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Astron. Astrophys. 334, 953-968 (1998)

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2. The convective model

2.1. Convection as a whole

A short discussion on the convective model adopted in the present computations is unavoidable, since the results obtained with diffusive mixing depend on the convective velocities and scale lengths.

Convection in stars is by far turbulent, the largest eddies having a size comparable to that of the entire convective region ([FORMULA] cm). They break down pumping (downscatter) and getting back (backscatter) kinetic energy into smaller eddies, until molecular viscosity takes over, and the kinetic energy stored in the whirlpools is thermally dissipated; billions of eddy scales are then present in stars. Navier-Stokes equations describing this process from first principles are both non-linear and non-local (Xiong 1985, Canuto 1992, Grossman 1996), and their analytical or numerical solution for a general stellar case is not yet available. In stellar modeling, we then still stick to local models to compute both the energy fluxes and the convective scale lengths.

As for the fluxes, MLT (Prandtl 1925) - originally developped for very viscous fluids - was first applied by Vitense (1953) to stellar modeling. In the MLT, the spectral distribution of eddies is mimicked by one "average" eddy only. More updated models provide now fluxes also for low-viscosity flows, in which the whole eddy spectrum is accounted for (Canuto & Mazzitelli 1991 and 1992, Canuto et al. 1996-CGM). Stothers & Chin (1995) defined Full Spectrum of Turbulence (FST) these latter models, as opposed to the one-eddy spectrum MLT. In the present work we adopt the FST scheme with the CGM fluxes.

The scale length [FORMULA] was originally chosen, in the MLT, coincident with the distance z from the convective boundary, consistently with Von Karman law for incompressible fluids (Prandtl 1925). Since however the one-eddy flux distribution is very crude, no realistic fit of the Sun could be obtained. It is then now customary to apply the MLT with [FORMULA], where [FORMULA] is a free parameter tuned on the solar model; depending on the micro-physical inputs, [FORMULA].

The more physically correct choice [FORMULA] close to the convective boundaries is instead allowed in an FST framework. However, far from the boundaries, also [FORMULA] must, in a star, approach the hydrostatic scale length [FORMULA]. To match these requirements, we consider both the upper and the lower convective boundary; we compute for each convective grid point [FORMULA] and [FORMULA] and assume z as their harmonic average. Close to the boundaries [FORMULA] or [FORMULA], which is the required result. For deeper layers, let us recall that, in a polytropic structure of index n, [FORMULA] (Lamb 1932). In convective regions [FORMULA], that is: [FORMULA]. The harmonic average gives [FORMULA], again close to the required value.

No perfect theory of convection, however, still exists, and the same is true for all other micro/macro-physical inputs. If we then ask for an exact fit of the Sun, some tuning of the models is required. With the MLT, all the physical uncertainties are usually reset by means of the completely free parameter [FORMULA]. At variance, in an FST environment we recall that [FORMULA] should also allow for overshooting ([FORMULA]), which is not yet known from first principles. We can then parametrize [FORMULA] (or [FORMULA]), such that [FORMULA] can be seen as a fine tuning parameter, constrained by [FORMULA] (Maeder & Meynet 1991, Stothers & Chin 1992). Note that FST tuning affects only layers close to the boundaries since, for inner layers, z fastly grows [FORMULA]. With CGM fluxes (Kolmogorov constant updated to [FORMULA]) the solar fit requires [FORMULA], the value slightly depending on the chemical abundance chosen. With [FORMULA] one would get an underestimate of the solar [FORMULA] around 1.5%; too much for helioseismological purposes, but almost negligible in the more general framework of stellar evolution. In the present paper we will always adopt

[EQUATION]

unless differently specified, where [FORMULA] is the distance from the top of convection increased by [FORMULA], and analogously for [FORMULA].

