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

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Appendix A: off disk-centre transformations

The analysis of magnetograms taken away from the center of the solar disk requires the elimination of projection effects. This implies the transformation of both the measured vector magnetic field (Hagyard 1987) and the image plane spatial coordinates (Venkatakrishnan et al., 1988). The total transformation from the image coordinates [FORMULA] (where [FORMULA] are parallel to the magnetogram axes and [FORMULA] is directed to the observer) to the heliographic coordinates [FORMULA] ([FORMULA] are the local horizontal coordinates, x directed to the solar West, y to the North and z is the local vertical) has been given by Gary & Hagyard (1990). These authors show that the solar curvature can be neglected in the analysis of an AR if it is closer than [FORMULA] from the centre of the disk. More precisely, for a region of 100 Mm width and located at [FORMULA] from disk centre neglecting the curvature leads to a maximum position error of 2 Mm at its border, that is the typical size of a magnetogram pixel. Because the distance between H [FORMULA] kernels is in general less than 100 Mm, the solar curvature has a negligible effect on the relative positioning of the brightenings and the field topology.

The QSLs determined with and without the previous transformation lie very close for all the studied configurations except for AR 2511 on June 13, 1980. This AR was located at E26 N20 on that date and it is the farthest one from disk centre with which we have dealt. When the observed region is far from the centre of the disk, [FORMULA] is a mixture of the vertical and horizontal field components, which can be separated only badly by the model because the photospheric field is not a force-free field. This fact is in particular important on June 13 region because the flare kernels (and QSLs also) associated to the following polarity are located in a weak field zone, while the ones associated to the preceding polarity lie on a more intense field. The resulting model shows a "laughing face" feature in [FORMULA] located towards the left of the preceding spots (Fig. 5b). This is clearly an artifact because the magnetic data obtained on June 14 are not consistent with the existence of this positive flux zone (see Démoulin et al., 1993). Supposing that the photospheric field is nearly vertical, so that [FORMULA] is proportional to [FORMULA], and transforming only the spatial coordinates we obtain a more satisfactory result where QSLs are found close to flare kernels and extrapolated coronal field-lines link them like in all the other flares studied (Fig. 5c,d). Clearly, when the AR is far from the centre of the disk (say by more than [FORMULA]) we need a precise measurement and calibration of the transverse field together with a good resolution of the [FORMULA] ambiguity so that the vertical field can be deduced (see Venkatakrishnan & Gary, 1989). When only the longitudinal component is available, it is better to suppose that the field is vertical than to use the longitudinal component as a boundary condition for the extrapolation.

Appendix B: numerical precision on QSLs

Added to the errors in the magnetic calibration (when transforming the observed polarization in Fe lines to the magnetic field components) and to the spatial resolution of the magnetograph ([FORMULA] Mm), there are errors intrinsic to the extrapolation method. First, the photospheric magnetic field is measured in a region where it is not force-free, second the derived electric currents appear more concentrated than the ones resulting from a linear force-free field assumption, and third there are numerical errors due to the numerical method used. In the absence of precise chromospheric magnetograms, the first limitation can partially be overcome because the magnetic field becomes nearly force-free just above the photosphere (at a height of [FORMULA] km; see Metcalf et al., 1995) and the flux of the vertical component is approximately preserved. The second problem could be avoided using a nonlinear force-free extrapolation, but this is still a research domain (see e.g. Amari & Démoulin, 1992). We describe below the effects of the third limitation.

Magnetograms have usually a magnetic flux unbalance of a few [FORMULA]. This unbalance can be taken into account in the Fourier transform extrapolation including the zero harmonic; but since this is a constant, implying an infinite energy in the field, we have decided to discard it. This change in the original data can be reduced by enlarging the size L of the computed region to several times the size of the magnetogram. In practice, an enlargement by a factor 2 typically leads to a uniform modification of the data of less than 10 G. This enlargement is at the detriment of the maximum shear that can possibly be considered ([FORMULA]). For [FORMULA] larger than [FORMULA], the large-scale harmonic solutions become periodic with height, implying again an infinite energy in the field. These solutions can be discarded but the boundary conditions are then changed. We rather choose [FORMULA] and, therefore, restrict our analysis to magnetic regions which are not highly sheared.

