2.1 The outer expansions

The outer layer in the asymptotic expansions is defined as the layer with $z$ being of the order of $z_{i}$, but below the capping inversion. We use $U_{m}$, $V_{g}$, $z_{i}$, $u_{\ast }^{2}$, $u_{\ast }^{2}w_{e}/(fz_{i})=-V_{g}w_{e}$, $w_{\ast }^{2}$, $Q$ and $z_{i}/w_{\ast }$ as the outer scales for the mean velocity components, height from the surface, the streamwise and lateral kinematic stress components, velocity variance, potential temperature flux, and time to define the dimensionless outer variables (with a subscript $_{o}$),

(2.5)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle U(z)=U_{m}U_{o}\left({\displaystyle \frac{z}{z_{i}}}\right),\quad V(z)=V_{g}V_{o}\left({\displaystyle \frac{z}{z_{i}}}\right),\quad \overline{uw}=u_{\ast }^{2}\overline{uw}_{o},\quad \overline{vw}={\displaystyle \frac{u_{\ast }^{2}w_{e}}{fz_{i}}}\overline{vw}_{o},\\ \displaystyle z=z_{i}z_{o},\quad \overline{u\unicode[STIX]{x1D703}}=Q\overline{u\unicode[STIX]{x1D703}}_{o},\quad t={\displaystyle \frac{z_{i}}{w_{\ast }}}\unicode[STIX]{x1D70F},\quad \overline{w^{2}}=w_{\ast }^{2}\overline{w_{o}^{2}},\quad \end{array}\right\}\end{eqnarray}$$

where $w_{e}$ is the entrainment velocity (at the capping inversion). These outer scales are given by the geometry ($z_{i}$), the external parameters ($f$, $w_{e}$), and the boundary conditions ($U_{g}$, $Q$ and $u_{\ast }$, although $u_{\ast }$ is not an independent one) of the problem. The scale of $\overline{vw}$ reflects the balance between the lateral pressure gradient and the lateral stress generated by entrainment (Lilly Reference Lilly1968; Deardorff Reference Deardorff1973; Wyngaard Reference Wyngaard2010). The mixed-layer scale is obtained from $Q$, $z_{i}$ and $g/\unicode[STIX]{x1D6E9}$, and is also an external parameter, because it is not a parameter in MOST or MMO. The scale of lateral stress and $V_{g}$ are obtained with the lateral mean velocity being of higher order in the outer layer. The outer-layer mean velocity scale $U_{m}$ will be determined later in this section (2.26), but is asymptotically close to the mean velocity in the mixed layer. Panton (Reference Panton2005) showed that the use of the free-stream velocity as the velocity scale leads to the velocity-defect law in neutral boundary layers. In the present study, the use of $U_{m}$ also leads to velocity-defect laws in the CBL.

These parameters (outer scales) and the budget equations (2.1)–(2.4) form the singular perturbation problem that describes the mean velocity of the CBL. In the following, the scaling laws of the mean velocity will be derived by analysing this problem using the method of matched asymptotic expansions. Therefore, our analysis is based on first principles. These scaling laws could potentially also be obtained less rigorously using dimensional analysis without involving first principles (the budget equations).

Writing equations (2.1) and (2.2) in terms of the outer variables,

(2.6)$$\begin{eqnarray}\displaystyle & \hspace{-6.0pt}\displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw}_{o}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}{\displaystyle \frac{u_{\ast }^{2}w_{\ast }}{z_{i}}}+w_{\ast }^{2}\overline{w_{o}^{2}}{\displaystyle \frac{U_{m}}{z_{i}}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{o}}{\unicode[STIX]{x2202}z_{o}}}-{\displaystyle \frac{g}{T}}Q\overline{u\unicode[STIX]{x1D703}}_{o}+{\displaystyle \frac{w_{\ast }u_{\ast }^{2}}{z_{i}}}\left({\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw^{2}}_{o}}{\unicode[STIX]{x2202}z_{o}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}}\right)\!_{o}+\left(\overline{u{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)\!_{o}\right)=0, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.7)$$\begin{eqnarray}\displaystyle & \hspace{-6.0pt}\displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw}_{o}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}{\displaystyle \frac{u_{\ast }^{2}w_{e}}{fz_{i}}}{\displaystyle \frac{w_{\ast }}{z_{i}}}+w_{\ast }^{2}\overline{w_{o}^{2}}{\displaystyle \frac{V_{g}}{z_{i}}}{\displaystyle \frac{\unicode[STIX]{x2202}V_{o}}{\unicode[STIX]{x2202}z_{o}}}-{\displaystyle \frac{g}{T}}Q\overline{v\unicode[STIX]{x1D703}}_{o}+{\displaystyle \frac{w_{\ast }}{z_{i}}}{\displaystyle \frac{u_{\ast }^{2}w_{e}}{fz_{i}}}\!\!\left(\!{\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw^{2}}_{o}}{\unicode[STIX]{x2202}z_{o}}}+\!\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}y}}}\right)_{\!o}\!+\!\left(\overline{v{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{\!o}\!\right)=0, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and dividing them by $w_{\ast }^{2}U_{m}/z_{i}$ results in the non-dimensional shear-stress budget equations for the outer variables,

(2.8)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{u_{\ast }}{w_{\ast }}}{\displaystyle \frac{u_{\ast }}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw}_{o}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}+\overline{w_{o}^{2}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{o}}{\unicode[STIX]{x2202}z_{o}}}-{\displaystyle \frac{w_{\ast }}{U_{m}}}\overline{u\unicode[STIX]{x1D703}}_{o}+{\displaystyle \frac{u_{\ast }}{w_{\ast }}}{\displaystyle \frac{u_{\ast }}{U_{m}}}\left({\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw^{2}}_{o}}{\unicode[STIX]{x2202}z_{o}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}}\right)_{\!o}+\left(\overline{u{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{\!o}\right)=0, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.9)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw}_{o}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}+\overline{w_{o}^{2}}{\displaystyle \frac{V_{g}}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}V_{o}}{\unicode[STIX]{x2202}z_{o}}}-{\displaystyle \frac{w_{\ast }}{U_{m}}}\overline{v\unicode[STIX]{x1D703}}_{o}+{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}\left({\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw^{2}}_{o}}{\unicode[STIX]{x2202}z_{o}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}y}}}\right)_{\!o}+\left(\overline{v{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{\!o}\right)=0. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

There are two independent small parameters $(u_{\ast }/w_{\ast })(u_{\ast }/U_{m})$ and $(w_{e}/w_{\ast })(V_{g}/U_{m})$ in (2.8) and (2.9), since in a CBL that obeys the Monin–Obukhov similarity $u_{\ast }\ll w_{\ast }$, $u_{\ast }\ll U_{m}$, $w_{e}\ll w_{\ast }$ and $V_{g}<U_{m}$. The streamwise temperature flux in the buoyancy term in (2.8) scales as $(-z/L)^{-2/3}$ in the local-free-convection layer (Wyngaard et al. Reference Wyngaard, Coté and Izumi1971), and is smaller in the outer layer as the turbulence becomes increasingly horizontally isotropic. The cross-wind temperature flux is much smaller. Therefore, the buoyancy terms are smaller than the terms containing $(u_{\ast }/w_{\ast })(u_{\ast }/U_{m})$ and $(w_{e}/w_{\ast })(V_{g}/U_{m})$. These small parameters are a result of the outer scales (2.5) and the terms in (2.8) and (2.9), reflecting the physics of the CBL. Therefore, in general, we can write the outer expansions for $U$ and $V$ as,

(2.10)$$\begin{eqnarray}\displaystyle & \displaystyle U_{o}(z_{o})=U_{o,1}(z_{o})+{\displaystyle \frac{u_{\ast }}{w_{\ast }}}{\displaystyle \frac{u_{\ast }}{U_{m}}}U_{o,2a}(z_{o})+{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}U_{o,2b}(z_{o}), & \displaystyle\end{eqnarray}$$

