4. The 3-D lmhs configuration for the filament channel at 12:14 UT
In this section we compare the reconstructed 3-D magnetic configuration with features in the filament channel observed on September 25 1996 at 12:14 UT (see Sect. 2). We show that for realistic values of the parameters of the model, the morphology of the body of the filament, as well as the location and shape of its lateral feet and some fine structures in the filament channel, are well correlated with the location of the dipped portion of upwards curved field lines.
4.1. The parameters of the model: a, H, , D and f
The first parameters that have to be fixed are the so-called "plasma parameters", namely a and H, which are defined in Sect. 3.1. We have given the value of in Sect. 3.2. Consequently, since we want to investigate the maximal effects of pressure and gravity on the reconstructed magnetic field configurations, we choose to use for all the following extrapolations.
We choose a typical value of Mm in order to have significant plasma effects in the prominence body, as typical prominences are known to be up to a few tens of Mm high. Moreover, filaments are composed of over-dense plasma with respect to the surrounding corona. Consequently it is worth to keep in mind that the plasma effects will be over-estimated in the corona surrounding the modeled filament.
An important parameter that needs to be fixed is the shear parameter . Using the results of Paper II where a preliminary successful comparison has been made with a value of Mm-1 between a lfff model and the observed filament, we keep the same value for all the following extrapolations. This value corresponds to a normalized value of , which is above the critical value above which an OX twisted flux tube is present.
We suppose that the dense absorbing plasma fills a dip up to a depth D which is equal to the pressure scale height . For typical physical values in a prominence, . Consequently, in the figures we only represent the dipped portion of field lines of km, at regular space locations.
The last parameter is the flux of the imposed twisted flux-tube, which is constrained by the factor f. In order to estimate the right value of f, we completed several extrapolations with a typical Mm-1 making:
(i) comparisons of the vertical flux of the observed large-scale bipolar field component (seen on Fig. 1d) with the one replacing it, which is introduced by the theoretical background field (as explained in Sect. 3.5),
(ii) comparisons of the H filament observed at 12:14 UT (see Fig. 1a) with the modeled configurations (using the method described in Sect. 4.2).
Satisfactory correlations of the model with the observations were found for . Thus we use a typical value of for all the following extrapolations.
4.2. Comparison between H dark features and dipped field lines in the filament channel
After modifying the SOHO/MDI magnetogram using the method described in Sect. 3.5, the magnetic field is extrapolated in lmhs in a 3-D box with horizontal sizes 125 Mm = 172". The parameters of the model are listed in Table 2. The magnetic dips computed from the lmhs extrapolation from the modified magnetogram shown in Fig. 1e are reported in Fig. 1b.
Table 2. Values of the parameters used for linear magnetohydrostatic (lmhs ) extrapolations: are the amplitudes of the harmonics which define the twisted flux-tube, and are the size of the computational box and define the periodicity for lmhs extrapolations, is the shear parameter, D is the depth of which the dipped field lines are presented, a is related to the strength of the plasma effects and H is the scale height on which the plasma effects are present.
4.2.1. Global shape of the filament
The global shape of the computed filament in Fig. 1b shows a satisfactory correlation with the H image in Fig. 1a. Firstly, the angle of the magnetic dips with the inversion line is in accordance with the orientation of the observed fine structures. Secondly, the initially straight twisted flux-tube forming the filament body is perturbed by the polarities of the filament channel, and it shows very similar features to the observed body of the filament. It is noteworthy that the filament body is also locally deformed in the vicinity of the strong positive polarity, near the largest foot at the middle. This polarity pushes the twisted flux-tube to the left, so that the filament seems to avoid this polarity, in accordance with the observations of Martin et al. (1994) and with the model of Priest et al. (1996).
