## 4. Results of the calculationsIn Fig. 1 solutions of the nonstationary problem are
presented for the model. The surface rotational
velocity km s
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 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. In Fig. 2 the same parameter distributions as those shown in
Fig. 1 are presented for the model
rotating with surface velocity
km s
A comparison of the numerical results for the two chosen 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 The meridional circulation velocity and the total diffusion
coefficient are found to scale approximately as
, for example, for
s 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
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 |