On the velocity, temperature, and humidity profiles in the lower atmosphere

On the basis of the assumption that the eddy viscosity, eddy conductivity, and eddy diffusivity are determined by the first and second differential coefficients of the mean velocity U , mean potential temperature Θ, and mean humidity X (specific humidity, mixing ratio, or water‐vapour pressure) the following formulae are obtained for the lower atmosphere over level ground (valid up to a height of about 25 m),\documentclass{article}\pagestyle{empty}\begin{document}$$ \frac{U}{{U_* }} = \frac{1}{k}\log \frac{{z - d}}{a},\frac{{\Theta - \Theta _0 }}{{\Theta _* }} = \frac{l}{l}\log \frac{{z - d}}{b},\frac{{x - x_0 }}{{x_* }} = \frac{1}{m}\log \frac{{z - d}}{c} $$\end{document}Here Θ, and X 0 denote potential temperature and humidity at the ground. U * is the friction velocity and Θ * and X * denote corresponding quantities for temperature and humidity. The above logarithmic formulae are in good agreement with Best's measurements when the parameters d, a, b, k, l are chosen to give the best fit with his observed velocity and temperature profiles which are grouped according to stratification and mean velocity. The zero‐plane displacement d is inferred to be the same for all profiles. Measurements by Sverdrup indicate that this is true for for temperatures whereas measurements by Rider and Robinson indicate that it is not always true for higher temperatures. Theoretical reasoning leads to the conclusion that d should increase with thermal instability and this is verified by Best's data. It is also inferred that k, l , and m should increase with instability, whereas a is found to decrease with instability.