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Astron. Astrophys. 325, 305-317 (1997)

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3. In what type of magnetic configurations do flares occur?

3.1. QSLs in flaring configurations

Compared to previous analyses of the flaring regions, we apply here two new methods both for the extrapolation of the observed field and for the topology computation. Because these methods use directly the magnetograms, a more accurate position of QSLs (as compared to the separatrices computed before) can be expected to lead to some progress in the understanding of the flaring mechanism. The performances of the new methods are illustrated below by application to four flaring regions (and a fifth one in Appendix A).

- AR 2372, located at N12 E08 on April 6, 1980, is a typical quadrupolar region where a small and less intense bipole emerged in between a larger one with a reversed polarity (Mandrini et al., 1991). Five H [FORMULA] flare kernels are present, labelled a-e in Fig. 1a. Kernels a and e are in regions where no strong deviation from a potential field is observed, while there is a strong localized shear in the region of kernels b and c. Kernel d is located where a new small bipole, that started emerging on this day, was observed on April 7 and 8. This emergence could have been the origin of flaring in this zone as discussed by Mandrini et al. (1993), but it is at the spatial-resolution limit of the magnetogram so we do not comment further on this kernel. Within the limitations of a linear force-free approach, the topology of AR 2372 is not very sensitive to [FORMULA] so we present the extrapolation with a potential approximation. It can be seen that the main four H [FORMULA] kernels (a, b, c and e) lie on QSLs or at a distance from them smaller than the accuracy in the relative positioning between magnetic data and H [FORMULA] filtergrams, considered of about 2 magnetogram pixels or [FORMULA] Mm. Besides, they can be linked by field lines (Figs. 1b-d).

[FIGURE] Fig. 1. Flaring AR 2372 on April 6, 1980: example of quadrupolar region formed by the emergence of a small bipole with a polarity reversed from the main one. a Observational data: H [FORMULA] kernels (hatched regions labelled a, b, c, d and e) and longitudinal field [FORMULA] (isocontours of [FORMULA] G, with positive and negative values drawn with solid and dashed lines respectively). b Intersection of the QSLs with the photosphere (thick isocontour lines of [FORMULA]) for a potential extrapolation of [FORMULA] (isocontours [FORMULA] G of [FORMULA] are added for reference). The coronal links between H [FORMULA] brightenings are given by four kinds of field lines. The regions where the vertical current density is greater than 10 mA m-2 are marked with horizontal (resp. vertical) hatching for positive (resp. negative) values. (c,d) Perspective view of Fig. 1b showing the coronal linkage at the borders of QSLs with field lines drawn as surfaces (for aesthetics the vertical scale has been multiplied by a factor 3 compared to the horizontal one). The isocontours [FORMULA] G of [FORMULA] (vertical field component) are added. e Same as in b but for a model with 12 magnetic sources. The sides of the Figs. 1a (resp. 1b-e) are parallel to the local [FORMULA] axes (resp. observer [FORMULA] axes) as defined in Appendix A.

- AR 2779, located at S13 W06 on November 12, 1980, is another example of quadrupolar region (see Bagalá et al., 1995 for a detailed study). The analyzed flare presents four kernels observed in OV (Fig. 2a). In this case the two bipoles in the configuration have approximately the same magnetic intensity. Strong magnetic shear is present along the longitudinal inversion line in the intermediate bipole and also at the location of the two most distant flare kernels. This departure from potential is taken into account, on average, in the coronal extrapolation of the observed photospheric field. Again the four kernels are located on QSLs and can be linked by magnetic field lines (Fig. 2b-d).

[FIGURE] Fig. 2. Flaring AR 2779 on November 12, 1980: example of quadrupolar region formed by two extended bipoles. The drawing convention is the same as in Fig. 1. a Observational data: OV kernels and longitudinal field [FORMULA]. b Intersection of the QSLs with the photosphere for a linear force-free extrapolation of [FORMULA] ([FORMULA] Mm-1) with field lines and current-density regions. c,d Perspective views of b, with field lines drawn as surfaces.

