that divide space into four magnetically independent regions around the current sheet.     

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.     

Flux linkages of bipolar sunspot groups: A computer study

Past studies of the structure of solar magnetic fields have used magnetograph data to compute selected field lines for comparison with the morphology of structures seen in various spectral wavelengths. While those analyses examine one of the integral properties of magnetic fields (field lines), they are not complete since they fail to determine the other important integral property: the boundaries of the flux of field lines of given connectivity. In the present analysis we determine such a system of boundaries, called separatrices, for the current free field of two p-f spot pairs so as to exhibit the line of self-intersection, called the separator. The analysis is compared with previous analytical work. These computer results, confirming earlier studies carried out using iron fillings, show that the separatrix has the form of two intersecting ovoids, defining four flux cells. New features which have emerged from this study include the observation that the projections of the separatrix in a plane perpendicular to the separator at its highest point do not intersect at 90° as has been widely believed, but rather closer to 60° in the case studied. The separator is very nearly circular over most of its length. The two neutral points (B = 0) which appear at the photospheric ends of the separator have the mixed radial-hyperbolic form as expected, a feature requiring every field line lying on the separatrix to connect with at least one of the two neutral points. The rotation of line direction with height (shear) is graphically illustrated in the potential field case studied here. We also exhibit a magnetic arcade.     

Multiply Connected Source and Null Pairs

The magnetic fields within the solar atmosphere have a complex topology owing to the fragmentary nature with which they thread the solar surface. The topologies of the potential magnetic fields containing only a few (up to four) point photospheric sources have been classified. For small numbers of sources determining the connectivity of source pairs is equivalent to counting the number of flux domains. As the numbers of sources increase this, however, is no longer the case. Instead, a pair of connected sources can have more than one distinct flux domain linking them. We call these multiply connected source pairs. Pairs of nulls connected by more than one separator are called multiply connected null pairs. Multiply connected source and null pairs go hand-in-hand such that two separators connecting the same pair of nulls immediately implies multiple flux domains linking the same source pair and vice versa. It is found that multiply connected source pairs are common not only in fairly complex potential magnetic fields but more interestingly in the resistive-MHD evolution of both simple and complex magnetic fields. Magnetic energy release is often significant around separators. Thus fields with multiply connected source pairs, which naturally have more separators, (i) have more sites for intense energy release and (ii) are likely to release energy more quickly than other magnetic fields. Moreover, the combination of multiply connected source and null pairs can give rise to a situation where flux is reconnected repeatedly between two flux domains.     

Magnetic nulls and topology in a class of solar flare models

We study the magnetic field-line topology in a class of solar flare models with four magnetic dipoles. By introducing a series of symmetry-breaking perturbations to a fully symmetric potential field model, we show that isolated magnetic nulls generally exist above the photosphere. These nulls are physically important because they determine the magnetic topology above the photosphere. In some special cases, there may be a single null above the photosphere with quasi two-dimensional properties. For such a model, aquasi null line connects the null to the photosphere. In the limit of small non-ideal effects, boundary layers and current sheetsmay develop along the quasi null line and the associated separatrix surfaces. Field lines can then reconect across the quasi null line, as in two-dimensional reconnection. In a more general force-free case, the field contains a pair of nulls above the photosphere, with a field line (theseparator) connecting the two nulls. In the limit of small non-ideal effects, boundary layers and current sheets develop along the separator and the associated separatrix surfaces. The system exhibits three-dimensional reconnection across the separator, over which field lines exchange identity. The separatrices are related to preferable sites of energy release during solar flares.     

Particle Acceleration Due to Coronal Non-null Magnetic Reconnection

Various topological features, for example magnetic null points and separators, have been inferred as likely sites of magnetic reconnection and particle acceleration in the solar atmosphere. In fact, magnetic reconnection is not constrained to solely take place at or near such topological features and may also take place in the absence of such features. Studies of particle acceleration using non-topological reconnection experiments embedded in the solar atmosphere are uncommon. We aim to investigate and characterise particle behaviour in a model of magnetic reconnection which causes an arcade of solar coronal magnetic field to twist and form an erupting flux rope, crucially in the absence of any common topological features where reconnection is often thought to occur. We use a numerical scheme that evolves the gyro-averaged orbit equations of single electrons and protons in time and space, and simulate the gyromotion of particles in a fully analytical global field model. We observe and discuss how the magnetic and electric fields of the model and the initial conditions of each orbit may lead to acceleration of protons and electrons up to 2 MeV in energy (depending on model parameters). We describe the morphology of time-dependent acceleration and impact sites for each particle species and compare our findings to those recovered by topologically based studies of three-dimensional (3D) reconnection and particle acceleration. We also broadly compare aspects of our findings to general observational features typically seen during two-ribbon flare events.     

