Overshooting from convective cores: Theory and numerical simulation

Convective overshooting increases the fraction of the star which is effectively mixed, thus altering models of stellar evolution. If the feed back of overshooting on the structure of the star is neglected the estimated extent of overshooting is very small. If the feed back is included in these estimates then the adiabatic core is extended by a distance comparable to a substantial fraction of the radius of the unstable region. An upper limit on convective overshooting is given by the integral constraint (Roxburgh 1978, 1989) with viscous dissipation neglected. If this constraint is applied to small convective cores then the maximum extent of the penetration region is shown to be at most about 0.18 times the radius of the core independent of the details of energy generation and opacity. The ratio of the maximum penetration distance to the scale height at the edge of the “classical boundary” varies very strongly with core size, and modelling overshooting by taking the penetration distance as a multiple of the scale height is likely to give misleading results. Numerical simulations of two-dimensional compressible convection in a fluid where the central regions are naturally convectively unstable, and the surrounding layers are stable, have been undertaken for different values of the Prandtl number. The results indicate that for low Prandtl numbers viscous dissipation is of decreasing importance and the simple integral condition gives a reasonable estimate of the extent of overshooting. Stellar seismology offers the possibility of detecting the location of the core — envelope interface through a periodic variation of the small frequency separation with frequency.