## 1. Introduction

The Monin–Obukhov similarity theory (MOST; Monin and Obukhov 1954; Obukhov 1946) is the foundation for our understanding of the atmospheric surface layer. It hypothesizes that the surface-layer (*Q*, the buoyancy parameter *β*, and the height from the surface *z*, where *κ* is the von Kármán constant. The theory predicts that for

While MOST has been successfully used in predicting the surface-layer scaling, it is nevertheless a hypothesis based largely on the phenomenology of the surface layer and dimensional arguments. Measurements can provide support to MOST, but it cannot positively prove it. In the present study, we derive MOST and the LFC scaling from first principles using the equations for the velocity and potential temperature variances and the method of matched asymptotic expansions (Bender and Orszag 1978; Van Dyke 1975; Cousteix and Mauss 2007). We also derive from the expansions the corrections to account for the departure from the LFC scaling for

In the following we first examine the variance equations for the velocity components and potential temperature to identify the mathematical structure of the problem (a singular perturbation problem). We then perform the method of matched asymptotic expansions to obtain MOST and the LFC scaling as well as the corrections to the latter for

## 2. The mathematical structure of the problem

*p*is the kinematic pressure. The upper- and lowercase letters denote the mean and fluctuating variables, respectively. The mean wind is aligned with the

*U*direction. When the shear production is absent (free convection), Eqs. (1)–(4) have the mixed-layer scaling; therefore, the resulting nondimensional solution depends on the nondimensional independent variable

*o*will be used to denote the outer variables defined in the next section. The mixed-layer scaling also holds in the presence of the mean shear production for

*z*), becomes a leading term in Eq. (2); that is, the presence of the shear production term results in a nonuniformly valid solution. This shear-production-dominated layer always exists, as long as it is above the roughness layer, for any small but nonzero mean shear. Therefore, zero mean shear is a singular limit for the solution; that is, the structure of the solution for a case with the mean shear approaching zero (but not equal to zero) is fundamentally different from that with the mean shear equaling zero. Consequently, the system described by Eqs. (1)–(4) has the structure of a singular perturbation problem, whose solution can be obtained using the method of matched asymptotic expansions. The layers with

## 3. Matched asymptotic expansions

In this section we use the method of matched asymptotic expansions to solve the singular perturbation problem to derive MOST and the LFC scaling for the vertical velocity (the horizontal components do not have this scaling) and potential temperature variances, as well as to obtain the second-order corrections to the LFC scaling. Matched asymptotic expansions are a method to solve a set of differential equations having a solution that has different scaling in different parts of the solution domain, that is, a nonuniformly valid solution. In this study they are the mixed-layer scaling and surface-layer scaling. The solution in each part of the domain is expressed as series expansions with their respective scaling. The expansions in the different parts are then asymptotically matched to obtain composite expansions (uniformly valid solution).

### a. Outer expansions

*τ*using the mixed-layer scales as follows:The nondimensional forms of Eqs. (1), (2), and (4) in terms of the outer variables areFor convenience, the equation for

*ϵ*is a small parameter whose order of magnitude has yet to be determined. However, it will become a leading term when

*z*is sufficiently small, and therefore results in a nonuniformly valid solution and a singular perturbation problem. Unlike this shear production term, the production of

*ϵ*can be written asThe equations for the leading-order terms

### b. Inner expansions and a proof of MOST

*z*is sufficiently small (in the inner layer), the term containing the mean shear in Eq. (8) becomes a leading term and the outer solution is no longer valid. A new scaling is needed in the inner layer. We define the dimensionless inner variables

*J. Fluid Mech.*). The nondimensional forms of Eqs. (1), (2), and (4) in terms of the inner variables areIn the inner layer

*J. Fluid Mech.*; Tong and Ding 2018, manuscript submitted to

*J. Fluid Mech.*). Therefore, the dynamics of the horizontal and vertical velocity components in the surface layer are M-O similar. We can therefore write the inner expansions for the vertical velocity and potential temperature variances asThe results in Eq. (22) are of fundamental importance: They show that the nondimensional vertical velocity and potential temperature variances are functions of

### c. Asymptotic matching to derive the LFC scaling

Since the outer and inner expansions describe the dynamics at the outer and inner scales, respectively, and are valid for

### d. Second-order corrections to the leading-order solutions (LFC scaling)

*ϵ*. Since the surface layer is a “constant flux” layer (e.g., Haugen et al. 1971), the turbulent flux

*t*by

*O*(1)] in Eq. (29). Thus,resulting inwhich is the same as that obtained using dimensional analysis (Carl et al. 1973). Therefore, the appropriate outer scale for

*ϵ*in Eq. (10) isSubstituting the outer expansions in Eq. (12) into Eqs. (7), (8), and (9) and collecting the terms of order

*ϵ*, we obtain the second-order equations for the outer variables,Since the first term on the right-hand side of Eq. (38) is now a leading term, we havewhere

### e. Comparison with measurements

The nondimensional coefficients *A*, *B*, *C*, *A* and *B* values the expansion for

The composite expansion for

The abovementioned comparisons show that by adding only the second-order corrections, the functional forms of the composite expansions already show very good agreement with the data, demonstrating the efficacy of the method of matched asymptotic expansions for analyzing the surface layer. Previously vertical profiles of turbulence statistics have been empirical expressions obtained by curve fitting field data [e.g., Caughey and Palmer (1979) for vertical velocity profiles]. Furthermore, the empirical curves for

## 4. Conclusions and discussions

In the reported study we used the method of matched asymptotic expansions to derive analytically Monin–Obukhov similarity theory for the vertical velocity and potential temperature variances and the local-free-convection scaling, which previously have been a hypothesis based on phenomenology. We focused on the vertical velocity and potential temperature variances. The equations for the horizontal velocity, vertical velocity, and potential temperature variances are used to derive MOST and the LFC scaling. The dominance of buoyancy and shear production terms in the outer and inner layers, which have different scaling properties, results in a nonuniformly valid solution and a singular perturbation problem, which is solved using the method of matched asymptotic expansions. We obtained

In deriving the inner equations [Eqs. (18)–(20)], we have used the surface-layer scaling of the terms in these equations, which is supported by observational evidence (e.g., Kaimal et al. 1976; Wyngaard et al. 1971). The surface-layer scaling of these terms can also be obtained from the surface-layer similarity of multipoint statistics (Tong and Nguyen 2015), which has also been derived mathematically using the method of matched asymptotic expansions (Tong and Ding 2018). Therefore, the derived scaling in the present study is a consequence of MMO, and the derivation is mathematically rigorous. The present work is also part of a comprehensive analytical derivation of MMO and MOST.

The present study uses the balance equations for the velocity and temperature variances to derive MOST and the LFC scaling for these variables, thereby providing strong analytical support to Monin–Obukhov similarity theory. The expansions go beyond the previous observation-based empirical formulas for turbulence statistics to provide physics-based, analytically derived expressions with clear physical origins and interpretations. These expressions and the understanding of the associated physics are also potentially important for a range of applications. The vertical velocity variance is often used in eddy viscosity and diffusivity models. For example, in numerical weather prediction models using column parameterization for the boundary layer, the analytical expression for the vertical velocity profile in convective boundary layers is important for improving the predicted temperature profile under convective conditions. The derived variance profiles can also benefit prediction of atmospheric dispersion and wave propagation.

This work was supported by the National Science Foundation through Grants AGS-1335995 and AGS-1561190.

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