Symmetry Analysis of Mean Velocity Distribution in Stratified Atmospheric Surface Layers

Abstract

The mean velocity distributions of unstably and stably stratified atmospheric surface layers (ASLs) are investigated here using the symmetry approach. Symmetry groups for the mean momentum and the Reynolds stress equations of ASL are searched under random dilation transformations, which, with different leading order balances in different flow regions, lead to a set of specific scalings for the characteristic length ${\ell }_{13}$ (defined by Reynolds shear stress and mean shear). In particular, symmetry analysis shows that in the shear-dominated region, ${\ell }_{13}$ scales linearly with the surface height z, which corresponds to the classical log law of mean velocity. In the buoyancy-dominated region, ${\ell }_{13}/L\sim {\left(z/L\right)}^{4/3}$ for unstably stratified ASL and ${\ell }_{13}/L\sim const$ for stably stratified ASL, where L is the Obukhov length. The specific formula of the celebrated Monin–Obukhov similarity function is obtained, and hence an algebraic model of mean velocity profiles in ASL is derived, showing good agreement with the datum from the QingTu Lake observation array (QLOA) in China.

1. Introduction

The atmospheric surface layer (ASL) is a specific turbulent boundary layer (TBL) in which atmosphere exchanges momentum and energy with the earth’s surface through nearly constant momentum and heat flux [1]. Due to its large dimension in the horizontal and surface-normal directions, ASL typically reaches a (friction) Reynolds number of several millions [2], making it a candidate for the study of high Reynolds number canonical wall turbulence. Also, as a result of the diurnal release and absorption of the heat of the earth’s surface, the turbulent flow states of ASL in the daytime and at night are significantly different [3]. Notable observation arrays, such as QLOA [4] and SLTEST (Surface Layer Turbulence and Environmental Science) [5], have been established to experimentally study the high Reynolds number properties of turbulent boundary layers [2,5,6,7,8,9,10,11,12,13]. Due to the limitation of grid spatial resolution, the numerical weather research and forecasting model (WRF) cannot solve the flow field in the ASL [14]. Therefore, studies on the unified description of mean velocity distribution (MVD) under various momentum and heat flux conditions are important for applications in the weather forecasting, pollutant prediction, and wind energy industry [1,3,15,16,17].

Since the boundary layer concept of Prandlt [18], the Monin–Obukhov (MO) similarity theory [19] has been a milestone for understanding the ASL and is regarded as the starting point of modern micrometeorology [16]. The MO theory relies on statistical stationarity and horizontal homogeneity, which results in the constant momentum and heat flux in the surface-height (or wall-normal) direction. In particular, four physical parameters are proposed in MO to depict the mean flow of the ASL: heat flux $H_w=\overline{w^{\prime }{T}^{\prime }}$, friction velocity $u_{\tau }=\sqrt{\overline{-u^{\prime }{w}^{\prime }}}$, buoyancy force factor $g/\overline{T}$, and the height z. Dimensional analysis leads to the Obukhov length scale $L=-u_{\tau }^3/\left[\kappa \overline{w^{\prime }{T}^{\prime }}g/\overline{T}\right]$, where $\overline{T}$ is the mean temperature, g is the acceleration of gravity, $\kappa$ is the von Kármán constant, and $w^{\prime }$ and $T^{\prime }$ are the fluctuating wall-normal velocity and temperature, respectively. In practical terms, L can be explained as a critical height, below which turbulence is dominated by the wall-induced shear effect, and above which it is dominated by buoyancy. Consequently, a non-dimensional similarity variable $\zeta =z/L$ is defined by MO to quantify the ratio between the shear production and buoyancy effect in the ASL. Since then, statistical quantities have all been expressed as certain functions of $\zeta$, validated by many atmospheric experimental studies [20,21].

Particularly for the mean velocity distribution, MO proposed investigating the following dimensionless similarity function:

$$\phi {}_m\left(\zeta \right)=\frac{\kappa z}{u_{\tau }}\frac{dU}{dz}.$$

(1)

To determine $\phi {}_m$, an implicit function equation ${\phi }_m^4-\gamma {\phi }_m^3\zeta =1$ has been derived under an approximation of the kinetic energy balance equation, known as the O’KEYPS equation (Obukhov, Kazansky, Ellison, Yamamoto, Panofsky, and Sellers) [22]. However, the constant eddy diffusivity assumed in the O’KEYPS equation remains a subject of doubt, and the empirical parameter $\gamma$, which varies from 5 to 18 for different data sets, indicates that the form of $\phi {}_m$ in the O’KEYPS equation may not be universal [23].

On the other hand, based on the Kansas experimental measurements, ${\phi }_m={(1-16\zeta )}^{-1/4}$ for $\zeta <0$ and ${\phi }_m=1+4.7\zeta$ for $\zeta >0$ are well-known as the Businger–Dyer (BD) function [24,25]. However, the asymptotic scaling of BD function at the free convection limit, i.e., ${\phi }_m\sim {\left(-\zeta \right)}^{-1/4}$ for $-\zeta \gg 1$, differs from the prediction by teh O’KEYPS equation. To address this, Carl et al. [26] proposed a modified ${\phi }_m={(1-15\zeta )}^{-1/3}$, to conform with the O’KEYPS equation. Moreover, Kader and Yaglom [27] argued that in the O’KEYPS equation, $\gamma \gg 1$, as the buoyancy acts only on the direction normal to the earth’s surface and hence contributes mostly to momentum transfer. They also proposed the same scaling ${\phi }_m\sim {(-\zeta )}^{1/3}$, validated by their measured data in the top convective sublayer of the ASL [27].

