## 4. The non linear development of the kink instabilityThe 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 resultsOur 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.
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.
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.
## 4.2. InterpretationIn a periodic geometry, the kink instability is linearly a
perturbation (
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 (
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 with
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 with a constant, which is the location of
the field lines at the photosphere, and 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
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
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 effectsFirstly, 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 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 © European Southern Observatory (ESO) 1997 Online publication: July 8, 1998 |