6.3 Separatrices and QSLs

In spite of the observational evidence of their significance, separatrices and separators in submerged poles models appear theoretically to have dubious relevance. Those separatrices which are the fan surfaces of sub-photospheric nulls have no characteristic in the photospheric or coronal fields which distinguish them as such. Since the sub-photospheric field is only an artifact of the modeling, it is unclear why topological features originating there would be physically meaningful.

The solution to this puzzle appears to lie in the concept of QSLs (see Section 5.2). Since a submerged poles model has a non-intermittent photospheric field it may be analyzed strictly as a pointwise mapping model, without regard to the sub-photospheric field. Doing so eliminates the separatrices from submerged nulls, and with them most of the photospheric regions. The footpoint mapping $X(x)$ may be computed from photospheric footpoint $x$ to conjugate footpoint $X$ . From that may be found the derivative norms $N± (x±)$ or squashing function $Q(x)$ discussed in Section 5.2. Large values of either indicate severe local distortion in the footpoint map. The locus of point where $Q(x)$ exceeds a threshold may be used to define the QSLs.

Titov et al. (2002 ) performed this analysis for a submerged pole model of Sweet’s configuration. They found two QSLs, shown in Figure 17 , whose location corresponded with separatrices from the two submerged null points. They repeated this process for models with poles at decreasing depths and found that the maximum values of $Q$ within the QSL became ever larger. They concluded that in the limit $d-- > 0$ , $Q --> oo$ , meaning the layers have become genuine discontinuities: they are separatrices.

Submerged poles models have proven particularly useful since they offer a fast and convenient means of predicting QSLs. At least with algorithms presently used, null points and fan surfaces can be found much faster than QSLs. Furthermore, the topology of submerged poles field will resemble that of an MCT field: separatrices partitioning the volume into a certain number of domains. Analysis of this partitioning gives an immediate indication of how many QSLs a field might therefore have.

The process of raising poles to $z = 0$ also serves as a bridge between the two classes of models. MCT models, described in Section 4, included one type of separatrix: fans from null points. Pointwise mapping models, on the other hand, had separatrices from coronal nulls and from BPs as well as QSLs and HFTs. In the process of starting with a continuous field generated by submerged poles, and raising the poles to the surface, the topological elements from one model are transformed into those of another model. Titov et al. (2002 ) showed that QSLs transform to genuine separatrices and Titov et al. (2003 ) showed that a HFT transforms to a separator and adjacent separatrices.

The two genuine separatrices in the pointwise mapping models suffer different fates in this limit. Nulls already in the corona when the poles are submerged will rise even higher into the corona as the poles are raised. Separatrices from fans of coronal nulls will therefore be the same in both models. Bald patches, however, are rooted in PILs which are absent from the intermittent fields of MCT models. As submerged poles are raised, the gradient $\~/ Bz$ in the vertical field at each PIL decreases in magnitude. In the limit $d --> 0$ the PILs expand and merge to form the field-free sea. The BP separatrices will, in general, disappear along with the PILs which they are rooted to. All separatrices in MCT models have closed footprints, and therefore enclose sub-volumes. The BP separatrices, on the other hand, do not enclose volumes so they have no counterpart in MCT models.