Linear adiabatic dynamics of a polytropic convection zone with an isothermal atmosphere - I. General features and real modes

To investigate and understand basic properties of non-radial solar p-modes with high wave numbers l, it is sufficient to consider only the outer layers of the sun. As an atmosphere, the upper part of the convection zone may be approximated by a plane layer with constant gravity. A simple standard model is a polytropic convection zone with an overlying isothermal atmosphere. In this case, the adiabatic wave equation of each layer can be solved analytically. However, the dispersion relation F(omega, k) = 0 of the acoustic and gravity modes of the whole layer is complicated and cannot be solved in closed form. In this paper, we present a model with a smooth transition between the poytropic convection zone and the isothermal atmosphere. For this model, using the column mass instead of the geometrical height, the adiabatic wave equation can be reduced to Whittakers differential equation. The geometrical height is a simple elementary function of the column mass. The dispersion relation F(omega, k) = 0 is a fourth order algebraic equation in omega(2). In the important case of an isentropically stratified polytropic convection zone, it reduces to a cubic equation in omega(2). In any case, the dispersion curves omega(k) can be given in closed form. As in the case of a purely polytropic convection zone, the z-dependence of the waves and the modes is represented by Whittaker functions. We analyze the behavior of the dispersion curves of modes with an adiabatic exponent gamma = 5/3 for layers with polytropic indices n = 3 and n = 3/2. Further, we investigate the appearance of resonances in the region of the continuous spectrum of acoustic waves. We find that these resonances are present only at frequencies slightly above the acoustic cutoff frequency of the isothermal atmosphere. The case of purely vertical wave propagation is considered separately. In the present paper, we deal only with real frequencies.