*Astron. Astrophys. 332, 127-134 (1998)*
## 3. Consistent formulation of convection
A unified model of stellar evolution including stellar atmospheres
as an outer boundary condition requires a consistent formulation of
convective energy transfer, because the fit point at
is well inside the solar hydrogen convection
zone, and any differences between formulations used for the model
atmosphere and the stellar interior will lead to *different
convective fluxes*. Further any description of solar convection has
to fulfill *observational* restrictions implied by radius and
luminosity of the present Sun, the Balmer line spectrum emerging from
the surface of the Sun and by solar oscillations. Two models of
convection are considered for this purpose.
### 3.1. The model of Böhm-Vitense
Mixing-length theory as developed by Vitense (1953, see also
Böhm-Vitense 1958) and formulated by Cox & Giuli (1968) seems
to fit the properties of the present Sun provided the mixing-length
parameter is greater than 1.5, a value that has
been used as a lower limit in many treatments of the interior
structure of the Sun. A mixing-length parameter of
as it would be required by the observations of
the Balmer lines is outside any acceptable error limit imposed by the
solar radius. However, the model of Böhm-Vitense has more free
parameters than only the mixing-length parameter
, and in principle it could be possible to find a
set of parameters that fits the present Sun *and* the Balmer
lines.
In the following *T*, , *g*,
, , *Q*, *v*,
, , ,
, and
have their usual meaning as given in the
references (see Vitense 1953, Böhm-Vitense 1958 or Henyey
et al. 1965). Besides a second parameter,
, is found in the expression for the convective
velocity *v*,
In 1953 Vitense used whereas in
Böhm-Vitense (1958) she changed to to
improve the description of the turbulent friction. Henyey et al.
(1965) argued that the actual value might even be somewhat greater
than . The equation determining the convective
efficiency factor has two more free parameters
replacing the volume to surface ratio of a convective element and
different expressions for different optical thickness
of the convective bubble. Vitense (1953, her
Eq. 8) formulated as
She represented the unknown geometry by a new parameter *y*
such that the volume to surface ratio can be written as
. The transition from optically thick to thin
bubbles is then described by an arithmetic mean with a weighting
factor *w*. With an expression for the optical thickness
Eq. (6) becomes
where now *y* and *w* are the two additional free
parameters.
Eq. (7) corresponds to Henyey et al. (1965, their
Eqs. 39 and 40) if *y* is set equal to 1. Eq. (14.39)
in Cox & Giuli (1968) is recovered from Eq. (7) for
and the expression in brackets reduced to
. All the formulations discussed above are based
on . Due to the different versions of the
Böhm-Vitense mixing-length theory it must be stressed that the
specification of the parameter is
*meaningless* if there is no reference to the corresponding
formulation of the theory.
### 3.2. The model of Canuto & Mazzitelli
While the mixing-length theory in its original formulation
(Böhm-Vitense 1958) has been the primary source of coding
convective energy transfer for more than three decades, it has also
been referred to as being essentially a one-parameter theory (Gough
& Weiss 1976). In fact it can be shown that under conditions such
as are found in convective envelopes of cool unevolved stars the
variation of the additional 'free' parameters ,
*y* or *w* is mostly compensated by the mixing-length
parameter . In order to improve the
representation of turbulent convective elements in inviscid flows such
as encountered in stellar interiors Canuto & Mazzitelli (1991)
have developed a different model of convection. In contrast to the
standard mixing-length theory they use in their description not only
*one* single type of convective elements with fixed geometric
proportions but a *spectrum* of eddies. They introduce a
distribution function for the turbulent kinetic
energy in the different eddies, in which *k* is related to the
scale size of an eddy by
. To calculate they solve
a set of coupled equations that describe their turbulent convection
model. Finally, they obtain new expressions for the convective flux,
and they propose two different formulations of the mixing-length
. One refers to the well known
with as pressure scale
height, the other introduces a parameter-free theory where
with *z* being the distance from the top
of the convection zone. Further improvements and discussions are found
in Canuto & Mazzitelli (1992) and Canuto (1996) who have
thoroughly investigated the differences between standard mixing-length
theory and their energy spectrum representation.
© European Southern Observatory (ESO) 1998
Online publication: March 10, 1998
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