## 2. The convective model## 2.1. Convection as a wholeA 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
( 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 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 was originally chosen, in
the MLT, coincident with the distance The more physically correct choice close to
the convective boundaries is instead allowed in an FST framework.
However, far from the boundaries, also must, in
a star, approach the hydrostatic scale length .
To match these requirements, we consider both the upper and the lower
convective boundary; we compute for each convective grid point
and and assume No unless differently specified, where is the distance from the top of convection increased by , and analogously for . ## 2.2. Diffusive mixing and overshootingAccording to first principles, in the presence of both nuclear reactions and turbulent mixing, the local temporal variation of the element follows the diffusion equation (Cloutman & Eoll 1976): stating mass conservation of element where , the turbulent diffusion timescale, is related to the one-point density-radial velocity correlation 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 In an MLT framework, the above approximation is highly disputable
since experiments show that turbulent chemical mixing is more
efficient at the Turbulent velocity 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: where and are
respectively turbulent velocity and pressure at the convective
boundary, 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, ## 2.3. Coupled mixing and nuclear evolutionWhen using Eq. (1) two alternatives choices could be made: a) numerically evaluate 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 as a function of
local abundances and cross sections, and solve it together with the
diffusion matrix for the whole stellar structure. If 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 - typically MB - well affordable according to safe and fast algorithms. For more details on the nuclear evolutionary scheme as a whole, see Sect. 9.3. © European Southern Observatory (ESO) 1998 Online publication: June 2, 1998 |