3. Reconstruction of the magnetic field in the filament channel
The aim of this section is to describe the model that we use for the reconstruction of the 3-D magnetic configurations of the observed filament channels. We use a magnetohydrostatic extrapolation method developed by Low (1992), which takes into account the effects of pressure and gravity on the magnetic field. Following the approach of Paper II, the observed inhomogeneous bipolar photospheric field component is replaced by a regular bipolar background field which constrains the presence of a twisted flux-tube.
3.1. Linear magnetohydrostatic equations
Let's use a cartesian system of coordinates, where z refers to the height and (x,y) to planes parallel to the photosphere. Low (1991) has solved these equations by defining the current density () of Eq. (1) with Euler potentials. For any given function of the altitude , and keeping constant as in the linear force-free field (lfff ), the linear magnetohydrostatic (lmhs ) equations can be re-written as:
where is a function which is independent of the magnetic field, and which only varies with z. This defines a background pressure (and density). Only the depletions and depend on the magnetic configuration. We choose as a decreasing exponential with height, as Low (1992) did in order to compute the general properties of lmhs solutions:
where a and H are the plasma parameters.
3.2. Physical interpretation of the plasma parameters
The influence of the plasma on the magnetic field is described by in Eq. (3). Its effect is to create currents in the (x,y) planes, parallel to the photosphere. In Eq. (6), a is the measure of the intensity of these currents at , where it is also the ratio between the pressure depletion and the magnetic pressure (Eq. ).
Low (1992) has used values of a up to 4, leading to the creation of low-lying twisted configurations. This was due to the change of sign of with low height. However this oscillating behavior of the solutions is unlikely to be physical. Aulanier et al. (1998b) have discussed the maximum value for a below which none of the harmonics oscillate due to the plasma effects. This value has been estimated to .
The vertical extension of the plasma effects is given by the scale-height H. Low (1992) explained in details how H can be in some cases equal or close to an isothermal pressure scale-height.
From Eq. (4), it is clear that the pressure depletion is larger as the magnetic field is more vertical. This is consistent with measurements of the pressure above strong fields, such as sunspots.
The properties of the density depletion are more complex, due to its expression (see Eq. ). Note that the last term is related to the curvature of the magnetic field, and that the density is higher in the presence of magnetic dips than in the case of arcades. This is consistent with finding dense plasma in magnetic dips, such as in filaments or dark fibrils (see Paper I and II). More details about the behavior of the pressure and density can be found in Low (1992) and Aulanier et al. (1998b).
It is noteworthy that the horizontal currents induced by the plasma have a component along the magnetic field when it is not vertical, so that the current density parallel to the field is no longer equal to . As a consequence, a slight non-linearity in the current distribution versus the magnetic field is brought by the plasma (see Low, 1992). Though this method still differs from the non-linear magnetohydrostatics (nlmhs ) where constant, thus we name the method as "linear" magnetohydrostatics (lmhs ).
3.3. Extrapolation method
From Eq. (3), can be expressed as a series of lmhs harmonics. Low (1992) has shown that the horizontal Fourier transformation of the vertical magnetic field component is in the form of Bessel functions:
where and . , where and are the wavenumbers with respect to x and y, being equal to and . (,) are the periodicities, and (,) the orders of the considered harmonics, in the (x,y) directions respectively. is the amplitude of the harmonic related the wavenumbers (,).
For the study of any particular active region, the boundary conditions (which is equal to the observed longitudinal component when the studied region is at the disc center) need to be imposed from an observed magnetogram. A fast Fourier transformation firstly permits to decompose these boundary conditions in the form of lmhs harmonics (see Eq. ), and secondly to calculate at any z. Consequently it is noteworthy that the lmhs method requires periodic boundary conditions. More details on the extrapolation method and on the transformation of coordinates can be found in Démoulin et al. (1997) and Aulanier et al. (1998b).
3.4. Imposing a twisted flux-tube
A direct extrapolation of the magnetic field of the SOHO/MDI magnetograms (shown in Fig. 1d) in the lfff assumption does not provide the existence of a twisted flux-tube. This point has already been discussed in Paper II. We have realized in the present study that direct lmhs extrapolations also failed to create twisted flux-tubes, imposing the physical constraints that the field strength decreases at large heights and that lmhs harmonics are not oscillating with height.
Following the justifications listed in Appendix A, we impose a background field which, when a high magnetic shear is imposed (see Paper I), constrains the presence of a twisted flux-tube. The OX flux-tube that was used in Paper II is kept in this study. This "OX" terminology has been introduced in Paper I, where "O" refers to an O-point at the center of the flux-tube, and "X" to an X-point at its bottom, just like the model proposed by Kuperus & Raadu (1974). The difference is that it satisfies the lmhs equations (Eq. ). The flux-tube field is expressed by harmonics of which amplitudes are:
where the parameter f is a multiplicative factor which has to be chosen in order to define the strength of the imposed background field with respect to the strength of the observed fields. The value of f is discussed in Sect. 4.1. The critical value of the normalized value of above which an OX flux tube appears in lfff with the parameters listed in Eq. (8) is .
3.5. Reconstruction of the magnetogram
The main bipolar component of the vertical field observed with SOHO/MDI has a mean width of 125 Mm (see Fig. 1d). We select such restrictive extension () since the lmhs , like the lfff , extrapolation is not able to describe areas having localized strong magnetic shears (as in filament channels) which are embedded in a region having a weaker shear (as in the large scale coronal loops overlaying the filament), which is the typical distribution of the magnetic shear in the environement of filaments (as proposed by Antiochos et al., 1994 and Schmieder et al., 1996). Then the background field (shown in Fig. 1f) can be combined with the observed field (shown in Fig. 1d) as follows:
In a first step, the inhomogeneous bipolar component of the SOHO/MDI magnetogram is cut, so that only the observed magnetic field that lies between Mm on each side of the inversion line (situated at ) is kept. By this way only the polarities in the filament channel are taken into account. In a second step, the theoretical background field (which ensures the presence of a twisted flux-tube with a large shear) is added to this cut magnetogram. Consequently, the observed dispersed bipolar component in the original magnetogram is replaced by a more regular one which is well aligned with the inversion line. Furthermore, a weak modulation of the field is brought by the theoretical background field in the region where Mm. But this only slightly modifies the polarities in the filament channel.
The constructed magnetogram is represented in Fig. 1e. It is noteworthy that the periodic boundary conditions required by the lmhs method lead to the fact that the positive flux on the right of Fig. 1e is only weakly connected to the negative flux on the left, allthough this still allows the presence of the twisted flux tube and some overlaying arcades.
At this point, lmhs or lfff extrapolations can be made using this constructed magnetogram (Fig. 1e) as boundary conditions.
© European Southern Observatory (ESO) 1999
Online publication: February 23, 1999