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Astron. Astrophys. 342, 300-310 (1999)

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1. Introduction

Magnetohydrodynamic (MHD) surface and body waves are known to be important in laboratory plasmas, in magnetospheric physics and in astrophysics (Lanzerotti et al. 1973; Chen & Hasegawa 1974; Lanzerotti & Southwood 1979; Takahashi & McPherron 1984; Bertin & Coppi 1985; Musielak & Suess 1988; Roberts 1991; Goedbloed & Halberstadt 1994; Goossens 1994; Narain & Ulmschneider 1996; Poedts & Goedbloed 1997; Poedts et al. 1997). The observations of solar magnetic field structures, such as magnetic flux tubes, sunspots, coronal loops and coronal holes, indicate that discontinuities exist on the Sun (Stenflo 1978; Zwaan 1989; Solanki 1993). These discontinuities can support MHD surface waves; the role of these waves in chromospheric and coronal heating has been extensively explored in the literature. It has been suggested that propagating MHD body and surface waves may supply large amounts of energy from the subphotospheric layers (generated there by turbulent convection) to the upper atmospheric layers. This wave energy is thought to be dissipated in the coronal plasma by mode-coupling, resonance as well as turbulent heating in the case of body waves and through resonant absorption and phase-mixing in the case of surface waves (see Narain & Ulmschneider 1996, and references therein).

Previous studies have been primarily concerned with three different models of structured magnetic field configurations, namely, magnetic interfaces, magnetic slabs and magnetic flux tubes. In this paper, the propagation of MHD waves along magnetic interfaces and slabs is considered, however, see Spruit (1981), Spruit & Roberts (1983), Herbold et al. (1985), Ulmschneider et al. (1991), and Ziegler & Ulmschneider (1997a, b) for discussions of the propagation of MHD waves along magnetic flux tubes. For linear MHD surface waves on a single magnetic interface, the resulting dispersion relations have been derived and studied by Roberts & Webb (1978), Wentzel (1979) and Roberts (1981a). The studies show that on such magnetic structures, body waves can either be longitudinal or transverse depending upon the type of perturbation imposed, and that surface waves can be either slow or fast depending on the relative magnitude of the temperatures on both sides of the interface. Time-dependent analytical solutions of the initial value problem for linear MHD surface waves on a single magnetic interface have been found by Lee & Roberts (1986) for the case of an incompressible background medium.

The propagation of MHD surface and body waves in magnetic slabs is much more complicated than that on a single magnetic interface due to the richness of modes that can exist in such magnetic field structures (Roberts 1981b), and due to the fact that these slabs can effectively interact with the external medium. The latter means that a two-dimensional treatment is required and, as a result, the mathematical description may become too complicated to obtain analytical solutions even for simple physical situations. Hence, in most cases a numerical approach will be necessary. Such an approach was developed by Wu et al. (1996) who investigated the propagation of linear MHD body and surface waves along magnetic slabs embedded in an unstratified medium by using a two-dimensional, time-dependent numerical model. Their approach did not allow, however, for large amplitude perturbations.

To understand the behavior of nonlinear MHD body and surface waves, it is necessary to incorporate both the nonlinearity and the magnetic field discontinuities in the numerical model. Such a model has been developed and is described in this paper (see Sect. 2; also Huang 1995, 1996). The model allows simulating both linear and nonlinear MHD body and surface waves in the presence of background magnetic field discontinuities, however, without gravity. The numerical model is implemented as two-dimensional and can incorporate either single or multiple magnetic interfaces such as a magnetic slab or multiple magnetic slabs with any type of magnetic field variation inside the slab. There are no gradients in the physical quantities other than the presence of discontinuities in the background magnetic field and in the gas pressure to satisfy the pressure balance across these discontinuities. The considered numerical model is based on ideal MHD, which means that there is no wave energy dissipation. The numerical procedure and two tests performed to verify it are described in Sect. 3. The numerical simulations with their physical interpretation of our three model cases are discussed in Sects. 4 to 6. The conclusions are summarized in Sect. 7.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998