Upward Overshooting in Turbulent Compressible Convection. III. Calibrate Parameters for One-dimensional Reynolds Stress Model

TL;DR

This study calibrates a one-dimensional Reynolds stress model using 3D simulation data for turbulent compressible convection, successfully reproducing key features like thermal and dissipation layers, and offers practical indicators for stellar modeling.

Contribution

It provides calibrated coefficients for the Reynolds stress model specific to upward overshooting in turbulent convection, validating its application in stellar structure models.

Findings

Calibrated coefficients match pure convection zone values for convection and isotropic terms.

Diffusive coefficients differ significantly, indicating boundary effects.

Reynolds stress model reproduces thermal and dissipation layers seen in 3D simulations.

Abstract

In this paper, we calibrate the coefficients for the one-dimensional Reynolds stress model with the data generated from the three-dimensional numerical simulations of upward overshooting in turbulent compressible convection. It has been found that the calibrated convective and isotropic coefficients are almost the same as those calibrated in the pure convection zone. However, the calibrated diffusive coefficients differ significantly from those calibrated in the pure convection zone. We suspect that the diffusive effect induced by the boundary is stronger than by the adjacent stable zone. We have checked the validity of the downgradient approximation. We find that the prediction of the downgradient approximation on the third-order moments is unsatisfactory. However, the prediction on their derivatives is much better. It explains why the performance of the Reynolds stress model is reasonable in application to the real stars. With the calibrated coefficients, we have solved the full set of nonlocal turbulent equations on Reynolds stress model. We find that the Reynolds stress model has successfully produced the thermal adjustment layer and turbulent dissipation layer, which were identified in the three-dimensional numerical simulations. We suggest to use the inflection point of the auto-correlation of temperature perturbation and the P\'eclet number as the indicators on measuring the extents of the thermal adjustment layer and turbulent dissipation layer, respectively. This result may offer a practical guidance on the application of the Reynolds stress model in 1D stellar structure and evolution models.