A Shear Stress Reynolds' Limit Formula

Historically, meteorological and climate studies have been prompted by the need for understanding precipitation to have better logistics in food production. Despite all efforts, nonlinearity in atmosphere dynamics is still a source of uncertainty. On the other hand, aeronautical science studies the boundary layer separation through the \emph{shear stress}. In this work, a mathematical interpretation of methods in classical aerodynamics theory in terms of successive layers of \emph{diffeomorphisms} over \emph{Lipschitz domains} allows us to estimate the boundary layer's \emph{shear stress}, $\tau^{*}_d$ and $\tau^{*}_m$, in dry and humid atmospheric conditions without assuming that there is not a convective derivative term in the conservation of momentum equation or that the gaseous boundary layer is incompressible: $$\tau^{*}_d = \frac{U}{h}\ \left(1-\frac{U^2}{2c_{pd}\ T_0}\right)^{19/25}, \hspace{7pt} \tau^{*}_m = \frac{U}{h}\ \left(1-\frac{U^2}{2c_{pm}\ T_0}\right)^{19/25},$$ where $h$ is the boundary layer's height, $T_0$ is the surface temperature, $U$ is the \emph{free stream velocity}; $c_{pd}$ is the \emph{specific heat at constant pressure for dry air} and $c_{pm}$ is the \emph{specific heat at constant pressure for moist air}. Furthermore, if $\hat{R}_m$ is a \emph{gas constant for moist air} and $p_0$ is the pressure at the surface, the density $\rho \hspace{2pt} \cong \hspace{2pt} p_0 \hspace{2pt} T_0^{\frac{2b}{b-1}-1} \hspace{2pt} \hat{R}_{m}^{-1} \hspace{2pt} \left[1-\left(U^2/2c_{ph}T_0\right)\right]^{\frac{b}{(b-1)}-1}$ for $b=1.405$. Moreover, this opens the possibility of finding a different deterministic family of atmosphere natural convection models.