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Astron. Astrophys. 330, 726-738 (1998)

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6. Numerical model

The boundaries of the simulation box are placed at [FORMULA], [FORMULA], and [FORMULA]  L, with [FORMULA] km the half-width of the coronal arcade. The computational box is divided into a non-uniform grid of 300 cells in the x -direction (with total length [FORMULA] km) and 300 cells in the z -direction (with total length [FORMULA] km). The Alfvén speed at the base of the corona is taken to be [FORMULA] (see Eq. [14]), which yields the Alfvén transit time [FORMULA] s.

For all numerical simulations we use outgoing boundary conditions. Since our subject of interest is concentrated around impulsively generated waves inside the coronal arcade we need to implement outgoing boundary conditions at all four boundaries. Obviously, other choices of boundary conditions would be more appropriate for different physics studied. For example, ingoing boundary conditions at the bottom boundary would be suited for studying foot-point motions or phenomena associated with emerging flux. However, we limit ourselves to impulsively generated waves launched at a certain height and consider their evolution before they reach the photosphere, so our choice of outgoing boundary conditions is justified as a mean of reducing numerical errors and has no influence on the results.

It is worth mentioning here that the proper treatment of the outgoing boundary conditions at the edges of the simulation region is not a trivial aspect of time-dependent numerical simulations. The outgoing boundary conditions must not reflect any disturbances going out of the numerical domain and must pass information about the physical system. A simple method for imposing boundary conditions is to generate image cells at the boundaries of the simulation region. In this code one row of image cells was generated at each side of the simulation box.

It is not easy to avoid reflections at the boundary without using some special approach, such as a characteristics-based extrapolation. One method we have used with success is to stretch the grid near the outflow boundaries, and to apply damping to the quantities over several zones near the boundary. This takes the form


where [FORMULA] is the ambient (unperturbed) value of the variable, [FORMULA] is the value after the flux corrected step, and [FORMULA] is the damped value kept at the end of the damping process. The damping coefficient d can vary from some small value [FORMULA] for the cell next to the boundary to [FORMULA] over n cells to smoothly damp out the oscillations. For example, d can be chosen as 0.1 and n set to 9. In this case the outgoing signal would be damped over 9 cells and d would take the value 0.9 in the ninth cell.

This way of implementing boundary conditions has been found to work well. Tests were performed with a larger number of grid points and no significant differences were found in the solutions. This fact ensures grid convergence and that the accuracy of results is not compromised by the grid size. Therefore, we conclude that the solution is adequately resolved by the original mesh of [FORMULA] points.

In order to maintain high spatial resolution in particular regions, we employ a nonuniform grid in which the computational cells vary in size. The cell size changes gradually, so that neighbouring cell dimensions are stretched by only a few tenths percent per cell (typically [FORMULA]). Cell sizes were set in such a way that the finest resolution is in the centre of the simulation box. For this purpose the grid generation routine (GGEN) has been adapted from the ZEUS2D code (Stone & Norman 1992).

Initially, at [FORMULA], the magnetic field and density are set to the initial state which is given by Eqs. (8) and (11), respectively, and an impulse localised in space is launched at the position [FORMULA], [FORMULA]. This pulse is specified by giving the initial components of the velocity parallel [FORMULA] and normal [FORMULA] to the equilibrium [FORMULA]. The transformation from cartesian components to field-related components can be determined from


where [FORMULA] and [FORMULA] are the normal and parallel unit vectors.

The purpose of this study is to explore the temporal development of pulses localised in space which are initially (at [FORMULA]) launched in the arcade. Instead of perturbations with complex spatial variations, we consider perturbations which have a symmetric spatial distribution and are localised in both the x - and z -directions. We consider the pulse launched at [FORMULA] and [FORMULA]  L as a representation of the various positions of initial perturbations. In what follows we will focus on a specific subset of all possible perturbations of the coronal plasma: we will assume that these perturbations are excited by a localised impulse which acts only at [FORMULA] but whose consequences are felt at later times.

Within the mentioned subset of possible perturbations, it is therefore sufficient to study the following model pulse


where [FORMULA], [FORMULA]  L, and [FORMULA] have been chosen.

The magnetic scale height [FORMULA] is fixed and held constant throughout the simulations. We will change the amplitude [FORMULA] of the initial impulse and the ratio of magnetic to pressure scale height, [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998