Numerical Analysis of Nonlocal Convection-Comparison with Three-dimensional Numerical Simulations of Efficient Turbulent Convection

We compare 1D nonlocal turbulent convection models with 3D hydrodynamic numerical simulations. We study the validity of closure models and turbulent coefficients by varying the Prandtl number, the Peclet number, and the depth of the convection zone. Four closure models of the fourth-order moments are evaluated with the 3D simulation data. The performance of the closure models varies among different cases and different fourth-order moments. We solve the dynamic equations of moments together with equations of the thermal structure. Unfortunately, we cannot obtain steady-state solutions when these closure models of fourth-order moments are adopted. In contrast, the numerical solutions of the down-gradient approximations of the third-order moments are robust. We calibrate the coefficients of the 1D down-gradient model from the 3D simulation data. The calibrated coefficients are more robust in cases of deep convection zones. Finally, we have compared the 1D steady-state solutions with the 3D simulation results. The 1D model has captured many features that appear in the 3D simulations: (1) del - del(a) has a U-shape with a minimum value at the lower part of the convection zone; (2) there exists a bump for del - del(a) near the top of the convection zone when the Peclet number is large; and (3) the temperature gradient can be sub-adiabatic due to the nonlocal effect. However, aside from these similarities, the prediction on the kinetic energy flux is unsatisfactory.