4. Three-dimensional configurations
The principal aim of the use of 3-D harmonics is the study of the magnetic configuration of the observed feet of prominences, that at periodic distance reach the photosphere. Démoulin et al. (1989) have attempted to understand prominence feet by using the 3-D harmonics of Eqs. (4- 6): there the prominence was modelled by a massive current line and its vertical equilibrium in the linear force-free field was studied. In the present paper the approach is totally different: we look for magnetic configurations with dips (without extra addition of concentrated current).
4.1. The choice of the 3-D harmonics
For simplicity, we will only use some having , so that all of the periodic feet under the main body of the prominence will have the same height. Different values for would mainly distinguish between suspended feet and feet that reach the photosphere. We use a set of six harmonics:
We assume that the periodicity in the y direction is Mm, which is a suitable value for the size of the supergranulation cells. As in Sect. 3, we use Mm. This gives us a ratio of . For this value, the 2.5-D configuration is only an approximate guide of what is happening in 3-D.
In order to have prominence feet, we choose the amplitude of harmonics to have bald patches, appearing periodically in the y direction. It is clearly seen in Fig. 2 that the BP region stands around the line given by Eq. (21). So we will have to choose and according more or less to this equation. The choice for the amplitude of the harmonics is also restricted by the polarity of the photospheric field we want to deal with, especially a globally bipolar region is present in many observations. Then, we choose and so that we stay around the bipolar region (see Fig. 1a and b). As expected from the 2.5-D approach, the underlying feet, as the lateral ones, are due to the presence of small parasitic polarities in this low field corridor. The amplitude of the 3D harmonics are limited to keep the flux of these polarities inferior to those of the principal bipolar component.
The last constraint, but not the least, is brought by measurements of the angle made by the magnetic field and the prominence axis, observed by Hanle effect (in this paper we name this angle ). Recent observationnal results show that for most prominences, is in the order of (Bommier & Leroy 1997). In Fig. 4 is shown the maximum value of this angle inside a fully 3-D prominence for fixed values of and and . One can see that this maximum value stands in the region of bald patches, where we expect the feet to be present. For low , we can get a quite large value of , whatever is the choice for and . For larger , the reverse line, that marks the transition from FX to OF, appears (see Fig. 2). At this place, . This means that there the field is completely aligned with the photospheric inversion line. On both sides of the reverse line, increases, in a normal (FX) or in an inverse (OF) polarity configuration. For higher values of , the highest values of can be found in the OBP region. This is an interesting result, as it shows than even with a high , the angle made by the magnetic field and the prominence axis can be still significant, especially when the twisted configuration reaches the photosphere. This effect is due to the fact that the larger is , the lower rotates the field in the y-direction and that at low height, for a large , the component of the second and third harmonics dominate the first one.
4.2. The commonly used OX configuration
It has been shown in previous sections that, with our assumptions, the FX configuration does not provide suitable characteristics for a prominence. So the magnetic configuration of prominences has to be modelled with a twisted flux tube. We first investigate the OX (see Fig. 3), by extending it to 3-D. This configuration has been introduced by Kuperus & Raadu (1974) as a suitable inverse polarity configuration for prominences. Then many studies have been built upon it in the field of prominences (see the review of Priest 1990) and coronal mass ejections (see Low 1996 and references therein), so that it has been commonly used as a 2.5-D configuration to describe a filament supported by a twisted flux tube. One has to note that the OX region in the (, ) diagram in Fig. 2 is very small, even for very high values of . Hence it is not easy to get this configuration. Nevertheless we describe one configuration with the following parameters:
Looking at the isocontours from Fig. 4, it is clear that there will be a strong gradient of the angle between the prominence and the magnetic field inside it, as the amplitudes and vary due to the 3-D effects. In the lower part of the feet of the 3-D configuration, we can show that, with a great care in the choice of the harmonics, we can get a quite suitable value of , around . If we want to have a greater value, the amplitude of the harmonics in should to be much larger. But then the photospheric magnetic field would be inconsistent with observations. Furthermore, whatever it is at low heights, falls rapidly to very small values in the main prominence body. There, its value is in the order of or less, which is totally inconsistent with observations made by Hanle effect (e.g. Bommier et al. 1994). This problem would not appear if the constraint due to the presence of periodic feet that reach the photosphere would not exist. The only way to get good values for for the OX configuration is to have a very low maximum height for the bottom of the prominence, so that it would be close to the bald patches regions almost everywhere. As this is not observed, the constraint is a very strong argument against these configurations in 3-D.