- AR 2776, located at N11 E07 on November 5, 1980, is a typical bipolar region where the longitudinal inversion line has an "S" shape (Fig. 3a). Strong magnetic shear is observed along the inversion line where the small southern polarity is located, while the direction of chromospheric fibrils indicates a departure from potential in the preceding and following polarities (see Démoulin et al., 1994). In order to explain the location of the two elongated H [FORMULA] flare ribbons, this shear should be included in the extrapolation because the location of QSLs in bipolar configurations is more sensitive to magnetic shear than in quadrupolar ones. Notice, in particular, the case of the longer ribbon to the left which has only a QSL counterpart if the right shear is considered in the model (compare Figs. 3a to 3b). The connectivity between flare brightenings, shown in Figs. 3c and 3d, is similar to that of quadrupolar regions.

[FIGURE] Fig. 3. Flaring AR 2776 on November 5, 1980: example of a bipolar region with an S-shaped inversion line. The drawing convention is the same as in Fig. 1. a Observational data: H [FORMULA] kernels and longitudinal field [FORMULA] b Intersection of the QSLs with the photosphere ([FORMULA]) for a linear force-free field extrapolation of the longitudinal field ([FORMULA] Mm-1). The numbers correspond to the decimal logarithm of the thickness (in meters) of the QSL at that location. Field lines and current-density regions are added. c,d Perspective view of Fig. 3b with the typical field lines (drawn as surfaces) on both side of QSLs. e Plot along a segment orthogonal to the QSL at point A (see Fig. 3b) of the function N (continuous curve), the distance D between the field line footpoints (dashed curve), the field line length L (dashed dotted curve) and the delay function [FORMULA] (dotted curve). The curves of [FORMULA] have been normalized to their minimum and maximum values in the range considered. f Same as Fig. 3b but for a model using 18 magnetic sources.

- AR 2511, located at N20 E00 on June 15, 1980, is a bipolar region where the inversion line is nearly straight and the observed magnetic field is nearly potential (at least in the strong field regions where B [FORMULA] G, see Démoulin et al., 1993). All the computed field lines are nearly parallel and this region seems at first sight to have a very simple topology, a magnetic arcade-like one. However, the concentration of the photospheric magnetic field makes the 3D connectivity more complex: QSLs are even present in such simple bipolar region; they are located on the two left H [FORMULA] kernels while the "U" QSL on the right is shifted by appromimativelly of 8 Mm and slightly rotated from the right H [FORMULA] kernel (Fig. 4a,b). In the present computation we are able to link the H [FORMULA] ribbons by field lines only if the starting point is located on the external part of the QSLs, while field lines starting from the internal part end-up rapidly inside (Fig. 4c). This is our worst studied example, principally because the two interacting bipoles are nearly parallel so that the QSL positions are strongly influenced by the location of coronal currents. Moreover, the internal bipole (emerging flux) has a low magnetic flux so the magnetic-field measurements are difficult there. The present computation refers rather to the relaxed state where the field is nearly potential. The difference in position between the H [FORMULA] ribbons and QSLs is then interpreted as due to the evolution of the magnetic configuration from a twisted-flux tube emerging in the large scale bipole to a more arcade-like configuration with weaker electric currents.

[FIGURE] Fig. 4. Flaring AR 2511 on June 15, 1980: example of a bipolar AR formed by the emergence of a small bipole having the same orientation as the main one. The drawing convention is the same as in Fig. 1. a Observational data: H [FORMULA] kernels and longitudinal field [FORMULA]. b,c Intersection of the QSLs with the photosphere ([FORMULA]) and some field lines for a linear force-free field extrapolation of [FORMULA] ([FORMULA] Mm-1). In b the numbers have the same meaning as in Fig. 3b; the region where the current-density is greater than 5 mA m-2 is marked. d Same as Fig. 4b but for a model using 28 magnetic sources.

