Higher-order mean velocity profile in the convective atmospheric boundary layer

Abstract

The higher-order mean velocity profile in the convective atmospheric boundary layer (CBL) is derived using the method of matched asymptotic expansions. The universal expansion coefficients are obtained using field measurement data. The profile accounts for the departures from the (leading-order) log law and local-free-convection scaling as well as the deviations from the Monin-Obukhov Similarity theory (MOST). Invoking MOST and the Multipoint Monin-Obukhov similarity theory, the perturbation equations are obtained from the Reynolds-stress, potential-temperature flux and potential temperature-variance budget equations and the mean momentum and mean potential temperature equations. The small parameters with the most impact in the equations are $(-z_i/L)^{-4/3}$, $(-z_i/L)^{-2/3}$ and $-h_0/L$, where $z_i$, $L$ and $h_0$ are the inversion height, the Obukhov length and the roughness height, respectively. Tong and Ding ({\it J.~Fluid Mech.} 2020) have identified the three-layer structure of the CBL In the present work, asymptotic matching between the outer and inner-outer layers also results in higher-order expansion terms. The expansion coefficients are obtained using measurement data from the recent M$^2$HATS field campaign. Comparisons between the expansions and the measurement show excellent agreement. The higher-order asymptotic expansions show that the convective logarithmic friction law derived by Tong and Ding (2020) is valid to at least the second order. The predicted friction law also agrees well with measurements. The higher-order mean velocity profile can provide improved accuracy over empirical profiles.

Figures (13)

• Figure 1: Schematic of the measurement set-up for obtaining the mean profile. Two met towers, 6 m and 32 m in height respectively, provided the mean velocity, the mean shear stress, flux, and gradients near the surface. A Doppler lidar performing Plan Position Indicator (PPI or cone) scan, was used to measure the mean horizontal velocity profiles throughout the boundary layer.
• Figure 2: Selected non-dimensional height range for matching between the outer and inner-outer layers
• Figure 3: Planar fit of the expansion coefficients to data in the matching region between the outer and inner-outer layers (local-free-convection layer). The variable, $Y = \dfrac{U}{u_*}-\dfrac{U_m}{u_*} - E(-\dfrac{z}{L})^{-5/3} - G(-\dfrac{z}{L})^{-3}$, is plotted on the vertical axis. The slopes of the planar fit yield the coefficients $A$ and $D$ associated with the $X_1 = (-\dfrac{z}{L})^{-1/3}$ and $X_2 = \epsilon'_3(-\dfrac{z}{L})^{1/3}$ terms, respectively. Darker and lighter symbols are above and below the plane respectively.
• Figure 4: Mean velocity profiles (velocity defect) in the local-free-convection layer. The measured data are shown as circles, and the black curves denote the predicted mean velocity profiles for a range of stability conditions characterized by different values of $-z_i/L$: 43.6, 74.3 and 146.2.
• Figure 5: Planar fit of the velocity-profile in the log-law layer. The fit yields the inverse of the von Kármán constant and the coefficient of the first higher-order term, which accounts for a part of the deviations from the logarithmic velocity profile. Symbols same as in figure \ref{['fig:planar_lfc']}.