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Astron. Astrophys. 351, 347-358 (1999)

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5. Summary and conclusions

Diffusion coefficients obtained in previous works (Schatzman & Montalbán 1995; Schatzman 1996) are based on Press' (1981) approximation which is not completely valid in the most central region of the Sun. In order to approach accurately the problem of mixing in the regions where the nuclear reactions producing neutrinos take place, it is necessary to introduce an adequate treatment of the non-adiabatic propagation of internal waves in those regions. We have taken the quasi-adiabatic approach given by Unno et al. 1989, valid in the whole of the solar radiative region (Sec. 2). This treatment implies a decrease of radiative damping in the central region, and therefore, an increase of the internal waves amplitude in the solar center. However, the effect of radiative damping diminution on the diffusion coefficient at the solar center is opposite to what we need in order to reproduce Morel & Schatzman (1996) modelizations. In other words, the diffusion coefficients computed by Press' approximation and by quasi-adiabatic approach using the same turbulence description are very different in the neutrino production region. Instead of obtaining a diffusion coefficient that increases rapidly in order to allow a maximum at [FORMULA], we obtain one that decreases more rapidly and begin to increase more deeply in the Sun.

Since the damping term depends strongly on wave frequency and wavenumber, we have analyse the effect of the perturbation distribution at the convective boundary on the amplitude of movements at the solar center. We have considered several forms for the source of internal waves, taking into account different distributions of energy of perturbations as function of wavelength and frequency (Sect. 4). As we saw, at the stellar center, the value of diffusion coefficient linked to the internal waves propagation depends strongly on the turbulence description. The coherence with light elements depletion could impose some constraints on the turbulence description.

In this work we have only considered the contribution to the diffusion process of the propagating gravity waves and we have neglected the contribution from g-modes. In fact, we have considered that radiative damping can be so strong that reflection at the center is avoided, and in consequence there is no production of eigen-g-modes. However, we have seen in Sect. 3 that for several couples of wave number and frequency the amplitude of perturbations near the center is not negligible, and eigen-g-modes are possible. Therefore, their contribution should be taken into account. As we noted in Sect. 1, g-modes could have other effect in addition to the macroscopic diffusion. Gough (1992) took into consideration the possibility of a change of the central temperature when taking into account the non-linear effect due to g-modes. In his paper, he analysed the dependence of the neutrino flux on the amplitude of the the g-mode [FORMULA], that displays the maximum of the amplitude at [FORMULA]. The result is that the neutrino flux is smaller when the amplitude of the mode is larger. A quantitative estimation of this effect implies to know the amplitude of these modes, and so, the characteristics of the excitation source.

We conclude that a complete analysis of the role of internal waves in the mixing in the central regions of the Sun should include not only the improvement of the treatment of internal waves propagation, but also a good characterization of the source function, the analyse of its ability to excitate g-modes, and the study of different effects of these modes.

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© European Southern Observatory (ESO) 1999

Online publication: November 2, 1999