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

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4. Conclusion

In this paper, we have considered the resistive evolution of a cylindrical coronal loop configuration that is kink unstable. We have assumed an initial axisymmetric flux tube carrying no net axial current with twisted magnetic field lines in the central region. We have then studied the change of the magnetic topology during the second stage of the evolution that is a reconnection process.

While previous results have shown the stationary character of the phenomenon in agreement with the Sweet-Parker model (Paper I), we have obtained that the internal field lines experiment reconnecting events when their trajectory intersects the current concentration region that has been formed during the early ideal phase.

We have examined the connectivity of field lines through the use of direct and inverse mappings, obtained by integrating a field line equation along the loop length from one photospheric end towards the other. The results can be summarized by drawing a schematic description of partial (direct and inverse) mappings to the loop apex for three states: the ideal kinked configuration (a), during the reconnection (b), and the final relaxed state (c). We have shown that the configuration evolves towards a relaxed state containing three topologically distinct regions. First, an internal region where the initially highly twisted field lines originating from closed circular arrangments with small enough radii [FORMULA] at each photospheric end, is progressively fully transformed into two interwoven flux tubes having a small amount of twist. We can then define a region I as one can see in Fig. 8c. The two flux tubes are labelled Ia and Ib in correspondence with initial conditions at [FORMULA] and [FORMULA], respectively. A second more peripheral region (II) is also obtained with field lines originating from non closed circular arrangements (with [FORMULA]) at each photospheric plane. Indeed, only a given range of azimuthal positions smaller than [FORMULA] at the photosphere must be taken into account to form the curves in this region. This second region is a weakly non axisymmetric annular flux tube, giving partial mappings to the loop apex which consist of closed curves with two branches (in correspondence with the two sets of initial conditions at [FORMULA] and [FORMULA]). The field lines obtained from the remaining azimuthal positions at the photosphere are in fact connected to region I or to inner lines of region II. Finally, these two regions remain embedded in the external potential region that is unaltered by the process. We also found that the critical radius [FORMULA] coincides with the site at which the axial component of the current density of the initial axisymmetric configuration reverses. It also corresponds to the radius defining the region within which the twist was maximum. We believe that this result is a general feature of loop configurations carrying no net axial current.

However, in cases of finite current loops, the final relaxed state should be probably less well ordered as suggested by the numerical results obtained by Einaudi et al. (1997).

A similar two flux tubes configuration has been recently obtained by Amari & Luciani (1999) using a 3D simulation of a curved twisted loop embedded in an external potential magnetic field. However, these authors have not investigated in details the magnetic topology change, making then the comparison with our results difficult. We can, nevertheless, conclude that the cylindrical geometry approximation used here probably gives the essential features of the phenomenon. We then hope that our results can serve as a useful guideline to understand the 3D reconnection phenomenon in the solar corona.

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© European Southern Observatory (ESO) 2000

Online publication: July 27, 2000