Issue 
A&A
Volume 538, February 2012



Article Number  A138  
Number of page(s)  6  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201117620  
Published online  15 February 2012 
Magnetictopology evolution in NOAA AR 10501 on 2003 November 18
Sternberg Astronomical Institute, Moscow State University, Universitetski prospekt, 13, 119991 Moscow, Russia
email: ivo@sai.msu.ru
Received: 1 July 2011
Accepted: 17 December 2011
Context. NOAA AR 10501 produced three flares on 2003 November 18. Two of them were associated with coronal mass ejections (CMEs).
Aims. We model the magneticfield structure of the active region, study the magnetictopology evolution, and propose a scenario of the observed events.
Methods. The coronal magnetic field is reconstructed using a topological model (also called magneticcharge model). We present an automatic method of choosing the magnetic charges for the case where the charges are located beneath the photosphere. The new method improves quantitative analysis of magnetograms and makes processing faster.
Results. We demonstrate that coronal conditions became more favourable for magnetic reconnection before the flaring events. It is also shown that the magneticfield configuration at the time of both CMEs was critical, close to what is called “topological trigger”. We assume that the topological trigger played a key role in the initiation of these CMEs.
Key words: Sun: magnetic topology / Sun: flares / Sun: coronal mass ejections (CMEs)
© ESO, 2012
1. Introduction
According to current understanding, a solar flare results from the reconnection of magnetic fields, i.e. the magnetic reconnection. There remains, however, considerable debate on which factors make some magnetic configurations more likely to flare than others. The exact relationship between flares and CMEs is also the subject of many works.
A powerful tool for answering these questions is a topological model, also called the magneticcharge model. A review of this topic can be found in Somov (2006). It assumes that a real magnetic field is modelled by the one created by “effective charges”, or “sources”. This approach allows us to reconstruct the coronal structure and to distinguish different magnetic fluxes and the places where these fluxes interact among themselves, i.e. to study magneticfield topology. The boundary surfaces of the fluxes are called the separatrix surfaces or “separatrices”. They intersect among themselves along field lines called the limiting field lines or “separators” (Sweet 1969; Syrovatskii 1981).
The charges can be located either beneath the photospheric plane or on this plane. The advantage of the models from the first category is a relatively realistic approximation of an observed photospheric field. These models are also good at representing the shapes of flare ribbons observed in Hα, hard Xray, and EUV ranges (Gorbachev & Somov 1989; Mandrini et al. 1991; Démoulin et al. 1993; Somov et al. 2005). They have allowed us to reveal some causes of the observed magneticfield imbalance in solar active regions (Oreshina & Somov 2006) and to interpret drastic changes in the magnetic fields associated with solar flares (Oreshina & Somov 2008).
The models from the second category assume that singular magnetic sources are located on the photospheric plane (Barnes et al. 2005; Des Jardins et al. 2009; Kazachenko et al. 2010; Longcope et al. 2010). These models are well suited to quantitative analysis because exactly half of the flux from a charge is located above the photosphere. On the other hand, the vertical component of a model magnetogram is nought everywhere except points where charges are located, so a visual comparison between an observed magnetogram and a model one is impossible for this category.
Démoulin et al. (1994) have developed an automatic algorithm for positioning sources beneath the photosphere at different depths. In comparison with models where all the sources are located on the same plane, the differences in the topology are not crucial; however, such an algorithm takes more computational time and makes the following analysis of the obtained topology more complicated. An automatic algorithm for partitioning magnetograms into charge regions was proposed by Barnes et al. (2005) for the models from the second category.
The main aim of our paper is to propose an automatic algorithm for choosing the magnetic charges located beneath the photosphere on one horizontal plane. It will allow us to make the model the clearest and the simplest and to join the advantages of both categories: (i) the model photospheric magnetic field will approximate the observed one closely, and (ii) a quantitative and fast analysis of a large number of magnetograms will be possible, too.
Then we apply the method for modelling a real active region in order to propose a scenario of observed eruptive events. We focus attention on the trigger of flares and CMEs by taking topological characteristics of the magneticfield structure into account.
