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Astron. Astrophys. 341, 560-566 (1999)

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2. Transport of chemicals

The conservation equation for a trace element of concentration c is

[EQUATION]

where [FORMULA] is the diffusivity tensor, [FORMULA] is the global motion of the fluid (in this case the meridional motions generated via the rotation of the star), and [FORMULA] describes any motion affecting only the contaminant. The concentration c is defined as the number density of ions compared to the number density of protons. Hereafter, the star is assumed to be axisymmetric, so that using a spherical coordinate system, for any X, [FORMULA]. Also, for rotating stars, the equipotential is not defined by [FORMULA]constant. Therefore the "horizontal" direction is taken to be perpendicular to the equipotential, and the "vertical" direction to be parallel to equipotential.

Following the work of Chaboyer & Zahn (1992) and Zahn (1992), Charbonneau (1992) examined the effect of anisotropic turbulent diffusion on the distribution of [FORMULA] (Zahn 1975, Tassoul & Tassoul 1983, Zahn 1987), and found that if the horizontal Reynold number is [FORMULA] then the equation above loses its bi-dimensional behaviour, and becomes one dimensional ie. [FORMULA]. Essentially the horizontal turbulence becomes vigorous enough that it may redistribute the concentration of the trace element faster than meridional motions can generate an anisotropy. This also makes the angular momentum independent of [FORMULA] and gives rise to "shellular" rotation of the star (Zahn 1992).

In the current study, however, the velocity field [FORMULA] may be non-zero due to the meridional forces on trace elements arising from the anisotropy of the radiation flux (see Sect. 2). To assess its likely effect in the presence of bulk meridional motions and diffusion, the ratio of the (turbulent) diffusive term to the radiation driven advective term across an equipotential [FORMULA] in Eq. 1 is examined:

[EQUATION]

where [FORMULA] is the horizontal diffusion coefficient, the extra velocity component is [FORMULA] and [FORMULA] is a function containing logarithmic derivatives of the concentration with respect to [FORMULA]. This is only an approximation as the unit vectors along an equipotential [FORMULA] and [FORMULA] are not parallel. However, this expression will suffice for the current study. Also it should be noted that there is no term in the angular momentum evolution equation equivalent to the radiation term [FORMULA] in Eq. 1. Therefore, shellular rotation will still be achieved in the star, although the distribution of the trace element may be anisotropic.

The discussion of turbulent diffusion coefficients [FORMULA] has a long history, and the simple approximation due to Chaboyer & Zahn (1992) is adopted here:

[EQUATION]

where [FORMULA] is a number of order unity and [FORMULA] is the radial component of the velocity of the bulk meridional circulation. With this replacement Eq. 2 becomes

[EQUATION]

The main controlling parameter here is the ratio of the radial component [FORMULA] of the bulk meridional flow and the velocity [FORMULA] of the trace element due to the radiation as it slips through the plasma.

If [FORMULA] then the turbulent diffusion term will dominate the movement of trace elements, and so the star will become chemically homogeneous on equipotentials. However, if [FORMULA] then the distribution of the elements is determined by the extra drift velocity [FORMULA]. The criterion adopted in this paper for chemically inhomogeneous stars is that the extra drift velocity [FORMULA] is larger than the typical velocity of meridional motions [FORMULA]. In a recent numerical study of the a 20[FORMULA] star, Urpin et al. (1996) found that for high rotational velocities, the circulation velocity is less than [FORMULA] cm s-1. This value is significantly lower than that from Eddington-Sweet theory, which fails close to the surface of the star. If velocities of [FORMULA] cm s-1 are produced due to the effects of rotation, then the star may develop metal-rich and metal-poor regions on the same equipotential.

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© European Southern Observatory (ESO) 1999

Online publication: December 4, 1998
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