3. Results for a two-dimensional configuration
We use a small number of harmonics to describe the magnetic field in the prominence to catch the main topological features. This also allows us to make a systematic study how the configuration evolves as we modify the various parameters involved. Démoulin & Priest (1989) have shown that it is possible to form a twisted configuration even with only two harmonics of the 2.5-D linear force-free field. For great enough, the component of the second harmonic can dominate the component of the first one at low height, creating an inverse configuration with a dip. With a third harmonic, Amari & Aly (1992) show that it is possible to form a twisted region detached from the lower boundary (photosphere). Passing alternatively between these two basic configurations permits intuitively to have an alternation of feet. However, before the analysis of fully 3-D configurations a preliminary 2.5-D study is necessary, in order to find every configuration satisfying the conditions for prominence support.
3.1. Combination of harmonics
In 2.5-D, the magnetic configuration is invariant along the y axis. Hence, the harmonics for the magnetic field given by Eq. (4), 5and 6can be expressed as:
We combine three harmonics, given by assuming that the amplitude of the first one is . Then the three components of the magnetic field are given by:
There are many parameters in such an approach. We decide to vary the amplitudes of the harmonics.
3.2. The polarity of the photospheric field
As we want to understand observations, we are looking for a photospheric vertical field which is mainly bipolar, with a low field region around the inversion line. In order to do so, we investigate the locations where at the level (photospheric level). In Fig. 1a and b the global behavior of the vertical component of the magnetic field at the photospheric level is reported. The choice we made for the used harmonics naturally leads to the three kinds of configurations shown in Fig. 1a and b. Within one period of the field along the x axis, between and , we find two sextupolar regions, two quadrupolar, and one bipolar for low values of and . They are bounded by two lines and one ellipse, which equations are:
This diagram is similar to the one found by Amari & Aly (1992); it differs only because we are using the magnetic field (rather than its potential) and the three first harmonics (rather than the first, third and fifth ones). This result gives us some first constraints on the amplitude of the harmonics we need for a suitable prominence, because we have to remain close to a bipolar configuration.
3.3. Basic topology for a 2.5-D configuration
Here we investigate the various typical vertical sizes for a 2.5-D prominence, such as the height of its top and bottom. In order to do so, we calculate the values of z where the field lines present a dip above the inversion line, at . For a configuration which is invariant along the y axis, the condition for having a dip in a field line at a given height is (from Eq. (16)):
The two first columns in Fig. 2 show respectively isocontours for the height of the top, and for the bottom, of the prominence, as , and vary. We assume that the prominence exists between these two values. One can see that whatever is , no prominence exists for large values of and . There are also some configurations for which the height of the bottom is zero. It can be guessed that the feet of the prominence will have such a configuration. From these figures, we can see that the prominence is higher for lower and . This is also true for its bottom, except for high values of , where the bottom of the prominence rises again with , after having crossed a region where its height is zero.
The topology of the field lines above the inversion line is shown in the third column in Fig. 2. There are nine possible topologies for the conditions we use in this paper. The explanation of the terminology that describes the different regions in Fig. 2 is given in Fig. 3. The prominence bottom is formed whether by a dip whose field line is tangent to the photosphere (defined as a bald patch by Titov et al. 1993), or an X-point, or a flat field line. Its top part is composed of an O-point or another flat field line. Such results are comparable with whose of Amari & Aly (1992). The corresponding notations are for Arcade, for FBP, for FF, for OBP, for OF, for FX and for OX. The combination of three harmonics highlights that if the bottom of the prominence is due to a flat field line belonging to a twisted flux tube, there are two X-points at some distance on both sides of the inversion line. This is due to the appearance of small parasitic polarities around the inversion line.
3.4. Modification of the topology with
The FX topology is dominant for low values of . But for (normalized value), the OF topology region appears (in the region shown in Fig. 2), and gradually extends in the FX region with increasing . In this shrinking FX region, the isocontours for the bottom and for the top of the prominence are almost parallel and have nearly equal value. Hence, FX prominences for high have a short vertical extent. They are either at very low altitude, or very thin. Matter supported in such a thin layer is unlikely to be observed as a dark filament in absorption on the disk. It follows that FX prominences, with a normal polarity configuration, necessarily have quite a small , as far as modeling in the linear force-free free field approximation shows.
Furthermore, at approximatively for the same critical value of , the little FF region disappears completely, leaving its place for a growing OX region. This is an important point, because it shows that O-points only appear for very high values of . They are the signatures of twisted flux tubes, leading to inverse polarity configuration for prominences.
For high values of , one can notice a transition region, going from FX to OF. This curve evolves with , gradually crossing the graph and it induces a discontinuity in the isocontour curves (see Fig. 2, first and second column). We call this curve the reverse line. Crossing this line from the FX to the OF configuration implies several changes. First, the height of the top of the prominence becomes equal to those of its bottom, which means that the vertical thickness of the prominence becomes zero. Then, the 2-D topology around evolves from one X-point to two X-point plus an O-point. The field becomes twisted. Finally, the configuration passes from normal polarity to inverse polarity.
3.5. The lateral dip structure
It has already been emphasized that the underlying feet structure of the prominence could be modelled by bald patches, but looking aside from the inversion line, one can notice that there are some secondary dips for the OF and the FF topologies (see Fig. 3). This is due to the presence of X-points (in the x-z plane) on both sides of the main prominence body. They are indeed created by some small parasitic polarities in a low field region around the inversion line. Then we can assume that some cold matter can also be supported here against gravity, leading to a lateral feet structure. Their three-dimensional aspect is explained in Sect. 4.4.
© European Southern Observatory (ESO) 1998
Online publication: December 16, 1997