Knowing the location of reconnection is a key to understanding a flare trigger. According to current understanding, the magnetic reconnection mechanism at a separator transforms magnetic energy into thermal and kinetic energy of plasma and accelerated particles. Accelerated electrons follow field lines crossing a region of primary energy release and descend to the chromosphere where they produce hard Xray (HXR) radiation. Des Jardins et al. (2009) notice a remarkable visual relationship between the spine lines and HXR footpoint tracks. They also demonstrate that there are spine lines that are not associated with HXR footpoints (Fig. 5 in their paper). Restante et al. (2009) have studied the relations between quasiseparatrix layers (QSLs), which are regions where there is a drastic change in field line linkage and in skeletons, which are composed of null points, spine field lines, and fans. They have shown that spines, as well as specific portions of fans are good predictors for the location of QSL footprints, and therefore of flare ribbons. They also pay attention to the fact that only some portions of the skeleton are found to be related to flare ribbons. In our paper we propose an explanation of these results using active region NOAA 10501, as an example. Moreover, we focus our attention on the trigger of flares and CMEs taking topological characteristics of magneticfield structure and evolution into account.
The paper is organised as follows. In Sect. 2, we describe the observational data of AR NOAA 10501. Then, in Sect. 3, a new automatic method is presented for choosing magnetic charges; it is applied to constructing the topological model of the AR under consideration. In Sect. 4, the changes in the magnetic field are analysed from 00:00 UT to 11:12 UT. The results are discussed in Sect. 5. In the last section, we formulate our conclusions.
2. Observations of NOAA AR 10501 on 2003 November 18
On 2003 November 18, AR 10501 was located near the centre of the solar disc (N 03, E 08). Figure 1 shows the temporal evolution of the solar Xray flux observed by Geostationary Operational Environmental Satellite (GOES). Five flares occurred on that day. Three of them were observed to originate in the AR under consideration: C3.8 at 05:25 UT, M3.2 at 07:52 UT, and M3.9 at 08:30 UT. The second and the third flares were associated with two CMEs, detected by Large Angle and Spectrometric Coronagraph (LASCO) on board SOHO. The first CME was detected in the C2 fieldofview at 08:06 UT and the second one was observed at 08:50 UT. Two days later, these CMEs produced the most powerful geomagnetic storm in solar cycle # 23 (Ermolaev et al. 2005; Srivastava et al. 2009; Chandra et al. 2010).
Fig. 1 Temporal evolution of the solar Xray flux observed by GOES 12 in 1.0 − 8.0 Å bandwidth on 2003 November 18. The flares occurred in AR 10501 are marked in grey. 
Figure 2 presents the lineofsight magnetogram obtained by the Michelson Doppler Imager (MDI) onboard SOHO at 00:00 UT. We show the five areas where magnetic field was the most intense: two areas of southern (negative) polarity and three areas of northern (positive) polarity. The maximal absolute value of the field for both polarities was about 3000 G. We have chosen 200 G as a boundary intensity level for these areas. Since our aim is to study the magneticflux evolution, this threshold allows us not to analyse the smallscale changes due to variations in atmospheric seeing and uncorrelated noise. We also do not consider small areas covering just a few pixels, because small fluxes seem to us unlikely to contribute much to the global topology, so we study the evolution of these five areas, which include a significant part of the global (whole AR) magnetic flux and which determine its largescale topology.
Fig. 2 Magnetogram obtained by MDI/SOHO at 00:00 UT on 2003 Nov. 18. The white (black) thick lines are 200 G ( − 200 G) levels. A length unit corresponds to one MDI pixel (1.45 × 10^{8} cm). 
3. Method for choosing charges for topological model
The 3D magnetic field created by charges is described by the formula (1)where N is the number of charges, e_{i} their intensities, and r_{i} = (x_{i},y_{i},z_{i}) their radius vectors. The xaxis is directed to the west, the yaxis to the north, and the zaxis is directed upwards from the solar surface. The z = 0 plane corresponds to the photosphere. We assume that all the charges are located beneath the photosphere at the same depth: z_{i} = const. = z_{0} for every i. As proposed in the previous section, N = 5 for this AR, so there are 16 unknowns: (x_{i},y_{i}), e_{i} where i = 1 ... 5, and z_{0}.