Energy Release Sites in Magnetic Fields Containing Single Or Multiple Nulls

An ongoing debate is how magnetic energy is released in solar flares, which type of magnetic instabilities are responsible for triggering the energy release, and which magnetic topologies are most likely to host the instabilities. In this connection magnetic reconnection has been a general ingredient, with most of the previous work focussing on 2D reconnection. A natural extension to this is to investigate reconnection in 3D topologies, in particular the behaviour of magnetic nulls and the magnetic topology associated with them. This paper investigates the difference in dynamical behaviour of a numerical domain that either contains a double null-point pair connected by a separator or only a fraction of the separator defined by the null-points. The experiments show that nulls can either accumulate current individually, or act together to produce a singular current collapse along the separator. The implication of these results for the interpretation of coronal data is discussed.     

Homologous Flares and Magnetic Field Topology in Active Region NOAA 10501 on 20 November 2003

We present and interpret observations of two morphologically homologous flares that occurred in active region (AR) NOAA 10501 on 20 November 2003. Both flares displayed four homologous Hα ribbons and were both accompanied by coronal mass ejections (CMEs). The central flare ribbons were located at the site of an emerging bipole in the centre of the active region. The negative polarity of this bipole fragmented in two main pieces, one rotating around the positive polarity by ≈ 110° within 32 hours. We model the coronal magnetic field and compute its topology, using as boundary condition the magnetogram closest in time to each flare. In particular, we calculate the location of quasi-separatrix layers (QSLs) in order to understand the connectivity between the flare ribbons. Though several polarities were present in AR 10501, the global magnetic field topology corresponds to a quadrupolar magnetic field distribution without magnetic null points. For both flares, the photospheric traces of QSLs are similar and match well the locations of the four Hα ribbons. This globally unchanged topology and the continuous shearing by the rotating bipole are two key factors responsible for the flare homology. However, our analyses also indicate that different magnetic connectivity domains of the quadrupolar configuration become unstable during each flare, so that magnetic reconnection proceeds differently in both events.     

Interpretation of Statistical Flare Data using Magnetic Reconnection Models

The ability of magnetic reconnection solutions to explain statistical flare data is discussed. It is assumed that flares occur at well-defined, isolated sites within an active region, determined by the null points and separators of the coronal magnetic field (Craig, 2001). Statistical flare observations then derive from a multiplicity of independent sites, flaring in parallel, that produce events of widely varying output (Wheatland, 2002). Given that the `separator length' at an individual site controls the event frequency and the mean energy release, it is shown that the observed frequency-energy spectrum N(E)can be inverted to yield a source function that relates directly to the distribution of separator lengths. It is also pointed out that, under the parallel flaring model, inferred waiting-time distributions are naturally interpreted as a superposition of individual point processes. Only a modest number of flaring separators is required to mimic a Poisson process.     

Effects of Complexity on the Flux-Tube Tectonics Model

The quiet-Sun magnetic field emerges through the solar photosphere in a multitude of mixed-polarity magnetic concentrations and is subsequently tangled up into intricate regions of interconnecting flux. Moreover, since these discrete concentrations are likely to be extremely small in size, with fluxes of around only 10¹⁷ Mx, the number of such flux sources in, say, a supergranule, will be extremely large. The flux-tube tectonics model of Priest, Heyvaerts, and Title (2002) demonstrated how the formation and dissipation of current sheets along the separatrices that separate the regions of different connectivity are likely to make an important contribution to coronal heating. Since the full complexity of the magnetic field is below present observable scales, this study examines the effect of having the magnetic flux emerge through configurations structured on smaller and smaller scales. It is found that, by fixing the amount of flux emerging into a given 2D region, the main factors influencing the current build-up along the separatrices are the number of sources through which the flux emerges and the spatial distribution of the sources on the photosphere. The free energy (i.e., that above potential) is stored lower and lower in the atmosphere as the complexity of the system increases. A simple comparison is then made between coronal heating by separator currents and by separatrix currents. It is found that both result in comparable amounts of energy release, with separatrix heating being the more dominant.     

Coronal Magnetic Topologies in a Spherical Geometry II. Four Balanced Flux Sources

The Sun’s magnetic field is the primary factor determining the structure and evolution of the solar corona. Here, magnetic topology is used in combination with a Green’s function method to model the global coronal magnetic field with a spherical photosphere. We focus on the case of three negative flux sources and one positive source, completing our previous categorisation of the topological states and bifurcations that are present in quadrupolar configurations in a spherical geometry. Three fundamental varieties of topological state are found, with three types of bifurcation taking one to the other. A comparison to the equivalent results for a planar photosphere is then carried out, and the differences between the two cases are explained.