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

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2. Description of the method used

2.1. Quasi-separatrix layers method

Following Priest & Démoulin (1995) and Paper I, the classical separatrices can be generalized to QSLs. These are regions where a drastic change in field-line linkage occurs, i.e. where field lines initially close separate widely over a short distance. Let us integrate over a distance s in both directions the field line passing at a point P [FORMULA] of the corona. The end points of coordinates [FORMULA] and [FORMULA] define a vector [FORMULA]. A drastic change in field-line linkage means that for a slight shift of the point P [FORMULA], [FORMULA] varies greatly. The function [FORMULA], defined by


allows us to locate the region with a drastic change in connectivity for a given value of s. In the case of solar flares, as discussed in Paper I, the distance s to be used is the distance to the photosphere (located at [FORMULA] here). Besides, since we are working in the approximation of an abrupt transition from a low to a high [FORMULA] plasma ([FORMULA], ratio of the plasma to the magnetic pressure) and of high magnetic Reynolds number, line-tying is imposed at the photospheric level and the location in that plane of field-line footpoints is a function of x and y only. Therefore, the function [FORMULA] simplifies to:


[FORMULA] is defined only on the boundary and is the norm of the displacement gradient tensor defined when mapping, by field lines, points from one section to another of the photosphere. The locations where [FORMULA] takes its highest values define the field lines involved in the QSLs. By following these lines we can locate the coronal portion of the QSLs. We refer the reader to Paper I for a discussion of the properties of [FORMULA] and of the basic characteristics of QSLs.

The SM, used previously to determine the locations of separatrices, requires the use of the magnetic field generated by sub-photospheric sources (because it is the change of the magnetic link betwen the sources which determine the separatrices). This is not compelling with the QSLM that can be used with any kind of magnetic field description. There is no longer a need either to precise how the observed field concentrations are grouped together or to integrate magnetic field lines below the photosphere. The QSLM permits to detect a separatrice as special case of QSL where its thickness is only limited by the spatial precision of the field-line integration. In particuliar, the QSLM permits to find separatrices associated to magnetic field-lines tangent to the boundary while the SM cannot (see Paper I).

2.2. Extrapolation of the photospheric field

The photospheric longitudinal field ([FORMULA]) has been extrapolated to the corona, under the linear force-free field assumption (constant [FORMULA]), using the discrete fast Fourier transform method as proposed by Alissandrakis (1981). This method takes into account the position of the active regions (ARs) on the solar disk (the transformation of coordinates from the observed frame to the local one is explained in Appendix A). The best value of [FORMULA] is determined by fiting the observed shear in the H [FORMULA] fibrils and transverse field in the flare region.

In previous studies (see references in the Introduction) subphotospheric magnetic poles or dipoles have been used to extrapolate the observed photospheric magnetic field to the corona. The SM method uses explicitly only sub-photospheric sources while the QSLM can use any representation of magnetic fields. We compare in this paper results obtain with Fourier transform with those obtain with sub-photospheric sources (both with a linear force-free field assumption). A model using a small number of sources is appropriate for computing QSLs when the magnetogram has well defined polarities, like in the case of AR 2372 (compare Fig. 1b to Fig. 1e). The method is still satisfactory when the magnetogram presents extended polarity regions as in AR 2776, though the best agreement between QSLs and H [FORMULA] brightenings has been found using the Fourier transform extrapolation (compare Fig. 3a,b and f).

The use of sources to extrapolate photospheric magnetograms has two main advantages. First, it allows us to take into account a large flux imbalance in the magnetic data. Second, the noise in the data is strongly decreased by a least-square fitting of the model to the observed magnetogram. The main disadvantages of using sources are that the model requires time to introduce interactively a great number of sources in the case of complex ARs, and that the description of the magnetic field by a series of sources is limited to linear force-free fields.

The Fourier transform is a more classical technique which takes into account the full magnetogram. The noise in the data can be easily decreased by filtering the higher harmonics, however this leads to a broadening of the magnetic polarities and this goes against a good computation of QSLs, since the distribution of the photospheric field in several concentrations is the origin of QSLs (in particular in bipolar regions). Without this filtering, strange connectivity patterns may be found. This is illustrated in Figs. 4, field lines link the external borders of QSLs that lie in zones where [FORMULA] in Fig. 4b, while in Fig. 4c field lines issued from the inner borders of QSLs reach the photosphere at places where the noise in the data is close to 50G. We choose to use the data without filtering and restrict our attention to magnetic field greater than 100 G. The main limitations of the Fourier-transform extrapolation are: imposed flux balance, periodicity of the solution in the horizontal (x, y) directions, proportionality between the current density and the magnetic field (through a constant called [FORMULA]) and the restriction of application to ARs with low magnetic shear.

The errors associated with the use of a discrete fast Fourier transform can be inferred by simulating a magnetogram using magnetic sources and, then, comparing the results of the extrapolation of the simulated photospheric field to the model field. We have found that the locations of QSLs differ by less than the magnetogram resolution. The main limitation of the extrapolation is therefore the linear force-free field hypothesis. The limitations of the extrapolation scheme and the implications of the numerical precision on the location of QSLs and their thickness are further described in Appendix B.

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© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998