First, we calculate charge positions (x_{i},y_{i}) as the “centre of mass” of every area R_{i}: (2)Here ψ_{k} is the observed magnetic flux through the area on the photosphere that corresponds to MDI pixel k, and x_{k} and y_{k} are the coordinates of this pixel. The summation is over all pixels of area R_{i}.
Then, we find the charge intensities e_{i} from the equality of the observed and the model magnetic fluxes for each area R_{i}: (3)Here Ψ_{i} is the observed magnetic flux in the area R_{i}, s the surface of an MDI pixel: s = 2.1 × 10^{16} cm^{2}. There are five equations with six unknowns in this system: e_{1} ... e_{5} and z_{0}. We then consider depth z_{0} as a free parameter. Different values of z_{0} will correspond to different sets of intensities e_{1} ... e_{5}.
Let us choose the set that approximates the global magnetic flux in the best way. In other words, we want to minimise the difference between model and observed fluxes on the AR scale: The summation in the first formula is over all magnetogram pixels with positive magnetic field, the one in the second formula is over all pixels with negative magnetic field, and Ψ_{ + }(Ψ_{ − }) is the observed positive (negative) magnetic fluxes in the whole AR.
Our experience shows that a model magnetogram approximates an observed one well if the charges are located at the depth of several units, not more than ten. This part of the method does not take much time.
The charge coordinates and intensities for the magnetogram from Fig. 2 are presented in Table 1. The corresponding model magnetogram is shown in Fig. 3. The output of this model provides the global magnetic flux in agreement with the observational values at about 90% (92% for positiveflux approximation and 88% for negativeflux approximation). We process other magnetograms of the AR keeping the same depth z_{0}. All we need to recalculate are the coordinates of charges (x_{i},y_{i}) (from formula 2) and their intensities (from system 3).
Intensities and coordinates of the magneticfield charges for modelling the magnetogram obtained at 00:00 UT.
Fig. 3 Model magnetogram corresponding to 00:00 UT. 
4. Magneticfield topology
4.1. Separators and separatrix surfaces
Determination of the magnetic charges allows the topology to be analysed. Figure 4 shows the vector field in the charge plane z = z_{0} = −4. Solid lines are the intersections of the separatrix surfaces with this plane. Des Jardins et al. (2009) and Restante et al. (2009) conclude that flaring ribbons correspond to some parts of these lines. There remains the question of which parts of such lines correspond to the location where flare ribbons might be found. The answer consists in the necessity of searching separators. It is here, in the vicinity of the separators, that the magneticfield structure makes the reconnection possible. Particles accelerated by reconnection come down to the photosphere where they produce hard Xray bremsstrahlung radiation.
Fig. 4 Vector field created by charges in the charge plane. Solid lines are the intersections of the separatrix surfaces with this plane, and X_{1}, X_{2}, X_{3}, and X_{4} are the null points. 
As is known, separator endpoints are null points in the magnetic field, i.e. the points where the magnetic field vanishes. However not every null point is a separator endpoint. As one can see in Fig. 4, there are four points where the field is nought in the charge plane: X_{1}, X_{2}, X_{3}, and X_{4}. Figure 5 demonstrates that there is a separator connecting the points X_{2} and X_{3}. Indeed, there is a line connecting these two points (Fig. 5, left), and the vector field in the vertical plane perpendicular to the straight segment (X_{2},X_{3}) contains an Xtype null point (Fig. 5, right). This is one of structures that enable magnetic reconnection (Oreshina & Somov 1998; Hornig & Priest 2003; De Moortel & Galsgaard 2006; Parnell et al. 2010).
Fig. 5 Left: vector field in the vertical plane containing points X_{2} and X_{3}; there is a field line connecting these points. Right: vector field in the vertical plane perpendicular to the straight segment (X_{2},X_{3}); there is an Xtype null point – a structure making reconnection possible. 
Analogically, it is possible to show the existence of the separator X_{3}X_{4}, so point X_{3} is a common endpoint of both separators. Point X_{1} does not correspond to any separator. Finally, there are two separators in this AR. The observed flares can be explained by the magnetic reconnection processes on them.
Searching separators is still a difficult problem that needs the investigation of a 3D magnetic structure. To develop an automatic and effective method for this part of the topological analysis is one of the aims of our future studies.
