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 () dynamics is governed by the kinematic surface stress (the square of the friction velocity) , the surface temperature flux Q, the buoyancy parameter β, and the height from the surface z, where is the boundary layer (inversion) height. Any nondimensional statistics therefore is a function of , where is the Obukhov length, and κ is the von Kármán constant. The theory predicts that for , the shear production of the turbulent kinetic energy dominates, and for , the buoyancy production dominates. In particular, it predicts that for the special case of (but ), the influence of mean shear becomes negligible, resulting in the so-called local-free-convection (LFC) scaling (Tennekes 1970; Wyngaard et al. 1971). In this layer the vertical velocity and potential temperature variances do not depend on , and have the forms and , respectively. However, when the conditions and are not satisfied (e.g., or ), we expect departure from the LFC scaling (Wyngaard et al. 1971). Field measurements support the prediction for the vertical velocity and potential temperature variances (section 3e).

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 or . The composite expansions include the influence of both and and are based on the physics of the surface layer. The former is an approximation of the Monin–Obukhov functions near , whose functional forms cannot be obtained analytically from dimensional analysis. Instead they must be obtained empirically from observations. The influence of is absent in MOST. Thus, the derivation is a major step toward an analytical proof of MOST. The analytical prediction of the variance profiles can benefit numerical weather prediction models under convective conditions. It will also be important for modeling atmospheric dispersion.

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 or . The analytical results for LFC are then compared with measurements and are followed by the conclusions.

2. The mathematical structure of the problem

The equations for the velocity components and potential temperature variances in a horizontally homogeneous atmospheric boundary layer are (e.g., Wyngaard and Coté 1971)

where , , , and are dissipation rates for , , , and , respectively; and 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 :

where is the mixed-layer velocity scale. The subscript 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 , since the effects of the shear production (of ) are small within this range of heights. Note that the terms in the and equations acquire the mixed-layer scaling (except the shear production term) as a result of the pressure–strain-rate correlation. However, the solutions in Eq. (5) are not valid for , where the shear production, which has the surface-layer scaling (as a result of the parameters and 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 and are the so-called outer and inner layers, respectively.

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

We define the dimensionless outer variables in the singular perturbation problem , , , , , , , , , and τ using the mixed-layer scales as follows:

The nondimensional forms of Eqs. (1), (2), and (4) in terms of the outer variables are

For convenience, the equation for is not given, as it is not explicitly used in the analysis. In the outer layer, the buoyancy production term and the pressure–strain-rate terms are of order one (leading terms), while the nondimensional shear production term ; therefore, it is a second-order term and can be written as

where 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 in Eq. (9) is a leading term and therefore is not the source of singularity as .

The outer expansions of the velocity components and potential temperature and their variances in terms of the power of ϵ can be written as

The equations for the leading-order terms , , and are obtained by substituting Eqs. (11) and (12) into Eqs. (7), (8), and (9) and collecting terms of order one ().

b. Inner expansions and a proof of MOST

As discussed above, when 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 , , , , , , , and as follows:

where is the inner length scale (thickness of the inner layer) and has yet to be determined. Here is used as the velocity scale, as shown in the derivation of the multipoint Monin–Obukhov similarity (MMO; Tong and Nguyen 2015; Tong and Ding 2018, manuscript submitted to J. Fluid Mech.). The nondimensional forms of Eqs. (1), (2), and (4) in terms of the inner variables are

In the inner layer , needs to be a leading-order term; thus, it must scale as

leading to . The fact that the pressure–strain-rate correlation terms are of the same order of magnitude and are leading-order terms in both the and equations is used in deriving Eq. (17). Therefore, the inner length scale is the Obukhov length. Equations (14), (15), and (16) become

where

is a small parameter. We note that for the vertical velocity and temperature variances to follow MOST—that is, to be Monin–Obukhov (M-O) similar, and to have a solution that scales with the inner variables—the variance equations must also be M-O similar. Although the horizontal velocity variances have mixed-layer scaling and are not M-O similar, their rate equations are (or have apparent M-O similarity; Ding et al. 2018, manuscript submitted to 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 as

The results in Eq. (22) are of fundamental importance: They show that the nondimensional vertical velocity and potential temperature variances are functions of only, thus providing a proof of MOST for these variables.

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 and , there exists an overlapping region where both conditions are satisfied and the expansions represent the same function. Therefore, if we write the outer expansion as a function of the inner variable and the inner expansion as a function of the outer variable, they should be equal.

The inner expansion of the outer expansion of is

keeping only one term.

The outer expansion of the inner expansion of is

Matching Eqs. (23) and (24) results in . Thus,

This is the LFC scaling for the vertical velocity variance, and is the velocity scale.

The inner expansion of the outer expansion of is

keeping only one term.

The outer expansion of the inner expansion of is

Matching Eqs. (26) and (27) results in . Thus,

This is the LFC scaling for the temperature variance.

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

When the conditions for the overlapping region ( and ) are not satisfied, departures from the LFC scaling [Eqs. (25) and (28)] are expected. Corrections to account for the departures can be made by including the higher-order terms in the expansions in Eqs. (12) and (22). To obtain the second-order terms, we need to first determine the scaling of the shear production term and the small parameter ϵ. Since the surface layer is a “constant flux” layer (e.g., Haugen et al. 1971), the turbulent flux is approximately independent of height from the surface and scales as . We consider the shear stress budget equation (e.g., Wyngaard et al. 1971),

Nondimensionalizing by , t by , and by results in

The shear production term must be a leading term [O(1)] in Eq. (29). Thus,

resulting in

which is the same as that obtained using dimensional analysis (Carl et al. 1973). Therefore, the appropriate outer scale for is . We can then define the outer variable as

