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

4. The non linear development of the kink instability

The initial magnetic equilibrium is expected to be driven towards a secondary bifurcated MHD equilibrium by the ideal kink (Baty & Heyvaerts 1996). Therefore, we have used a numerical procedure where viscous dissipation has been progressively added to rapidly find this kinked equilibrium, avoiding numerical oscillations. Typically, this artificial dissipation time was at the beginning of the simulation, and at the end. We have selected 7 unstable magnetic equilibrium configurations corresponding to aspect ratios between 2 and 10. In order to simplify the interpretation of the results, we have chosen the value of the twist in order to have cases with linear growth rates of order .

4.1. Simulation results

Our results show the development of a current concentration during the non linear development of the kink instability. At the axial midplane of the loop, this current concentration is localized at the "resonant point" described above. And, it extends all along the loop, in the form of an helical ribbon of intense current. Its radial location and its amplitude exhibit an axial modulation along the loop because of the line-tying effect. The typical structure of the axial component of the current density can be seen in Fig. 5.

Fig. 5. The axial component of the current density for the two axial planes: a and b , and the corresponding iso-contours: c for and d for

The secondary bifurcated equilibrium is considered to be reached when the flow associated with the instability is fully dissipated by viscosity, and when the displacement of the magnetic axis has stopped. A typical time evolution (for the case ) of the radial location of the magnetic axis as a function of time is plotted in Fig. 6. In this Figure, by comparing the values for and , one should also note the axial modulation of the displacement.

Fig. 6. The time evolution of the normalized radial location of the magnetic axis, for three axial planes , , and

Fig. 7. The amplitude of the normalized current concentration (circles) and the normalized thickness of the current layer (squares) as a function of the aspect ratio L / a, measured at the axial midplane

We have measured the maximum amplitude of the current concentration , which is reached at the axial midplane of the loops for the bifurcated equilibria. The results are plotted as a function of the aspect ratio in Fig. 7, showing an increasing linear-like dependence. We have also measured the radial thickness of the current layer, which appears to have a decreasing linear-like behaviour. We stress that it is necessary to have a good enough spatial resolution in the computations, in order to be able to distinguish between a genuine current sheet and an intense current layer of non-zero thickness. We have been able to do so by using a radial accumulation with a grid resolution of order . The complicated three dimensional current layer structure makes it difficult to measure its thickness for the other axial midplanes with . Fortunately, the amplitude of the current can be precisely measured, and typical axial variations are plotted in Fig. 8. Let us now turn to the interpretation of these results.

Fig. 8. The amplitude of the normalized current concentration as a function of the axial coordinate z, for the case

4.2. Interpretation

In a periodic geometry, the kink instability is linearly a perturbation (n being the axial mode number) which is associated to a quasi-rigid shift of a central column. The non linear evolution of the ideal kink mode leads to a secondary kinked MHD equilibrium containing an helical singular current sheet located at the magnetic surface. This singularity corresponds to a discontinuity in the magnetic field and appears as a negative current spike with a vanishing thickness (or equivalently an infinite amplitude) (Park et al. 1980; Rosenbluth et al. 1973). In particular, the kinked configuration is characterized by a discontinuity at the radius location of the helical field component , which is defined as the projection of the magnetic field on the wave vector . Then, at the magnetic surface, the lines of force have the same pitch as the kink perturbation. A simplified version of this configuration in slab geometry is shown in Fig. 9 (a) and (b), replacing by x and by . When crossing the surface, the helical field undergoes a jump, the other components of the magnetic field being approximately constant. Although this discontinuity induces a singularity in the current density, the equilibrium is achieved because the jump is antisymmetric (Waelbroeck 1989).

Fig. 9a. A simplified slab un-tied magnetic configuration in the vicinity of

Fig. 9b. The corresponding variation of the component as a function of x

It has been seen that the essential difference between line-tied and periodic geometries is an axial modulation, sometimes called a ballooning effect since it bears some resemblance with ballooning instabilities in tokamaks. In Fig. 10, we have plotted typical projections of the field lines on the (r,z) plane, as obtained in our simulations for a kinked equilibrium. One can observe the curvature of the perturbed magnetic field, due to the line-tying effect. As the radial location of the current concentration is at in this case, the magnetic field appears to be mostly curved in the innermost part before the current layer, and the curvature radius varies with the field line considered (i.e. there is a "diffuse" curvature effect). Now, if one simply superposes this diffuse curvature effect to the simple model shown in Fig. 9, a line-tied model in slab geometry can be deduced as illustrated in Fig. 11a. In the vicinity of the current layer and near the axial midplane of the loop, this configuration would be in radial equilibrium if:

