## Appendix A: off disk-centre transformationsThe 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 (where
are parallel to the magnetogram axes and is
directed to the observer) to the heliographic coordinates
( are the local
horizontal coordinates, 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, 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 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 is proportional to , 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 ) we need a precise measurement and calibration of the transverse field together with a good resolution of the 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 QSLsAdded 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 ( 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 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
. 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 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 (). 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 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 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 - and increases geometrically by a
factor towards the borders. The same kind of
grid is used in the vertical direction ( 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 The evolution of the decimal logarithm of the QSL thickness
() in meters as a function of
(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:
Mm, being the radius of the flux concentrations
Mm). For the non-uniform grid with a number of
points and (leading to
with Mm) we have only
approximate results, but and
() are large enough and
no significant differences are found with respect to a mesh twice
finer ( and , leading to
; that is to say, a nearly uniform mesh). The
influence of the flux-concentration radius is shown in Fig. 6b for the
largest grid ( and ). The
QSL thickness can still be obtained with a good precision for a flux
concentration with a radius of 2 Mm ( Mm for the
figure) but not for a radius of 0.5 Mm ( Mm),
because is not large enough (even for
,
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. © European Southern Observatory (ESO) 1997 Online publication: May 5, 1998 |