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

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1. Introduction

Since the X-ray and UV-imaging made by the Skylab mission in 1973, it has become clear that the solar corona is highly structured (Rosner et al. 1978). Recent observations of the X-ray emission from the solar corona by the Yohkoh mission (Acton et al. 1992) and by white-light from the eclipse of 1991 (November & Koutchmy 1996) have revealed the complex nature of this region in finer detail. In particular, these observations show that the solar corona is permeated by a magnetic field which can be structured in coronal arcades with their footpoints tightly rooted in the much denser photosphere. Coronal arcades support waves which can be excited by e. g. localised perturbers (ade et al. 1995).

The study of wave propagation in coronal arcades has an intrinsic importance stemming from our wish to understand the physical processes that occur in an arcade, and the possibility that such a study may supply information about the wave heating of the solar corona (e. g., Musielak 1992; Zirker 1993). It is natural to believe that including the curved magnetic field topology in a coronal model is an important ingredient towards an understanding of coronal heating. In the context of wave heating theories, one expects a localised source of energy, such as a flare or instability, to shake the bottom of an arcade so that MHD waves will be launched upwards into the coronal part of the arcade.

Wave propagation in coronal arcades is complex because of the nonuniformity of the plasma and the curved magnetic field topology. As a consequence of that, the first analytical treatments were concerned with simple structures with straight magnetic field lines. Roberts et al. (1984) studied impulsively generated waves in coronal loops represented by density enhancements. Considering a top hat density profile, they showed that an impulsively generated fast sausage wave exhibits three phases in its temporal signature. On the other hand, impulsively generated nonlinear waves in solar coronal loops represented by slabs of enhanced density, with arbitrary plasma- [FORMULA], have been studied numerically by Murawski & Roberts (1994) and Murawski (1994). The results show that the total reflection which occurs in the region of low Alfvén speed leads to trapped waves. These waves propagate along the slab and exhibit the three phases in their temporal signatures; a periodic phase is followed by a quasi-periodic phase and finally by a decay phase. The results of the numerical simulations are in general agreement with the analytical predictions of Roberts et al. (1984).

A rigorous treatment of wave propagation in a curved magnetic field geometry has been performed by Oliver et al. (1993), Oliver et al. (1996), and Smith et al. (1997), who obtained the spectrum of MHD waves in potential and non-potential coronal arcades. Also, combined Fourier and Laplace transformations have been used by ade et al. (1995) to solve the two-dimensional initial value problem of wave excitation in a coronal arcade. In particular, this approach allowed them to conclude that high-frequency perturbers of fast magnetosonic waves disturb only the region just above the perturbers. Low-frequency perturbers, on the other hand, are able to disturb the whole coronal magnetic configuration above the active region (ade et al. 1995).

Although much attention has been paid to the propagation of linear waves in arcades (e. g., Oliver et al. 1993; ade & Ballester 1995a; ade et al. 1995), so far there has been no satisfactory in depth study of the propagation of nonlinear waves. The need for time-dependent calculations in coronal arcades is stimulated by observations and theory which show the existence of time-dependent behaviour. In this paper we want to study the transient phenomena which are excited by a localised motion in an initially static coronal arcade. We will do so by examining the time evolution of plasma parameters described by the nonlinear MHD equations. As a consequence of the vertical inhomogeneity in the plasma profiles and the curved topology of the magnetic field, this problem does not seem to be amenable to analytical treatment. In the present paper we seek to remedy this shortfall by carrying out numerical simulations. The development of supercomputers makes detailed and accurate calculations with relatively short calculation times possible. Such treatment is now feasible as the increasing power of modern computers makes realistic numerical experiments and simulations accessible. There are also powerful numerical techniques suitable for this purpose. One of them is used in this study.

The format of the paper is as follows. The physical model used in the present study is described in the next section. The equilibrium plasma variables and the linear wave equation are presented in Sects. 3 and 4, respectively. The numerical simulations of MHD equations are described briefly in Sect. 5 and the numerical model is shown in Sect. 6. We present and discuss our detailed results in Sect. 7. The paper concludes with a short summary and conclusions.

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© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998
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