4. Results of the calculations
In Fig. 1 solutions of the nonstationary problem are presented for the model. The surface rotational velocity km s-1 has been chosen as large as those observed in the most rapidly rotating OB-stars. It corresponds to s-1 and to the ratio of the centrifugal acceleration to the gravity on the stellar equator . The solid line in Fig. 1a gives the steady-state asymptotic solution (Eq. (12 )). One can see that in this particular case the nonstationary solutions approach the stationary one very quickly (numbers along the lines show their respective ages in years). The MS life-time for the model is years, so, for s-1 it takes only about 1% of the MS life-time for the star to settle in the steady-state rotation. This "relaxation" time can be roughly estimated a priori as the ratio (see below) which is approximately proportional to (Eqs. (1-2 )) for a fixed stellar mass. Obviously, the more rapidly rotating stars reach the state of asymptotic rotation quicker than stars rotating slower.
In panels b and c the evolution of the meridional circulation velocity and of the total diffusion coefficient are shown. For the initial uniform rotation law the quantity U changes sign near the stellar surface. This is a well-known result being due to the negative term in brackets in the first row of expression (2) (Gratton 1945; Öpik 1951). In the classical description with the uniform rotation law applied, the meridional circulation currents consist of two zones, the mass of the outer zone being much smaller than that of the inner zone (Pavlov & Yakovlev 1978). Slow mixing across the "quiet" layer separating these zones can occur only as a purely microscopic diffusion process (Charbonnel & Vauclair 1992). In the past there were even some speculations that the quiet layer could prevent the radiative envelope of a single rotating MS star from being fully mixed (Vauclair 1988; Leushin et al. 1989). However, in Zahn's scheme, as our calculations show (see also Urpin et al. 1996; Talon et al. 1997), the position of the quiet layer shifts towards the stellar surface as the rotation profile evolves. As a result, the outer zone gradually (and, for a large enough value of , rather quickly) becomes smaller in mass and then disappears completely. Simultaneously with these changes a characteristic value of U (say, in the middle of the envelope) decreases by a factor of about 50. As long as the rotation profile stays close to the initial flat one, (i.e. meridional circulation) remains the main contributor to the total diffusion coefficient. On the other hand, for approaching the asymptotic distribution, (turbulent diffusion) begins to play a dominant role. Finally, the sum is found to decrease near the convective core border by dex compared to the case of uniform rotation (Fig. 1 c).
In Fig. 1d we have plotted the ratio against the relative mass coordinate, the mixing time being here defined as . We see that even for the rotation profiles lying very close to the flat one, exceeds in the greater part of the inner envelope. This result strongly contrasts with the classical (Eddington-Sweet) estimate of the mixing time
where is the Kelvin-Helmholtz time. For our ZAMS model years, so, for and years we get ! A main reason for such the large difference in the mixing time-scales (Zahn's scheme versus the classical description) is the horizontal erosion responsible for the factor appearing in (8) which considerably reduces the product , the classical estimate of the mixing rate. It is this factor that causes a substantial diminution of even for the nearly uniform rotation.
It should be noted that in real evolving stellar models the advection and diffusion will never reach a stage where they exactly balance each other so as to make the time derivative in (4) equal to zero. Therefore, it would be more correctly to consider not as a time to achieve a stationary regime (which is never met) but instead as a characteristic time for the change of the inner rotation distribution.
This time is much shorter than the mixing time defined above because the angular momentum transport by meridional circulation is not affected by the horizontal erosion (Eq. (4 )). It is the difference in the rates of chemical mixing and of angular momentum transport that is considered as the main advantage of Zahn's new scheme, particularily, for the interpretation of a rather quick spin-down of the Sun and solar-like stars accompanied by a much slower depletion of their surface Li abundance (Zahn 1997).
If we take the mass-radius relation of Beech & Mitalas (1994) valid for MS stars in the mass range , then and, consequently, in the first approximation . This relation holds (within an uncertainty factor of ) in our numerical calculations. For example, the model has years (the Kelvin-Helmholtz time increases with decreasing M on the MS), and for and s-1 the calculations give and years, respectively (compared to and years for the model rotating with s-1).
The meridional circulation velocity and the total diffusion coefficient are found to scale approximately as , for example, for s-1 distributions of versus go about 0.6 () above the curves plotted in Fig. 2c for s-1.
From Figs. 2c and 2d we infer that there is no a big difference (if one does not consider very deep layers adjacent to the convective core) between a state of rotation close to the uniform one and that approaching the asymptotic regime with respect to their ability (or, better to say, disability) to mix chemical elements. In particular, complete mixing of the radiative envelope is reached in neither case. Therefore, searching for an additional angular momentum transport mechanism (for instance, internal gravity waves; Zahn et al. 1997; see also the next section), which could support a state of nearly uniform rotation will hardly help to speed up mixing of chemical elements unless this new mechanism can effectively mix them itself. But it is important to note that even in the most unfavourable considered case of , s-1 and years the turbulent diffusion (we have mentioned above that for rotation close to the steady-state one is much smaller than ) turns out to be fast enough for some mixing, as diffusion-like abundance profiles in the envelope are built up and the surface abundance of 14 N is altered significantly by the end of the star's MS life (Fig. 3). To plot Fig. 3 we solved Eqs. (7 ) by the method and with the input physics described in Denissenkov et al. (1998, the nuclear kinetics network for 26 nuclides) for the fixed temperature and density distributions taken from our present ZAMS models. Note that the surface N abundance has begun to increase only after some delay time required for the diffusion wave to reach the surface (Fig. 3b, see also Talon et al. 1997).
Fig. 4 demonstrates that shortly after the rotation profile has begun to evolve, all the consistency criteria (14-17 ) are fulfilled, and only during the very early evolution we meet a situation when , but we ignored that.
© European Southern Observatory (ESO) 1999
Online publication: November 26, 1998