The vertical meridional circulation velocity in stars may be written

∇ad µ„¦

ur = , (15.1)

∇ad ’ ∇ + ∇µ g

where g is the local gravitational acceleration, ∇ad and ∇ are respectively

the usual adiabatic and actual logarithmic gradients (d ln T /d ln P ) of tem-

perature T with respect to pressure P , and ∇µ the mean molecular weight

gradient (d ln µ/d ln P ). The factor µ„¦ is de¬ned below (equation 15.4).

Writing ur as

ur = Ur P2 (cos θ) , (15.2)

Element settling and rotation-induced mixing in slowly rotating stars 225

where P2 is a Legendre polynomial and θ is co-latitude, the horizontal merid-

ional velocity is given by

1d

uθ = ’ (ρr 2 Ur ) sin θ cos θ , (15.3)

2ρr dr

ρ being the density. The expression for µ„¦ is then obtained as a function of

the „¦ and µ-currents:

L

µ„¦ = (E„¦ + Eµ ) P2 (cos θ) , (15.4)

M

with

8 „¦2 r 3 „¦2

1’

E„¦ = , (15.5)

3 GM 2πGρ

ρm r d dΛ 2HT Λ

’ (χµ + χT + 1)Λ ’

Eµ = HT .(15.6)

ρ 3 dr dr r

Here L and M are the luminosity and mass of the star; ρ represents the

density average on the level surface ( ρ) while ρm is the mean density inside

the sphere of radius r; HT is the temperature scale height; Λ represents the

horizontal relative µ ¬‚uctuations; and χµ and χT represent the derivatives

‚ ln χ ‚ ln χ

χµ = ; χT = . (15.7)

‚ ln µ ‚ ln T

P,T P,µ

In these expressions, the deviations from perfect gas law have been neglected,

as have the energy production terms which are completely negligible in the

regions of the star where the process takes place. We have also assumed

di¬erential rotation to be negligible, as observed in the radiative interior of

the Sun from helioseismic studies. The corresponding condition on „¦ is

„¦2 r 3

‚ ln „¦

< . (15.8)

‚ ln r GM

15.3.2 Self-regulating process

The hydrodynamical situation inside the star when the two currents be-

come of the same order of magnitude is di¬cult to treat numerically, as

the meridional circulation velocity nearly vanishes while two opposite “big

numbers” cancel each other. A 2D numerical simulation of this process has

been done in a static stellar model and gives interesting results, which will

be extensively described by Th´ado & Vauclair (2002).

e

In the critical situation, the horizontal µ-gradient Λ remains everywhere

Vauclair

226

close to the critical value Λc , except just below the convective zone. As mi-

croscopic di¬usion does proceed, Λ is forced to remain small in the boundary

layer, because the chemical composition is homogeneous inside the convec-

tive zone. As a consequence, the vertical velocity ur strongly increases with

radius as it goes from regions where |Eµ | |E„¦ | to regions where Eµ 0.

The horizontal ¬‚ow, which is derived from the mass continuity equation, is

related to the induced negative divergence of the vertical ¬‚ux:

div(ur ρr 2 ) = ’u‘ ρr . (15.9)

Contrary to “normal” circulation, the direction of u‘ is reversed: it now goes

from the downgoing ¬‚ow towards the upgoing one while it normally proceeds

the other way round. A loop forms below the convective zone so that the

matter which goes down at the equator is brought back up to the convective

zone in the upwards ¬‚ow. The horizontal velocity in this boundary region

is much larger than the vertical velocity because of the large variation of

(ur ρr 2 ) (equation 15.9). Its order of magnitude is given by

u‘ HΛ ∼ ur r , (15.10)

where HΛ is the di¬usion scale height, needed for Λ to increase from zero to

Λc . From numerical simulation, HΛ is of order r/100, so that the horizontal

velocities are about 100 times larger than the vertical ones. Below this

region, the mixing velocities are very small in all the layers where the two

currents are of the same order of magnitude.

15.4 Conclusion

The e¬ect of the di¬usion-induced µ-gradients on the meridional circulation,

which were not introduced in previous computations, seems of fundamental

importance in computing abundance variations in stellar surfaces. In the

past, this e¬ect was overcome by some parametrized inertial term introduced

to avoid numerical instabilities. The di¬culty comes from the fact that

two large terms nearly cancel, which is di¬cult to treat numerically in the

process of stellar evolution. The preliminary results presented here have

been obtained with some assumptions. Two cases have been treated:

1) For lithium in halo stars, computations have been done in a stellar evo-

lution code, with the assumption that, as soon as |Eµ | |E„¦ |, the horizontal

and vertical µ-gradients adjust themselves to keep close to the critical regime

(Th´ado & Vauclair 2001). We have shown that, under this assumption, the

e

lithium plateau is well reproduced.

Element settling and rotation-induced mixing in slowly rotating stars 227

2) To test the process more precisely, we have done a 2D simulation in

a static model, without any such assumption. We ¬nd that, as soon as

|Eµ | |E„¦ |, a new loop takes place just below the convective zone, while

the µ-gradients remain close to the critical values.

In the near future, the consequences of this process will be tested for

the Sun and solar-type stars. The basic interest is that, for the ¬rst time,

we have evidence of a mixing process which is directly modulated by the

element settling.

References

Antia, H.M. & Chitre, S.M., 1998, A&A 339, 239

Balachandran, S.C. & Bell, R.A., 1997, American Astron. Soc. Meeting 191, 7408

Basu, S., 1997, MNRAS 288, 572

Basu, S., 1998, MNRAS 298, 719

Bonifacio, P. & Molaro, P., 1997, MNRAS 285, 847

Brun, A.S., Turck-Chieze, S. & Morel, P., 1998, ApJ 506, 113

Christensen-Dalsgaard, J., Pro¬tt, C.R. & Thompson, M.J., 1993, ApJ 408, L75

Christensen-Dalsgaard, J., Dappen, W., et al., 1996, Science 272, 1286

Eddington, A.S., 1926, The internal constitution of the stars, Cambridge Univ.

Press, Cambridge

Geiss, J. 1993, in Prantzos, N., Vangioni-Flam, E. & Cass´, M., eds, Origin and

e

Evolution of the Elements, (Cambridge Univ. Press), p. 90

Geiss, J. & Gloecker, G., 1998, Space Sci. Rev. 84 , 239

Gough, D. O., Kosovichev, A. G., Toomre, J., et al., 1996, Science 272, 1296

Maeder, A. & Zahn, J.-P., 1998, A&A 334, 1000

Mestel, L., 1953, MNRAS 113, 716

Mestel, L., 2003, in this volume

Michaud, G., 1970, ApJ 160, 641

Michaud, G., Charland, Y., Vauclair, S. & Vauclair, G., 1976, ApJ 210, 447

Michaud, G., Fontaine, G. & Beaudet, G., 1984, ApJ 282, 206

Pinsonneault, M. H., Deliyannis, C. P. & Demarque, P., 1992, ApJS 78, 179

Pinsonneault, M. H., Walker, T. P., Steigman, G. & Narayanan, V. K., 1999, ApJ

527, 180

Richard, O., Vauclair, S., Charbonnel, C. & Dziembowski, W.A., 1996, A&A 312,

1000

Schatzman, E., 1969, A&A 3, 331

Schatzman, E., 1977, A&A 56, 211

Th´ado, S. & Vauclair, S., 2001, A&A 375, 70

e

Th´ado, S. & Vauclair, S., 2002, ApJ, in press

e

Vauclair, G., Vauclair, S. & Michaud, G., 1978a, ApJ 223, 920

Vauclair, S., 1988, ApJ 335, 971