More recently, Katul et al. developed a heuristic model of MVD in ASL under the attached eddy hypothesis. The characteristic velocity $u_{\tau }$ is derived from the assumed inertial-range spectrum, which further leads to a relation between buoyancy and momentum flux [23,28]. This model yields a similar prediction to that of the O’KEYPS equation, but it incorporates more physical considerations. For example, the large value of $\gamma$ is attributed to the effects of turbulent transport, pressure redistribution, and the anisotropy of turbulent eddies. However, it should also be noted that the functional form of the similarity function becomes more complicated, and its prediction for the stable ASL deviates from observations. To account for this, Li et al. revisited Katul’s model by introducing the Ozmidov scale [29] and obtained a better description of the data. Liu et al. developed an analytical model describing the vertical structure of conventionally neutral atmospheric boundary layers, providing predictions of wind and turbulent shear stress profiles [17,30]. In addition to the works mentioned, there are also efforts to extend the MO theory by further consideration of large-scale coherent structures [31], turbulence anisotropy [32], and non-zero vertical turbulent transport effects [33].

This paper aims to understand the MVD from a symmetry perspective. It is worth noting that symmetry analysis is a well-developed method for finding the similarities or invariant solutions of differential equations. A typical example is the Blasius similarity solution of a laminar boundary layer flow, the symmetry analysis of which involves three steps. First, search the dilation symmetry transformation that keeps the governing equation unchanged; second, obtain the dilation invariants for the independent and dependent variables; finally, use these invariants as variables to transfer the partial differential equation to an ordinary differential equation. When dealing with turbulence, there is a challenge in these steps because of the unknown Reynolds stresses. However, by searching the symmetry of characteristic length scales, one can construct candidate invariant solutions by using group invariants as similarity variables. This approach has been utilized to describe turbulent mean flows in wall flows, with details provided in [34,35,36].

Specifically, for canonical turbulent boundary layers, a multi-layer description of the mean velocity profiles has been obtained in [34,35,36] through a random dilation analysis of Reynolds-averaged Navier–Stokes equations. In comparison to previous works, the novelty in [34,35,36] is that different leading order balances are considered in the symmetry analysis, and a general ansatz is proposed to connect the local dilation invariants. In the past few years, a large set of experimental and direct numerical simulation (DNS) data of canonical wall flows (channel, pipe, and turbulent boundary layer—TBL) have verified the multi-layer description, with successful extensions to complex boundary conditions, including pressure gradient effects, heat flux effects, and surface roughness [37,38]. Note that, as ASL is also a wall-induced shear flow, the dilation symmetry proposed in [34,35,36] along the wall-normal direction (z) may also exist. Thus, we plan to develop a similar modeling of the ASL in this paper based on the symmetry analysis approach.

Before proceeding further, it is important to note that a theoretical description of thermally stratified ASL flow is crucial for climate modeling of the near-surface wind fields. For instance, in the Weather Research and Forecasting (WRF) model, due to computational power cost, the WRF cannot resolve the near wall flow details. Therefore the Monin–Obukhov similarity theory (MOST) is used to estimate the exchanges of heat, momentum and humidity between the earth’s surface and the ASL. Specially for the wind speed at approximately 10 m above the earth’s surface, the accuracy of MOST is crucial for WRF predictions. Various corrections of MOST have been proposed in the WRF [14]. In this context, we present a theoretical framework for describing thermally stratified ASLs based on the symmetry approach, which offers a more comprehensive understanding compared to the models and corrections based on MOST. The results can be integrated into the WRF to enhance the prediction of near-surface wind fields.

The rest of this paper is organized as follows. The balance equations and the dilation symmetry analysis are introduced in Section 2. Section 3 introduces the experimental data, the comparison of which with our theory is provided in Section 4. Final conclusions are presented in Section 5.

2. Methods

This study is inspired by the symmetry approach for canonical TBL developed by She et al. and Chen et al. [35,36]. Note that to obtain the mean velocity distributions in ASL, one needs to address the closure problem of the unknown Reynolds shear stress. In the literature, this is usually resolved by the hypothesis of eddy viscosity or mixing length (the so-called stress length here), which builds a relation between the mean shear and Reynolds stress. However, these hypotheses have no link with the balance equations. The current paper aims to develop a procedure to determine the mixing length (or stress length) based on the balance equations. As we demonstrate below, the balance equations allow for a set of random dilations, which define the dilation invariant of the stress length. By assuming a constant dilation invariant, we can thus obtain the power-law scaling exponent for the stress length function, and hence the mean velocity distributions in ASL. Therefore, the key contribution of our paper is to present the rationale behind the scaling exponent of stress length from the symmetry consideration of the balance equations. This is reminiscent of Monin and Obukhov, who also derived the logarithmic law of wind distribution based on the concept of dilation symmetry [19]. However, our approach contains more mathematical details and extends to different flow regions.

