Some 60 years ago, six researchers obtained a semi-empirical equation that describes how the stability correction function for the mean velocity profile (ϕm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _\mathrm{m}$$\end{document}) 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 ϕm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _\mathrm{m}$$\end{document}. 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/κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1{/}\kappa $$\end{document}, where κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} is the von Kármán constant, and is modulated by the turbulent Prandtl number.

中文翻译：

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O'KEYPS 方程和 60 年后

大约 60 年前，六名研究人员获得了一个半经验方程，该方程描述了平均速度剖面的稳定性校正函数 (ϕm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _\mathrm{m}$$\end{ document}) 在大气表面层随稳定性参数而变化——著名的 O'KEYPS 方程。它们的推导基本上基于中性和对流条件之间湍流涡粘性的插值。将 O'KEYPS 方程与新的理论发展——例如现象学和共谱预算模型——进行比较，表明海森堡的涡粘性提供了一个统一的框架来解释 ϕm\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{ wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _\mathrm{ m}$$\end{文档}。O'KEYPS 方程中的经验系数基于对观测数据的拟合，约为 10，被发现主要与随着不稳定性增加而增加的湍流涡流大小有关。