(2.11)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{V_{g}}{U_{m}}}V_{o}(z_{o})={\displaystyle \frac{V_{g}}{U_{m}}}V_{o,1}(z_{o})+{\displaystyle \frac{u_{\ast }}{w_{\ast }}}{\displaystyle \frac{u_{\ast }}{U_{m}}}V_{o,2a}(z_{o})+{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}V_{o,2b}(z_{o}). & \displaystyle\end{eqnarray}$$

Thus,

(2.12)$$\begin{eqnarray}V_{o}(z_{o})=V_{o,1}(z_{o})-{\displaystyle \frac{fz_{i}}{w_{\ast }}}V_{o,2a}(z_{o})+{\displaystyle \frac{w_{e}}{w_{\ast }}}V_{o,2b}(z_{o}).\end{eqnarray}$$

The last terms in the outer expansions result from the Coriolis effects. The expansions for the velocity derivatives are

(2.13)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}U_{o}}{\unicode[STIX]{x2202}z_{o}}}={\displaystyle \frac{\unicode[STIX]{x2202}U_{o,1}}{\unicode[STIX]{x2202}z_{o}}}+{\displaystyle \frac{u_{\ast }}{w_{\ast }}}{\displaystyle \frac{u_{\ast }}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{o,2a}}{\unicode[STIX]{x2202}z_{o}}}+{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{o,2b}}{\unicode[STIX]{x2202}z_{o}}}, & \displaystyle\end{eqnarray}$$

(2.14)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}V_{o}}{\unicode[STIX]{x2202}z_{o}}}={\displaystyle \frac{\unicode[STIX]{x2202}V_{o,1}}{\unicode[STIX]{x2202}z_{o}}}-{\displaystyle \frac{fz_{i}}{w_{\ast }}}{\displaystyle \frac{\unicode[STIX]{x2202}V_{o,2a}}{\unicode[STIX]{x2202}z_{o}}}+{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{\unicode[STIX]{x2202}V_{o,2b}}{\unicode[STIX]{x2202}z_{o}}}. & \displaystyle\end{eqnarray}$$

Inserting equations (2.13) and (2.14) into (2.8) and (2.9), we find that the only leading-order terms are the mean-shear production terms (those that do not contain small parameters). Thus,

(2.15a,b)$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}U_{o,1}}{\unicode[STIX]{x2202}z_{o}}}=0,\quad {\displaystyle \frac{\unicode[STIX]{x2202}V_{o,1}}{\unicode[STIX]{x2202}z_{o}}}=0,\end{eqnarray}$$

resulting in

(2.16)$$\begin{eqnarray}\displaystyle & \displaystyle U_{o,1}=1, & \displaystyle\end{eqnarray}$$

(2.17)$$\begin{eqnarray}\displaystyle & \displaystyle V_{o,1}=0,\quad \text{since }V=0\text{ at some height in the mixed layer}. & \displaystyle\end{eqnarray}$$

Setting the leading order $U$ to $U_{m}$ results in $U_{o,1}=1$. Thus, from the outer expansions (2.10) and (2.12), we have

(2.18)$$\begin{eqnarray}\displaystyle & \displaystyle U=U_{m}U_{o}=U_{m}+{\displaystyle \frac{u_{\ast }}{w_{\ast }}}u_{\ast }U_{o,2a}+{\displaystyle \frac{w_{e}}{w_{\ast }}}V_{g}U_{o,2b}, & \displaystyle\end{eqnarray}$$

(2.19)$$\begin{eqnarray}\displaystyle & \displaystyle V=V_{g}V_{o}={\displaystyle \frac{u_{\ast }}{w_{\ast }}}u_{\ast }V_{o,2a}+{\displaystyle \frac{w_{e}}{w_{\ast }}}V_{g}V_{o,2b}. & \displaystyle\end{eqnarray}$$

The leading-order expansions (2.16) and (2.17) are not valid in the roughness layer, confirming that the system described by (2.8) and (2.9) is a singular perturbation problem. The second-order terms for $U$

(2.20)$$\begin{eqnarray}U-U_{m}={\displaystyle \frac{u_{\ast }}{w_{\ast }}}u_{\ast }U_{o,2a}+{\displaystyle \frac{w_{e}}{w_{\ast }}}V_{g}U_{o,2b}\end{eqnarray}$$

are the difference between the mean velocity and $U_{m}$. Inserting $\unicode[STIX]{x2202}U_{o}/\unicode[STIX]{x2202}z_{o}$ (2.13) into (2.8) leads to $\unicode[STIX]{x2202}U_{o,2b}/\unicode[STIX]{x2202}z_{o}=0$, since at the second order, equation (2.8) does not contain the parameter $(w_{e}/w_{\ast })(V_{g}/U_{m})$ (§ 2.3 further shows that $U_{o,2b}=0$). Equation (2.20) is a new velocity-defect law, which we term the mixed-layer velocity-defect law. The scale of the defect is $u_{\ast }(u_{\ast }/w_{\ast })$, which is asymptotically smaller than that in a neutral boundary layer ($u_{\ast }$). Physically this is because the larger vertical velocity variance (${\sim}w_{\ast }^{2}$) reduces the mean shear to maintain the same shear stress. Some previous works (e.g. Zilitinkevich & Deardorff Reference Zilitinkevich and Deardorff1974, Garratt et al. Reference Garratt, Wyngaard and Francey1982) defined the velocity defect as $U-U_{g}$, which also depends on $w_{e}$ and $f$ as discussed later in this subsection. With such a definition, the variations of $U$ in the outer layer, which is asymptotically smaller than $U-U_{g}$, is obscured. Consequently, this definition does not reflect the velocity defect in the outer layer.

The new velocity-defect law further indicates the importance of the mixed-layer mean velocity scale $U_{m}$. This defect law provides the scaling of the variations of the mean velocity in the mixed layer, and therefore is different in nature from that of Garratt et al. (Reference Garratt, Wyngaard and Francey1982), which describes the difference between the height-averaged mean velocity and the geostrophic wind.

Similarly, inserting $\unicode[STIX]{x2202}V_{o}/\unicode[STIX]{x2202}z_{o}$ (2.14) into (2.9) leads to $\unicode[STIX]{x2202}V_{o,2a}/\unicode[STIX]{x2202}z_{o}=0$; since at the second order, equation (2.9) does not contain the parameter $fz_{i}/w_{\ast }$ (§ 2.3 further shows that $V_{o,2a}=0$). We can evaluate the magnitudes of $V$ for typical values of $w_{\ast }$, $w_{e}$ and $V_{g}$ using (2.19) as

(2.21)$$\begin{eqnarray}V\sim {\displaystyle \frac{w_{e}}{w_{\ast }}}V_{g}\sim {\displaystyle \frac{0.03}{2.0}}\times 1\sim 0.01~\text{m}~\text{s}^{-1}.\end{eqnarray}$$

Thus, the change in the direction of the mean velocity in the outer layer is negligible compared to that across the inversion layer (${\sim}11^{\circ }$ for $U_{g}=5~\text{m}~\text{s}^{-1}$ and $V_{g}=1~\text{m}~\text{s}^{-1}$).

The equation for the second-order term $U_{o,2a}$ is,

(2.22)$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw}_{o,1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}+\overline{w_{o,1}^{2}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{o,2a}}{\unicode[STIX]{x2202}z_{o}}}-{\displaystyle \frac{w_{\ast }^{2}}{u_{\ast }^{2}}}\overline{u\unicode[STIX]{x1D703}}_{o,1}+{\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw^{2}}_{o,1}}{\unicode[STIX]{x2202}z_{o}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}}\right)_{\!o}+\left(\overline{u{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{\!o}=0.\end{eqnarray}$$

This equation does not contain a small parameter (the buoyancy production term is also of order one as the horizontal flux is of order $u_{\ast }^{2}/w_{\ast }^{2}$). However, the scaling of $\overline{w_{o}^{2}}$ changes when $z$ decreases to $z<-L$ (from $u_{f}^{2}$ to $u_{\ast }^{2}$, Tong & Ding Reference Tong and Ding2018), also resulting in a non-uniformly valid solution and a singular perturbation problem, where $u_{f}=(\unicode[STIX]{x1D6FD}Qz)^{1/3}$ is the local-free-convection velocity scale. Although equation (2.22) is not explicitly used, it helps us identify the source of the singularity.

