*Astron. Astrophys. 344, 973-980 (1999)*
## Linear adiabatic dynamics of a polytropic convection zone with an isothermal atmosphere
### I. General features and real modes
**
F. Schmitz and
S. Steffens
**
Astronomisches Institut der Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany
*Received 18 September 1998 / Accepted 11 January 1999*
**Abstract**
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 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
is a fourth order algebraic equation
in . In the important case of an
isentropically stratified polytropic convection zone, it reduces to a
cubic equation in . In any case, the
dispersion curves 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
for layers with polytropic indices
and . 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.
**Key words:** hydrodynamics
Sun: atmosphere
Sun: oscillations
*Send offprint requests to:* F. Schmitz (Schmitz@astro.uni-wuerzburg.de)
This article contains no SIMBAD objects.
### Contents
© European Southern Observatory (ESO) 1999
Online publication: March 29, 1999
helpdesk.link@springer.de |