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Astron. Astrophys. 360, 707-714 (2000)

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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 [FORMULA] is given by the following dispersion relation


where [FORMULA] is the horizontal wavevector, [FORMULA] 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 ([FORMULA]%) 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.

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© European Southern Observatory (ESO) 2000

Online publication: August 17, 2000