Given that the scale of $\overline{vw}$ is $u_{\ast }^{2}w_{e}/(fz_{i})=-V_{g}w_{e}$, we can write the mean momentum equation for $V$ as,

(2.23)$$\begin{eqnarray}V_{g}{\displaystyle \frac{w_{\ast }}{z_{i}}}{\displaystyle \frac{\unicode[STIX]{x2202}V_{o}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}+{\displaystyle \frac{u_{\ast }^{2}w_{e}}{fz_{i}}}{\displaystyle \frac{1}{z_{i}}}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw}_{o}}{\unicode[STIX]{x2202}z_{o}}}=fU_{m}{\displaystyle \frac{U_{g}-U}{U_{m}}}.\end{eqnarray}$$

Dividing this equation by $U_{m}w_{\ast }/z_{i}$ results in

(2.24)$$\begin{eqnarray}{\displaystyle \frac{V_{g}}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}V_{o}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}+{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw_{o}}}{\unicode[STIX]{x2202}z_{o}}}={\displaystyle \frac{u_{\ast }}{V_{g}}}{\displaystyle \frac{u_{\ast }}{w_{\ast }}}\left({\displaystyle \frac{U_{g}-U_{m}}{U_{m}}}-{\displaystyle \frac{u_{\ast }}{w_{\ast }}}{\displaystyle \frac{u_{\ast }}{U_{m}}}U_{o,2a}\right).\end{eqnarray}$$

For $z<z_{i}$, the mean velocity is quasi-steady, the time rate of change is dropped. Equation (2.23) becomes

(2.25)$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw}_{o}}{\unicode[STIX]{x2202}z_{o}}}={\displaystyle \frac{U_{g}-U}{(u_{\ast }^{2}w_{e})/(f^{2}z_{i}^{2})}}.\end{eqnarray}$$

Both sides of (2.25) are of order one. It follows that $U_{g}-U_{m}$ scales as $u_{\ast }^{2}w_{e}/(f^{2}z_{i}^{2})$, which is the jump in $U$ across the inversion layer needed to balance the vertical derivative of the lateral stress in the outer layer. Thus, we can define the mean velocity scale $U_{m}$ as

(2.26)$$\begin{eqnarray}U_{m}\equiv U_{g}-{\displaystyle \frac{u_{\ast }^{2}w_{e}}{f^{2}z_{i}^{2}}}.\end{eqnarray}$$

Garratt et al. (Reference Garratt, Wyngaard and Francey1982) obtained a similar expression for the height-averaged mixed-layer velocity. Thus,

(2.27)$$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw}_{o}}{\unicode[STIX]{x2202}z_{o}}} & = & \displaystyle {\displaystyle \frac{U_{g}-U_{m}}{(u_{\ast }^{2}w_{e})/(f^{2}z_{i}^{2})}}-{\displaystyle \frac{(u_{\ast }/w_{\ast })u_{\ast }U_{o,2a}}{(u_{\ast }^{2}w_{e})/(f^{2}z_{i}^{2})}},\nonumber\\ \displaystyle & = & \displaystyle 1-{\displaystyle \frac{f^{2}z_{i}^{2}}{w_{\ast }w_{e}}}U_{o,2a},\end{eqnarray}$$

where,

(2.28)$$\begin{eqnarray}{\displaystyle \frac{f^{2}z_{i}^{2}}{w_{\ast }w_{e}}}\sim {\displaystyle \frac{0.01}{2\times 0.03}}\sim 0.15.\end{eqnarray}$$

Thus, there are small but significant variations of $\unicode[STIX]{x2202}\overline{vw_{o}}/\unicode[STIX]{x2202}z_{o}$ in the outer layer.

Using (2.19), the mean momentum equation for $U$ can be written as

(2.29)$$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw_{o}}}{\unicode[STIX]{x2202}z_{o}}} & = & \displaystyle {\displaystyle \frac{fz_{i}}{u_{\ast }^{2}}}(V-V_{g})={\displaystyle \frac{V_{g}-V}{V_{g}}},\nonumber\\ \displaystyle & = & \displaystyle {\displaystyle \frac{V_{g}-V}{U_{m}}}{\displaystyle \frac{U_{m}}{V_{g}}}=1-{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}{\displaystyle \frac{U_{m}}{V_{g}}}V_{o,2b},\nonumber\\ \displaystyle & = & \displaystyle 1-{\displaystyle \frac{w_{e}}{w_{\ast }}}V_{o,2b}.\end{eqnarray}$$

The parameter $w_{e}/w_{\ast }$ is estimated as

(2.30)$$\begin{eqnarray}{\displaystyle \frac{w_{e}}{w_{\ast }}}\sim {\displaystyle \frac{0.03}{2}}\sim 0.015,\end{eqnarray}$$

indicating that the second-order terms on the right-hand side of (2.29) is of higher order, i.e. to the leading order, the $\overline{uw}_{o}$ varies linearly with $z_{o}$. Therefore the Coriolis term is of higher order and at the leading order, Earth’s rotation does not affect the fundamental characteristics of the convective atmospheric boundary layer.

We note that if $w_{\ast }^{2}$ is used as the scale for the kinematic stress ($\overline{uw}=w_{\ast }^{2}\overline{uw}_{o}$) to focus on the dynamics of $\overline{u_{o}^{2}}$ and $\overline{w_{o}^{2}}$, the non-dimensional shear-stress transport equation becomes,

(2.31)$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw}_{o}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}+{\displaystyle \frac{U_{m}}{w_{\ast }}}\overline{w_{o}^{2}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{o}}{\unicode[STIX]{x2202}z_{o}}}-\overline{u\unicode[STIX]{x1D703}}_{o}+{\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw^{2}}_{o}}{\unicode[STIX]{x2202}z_{o}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}}\right)_{\!o}+\left(\overline{u{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{\!o}=0.\end{eqnarray}$$

All the terms in this equation are of higher order, indicating that in the outer layer, the effects of the shear-stress dynamics on that of $\overline{u_{o}^{2}}$ and $\overline{w_{o}^{2}}$ are of higher order, whereas the leading budget terms in the equations of $\overline{u_{o}^{2}}$ and $\overline{w_{o}^{2}}$ are of order one (Tong & Ding Reference Tong and Ding2018).

2.2 The inner-outer expansion

Two inner layers have been previously identified in the convective boundary layer (Tong & Ding Reference Tong and Ding2018, Reference Tong and Ding2019): one has a length scale of $-L$ , which we term the inner-outer layer, and the other has a scale of $h_{0}$, which we term the inner-inner layer. This structure is in contrast with the neutral boundary layer, which has only one inner layer, and is a key difference between a convective boundary layer and a neutral one.

