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Astron. Astrophys. 331, 392-404 (1998)

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1. Introduction

The linear MHD spectrum of a system consisting of a flux tube and surroundings has been investigated before, with various assumptions and simplifications. The thin tube approximation is a regularly used approximation. It enables to consider other generally excluded phenomena as gravity (Defouw 1976; Roberts & Webb 1978) or complex frequencies (Roberts & Webb 1979; Spruit 1982). Papers which admit arbitrary radius include Wentzel (1979), Edwin & Roberts (1983), and Meerson et al. (1978). Wentzel (1979) and Edwin & Roberts (1983) are concerned with real [FORMULA]. While Meerson et al. (1978) allows for complex [FORMULA], but makes other restrictive assumptions. Cally (1986) allows for complex [FORMULA] and calculates the leaky and non-leaky modes for uniform tubes with arbitrary radius. Other references where external propagating solutions were allowed include Cadez & Okretic (1989) for a double step profile and Rae & Roberts (1983) for slab and interface.

A leaky mode is characterized by an external solution which carries energy away from the tube. These modes are damped due to this acoustic wave leakage into the surroundings and have therefore a complex eigenfrequency. Due to this damping in time, the eigenfunctions show a growth in the external solutions with distance from the tube. Amplitudes at large distances correspond to earlier and therefore larger amplitudes within the tube.

In this paper gravity is neglected. The tube radius is arbitrary and [FORMULA] is regarded as being complex, so that there can be outward travelling external waves. The method used in this paper to find eigenmodes gives us the opportunity to extend the investigation of leaky and non-leaky oscillations done by Cally (1986) to inhomogeneous flux tubes. His results for homogeneous tubes will serve thereby as a testcase for our method.

The wave fields inside and outside the flux tube can be divided into an exciting and scattered and a transmitted part. Or else in a radially incoming, radially outgoing and a transmitted field. For each of these wave fields, an impedance can be defined at the tube boundary. This (normal acoustic) impedance is the ratio of the total linear pressure perturbation and the normal velocity perturbation and is defined for a given frequency. The acoustic impedance of a medium contains information on the allowed perturbations in the medium, since it prescribes how the amplitude ratio and the phase difference between the linear pressure perturbation and the velocity field vary in space.

The boundary conditions for the total pressure perturbation and the normal velocity component may be transformed into an impedance matching criterion which we can use to select eigenfrequencies by assuming that there is no incoming, exciting or driving field. Analogously we can find the optimal driving frequencies which are defined by the condition that when the system is driven at this (complex) frequency the incoming wave is totally absorbed by the tube. The spectrum of optimal driving frequencies is produced by inserting the assumption that there is no scattered field. These impedances and the matching criteria will be addressed in Sect. 2.

A connection between these optimal driving frequencies and the eigenfrequencies of the system was established quite recently (Goossens & Hollweg 1993; Keppens 1996). Keppens drew the attention on the difference between the real driving frequency at which maximal (not necessarily 100%) absorption occurs and the complex optimal driving frequency for which total absorption is found. Eigenfrequencies, real maximal driving frequencies and complex optimal driving frequencies are all closely related. The relation between them is more transparent since Keppens (1996) formulated it in terms of impedances.

The solving for eigenfrequencies and optimal driving frequencies will be done numerically. We therefore extend the simple numerical scheme, discussed in Stenuit et al. (1995), with the impedance matching criteria.

When we consider inhomogeneous tubes, resonant absorption may occur. The spatial variation of the equilibrium quantities induces an Alfvén and a slow continuum. When the system is driven by an impinging wave with a frequency lying in one of these continua, (a) singularity(ies) occur(s). This implies very large gradients to build up towards an infinite amplitude which can only be stopped by any sort of dissipation, which converts some of the energy into heat.

The simple scheme mentioned above treats the possible resonances by the use of the SGHR method (Sakurai et al. 1991; Goossens et al. 1995). This method derives its computational simplicity from the fact that it circumvents the numerical integration of the full dissipative equations. The method is based on jump conditions over the dissipative layer surrounding the resonance(s). These jump conditions are obtained from an asymptotic analysis of analytical solutions to simplified versions of the linear non-ideal MHD equations in this dissipative layer. The equations and the treatment of the resonances are explained in Sect. 3.

Sect. 4 gives the comparison of the results for uniform tubes with the results of Cally (1986). The results for inhomogeneous tubes are discussed in Sects. 5 and 6 respectively. Sect. 7 summarizes the main findings.

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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998
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