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Astron. Astrophys. 329, 315-318 (1998)

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3. Numerical results

We solved the fourth order differential system describing the evolution of the rotation profile in the radiate zone using an implicit scheme, the internal structure being given by our stellar evolution code (a modified version of the Geneva code). The convective zones are assumed to rotate as solid bodies2. Here, we did not calculate the diffusion of chemicals associated with circulation and shear turbulence. More complete results will be presented in a forthcoming paper. Furthermore, we used a perfect gas law for the equation of state since the non ideal effects will not influence the evolution of the rotation profile.

Let us first examine the role played be each transport mechanism in the evolution of the rotation profile. If only circulation and shear turbulence are considered, the resulting [FORMULA] profile will have a very large slope, as the surface loses momentum through the wind much more rapidly than it can be transported by those two mechanisms (see Matias & Zahn 1997). The asymptotic regime described by Zahn (1992) in the case of a moderate wind is thus never achieved.

The solution for the wave transport (in the linear regime, when neglecting all variations except those of [FORMULA] ) is

[EQUATION]

where

[EQUATION]

We thus expect to see the rotation profile move inwards, with a velocity which is inversely proportional to the damping integral, leaving behind a region rotating at the velocity of the convective zone. However, since we know that the linear regime is valid only as long as differential rotation is not too strong, we should expect the rotation profile to be modified as the "front" moves inwards.

When considering all three mechanisms together, we may recognize the characteristics of both solutions. This is illustrated in Fig. 1, which is the result of an evolution for a solar mass star, starting as a fully convective object on the Hayashi line. The advance of the "synchronized" portion is manifest, whereas the steep [FORMULA] profile below is governed by the circulation and shear turbulence in regions where the internal waves cannot penetrate. One verifies that the circulation, being of advective nature, is far more efficient in the transport of angular momentum than the shear turbulence. Note however that, due to the erosion by anisotropic turbulence, the circulation is far less effective in the transport of chemicals, as was shown by Chaboyer & Zahn (1992).

[FIGURE] Fig. 1. Evolution of the solar rotation profile with time. The ages of the corresponding models are indicated in Gyrs. One sees clearly the advance of the front resulting from the action of gravity waves.

Fig. 2 illustrates the evolution of the rotation period. During the pre-main sequence evolution, the convective zone is quite deep and gravity waves are efficient enough to maintain a state of quasi-solid body rotation. The overall contraction leads to an acceleration of the surface. Reaching the main sequence phase, the wave production efficiency diminishes as the convection zone regresses and the stellar surface decelerates now more rapidly than the interior under the action of the magnetic torque. The retrograde waves are then not able to travel deeply into the radiative zone. At about [FORMULA] yrs, the wave front begins to penetrate into the radiative zone, from where it extracts angular momentum faster than the wind can carry it away. This leads to an increase of the surface velocity, at an age which depends on the magnitude of the total wave luminosity as well as on the initial angular momentum contained in the model. Unfortunately, no observations seem to exist that could confirm the existence of such a feature.

[FIGURE] Fig. 2. Evolution of the rotation period. The solid line represents the actual calculation, the dashed line, the evolution of a sun in solid body rotation with the same initial angular momentum and the dotted line, Skumanich's law for the same rotation period at the beginning of the main sequence.

Let us stress once more that the wave luminosity is still very unsure. The existence of a rapidly rotating core in the Sun could help us estimate that value. However, the efficiency of the wave production will be in competition with the initial angular momentum of the model (that is, the core can be rotating rapidly either due to the poor wave transport or to the very high initial angular momentum of the model), and other observational constraints will be required to sort out all these effects.

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© European Southern Observatory (ESO) 1998

Online publication: November 24, 1997
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