3D simulations and MLT: II. RA-ILES results.

In a previous paper (Arnett, et al., 2019) we introduced the use of Reynolds averaged implicit large eddy simulations (Moc\'ak, et al., 2019) to the classical problem of stellar convection(B\" ohm-Vitense, 1958; mixing length theory, MLT). We explored the structure of turbulent boundary layers, multi-modal behavior, intermittency, fluctuations, and composition gradients, and found that the Kolmogorov dissipation length played a role in some respects akin to the B\" ohm-Vitense mixing length. We now extend our analysis by extracting the sub-grid dissipation of our method (the "mixing length"), and by quantifying errors in resolution of boundary layers. The results for weakly-stratified convectionshow quantitative agreement with the four-fifths law of Kolmogorov. We examine the differences between weakly and strongly stratified convection(i.e., core convectionand surface convection zones, respectively). We find that MLT is a weak-stratification theory (which ignores turbulent kinetic energy), and for precise work should be modified for strong-stratification cases like the solar and stellar atmospheres. We derive the `effective mixing length' for strong-stratification; it is the density scale height, so $\alpha \approx \Gamma \sim 5/3$, in surprising agreement with many stellar evolutioncalibrations, but smaller than the preferred values for the Standard Solar Model(SSM), an error we attribute in part to the lack of a turbulent boundary layer, which we find at the bottom of the convection zonebut missing in MLT and SSM.