The ensemble averaged momentum and Reynolds stresses equations are [27,36]

$$\begin{array}{c}\hfill \frac{\partial \overline{-u^{\prime }{w}^{\prime }}}{\partial z}+\nu \frac{{\partial }^{2}U}{\partial z^2}=0,\end{array}$$

(2)

$$\begin{array}{c}\hfill \underset{SP}{\underbrace{\overline{-u^{\prime }{w}^{\prime }}\frac{\partial U}{\partial z}}}+\underset{R_u}{\underbrace{\overline{p^{\prime }\frac{\partial u^{\prime }}{\partial x}}}}-\underset{T_u}{\underbrace{\frac{1}{2}\frac{\partial \overline{u^{\prime 2}{w}^{\prime }}}{\partial z}}}-\underset{{ϵ}_u}{\underbrace{\nu \overline{|\nabla u^{\prime }{|}^2}}}+\underset{D_u}{\underbrace{\nu \frac{{\partial }^2}{\partial z^2}\left(\frac{1}{2}\overline{u^{\prime 2}}\right)}}=0,\end{array}$$

(3)

$$\begin{array}{c}\hfill \underset{R_v}{\underbrace{\overline{p^{\prime }\frac{\partial v^{\prime }}{\partial y}}}}-\underset{T_v}{\underbrace{\frac{1}{2}\frac{\partial \overline{v^{\prime 2}{w}^{\prime }}}{\partial z}}}-\underset{{ϵ}_v}{\underbrace{\nu \overline{|\nabla v^{\prime }{|}^2}}}+\underset{D_v}{\underbrace{\nu \frac{{\partial }^2}{\partial z^2}\left(\frac{1}{2}\overline{v^{\prime 2}}\right)}}=0,\end{array}$$

(4)

$$\begin{array}{c}\hfill \underset{B}{\underbrace{\overline{w^{\prime }{T}^{\prime }}\frac{g}{\overline{T}}}}+\underset{R_v}{\underbrace{\overline{p^{\prime }\frac{\partial w^{\prime }}{\partial z}}}}-\underset{T_w}{\underbrace{\frac{1}{2}\frac{\partial \overline{w^{\prime 3}+2p^{\prime }{w}^{\prime }}}{\partial z}}}-\underset{{ϵ}_w}{\underbrace{\nu \overline{|\nabla w^{\prime }{|}^2}}}+\underset{D_w}{\underbrace{\nu \frac{{\partial }^2}{\partial z^2}\left(\frac{1}{2}\overline{w^{\prime 2}}\right)}}=0,\end{array}$$

(5)

$$\begin{array}{c}\hfill -\frac{1}{2}\overline{w^{\prime }{w}^{\prime }}\frac{\partial U}{\partial z}-\frac{1}{2}\overline{u^{\prime }\frac{\partial p^{\prime }}{\partial z}}-\frac{1}{2}\overline{w^{\prime }\frac{\partial p^{\prime }}{\partial x}}-\frac{1}{2}\frac{\partial \overline{u^{\prime }{w}^{\prime }{w}^{\prime }}}{\partial z}-\nu \overline{|\nabla u^{\prime }•\nabla w^{\prime }|}+\nu \frac{{\partial }^2}{\partial z^2}\left(\frac{1}{2}\overline{u^{\prime }{w}^{\prime }}\right)=0,\end{array}$$

(6)

where overline denotes the time-spatial ensemble average; $p^{\prime }$ is the pressure fluctuation and $u^{\prime },w^{\prime },v^{\prime }$ are the streamwise, vertical (or surface-height), and spanwise velocity fluctuations with U the mean streamwise velocity; SP is the shear production; $R_{u,w,v}$ are pressure redistribution terms; $T_{u,w,v}$ are vertical spatial turbulent transports; ${ϵ}_{u,w,v}$ are turbulent dissipation rates; $D_{u,w,v}$ are diffusion terms; subscript $u,w,v$ denote three velocity fluctuation components.

The above equations for ASL are similar to the canonical boundary layer flows except for the additional buoyancy term (i.e., the heat flux) $B=\overline{w^{\prime }{T}^{\prime }}g/\overline{T}$ in Equation (5). The buoyancy term could be an energy source for upward heat flux ($B>0$), or a sink for downward heat flux ($B<0$). Since the shear production (SP) decreases as $z^{-1}$ and the buoyancy (B) is invariant as the height Z increases, the shear production dominates the balance equations for small z while the buoyancy dominates for large z. As shown below, the alteration of dominant balance would result in different dilation transformations, leading to different scalings in different flow regions.

To proceed, normalizing the above Equations (2)–(6) by the friction velocity $u_{\tau }$, the wall heat flux $H_w$, and the Obukhov length L yields

$$\begin{array}{c}\hfill \frac{\partial \overline{-u^{\prime }{}^{+}{w}^{\prime }{}^{+}}}{\partial {\zeta }_z}+\frac{1}{L{}^{+}}\frac{{\partial }^{2}U{}^{+}}{\partial {\zeta }_z^2}=0,\end{array}$$

(7)

$$\begin{array}{c}\hfill \overline{-u^{\prime }{}^{+}{w}^{\prime }{}^{+}}\frac{\partial U{}^{+}}{\partial {\zeta }_z}+\overline{p^{\prime }{}^{+}\frac{\partial u^{\prime }{}^{+}}{\partial {\zeta }_x}}-\frac{1}{2}\frac{\partial \overline{u^{\prime }{{}^{+}}^2{w}^{\prime }{}^{+}}}{\partial {\zeta }_z}-\frac{1}{L{}^{+}}\overline{|{\nabla }_L{u}^{\prime }{}^{+}{|}^2}+\frac{1}{L{}^{+}}\frac{{\partial }^2}{\partial {\zeta }_z^2}\left(\frac{1}{2}\overline{u^{\prime }{{}^{+}}^2}\right)=0,\end{array}$$

(8)

$$\begin{array}{c}\hfill \overline{p^{\prime }{}^{+}\frac{\partial v^{\prime }{}^{+}}{\partial {\zeta }_y}}-\frac{1}{2}\frac{\partial \overline{v^{\prime }{{}^{+}}^2{w}^{\prime }{}^{+}}}{\partial {\zeta }_z}-\frac{1}{L{}^{+}}\overline{|{\nabla }_L{v}^{\prime }{}^{+}{|}^2}+\frac{1}{L{}^{+}}\frac{{\partial }^2}{\partial {\zeta }_z^2}\left(\frac{1}{2}\overline{v^{\prime }{{}^{+}}^2}\right)=0,\end{array}$$

