Astron. Astrophys. 342, 854-862 (1999)

2. Saddle point and separator

Since structures of the field near the separatrices or separators are rather peculiar, some observational characters such as the arrangement of the chromospheric fibril or the photospheric transverse field can serve to search for these structures. One example is demonstrated by Filippov (1995) by virtue of the H filtergrams, in which the stable saddle points and the imprint of separatrices, looking like a fir-tree branch, were clearly outlined by the large-scale fibrils. In another example, using the Poincaré index, Wang & Wang (1996) determined the saddle points and magnetic lanes in the computed transverse field in the chromospheric plane. Their results encourage us to further explore such topological features directly from the observed photospheric transverse field, which may bear non-potential features favoring the occurrence of flares.

In general, for a 2D field, if magnetic poles are assumed to be singular points of node type, the geometrical properties near singular points can be mathematically described with Poincaré index with the value of 1 or -1. The value of -1 indicates the case of a saddle point (Arnold 1973; Bogoliubov & Mitropolsky 1961). Poincaré index of an isolated singular point is defined as the number of rotations of the 2D field vector along a closed curve surrounding the point with a finite diameter, i.e.

where .

It should be pointed out that the topology of a 3D field cannot always be determined merely with the 2D information (Démoulin et al. 1996; Amari 1997). That is to say, if a special structure around a certain point is found in a 2D field, it may be only due to the perspective effect of a 3D field, but does not necessarily indicate the 3D singularity at this point (say, a separator passing through it). Such cases are shown by Priest & Forbes (1989) and Filippov (1995). Therefore, we should be much careful in the study when the 2D topological properties are referred to as the 3D ones.

Nevertheless, when we discuss the properties of the 2D field which are confined on the photospheric or chromospheric surface, the validity of the 2D method may be justified in studying the relation of magnetic topology to energy release in solar flares. The reasons are:

Firstly, when the vertical field component at the saddle point is zero, the saddle point of 2D field gives the true position of a separator intersecting with a horizontal plane; in another word, the saddle point corresponds to a 3D null (Sweet 1958, 1969; Baum & Bratenahl 1980; Hénoux & Somov 1987). In this case, the origin of the saddle point associated with the separator does not need to lie strictly in a plane normal to the separator, but can be in any plane passing through or near the null parallel to its eigen field lines (Lau & Finn 1990). These properties can be used to set up a set of procedures to confirm the relation of the saddle points to the location of the separators (Wang 1997).

Secondly, from the model of magnetic sources, we can deduce that the saddle points are close to the possible intersections of the separators with the photospheric (or chromospheric) surface. It is known that observations are in agreement with the idea that magnetic flux converges at sub-photospheric layers. Consequently, the observed field concentrations can be modeled by the magnetic sources, which are situated below the photosphere usually in a depth much smaller than the horizontal size of the active region. Using 4 magnetic charges, Démoulin et al. (1994b) modeled 10 datasets for 5 active regions, giving the average depth of 0.130.06 for the main bipole and 0.060.02 for the parasitic bipole if the size of the main bipole is normalized to unity. They found that the nulls were also present below the photosphere (with the depths less than 0.1) for most cases (nearly 90%). In other words, all the magnetic charges and the major nulls are supposed to be situated in a thin horizontal layer. With this assumption, the separatrices and separators are nearly vertical near this thin layer (Baum & Bratenahl 1980; Gorbachev & Somov 1988; Filippov 1995), thus we believe that the 2D magnetic lanes or the saddle points are not far from the `true' location of the separatrices or separators in the photospheric (or chromospheric) surface.

Thirdly, since the separators or the separatrices are preferable places for energy release and at the same time are sites of structural stability, we may expect them to be morphologically associated with the observed manifestations such as homogeneous flares or repeated occurrences of energy releases. This also provides us an important referable evidence for the relation between the 2D topology and the separators or separatrices.

© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999
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