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Astron. Astrophys. 363, 779-788 (2000)

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4. The magnetic topology of the complex AR 7031 - AR 7038

4.1. The magnetic field model

The photospheric longitudinal field ([FORMULA]) has been extrapolated to the corona, under the linear force-free field assumption ([FORMULA], with constant [FORMULA]), using the discrete fast Fourier transform (FFT) method as proposed by Alissandrakis (1981). The main limitations of our extrapolation method are: imposed flux balance, periodicity of the solution in the horizontal (x, y) directions, proportionality between the current density and the magnetic field (through the constant [FORMULA]).

We have modeled the magnetic field using as boundary condition the line of sight component of the map builded combining both regions on January 30, 1992. In our model we take into account the projection effects due to the location of the ARs on the solar disc, as described in Démoulin et al. (1997). Using a linear force-free model for the field, we are not able to fit locally the transverse field measurements (the higher transverse field regions are too localized, as mentioned in Sect. 2.1). We have determined a global value of [FORMULA] by comparing computed field lines with the best observed SXT loops during the flares. This value turned out to be [FORMULA] = 0.009 Mm-1. The model of the field is shown in Fig. 5. Notice also, that the direction of the computed field lines at low heights agrees well with the direction of the observed transverse field. However, in order to fit the shape of the interconnection arc it was necessary to increase the value of [FORMULA] to 0.014 Mm-1. The resulting model is shown in Fig. 4d. The coronal magnetic field is most probably non-linear force free; therefore, this higher value of [FORMULA] should be related to the high localized magnetic shear observed in AR 7038.

[FIGURE] Fig. 5a-f. Magnetic model and topology of the magnetic field of AR 7031. We have drawn sets of computed field lines that follow the shape of the soft X-ray loops and that have their footpoints at both sides of the QSLs, where the H[FORMULA] brightenings are located. a and b correspond to flare A ; while c -f correspond to flare B (see text). The magnetic field isocontours are the same as in Fig. 4. The box axes are in Mm and the photospheric locations of QSLs are shown with thick continuous lines.

4.2. QSLs and their relation with the observed events

Following Démoulin et al. (1997), we use here the QSLM to find the locations of QSLs and analyze their relation to the observed flares. QSLs are regions where field lines initially close together separate widely when we follow them. QSLs are defined in terms of a dimensionless function N, which is:


being [FORMULA], [FORMULA] and [FORMULA] the photospheric footpoints of a given field line; while [FORMULA] are the coordinates along the photospheric plane which lies at [FORMULA]. The locations where [FORMULA] takes its highest values define the field lines involved in the QSLs. We refer the reader to Démoulin et al. (1996) for a discussion of the properties of [FORMULA] and of the basic characteristics of QSLs. Applications of the QSLM to different observed phenomena can be found in Mandrini et al. (1996), Schmieder et al. (1997), Démoulin et al. (1997) and Gaizauskas et al. (1998).

AR 7031 is essentially a bipolar region, except for the several small flux concentrations that emerged between January 29 and 30; in particular the small bipole at the North of the main positive polarity. The topology of the field is complex since the flux of the very intense preceding polarity is shared by all the surrounding negative fields. In Figs. 5 we show the locations of the QSLs computed for the magnetic field model of the combined ARs. For the sake of clearness, we have drawn in Fig. 5a-b the QSLs associated to the flares A -A' (which we will call the northern QSLs), and in Fig. 5c-d those associated to flare B (which we will call the southern QSLs). In Fig. 5e-f we have combined both sets of QSLs together with a central one to where the H[FORMULA] emission of flare B extends during its late phase. The location of flare brightenings in H[FORMULA] agrees with the intersection of the QSLs with the photosphere for all the observed flares.

According to the evolution of the flares described in Sect. 3, we see that the magnetic structures associated to the northern QSLs do not interact with the ones related to the southern QSLs during flares A and A' ; no brightenings are observed either in H[FORMULA] or soft X-rays related to field lines having their footpoints at the southern QSLs. However, during flare B soft X-ray emission is observed (see Fig. 4c) following the shape of field lines related to the northern QSLs (see Figs. 5a-b). This implies that during this large event energy release occurring at one site of the AR might induced energy release at a different one. We now discuss the different events in particular.