(9)

$$\begin{array}{c}\hfill \overline{w^{\prime }{}^{+}{T}^{\prime }{}^{+}}+\overline{p^{\prime }{}^{+}\frac{\partial w^{\prime }{}^{+}}{\partial {\zeta }_z}}-\frac{1}{2}\frac{\partial \overline{w^{\prime }{{}^{+}}^3+2p^{\prime }{}^{+}{w}^{\prime }{}^{+}}}{\partial {\zeta }_z}-\frac{1}{L{}^{+}}\overline{|{\nabla }_L{w}^{\prime }{}^{+}{|}^2}+\frac{1}{L{}^{+}}\frac{{\partial }^2}{\partial {\zeta }_z^2}\left(\frac{1}{2}\overline{w^{\prime }{{}^{+}}^2}\right)=0,\end{array}$$

(10)

$$\begin{array}{c}-\frac{1}{2}\overline{w^{\prime }{}^{+}{w}^{\prime }{}^{+}}\frac{\partial U{}^{+}}{\partial {\zeta }_z}-\frac{1}{2}\overline{u^{\prime }{}^{+}\frac{\partial p^{\prime }{}^{+}}{\partial {\zeta }_z}}-\frac{1}{2}\overline{w^{\prime }{}^{+}\frac{\partial p^{\prime }{}^{+}}{\partial {\zeta }_x}}-\frac{1}{2}\frac{\partial \overline{u^{\prime }{}^{+}{w}^{\prime }{}^{+}{w}^{\prime }{}^{+}}}{\partial {\zeta }_z}-\\ \frac{1}{L{}^{+}}\overline{|{\nabla }_L{u}^{\prime }{}^{+}•{\nabla }_L{w}^{\prime }{}^{+}|}+\frac{1}{L{}^{+}}\frac{{\partial }^2}{\partial {\zeta }_z^2}\left(\frac{1}{2}\overline{u^{\prime }{}^{+}{w}^{\prime }{}^{+}}\right)=0,\end{array}$$

(11)

where superscript plus + means being normalized by wall variables, i.e.,

$$[u^{\prime }{}^{+},v^{\prime }{}^{+},w^{\prime }{}^{+},U{}^{+},p^{\prime }{}^{+},L{}^{+},T^{\prime }{}^{+}]=\left[\frac{u^{\prime }}{u_{\tau }},\frac{v^{\prime }}{u_{\tau }},\frac{w^{\prime }}{u_{\tau }},\frac{U}{u_{\tau }},\frac{p^{\prime }}{u_{\tau }^2},\frac{L}{\nu /u_{\tau }},\frac{T^{\prime }}{\kappa H_w/u_{\tau }}\right],$$

and ${\nabla }_L=\partial /\partial {\zeta }_x+\partial /\partial {\zeta }_y+\partial /\partial {\zeta }_z$ is gradient operator, with ${\zeta }_x=x/L$, ${\zeta }_y=y/L$, and ${\zeta }_z\equiv \zeta =z/L$.

Following the step in [35,36], we define the Reynolds shear stress length function ${\ell }_{13}^{\wedge }=\frac{\sqrt{\overline{-u^{\prime }{}^{+}{w}^{\prime }{}^{+}}}}{\partial U{}^{+}/\partial \zeta }$ , which characterizes the size of eddies responsible for vertical momentum transport. This enables us to solve the mean velocity profile from Equation (7):

$$U{}^{+}(h/L)={\int }_{h_0/L}^{h/L}\frac{\sqrt{\overline{-u^{\prime }{}^{+}{w}^{\prime }{}^{+}}}}{{\ell }_{13}^{\wedge }}d\zeta .$$

(12)

Here, h is the height of interest, $h_0$ is the typical roughness height, and ${\ell }_{13}^{\wedge }={\ell }_{13}/L$ with ${\ell }_{13}=\frac{\sqrt{-u^{\prime }{w}^{\prime }}}{\partial U/\partial z}$. Note that ASL is a constant momentum flux layer, which means $\overline{-u^{\prime }{}^{+}{w}^{\prime }{}^{+}}=1$; hence, Equation (12) is written

$$U\left(h\right)=u_{\tau }{\int }_{h_0}^h\frac{dz}{{\ell }_{13}}.$$

(13)

Once ${\ell }_{13}$ is known, the mean velocity $U\left(h\right)$ can also be determined.

To obtain the formula of ${\ell }_{13}$, a random dilation transformation [36] is introduced as follows:

$$\begin{array}{cccccc}{\zeta }_i^{*}=q_i{\zeta }_i,& U^{*}={\lambda }_{U}U& L{}^{*}={\lambda }_{L}L& u_i^{\prime *}={\lambda }_i{u}_i^{\prime },& p^{\prime *}={\lambda }_p{p}^{\prime },& T^{\prime *}={\lambda }_T{T}^{\prime },\end{array}$$

