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Astron. Astrophys. 360, 345-350 (2000)

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3. The numerical results

In order to investigate the change of the magnetic topology that arises during the reconnection process, we have examined the connectivity of different field lines. As the coronal perturbations are assumed to vanish at the photosphere (inertial anchoring), each line can be indentified by its radial and azimuthal coordinates at one photospheric end-plate. The two photospheric planes are situated at [FORMULA]. As the axial component of the magnetic field remains positive everywhere in the coronal medium (for the present configuration) during the evolution of the system, each field line can be obtained by numerically integrating the following field line equation from one photospheric end towards the other:

[EQUATION]

where [FORMULA] defines the intersection of the line with a z = constant coronal plane. [FORMULA] and [FORMULA] are the perpendicular and axial magnetic field components respectively. Using an initial condition [FORMULA] at the photospheric plane situated at [FORMULA], the solution to Eq. (1) gives a point [FORMULA] at a given altitude z and a final point [FORMULA] at the other photospheric end. Consequently, [FORMULA] and [FORMULA] define (direct) partial and final mappings respectively. Inverse partial and final mappings given by [FORMULA] and [FORMULA] can be also obtained by integrating backward Eq. (1) and using an initial condition [FORMULA] at the other photospheric plane situated at [FORMULA].

First, we have followed the final mapping [FORMULA] of a single field line, anchored at [FORMULA], as a function of time. The results are plotted in Fig. 3. One can easily see the drastic change of the coordinates at [FORMULA] indicating the occurrence of the reconnection process through this change of field line connectivity. The beginning of the reconnecting event is determined by the time at which the field line trajectory intersects the current layer.

[FIGURE] Fig. 3. The final mapping [FORMULA] as a function of time, giving the radial (circles) and azimuthal (stars) coordinates at [FORMULA] of a given field line originating from [FORMULA] = [FORMULA]. The results are reported in arbitrary units in order to be plotted on the same graph.

Second, we have investigated the direct partial mapping [FORMULA] (i.e. to the loop apex) of three sets of field lines arranged in three photospheric circles given by [FORMULA], [FORMULA], and [FORMULA], for the ideal kinked configuration. 1000 equidistant [FORMULA] values situated in the range [FORMULA] have been selected. The results plotted in Fig. 4, show that the two internal circles are transformed into two ovoid curves with an off-axis displacement of the magnetic axis. This is not the case for the more external lines which remain on an almost unaltered circle. This distorted topology is in agreement with surfaces shown in Fig. 2. Note that the inverse partial mapping [FORMULA] obtained with the same initial conditions at [FORMULA] would lead to points lying on the same curves as in Fig. 4, and the final mapping [FORMULA] would give circular curves corresponding to these initial conditions. This is due to the impossibility for the topology to change in ideal MHD.

[FIGURE] Fig. 4. The direct partial mapping to the loop apex [FORMULA] (in cartesian coordinates) of three sets of magnetic field lines originating from [FORMULA], [FORMULA], and [FORMULA] for the ideal kinked configuration.

Focusing now on the set of field lines originating from [FORMULA] (again 1000 equidistant [FORMULA] values between 0 and [FORMULA] are considered), we have examined (direct and inverse) partial mappings [FORMULA] during the evolution of the configuration. The results are plotted in Fig. 5 at four times including the saturation. One can easily see that the points obtained from the direct and inverse mappings lie now on two different curves which do not coincide. This means that the initial topology is destroyed and the field lines are re-arranging due to the reconnection. Moreover, the two curves encompass two regions that are separating as time goes on. This result is confirmed by the 3D view of Fig. 6, which shows that two well separated interwowen surfaces are finally obtained for the relaxed state. We have checked that similar results are obtained for other sets of field lines starting from more internal circular curves (with radii [FORMULA]) at the photospheric end-plates, and that the resulting mappings give then two interwowen tubes of different nested surfaces. We have found that the two flux tubes wrap one around each other as well as around the original axis with an angle of order [FORMULA]. This angle represents the helicity of each tube's axis, and it is also known as the writhe (Berger & Field 1984). This writhe value results from the deformation induced by the kink (see Fig. 2). As concerns the twist of the field lines within each tube, small values situated in the range [FORMULA] have been obtained. Therefore, the field lines have lost most of their initial amount of twist during the reconnection process, in agreement with previous results (Einaudi et al. 1997, Amari & Luciani 1999).

[FIGURE] Fig. 5a-d. The partial mappings to the loop apex [FORMULA] of a set of field lines originating from a circular arrangment [FORMULA] at the two photospheric ends [FORMULA] (green circles) and [FORMULA] (blue stars), for an early resistive phase [FORMULA] a , during the reconnection at [FORMULA] b and [FORMULA] c , and for the final relaxed state [FORMULA] d .

[FIGURE] Fig. 6. A 3D view of field lines corresponding to mappings of Fig. 5d.

The preceeding result on the transformation into two separated flux tubes cannot be true for all the field lines of the numerical flux tube, as at least the external potential region should remain unchanged (Einaudi et al. 1997). Therefore, we have investigated the partial mappings [FORMULA] using the initial conditions [FORMULA] at [FORMULA], again with 1000 equidistant [FORMULA] values lying on a closed circular curve. The results that are plotted in Fig. 7 for the final relaxed configuration, show that the two curves do not encompass (now) two well separated regions. However, some of the field lines are clearly connected to the region occupied by the two interwoven flux tubes. Selecting only a given range of [FORMULA] values smaller than [FORMULA], it is then possible to obtain two non closed curves that form a single closed one as one can see in the schematic mappings of Fig. 8c (region II). The corresponding field lines form a single weakly non axisymmetric surface that surrounds the double flux tubes structure. This means that this second region is an annular flux tube. We have also found that the field lines situated in this second region have no twist.

[FIGURE] Fig. 7. Same as in Fig. 5d for [FORMULA]

[FIGURE] Fig. 8a-c. Schematic partial mappings to the loop apex of 5 set of field lines originating from circular arrangments at the two photospheric ends [FORMULA] for the ideal kinked state a , during the resistive phase b , and for the final relaxed state c . Closed and non closed circles are used for lines situated in regions I and II respectively.

The separation of these two topologically distinct regions can be made by using the initial conditions [FORMULA] for the direct and inverse mappings. Indeed, we have found that the internal region containing the two well separated flux tubes corresponds to field lines obtained with footpoints arranged in circles with small radii at one photospheric end. More precisely, it is determined with mappings using initial conditions [FORMULA] and all the [FORMULA] values in the range [FORMULA] at [FORMULA]. A separatrix surface can then be obtained from the critical radius [FORMULA]. We have found that this value is correlated to the current distribution of the initial axisymmetric equilibrium. Indeed, it corresponds to the radius at which the axial current density reverses sign. We have also found that this value is correlated to the radius defining the region within which the initial field lines are highly twisted. We believe that this result remains true for a general axisymmetric configuration carrying a zero axial electric current, meaning that it is possible to predict the gross features of the magnetic topology of the relaxed state in terms of the equilibrium parameters defining the initially unstable twisted loop. We have finally checked that the field lines situated in the external potential region are not affected by the process, defining then a third region.

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© European Southern Observatory (ESO) 2000

Online publication: July 27, 2000
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