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Astron. Astrophys. 359, 759-765 (2000)

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1. Introduction

Recently in solar physics a certain interest has arisen to the problem of interaction of waves and steady flows. In particular, flows can affect MHD waves trapped by magnetic flux tubes, modifying dispersion relations and changing wave cut-offs (Nakariakov & Roberts 1995). Also, it was shown (Joarder et al. 1997) that in the photosphere and below, observationally registered down-flows can cause negative energy wave effects, leading to instability and over-stability of the waves due to their interaction with the flows. In Nakariakov et al. (1998), it was found that even very weak flows, as long as they had sufficiently sharp gradients, can dramatically affect propagation of the waves, causing enhanced coupling of different wave modes. Thus, it is believed that the wave-flow interactions play a very important role in the dynamics of the solar plasma.

The Sun exhibits global oscillations which are influenced by a turbulent flow in the convection zone. The first evidence of these oscillations was provided by Leighton et al. (1962) who measured the Doppler shifts of photospheric spectral lines. Subsequently, it was proposed by Ulrich (1970) that acoustic waves were the cause of the Doppler shifts. As pressure provides the restoring force for these oscillations, they are called p -modes. Since the observable properties of solar global oscillations depend on the underlying solar structure and dynamics, they provide a potential tool for probing the solar interior (Gough 1994).

Although recent investigations prove that new solar models closely describe the Sun, there are still significant discrepancies between the predictions and the observations of the p -modes. As these discrepancies depend on frequency but are nearly independent of the spherical harmonic degree l it is supposed that the discrepancies are caused by effects that occur close to the solar surface. In this paper we investigate a possible cause of these discrepancies: stochastic flow as it affects p -mode frequencies and their line-widths.

The influence of convection on solar oscillations was discussed by several authors. In particular, Brown (1984), while considering vertical velocity perturbations, suggested that such perturbations cause a frequency decrease of sufficient magnitude to produce an observable effect. Lavely & Ritzwoller (1992) used perturbation theory to investigate the effects of steady-state convection on the p -mode line-widths and frequencies. They showed that large-scale convective flows have a systematic effect on the line-widths of the p -modes of a low spherical harmonic degree [FORMULA]. It has been shown by Kosovichev (1995) that the turbulent pressure decreases the sound speed in the external regions of the convection zone, leading to reduction of the p -mode frequencies. Gruzinov (1998) applied perturbation theory, but with an emphasis on finding analytical expressions for the frequency shift. Zhugzhda (1998) considered corrections arising from a sinusoidal perturbation to the sound speed and the vertical velocity, ignoring any horizontal flows. He found that the result is a frequency reduction which is proportional to both frequency and spherical harmonic l. Swisdak & Zweibel (1999) implemented an approach for determining the frequencies of the p -modes in a convective envelope. This approach is based on the ray approximation and the method of adiabatic switching to seek the eigenfrequencies of a Hamiltonian system. As a result, large-scale convective perturbations can generate downward frequency shifts which are second-order in the perturbation strength. Bömer & Rüdiger (1999) examined the influence of turbulence by the Reynolds stress on a reduction of the frequencies of the p -modes.

This paper is organized as follows. The next section presents linear equations which describe the interaction between the flow and the acoustic waves. Sect. 3 presents the random dispersion relation which is solved numerically for the case of a space-dependent random flow as well as for a time-dependent flow. Sect. 4 is devoted to numerical simulations of linear hydrodynamic equations. The paper is concluded by a short summary.

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© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000