4.2.2. The lateral feet
The largest foot observed on the right side of the filament (F1) is formed by two main negative parasitic polarities (seen in Figs. 1d,e). The one closest to the filament (N1) forms a broad lateral extension of the dip pattern from the twisted flux-tube, and the other one (N2) creates another set of dips which forms a nearly continuous pattern with the first one. Some field lines present two dips, in each of these regions. Consequently pressure-driven mass flows along the field lines can easily occur between the broad part of the foot related to N1, which is closer to the filament body, and the thin part related to N2, which is further away from it. It is clear from Fig. 1b that this dip pattern shows a slight interruption between the two parasitic polarities, while this does not appear in H (see Fig. 1a). Though we believe that this is a minor difference that may be due to some of reasons given in Appendix B.
Another large lateral foot is observed in H at the top of the left side of the filament (F2). However the model does only poorly recover this foot from the dips distribution (compare Fig. 1a and Fig. 1b). A group of very low lying (below 3 Mm) dips are present above a weak positive parasitic polarity, though it does not form a continuous pattern with the filament. This can be due to some of the reasons given in Appendix B.
4.2.3. The chromospheric fine structures
Some dips appear away from the filament and its feet in the filament channel (see Fig. 1b). These are also created by parasitic polarities, which are too far from the twisted flux-tube to form a continuous dip pattern with it as in the case of feet. These dips are located very low in the atmosphere (below 4 Mm). Such typical isolated and low-lying dips have already been associated with observed dark H fibrils in Paper II and in Aulanier et al. (1998b). In this study, some of them show a correlation with some dark elongated fine structures observed in H (Fig. 1a): For example, the dark half-circle at the left bottom of the filament (S1), the other one, higher on the left side between the two largest feet (S2), a dark feature located on the right side, oppositely to the upper large foot (S3), and the "M-shaped" group of fibrils on the left of the filament (S4). However, overlaying Fig. 1a and Fig. 1b reveals that this correlation is not as clear as for the filament itself, even if the orientation and location of some of the observed fine structures are well matched by some dip patterns. These differences are probably due to some reasons given in Appendix B.
The orientation of the computed magnetic field in the filament channel is globally parallel to the photospheric inversion line. On both sides of the filament (around Mm), it shows a typical fishbone structure discovered in H fibrils by Filippov (1994). This global organization is characteristic of a twisted flux tube located in a bipolar region.
4.3. The filament as a prominence
The computed prominence is composed by a continuous dipped field line pattern. For viewing purpose the dips are only shown at fixed intervals ( Mm), leading to the horizontal structures seen on Fig. 2. Reducing the imposed vertical interval between the computed dips would lead to a dark continuous feature, which would not show these horizontal structures anymore, but which would not give a general idea of its 3-D geometry. It is noteworthy in Fig. 2a-d that some very dark regions still appear in the prominence, of which shape and location strongly depend on the projection views. These dark areas are due to the projection effects on the shape of the computed dips. We propose that such variation of the geometrical length along the line-of-sight affects the measurements of the line-of-sight integrated densities in filaments and prominences using radiative transfer models.
The computed feet (on Fig. 2) are in good qualitative agreement with typical observed feet of prominences and filaments which are close to the limb (e.g. Zirin 1988 pages 267, 268 and Martin, 1990). It is clear from Fig. 2b,c that the series of dips in the largest lateral feet reach the photosphere, forming bald patches (i.e. field lines tangent to the photosphere). The results of Paper II are confirmed here: the feet are not formed by magnetic arcades, but rather by dipped field lines which form a continuous pattern from the prominence body to the photosphere. In this context the magnetic field in the feet is nearly horizontal. Though Fig. 2a shows that projection effects can lead dipped field lines to be apparently nearly vertical in the lower parts of the prominence (e.g. in the foot F1). This happens when the orientation of the magnetic dips is nearly parallel to the line-of-sight. Such false conclusion of nearly vertical structures can also be derived in interpreting observations, where the observed fine structures and the mass flows are projected on the observing plane.
© European Southern Observatory (ESO) 1999
Online publication: February 23, 1999