A simple scheme to implement a nonlocal turbulent convection model for   the convective overshoot mixing
  The classical 'ballistic' overshoot models show some contradictions and are not consistence with numerical simulations and asteroseismic studies. Asteroseismic studies imply that overshoot is a weak mixing process. Diffusion model is suitable to deal with it. The form of diffusion coefficient in a diffusion model is crucial. Because the overshoot mixing is related to the convective heat transport (i.e., entropy mixing), there should be a similarity between them. A recent overshoot mixing model shows consistence between composition mixing and entropy mixing in overshoot region. A prerequisite to apply the model is to know the dissipation rate of turbulent kinetic energy. The dissipation rate can be worked out by solving turbulent convection models (TCMs). But it is difficult to apply TCMs because of some numerical problems and the enormous time cost. In order to find a convenient way, we have used the asymptotical solution and simplified the TCM to be a single linear equation for turbulent kinetic energy. This linear model is easy to be implemented in the calculations of stellar evolution with ignorable extra time cost. We have tested the linear model in stellar evolution, and have found that the linear model can well reproduce the turbulent kinetic energy profile of full TCM, as well as the diffusion coefficient, abundance profile and the stellar evolutionary tracks. We have also studied the effects of different values of the model parameters and have found that the effect due to the modification of temperature gradient in the overshoot region is slight. 
Q. S. Zhang
astro-ph.SR