Fig. 10. The projections of different magnetic field lines in the (r,z) plane for an aspect ratio value , the axis is represented by the hatched line

Fig. 11a. A simplified slab tied magnetic configuration in the vicinity of

The main effect is to prevent the formation of the discontinuity in the magnetic field component , and to lead to an equilibrium between gradients in and curvature force in the vicinity of the layer essentially in a region between and . In order to characterize the configuration in more details, in the above described region, we assume a simple linear dependence with x of the magnetic field component :

with A to be determined later (see Fig. 11b). The magnetic field component can be obtained from the field line equation:

Fig. 11b. The corresponding variation of the component as a function of x

Then, replacing Eqs.(10) and (11) in Eq.(9), one obtains:

where has been assumed constant in this region. The expression (12) can be re-written as:

with the local curvature radius defined by:

which depends on the field line considered (see Fig. 10). Now, applying the force balance given by Eq. (13) at the middle of the region (i. e. for ), one obtains:

which gives an expression for the length scale of the current gradient. To facilitate comparison with the simulation results, we assume a simple sinusoidal form for the field lines in the x -z plane:

with a constant, which is the location of the field lines at the photosphere, and C an amplitude factor which depends on the field line considered. At the axial midplane of the loop (), the curvature radius can then be reduced to the following expression:

and the expression (15) can be written:

We have measured the curvature radius (at the axial midplane , and in the middle of the maximum current gradient) as a function of the aspect ratio of the loop in the simulations. The results are plotted in Fig. 12, and show very good agreement with the quadratic dependence given by the expression (17) if one takes . This value of the amplitude also corresponds very well with the value of the radial displacement, as represented in Fig. 6. Then, we have measured the radial thickness e in our simulations using the following definition:

with the maximum value of the initial equilibrium current density. The results are plotted in Fig. 13, together with evaluations using Eq. (18), estimating in normalized units. The physical meaning of A is given by the shear of field lines for the kinked equilibrium. We have checked that its value is of order of the above estimate, whatever the aspect ratio. Indeed, the value of A is mainly determined by the quantity of unstable magnetic energy, or equivalently by the growth rate which was taken to be approximately the same for all the simulations. Therefore, we have obtained a rather good agreement between the results of the simulations and the above simple model. The resulting current amplitudes displayed in Fig. 7 can also easily be explained by making the approximation . It is important to note that the quadratic dependence of the radial thickness of Eq. (18) is weakened by the slow increase of as a function of the aspect ratio, because of the increase of the critical twist (see Fig. 4). Then, at the axial midplane of the loop, the scaling law of the characteristics (thickness and amplitude) of the current concentration as a function of the aspect ratio is an algebraic linear-like dependence.

Fig. 12. The normalized curvature radius at at the axial midplane as a function of the aspect ratio value , measured in the simulations (circles), and values obtained using Eq. (17) with (plain line)

Fig. 13. The normalized thickness of the current layer at the axial midplane , measured in the simulations (circles), and calculated using Eq. 17 (squares) with and

As concerns the current gradients obtained at other axial planes (), we must emphasize the difficulty of obtaining a simple expression like Eq. (18) in order to describe the thickness of the current layers. However, as can be seen in Fig. 8, current gradients are less intense than in the axial midplane, and their axial variation can be described by a simple sinusoidal modulation.

4.3. Initial equilibrium profiles effects

Firstly, we have investigated the effect of the linear growth rate on the kinked equilibrium structure. When the linear growth rate is increased, the main effect is to increase the value of A, and then to lead to a smaller thickness in correlation with a higher amplitude of the current concentration. We must point out that in this study, we have chosen a given value of the linear growth rate of order , as it corresponds to a characteristic time scale for the formation of current gradients sufficiently rapid compared to the typical photospheric time scale.

Secondly, we have used equilibria with different twist profiles. Particularly, we have varied the shear in the centre of the flux tube. We have observed a weak effect. The results are only changed by 20 percent when the shear is increased or reduced by a factor of two. Indeed, the effect of the initial shear is mainly to lead to a different field line bending in the region considered. In the simple model considered in this study, we have neglected the field line bending, assuming the bending to be dominated by the axial one, due to the line-tying effect. Therefore, this result is in agreement with our earlier assumptions.

Finally, we have considered non force free initial magnetic equilibria with a gradient pressure term. The main effect is to give an additional pressure contribution for the radial force balance in Eq. (18). The simulation results have shown a weak effect when the plasma pressure is much smaller than the magnetic pressure. Hood et al. (1994) have shown that MHD instabilities with higher m mode numbers can be destabilized when pressure becomes important. However, these instabilities are beyond the scope of this paper, as we focus on low-beta kink instabilities.

© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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