Tong & Ding (Reference Tong and Ding2018) have shown analytically using the budget equations for $\overline{w^{2}}$ and $\overline{\unicode[STIX]{x1D703}^{2}}$ that their scaling changes as $z$ decreases across $-L$, going from buoyancy dominated to shear dominated. Therefore $-L$ and $u_{\ast }$ are the length scale and vertical velocity scale, respectively. Accordingly, we define the dimensionless inner-outer variables as,

(2.32)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle U(z)=U_{m}U_{io}\left(-{\displaystyle \frac{z}{L}}\right),\quad V(z)=V_{g}V_{io}\left(-{\displaystyle \frac{z}{L}}\right),\quad \overline{uw}=u_{\ast }^{2}\overline{uw}_{io},\\[5.0pt] \displaystyle \overline{vw}={\displaystyle \frac{u_{\ast }^{2}w_{e}}{fz_{i}}}\left(-{\displaystyle \frac{L}{z_{i}}}\right)\overline{vw}_{io},\quad \displaystyle z=-Lz_{io},\quad \overline{u\unicode[STIX]{x1D703}}=Q\overline{u\unicode[STIX]{x1D703}}_{io},\quad \overline{w^{2}}=u_{\ast }^{2}\overline{w_{io}^{2}}.\end{array}\right\}\end{eqnarray}$$

The scale for $\overline{vw}$ is such because its gradient is of the same order of magnitude as in the outer layer ($U_{g}-U$ is constant to the leading order). Writing equations (2.1) and (2.2) in terms of the inner-outer variables,

(2.33)$$\begin{eqnarray}\displaystyle & & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw}_{io}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}{\displaystyle \frac{u_{\ast }^{2}w_{\ast }}{z_{i}}}+u_{\ast }^{2}\overline{w_{io}^{2}}{\displaystyle \frac{U_{m}}{-L}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{io}}{\unicode[STIX]{x2202}z_{io}}}-{\displaystyle \frac{g}{T}}Q\overline{u\unicode[STIX]{x1D703}}_{io}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,{\displaystyle \frac{u_{\ast }^{3}}{-L}}\left({\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw^{2}}_{io}}{\unicode[STIX]{x2202}z_{io}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}}\right)_{io}+\left(\overline{u{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{io}\right)=0,\end{eqnarray}$$

(2.34)$$\begin{eqnarray}\displaystyle & & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw}_{io}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}{\displaystyle \frac{u_{\ast }^{2}w_{e}}{fz_{i}}}\left(-{\displaystyle \frac{L}{z_{i}}}\right){\displaystyle \frac{w_{\ast }}{z_{i}}}+u_{\ast }^{2}\overline{w_{io}^{2}}{\displaystyle \frac{V_{g}}{-L}}{\displaystyle \frac{\unicode[STIX]{x2202}V_{io}}{\unicode[STIX]{x2202}z_{io}}}-{\displaystyle \frac{g}{T}}Q\overline{v\unicode[STIX]{x1D703}}_{io}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,{\displaystyle \frac{u_{\ast }}{z_{i}}}{\displaystyle \frac{u_{\ast }^{2}w_{e}}{fz_{i}}}\left({\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw^{2}}_{io}}{\unicode[STIX]{x2202}z_{io}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}y}}}\right)_{io}+\left(\overline{v{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{io}\right)=0,\end{eqnarray}$$

and dividing by $u_{\ast }^{2}U_{m}/(-L)$ results in the non-dimensional shear-stress budget equations in the inner-outer layer,

(2.35)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle {\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{u_{\ast }}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw}_{io}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}+\overline{w_{io}^{2}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{io}}{\unicode[STIX]{x2202}z_{io}}}-{\displaystyle \frac{u_{\ast }}{U_{m}}}\overline{u\unicode[STIX]{x1D703}}_{io}+{\displaystyle \frac{u_{\ast }}{U_{m}}}\left({\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw^{2}}_{io}}{\unicode[STIX]{x2202}z_{io}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}}\right)_{io}+\left(\overline{u{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{io}\right)=0,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.36)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle {\displaystyle \frac{u_{\ast }^{4}w_{e}}{w_{\ast }^{5}}}{\displaystyle \frac{V_{g}}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw}_{io}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}+\overline{w_{io}^{2}}{\displaystyle \frac{V_{g}}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}V_{io}}{\unicode[STIX]{x2202}z_{io}}}-{\displaystyle \frac{u_{\ast }}{U_{m}}}\overline{v\unicode[STIX]{x1D703}}_{io}\nonumber\\ \displaystyle & & \displaystyle \qquad +{\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}\left({\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw^{2}}_{io}}{\unicode[STIX]{x2202}z_{io}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}y}}}\right)_{io}+\left(\overline{v{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{io}\right)=0.\end{eqnarray}$$

The scaling of the $\overline{u\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z}$ can be obtained using the MMO scaling of the $u$–$\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z$ cospectrum. The results (Tong & Ding Reference Tong and Ding2019, (2.16) and § 3.5) show from first principles that the spectra of $u$ and $\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z$ in the convective range ($1/z_{i}<k<-1/L$) have scaling exponents of $-5/3$ and $-1/3$, respectively, the latter is caused by buoyancy fluctuations, where $k$ is the horizontal wavenumber. Their cospectrum therefore has a scaling exponent no less than $-1$, i.e. the cospectrum cannot be steeper that $k^{-1}$, because $u$ and $\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z$ may be less correlated at scales larger than $-L$ as it is an off-diagonal component of the velocity-pressure gradient tensor. Thus $\overline{u\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z}$ at $z\sim -L$ is dominated by the scales near $-L$, and therefore $u_{\ast }^{3}/L$ is its proper scale. The spectral scaling exponents of $w$ and $\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}x$ are $1/3$ and $-1/3$, respectively, the latter has the same scaling as $\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z$. Therefore $\overline{w\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}x}$ is also dominated by the scales near $-L$ and scales as $u_{\ast }^{3}/L$. However, this term is smaller than $\overline{u\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z}$ because $w$ is smaller than $u$ at $z\sim -L$. The scaling of the velocity-pressure gradient terms is consistent with the role of these terms (partly) as the destruction rate of the shear stress. The transport term $\unicode[STIX]{x2202}\overline{uw^{2}}_{io}/\unicode[STIX]{x2202}z_{io}$ is small above the convection-induced stress layer (Businger Reference Businger1973; Sykes, Henn & Lewellen Reference Sykes, Henn and Lewellen1993; Grachev, W. & Zilitinkevich Reference Grachev, Fairall and Zilitinkevich1997; Zilitinkevich et al. Reference Zilitinkevich, Hunt, Esau, Grachev, Lalas, Akylas, Tombrou, Fairall, Fernando and Baklanov2006; Tong & Ding Reference Tong and Ding2019). The different scaling of $\overline{u\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z}$ in the outer and inner-outer layers also contributes to the singular nature of the problem.

There are two small parameters $u_{\ast }/U_{m}$ and $(u_{\ast }^{2}/w_{\ast }^{2})(w_{e}/w_{\ast })(V_{g}/U_{m})$ in (2.35) and (2.36), which are a result of the inner-outer scales (given in 2.32) and the terms in the budget equations. Thus, in general, we can write the inner-outer expansions of $U$ and $V$ as

(2.37)$$\begin{eqnarray}\displaystyle & \displaystyle U_{io}(z_{io})=U_{io,1}(z_{io})+{\displaystyle \frac{u_{\ast }}{U_{m}}}U_{io,2a}(z_{io})+{\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}U_{io,2b}(z_{io}), & \displaystyle\end{eqnarray}$$

(2.38)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{V_{g}}{U_{m}}}V_{io}(z_{io})={\displaystyle \frac{V_{g}}{U_{m}}}V_{io,1}(z_{io})+{\displaystyle \frac{u_{\ast }}{U_{m}}}V_{io,2a}(z_{io})+{\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}V_{io,2b}(z_{io}), & \displaystyle\end{eqnarray}$$

thus,

(2.39)$$\begin{eqnarray}V_{io}(z_{io})=V_{io,1}(z_{io})+{\displaystyle \frac{u_{\ast }}{V_{g}}}V_{io,2a}(z_{io})+{\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}V_{io,2b}(z_{io}).\end{eqnarray}$$

The velocity gradients are,

(2.40)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}U_{io}}{\unicode[STIX]{x2202}z_{io}}}={\displaystyle \frac{\unicode[STIX]{x2202}U_{io,1}}{\unicode[STIX]{x2202}z_{io}}}+{\displaystyle \frac{u_{\ast }}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{io,2a}}{\unicode[STIX]{x2202}z_{io}}}+{\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{io,2b}}{\unicode[STIX]{x2202}z_{io}}}, & \displaystyle\end{eqnarray}$$

(2.41)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}V_{io}}{\unicode[STIX]{x2202}z_{io}}}={\displaystyle \frac{\unicode[STIX]{x2202}V_{io,1}}{\unicode[STIX]{x2202}z_{io}}}+{\displaystyle \frac{u_{\ast }}{V_{g}}}{\displaystyle \frac{\unicode[STIX]{x2202}V_{io,2a}}{\unicode[STIX]{x2202}z_{io}}}+{\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{\unicode[STIX]{x2202}V_{io,2b}}{\unicode[STIX]{x2202}z_{io}}}. & \displaystyle\end{eqnarray}$$

In general, there are two inner-outer expansions, one to be matched with the outer expansion and other one to be matched with the inner-inner expansion. However, the expansions in (2.37)–(2.41) (up to the second order) can be matched with both the outer expansion and the inner expansion because it is the leading-order expansion for the velocity derivative.

