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

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2. Physical model

As pointed out in Sect. 1, we restrict our attention to the usual cylindrical geometry approximation. As seen in Fig. 1, the solar loop is modelled by a flux tube of length L, with foot points of magnetic field lines anchored in the photosphere, at two end-plates at [FORMULA]. To start the numerical simulation, we have used magnetic equilibria twisted locally by the photospheric flow. Adding an arbitrary small [FORMULA] perturbation (m being the azimuthal mode number), we followed the dynamic evolution of the system on a fast time scale as compared with the slow photospheric time scale. The coronal perturbations are forced to vanish at the ends of the loop because of the inertial anchoring that gives rise to the line-tying conditions. We impose all the components of the perturbations vanish at the photosphere, i.e. we use the usual rigid plates conditions (Velli et al. 1990a). The photosphere is supposed to have a finite axial extent [FORMULA], to simplify the numerical approach. More details about the effects of the numerical extent of the photospheric part are given by Baty & Heyvaerts (1996), who found that the results are not sensitive to the exact value of [FORMULA] as long as it remains large enough compared to the smallest axial scale length included in the simulation.

[FIGURE] Fig. 1. Cylindrical model of the twisted flux tubes. The corona extends beween [FORMULA] and [FORMULA], and the photosphere has an axial extent [FORMULA]

An arbitrary radial boundary condition is also assumed, which is a perfectly conducting wall placed at [FORMULA] in order to simplify the computation. The position of this wall will produce negligible effects provided the wall is far enough away from the loop axis (Hood & Priest 1981; Craig et al. 1988; Mikic et al. 1990).

2.1. Equations

Numerical computations are carried out using our code, SCYL, which solves the following full set of compressible and dissipative MHD equations, which can be written as (in non dimensional form):

[EQUATION]

Here, [FORMULA] is the mass density, P the plasma pressure, [FORMULA] the fluid velocity, [FORMULA] the magnetic field, [FORMULA] the electric current density. [FORMULA] and µ are the magnetic diffusivity and the kinematic viscosity respectively, and [FORMULA] is the ratio of specific heats (a value of 5/3 is used). The Lundquist number S, which is the ratio of the diffusion time [FORMULA] to the Alfvén time [FORMULA], is [FORMULA], [FORMULA] being the value of the resistivity on the magnetic axis. The energy equation is as simplified as possible, describing only energy convection (Eq. (3)) because the aim of the simulation is primarily to understand the dynamics. It would be easy to modify Eq. (3) to include thermal conductivity, and we plan to do so in future work.

The variables are normalized as follows:

[EQUATION]

where a is the radius of the flux tube, [FORMULA] the density at the radial boundary [FORMULA] and [FORMULA] a reference magnetic field value given by [FORMULA] ([FORMULA] being the magnetic field value on the axis and [FORMULA] the full length of the cylinder, [FORMULA]).

Again, we discuss only ideal MHD dynamics (i. e. [FORMULA]).

2.2. The numerical procedure

The MHD equations (1-5) are integrated in time with an efficient semi-implicit method allowing large time steps, limited by the non linear physical plasma phenomena, and ensuring a high spatial resolution (Lerbinger & Luciani 1991). The scheme is linearly fully implicit, and the stability is ensured by adding a small elliptic operator in the non linear phases. This elliptic operator dominates only the non linear magnetic perturbation, and is much smaller in size than elliptic operators usually used in semi-implicit methods where it has to dominate the equilibrium magnetic field contribution to ensure numerical stability. The diffusive terms are treated fully implicitly, and a second order predictor-corrector scheme has been developed applying the semi-implicit operator both at the predictor and corrector levels.

The periodic version of the code, XTOR, has been used previously to study the non linear magnetic reconnection process associated with the [FORMULA] internal kink instability in tokamak, in cylindrical geometry (Baty et al. 1991) and in toroidal geometry (Baty et al. 1992, 1993). The line-tied cylindrical version, XCYL, has proved its ability to simulate long time MHD evolution together with a high spatial resolution in a line-tied solar context (Baty & Heyvaerts 1996).

Cylindrical coordinates (r, [FORMULA],z) are used, with r the radial coordinate (labelling equilibrium magnetic surfaces), [FORMULA] the azimuthal angle, and z the axial coordinate. Radially, we use finite differences on two staggered meshes. Variables are expanded in double Fourier series in [FORMULA] and [FORMULA] ([FORMULA]) and operations are performed using a fast Fourier transform (FFT). We use 100 grid points in the radial direction with the possibility of accumulation in the vicinity of current layers. For the other directions, [FORMULA] grid points are used in [FORMULA].

In order to get a precise description of the modes, we adopt a "de-aliasing" procedure in the axial direction as well as in the [FORMULA] direction. Indeed, when FFT's are used, "aliasing" errors introduced by the discrete transforms quickly degrade the solution, especially in the higher harmonics (Aydemir & Barnes 1985). In this work, we use up to 7 modes, each with a band of 80 axial harmonics centered around the fastest growing axial harmonic (Einaudi & Van Hoven 1983).

At the outer perfectly conducting boundary wall at [FORMULA], we imposed a vanishing radial component of the velocity together with the usual electromagnetic boundary conditions on the electric and magnetic fields.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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