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

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3. Diffusion process due to internal waves

A 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 [FORMULA] plumes,

[EQUATION]

each of the velocities [FORMULA] having the same auto-correlation properties,

[EQUATION]

We shall consider later (appendix) the problem of the auto-correlation function.

The velocity field at point ([FORMULA]) is the superimposition of all the contributions starting from [FORMULA] and arriving at points [FORMULA]

[EQUATION]

The vertical diffusion coefficient produced by this velocity field is given in Schatzman (1996) using the results from Knobloch (1977):

[EQUATION]

where v is the Lagrangian velocity, and the relationship with our Eulerian velocity u is:

[EQUATION]

Then, the diffusion coefficient is given by Eq. (4.13) in Schatzman (1996)

[EQUATION]

Later on, we shall use [FORMULA].

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 ([FORMULA]). 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 [FORMULA]. [FORMULA], being the horizontal amplitude, [FORMULA] 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 z is introduced using Eqs. 18 and 24.

The inverse Fourier transform at depth z, including this time [FORMULA]-dependent terms, introduced by the z derivatives present in Eq. (37), provides the description of the motion at depth z:

[EQUATION]

[EQUATION]

where [FORMULA] is the Fourier transform of u.

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:

[EQUATION]

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, [FORMULA] 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). [FORMULA] is defined as [FORMULA]. The values of these parameters are: [FORMULA], [FORMULA], and [FORMULA], or [FORMULA]. 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 [FORMULA] is much smaller than in Schatzman & Montalbán (1995).

[FIGURE] Fig. 3. Diffusion coefficient due to internal waves generated by a turbulence distribution characterized by [FORMULA] and [FORMULA] (Schatzman & Montalbán 1995, Schatzman 1996). Solid line correspond to quasi-adiabatic approach, dashed line correspond to Press' (1981) approach.

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 [FORMULA] (Eq. A9), and this magnitude has also decreased below [FORMULA] 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.

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© European Southern Observatory (ESO) 1999

Online publication: November 2, 1999
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