## Date of Award

August 2020

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Department

Mechanical Engineering

## Committee Member

Chenning Tong

## Committee Member

Joshua B. Bostwick

## Committee Member

Nigel B. Kaye

## Committee Member

Richard S. Miller

## Committee Member

Leo G. Rebholz

## Abstract

The multi-point Monin-Obukhov similarity theory (MMO) scaling properties of the surface-layer statistics were derived analytically, including one-point ﬂuctuation statistics, multi-point statistics, and the mean velocity proﬁle. The Monin-Obukhov similarity theory (MOST) is the foundation for understanding the atmospheric surface layer. While many surface-layer statistics has been shown to follow MOST, a number of important statistics do not conform to MOST due to the incomplete similarity. MMO was proposed by Tong & Nguyen (2015) to address the issue of the incomplete similarity and has been successfully used in predicting turbulence spectra. However, both MOST and MMO were proposed as hypotheses based on phenomenology. Measurements can provide support to them, but cannot positively prove them. In this work, starting from the singularity nature of the convective atmospheric surface layer (CBL), we employ the method of matched asymptotic expansions to analytically derive them.

We derive MOST and the local-free-convection (LFC) scaling from the equations for the velocity and potential temperature variances. The different dominance of the buoyancy and shear effects in the outer and inner layers results in a nonuniformly valid solution and a singular perturbation problem. The Obukhov length L is shown to be the length scale of the inner layer and the inner expansions are functions of z/L, where z is the height from the ground, providing a proof of MOST. The LFC scaling is obtained by matching the leading-order terms between the two layers. We also derive the second-order corrections to the leading-order terms. The resulting composite solutions show very good agreement with ﬁeld data. We derive MMO analytically for the case of the horizontal Fourier transforms of the velocity and potential temperature ﬂuctuations, using the spectral forms of the Navier-Stokes and the potential temperature equations. We show that for the large-scale motions (wavenumber k < 1/z) in a convective surface layer the solution is uniformly valid with respect to z (i.e., as z decreases from z > −L to z < −L). However, for z < −L the solution is not uniformly valid with respective to k as it increases from k < −1/L to k > −1/L, resulting in a singular perturbation problem. We show that (1) −L is the characteristic horizontal length scale; (2) The Fourier transforms satisfy MMO with the nondimensional wavenumber −k L as the independent similarity variable. Two scaling ranges, the convective range and the dynamic range, discovered for z << −L in Tong & Nguyen (2015) are obtained. We derive the leading-order spectral scaling exponents for the two scaling ranges and the corrections to the scaling ranges for ﬁnite ratios of the length scales. The analysis also reveals the dominant dynamics in each scaling range. The analytical derivations of the characteristic horizontal length scale (L) and the validity of MMO for the case of two-point horizontal separations provide strong support to MMO for general multi-point velocity and temperature differences. The mean velocity proﬁle in the CBL is also derived using matched asymptotic expansions with the scaling properties needed provided by MMO. The shear-stress budget equations and the mean momentum equations are employed in the derivation. Three scaling layers are identiﬁed: the outer layer, which includes the mixed layer, the inner-outer layer and the inner-inner layer, which includes the roughness layer. There are two overlapping layers: the local-free-convection layer and the log layer, respectively. Two new velocity-defect laws are discovered: the mixed-layer velocity-defect law and the surface-layer velocity-defect law. The LFC mean proﬁle is obtained by asymptotically matching the expansions in the ﬁrst two layers. The log law is obtained by matching the expansions in the last two layers. The von Kármán constant is obtained using velocity and length scales, and therefore has a physical interpretation. A new friction law, the convective logarithmic friction law, is obtained. The present work provides an analytical derivation of the mean velocity proﬁle hypothesized in MOST.

## Recommended Citation

Ding, Mengjie, "Multi-point Monin–Obukhov Similarity Theory in the Convective Atmospheric Surface Layer Using Matched Asymptotic Expansions" (2020). *All Dissertations*. 2679.

https://open.clemson.edu/all_dissertations/2679