Astron. Astrophys. 318, 621-630 (1997)

3. Equilibrium, stability, and linear phase

3.1. Equilibrium

Mikic et al. (1990) have calculated the equilibrium magnetic configuration which result from an imposed localized photospheric flow. Such equilibria are two dimensional with radial and axial dependences. However, it is apparent from their results (see Fig. 4) that the equilibrium quantities are independent of the axial coordinate z over most of the loop length except in narrow boundary layers near the photosphere. Neglecting the z dependence of the initial equilibrium, Baty & Heyvaerts (1996) have shown that the effects on the stability are negligible. Therefore, for the sake of simplification, we consider one dimensional (radial) force-free equilibria. Such equilibria are fully determined by the radial profile of a function which can be taken to be the magnetic twist angle . For a flux tube of length L, the twist angle measured from to is given by the following expression:

with and the azimuthal and axial components of the equilibrium magnetic field. In this study, we adopt localized twist profile as shown in Fig. 2. The resulting axial profile of the current density is also plotted in Fig. 2. The current density appears to be mainly confined within .

Fig. 2. The twist profile as a function of the normalized radius (in arbitrary units), and the resulting axial component of the current density

Fig. 3. The linear growth rate for an aspect ratio value , as a function of the twist

Fig. 4. The critical twist value as a function of the aspect ratio

3.2. Stability

For a given loop length, the equilibrium is kink unstable when the twist exceeds a critical value (Hood & Priest 1979; Einaudi & Van Hoven 1983; Mikic et al. 1990; Velli et al. 1990; Baty & Heyvaerts 1996). Indeed, in Fig. 3, we have plotted the linear growth rate of the ideal kink mode as a function of the twist angle , for an aspect ratio . is the maximum twist value, obtained near , as can be seen in Fig. 2. Each point has been obtained for a vanishing viscosity (a dissipation time is used), and after optimization of the time step and the radial grid resolution. Then, we have investigated the dependence of the critical twist value as a function of the aspect ratio of the loop. The results are plotted in Fig. 4, showing a good agreement with the typical dependence found by Hood & Priest (1979).

3.3. Linear properties of the kink mode

The linear structure of the kink mode can be said to be fairly well established (Mikic et al. 1990; Velli et al. 1990; Baty & Heyvaerts 1996). At the axial midplane of the loop, the mode takes the form of a cellular convection pattern with a continuous radial displacement. As far as the axial structure is concerned, the mode amplitude vanishes at the photosphere and balloons out with a maximum value at the axial midplane. Contrary to periodic geometry (Rosenbluth et al. 1973), resonant magnetic surfaces are absent due to the line tying effect. As a consequence singular layers are suppressed. The only "resonant point" is a point located at the axial midplane of the loop where the radial component of the perturbed magnetic field vanishes, and corresponds to a resonance condition ( being the local wave vector of the mode and the equilibrium magnetic field). It has been shown that this point coincides with the location where the current concentration develops during the further evolution of the configuration (Baty & Heyvaerts 1996).

© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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