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

TL;DR

This paper derives a higher-order mean velocity profile for the convective atmospheric boundary layer using asymptotic expansions and field data, improving accuracy over existing empirical models.

Contribution

It introduces a higher-order asymptotic expansion of the velocity profile accounting for deviations from classical theories, validated with recent field measurements.

Findings

Excellent agreement between the expansion and measurement data.

The derived friction law is valid to at least second order.

Higher-order profiles offer improved accuracy over empirical models.

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.