4.2. Topology evolution
The aim of this section is to investigate the conditions on separators in order to answer the question of why one magnetic configuration is more likely to reproduce a flare than another one. Somov & Titov (1985a,b) and Somov (2006, Sect. 6.2.2) demonstrated that a strong longitudinal (along the separator) component of the magnetic field can considerable limit energy release in a reconnecting current layer (RCL). Indeed, decreasing the longitudinal magnetic field increases the compressibility of plasma flowing into the RCL. As a result, the width of the layer decreases, which leads to increasing the reconnection rate. The procedure aiming to find where the flare ribbons might be located should therefore contain two steps: (1) searching separators and their footprints in the photospheric plane, (2) analysing physical conditions on separators (estimation of magneticstrength value).
Oreshina & Somov (2009) proposed a topologicalmodel parameter that allows the longitudinal magnetic field on a separator to be judged. The lower its value, the better the conditions for magnetic reconnection. As an example, they constructed the topological models of AR NOAA 9077 for 2000 July 12 and 14, i.e. two days before the Bastille Day flare and the flare day. They showed that there were two separators and that the coronal conditions for reconnection on both of them improved significantly before the flare. In the present paper, we investigate the evolution of this parameter on the scale of several hours before and after the flares and CMEs.
The parameter under consideration is λ_{z}. It is calculated at the null points, which are the separator endpoints as follows. One of the characteristics of an arbitrary null point X_{0} is the matrix with elements (e.g. Somov 2008a) (4)Here α and β are indexes that can take values x, y, and z, and ϕ is the potential of the field (1):
As a result, (4) is the matrix of magneticfield gradients at the null point X_{0}. We are interested in the eigenvalue λ_{z} corresponding to the vertical eigenvector of (4), i.e. in the gradient value along a separator. If the separator length does not significantly change with time, the lower the gradient absolute value at separator endpoints, the lower the absolute value of the magnetic field at the separator, the better the conditions for reconnection.
We let λ_{z} (X_{1}), λ_{z} (X_{2}), λ_{z} (X_{3}), and λ_{z} (X_{4}) be the parameters characterising the points X_{1}, X_{2}, X_{3}, and X_{4}, respectively. We study a set of eight magnetograms of the AR NOAA 10501 obtained by MDI/SOHO on 2003 November 18 between 00:00 UT and 11:12 UT with 96 min intervals. The changes at the null points are presented in Fig. 6, with the moments of the three flares under consideration. These are the same flares as in Fig. 1. Parameter λ_{z} (X_{3}) stayed negative during the reporting period; the figure shows its absolute values.
Fig. 6 Changing values of the parameters λ_{z} (X_{1}), λ_{z} (X_{2}), λ_{z} (X_{3}), and λ_{z} (X_{4}) with time. The moments of the three flares under consideration are marked in grey. These are the same flares as in Fig. 1. 
Our calculations show that the parameter λ_{z} (X_{1}) remained almost constant. There were oscillations of the parameter λ_{z} (X_{2}) with an amplitude of less than two units. Analogically, the parameter λ_{z} (X_{4}) varied within the limits of 2.5 units. The most significant changes were observed at point X_{3}. Here the parameter λ_{z} (X_{3}) decreased by 6.4 units (from 13.0 to 6.6). Note that X_{3} is a common endpoint of both separators. The first flare (05:25 UT) occurred on the decreasing phase of this parameter. The second and the third flares (07:52 UT and 08:30 UT) coincide with its deep minimum.
Figure 7 presents the maximum values of magnetic field at both separators for each of the eight time points. We have connected the points by two lines to separate the processes on two separators better. One can see that two flares occurred in the vicinity of the deepest magnetic field minimum on the separator X_{3}X_{4}. These two events also coincide with a minimum on the separator X_{2}X_{3} even if not as deep. It allows the stronger energy to be released during flares. Indeed, the first flare was that of C class, while the second and the third ones were M class.
Comparing Figs. 6 and 7 shows some similarity between the evolution of the parameters λ_{z} and that of the magnetic field values on separators. This confirms our suggestion that the lower the parameter λ_{z}, the better the conditions for the reconnection. This similarity is not a perfect correlation because the parameters λ_{z} at two endpoints of a separator change differently. But we propose to pay special attention to the separators for which the parameters λ_{z} decreases significantly.
