4. Role of turbulence description
Following the first paper of Press (1981), we consider that the internal gravity waves propagating in the radiative region are generated by the convective motions in the overshooting region, at the boundary between the convective and radiative regions. The problem is how to describe this "convective motion". Press (1981) considered that the motion could be described by only one eddy with a size given by the mixing length theory (MLT) of convection. Later on, some improvements were introduced (Garcia López & Spruit 1991; Schatzman 1993; Montalbán 1994) which did consist in considering a Kolmogorov spectrum of turbulence with characteristic scales given by the MLT. Laboratory experiments and numerical simulations have shown that in fact, the MLT is not appropriate for describing the convection movement, and that the convective transport of energy occurs through large structures. Therefore we have taken the model of plumes by Rieutord & Zahn (1995) to describe the convection. We consider that the gravity waves have been generated by the turbulent flow inside these plumes. Laboratory experiments provide the spatial evolution of these structures (see Turner 1986 for review) and their dimensions and velocity as they go down, as well as the relationship between the velocity of the mean flow and the mean square velocity of the turbulent flow inside the plume (List 1982).
However, the source of the gravity waves is the turbulent flow in the penetrative region, where the turbulent flow concentrated in the vertical plumes becomes horizontal and where the plumes disappear because of the strong stratification of the radiative zone. How can we describe this turbulent flow in that thin layer? What is the relationship with the turbulent flow in the plumes? We do not have the answer to these questions, so we shall try to approach the problem using some simple forms for the turbulence description.
In order to suggest forms of the velocity correlation, we guide ourselves by the results of isotropic, homogeneous, incompressible turbulence. These conditions certainly do not apply to our physical case, but it is only for this simple case that we have any guide as to the form of the correlation.
For incompressible, homogeneous, isotropic turbulence, the Fourier transform of the velocity correlation has the form (Batchelor 1953):
As we did above (appendix, Eq. A17), we consider that the turbulence energy spectrum can be factorized into a spectrum and a frequency-dependent factor . For the k-spectrum we take a Kolmogorov spectrum , with a given minimum wave number of the distribution. And for the frequency-dependent factor we have chosen an exponential distribution (Eq. A20) as indicated by some experimental studies (Anselmet et al., 1984). Furthermore, we consider that this frequency factor is independent of the wave number. Stein (1967) considered three different frequency-factors with the form of a smoothed delta function centered about , with . He considered an exponential distribution, and two Gaussian ones:
which has a maximum at and falls off for both lower and higher frequencies.
In order to explore the effects on the diffusion coefficient of the frequency dependence of the turbulent spectrum, we have represented the diffusion coefficient obtained with these four frequency-spectrum.
Let us now explore the form of the diffusion coefficient obtained with the different frequency dependences mentioned above. In Fig. 4 we represent the results obtained from Eq. A20, labeled by "0", and from Eqs. 39,40 and 41, labeled by "1", "2" and "3" respectively. We see that the fact of neglecting the dependence on k of frequency distribution is not very important, and the result is similar to the other exponential distribution (Eq. 39). However the behavior is very different when we take the Gaussian distributions. The diffusion coefficient decreases more rapidly in these cases, and its magnitude near the center of the Sun is very different for the three distributions proposed by Stein (1967).
Concerning the k-spectrum, we consider only the Kolmogorov spectrum, but there are uncertainties with respect to the characteristics scales. In previous works we considered the turbulent flow inside the plumes just before they find the boundary. Then, the largest eddy was determined by the radial dimension of the plume and the mean square velocity was related with the velocity at the center of the plume (see Schatzman & Montalbán 1995; Schatzman 1996). However, as we mentioned above, we do not know how large is the turbulent spectrum in the penetrative region. It is possible that the spatial dimension change with respect to the one inside the plume. In fact we could hope that the minimum k were given by the horizontal dimension, in that case, by the surface at the boundary of convective zone (). We have made the computing of diffusion coefficient taking two different values of and the corresponding and with the frequency spectrum labeled "0".
In Fig. 5. we show the effect of changing for a given value of , and the effect of changing the maximum spatial dimension of the distribution of energy. We see that increasing the spatial scale from that one given by the plume to the one of the order of the radius of the convective zone, the diffusion coefficient decreases significatively, and that effect is even more important at the central region. The sensitivity of the diffusion coefficient to the minimum horizontal wavenumber of the turbulence distribution, and the fact that the light element abundances (sensitive to the mixing near the convective zone) were well justified by this transport process, taking into account the spatial scale of the perturbations associated to plumes dimension, could allow us to impose some constraints on spatial characteristics of the source functions of gravity waves. Concerning the time scale, the diffusion coefficient is much more sensitive to the frequency near the center than near the surface. So far we consider that the maximum of the frequency distribution was related also with the spatial scale by , but another relation is not impossible. In Fig. 5 it is also shown that when we change the value of from it value by a factor two, the diffusion coefficient change by a factor 50 at the depth .
© European Southern Observatory (ESO) 1999
Online publication: November 2, 1999