Astron. Astrophys. 356, 319-326 (2000)

Linear adiabatic dynamics of a polytropic convection zone with an isothermal atmosphere

II. Quasi-stationary solutions

F. Schmitz and S. Steffens

Astronomisches Institut der Universität Würzburg, Am Hubland, 97074 Würzburg, Germany

Received 7 April 1999 / Accepted 3 February 2000

Abstract

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 is a third-order algebraic equation in 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 quasi-stationary solutions.

Key words: hydrodynamics Sun: atmosphere Sun: chromosphere

Send offprint requests to: F. Schmitz (schmitz@astro.uni-wuerzburg.de)

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Contents

• 1. Introduction
• 2. Basic relations
• 3. Solutions of the wave equation with complex frequencies and their dispersion relation
• 4. Dispersion curves and properties of quasi-stationary waves
  • 4.1. Outgoing progressive waves
  • 4.2. The case
  • 4.3. The general case
• 5. The physical and mathematical meaning of the quasi-stationary waves
  • 5.1. Quasi-stationary solutions of the ordinary wave equation
  • 5.2. The general case
  • 5.3. A two-layer model
• 6. Conclusions
• Acknowledgements
• Appendices
  • A: Calculation of an instationary solution the Laplace transformation
• References

© European Southern Observatory (ESO) 2000

Online publication: March 28, 2000
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