(14)

where $q_i$ and ${\lambda }_U$, ${\lambda }_L$ are regular dilation factors, while ${\lambda }_i$, ${\lambda }_p$, and ${\lambda }_T$ are random factors with zero means; and $i=(1,2,3)$ denotes $(x,y,z)$. Note that superscript ‘+’ is neglected here. Substituting (14) into the balance Equations (7)–(11), the symmetry requires that equations under dilation remain invariant, which leads to the following relationships among dilation parameters:

$$\begin{array}{cc}\hfill \frac{\overline{{\lambda }_1{\lambda }_3}}{q_3}& =\frac{1}{{\lambda }_L}\frac{{\lambda }_U}{q_3^2},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \overline{{\lambda }_1{\lambda }_3}\frac{{\lambda }_U}{q_3}=\frac{\overline{{\lambda }_p{\lambda }_1}}{q_1}=\frac{\overline{{\lambda }_1{\lambda }_1{\lambda }_3}}{q_3}=\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_1{\lambda }_1}}{q_1^2}=\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_1{\lambda }_1}}{q_2^2}& =\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_1{\lambda }_1}}{q_3^2},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \frac{\overline{{\lambda }_p{\lambda }_2}}{q_2}=\frac{\overline{{\lambda }_2{\lambda }_2{\lambda }_3}}{q_3}=\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_2{\lambda }_2}}{q_1^2}=\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_2{\lambda }_2}}{q_2^2}& =\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_2{\lambda }_2}}{q_3^2},\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \overline{{\lambda }_3{\lambda }_T}=\frac{\overline{{\lambda }_p{\lambda }_3}}{q_3}=\frac{\overline{{\lambda }_3{\lambda }_3{\lambda }_3}}{q_3}=\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_3{\lambda }_3}}{q_1^2}=\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_3{\lambda }_3}}{q_2^2}& =\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_3{\lambda }_3}}{q_3^2},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \overline{{\lambda }_3{\lambda }_3}\frac{{\lambda }_U}{q_3}=\frac{\overline{{\lambda }_p{\lambda }_1}}{q_3}=\frac{\overline{{\lambda }_p{\lambda }_3}}{q_1}=\frac{\overline{{\lambda }_1{\lambda }_3{\lambda }_3}}{q_3}=\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_1{\lambda }_3}}{q_1^2}=\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_1{\lambda }_3}}{q_2^2}& =\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_1{\lambda }_3}}{q_3^2}.\hfill \end{array}$$

(19)

An important fact is that in different flow regions, there are different leading order balances in (7)–(11) so that we can define a locally valid dilation group by neglecting unimportant terms. This provides extra freedoms for the group parameters in Equations (15)–(19), as practised below in different flow layers.

2.1. Homogeneous Dilations in the Shear Dominated Layer

In the shear dominated layer, the shear production (SP) is balanced by dissipation from all three spatial directions. Hence, all the spatial derivatives in (3)–(6) are important, leading to a homogeneous dilation in three directions, i.e., $q_1=q_2=q_3=q$ in (15)–(19). Furthermore, from $\overline{{\lambda }_1{\lambda }_3}\frac{{\lambda }_U}{q}=\frac{\overline{{\lambda }_p{\lambda }_1}}{q}=\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_1{\lambda }_3}}{q^2}$, we obtain ${\lambda }_U{\lambda }_L=q^{-1}$; considering $\overline{{\lambda }_3{\lambda }_T}=1$ for constant heat flux, from Equations (15), (18), and (19), we obtain ${\lambda }_L^2/{\lambda }_U=q^{-3}$, and hence

$$\begin{array}{cc}\hfill {\lambda }_L& =q^{-4/3},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\lambda }_U& =q^{1/3},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \overline{{\lambda }_1{\lambda }_1}=\overline{{\lambda }_1{\lambda }_3}=\overline{{\lambda }_3{\lambda }_3}& =q^{2/3},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \overline{{\lambda }_p{\lambda }_1}=\overline{{\lambda }_p{\lambda }_3}=\overline{{\lambda }_1{\lambda }_1{\lambda }_3}=\overline{{\lambda }_2{\lambda }_3{\lambda }_3}=\overline{{\lambda }_3{\lambda }_3{\lambda }_3}& =q.\hfill \end{array}$$

(23)

Therefore, the dilation factor for length function ${\ell }_{13}^{\wedge }$ is

$${\lambda }_{13}=\frac{{\overline{{\lambda }_1{\lambda }_3}}^{1/2}}{{\lambda }_U/q}=\frac{q^{1/3}}{q^{1/3}/q}=q,$$

(24)

whose corresponding dilation invariant is

$${\mathrm{I}}_{13}=\frac{{\ell }_{13}^{\wedge }}{\zeta }.$$

(25)

It is a normal result that length function takes its ordinary dimension if no direction is preferred, the same as the classical dimensional analysis result. In [36], the dilation invariant for stress length is found to be ${\mathrm{I}}_{13}=0.45$, and hence ${\ell }_{13}^{\wedge }=0.45\zeta$ for canonical boundary layer flows. In ASL, the value of ${\mathrm{I}}_{13}$ may slightly change, but the functional form is the same.