The shear production term in Eq. (8) can then be written as

Its nondimensional form is

Therefore, the small parameter ϵ in Eq. (10) is

Substituting 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 have

where is the correlation between and , and , , and are the dissipation rates for , , and , respectively. Here we assume that the correlation coefficient between and is also . The fact that the pressure–strain-rate correlation terms are of the same order of magnitude and are leading terms in Eqs. (37) and (38) is used in deriving Eq. (40). In the last step the estimate is used. Therefore, from Eqs. (33) and (25)

giving

Thus, the second-order correction term is . From Eqs. (39) and (42),

and

where is the correlation coefficient between and . Therefore, the second-order term . Thus, and with the second-order corrections for are

and

respectively. Rewriting the last two equations in dimensional form and dropping the terms, we have

and

respectively. Each of these expressions contains two nondimensional coefficients, which will be determined using measurements (section 3e). Similarly, substituting the inner expansions in Eq. (22) into Eqs. (18) and (20) and collecting the order terms, we have the equations for the second-order inner variables,

where and are the dissipation rates for and . Now the term on the left-hand side of Eq. (49) is a leading term, thus

In the last step the estimate is used, where is the correlation between and , and we assume that the correlation coefficient between and is also . Therefore, from Eq. (25),

resulting in

Thus, the second-order correction term is . From Eqs. (50) and (25),

and

where is the correlation between and . Therefore, the second-order term . Thus, and with the second-order corrections are

and

respectively. Rewriting the last two equations in dimensional form and dropping the terms, we have

and

respectively. The nondimensional coefficients in the abovementioned equations will be determined below. Summing the two asymptotic expansions and subtracting the common parts, we obtain the composite (uniform) expansions

and

which are valid in both the inner and outer layers.

e. Comparison with measurements

The nondimensional coefficients A, B, C, , , and are now determined using measurements from the Kansas (Wyngaard et al. 1971), Minnesota (Kaimal et al. 1976; Izumi and Caughey 1976), Atmospheric Radiation Measurement (ARM; Mather and Voyles 2013; Berg et al. 2017), and Ashchurch (Caughey and Palmer 1979) field programs. By fitting the LFC scaling term of to the Kansas data (Fig. 1) for , we find (the solid line in Fig. 1). Fitting Eq. (47) for down to 0.1 (dashed line in Fig. 1), we obtain . Fitting the LFC scaling term of to the Kansas data (Fig. 2) for gives (the solid line in Fig. 2). Fitting Eq. (48) to the Kansas data for down to 0.03 (dashed line in Fig. 2), we obtain . Here the second-order corrections account for departure from the LFC scaling caused by the mean shear production for . Applying the Minnesota, Ashchurch, and ARM data (Figs. 3 and 4) to Eqs. (58) and (59), we find and . Using these values for the coefficients, the composite expansions show very good agreement with the field data. With these A and B values the expansion for also fits reasonably well the Northern Hemisphere Climate Processes Land Surface Experiment (NOPEX) data (Johansson et al. 2001; not shown), which have larger scatters. For the uncertainty levels for the NOPEX data (16% for and 10% for ), it is clear from Figs. 1 and 3 that self-correlation effects do not alter the trends of the data in any significant way. Furthermore, judging from the scatter of the data points, the uncertainties in the Kansas data are lower. As a result, any effects of self-correlation would be even smaller. Therefore, the observed LFC scaling and departure from it for are due to the surface-layer physics, not self-correlation effects.

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Comparison of the composite expansion for the vertical velocity variance with the Kansas (1968) data in terms of the inner (surface layer) variables. The LFC limit (solid line) and LFC and the second-order correction (dashed line) are marked.

Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0016.1

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Comparison of the composite expansion for the temperature variance with the Kansas (1968) data in terms of the inner (surface layer) variables. Line styles are the same as in Fig. 1.

Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0016.1

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Comparison of the composite expansion for the vertical velocity variance with the Minnesota (circles), Ashchurch (triangles), and ARM (asterisks) data in terms of the outer (mixed layer) variables. Line styles are the same as in Fig. 1.

Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0016.1

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Comparison of the composite expansion for the temperature variance with the Minnesota data in terms of the outer (mixed layer) variables. Line styles are the same as in Fig. 1.

Citation: Journal of the Atmospheric Sciences 75, 10; 10.1175/JAS-D-18-0016.1

The composite expansion for is valid up to , well beyond the surface layer. Here the second-order corrections account for the influence of the inversion height (atmospheric boundary layer depth), which is reflected in the time derivative terms in Eq. (18). Unlike in the surface layer where , the turbulence is no longer in equilibrium with the external conditions (influences) when . The composite expansion for is valid up to , because at the inversion layer there is a production source with different scaling, which is beyond our surface-layer analysis. From the point of view of singular perturbation problems, there is a second inner layer near , which needs to be considered in the matched asymptotic expansions if we want to predict the behavior there. Therefore, the current correction cannot capture the trend of near . For , the inversion damps the fluctuations, which does not necessarily result in a second inner layer.

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 and are separate curves. The composite expansions obtained in the present study provide unified expressions for the vertical velocity and potential temperature variances from to . Equally important, each part of the expansions has a clear physical interpretation (origin).

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 as the thickness of the inner layer. The inner expansions were found to depend on only, providing a proof of MOST for the vertical and potential temperature variances. The LFC scaling was obtained by matching the leading-order inner and outer expansions. Corrections for the departure from the scaling for and , which cannot be obtained analytically using dimensional analysis, are also derived by including the second-order expansions. The composite expansions obtained show very good agreement with the Kansas, Minnesota, Ashchurch, and Atmospheric Radiation Measurement (ARM) field data, achieved with only leading- and second-order expansions, demonstrating that matched asymptotic expansions provide an effective method for analyzing and understanding the atmospheric boundary layer.

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.