## 3. Diffusion process due to internal wavesA first approach of the diffusion problem is due to Press (1981). As mentioned by Bretherton (1969), adiabatic oscillations do not produce any diffusion process, as any piece of the fluid, in its vertical motion, comes always back to the same place. The diffusion process induced by random gravity waves results from radiative damping of waves. As already mentioned, we deal with the problem of the generation of internal waves produced by the turbulent motion present inside the plumes, at the boundary of the convective zone. A description of the plumes, based on the picture given by Rieutord & Zahn (1995), is given by Schatzman (1996). We shall describe the motion at the boundary by the sum of the motions of plumes, each of the velocities having the same auto-correlation properties, We shall consider later (appendix) the problem of the auto-correlation function. The velocity field at point () is the superimposition of all the contributions starting from and arriving at points The vertical diffusion coefficient produced by this velocity field is given in Schatzman (1996) using the results from Knobloch (1977): where Then, the diffusion coefficient is given by Eq. (4.13) in Schatzman (1996) Later on, we shall use . The field of velocity is described by the superimposition of
"monochromatic" internal waves which propagate in the radiative
interior of the star (Eqs. 18 and 24). The source function of
these waves will be described using the the Fourier transform of the
motion taking place at the boundary of the radiative zone
(). This allows us to introduce the
spectrum of the turbulence generating the internal waves. In order to
relate the amplitude of the movement at each side of the boundary
convection/stratification, we chose the solution given by Press
(1981), assuming a discontinuous behaviour of the
Brunt-Väisälä frequency at the transition layer. It is
well justified (Montalbán 1994) for stellar models using the
treatment of the overshooting by Zahn (1991). Therefore, the vertical
amplitude of a monochromatic component of the field is given by
.
, being the horizontal amplitude,
is taken equal to the turbulent
mean square velocity in the convective zone generating the waves. It
is equivalent to the treatment of the transition used by García
López & Spruit (1991) in considering the continuity of
pressure fluctuations on both sides of the boundary. The variation of
amplitude of monochromatic waves as a function of depth The inverse Fourier transform at depth where is the Fourier transform
of Appendix gives the details of the way to obtain the expression of the diffusion coefficient. It is first expressed as a function of the variables in Fourier space. We follow the rule given by Knobloch(1977), which consists in replacing the ensemble avera ge of four terms by the product of two terms. We use the kinetic spectrum given by Lesieur and finally obtain the diffusion coefficient: In Fig. 3 we plot this diffusion coefficient and the curve obtained in Schatzman & Montalbán (1995) using Press approach. Both diffusion coefficients have been calculated considering the same model of plumes (Schatzman 1996) which generate the internal waves. The spatial scale of the turbulence inside the plumes, is defined by the plume radius at the boundary of the convective zone (Eqs. 2.25, 3.11 of Schatzman 1996), and the velocity of turbulent flow in the plumes is given by the experimental relationship (List 1982) with the maximum velocity in the plume (Schatzman 1996 Eq. 2.26). is defined as . The values of these parameters are: , , and , or . We note that the new expression for the diffusion coefficient takes almost the same value near the boundary but decreases more rapidly as the depth increases than the old one. Furthermore, more significant for our present question, its value below is much smaller than in Schatzman & Montalbán (1995).
The damping of internal waves has been decreased with respect to the old model and therefore, the waves go down more deeply. However, in the expression for the diffusion coefficient, appears also the radiative damping derivative as a factor (Eq. A9), and this magnitude has also decreased below as a consequence of new gradients appearing in the new expression of radiative damping. That behaviour implies that, when we take into account the effect of mean molecular weight gradient, the parameters of the maximum obtained by Schatzman & Montalbán (1995) will not correspond to the values required in Morel & Schatzman (1996) and Schatzman (1997). We would obtain a maximun located at lower radius and with l ower amplitude. © European Southern Observatory (ESO) 1999 Online publication: November 2, 1999 |