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

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7. P modes and convection

The current study was undertaken in connection with the theory of p modes in the solar convection zone. As a sequel to Paper 1, a new approach has been developed, but specific problems will require separate studies. In this section we just outline the basic effects that in our view are essential for helioseismology. The slow turbulence limit, which is used in the current study is valid for p modes except upper thin layer of the atmosphere, where the turnover time of convective cells 2-3 times less than the wave period. The fast turbulence limit, which has been developed so far only for the case of [FORMULA] (Rüdiger et al 1997), is not valid for p modes in this layer as well. The reason to believe, that the applications of the developed methods to the treatment of p modes (Zhugzhda 1994, Rüdiger et al 1997) are reasonable, is that the effect of the sound dispersion appears in both of the limiting cases of slow and fast turbulence.

Dispersion of acoustic waves and the eigenfrequencies of p modes. For a given model of the convection zone, the phase velocity of acoustic waves in a convective atmosphere depends on frequency and degree l. For [FORMULA] it was shown in Paper 1 that the corrections to the eigenfrequencies due to the dispersion of the acoustic waves are essential and have the right sign and order of magnitude to remove the discrepancy between theory and observation. The current treatment allows to calculate the phase velocity of these waves for any value of l. Fig. 6 shows the relative correction to the vertical phase velocity as a function of [FORMULA] for different values of [FORMULA].

[FIGURE] Fig. 6. Difference between the vertical phase velocities in a moving thermally structured ([FORMULA], [FORMULA]) and a uniform atmosphere, [FORMULA], in units of [FORMULA], for [FORMULA] (solid), [FORMULA] (dotted), [FORMULA] (dashed), [FORMULA] (dash-dotted).

The phase velocity decreases with decreasing [FORMULA] for [FORMULA] because for the case of vertical propagation it exceeds the sound velocity in the low frequency limit by approximately [FORMULA] due to the flow effect as expressed by Eq. (32); in the case of horizontal propagation the phase velocity is smaller than the sound speed by [FORMULA], cf. Eq. (45), due to the temperature fluctuations. The sharp increase of the dispersion for [FORMULA] and [FORMULA] is due to the approach to the Brillouin zone border, where the acoustic branches in the ([FORMULA])-diagram change their direction (Figs. 1, 2, 3).

Vibrational waves. The vibrational waves, which appear as an exact analytical solution of Eq. (15), are local acoustic oscillations in a convective atmosphere whose frequencies fall within band above [FORMULA] (47). But the "eigenoscillations" of adjacent turbulent elements are strongly coupled, like the oscillations of the atoms in a crystal lattice. This enables these local oscillations to propagate, which is why they are called vibrational waves. Vibrational waves in the convection zone transport acoustic energy, while vibrational waves (phonones) in a solid are responsible for heat transfer.

Brillouin zones. Partial reflection occurs when the waves meet a Brillouin zone border on their path in the convection zone. This additional partial scattering at certain levels in the convection zone exists only for waves of a definite frequency, which depends on the size of the convective elements and on the sound velocity. Such selective reflection could produce special features in the spectrum of p modes, which may hardly be explained by peculiarities in the temperature gradient at that level.

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998