2.2. Inhomogeneous Dilation in the Convective Layer ($B>0$)

Let us consider the region where height $z>L$ and $B>0$, and hence the buoyancy term overtakes the shear production to balance dissipation in Equations (16)–(18). Subsequently, the shear production and turbulent transport terms could be neglected in Equations (16)–(18). On the other hand, in the Reynolds shear stress Equation (11) or (19), the dominant balance is still between the first term of production and the fifth term of dissipation. Based on these considerations, Equations (15), (18) and (19) could be, respectively, simplified to

$$\begin{array}{cc}\hfill \frac{\overline{{\lambda }_1{\lambda }_3}}{q}& =\frac{1}{{\lambda }_L}\frac{{\lambda }_U}{q^2},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \overline{{\lambda }_3{\lambda }_T}=& =\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_3{\lambda }_3}}{q^2},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill \overline{{\lambda }_3{\lambda }_3}\frac{{\lambda }_U}{q}& =\frac{1}{{\lambda }_L}\frac{\overline{{\lambda }_1{\lambda }_3}}{q^2}.\hfill \end{array}$$

(28)

Furthermore, considering that momentum and heat flux are constant in the buoyancy dominant layer, we obtain $\overline{{\lambda }_3{\lambda }_T}=1$ and $\overline{{\lambda }_1{\lambda }_3}=1$, and hence ${\lambda }_U=q^{-1/3}$, ${\lambda }_T=q^{-4/3}$ and $\overline{{\lambda }_3{\lambda }_3}=q^{2/3}$. Thus, the dilation factor of ${\ell }_{13}^{\wedge }$ is

$$\begin{array}{c}\hfill {\lambda }_{13}=\frac{1}{{\lambda }_U/q}=q^{4/3}.\end{array}$$

(29)

Therefore, in the convective layer, the stress length satisfies ${\ell }_{13}^{\wedge }\propto {\zeta }^{4/3}$.

2.3. Inhomogeneous Dilation in the Stably Stratified Layer ($B<0$)

When $B<0$, buoyancy, acting as a sink, would absorb kinetic energy. Hence, the dominant balance for the Reynolds normal stresses are between SP and B (instead of dissipation). In other words, when summing Equations (8)–(10) all together, the balance is between SP and B. Under this condition, the dilation parameters are simplified to be

$$\overline{{\lambda }_3{\lambda }_T}=\overline{{\lambda }_1{\lambda }_3}\frac{{\lambda }_U}{q}.$$

(30)

Again, as $\overline{{\lambda }_3{\lambda }_T}=1$ and $\overline{{\lambda }_1{\lambda }_3}=1$, we obtain ${\lambda }_U=q$. Therefore, the dilation parameter of ${\ell }_{13}$ is

$${\lambda }_{13}=\frac{1}{{\lambda }_U/q}=1,$$

(31)

which means that ${\ell }_{13}^{\wedge }$ is a constant.

2.4. Composite Formula of ${\ell }_{13}^{\wedge }$ in ASL

According to the dilation analysis above, the scaling of ${\ell }_{13}^{\wedge }$ in different flow layers is obtained. Following the same matching procedure introduced in [35], a two-layer formula of stress length connecting the adjacent power-law scalings can be obtained:

$$\begin{array}{c}\hfill {\ell }_{13}^{\wedge }\propto \left\{\begin{array}{cc}\zeta {\left(1-\frac{\zeta }{{\zeta }_{UC}}\right)}^{1/3},\hfill & \zeta <0\hfill \\ \zeta {\left(1+\frac{\zeta }{{\zeta }_{SC}}\right)}^{-1},\hfill & \zeta >0\hfill \end{array}\right..\end{array}$$

(32)

Here, ${\zeta }_{UC}$ and ${\zeta }_{SC}$ are empirical parameters to be determined from data. In [33], they are given as ${\zeta }_{UC}=1/6.3$ and ${\zeta }_{SC}1/2$, respectively. For $\zeta$ below the critical height ${\zeta }_{UC}$ and ${\zeta }_{SC}$, the linear scaling of ${\ell }_{13}^{\wedge }$ is consistent with wall-attached eddy size in the log layer [39]. For $\zeta$ above ${\zeta }_{UC}$, the scaling ${\zeta }^{4/3}$ for unstably stratified ASL indicates that the momentum transport eddies are stretched in the vertical direction by the buoyancy force. On the contrary, for stably stratified ASL, the buoyancy force depresses eddy size in a vertical direction, resulting in a finite value of ${\ell }_{13}^{\wedge }$.

2.5. Composite Formula of ${\varphi }_m$ in ASL

According to the definitions of ${\varphi }_m$ and ${\ell }_{13}$, we obtain the relation between ${\varphi }_m$ and ${\ell }_{13}$ as:

$${\varphi }_m\left(\zeta \right)=\frac{\sqrt{\overline{-u^{\prime }{w}^{\prime }}}}{{\ell }_{13}}\times \frac{\kappa z}{u_{\tau }}.$$

(33)

Considering that $\sqrt{\overline{-u^{\prime }{w}^{\prime }}}/u_{\tau }=1$, we have

$$\begin{array}{c}\hfill {\varphi }_m\propto \left\{\begin{array}{cc}{\left(1-\zeta /{\zeta }_{UC}\right)}^{-1/3},\hfill & \zeta <0\hfill \\ 1+\zeta /{\zeta }_{SC},\hfill & \zeta >0\hfill \end{array}\right..\end{array}$$

(34)

3. Data

Data collected for the verification of our derivations include the Kansas measurements, the AHATS (advection horizontal array turbulence study) measurements, and the QLOA measurements. AHATS investigated surface-layer turbulence in the San Joaquin Valley, California, during the summer of 2008, while details on Kansas and AHATS are referred to references [24,40].

For QLOA, it is conducted on the dry lake bed located in Minqin County, Gansu Province, in the northwestern region of China. QLOA has one main tower that is 32 m high, surrounded by twenty lower towers that are 5 m high and shaped like the character ‘T’. Data measured from the main tower are examined here, which are acquired from eleven sonic anemometers. Wind speed vectors and virtual temperature are sampled at a frequency of 50 Hz. QLOA has collected day and night data lasting about ten years, covering various weather conditions, and has been used to study the large-scale motion, energy spectrum, amplitude modulation between multi-scale turbulent motions, and two-phase flows during sand storms [4,41,42]. The observations of QLOA are divided into time-ensemble blocks of 1 h. This study uses 12 sets of unstably stratified data and 11 sets of stably stratified data, with details shown in

Table 1. The information of QLOA data used in this paper.