3.2. Thickness of quasi-separatrix layers

Inside the QSLs, shown as an isocontour of N in Figs. 1 to 4, this function has much higher values in very thin and elongated regions. As it is numerically very expensive to resolve the variations of N for the whole AR, we study the behaviour of this function cutting the QSL with a segment orthogonal to it at some localized points. The thickness, [FORMULA], of a QSL is defined as the width of the function N at half-height. In the absence of magnetic null points or of field lines touching tangentially the photosphere, a QSL has a finite thickness. How thin can it be in a flaring configuration? We have found in all the cases, QSLs thinner than the magnetogram resolution ([FORMULA] Mm). Does this numerical "super-resolution" have a meaning? The thickness of QSLs, like their locations, come from global properties of the magnetic configuration derived from field-line linkage. In Appendix B we show that the thickness magnitude of a QSL can be determined only if the magnetogram and the numerical grid for the extrapolation resolves the photospheric field concentrations.

Our analysis of the thickness at different places along the QSLs shows that, for the two quadrupolar regions (AR 2372 and AR 2779), the values of [FORMULA] go down to the computer precision at the location of flare brightenings (with thickness lower than 1 m !). This means that in these ARs QSLs behave physically like separatrices. For bipolar ARs (AR 2776 and AR 2511) the values of [FORMULA] turn out to be much higher (see Figs. 3b and 4b), in agreement with the results found for simple theoretical configurations in Paper I. The largest value of [FORMULA] ([FORMULA] m) corresponds to the left extreme of the left ribbon in AR 2776. Fig. 3e shows the typical behaviour of N when the QSL thickness can be numerically resolved. In this figure we have also included, for completeness, three other functions that have a theoretical interest in understanding 3D magnetic reconnection (see Paper I for further comments). These are: the distance (D) between the photospheric footpoints of field lines, the field-line length (L) and the delay function or flux tube volume ([FORMULA], where the integration is performed along a magnetic field line). For the flaring configurations studied, like for the theoretical ones considered in Paper I, these three functions have similar shapes and sharp variations on the scale length of the QSL thickness; however, QSLs are better defined in terms of N (see Paper I).

3.3. Electric currents

The vertical photospheric currents have beeen computed from the transverse magnetic field measurements (see references in Sect. 3.1). For almost all the studied ARs two photospheric current kernels of opposite sign, linked in the corona by field lines lying close to the computed separatrices, were found. This result remains with the Fourier transform extrapolation and the QSLs computation, as shown in Figs. 1b, 2b, 3b and 4b (in this last case only the current in the preceding polarity is detected above the noise level because the transverse field in the following polarity is too low). This indicates that the energy is presumably stored in these field-aligned currents and released during the flares. This result is remarkable since with both type of extrapolation are linear force-free so they do not take into account such concentration of the electric currents. Due to the Ampere's law the magnetic field created outside a current does not depend precisely on the current distribution so that the QSL deduced here are expected to be located close to the one deduced by a non-linear force-free field extrapolation.

However, we cannot deduce the local behaviour of field lines associated to the concentrated currents from linear force-free field extrapolation. The total twist of the field lines surrounding the current loop can be estimated by neglecting the curvature of the loop (because of a low aspect ratio) and by assuming a cylindrical symmetry. Then, following Amari et al. (1991, Sect. 6.3) the number of turns [FORMULA] of the field lines at the periphery of the current loop can be estimated as:


where I is the total current, F the magnetic flux and L the length of the loop. For the regions studied we have estimated [FORMULA] and L for each magnetogram, what gives [FORMULA] in the range [0.2, 0.5]. Of course these values have to be taken with caution because of the simple estimate made, but also because the calculation of the electric currents using transverse field measurements is problematic with the present magnetographs (see e.g. Gary and Démoulin, 1995, for a detailed description of the difficulties). Nevertheless, these low twist values are consistent with the absence of prominences in the studied flares (which are likely to be supported at the bottom of magnetic twisted flux tubes) and with the absence of ribbons with an "umbrella hand" shape which appear only when the twist is greater than one turn (Démoulin et al., 1996a).

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© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998