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Astron. Astrophys. 332, 314-324 (1998)

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1. Introduction

The exploration of hydrodynamical waves in a convective atmosphere is initiated especially by helioseismology. The accuracy of the measured frequencies and line profiles of p modes is sufficient to reveal the effects of convection on the waves. A general theory of hydrodynamical waves in a turbulent convective atmosphere does not exist so far. There are several ways of attack, connected with different aspects. Wave generation by turbulence, wave absorption due to turbulent viscosity, the effect of turbulence on the effective phase velocity, the interaction of the wave turbulence and turbulence itself belong to these aspects. All these problems are coupled physical effects; their separate treatment is a causal approach. Conceptually, the limiting cases of slow and fast turbulence are used to simplify the problem. The approximation of fast fast turbulence is valid when the lifetime or turnover time of the turbulent elements is small enough in comparison with the wave period so that time averaging becomes possible. The simplest approach of this kind is to consider the effect of turbulent pressure on the sound velocity (Kosovichev 1995, Rosenthal 1997). Mean-field theory permits an easy treatment of the turbulent viscosity (Stix et al. 1993) and of the corrections to the sound velocity (Rüdiger et al 1997). In the limiting case of slow turbulence the waves are affected only by the space pattern of the random velocity and temperature fluctuations, which are so slow that the waves can follow the changes of the pattern. This approach has been used by Murawski & Roberts (1993a,b), Murawski & Goossens (1993) for f modes and by Rosenthal et al. 1995a, b for p modes. Their treatment of the f and p modes was based implicitly on the assumption that there are only slight changes of the waves that are typical for a uniform atmosphere. Consequently, perturbation theory has been applied to the problem. Such an approach is sound but not universally true.

A first attempt of an exact derivation of the effects of the velocity and temperature fluctuations on acoustic waves in an atmosphere has been undertaken by Zhugzhda & Stix (1994, hereafter Paper 1). In their simple 1D model of stationary convection a variety of wave modes has been found. These new effects can be understood in terms of solid state physics. A good example is the difference in the behaviour of electrons in vacuum and in the potential field of a metal lattice. In the latter case the motion of the electrons is described in terms of Bloch wave propagation (Bloch 1928), which is not a perturbation of the free electron state. Acoustic waves in a structured atmosphere are similar: They are affected by the fluctuations in temperature and velocity in such a way that their properties qualitatively differ from the properties of waves in a uniform atmosphere. Moreover, as will be shown, new wave modes appear in the structured atmosphere.

The analysis of Zhugzhda & Stix had been restricted to longitudinal waves, and the clues provided by solid state physics had not been appreciated. In the present paper the theory is developed in order to understand some of the properties of waves in a nonuniform atmosphere in terms of solid state concepts such as Brillouin zones, vibrational waves and band gaps. Although this terminology is used, it will be clear that the analogy to, for example, the theory of acoustic waves in a crystal lattice can be applied only in a general sense. The governing equation (Eq.  15below) is substantially more complicated than, for example, the equation for electrons in a periodic potential, which appears in the theory of Bloch waves. This precludes the use of the theory of Mathieu functions. Instead, the exact solution of the problem will be obtained by means of the general theory of equations with periodic coefficients. In this paper I shall treat the simplest cases step by step in order to introduce the concepts that so far have not been discussed in the astrophysical literature.

The stationary 1D model of regular hot and cold flows could be applied to waves propagating in a stationary convective atmosphere. It is discussed whether the same effects appear in a turbulent convective atmosphere with random temperature and velocity fluctuations. Finally, the effects of convection on p modes are outlined.

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998