## 4. Role of turbulence descriptionFollowing 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 ( 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
which is large for all frequencies smaller than and decreases rapidly for higher frequencies; which falls off faster at high frequencies, and 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
Concerning the 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 |