No.	Time and Date	${\mathit{u}}_{\mathit{\tau }}$ (m/s)	${\mathit{H}}_{\mathit{w}}$ (K·m/s)	L (m)
1	2014-5-23 7:00–8:00	0.28	0.016	−96.4
2	2014-5-23 8:00–9:00	0.31	0.087	−22.8
3	2014-5-23 9:00–10:00	0.32	0.155	−13.5
4	2014-5-23 10:00–11:00	0.33	0.203	−11.4
5	2014-5-23 11:00–12:00	0.34	0.222	−11.4
6	2014-5-23 12:00–13:00	0.29	0.209	−7.7
7	2014-5-23 13:00–14:00	0.33	0.230	−10.4
8	2014-5-23 14:00–15:00	0.29	0.279	−6.1
9	2014-5-23 15:00–16:00	0.30	0.180	−9.8
10	2014-5-23 16:00–17:00	0.34	0.161	−16.9
11	2014-5-23 17:00–18:00	0.39	0.147	−27.2
12	2014-5-23 18:00–19:00	0.35	0.080	−36.8
13	2014-3-27 0:00–1:00	0.26	−0.036	31.0
14	2014-3-27 1:00–2:00	0.59	−0.057	237.6
15	2014-3-27 2:00–3:00	0.67	−0.055	351.0
16	2014-3-27 3:00–4:00	0.59	−0.039	350.5
17	2014-3-27 4:00–5:00	0.48	−0.026	263.9
18	2014-3-27 5:00–6:00	0.41	−0.018	259.4
19	2014-3-27 6:00–7:00	0.31	−0.014	134.9
20	2014-3-27 7:00–8:00	0.25	−0.007	143.8
21	2014-5-23 2:00–3:00	0.22	−0.025	27.5
22	2014-5-23 6:00–7:00	0.21	−0.013	47.2
23	2014-5-23 20:00–21:00	0.23	−0.019	40.6

. The pretreatment of the data is conducted to transform wind signals into a streamwise direction [4]. The friction velocity $u_{\tau }$ is estimated using Reynolds shear stress, $u_{\tau }=\sqrt{{\sum }_{i=1}^{11}{〈-u^{\prime }{w}^{\prime }〉}_i/11}$. Here i denotes sonic anemometer at various heights and ${〈*〉}_i$ means averaging one hour at ith height. Similarly, the wall heat flux is estimated by $H_w={\sum }_{i=1}^{11}{〈w^{\prime }{T}^{\prime }〉}_i$/11. To obtain shear production, $dU/dz$ is calculated from mean velocity data using the log-polynomial fitting [43], specifically, $U=c_0+c_{1}logz+c_2{(logz)}^2$. At each height z, five adjacent points are used to determine the coefficients $c_0$, $c_1$, and $c_2$, so that $dU/dz=c_1/z+2c_{2}logz/z$. The turbulent dissipation rate is estimated using $ϵ\approx \frac{1}{K_1-K_0}{\int }_{K_0}^{K_1}{(E_{11}\left(k\right)k^{5/3}/C_{11})}^{3/2}dk$, where $E_{11}$ is the spectrum of streamwise velocity fluctuation, $C_{11}=0.5$ is the longitudinal Kolmogorov constant, while $K_0$ and $K_1$ indicate the start and the end of the Kolmogorov initial range, and wavenumber k is calculated from frequency and local mean velocity by using the Taylor freezing hypothesis, $k=2\pi f/U$ [44].

4. Results

In Section 2, we demonstrate that different leading order balances of the budget equations lead to different dilation symmetries. However, due to a scarcity of data, it is impossible to check every equation for Reynolds stresses. To address this, we sum together Equations (3)–(5) and obtain the turbulent kinetic energy equation (TKE), which could also be used to verify the leading order balances in different flow conditions. Accordingly, Figure 1 shows the wall-normal (or surface-normal) variation of shear production ($SP$), buoyancy effect (B), dissipation rate ($ϵ$), and all other terms (including pressure and spatial transport effects) as the residue (i.e., $SP+B-ϵ$). In particular, Figure 1a presents the neutrally stratified ASL where heat flux is small and the Obukhov length L is about $-96.4$ m. It is clear that for lower heights (e.g., $z<10$ m), the dominant balance is between the shear production and dissipation, which is consistent with the analysis in Section 2.1. Conversely, for $z>10$ m, the residue is comparable with the dissipation, but the buoyancy term is always smaller than others, hence indicating the neutral stratification condition.

In contrast, Figure 1b shows the unstably stratified case in which $L=-6.1$ m. While the dominant balance is between the shear production and the dissipation for $z<1$ m, the heat flux is much larger than the neutral case. For $z>5$ m, the heat flux is comparable to the dissipation and the shear production, indicating a strong buoyancy effect that plays a role as an energy source—consistent with the analysis in Section 2.2. Meanwhile, the pressure and transport effect (indicated by the residue) is negative, drawing out the local kinetic energy to other flow regions.

Moreover, the TKE budget for the stably stratified ASL is shown in Figure 1c for which $L=27.5$ m. It is obvious that when $z<20$ m, the shear production and dissipation are the two dominant terms. However, for $z>20$ m, the shear production, buoyancy effect, dissipation, and the transport effect are all comparable with each other. The notable point is that the buoyancy term B is negative (consistent with the analysis in Section 2.3), indicating that it drains out flow energy as the role of dissipation $ϵ$, a distinct feature for the stably stratified condition.