Another problem with this configuration is that the lateral dips that are present for the FF and the OF configuration (see Sect. 3.5) do not appear for the OX topology. This is due to the morphology of the parasitic polarities in the low field region. Hence, a prominence supported by an OX topology is very thin, and does not have lateral feet structures. Some small filaments in active regions are observed to be very thin with surrounding fibrils oriented parallel to the inversion line (e.g. Martin 1990). Though, this is probably due to the small thickness of the low field region. High resolution observations still show lateral feet structures very close to the filament, which is consistent with a thin low field corridor.
In the present model, the OX topology does not fit for suitable observable parameters in a prominence. The presence of parasitic polarities in the region close to the inversion line naturally leads to the OF configuration, which is after all the biggest region in the parameters space for high (see Fig. 2). On the other hand, one has to remember that if a OF twisted flux tube erupts, the effect of the parasitic polarities will decrease as the configuration rises and expands in the corona. So the topology could evolve to OX. The lateral feet will then shrink (as observed by Martin, private communication). Finally, if the so commonly used OX configuration does not stand for a stable prominence, it is still the best one that can be used for the description of eruptive prominences and coronal mass ejections. It can be formed by the eruption of a non-symmetrical OF configuration in which one lateral X point rises more rapidly than the other.
4.3. Filaments in OF twisted flux tubes
Using the limitations given in Sect. 4.1, one can easily find many combinations of parameters that match the observed parameters for a suitable prominence. Here we only give one. We take a set of parameters that gives us a 2.5-D OF topology, close to the bald patches (for the presence of feet and large value of ) and close to the bipolar region. We restrict the amplitude of 3-D harmonics (to get small parasitic polarities). The results with the parameters as above for the OX configuration, except that , are shown in Fig. 5a-d.
The computed magnetogram shown in Fig. 5a gives the value of , as it would be observed by a magnetograph that measures the normal component of the magnetic field. One can see a main bipolar region, with a low field corridor extending 20 Mm away on each sides of the inversion line, at . The parasitic polarities inside this corridor naturally appear in an hexagonal pattern, which matches well with observed concentrated fields located at the edges of supergranules.
The location of the dips in the y-z plane is shown in Fig. 5b. It represents a low height prominence (around 20 Mm), with underlying feet that reach the photosphere every 30 Mm. A higher prominence can be achieved, either by increasing or . This would respectively lead to quiescent prominence, or highly sheared active region prominence.
As measurements of the magnetic field inside the prominence give a strong constraint on the model, the norm of and the angle inside the prominence are represented in Fig. 5d and c. The magnetic field slightly increases with height in the prominence, which is coherent with the presence of dips in the field lines. The value of the angle decreases with height for this kind of twisted topology.
Up to now, we have only considered the presence of dips above the inversion line, in particular the underlying feet, despite of the presence of lateral dips in 2.5-D for the OF and OBP configurations (see Sect. 3.5). We compute the magnetic field in a cube of , and we look for the dips everywhere in this region, using Eq. (16) (see Fig. 6a and b). Viewed from the top, these secondary dips are more or less located above the secondary inversion lines, that stand around the nearest parasitic polarities of the low photospheric field region. These lateral dips are located in between the prominence underlying feet and the parasitic polarities. They are not suspended features in the corona but reach the photosphere, forming bald patches. Another interesting feature that did not appear at all in 2.5-D is the presence of other smaller dip structures, further away from the prominence. They are also related to other small magnetic polarities, but they do not join the prominence.
4.4. Fine structures in OF prominences
Assuming that the shape of the prominence is characterized by the presence of cold plasma contained in magnetic field-line dips, it is necessary to know which portion of these is likely to be filled with matter. A good approximation is to suppose that the cold plasma fills a dip up to certain height which corresponds to the pressure height scale, given by:
where the ionisation degree, the Boltzmann constant, , and the proton, neutral hydrogen and the electron density, the solar surface gravity, and T the temperature. For typical values for in a prominence, H varies between 0.2 and 0.5 Mm. But, with macroscopic kinetic energy, H is likely to be higher. Taking this fact into account, we represent the portion of the field lines which can support plasma up to a height of Mm, at regular space locations where a dip is present in the magnetic configuration.
