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Astron. Astrophys. 318, 289-292 (1997)

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3. Results of numerical modelling

As the initial magnetic field of a simplified active region (i.e. without computed currents), we take the dipole magnetic field:

[EQUATION]

[EQUATION]

[EQUATION]

where C is a constant, z0 is the position of the magnetic source below the photosphere (z0 = - 5 [FORMULA] 107 m in our case), r is the distance from the magnetic source position to the position where the magnetic field is calculated. The constant C is chosen to give the maximum of this field as 100 G. This maximum is situated in the bottom-surface centre of the computational box. Then using our model with the increasing electric current we computed force-free current paths for two cases: a) the current is represented by a single current path with [FORMULA] = 4 [FORMULA] m, and b) the current is represented by four current paths having the distances of 4 [FORMULA] m at injection positions with [FORMULA] = 2 [FORMULA] m. It means that in both cases the current cross-section area is 5 [FORMULA] m2. The distance between particles is 5 [FORMULA] 105 m, the particle "velocity" is 5 [FORMULA] 105 m s-1, and the "time" step is 1 s. The results for a single current path with the electric currents: 5 [FORMULA] 1010, 1 [FORMULA] 1011, 2 [FORMULA] 1011, and 3 [FORMULA] 1011 A are shown in Fig. 1. In computations the electric current was increased up to a specific value and then kept constant for some relaxation time. The corresponding magnetic field is included for an illustration. You can see strong deviations from the initial magnetic field. On the other hand, results with four current paths for total electric currents: 5 [FORMULA] 1010, 2 [FORMULA] 1011, 3 [FORMULA] 1011, and 3.5 [FORMULA] 1011 A are depicted in Fig. 2.

[FIGURE] Fig. 1a-d. The single electric current path (thick line) and magnetic field (thin lines) for I = 5 [FORMULA] 1010 A a, I = 1 [FORMULA] 1011 A b, I = 2 [FORMULA] 1011 A c, and 3 [FORMULA] 1011 A d. The distances are expressed in 104 km.
[FIGURE] Fig. 2a-d. The four electric current paths for [FORMULA] = 5 [FORMULA] 1010 A a, [FORMULA] = 2 [FORMULA] 1011 A b, [FORMULA] = 3 [FORMULA] 1011 A c, and 3.5 [FORMULA] 1011 A d. The distances are expressed in 104 km.

In both cases, for low electric currents the current path is close to the dipole magnetic field line (Fig. 1a), while for higher currents the current path becomes sheared (Fig. 1b) and then the current path is screwed into the helical structure (Fig. 1c). Simultaneously with the current increase the current paths move upwards (Fig. 1 and 2).

We studied the stability of computed structures. It was found that the configurations in Figs. 1a, 1b, and 2a, 2b, are quite stable in time. As concerns Figs. 1c and 2c, the global arch is stable, the helical structure keeps its form, but it shows some oscillatory motion along the arch axis. On the other hand, those in Figs. 1d and 2d are unstable, their internal structure is strongly variable, the global shape shows some stability, distances between the particles do not remain the same, i.e. the correct description of the electric current is disturbed. It seems that there is some critical electric current, above which no force-free electric current structures exist.

Comparing the single current path case with that of four paths, we can see that the global features remain the same. It can also be seen that the stability of the current paths is better in the four path case in comparison with the single path case.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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