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Astron. Astrophys. 334, 1000-1006 (1998)

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4. A revised expression for the velocity of meridional circulation

The velocity of meridional circulation is derived from the equation of energy conservation (cf. Mestel 1965)

[EQUATION]

where S is the entropy per unit mass and [FORMULA] the thermal conductivity. The term [FORMULA] refers to the nuclear energy only. We include the flux [FORMULA] of thermal energy due to horizontal turbulence; it can be approximated by [FORMULA]. [FORMULA] is the coefficient of horizontal turbulent diffusion, which we have introduced above in x3; it is large with respect to vertical diffusivity, a property which ensures shellular rotation and tends to smoothen the chemical composition over horizontal surfaces (cf. Zahn 1992).

The problem of meridional circulation has been studied for 3/4 of a century (Eddington 1925); it is rather surprising then that it is still in debate nowadays. However, there are four points which lead us to re-examine that old, important astrophysical problem.

1. The first point is rather minor. Usually, only the case of a perfect gas is considered, but for massive stars, in particular, a more general equation of state is needed, which is easy to implement.

2. In general the approximation of a stationary circulation is made, which is certainly valid for main sequence stars. However, in the shell H-burning phase, in the He-burning and subsequent phases, it is not applicable and we need to remove this hypothesis.

3. The presence of the term containing [FORMULA], the thermal flux due to the horizontal turbulence in expr. (4.13), introduces some changes which need to be accounted for.

4. Some effects of the gradients of the mean molecular weight µ, in particular horizontal homogeneities, have already been taken into account by Chaboyer and Zahn (1992) and Zahn (1992), who considered mainly stationary stellar models. However, some rather large effects due to the radial µ-gradients in the equation of energy conservation must also be considered in the case of evolutionary models.

In the following we try to be as short as possible and we mainly refer to Zahn (1992) whenever the developments are similar. All quantities are expanded linearly around their average on a level surface or isobar, for example

[EQUATION]

Later on, we shall need also a similar expansion for the divergence of the centrifugal force:

[EQUATION]

To express the non-stationarity we may develop (4.13) in

[EQUATION]

and note that the horizontal average in lagrangian coordinates is (cf. Kippenhahn and Weigert 1990)

[EQUATION]

where [FORMULA] is the release rate of gravitational energy.

From now on, we concentrate on the horizontal perturbations, and proceed to linearize Eq. (4.13), putting for convenience the horizontal diffusion term on the left hand side:

[EQUATION]

The horizontal average [FORMULA] is zero on a level surface, which means that the star is in average radiative equilibrium including nuclear and gravitational energy production. The second last term involving [FORMULA] in square brackets was missing in Zahn (1992), as pointed out by Urpin et al. (1995).

Since

[EQUATION]

with [FORMULA], we get

[EQUATION]

The terms are ordered in the same way as in Zahn (1992) to facilitate the comparison. Note in particular the term [FORMULA] associated with [FORMULA]. The last term in [FORMULA] will be neglected later on because it is of the order of [FORMULA] smaller than the other terms in [FORMULA].

The derivative [FORMULA] is replaced by [FORMULA] and we introduce the auxiliary variables [FORMULA] and [FORMULA]. By using a general equation of state (cf. Kippenhahn and Weigert 1990) of the form

[EQUATION]

we get for the subsonic perturbations on the isobar

[EQUATION]

with [FORMULA] and [FORMULA]. For a mixture of perfect gas and radiation, [FORMULA] and [FORMULA]. Thus one has

[EQUATION]

The horizontal linear expansion of the radiative conductivity and nuclear energy generation rate can be written

[EQUATION]

and

[EQUATION]

where the indices T and µ refer to the derivatives with respect to T and µ, while [FORMULA] is the internal average [FORMULA]. The term [FORMULA] refers indeed to the sum of nuclear and gravitational energy.

The fluctuation of gravity may be calculated by solving the perturbed Poisson equation, as explained in Zahn (1992); an approxmiate expression is

[EQUATION]

The density fluctuation is drawn from the hydrostatic equation

[EQUATION]

These expressions imply that the horizontal variations of [FORMULA] and g are functions of [FORMULA]. We may also write expression (4.18) by separating the terms which depend on [FORMULA], either explicitely or through [FORMULA] or [FORMULA], from the terms which depend on the horizontal variation of µ:

[EQUATION]

(we use an asterisc here because we anticipate that this will not be the final form of [FORMULA]).

Without entering into details, we get, by defining [FORMULA], [FORMULA] the temperature scale height, [FORMULA] the average density inside the mass [FORMULA]

[EQUATION]

This writing has been chosen in order to facilitate the comparison with Zahn (1992) with whom we notice some slight differences, mainly due to the use of a more general equation of state.

