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

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2. The loop model

The cylindrical flux tube as well as the numerical procedure used in this study are described in details in Paper I. However, we briefly recall here the main features of the coronal loop model.

An initially axisymmetric equilibrium with a force-free magnetic field and a vanishing plasma pressure is assumed. The twist, that has been numerically calculated, is a decreasing function of the radial coordinate with a maximum value of [FORMULA] on the axis (see Fig. 2 in Paper I). As one can see in Fig. 1, an internal twisted flux tube of length L situated inside [FORMULA] is embedded into a potential region having a vanishing azimuthal magnetic field component. A perfectly conducting wall is also placed at the outer radial boudary [FORMULA] for the numerical procedure requirements. This loop configuration is kink unstable because the amount of twist exceeds a critical value (of [FORMULA] on the axis, see Paper I). The first stage of the development of the kink mode has been then followed (see the results reported in paper I) using our code SCYL with a vanishing resistivity coefficient in the MHD equations. This ideal phase is characterized by a strong deformation of the internal region of the loop while the more external (potential) region is unaltered (see Fig. 2). An intense helical-like current concentration is also forming along the loop. When the smaller magnetic length scale reaches the grid resolution, the ensuing evolution becomes resistive and the use of a non zero resitivity coefficient (in MHD equations) dominating the residual numerical resistivity is necessary to pursue the simulation (Arber et al. 1999, Paper I).

[FIGURE] Fig. 1. Projection on the [FORMULA] plane of several magnetic field lines anchored at different positions [FORMULA] at the photosphere (at [FORMULA]), for the initial twisted (axisymmetric) configuration. The cylindrical coordinate system [FORMULA] is used.

[FIGURE] Fig. 2. A 3D view of the ideal kinked configuration (starting point of the present simulation) showing many field lines anchored at two radial positions [FORMULA] and 0.4 at the photosphere.

Therefore, as the starting point of the present work, a constant and uniform resistivity coefficient [FORMULA] (in our dimensionless units) is introduced at a given time [FORMULA] ([FORMULA] being the radial Alfvén crossing time defined in Paper I) corresponding to a kinked state that gives a (maximum) current density amplitude of two times the initial equilibrium one. This resistivity value is an optimal one as higher values would lead to the unrealistic diffusion of the whole equilibrium while smaller values lead to a considerable amount of time of cpu computation (Paper I). A viscosity coefficient equal to the resistivity one is also used. We have then completed the time evolution of the system until a relaxed state of lower magnetic energy was obtained. As explained in Paper I and by Arber et al. (1999), this final state is characterized by the dissipation of the current concentration that is representative of the available free magnetic energy.

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© European Southern Observatory (ESO) 2000

Online publication: July 27, 2000
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