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Astron. Astrophys. 332, 127-134 (1998)

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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 [FORMULA] 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 [FORMULA] 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 [FORMULA] 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 [FORMULA], 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, [FORMULA], g, [FORMULA], [FORMULA], Q, v, [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA] have their usual meaning as given in the references (see Vitense 1953, Böhm-Vitense 1958 or Henyey et al.  1965). Besides [FORMULA] a second parameter, [FORMULA], is found in the expression for the convective velocity v,


In 1953 Vitense used [FORMULA] whereas in Böhm-Vitense (1958) she changed to [FORMULA] to improve the description of the turbulent friction. Henyey et al.  (1965) argued that the actual value might even be somewhat greater than [FORMULA]. The equation determining the convective efficiency factor [FORMULA] has two more free parameters replacing the volume to surface ratio of a convective element and different expressions for different optical thickness [FORMULA] of the convective bubble. Vitense (1953, her Eq. 8) formulated [FORMULA] as


She represented the unknown geometry by a new parameter y such that the volume to surface ratio can be written as [FORMULA]. 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 [FORMULA] 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 [FORMULA] and the expression in brackets reduced to [FORMULA]. All the formulations discussed above are based on [FORMULA]. Due to the different versions of the Böhm-Vitense mixing-length theory it must be stressed that the specification of the parameter [FORMULA] 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 [FORMULA], y or w is mostly compensated by the mixing-length parameter [FORMULA]. 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 [FORMULA] for the turbulent kinetic energy in the different eddies, in which k is related to the scale size [FORMULA] of an eddy by [FORMULA]. To calculate [FORMULA] 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 [FORMULA]. One refers to the well known [FORMULA] with [FORMULA] as pressure scale height, the other introduces a parameter-free theory where [FORMULA] 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.

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998