2. Overshooting as a diffusive process
In a seminal paper Freytag et al. (1996) have investigated the role of overshoot from the lower boundary of outer stellar convective layers. Based on two-dimensional hydrodynamical models they showed convincingly how the dynamics of turbulent convection is governed by the fast narrow downdrafts, which give rise to overshooting beyond the formal convective boundary. The simulations showed that the assumption of a finite overshooting distance does not represent the situation appropriately. Rather, the velocity field of the downward motions extends beyond the region with significant convective flux and declines exponentially. The scale length is correlated with the pressure scale height, but, and this is an important point, depends on the particular stellar parameters. Although the velocities decline rapidly, they are still of order of typical diffusion speeds, such that convective motions can counteract the sedimentation process. The temperature structure below the formal convective boundary remains dominated by radiative diffusion, as is necessary for SSM, which already without overshooting reproduce the depth of the solar convective envelope extremely well. This is the important difference to other overshooting approaches, in which the temperature gradient below the formally stable boundary is changed as well.
The complete convective envelope and the adjacent overshooting region fit into the simulation box only for hot stars (A-type and DA white dwarfs). From this, Freytag et al. (1996) derived a relation for the diffusion coefficient appropriate for the overshooting:
Here, z is the radial distance from the formal lower boundary of the convective zone, the velocity scale height, and is a "typical" diffusion coefficient for convection at , e.g. resulting from the convection speed derived in the mixing-length picture. Freytag et al. (1996) give examples for : in terms of the pressure scale height it varies from for an A-type star to for the white dwarfs. Formulated this way, overshooting can easily be incorporated into SSM-codes, which already have implemented particle diffusion and treat convection as a fast diffusive process, too. The varying proportionality factor is then expressed a free parameter, .
Since the solar convective envelope could not be simulated completely, Freytag et al. (1996) could argue only qualitatively about the effect on the photospheric 7Li and 9Be abundance. They concluded that Mm ("with appreciable uncertainty") could be a reasonable choice to destroy 7Li during the solar main-sequence evolution. This value for corresponds to .
Bl"ocker et al. (1998) have calculated solar models using the qualitative results of Freytag et al. (1996) to investigate the effect of such an overshooting approach on the solar 7Li. Although the bottom-line of this work is that for - applied from the ZAMS on - the observed depletion of 7Li is obtained, the authors mention some potential problems: (i) for the PMS, f had to be adjusted (no number is quoted) to obtain the depletion by -0.3 dex observed in young open clusters; (ii) the MS-value for f is larger than that derived by Herwig et al. (1997) for overshooting in the same description from convective cores and deep convective envelopes (); (iii) no other mixing/diffusion process had been taken into account. As such, Bl"ocker et al. (1998) concluded correctly that their study is a demonstration of the potential effect of the overshooting prescription on 7Li, deserving more detailed follow-up investigation.
© European Southern Observatory (ESO) 1999
Online publication: June 18, 1999