When using a Green extrapolation method, the magnetic field is usually assumed to be zero outside the magnetogram. As mentioned in Sect. 2.2, with a discrete fast Fourier transform method the magnetic field is implicitly supposed to be periodic along two orthogonal horizontal directions ([FORMULA]). In order to avoid aliasing errors (Alissandrakis 1981; see also Schmieder et al., 1990), due to the interaction between the studied region and its neighbour images, we take an integration area of side [FORMULA] Mm; that is to say twice the size of any typical magnetogram used here, the field being put to zero outside the observed region. As mentioned in the previous paragraph, this value of L restricts the maximum shear that can be considered to [FORMULA] Mm-1. For further properties of the method and comparison to others see Gary (1989).

Besides, when using the Fourier transform method, the magnetic field is computed on a uniform horizontal mesh at any height. In order to save computer memory, we keep the results in a non-uniform grid. The grid size is minimum at the centre of the integration area - let us call it [FORMULA] - and increases geometrically by a factor [FORMULA] towards the borders. The same kind of grid is used in the vertical direction (z). In this direction the minimum spacing, taken as [FORMULA], starts at the photospheric level and the grid increases with height by a factor [FORMULA]. The height of the integration volume is [FORMULA]. With this non-uniform mesh, we keep only a total number of points [FORMULA] much lower than [FORMULA].

We have seen in Sect. 3.2 that the thickness of the computed QSLs can be much smaller than the magnetogram resolution. We analyse now the influence of the discretization used on the properties of QSLs. With this aim, we build up a magnetogram formed by a bipolar background flux and by four flux concentrations of radius R (we simply use a uniform field for both components). The ([FORMULA]) coordinates in Mm for the positive concentrations are (40.,0.) and (60.,0.), while for the negative ones they are (0.,10.) and (0.,-10.). We take for this test [FORMULA] Mm, so that aliasing errors are negligible in the determination of QSLs. This theoretical configuration, still close to the observed ones, allows us to calculate the thicknesses of QSLs down to the precision of the computer (with a relative value lower than [FORMULA]).

The evolution of the decimal logarithm of the QSL thickness ([FORMULA]) in meters as a function of [FORMULA] (Fig. 6a) shows that more than 200 Fourier modes (in one direction) are required to obtain meaningful results (i.e., we need to resolve the photospheric polarities: [FORMULA] Mm, being the radius of the flux concentrations [FORMULA] Mm). For the non-uniform grid with a number of points [FORMULA] and [FORMULA] (leading to [FORMULA] with [FORMULA] Mm) we have only approximate results, but [FORMULA] and [FORMULA] ([FORMULA]) are large enough and no significant differences are found with respect to a mesh twice finer ([FORMULA] and [FORMULA], leading to [FORMULA] ; that is to say, a nearly uniform mesh). The influence of the flux-concentration radius is shown in Fig. 6b for the largest grid ([FORMULA] and [FORMULA]). The QSL thickness can still be obtained with a good precision for a flux concentration with a radius of 2 Mm ([FORMULA] Mm for the figure) but not for a radius of 0.5 Mm ([FORMULA] Mm), because [FORMULA] is not large enough (even for [FORMULA], R is two times smaller than [FORMULA]).

[FIGURE] Fig. 6. Evolution of the logarithm of the QSL thickness [FORMULA] (in m) with the number of Fourier modes [FORMULA] for a theoretical configuration (see Appendix B). The three curves correspond to: a different resolution of the final non-uniform grid; b different radius R of the photospheric flux concentration.

The tests described above show that a QSL thickness much lower than the pixel size has a meaning. The numerical grid needs only to be fine enough to resolve the magnetic field spatial variations, so as to calculate the magnetic field lines correctly. In our computations we have found that, taking the maximum number of points allowed by the computer memory, we have a numerical grid fine enough compared to the magnetogram resolution. It is mainly the conversion from polarization to magnetic field, the magnetogram resolution and the physical equations used to model the coronal field, that limit the accuracy in the computation of the properties of QSLs. With these limits kept in mind, a more sophisticated extrapolation will give the coronal field on a numerical mesh, so it is important that the QSLs thickness can indeed be estimated for such numerical models.

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© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998