## 2. Physical modelAs 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
An arbitrary radial boundary condition is also assumed, which is a perfectly conducting wall placed at 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. EquationsNumerical 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): Here, is the mass density, The variables are normalized as follows: where Again, we discuss only ideal MHD dynamics (i. e. ). ## 2.2. The numerical procedureThe 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 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 ( 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 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 , we imposed a vanishing radial component of the velocity together with the usual electromagnetic boundary conditions on the electric and magnetic fields. © European Southern Observatory (ESO) 1997 Online publication: July 8, 1998 |