## 1. IntroductionThe recently observed standing waves in the chromospheric and low transition regions of the sun by Bocchialini & Baudin (1995) provide new support for wave heating theories of the solar corona. Like all proposed heating mechanisms of the solar corona, see the review by Zirker (1993), magnetic wave heating depends on the creation of small length scales to obtain effective magnetic diffusion times. In wave heating of closed loop systems, the creation of these small length scales is tightly connected to another important phenomenon: resonances. These resonances occur due to the large density gradients in the low chromosphere and photosphere, which effectively turn coronal loops into resonant cavities (Hollweg 1984). The resonances yield standing waves which are dissipated by non-ideal mechanisms and, hence, heat the plasma. The conversion of magnetic energy into heat is efficient for waves with frequencies that lie within certain intervals, the so-called continuum frequency ranges. Resonant absorption was first introduced as a possible heating mechanism for the solar corona by Ionson (1978). Since then many papers on the subject have appeared; see, e.g., Kuperus et al. (1981), Heyvaerts & Priest (1983), Davila (1987), Grossmann & Smith (1988), Poedts et al. (1989), Poedts & Kerner (1992), Halberstadt & Goedbloed (1995a, 1995b), Ofman et al. (1995), Poedts & Boynton (1996), Ofman & Davila (1996). In linearized ideal magnetohydrodynamics (MHD) two types of continua exist which are commonly referred to as Alfvén and slow magnetosonic although this classification is, strictly speaking, only valid for one-dimensional equilibrium configurations. Because each magnetic surface has its own characteristic set of discrete frequencies, due to the assumed inhomogeneity of the equilibrium, corresponding sets of continua are traced out when one scans over these surfaces. The continuum eigenoscillations are singular at the resonant surfaces. It is really this singular behavior that is responsible for the heating mechanisms based on resonant absorption of Alfvén waves. When non-ideal effects are taken into account, continua no longer exist and they are replaced by closely spaced damped discrete eigenmodes with frequencies located in the ideal continuum ranges. These modes still possess very large gradients which are responsible for short magnetic diffusion times and, hence, for efficient heating. Continua and the associated singular behavior of magnetic field lines and surfaces have been the subject of a lot of research (see, e.g., Tataronis & Grossmann 1973, Chen & Hasegawa 1974, Goedbloed 1975, Krylov et al. 1979, Hameiri 1981, Krylov & Lifschitz 1984, Southwood & Kivelson 1986, Mond et al. 1990, Poedts & Goossens 1991, Halberstadt & Goedbloed 1993, Goedbloed & Halberstadt 1994) and the results have found applications not only in coronal but also in nuclear fusion and magnetospheric contexts. In most studies of MHD wave activity and heating, the effects of the huge difference in the physical properties between the corona and the underlying chromosphere and photosphere are lumped into the boundary conditions. The last decade has seen a continuing debate about the precise form of these boundary conditions. The debate is mainly about whether line-tied or flow-through boundary conditions are the most appropriate ones. Line-tied boundary conditions correspond with rigid wall conditions, i.e., all velocity components vanish at the endpoints, whereas flow-through boundary conditions only prohibit motions perpendicular to the magnetic field. Recently, however, it has been recognized that both types of boundary conditions may be too severe and that modeling of the underlying atmosphere is important. See, for example, Van der Linden et al. (1994) who studied the influence of the stratification of the solar atmosphere on the stability of magnetic structures and who showed that near marginal stability both types of boundary conditions are violated. Mok & Van Hoven (1995) have performed interesting numerical calculations of the dynamical properties of a realistic thermal-structure interface between a coronal loop and the chromosphere/photosphere and qualitatively confirmed the discontinuous density and temperature model of Hollweg (1984). The aim of this paper is to describe the influence of the chromospheric/photospheric layers on the continuous spectrum of coronal loops. This is of fundamental interest for wave heating theories of the solar corona. Inclusion of the chromospheric/photospheric layers automatically leads to two-dimensional equilibria. Of special interest is the coupling between Alfvén and slow magnetosonic continuum waves induced by this two-dimensionality. Poedts et al. (1985) and Poedts & Goossens (1991) were the first to study two-dimensional continua in coronal context. We derive the equations governing the continuous spectra for two-dimensional equilibria including gravity in general curvilinear coordinates. Next, this set of equations is solved numerically for coronal magnetic flux tubes with varying cross-sections and with the ends located in the high-beta, flux concentrated chromospheric/photospheric regions. This calculation requires an accurate, explicit, two-dimensional equilibrium of the expanding flux tube. Such equilibria are obtained by means of the numerical equilibrium program PARIS, introduced in a previous paper (Beliïn et al. 1996b). This paper is organized as follows. In Sect. 2, the model of a coronal magnetic flux tube is described. In Sect. 3, the equations for the continuous spectrum in such tubes are derived and the properties of these equations are discussed. In Sect. 4, we show some typical results. Finally, in Sect. 5 our conclusions are formulated. An appendix is devoted to the derivation of the equations for the continuous spectrum. © European Southern Observatory (ESO) 1997 Online publication: June 5, 1998 |