*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
In formula (1 ) *L* is the luminosity at radius *r*,
*M* the mass enclosed within a sphere of radius *r* (the
Lagrangian mass coordinate), the local gravity,
the total pressure, the
specific heat, the adiabatic temperature
gradient and is the actual gradient, which in a
stellar radiative zone equals the radiative one
, where 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 and of the mean
molecular weight as in Zahn's paper.
is the temperature scale height and
and give the mean energy
production rate and the mean density, respectively.
is the local nuclear energy generation rate,
the radiative conductivity, symbols
, and
, being used to denote
their logarithmic derivatives with respect to *T* and
*µ*. The quantity , with
, approaches the limit 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
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
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