Fig. 7 Maximum values of magnetic field (in Gauss) on both separators for each of the eight time points. Black and white circles correspond to the separators X_{2}X_{3} and X_{3}X_{4}, respectively. 
This result can be used to develop a reliable and physically justified method for predicting solar flares. A more detailed study of the relationship between parameters λ_{z}, the magnetic field on separators, and flare trigger will be the focus of our future work. Parameter λ_{z} is calculated from a 2D magnetic field in the plane of charges. In comparison with those in a 3D magnetic structure, such calculations are thus simpler and faster. One can first calculate values λ_{z} at all null points and then search for the separators connecting those that show the evolution towards the reconnection. It will be faster than searching all the separators of an active region.
4.3. Topological trigger
Syrovatskii & Somov (1980) showed that slow motions of the photospheric field can lead to a rapid change in the coronal magnetic structure. The term “topological trigger” was introduced by Gorbachev et al. (1988). They used the simplest 3D model of potential magnetic field and pointed out the appearance of a new null point in the corona and its fast motion along the separator above the photosphere. Inverarity & Priest (1999) demonstrate the theoretical existence of stable coronal nulls, but do not discuss in detail the mechanisms which cause them to be born through a bifurcation. Brown & Priest (2001) explored the topology of the simplest configuration due to four sources with a coronal null and a bifurcation that causes such a null point to be born or to die.
Meanwhile, the question of precisely how the topological trigger materialises under actual conditions in solar active regions remained unclear. Somov (2008a,b) showed that the topological trigger effect should be taken into account when large eruptive flares are modelled. This possibility was illustrated by the example of an extended 1B/M4 flare on 1980 November 5 in AR 2776. The 3D topological trigger effect is not a resistive instability that leads to changing the topology of the field configuration from pre to postreconnection state. On the contrary, the topological trigger is a quick change in the global topology that dictates the fast reconnection in large eruptive flares (Somov 2008b; Oreshina & Somov 2009).
We now come back to the solar flares under consideration and investigate in detail the magneticfield structure on the separators at 08:00 UT, i.e. between the two last flares. The locations of the charges and their intensities corresponding to this moment are presented in Table 2. The comparison with the initial moment (00:00 UT, Table 1) shows that the charges moved within two pixels. The intensities of the first four charges increased, while that of the fifth charge decreased. In this section, we demonstrate that the magneticfield configuration became critical so that even minor changes in the charge characteristics could lead to drastic changes in the coronal magnetic field.
Intensities and coordinates of the magneticfield charges for modelling the magnetogram obtained at 08:00 UT.
Figure 8 (top) shows the vector field in the plane of the separators X_{2}X_{3} and X_{3}X_{4} at 08:00 UT. If we only move the charge 3 along the xaxis from the point x_{3} = 29.3 to the point x_{3} = 29.4, keeping the coordinates and intensities of other charges, the sign of the parameter λ_{z} (X_{3}) will be changed. It means changing the magneticfield direction along both separators at the endpoint X_{3} and, therefore, changing the magnetic structure in the vicinity of both separators. These changes are presented in Fig. 8 (middle). A new null point is born from the point X_{3} on the separator X_{2}X_{3} at the height of z = 0.06, i.e. at the photospheric level.
Possible changes in charge characteristics, even the most minor ones, lead to the fast motion of this null point along the separator. As an example, Fig. 8 (bottom) presents the magneticfield configuration in the vicinity of separators for the case when charge 3 is moved to the point (x_{3},y_{3}) = (30.0,37.0). The new null point moved up to the height of z = 4.5. Here, in the corona, a null point on a separator implies the most favourable conditions for magneticfield reconnection and makes this process the most powerful (Somov 2006). However, this is not the whole story. As seen from Fig. 8, the presence of a coronal null point makes the magneticfield structure instable. Even small variations in the magnetic field in the photosphere involve drastic changes in the coronal magnetic field on both separators, which can cause CMEs.
