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Astron. Astrophys. 343, 615-623 (1999)

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1. Introduction

Quiescent solar prominences are cool, dense clouds in the solar corona which are suspended against gravity by the curvature-forces of magnetic fields penetrating them. The lifetimes of quiescent prominences are up to a few month which justifies the theoretical modeling on the basis of mechanical and thermal equilibria.

A local description of the mechanical properties was first given by Kippenhahn & Schlüter (1957), who gave the intuitive picture of a magnetic hammock. In order to get a global model for a prominence it is necessary to extend this local description to a field structure which is rooted down in the solar photosphere. There are mainly two different types of models which are characterized by the directions of the transverse magnetic field in the prominence and the photosphere. The same direction corresponds to the normal polarity models, the opposite direction to the inverse polarity models, where the prominence rests in a current sheet above a X-type neutral point. A model of the latter case was first proposed by Kuperus & Raadu (1974). Observational data gives evidence, that the normal polarity prominences are mainly prominences with heights less than about [FORMULA], whereas the higher prominences are of the inverse polarity type (Leroy et al. 1984). Alternatively a prominence can be described by twisted field structures which was modeled by Priest et al. (1989).

Different attempts has been made to include an energy equation. Hood & Anzer (1988) or Steele & Priest (1990) solved the energy equation for a given field line without treating the full mechanical equilibrium. The mechanical and thermal equilibrium was solved consistently by Lerche & Low (1977) for a simplified energy equation or by Milne et al. (1979) for a one-dimensional Kippenhahn-Schlüter model.

Stability investigations performed by Galindo-Trejo & Schindler (1984) and Galindo-Trejo (1987) show the stability of the models developed by Menzel (1951), Dungey (1953), Kippenhahn & Schlüter (1957) and Lerche & Low (1980). DeBruyne & Hood (1993) analysed the stability of the model developed by Hood & Anzer (1990) with help of the energy principle of Bernstein et al. (1958) and showed instability for many cases. Longbottom et al. (1994) give stability conditions for 2D current sheet models. Schutgens (1997a), (1997b) discusses the different influences of the photospheric boundary conditions on prominences of normal and inverse type.

Stability analysis gives also information about possible prominence oscillations. The different modes are discussed on the basis of idealized models by Joarder & Roberts (1992, 1993), Oliver et al. (1993), Oliver & Ballester (1995) and Joarder et al. (1997).

This work is based on a flux tube model for quiescent solar prominences developed first by Ballester & Priest (1989). Degenhardt & Deinzer (1993), Degenhardt (1995) and Cramphorn (1996) extended this model by self-consistently including an energy balance (Schmitt & Degenhardt 1995). A quiescent prominence is modeled as a sequence of static slender flux tubes arranged behind each other and embedded in an isothermal corona as shown in a sketch in Fig. 1. This model belongs to the normal polarity type. The prominence rests as cool, dense plasma in a dip at the summit of the arch like flux tubes which reach out far into the corona and are rooted down in the chromosphere along lines of opposite magnetic polarity. For each flux tube the magnetohydrostatic force equilibrium and an energy balance between radiative losses, heat conduction and coronal heating is solved numerically.

[FIGURE] Fig. 1. Sketch of the described model

We tested these models for their dynamical stability by applying the stability formalism for thin magnetic flux tubes derived by Schmitt (1995, 1998). He obtained a canonical form of the linear stability equations for slender flux tubes which leads to a self-adjoint force operator for adiabatic perturbations. The analysis yields the conditions for stability and the frequencies of the oscillations in the (neutrally) stable case or growth rates of the disturbances in the unstable case, respectively. The eigenfunctions give hints on the possible instability mechanism. We remark that we only investigate the global stability of an individual thin flux tube of the prominence. The interaction with the environment is given by the lateral pressure balance and a retroaction of the flux tube on the external plasma is neglected.

The eigenvalue problem is solved numerically. As a first step we used a matrix eigenvalue formulation with expansions into fourier modes for a complete spectrum of approximate eigenvalues. For a refined treatment of particular modes we applied the Riccati shooting method (see Gautschy & Glatzel 1990) with adaptive step size integration for the corresponding initial value problem. The use of this method was necessary due to vastly different values of the stability coefficients inside and outside the prominence along the tube. Moreover, we applied the energy principle for a test of the results.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999