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

## 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

where

and

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

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

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

with a free parameter . 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, 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)

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

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

© European Southern Observatory (ESO) 1999

Online publication: November 26, 1998