Radial stellar oscillations under the influence of the dynamics of the atmosphere — a one-dimensional approach I. Linear adiabatic oscillations of a special model

The dynamics of the quiet solar atmosphere are highly nonlinear. Both the standing waves of solar oscillations and acoustic waves generated in the upper convection zone become nonlinear in the atmosphere and transform into shock waves. Interactions of shock waves, the formation of contact discontinuities, and interactions of shocks with these discontinuities will occur. The strong nonlinear dynamics of the atmosphere should influence high order p-modes of the Sun. In this series of papers we shall deal with fundamental properties of the interaction of the interior of a star with its atmosphere. According to the state of numerical techniques, we must restrict ourselves to radial oscillations or to the vertical dynamics of the atmosphere, respectively. As the nonlinear dynamics of the atmosphere governs the problem, we use a simple equilibrium model of the Sun or a star. For simplicity, we do not take a radial model but a plane layer model. Our particular “standard model” is a layer with nearly constant density in the interior and a smoothly matched isothermal atmosphere. The structure of this configuration is fitted to the structure of the Sun. In the present paper we present the equilibrium model and solutions of its linear adiabatic wave equation. The equilibrium configuration has been selected so, that the wave equation can be transformed to the equation of the associated Legendre functions. We determine the discrete eigenfrequencies, the modes, and the eigenfunctions of the continuous frequency spectrum. Resonances of the continuum are discussed. Also a set of discrete complex frequencies exists. The corresponding waves are not damped modes but limiting cases of instationary waves. The influence of an isothermal corona with a discontinuous transition layer on the frequency spectrum is investigated. We find strong resonances at frequencies between the discrete frequencies of the corona-free model.