Inserting $\unicode[STIX]{x2202}U_{io}/\unicode[STIX]{x2202}z_{io}$ and $\unicode[STIX]{x2202}V_{io}/\unicode[STIX]{x2202}z_{io}$ into (2.35) and (2.36), the only leading-order terms are the mean-shear production terms. Thus,

(2.42a,b)$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}U_{io,1}}{\unicode[STIX]{x2202}z_{io}}}=0,\quad {\displaystyle \frac{\unicode[STIX]{x2202}V_{io,1}}{\unicode[STIX]{x2202}z_{io}}}=0.\end{eqnarray}$$

By matching with leading-order term of the outer solution (2.18) and (2.19) (formal matching is given in (2.54)), we have

(2.43a,b)$$\begin{eqnarray}U_{io,1}=1,\quad V_{io,1}=0.\end{eqnarray}$$

Thus, from the inner-outer expansions (2.37) and (2.39), the inner-outer solution is,

(2.44)$$\begin{eqnarray}\displaystyle & \displaystyle U=U_{m}U_{io}=U_{m}+u_{\ast }U_{io,2a}+{\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}V_{g}U_{io,2b}, & \displaystyle\end{eqnarray}$$

(2.45)$$\begin{eqnarray}\displaystyle & \displaystyle V=V_{g}V_{io}=u_{\ast }V_{io,2a}+{\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}V_{g}V_{io,2b}. & \displaystyle\end{eqnarray}$$

From the inner-outer solution (2.44), we have,

(2.46)$$\begin{eqnarray}U-U_{m}=u_{\ast }U_{io,2a}+{\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}V_{g}U_{io,2b}.\end{eqnarray}$$

Inserting $\unicode[STIX]{x2202}U_{io}/\unicode[STIX]{x2202}z_{io}$ (2.40) into (2.35) leads to $\unicode[STIX]{x2202}U_{io,2b}/\unicode[STIX]{x2202}z_{io}=0$, since at the second order, equation (2.35) does not contain the parameter $(u_{\ast }^{2}/w_{\ast }^{2})(w_{e}/w_{\ast })(V_{g}/U_{m})$ (§ 2.3 further shows that $U_{io,2b}=0$). Equation (2.46) is another new velocity-defect law, the surface-layer velocity-defect law, in the convective atmospheric boundary layer. Thus the $z/L$ dependence of $U$ is in the form of a velocity-defect law, not the law of the wall. The term ‘surface layer’ here is used to indicate that the range of heights within which this velocity-defect law is valid is inside the surface layer, not that the defect law is valid in the entire surface layer. Equation (2.43) shows that $U_{m}$ is also the upper boundary condition for the leading-order velocity in the inner-outer layer (due to the asymptotically small velocity defect in the outer layer), indicating that the mean velocity in this layer is coupled to the mixed layer rather than to the surface. Again, previous works (e.g. Zilitinkevich & Deardorff Reference Zilitinkevich and Deardorff1974; Garratt et al. Reference Garratt, Wyngaard and Francey1982) used $U-U_{g}$ as the defect, which does not reflect the velocity defect in the surface layer.

We note that the surface-layer velocity-defect law cannot be obtained using MOST, as the velocity scale $U_{m}$ is not a parameter in the Monin–Obukhov similarity. Since MOST only involve the surface-layer parameters, it can only provide the Monin–Obukhov similarity functions. The surface-layer velocity-defect law, on the other hand, also needs to include $U_{m}$.

The inner-outer layer also limits the extent of the outer layer. In the neutral ABL the outer layer extends down to heights $z\gg h_{0}$. In the CBL, on the other hand, it only extends down to heights of order $-L$ due to the small velocity defect in the outer layer. Note that the inner-outer solution (2.43) is not valid in the roughness layer, indicating that there is still another singular (inner) layer.

Similarly, inserting $\unicode[STIX]{x2202}V_{io}/\unicode[STIX]{x2202}z_{io}$ (2.41) into (2.36) leads to $\unicode[STIX]{x2202}V_{io,2a}/\unicode[STIX]{x2202}z_{io}=0$, since at the second order, equation (2.36) does not contain the parameter $u_{\ast }/V_{g}$ (§ 2.3 shows that $V_{io,2a}=0$). We evaluate the magnitude of $V$ using typical values of $u_{\ast }$, $w_{\ast }$, $w_{e}$ and $V_{g}$,

(2.47)$$\begin{eqnarray}V\sim {\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}V_{g}\sim {\displaystyle \frac{0.3^{2}}{2^{2}}}{\displaystyle \frac{0.03}{2}}\times 1\sim 4\times 10^{-4}~\text{m}~\text{s}^{-1},\end{eqnarray}$$

indicating that the change in the direction of the mean velocity direction in this layer is also negligible.

The equation for the second-order term $U_{io,2a}$ is,

(2.48)$$\begin{eqnarray}{\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw}_{io,1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}+\overline{w_{io,1}^{2}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{io,2a}}{\unicode[STIX]{x2202}z_{io}}}-\overline{u\unicode[STIX]{x1D703}}_{io,1}+{\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw^{2}}_{io,1}}{\unicode[STIX]{x2202}z_{io}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}}\right)_{io}+\left(\overline{u{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{io}=0.\end{eqnarray}$$

Equation (2.48) contains a small parameter $u_{\ast }^{2}/w_{\ast }^{2}$ in the time derivative term, which becomes order one in (2.22) (the outer layer). In addition, the scaling of $\overline{w^{2}}$ and $\overline{u\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z}$ changes when $z$ increases from $z<-L$ to $z\gg -L$, as discussed in section (2.1). Therefore (2.22) and (2.48) form a singular perturbation problem. The leading-order expansions $U_{o,2a}$ and $U_{io,2a}$ can be obtained by the method of matched asymptotic expansions.

From the mean momentum equation for $V$, we can write the inner-outer expansion of $\unicode[STIX]{x2202}\overline{vw}/\unicode[STIX]{x2202}z$ as

(2.49)$$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw}_{io}}{\unicode[STIX]{x2202}z_{io}}} & = & \displaystyle {\displaystyle \frac{U_{g}-U_{m}}{(u_{\ast }^{2}w_{e})/(f^{2}z_{i}^{2})}}-{\displaystyle \frac{u_{\ast }U_{io,2a}}{(u_{\ast }^{2}w_{e})/(f^{2}z_{i}^{2})}},\nonumber\\ \displaystyle & = & \displaystyle 1-{\displaystyle \frac{f^{2}z_{i}^{2}}{u_{\ast }w_{e}}}U_{io,2a},\end{eqnarray}$$

where,

(2.50)$$\begin{eqnarray}{\displaystyle \frac{f^{2}z_{i}^{2}}{u_{\ast }w_{e}}}\sim {\displaystyle \frac{0.01}{0.3\times 0.03}}\sim 1.\end{eqnarray}$$

Thus there are also significant variations of the gradient of $\overline{vw}_{io}$ in the inner-outer layer. From the mean momentum equation for $U$, we can write the inner-outer expansion of $\unicode[STIX]{x2202}\overline{uw}/\unicode[STIX]{x2202}z$,

(2.51)$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw_{io}}}{\unicode[STIX]{x2202}z_{io}}}=-{\displaystyle \frac{fL}{u_{\ast }^{2}}}(V-V_{g})=-{\displaystyle \frac{L}{z_{i}}}+{\displaystyle \frac{L}{z_{i}}}{\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}V_{io,2b}\ll 1.\end{eqnarray}$$

Thus, the variations of the gradient of $\overline{uw}_{io}$ in the inner-outer layer are of higher order, (i.e. the stress is approximately constant). Although the variations of the $\overline{vw_{io}}$ gradient are of order one, $\overline{vw}$ itself is of higher order compared to $\overline{uw}$. Thus, the behaviour of the inner-outer layer at the leading order is not affected by Earth’s rotation.