In the case of uniform rotation [FORMULA], and only the first term in [FORMULA] remains in [FORMULA]. It is then positive, meaning that angular momentum is transported inwards, except close enough to the surface, as pointed out already by Gratton (1945) and Öpik (1951). When evolutionary effects are taken in account, the term [FORMULA] is also positive, and is the dominant one; the 2nd and 3rd terms in [FORMULA] can be significant near the surface, whereas all terms in [FORMULA] are completely negligible in general.

For [FORMULA] we have likewise

[EQUATION]

Let us now turn to the first member of our equation of energy conservation (4.28). Keeping only the first order term in the horizontal fluctuations, we can write

[EQUATION]

where [FORMULA] has been defined in Sect. 2: it is the amplitude of the radial component of the meridional velocity, [FORMULA].

We will see in the Appendix that in a medium of varying composition the entropy of mixing must be taken into account. In the simplest case, where the stellar material can be approximated by a simple mixture of H, He with a fixed abundance of metals, the entropy of mixing may be expressed in terms of the molecular weight only, and then

[EQUATION]

Since there are no pressure fluctuations on the isobar, we have (cf. 4.11)

[EQUATION]

Likewise, for the entropy gradient we get

[EQUATION]

with [FORMULA] and [FORMULA] the adiabatic and actual T -gradients, and [FORMULA]. At this point, we replace the time derivative of [FORMULA] by the expression derived in x3 (3.9):

[EQUATION]

to cast (4.28) in its final form

[EQUATION]

where we have replaced [FORMULA] by

[EQUATION]

The amplitude of the meridional velocity [FORMULA] can thus be written as

[EQUATION]

In a stationary situation, as is usually considered, the term [FORMULA] is zero, and one has [FORMULA] and [FORMULA] in the expressions for [FORMULA] and [FORMULA]. If, in addition, one takes a perfect gas with [FORMULA], one finds again the expressions given by Zahn (1992), except for the horizontal diffusion term which has been added here to [FORMULA] and, even more importantly, for the gradient of molecular weight [FORMULA] which appears in the superadiabatic gradient. The latter now takes the form which enters in the Ledoux criterion for convective instability and in the Brunt-Väisälä frequency:

[EQUATION]

In case of stationarity, one would also have from (3.9)

[EQUATION]

If as supposed by Zahn (1992) and by Chaboyer and Zahn (1992), there is some proportionality between [FORMULA] and [FORMULA], we get

[EQUATION]

with a factor [FORMULA] of the order of unity. This means that, in a situation of equilibrium, the horizontal fluctuations of µ are some fixed fraction of the vertical µ-gradient. It is noteworthy that this fraction does not depend on rotation to the first order. This can be understood in the following way: a larger rotation leads to a faster meridional circulation, thus building larger horizontal µ-fluctuations; however, at the same time, the larger rotation creates a stronger horizontal turbulence, which smoothens the horizontal µ-fluctuations, and the two effects, as far as they both depend on [FORMULA], tend to compensate each other.

However, the hypothesis of stationarity is not applicable after main-sequence evolution, because the timescales of the variations of entropy, i.e. the Kelvin-Helmholtz timescale and the evolutionary timescale, may be of the same order of magnitude, or the second one may even be shorter than the first one in the advanced stages. In that case the term in [FORMULA] will play an important role; it may then be derived from the time derivative of the [FORMULA] -gradient (see 4.27).

Before leaving this section, it may be useful to spell out the revised expression for [FORMULA]:

[EQUATION]

Finally, let us stress that the meridional velocity is insensitive to the specific dependence of entropy on the chemical composition, as illustrated by the absence of [FORMULA] in (4.38). This result can be easily extended to the general case where S is a function of the concentration of more than two species. A term containing [FORMULA] is remaining only in [FORMULA],

[EQUATION]

but such a µ-contribution is usually neglected in stellar evolution.

The origin of the term [FORMULA] in expression (4.38) for the velocity of the meridional circulation is interesting to trace back. This term is coming from the horizontal fluctuations [FORMULA] of entropy in (4.28), which depend on [FORMULA] an in turn on [FORMULA] through the equation of state. Then through the advection/diffusion equation (3.8) for the µ-variations, the horizontal fluctuations [FORMULA] also depend on the vertical µ-gradient [FORMULA]. Athough the effect of horizontal smoothing has been taken in account in the derivation of (4.28), the result is independent of [FORMULA] ; however the role of that smoothing is crucial, for it prevents the fluctuations of µ to become too large, thus allowing this linear treatment of the meridional circulation.

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998

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