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The O’KEYPS Equation and 60 Years Beyond

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Abstract

Some 60 years ago, six researchers obtained a semi-empirical equation that describes how the stability correction function for the mean velocity profile (\(\phi _\mathrm{m}\)) in the atmospheric surface layer varies with the stability parameter—the famous O’KEYPS equation. Their derivations are essentially based on interpolation of the turbulent eddy viscosity between neutral and convective conditions. Comparing the O’KEYPS equation with new theoretical developments—such as phenomenological and cospectral budget models—suggests that Heisenberg’s eddy viscosity provides a unifying framework for interpreting the behaviour of \(\phi _\mathrm{m}\). The empirical coefficient in the O’KEYPS equation, which is on the order of 10 based on data fitting to observations, is found to be primarily linked to the increase of the size of turbulent eddies as instability increases. The ratio of the sizes of turbulent eddies under convective and neutral conditions is on the order of \(1{/}\kappa \), where \(\kappa \) is the von Kármán constant, and is modulated by the turbulent Prandtl number.

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Acknowledgements

This material is based upon work supported by the U.S. National Science Foundation under Grant AGS-1853354. This paper was completed when I was visiting Leibniz University Hannover, while supported by the Alexander von Humboldt Foundation. I thank Professor Marc Parlange and Professor Heping Liu for allowing me to use the lake and dryland datasets.

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    Dan Li

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Li, D. The O’KEYPS Equation and 60 Years Beyond. Boundary-Layer Meteorol 179, 19–42 (2021). https://doi.org/10.1007/s10546-020-00585-y

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