ons).

Before closing, we would like to emphasize that the proposed shear-based parameterizations are only applicable away from the surface. Near the surface, due to the blocking effect [see @hunt88; @hunt89], $L_C$ or $L_H$ cannot be a representative length scale. They should be properly combined with an explicit parameterization involving height above ground (e.g., the harmonic mean of $0.4z$ and $L_H$).

Data and Code Availability {#data-and-code-availability .unnumbered}
==========================

The DNS code (HERCULES) is available from: <https://github.com/friedenhe/HERCULES>. Upon acceptance of the manuscript, all the analysis codes and processed data will be made publicly available via [zenodo.org](zenodo.org). Given the sheer size of the raw DNS dataset, it will not be uploaded on to any repository; however, it will be available upon request from the authors.

The first author thanks Bert Holtslag for thought-provoking discussions on this topic. The authors acknowledge computational resources obtained from the Department of Defense Supercomputing Resource Center (DSRC) for the direct numerical simulations. The views expressed in this paper do not reflect official policy or position by the U.S Air Force or the U.S. Government.

Appendix 1: Derivation of Length Scales {#appendix-1-derivation-of-length-scales .unnumbered}
=======================================

#### Integral Length Scale:

Based on the original ideas of Taylor![@taylor35], both Tennekes and Lumley [@tennekes72] and Pope [@pope00] provided a heuristic derivation of the integral length scale. Given TKE ($\overline{e}$) and mean energy dissipation rate ($\overline{\varepsilon}$), an associated integral time scale can be approximated as $\overline{e}/\varepsilon$. One can further assume $\sqrt{\overline{e}}$ to be the corresponding velocity scale. Thus, an integral length scale ($\mathcal{L}$) can be approximated as $\overline{e}^{3/2}/\varepsilon$.

In the literature, the autocorrelation function of the longitudinal velocity series is commo