*Astron. Astrophys. 360, 707-714 (2000)*
## 1. Introduction
The classical *f* -mode (*f* stands for fundamental) is
recognized as a compressionless wave that propagates in an inviscid
atmosphere that is permeated by constant gravity field (e.g., Campbell
& Roberts 1989). Its frequency is
given by the following dispersion relation
where is the horizontal
wavevector, Mm is the solar radius,
and *l* is the spherical degree. This dispersion relation shows
that the classical *f* -mode frequency is independent of the
internal structure of the Sun.
The high accuracy (%) observations
of the *f* -mode by Libbrecht et al. (1990), Rhodes et al.
(1991), Fernandes et al. (1992), Bachmann et al. (1995), and Duvall et
al. (1998) have shown that its frequency for high value of *l* is
substantially lower than follows from parabolic dispersion relation
(1).
Murawski & Roberts (1993b), Rosenthal & Gough (1994), and
Rosenthal & Christensen-Dalsgaard (1995) have suggested that the
*f* -mode is a surface gravity wave and attempted to explain
observed *f* -mode frequency shifts. The frequencies of the
interfacial *f* -mode differ from those of the classical *f*
-mode only at very high spherical degree. These shifts can be used to
verify the structure of the atmosphere. As a consequence of that the
*f* -mode can serve as a diagnostic tool of the solar
atmosphere.
The *f* -mode is also influenced by other effects. For
example, Pinter & Goossens (1999) have shown that the *f*
-mode frequencies are increased by a horizontal magnetic field in the
solar chromosphere. On the other hand, Vanlommel & Cadez (1998)
and Vanlommel & Goossens (1999) discussed the effect of frequency
shift due to variations in the temperature profile. Ghosh et al.
(1995) have proved that flows produce decreases of the *f* -mode
frequency. Murawski & Roberts (1993a,b), Murawski & Goossens
(1993), Gruzinov (1998), and Murawski et al. (1998) discussed the
models in which the *f* -mode is scattered by granulation,
modeled as a turbulent velocity field that is located in the
convection zone. This process makes transfer of coherent energy into
incoherent energy by exciting random waves and results in attenuation
of the *f* -mode and consequently in line broadening. The random
scattering will also affect the phase of the mode; hence the phase
speed is changed (Pelinovsky et al. 1998).
Murawski et al. (1998) and Medrek et al. (1999) generalized the
above mentioned models for the case of the complex frequency.Medrek
& Murawski (2000) considered the effect of various energy spectra
on the frequency and line-width of the *f* -mode. In these
models, the calculations were carried out for a plane-parallel
equilibrium consisting of two layers in which mass densities were
assumed constants, while the realistic model of the solar atmosphere
should take the stratification into account.
The main goal of this paper is to examine the influence of
stratification and turbulence on frequencies and line-widths of the
solar *f* -mode. To do so, we present a generalization of the
model developed by Murawski & Roberts (1993a,b) to the case of a
stratified atmosphere and convection zone, and explain the frequency
reduction and wave damping of the *f* -mode.
We start by setting up the problem in Sect. 2, where we
describe the physics included in our equations. In Sect. 3, we
derive the dispersion relation for the non-turbulent *f* -mode
and consider as an illustrative example the case of isothermal plasma.
Sect. 4 presents the dispersion relation for the turbulent
*f* -mode. In the following section, we investigate the influence
of turbulence and equilibrium structure on the frequencies and
line-widths of the solar *f* -mode. We compare these frequencies
and line-widths with the results of recent observations by the
SOHO/MDI.
© European Southern Observatory (ESO) 2000
Online publication: August 17, 2000
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