2.3 Matching between the outer and the inner-outer expansion

The mean velocity in terms of the outer expansion (up to the second order) is

(2.52)$$\begin{eqnarray}U=U_{m}U_{o}=U_{m}\left(1+{\displaystyle \frac{u_{\ast }}{w_{\ast }}}{\displaystyle \frac{u_{\ast }}{U_{m}}}U_{o,2a}+{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}U_{o,2b}\right).\end{eqnarray}$$

In terms of the inner-outer expansion, it is

(2.53)$$\begin{eqnarray}U=U_{m}U_{io}=U_{m}\left(1+{\displaystyle \frac{u_{\ast }}{U_{m}}}U_{io,2a}+{\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}U_{io,2b}\right).\end{eqnarray}$$

Matching the two expansions we have

(2.54)$$\begin{eqnarray}U_{m}\left(1+{\displaystyle \frac{u_{\ast }}{w_{\ast }}}{\displaystyle \frac{u_{\ast }}{U_{m}}}U_{o,2a}+{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}U_{o,2b}\right)=U_{m}\left(1+{\displaystyle \frac{u_{\ast }}{U_{m}}}U_{io,2a}+{\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}{\displaystyle \frac{V_{g}}{U_{m}}}U_{io,2b}\right).\end{eqnarray}$$

For the parameter $(u_{\ast }/w_{\ast })(u_{\ast }/U_{m})$, we have

(2.55)$$\begin{eqnarray}{\displaystyle \frac{u_{\ast }}{w_{\ast }}}u_{\ast }U_{o,2a}\left({\displaystyle \frac{z}{z_{i}}}\right)=u_{\ast }U_{io,2a}\left(-{\displaystyle \frac{z}{L}}\right).\end{eqnarray}$$

Thus,

(2.56)$$\begin{eqnarray}{\displaystyle \frac{u_{\ast }}{w_{\ast }}}\left({\displaystyle \frac{z}{L}}{\displaystyle \frac{L}{z_{i}}}\right)^{\unicode[STIX]{x1D6FC}}=\left(-{\displaystyle \frac{z}{L}}\right)^{\unicode[STIX]{x1D6FC}},\quad \unicode[STIX]{x1D6FC}=-{\displaystyle \frac{1}{3}},\end{eqnarray}$$

because $L/z_{i}$ has to cancel on the left-hand side (note that $-L/z_{i}\sim (u_{\ast }/w_{\ast })^{3}$). This leads to

(2.57a,b)$$\begin{eqnarray}U_{o,2a}=A_{u}\left({\displaystyle \frac{z}{z_{i}}}\right)^{-1/3},\quad U_{io,2a}=A_{u}\left(-{\displaystyle \frac{z}{L}}\right)^{-1/3}.\end{eqnarray}$$

For the parameter $(w_{e}/w_{\ast })(V_{g}/U_{m})$, we have

(2.58)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{w_{e}}{w_{\ast }}}V_{g}U_{o,2b}\left({\displaystyle \frac{z}{z_{i}}}\right)={\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}V_{g}U_{io,2b}\left(-{\displaystyle \frac{z}{L}}\right), & \displaystyle\end{eqnarray}$$

(2.59)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}\left(-{\displaystyle \frac{z_{i}}{L}}{\displaystyle \frac{z}{z_{i}}}\right)^{\unicode[STIX]{x1D6FD}}=\left({\displaystyle \frac{z}{z_{i}}}\right)^{\unicode[STIX]{x1D6FD}},\quad \unicode[STIX]{x1D6FD}={\displaystyle \frac{2}{3}}, & \displaystyle\end{eqnarray}$$

leading to

(2.60a,b)$$\begin{eqnarray}U_{o,2b}=B_{u}\left({\displaystyle \frac{z}{z_{i}}}\right)^{2/3},\quad U_{io,2b}=B_{u}\left(-{\displaystyle \frac{z}{L}}\right)^{2/3}.\end{eqnarray}$$

However, since $\unicode[STIX]{x2202}U_{o,2b}/\unicode[STIX]{x2202}z_{o}=0$, we have $B_{u}=0$; thus $U_{o,2b}=U_{io,2b}=0$. From (2.57) and (2.60) we can write

(2.61)$$\begin{eqnarray}U=U_{m}+u_{\ast }A_{u}\left(-{\displaystyle \frac{z}{L}}\right)^{-1/3}.\end{eqnarray}$$

This is the velocity profile in the local-free-convection scaling layer. Monin & Obukhov (Reference Monin and Obukhov1954) predicted the $(-z/L)^{-1/3}$ scaling, but did not predict it as a velocity defect.

Similarly, for the $V$ component of the mean velocity, matching results in

(2.62a,b)$$\begin{eqnarray}V_{o,2a}=A_{v}\left({\displaystyle \frac{z}{z_{i}}}\right)^{-1/3},\quad V_{io,2a}=A_{v}\left(-{\displaystyle \frac{z}{L}}\right)^{-1/3},\end{eqnarray}$$

and

(2.63a,b)$$\begin{eqnarray}V_{o,2b}=B_{v}\left({\displaystyle \frac{z}{z_{i}}}\right)^{2/3},\quad V_{io,2b}=B_{v}\left(-{\displaystyle \frac{z}{L}}\right)^{2/3}.\end{eqnarray}$$

Since $\unicode[STIX]{x2202}V_{o,2a}/\unicode[STIX]{x2202}z_{o}=0$, we have $A_{v}=0$; thus $V_{o,2a}=V_{io,2a}=0$. Thus we can write

(2.64)$$\begin{eqnarray}V=B_{v}V_{g}{\displaystyle \frac{u_{\ast }^{2}}{w_{\ast }^{2}}}{\displaystyle \frac{w_{e}}{w_{\ast }}}\left(-{\displaystyle \frac{z}{L}}\right)^{2/3}.\end{eqnarray}$$

2.4 The inner-inner expansion

The inner-inner layer has a length scale of $h_{0}$ and a velocity scale of $u_{\ast }$ (Tong & Ding Reference Tong and Ding2018). We use $U_{m}$, $V_{g}$, $u_{\ast }^{2}$, $h_{0}$ and $Q$ to define the dimensionless inner-inner variables as follows:

(2.65)$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle U(z)=U_{m}U_{ii}\left({\displaystyle \frac{z}{h_{0}}}\right),\quad V(z)=V_{g}V_{ii}\left({\displaystyle \frac{z}{h_{0}}}\right),\quad \overline{uw}=u_{\ast }^{2}\overline{uw}_{ii},\\[5.0pt] \quad \overline{vw}=fU_{g}(-L)\overline{vw}_{ii},\quad z=h_{0}z_{ii},\quad \overline{u\unicode[STIX]{x1D703}}=Q\overline{u\unicode[STIX]{x1D703}}_{ii},\quad \overline{w^{2}}=u_{\ast }^{2}\overline{w_{ii}^{2}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The scale for $\overline{vw}$ is obtained by integrating the leading-order mean momentum equation $\unicode[STIX]{x2202}\overline{vw}/\unicode[STIX]{x2202}z=fU_{g}$ from $h_{0}$ to $-L$, with the upper limit determined by the fact that the vertical derivative of the lateral stress matches that due to the entrainment. The shear-stress budget equations are as follows:

(2.66)$$\begin{eqnarray}\displaystyle & & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw}_{ii}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}{\displaystyle \frac{u_{\ast }^{2}w_{\ast }}{z_{i}}}+u_{\ast }^{2}\overline{w_{ii}^{2}}{\displaystyle \frac{U_{m}}{h_{0}}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{ii}}{\unicode[STIX]{x2202}z_{ii}}}-{\displaystyle \frac{g}{T}}Q\overline{u\unicode[STIX]{x1D703}}_{ii}+{\displaystyle \frac{u_{\ast }^{3}}{h_{0}}}\left({\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw^{2}}_{ii}}{\unicode[STIX]{x2202}z_{ii}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}}\right)_{ii}+\left(\overline{u{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{ii}\right)=0,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.67)$$\begin{eqnarray}\displaystyle & & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw}_{ii}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}fU_{g}(-L){\displaystyle \frac{w_{\ast }}{z_{i}}}+u_{\ast }^{2}\overline{w_{ii}^{2}}{\displaystyle \frac{V_{g}}{h_{0}}}{\displaystyle \frac{\unicode[STIX]{x2202}V_{ii}}{\unicode[STIX]{x2202}z_{ii}}}-{\displaystyle \frac{g}{T}}Q\overline{v\unicode[STIX]{x1D703}}_{ii}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,{\displaystyle \frac{u_{\ast }}{h_{0}}}fU_{g}(-L)\left({\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw^{2}}_{ii}}{\unicode[STIX]{x2202}z_{ii}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}y}}}\right)_{ii}+\left(\overline{v{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{ii}\right)=0.\end{eqnarray}$$

Dividing (2.66) and (2.67) by $u_{\ast }^{2}U_{m}/h_{0}$ results in the non-dimensional shear-stress budget equations in the inner-inner layer,

(2.68)$$\begin{eqnarray}\displaystyle & & \displaystyle {\displaystyle \frac{h_{0}}{z_{i}}}{\displaystyle \frac{w_{\ast }}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw}_{ii}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}+\overline{w_{ii}^{2}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{ii}}{\unicode[STIX]{x2202}z_{ii}}}-{\displaystyle \frac{(g/T)Qh_{0}}{u_{\ast }^{2}U_{m}}}\overline{u\unicode[STIX]{x1D703}}_{ii}+{\displaystyle \frac{u_{\ast }}{U_{m}}}\left({\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw^{2}}_{ii}}{\unicode[STIX]{x2202}z_{ii}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}}\right)_{ii}+\left(\overline{u{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{ii}\right)=0,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(2.69)$$\begin{eqnarray}\displaystyle & & \displaystyle {\displaystyle \frac{h_{0}}{z_{i}}}{\displaystyle \frac{f(-L)w_{\ast }}{u_{\ast }^{2}}}{\displaystyle \frac{U_{g}}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw}_{ii}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}+\overline{w_{ii}^{2}}{\displaystyle \frac{V_{g}}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}V_{ii}}{\unicode[STIX]{x2202}z_{ii}}}-{\displaystyle \frac{(g/T)Qh_{0}}{u_{\ast }^{2}U_{m}}}\overline{v\unicode[STIX]{x1D703}}_{ii}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,{\displaystyle \frac{f(-L)}{u_{\ast }}}{\displaystyle \frac{U_{g}}{U_{m}}}\left({\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw^{2}}_{ii}}{\unicode[STIX]{x2202}z_{ii}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}y}}}\right)_{ii}+\left(\overline{v{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{ii}\right)=0.\end{eqnarray}$$

Similar to the inner-outer layer, the scaling of the $\overline{u\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z}$ can be obtained using the MMO scaling of the $u$–$\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z$ cospectrum. Tong & Ding (Reference Tong and Ding2019) have shown that the spectrum of $u$ in the dynamic range have scaling exponents of $-1$. Near $kz=1$, $\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z$ is balanced by the nonlinear term (N3) in equation (3.19) in Tong & Ding (Reference Tong and Ding2019), which has a spectral scaling exponent of 3. Therefore the $u-\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z$ cospectrum has a scaling exponent no less than $1$. Thus $\overline{u\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z}$ is dominated by the scales near $z$, and $u_{\ast }^{3}/z$ is its proper scale. Similarly, the spectral scaling of $w$ and $\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}x$ are both 1. Thus, $\overline{w\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}x}$ is also determined by the scales near $z$, thus having a scale of $u_{\ast }^{3}/z$. For $z\rightarrow h_{0}$, we expect that $u_{\ast }^{2}/h_{0}$ is the proper scale for these quantities. Again, the transport term is small.

Similar to (2.35) and (2.36) in the inner-outer layer, there are also two small (independent) parameters $u_{\ast }/U_{m}$ and $(f(-L)/u_{\ast })(U_{g}/U_{m})$ in the (2.68) and (2.69). Thus, we can write the inner-inner expansions as

(2.70)$$\begin{eqnarray}\displaystyle & \displaystyle U_{ii}(z_{ii})=U_{ii,1}(z_{ii})+{\displaystyle \frac{u_{\ast }}{U_{m}}}U_{ii,2}(z_{ii}), & \displaystyle\end{eqnarray}$$

(2.71)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{V_{g}}{U_{m}}}V_{ii}(z_{ii})={\displaystyle \frac{V_{g}}{U_{m}}}V_{ii,1}(z_{ii})+{\displaystyle \frac{f(-L)}{u_{\ast }}}{\displaystyle \frac{U_{g}}{U_{m}}}V_{ii,2}(z_{ii}), & \displaystyle\end{eqnarray}$$

thus,

(2.72)$$\begin{eqnarray}V_{ii}(z_{ii})=V_{ii,1}(z_{ii})+{\displaystyle \frac{f(-L)}{u_{\ast }}}{\displaystyle \frac{U_{g}}{V_{g}}}V_{ii,2}(z_{ii}).\end{eqnarray}$$

The velocity gradients are

(2.73)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}U_{ii}}{\unicode[STIX]{x2202}z_{ii}}}={\displaystyle \frac{\unicode[STIX]{x2202}U_{ii,1}}{\unicode[STIX]{x2202}z_{ii}}}+{\displaystyle \frac{u_{\ast }}{U_{m}}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{ii,2}}{\unicode[STIX]{x2202}z_{ii}}}, & \displaystyle\end{eqnarray}$$

(2.74)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}V_{ii}}{\unicode[STIX]{x2202}z_{ii}}}={\displaystyle \frac{\unicode[STIX]{x2202}V_{ii,1}}{\unicode[STIX]{x2202}z_{ii}}}+{\displaystyle \frac{f(-L)}{u_{\ast }}}{\displaystyle \frac{U_{g}}{V_{g}}}{\displaystyle \frac{\unicode[STIX]{x2202}V_{ii,2}}{\unicode[STIX]{x2202}z_{ii}}}. & \displaystyle\end{eqnarray}$$

Inserting $\unicode[STIX]{x2202}U_{ii}/\unicode[STIX]{x2202}z_{ii}$ and $\unicode[STIX]{x2202}V_{ii}/\unicode[STIX]{x2202}z_{ii}$ into (2.68) and (2.69), the only leading-order terms are the mean-shear production terms. Thus,

(2.75a,b)$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}U_{ii,1}}{\unicode[STIX]{x2202}z_{ii}}}=0,\quad {\displaystyle \frac{\unicode[STIX]{x2202}V_{ii,1}}{\unicode[STIX]{x2202}z_{ii}}}=0,\end{eqnarray}$$

and

(2.76a,b)$$\begin{eqnarray}U_{ii,1}=0,\quad V_{ii,1}=0,\end{eqnarray}$$

since $U_{ii,1}$ and $V_{ii,1}$ must equal zero at some $z$ value, which can be defined as $h_{0}$. Thus, we can write the inner-inner solution as,

(2.77)$$\begin{eqnarray}\displaystyle & \displaystyle U=U_{m}U_{ii}=u_{\ast }U_{ii,2}(z_{ii}), & \displaystyle\end{eqnarray}$$

(2.78)$$\begin{eqnarray}\displaystyle & \displaystyle V=V_{g}V_{ii}={\displaystyle \frac{f(-L)}{u_{\ast }}}U_{g}V_{ii,2}(z_{ii}). & \displaystyle\end{eqnarray}$$

For a strongly convective boundary layer, e.g. $-L=20$ m, $u_{\ast }=0.3~\text{m}~\text{s}^{-1}$, $U_{g}=5~\text{m}~\text{s}^{-1}$, the lateral velocity is of order $f(-L)U_{g}/u_{\ast }=0.03~\text{m}~\text{s}^{-1}$, which is relatively small.

