3.3 Multi sources
3.3.3 The role of fan eld lines
QSLs have already been found in simple bipolar congurations, in which the cor- responding potential elds calculated from point- or line-sources would not possess null points, and therefore no spines. But this has only been reported in highly non- potential elds, in which large-scale shear or twist either creates an S shape bald patch separatrix (Titov and Démoulin, 1999) or a double-J shape QSL footprint pattern (Démoulin et al., 1996, Titov, 2007). But in potential source models, one can hardly imagine how QSLs could not be related to any topological property of the potential magnetic eld conguration.
Apart from the spine eld lines, the only topological elements in source models are the separatrix (or fan) surfaces. Fan eld lines calculated with the SM are plotted in Figure 3.4 d. Note that for most of the fans, we have not plotted eld lines covering their whole surface. This is visible, for example in Figure 3.4 d, for the red fan associated with the null point between N1-3, for the dark-blue fan associated with the null point between P4-6, and for two very large-scale red and green fans associated with the null points between N1-2 and N1-5 respectively. The reason is that for most of the null points, the fans are far from being axisymmetric around the axis of the spine, because the complex distribution of sources results in generating fan-related eigenvectors of dierent amplitudes. For the 4 nulls described above, the ratio between the pair of fan eigenvalues of each null is 3.5, 4.9, 8.1
and 24, respectively. In these cases, fan eld lines starting from the null bend in directions parallel or anti-parallel to the eigenvector associated with the largest eigenvalue, as described by Parnell et al. (1996) and found by Mandrini et al. (2006) for photospheric null points and by Masson et al. (2009) for coronal (i.e. high altitude) null points.
The sum of spines and fans (Figure 3.4 d) now show more structures than the QSL footprints (Figure 3.4 a). Still, one can surprisingly associate the V shape and the spine extension branches, with the two red and green fans of very large eigenvalue ratios described above. Those match very well with one another. This fan-branch relation was not expected from the results from our 4 source models, and then could not be predicted with our rst explanation of QSL footprint locations on plane above point sources. However, this relation is far from being obvious elsewhere in the model. This shows that, in general, only a subset of the fan eld lines can transform into QSL footprints.
On one hand, there is not a single branch that starts from the null points, even though Q there peaks to large values. On the other hand, only the parts of fans that start from the sources seem relate to branches. To clarify this, we plot in Figure 3.4 c the spines at z = −0.1 as well as the portions of fan eld lines only, starting from the sources atz =−0.1and ending in the photospheric plane atz = 0
Figure 3.6: This cartoons show the behaviour ofQ when we have a spine (S) and a fan (F). (a) showsQfactor follows the spine shape (S) until it has a fan contribution (F) that is given by the end points of fan eld lines starting from the source (+) on the photospheric plane. In this case it curves on that shape. (c) shows that (a) case is true, but when we have also the presence of a consecutive spine, theQfactor smoothly follow it and also a branch following the fans contribution.
photospheric ending points all match with the branches. As can be seen, for example, with the fuchsia fan portions that start from the sources P3-4-6, whose photospheric endpoints t the horse-shoe shape branch that surrounds the source N4. A similar association can be seen around N7, where the spine-related QSL footprint does not end in the source, but rather splits into dierent branches that correspond to the endpoints of the blue, cyan and green fan eld lines. The topological transitions from separatrices to QSLs in these two examples are sketched in Figures 3.6. This nding explains the reason why the long V shape and spine extension branches were readily matched by the drawn red and green fan eld lines. Indeed, the very small relative amplitude of thez-aligned fan eigenvector in these two fans permits a very
small amount of magnetic ux to cross the z = 0 plane. So the majority of eld
lines which we plotted in Figure 3.4 d were actually conned belowz = 0.
The full curve that joins the photospheric endpoints of eld lines which belong to a given fan, and which start from a given source, extend from close to this source atz = 0 up to the inversion line atz = 0. If this whole curve would turn into a QSL footprint in all cases, then all branches should stop just before an inversion line. Figure 3.4 a shows that this indeed happens (see, e.g., the long branches which link the spine-related QSL footprints from P1, P2 and P7), but that it is not the case in general (see e.g. the branches which emanate from N2 and N7). These dierent behaviours actually relate to the lower limit forQ(z = 0) for which one considers a QSL to be signicantly narrow (i.e. important).
So, qualitatively, our results show that apart from the main spine-related QSL footprints, QSL branches correspond to the curve dened by the photospheric end- points of all fan eld lines which start from a subphotospheric source. This gives us an improved estimator for QSL footprints in source models.