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Astron. Astrophys. 341, 181-189 (1999)

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2. Basic equations

We have modified the expression for the amplitude of the radial component of the meridional circulation velocity U derived by Zahn (1992, Eqs. (3.37-39)) to take into account effects of the radiation pressure which are important in massive MS stars. Following a procedure similar to that used by Zahn we find

[EQUATION]

where

[EQUATION]

and

[EQUATION]

In formula (1 ) L is the luminosity at radius r, M the mass enclosed within a sphere of radius r (the Lagrangian mass coordinate), [FORMULA] the local gravity, [FORMULA] the total pressure, [FORMULA] the specific heat, [FORMULA] the adiabatic temperature gradient and [FORMULA] is the actual gradient, which in a stellar radiative zone equals the radiative one [FORMULA], where [FORMULA] is the opacity. Other quantities have their usual meaning. In expressions (2) and (3 ) we have used the same notations for the measure of the horizontal fluctuations of density [FORMULA] and of the mean molecular weight [FORMULA] as in Zahn's paper. [FORMULA] is the temperature scale height and [FORMULA] and [FORMULA] give the mean energy production rate and the mean density, respectively. [FORMULA] is the local nuclear energy generation rate, [FORMULA] the radiative conductivity, symbols [FORMULA], [FORMULA] and [FORMULA], [FORMULA] being used to denote their logarithmic derivatives with respect to T and µ. The quantity [FORMULA], with [FORMULA], approaches the limit [FORMULA] as soon as a contribution of the ideal gas pressure becomes dominating over that of the radiation one. In this case our Eqs. (2) and (3 ) turn into Zahn's Eqs. (3.37) and (3.38) with necessary small corrections applied to (3.37) (Zahn's private communication; see also Maeder & Zahn (1998)).

The evolution of the angular velocity profile can be calculated by solving the angular momentum transport equation

[EQUATION]

where [FORMULA] is the vertical component of the turbulent viscosity (Zahn 1992, 1997). This is a nonlinear fourth order PDE . For [FORMULA], which is assumed to be of the same order of magnitude as the turbulent diffusivity in the vertical direction [FORMULA], we take the expression used by Talon et al. (1997)

[EQUATION]

This was derived for the first time by Talon & Zahn (1997). In formula (5) [FORMULA] is the critical Richardson number, [FORMULA] the radiative diffusivity, [FORMULA] and [FORMULA] are the T- and µ- components of the squared Brunt-Väisälä (buoyancy) frequency, [FORMULA] being the pressure scale height, and [FORMULA] for a mixture of ideal gas and radiation. Unfortunately, there is no expression for the horizontal component of the turbulent diffusivity [FORMULA] deduced from first principles. Zahn (1992) has proposed to use the estimate

[EQUATION]

with a free parameter [FORMULA]. Making use of formula (6 ) we explicitly assume that it is the meridional circulation which is responsible for producing and sustaining a state of differential rotation on the level surfaces and, therefore, [FORMULA] is likely to be proportional to U.

The right hand side of Eq. (4) includes two terms, the first one describing advection by meridional circulation and the second one modelling vertical turbulent diffusion of the angular momentum. Due to effects of horizontal erosion the contribution of meridional circulation to the rhs of the equation of nuclear kinetics takes a diffusion form (Chaboyer & Zahn 1992)

[EQUATION]

where [FORMULA] is a source/sink term from nuclear reactions, [FORMULA] the relative abundance of the i-th nuclide (by particle number), and

[EQUATION]

Eqs. (1 -8) supplemented with appropriate initial and boundary conditions (see the next section) make a completely self-consistent set.

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© European Southern Observatory (ESO) 1999

Online publication: November 26, 1998
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