4.3. Emerging bipole: flares A and A'

The homologous flares A and A' occurred in a recurrent flaring place for AR 7031. In Figs. 5a-b we have drawn sets of field lines with footpoints lying at both sides of the photospheric portion of QSLs. These field lines are formed by magnetic reconnection at the QSLs during flares A and A' . The magnetic connectivity, before reconnection occurs, corresponds to the link between the positive and negative field in the main large bipole and in the small emerging bipole (the four kinds of connectivities and the two interacting bipoles are shown in Fig. 6a). Notice that the set of reconnected field lines (see Fig. 5a-b) clearly follows the shape of the soft X-ray arcade connecting the big preceding positive polarity to the northern portion of the following elongated negative polarity (Fig. 3a-b).

[FIGURE] Fig. 6a-c. Magnetic connectivities and interacting bipoles. a , b and c outline the connectivities before/after (thin/thick continuous coronal lines) magnetic reconnection for flares A -A' , B and the interconnecting arc, respectively. The numbers in the figures indicate which are the interacting polarities in each case. The field line sketch is drawn on the photospheric field model where we have also included the locations of QSLs (thicker continuous lines), the convention for the field and spatial dimensions are the same as in Fig. 5.

Three bright kernels can be seen in Fig. 2b, while a fourth fainter brightening is seen at the footpoints of the field lines fitting the shape of the soft X-ray arcade. This very faint kernel can be observed only in a few images of the set corresponding to both flares (A and A' ) and it can be easily over-seen when looking at the movies. However, since it is located on the QSL lying on the following polarity, and its shape matches the shape of that portion of the QSL, we were able to find it using the QSL as a guide. This is not the first time that the analysis of the topology of the magnetic field let us locate a faint brightening not visible at first sight in the flare data (Bagalá et al. 1995).

Summarizing, we conclude that the emergence of the northwestern bipole lead to the interaction between loops associated to it with the northern loops belonging to the AR main bipolar field. This interaction was the origin of A and A' flares.

4.4. Bipolar configuration: flare B

Flare B is a two-ribbon event. In this case the brightenings are located on the southern QSLs. In Figs. 5c and d we show sets of reconnected field lines having footpoints at both sides of the QSLs at the place of the H[FORMULA] ribbons. The topology of the field related to this event is typical of bipolar configurations, where two elongated QSLs are found; however, two sets of interacting field lines can be identified as in a quadrupolar field (Démoulin et al. 1996). Fig. 6b illustrates the connectivities before and after reconnection and indentifies the polarities of the interacting bipoles. We suggest that the evolution of the following polarity (compare Fig. 1b-c) induces the interaction between the magnetic structures associated to the southern QSLs. Energy release occurring at the QSLs is the origin of flare B. As the flare evolves (see Figs. 2c-d), the ribbons expand but they still lie close to QSLs. Notice that the originally more concentrated ribbon extends towards a portion of a different QSL at the North during the gradual phase of the flare (see Fig. 5e-f). This implies that close by magnetic structures become involved in the energy release process as discussed in Sect. 5.

4.5. Interaction between ARs: interconnecting arc

Concerning the large scale interconnection arc, Fig. 4 shows that field lines, issued from some of the QSLs lying on the main positive polarity, extend towards AR 7038. These field lines match the shape of the arc, as observed during the occurrence of flare B , and have their opposite footpoints at the QSLs located on the negative polarity in AR 7038. These results let us conclude that energy release at the QSLs, associated with AR 7031, produces the enhancement of the interconnection arc and may induce the occurrence of flares in AR 7038 (see Sect. 3.1).

What is the physical origin of the interconnecting arc ? Compared to flares A , A' and B , the observations of the arc are much less proving since the changes induced in the atmospheric plasma after energy release are less evident (e.g. no significant H[FORMULA] brightenings and a low soft X-ray emissivity). Nevertheless, the topology shown in Fig. 4d, together with the theoretical knowledge of the magnetohydrodynamic evolution of two interacting bipoles, let us propose the following scenario. First, free magnetic energy is present in both regions, as indicated by the transverse magnetic field, and also probably a current layer develops between the two ARs (formed by the interaction of the two initially independent configurations represented in Fig. 6c by the two thin lines). The relative slow evolution of the ARs (in particular the rotation of AR 7038) is expected to induce a low reconnection rate between sets of loops belonging to them. This is probably the origin of the weak interconnecting arc seen before flare B . During flare B , fast modifications in the field of AR 7031 induce a higher reconnection rate. In both cases, reconnection is forced by the magnetic evolution of the individual ARs. The multi-loop system forming the interconnecting arc is then thought to be a set of reconnected loops filled by plasma, which have the same physical origin of "post" flare loops. As it is observed for some flares, the second set of reconnected field lines does not show up in the observations because its size (and in particular its volume) is too large, so that the increase in plasma density (and so in emission) is too low.

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Online publication: December 11, 2000