It is striking to see in Fig. 7 that some field lines not only appear inside the prominence body, but also aside from the prominence. Viewed from above, the lateral dip structures are composed of some portions of field lines, that look like what has been introduced as barbs by Martin et al. (1994). These structures match quite well those of observed regular filaments (see Fig. 8). They naturally appear periodically, in pairs, at every 30 Mm. Each pair corresponds to the presence of small parasitic polarities close to the prominence, and hence, to the void between two underlying feet (see Fig. 6b). Now let's focus on the shape of one pair of feet. The orientation, the shape and the proximity of the represented field lines give the first impression that some field lines can cross the prominence. This is not so. One can see in Fig. 9a and b that these two lateral features are in fact composed of different field lines, each having a dip that can contain plasma.
There are also other dips at the edge of the photospheric low field region (in the filament channel). They are located at low height (Fig. 6a and b) and the associated fibrils seem to converge in the direction of the filament. This is due to an inversion of the component. We got the fishbone structure present at the border of a flux tube embedded in an arcade as observed by Filippov (1994). We propose these structures as a possible explanation for plagettes or dark fibrils around prominences (see Martin et al. 1994).
It is noteworthy that we focus here only on the dips, but at the chromospheric level dense material can be injected along field lines without dips. Then a representation like Fig. 7 may be completed by fibril-like objects which visualize the lower part of field lines filled dynamically with cold plasma. This have been realized by Low & Hundhausen (1995) in their Fig. 7. A similar pattern is present in our model, but with a modulation in the y direction; we however choose to focus here only on the new aspect: the lateral dips.
It is obvious in Fig. 7 that a simple reversal of the sign of changes the orientation of the lateral feet: A positive value of gives a sinistral and left-bearing prominence, with overlying arcades having a right skew, and a right-handed helical field for the flux tube that supports the prominence. A negative gives a dextral and right-bearing prominence, with overlying arcades having a left skew, and a left-handed helical field twist for the flux tube. This naturally links together the observed different patterns of chirality, in relation with the hemispheric helicity segregation (Martin et al. 1994; Rust & Kumar 1994; Pevtsov et al. 1995).
4.5. Evolution with the shear
Let suppose that the magnetic helicity slowly accumulates in the magnetic configuration (see e.g. Rust & Kumar 1994). We now investigate the quasi-stationary evolution of a given structure defined as in Sect. 4.3, as we vary from 0 (potential case) to 1 (). It can be clearly seen in Fig. 10 that for low , the configuration is FX, normal polarity, and that the prominence is very thin in height (a few Mm). The orientation of dark fibrils in the feet has not the observed direction (they are nearly orthogonal to the foot rather than along it). Furthermore, the prominence is interrupted along its axis. It is formed by a succession of feet without prominence "body". Here we see that it is not easy to build a suitable FX configuration for a well developed prominence.
The interesting part of this evolution is when gets its critical value , in other words when the reverse line crosses the point defined by (). We already know the properties of such an event for the 2.5-D case (see Sect. 3.3). Fig. 10 confirms that the magnetic field in the prominence aligns itself with the inversion line, as the prominence vertical extent decreases to zero. But what is new here is that this is also true in the full prominence corridor. From an observationnal point of view, this configuration will be composed of elongated fibrils, parallel to the inversion line, with no observable prominence, in a low field corridor around two major magnetic polarities. This is the signature of filament channels (e.g. Gaizauskas et al. 1997).
For higher values than , the prominence becomes OF. It has the right observed chirality characteristics and the individual fibrils in the feet are approximately along their axis, as observed. As keeps increasing, we get a prominence whose top rises (see Démoulin & Priest 1989). With a value of close enough to , one can get dips up to heights as large as 50 Mm. The effect of is smaller at low height, where the first harmonic has a negligible contribution in Bx. Hence the feet are almost unaffected by in OF configurations (the properties of feet are mainly determined by the local parasitic polarities).
To conclude with this evolution, one has to stay aware that what we have shown here does not come from a full MHD treatment, which is far beyond the scope of this paper. The step from FX to OF is assumed to be due to reconnection processes. Does this evolution really take place on the sun or does the configuration emerged twisted (so mainly in the OF configuration)? The first scenario can be validated only if one can observe the rotation of the fibrils in the feet as the prominence is forming or if one can justify the absence of significant dense material in the FX case so that this kind of prominence would be only weakly visible in projection on the solar disk. The small extension in height contributes to this, but is it sufficient?
© European Southern Observatory (ESO) 1998
Online publication: December 16, 1997