Therefore, different leading order balances specified in the above Section 2 have all been verified by ASL data. Now, let us validate the scaling of stress length derived in Section 2. Figure 2 shows the comparison between Equation (32) and the data, with panel (a) for the stable stratification and (b) for the unstable stratification of ASL. Note that ${\varphi }_m$ are extracted from Kansas and AHATS data and then translated to ${\ell }_{13}$ using ${\ell }_{13}=\kappa z/{\varphi }_m$. Dashed lines indicate the linear scaling, while solid lines are Equation (32) derived from our symmetry analysis. For $\left|\zeta \right|=|z/L|<0.1$, the stress length ${\ell }_{13}^{\wedge }$ displays a linear variation (dashed line), indicating the dominance of the shear production and hence the log-law of mean velocity. However, for large $\left|\zeta \right|$, ${\ell }_{13}^{\wedge }$ deviates from linear behavior. On the one hand, for stable stratification in Figure 2a, ${\ell }_{13}^{\wedge }$ tends to be a constant. A close examination of the data shows the best fit of ${\ell }_{13}^{\wedge }=0.35\zeta {(1+2.0\zeta )}^{-1}$ for QLOA measurements, while ${\ell }_{13}^{\wedge }=0.35\zeta {(1+4.0\zeta )}^{-1}$ for the Kansas and AHATS measurements. The different value of ${\zeta }_{SC}=0.5$ for QLOA compared to ${\zeta }_{SC}=0.25$ for Kansas and AHATS is understandable because a non-zero pressure and spatial transport effect may bring in different heat flux, subsequently altering ${\zeta }_{SC}$. Such a point has been observed and explained in [33]. On the other hand, as shown in Figure 2b for the unstable stratification case, all the data from QLOA, Kansas, and AHATS align nicely. They follow the trend of Equation (32) with ${\ell }_{13}^{\wedge }=0.40\zeta {(1-6.3\zeta )}^{1/3}$, which is a general expression for the unstable stratification of ASL.

Finally, the mean velocity profile is obtained using Equation (13) with the stress length provided in Equation (32). While MVD data of Kansas and AHATS measurements are not available, the comparison is presented here only for the QLOA measurements, as shown in Figure 3. In total, there are twelve mean velocity profiles presented here; they are measured in different time. At $h=30$ m (the highest data point for experimental observation), the friction Reynolds number $Re$_30m$=\frac{30\mathrm{m}\ast u_{\tau }}{\nu }$ is calculated, along with the value of the Obukhov length L, both marked on top of the labels for each of the subplots. Notably, subplots (a), (g), and (h) pertain to the neutrally stratified ASL indicated by the very large value of $\left|L\right|$, approximately 100 m. The latter condition means that most of the data points are measured in the flow region $|z/L|<0.1$, in which buoyancy is insignificant. Moreover, subplots (b), (c), (d), (e), and (f) are for unstably stratified ASL, where the heat flux is upward and L is negative. The rest of the subplots are for stably stratified ASL where L is positive. The dashed lines in Figure 3 indicate the log-law, which agrees with most flow data for neutrally stratified cases (subplots (a), (g), and (h)), but only depicts data at small z for other flow cases. The red lines in Figure 3 indicate the predictions using ${\ell }_{13}^{\wedge }=0.35\zeta {(1+2.0\zeta )}^{-1}$ for unstable cases, while blue lines indicate the predictions using ${\ell }_{13}^{\wedge }=0.40\zeta {(1-6.3\zeta )}^{1/3}$ for stable cases. It is evident that data deviation from the log-law is well captured by the formula of ${\ell }_{13}$ derived from our symmetry analysis.

It should be mentioned that, to obtain the mean velocity profile, an integration parameter $h_0$ is needed in Equation (13), which indicates the surface roughness height in QLOA. According to our study, we find that $h_0$ varies slightly for the above twelve mean velocity profiles. That is, for subplots (a)–(f), $h_0=$ (0.08, 0.23, 0.57, 0.42, 0.32, 0.20) mm; for subplots (g)–(l), $h_0=$ (0.60, 0.20, 0.28, 1.30, 1.60, 1.90) mm. The mechanism for the slight variation of these $h_0$ heights deserves future studies.

5. Conclusions

The symmetry approach developed for canonical wall turbulence [35,36] has now been extended to describe the mean velocity distribution in the stratified atmospheric surface layer. By performing the random dilation on the governing equations and further considerations on different leading order balances, the scaling of Reynolds stress length is specified in different flow layers. That is, in the shear-dominated log layer, ${\ell }_{13}\propto \zeta$; in the buoyancy dominated layer, ${\ell }_{13}\propto {\zeta }^{4/3}$ for unstably stratified ASL while a constant ${\ell }_{13}$ for stably stratified ASL. Using the matching procedure in [35,36], a composite formula of ${\ell }_{13}$ is obtained, i.e., $0.40\zeta {(1-6.3\zeta )}^{1/3}$ for unstable stratification and $0.35\zeta {(1+2.0\zeta )}^{-1}$ for stable stratification, which leads to a close representation for the mean velocity distributions measured in QLOA.

It should be noted that the linear coefficients are 0.40 for unstable stratification and 0.35 for stable stratification. Additionally, the value of ${\zeta }_{UC,SC}$ cannot be explained by the symmetry view. More effort should be devoted to determining these values from a theoretical perspective. Furthermore, the symmetry approach can also be applied to describe the intensity profiles in the streamwise, wall-normal, and spanwise directions, which will be investigated in the future.