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

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5. Discussion

We have presented full three dimensional MHD simulations of the non linear evolution of the ideal [FORMULA] kink mode in cylindrical line-tied coronal loops. In particular, we have provided a study of the characteristics (thickness and amplitude) of the current concentration, which is known to develop when the kink instability drives the initial magnetic configuration towards a secondary bifurcated MHD equilibrium (Baty & Heyvaerts 1996).

Firstly, we have investigated the effect of the aspect ratio of the loop. At the axial midplane of the loop, the simulation results have shown a decreasing linear-like scaling of the thickness of the current layer as a function of the aspect ratio [FORMULA], for values of [FORMULA] in the range 2-10. The amplitude of the corresponding current density is correspondingly linearly increasing. A simple model is proposed to interpret this, with rather good agreement with the numerical results. Indeed, the line-tying effect leads to an axial field line bending which depends on the field line considered and prevents the formation of a discontinuity of the magnetic field. Then, contrary to the un-tied configuration where a genuine singular current sheet forms, the bifurcated kinked equilibrium is characterized by a force balance between sharp pressure gradients in the magnetic field and the curvature force due to the axial line-bending.

This effect is intimately linked to the absence of magnetic resonance in line-tied magnetic configurations and to the existence of a non-zero minimum of the Alfvén frequency (Goedbloed & Halberstaedt 1994) given by:


Indeed, one must note in Eq. (20) that the [FORMULA] term, which is also the line-bending term arising in the expression (17), determines the thickness of the current layer.

Our result is also consistent with the recent analytical work of Longcope & Strauss (1994). These authors have found that current sheets do not form in the framework of the line-tied ideal coalescence instability. They have shown that the line-tying constraint is inconsistent with a genuine singular current sheet, but allows the formation of current layers, the thickness of which is five or six orders of magnitude smaller than the initial scale length of the equilibrium. This thickness depends exponentially on the length L between the two photospheric axial boundaries, and vanishes in the limit of infinite length L. Applying the same arguments to the line-tied ideal kink instability, they predicted a linear dependence of the thickness of current layers on the length of the loop.

We have also investigated the effect of the initial equilibrium parameters: the shear of the initial equilibrium, and the pressure gradient effect. The results indicate only weak effects on the features of the current concentration as long as the plasma pressure is dominated by the magnetic one.

Given typical values of loop parameters in active regions, it can be seen that the amount of available magnetic energy stored is sufficient to feed the observed heating flux of [FORMULA]. Indeed, as the result of photospheric motions, the loop acquires an azimuthal magnetic field component [FORMULA] corresponding to a twist angle [FORMULA]. When [FORMULA] exceeds the critical value [FORMULA], the initial configuration is rapidly driven towards a kinked configuration where [FORMULA] gives an estimate of the magnetic field to be further dissipated. The corresponding stored magnetic energy is then:


and the rate of energy transfer to the loop is given by:


where [FORMULA] is the photospheric time scale necessary to induce the twist in the loop. The expression (22) leads to the average flux:


For [FORMULA] G, [FORMULA] m, [FORMULA] s, we get [FORMULA]. The problem is how to dissipate this energy quickly enough.

The subsequent conversion of the magnetic energy into heat by ohmic dissipation is a challenging aspect of the theory. Indeed, because of the smallness of classical values of the resistivity [FORMULA], the resistive dissipation power [FORMULA] needed to explain the observed heating rate requires extremely large current densities [FORMULA]. This corresponds to a magnetic field structured on a very fine scale of order of a few meters. For typical observed loops, a scale length of about five orders of magnitude smaller than the gross features of the magnetic configuration is necessary. With our adopted parameters, the results have indicated that the gradient length scale in the kinked configuration is only two or three orders of magnitude smaller than the initial magnetic configuration one. Such magnetic features certainly lead to insufficient ohmic dissipation when classical resistivity effect is considered. However, in this work, we have restricted our computations to aspect ratio smaller than 10, mainly because of memory storage constraint of the computer. Consequently, it is necessary to explore higher values of the aspect ratio in order to investigate if the linear-like scaling law remains valid or not, or to explore whether there exists a critical length beyond which singular current sheets do form. In this study, we have also used the cylindrical geometry approximation which seems reasonable for studying the stability of loops of large aspect ratio. However, the toroidal field line bending could change the non linear force balance given by the expression (12), eventually leading to a different scaling law. These aspects remain to be investigated. Moreover, when the current density exceeds a critical value, current-driven microinstabilities (Rosner et al. 1978; Beaufumé et al. 1992) are excited leading then to an anomalous resistivity enhanced by several orders of magnitude over its classical value.

Another interesting question concerns the nature of the ohmic dissipation. It is of particular interest to determine if a reconnection mechanism takes place, as is the case in periodic magnetic configurations. It is also possible that the diffusive processes could generate some kind of MHD turbulence. We plan to study the further resistive evolution of kinked line-tied magnetic structures in future work.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998