Turbulent convection model in the overshooting region: II. Theoretical analysis

TL;DR

This paper provides a semi-analytic analysis of turbulent convection models in stellar overshooting regions, revealing the structure, decay properties, and key parameters influencing turbulent heat transfer and kinetic energy distribution.

Contribution

It derives approximate and asymptotic solutions for turbulent convection in overshooting zones, clarifying the physical processes and simplifying model application in stellar evolution calculations.

Findings

The overshooting region has three distinct parts with different turbulence characteristics.

The turbulent heat flux overshooting length is about one turbulent scale height.

Turbulent correlations can be estimated using the boundary kinetic energy and exponential decay functions.

Abstract

Turbulent convection models are thought to be good tools to deal with the convective overshooting in the stellar interior. However, they are too complex to be applied in calculations of stellar structure and evolution. In order to understand the physical processes of the convective overshooting and to simplify the application of turbulent convection models, a semi-analytic solution is necessary. We obtain the approximate solution and asymptotic solution of the turbulent convection model in the overshooting region, and find some important properties of the convective overshooting: I. 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 $k$, $\bar{u_{r}'T'}$, and $\bar{T'T'}$ are only determined by the parameters of the TCM, and there is an equilibrium value of the anisotropic degree $\omega$. II. The overshooting length of the turbulent heat flux $\bar{u_{r}'T'}$ is about $1H_k$($H_k=|\frac{dr}{dlnk}|$). III. The value of the turbulent kinetic energy at the convective boundary $k_C$ can be estimated by a method called \textsl{the maximum of diffusion}. Turbulent correlations in the overshooting region can be estimated by using $k_C$ and exponentially decreasing functions with the decaying indices.