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

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.