2.2. Diffusive mixing and overshooting

According to first principles, in the presence of both nuclear reactions and turbulent mixing, the local temporal variation of the [FORMULA] element follows the diffusion equation (Cloutman & Eoll 1976):

[EQUATION]

stating mass conservation of element i if molecular diffusion be negligible with respect to turbulent diffusion (almost always true). The diffusion coefficient D is:

[EQUATION]

where [FORMULA], the turbulent diffusion timescale, is related to the one-point density-radial velocity correlation

[EQUATION]

Unfortunately, knowledge of the second order momentum in Eq. (3) requires previous solution of the Navier-Stokes Eqs. for a compressible stellar fluid, not yet available in the huge variety of cases of astrophysical interest. Also for D a local approximation is then customarily used, that is: [FORMULA], where u is the average turbulent velocity and [FORMULA] ([FORMULA] in the formally convective region) is the convective scale length.

In an MLT framework, the above approximation is highly disputable since experiments show that turbulent chemical mixing is more efficient at the smallest scales, at variance with the unique (largest) MLT scale. More physically sound is the FST description: the average velocity accounts also for the smallest eddies, and the scale length is always the more correct one. In our case we then expect to get a better approximation to the "true" diffusion coefficient.

Turbulent velocity u is computed according to Eqs. (88), (89) and (90) by CGM. One problem arises because of the "locality" of the FST model (u at the convective boundaries vanishes). However, u sharply goes to zero only very close to the boundaries; a little deeper in the convective region, the velocity profile approaches to a plateau. It is then possible to extrapolate a "reasonable" non-vanishing velocity at the formal boundary, and test computations starting from a pressure either 2%, 5% or 10% inside the boundary gave almost identical results. We then extrapolate [FORMULA] vs. [FORMULA] to get the turbulent velocity at the formal boundary starting from a pressure 5% inside.

As for diffusive overshooting (if any) we recall that, according to Xiong (1985), turbulent velocity exponentially vanishes outside formally convective regions. We then write, in overshooting regions:

[EQUATION]

where [FORMULA] and [FORMULA] are respectively turbulent velocity and pressure at the convective boundary, P is the local pressure, [FORMULA] is a free parameter to be tuned as discussed in the following, and [FORMULA] is the thickness of the convective region in fractions of [FORMULA] (maximum value of [FORMULA]), to maintain a non-locality flavor, since [FORMULA] scales overshooting according to the thickness of the convective region. While the diffusive scale length in formally convective regions is obviously [FORMULA] as in Sect. 2.1, in the overshooting regions we use [FORMULA] to maintain continuity of the diffusive coefficient D at the boundaries.

In a first approximation, then, our framework for diffusive mixing and overshooting shows similarities to the one by Deng et al. (1996a,b). However:

a) use of FST instead of MLT, with different diffusive velocities and scale lengths, leads to different results, and:

b) the chemical evolutionary scheme is far different in our case, since we fully couple nuclear evolution and diffusive mixing (hereafter to: "coupled-diffusion").

Of course, ATON 2.0 code also allows computations with instantaneous mixing and overshooting (in this latter case, the overshooting distance is simply [FORMULA] as usual). In the following, comparisons between results obtained with instantaneous mixing and coupled-diffusion will be presented.

2.3. Coupled mixing and nuclear evolution

When using Eq. (1) two alternatives choices could be made:

a) numerically evaluate [FORMULA] as a function of local abundances and cross sections, substitute in Eq. (1) and only then apply the diffusive algorithm. The bonus of this choice is that the problem is reduced to a "semi-local" one: no large matrices must be stored and computations are relatively fast. The drawback is that nuclear evolution is completely decoupled from mixing, and some mechanisms -like hot bottom burning lithium production in the envelopes of AGB stars (Sackmann et al. 1995), or effect of mixing on the CNO equilibria - cannot even be addressed;

b) analytically expand [FORMULA] as a function of local abundances and cross sections, and solve it together with the diffusion matrix for the whole stellar structure. If chem is the number of chemical species and grid is the number of grid points, a matrix of rank [FORMULA] - typically [FORMULA] MB - must be stored and inverted. Method b) severely threatens any workstation, but correctly couples nuclear evolution to the contemporary change of composition due to turbulent transport of matter (coupled-diffusion).

We then made choice b) which, after some algebraic manipulations fully described in Sect 9.4, can be reduced to the storage and inversion of a matrix of rank [FORMULA] - typically [FORMULA] MB - well affordable according to safe and fast algorithms. For more details on the nuclear evolutionary scheme as a whole, see Sect. 9.3.

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998

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