Linear adiabatic dynamics of a polytropic convection zone with an isothermal atmosphere II. Quasi-stationary solutions

For a plane model of the exterior parts of the sun, the behavior of adiabatic waves with complex frequencies is investigated. The equilibrium configuration is a one-layer model with isentropic stratification at great depth and an asymptotically isothermal atmosphere. The wave equation reduces to Whittaker’s equation with complex parameters. By the assumption that only outgoing progressive waves are present in the atmosphere, we obtain a discrete spectrum of complex frequencies. The dispersion relation F (ω, k) = 0 is a third-order algebraic equation in ω2 with real coefficients. There are no connections of the ridges of the eigenmodes with the ridges of the quasi-stationary waves. Instead, there are striking gaps, and the ridges of quasi-stationary waves extend into the region below the acoustic cut-off frequency. The findings indicate that the ridges of the quasi-stationary solutions cannot explain the ridges of the observed pseudo-modes. As the solutions are not quadratic integrable and form no basis, they do not represent eigenmodes. The behavior of the quasi-stationary solutions is related to the behavior of quasi-stationary states of certain quantum mechanical systems. To answer the question whether quasi-stationary waves are limiting cases of instationary waves, we consider a simple one-dimensional two-layer model. For this case, instationary solutions are compared with the corresponding quasistationary solutions.