TURBULENT CONVECTION MODEL IN THE OVERSHOOTING REGION. II. THEORETICAL ANALYSIS

Turbulent convection models (TCMs) are thought to be good tools to deal with the convective overshooting in the stellar interior. However, they are too complex to be applied to calculations of stellar structure and evolution. In order to understand the physical processes of the convective overshooting and to simplify the application of TCMs, a semi-analytic solution is necessary. We obtain the approximate solution and asymptotic solution of the TCM in the overshooting region, and find some important properties of the convective overshooting. (1) The overshooting region can be partitioned into three parts: a thin region just outside the convective boundary with high efficiency of turbulent heat transfer, a power-law dissipation region of turbulent kinetic energy in the middle, and a thermal dissipation area with rapidly decreasing turbulent kinetic energy. The decaying indices of the turbulent correlations k, (u'T-r') over bar, and (T'T') over bar are only determined by the parameters of the TCM, and there is an equilibrium value of the anisotropic degree.. (2) The overshooting length of the turbulent heat flux (u'T-r') over bar is about 1H(k) (H-k = vertical bar dr/d ln k vertical bar). (3) The value of the turbulent kinetic energy at the convective boundary k(C) can be estimated by a method called the maximum of diffusion. Turbulent correlations in the overshooting region can be estimated by using kC and exponentially decreasing functions with the decaying indices.