Logarithmic profiles of velocity in stably stratified atmospheric boundary layers

The universal velocity log law proposed by von Kármán in wall-bounded turbulent flows is one of the cornerstones of turbulence theory. When buoyancy effects are important, the universal velocity log law is typically believed to break down according to Monin-Obukhov similarity theory (MOST), which has been used in almost all global weather and climate models to describe the dependence of the mean velocity profiles on buoyancy near the earth's surface and to characterize the surface-atmosphere exchange of momentum, heat, water vapor, and carbon dioxide. In contrast to MOST, we propose logarithmic profiles of near-wall mean velocity in the stably stratified atmospheric boundary layers based on direct numerical simulations and field observations across a wide range of buoyancy effects. We find that buoyancy does not seem to change the logarithmic nature of velocity profiles but instead modifies the slope of the log law in stably stratified conditions. This paper provides a perspective on wall turbulence and can be applied to numerical simulations of turbulence, weather, and climate.

Figure 1

Normalized velocity $U/u_{\tau }$ and momentum flux $-\overline{w^{\prime }{u}^{\prime }}/u_{\tau }^2$ in the vertical direction of the DNS experiments. The blue line denotes the normalized momentum flux $-\overline{w^{\prime }{u}^{\prime }}/u_{\tau }^2$ in the velocity log law region.

Figure 2

Vertical profiles of normalized velocity gradient $(z/u_{\tau }\partial u/\partial z)$ (equal to $1/{\kappa }_u$) in DNS data sets. The Monin-Obukhov similarity functions of Businger et al. [68] denoted by MOST: BD and Gryanik et al. [70] denoted by MOST: GLGS are also shown. The Businger et al. [68] formula: $\frac{z}{u_{\tau }}\frac{\partial U}{\partial z}=\frac{1}{\kappa }(1+4.7\frac{z}{L})$. The Gryanik et al. [70] formula: $\frac{z}{u_{\tau }}\frac{\partial U}{\partial z}=\frac{1}{\kappa }(1+\frac{5.0z/L}{{(1+0.3z/L)}^{2/3}})$.

Figure 3

Normalized velocity $\frac{U-U_{h1}}{u_{\tau }}$ in the vertical direction of Cabauw observations. $R^2$ denotes the coefficient of determination for $\frac{U-U_{h1}}{u_{\tau }}$ and $ln\left(z^{+}\right)$, and $z_{10m}/L$ denotes the stability parameter $z/L$ at the height 10 m. MOST: BD and MOST: GLGS denote the computed velocity profiles based on Businger et al. [68] and Gryanik et al. [70], respectively.

Figure 4

The ratio $\frac{{\kappa }_u}{\kappa }$ plotted against (a) $\frac{z_i}{L}$, and (b) $\frac{z_i}{{\delta }_v}$ under various stably stratified conditions in DNS experiments and Cabauw observations. The linear fit (with a coefficient of determination $R^2=0.7$) for the field observations is also shown in (a).