Fig. 8 Vector field in the plane of the separators X_{2}X_{3} and X_{3}X_{4}. Top: configuration corresponding to the system parameters presented in Table 2. Middle: the same for the case when charge 3 is moved to the point x_{3} = 29.4. X_{c} is a new null point. Bottom: the same for the case when charge 3 is moved to the point (x_{3}, y_{3}) = (30.0, 37.0). 
Also the existence of a coronal null point plays a key role in a number of models for eruptive events, for example, breakout model for CMEs (Antiochos et al. 1999). Our method for analysing the topology of the largescale magnetic field of an AR allows us to find coronal null points situated on separators, i.e., precisely where the magnetic energy is accumulated before a flare, and is converted into the particle energy during a flare.
We do not claim that the movement of exactly charge 3 caused the topological trigger, hence CMEs in this AR. We just demonstrate that some critical state could be reached easily and trigger drastic changes in the coronal magneticfield structure in the vicinity of two separators.
5. Conclusions
A method was proposed for automatically choosing the magnetic charges for topological modelling solar ARs. It improves quantitative analysis of magnetograms and makes processing faster,
which in turn allows the magneticfield evolution to be analysed on small timescales.
The method is applied to NOAA AR 10501, which produced three flares and two CMEs on 2003 November 18. Our model suggests the following possible scenario for these events. The magnetic field structure was characterised by the presence of two separators. The conditions on them became more favourable for magnetic reconnection between 00:00 UT and 08:00 UT, which caused three flares. The longitudinal magnetic field decreased significantly along both separators at about 08:00 UT when two flares occurred. As a consequence, the second and the third flares were more powerful than the first one (Mclass in contrast with Cclass). At 08:00 UT, the magneticfield structure was very close to a critical state. Even minor changes in photospheric magnetic field could lead to rapid changes in the corona in the vicinity of both separators, i.e. to a topological trigger that could cause two CMEs.
Acknowledgments
We are grateful to the anonymous referee, whose constructive comments have improved this paper. We thank the SOHO team for the availability of the satellite data. We took the magnetograms from the web site http://soi.stanford.edu/data/. This work was supported by the RFBR grant No. 110200843a.
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All Tables
Intensities and coordinates of the magneticfield charges for modelling the magnetogram obtained at 00:00 UT.
Intensities and coordinates of the magneticfield charges for modelling the magnetogram obtained at 08:00 UT.
All Figures
Fig. 1 Temporal evolution of the solar Xray flux observed by GOES 12 in 1.0 − 8.0 Å bandwidth on 2003 November 18. The flares occurred in AR 10501 are marked in grey. 

In the text 
Fig. 2 Magnetogram obtained by MDI/SOHO at 00:00 UT on 2003 Nov. 18. The white (black) thick lines are 200 G ( − 200 G) levels. A length unit corresponds to one MDI pixel (1.45 × 10^{8} cm). 

In the text 
Fig. 3 Model magnetogram corresponding to 00:00 UT. 

In the text 
Fig. 4 Vector field created by charges in the charge plane. Solid lines are the intersections of the separatrix surfaces with this plane, and X_{1}, X_{2}, X_{3}, and X_{4} are the null points. 

In the text 
Fig. 5 Left: vector field in the vertical plane containing points X_{2} and X_{3}; there is a field line connecting these points. Right: vector field in the vertical plane perpendicular to the straight segment (X_{2},X_{3}); there is an Xtype null point – a structure making reconnection possible. 

In the text 
Fig. 6 Changing values of the parameters λ_{z} (X_{1}), λ_{z} (X_{2}), λ_{z} (X_{3}), and λ_{z} (X_{4}) with time. The moments of the three flares under consideration are marked in grey. These are the same flares as in Fig. 1. 

In the text 
Fig. 7 Maximum values of magnetic field (in Gauss) on both separators for each of the eight time points. Black and white circles correspond to the separators X_{2}X_{3} and X_{3}X_{4}, respectively. 

In the text 
Fig. 8 Vector field in the plane of the separators X_{2}X_{3} and X_{3}X_{4}. Top: configuration corresponding to the system parameters presented in Table 2. Middle: the same for the case when charge 3 is moved to the point x_{3} = 29.4. X_{c} is a new null point. Bottom: the same for the case when charge 3 is moved to the point (x_{3}, y_{3}) = (30.0, 37.0). 

In the text 
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Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.