The mean momentum equation for $V$ is

(2.79)$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{vw}_{ii}}{\unicode[STIX]{x2202}z_{ii}}}={\displaystyle \frac{h_{0}}{-L}}-{\displaystyle \frac{h_{0}}{-L}}{\displaystyle \frac{u_{\ast }}{U_{g}}}U_{ii,2}+{\displaystyle \frac{\text{d}\unicode[STIX]{x1D70F}_{ryii}}{\text{d}z_{ii}}}.\end{eqnarray}$$

The mean momentum equation for $U$ is

(2.80)$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw_{ii}}}{\unicode[STIX]{x2202}z_{ii}}}={\displaystyle \frac{h_{0}}{z_{i}}}-{\displaystyle \frac{h_{0}}{z_{i}}}{\displaystyle \frac{V}{V_{g}}}+{\displaystyle \frac{\text{d}\unicode[STIX]{x1D70F}_{rxii}}{\text{d}z_{ii}}}={\displaystyle \frac{h_{0}}{z_{i}}}-{\displaystyle \frac{h_{0}}{z_{i}}}{\displaystyle \frac{f(-L)}{u_{\ast }}}{\displaystyle \frac{U_{g}}{V_{g}}}V_{ii,2}+{\displaystyle \frac{\text{d}\unicode[STIX]{x1D70F}_{rxii}}{\text{d}z_{ii}}}.\end{eqnarray}$$

Integrating these equations from $h_{0}$ to $z_{ii}$ and keeping the leading-order results in

(2.81)$$\begin{eqnarray}\displaystyle & \displaystyle \overline{vw}_{ii,1}=1-\unicode[STIX]{x1D70F}_{ryii,1}, & \displaystyle\end{eqnarray}$$

(2.82)$$\begin{eqnarray}\displaystyle & \displaystyle \overline{uw}_{ii,1}=1-\unicode[STIX]{x1D70F}_{rxii,1}. & \displaystyle\end{eqnarray}$$

Again, since $\overline{vw}$ is much smaller than $\overline{uw}$, the behaviours of the inner-inner layer at the leading order is not affected by Earth’s rotation at the leading order.

The equation for the second term $U_{ii,2}$ is,

(2.83)$$\begin{eqnarray}{\displaystyle \frac{h_{0}}{z_{i}}}{\displaystyle \frac{w_{\ast }}{u_{\ast }}}{\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw}_{ii}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}}+\overline{w_{ii}^{2}}{\displaystyle \frac{\unicode[STIX]{x2202}U_{ii,2}}{\unicode[STIX]{x2202}z_{ii}}}-{\displaystyle \frac{(g/T)Qh_{0}}{u_{\ast }^{3}}}\overline{u\unicode[STIX]{x1D703}}_{ii}+\left({\displaystyle \frac{\unicode[STIX]{x2202}\overline{uw^{2}}_{ii}}{\unicode[STIX]{x2202}z_{ii}}}+\left(\overline{w{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x}}}\right)_{ii}+\left(\overline{u{\displaystyle \frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}}}\right)_{ii}\right)=0.\end{eqnarray}$$

Equation (2.83) contains a small parameter $(g/T)Qh_{0}/u_{\ast }^{3}$ in the $\overline{u\unicode[STIX]{x1D703}}_{ii}$ term, which becomes order one in (2.48) (the inner-outer layer), and is a correction due to buoyancy production as $z\rightarrow -L$. The roughness contribution (as part of the pressure terms) in (2.48) is of higher order, which becomes order one in (2.83). Thus (2.48) and (2.83) also form a singular perturbation problem. The leading-order expansions $U_{io,2a}$ and $U_{ii,2}$ can be obtained by the method of matched asymptotic expansions.

2.5 Matching between the inner-outer and the inner-inner expansion

The mean velocity in terms of the inner-outer expansion is

(2.84)$$\begin{eqnarray}U=U_{m}U_{io}=U_{m}+u_{\ast }U_{io,2a}(z_{io}).\end{eqnarray}$$

In terms of the inner-inner expansion, it is

(2.85)$$\begin{eqnarray}U=U_{m}U_{ii}=u_{\ast }U_{ii,2}(z_{ii}).\end{eqnarray}$$

Matching the two expansions results in

(2.86)$$\begin{eqnarray}U_{ii,2}\left(-z_{io}{\displaystyle \frac{L}{h_{0}}}\right)={\displaystyle \frac{U_{m}}{u_{\ast }}}+U_{io,2a}(z_{io}).\end{eqnarray}$$

This equation shows that $U_{m}/u_{\ast }$ is a function of $-L/h_{0}$. Differentiating equation (2.86) with respect to $z_{io}$ results in

(2.87)$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}U_{ii,2}}{\unicode[STIX]{x2202}z_{io}}}=-{\displaystyle \frac{\text{d}U_{ii,2}}{\text{d}z_{ii}}}{\displaystyle \frac{L}{h_{0}}}={\displaystyle \frac{\text{d}U_{io,2a}}{\text{d}z_{io}}}.\end{eqnarray}$$

Thus

(2.88)$$\begin{eqnarray}{\displaystyle \frac{\text{d}U_{ii,2}}{\text{d}z_{ii}}}z_{ii}={\displaystyle \frac{\text{d}U_{io,2a}}{\text{d}z_{io}}}z_{io}={\displaystyle \frac{1}{\unicode[STIX]{x1D705}}},\end{eqnarray}$$

where $\unicode[STIX]{x1D705}$ does not depend on either $z_{ii}$ or $z_{io}$, and therefore does not depend on $-L/h_{0}$. Differentiating equation (2.86) with respect to $-L/h_{0}$ results in

(2.89)$$\begin{eqnarray}-{\displaystyle \frac{\unicode[STIX]{x2202}U_{ii,2}}{\unicode[STIX]{x2202}L/h_{0}}}={\displaystyle \frac{\text{d}U_{ii,2}}{\text{d}z_{ii}}}z_{io}=-{\displaystyle \frac{\text{d}U_{ii,2}}{\text{d}z_{ii}}}z_{ii}{\displaystyle \frac{h_{0}}{L}}=-{\displaystyle \frac{\text{d}U_{m}/u_{\ast }}{\text{d}L/h_{0}}}.\end{eqnarray}$$

The last term in (2.89) does not depend on $z_{ii}$. From (2.89) we obtain

(2.90)$$\begin{eqnarray}U_{ii,2}={\displaystyle \frac{\text{d}U_{m}/u_{\ast }}{\text{d}L/h_{0}}}{\displaystyle \frac{L}{h_{0}}}\ln z_{ii}={\displaystyle \frac{1}{\unicode[STIX]{x1D705}}}\ln z_{ii}={\displaystyle \frac{1}{\unicode[STIX]{x1D705}}}\ln {\displaystyle \frac{z}{h_{0}}}.\end{eqnarray}$$

Note that this expression is valid only in the matching layer. This $h_{0}$ is defined by extrapolating the profile to $U_{ii,2}=0$. From (2.88) we obtain

(2.91)$$\begin{eqnarray}U_{io,2a}={\displaystyle \frac{\text{d}U_{m}/u_{\ast }}{\text{d}L/h_{0}}}{\displaystyle \frac{L}{h_{0}}}\ln z_{io}+C={\displaystyle \frac{1}{\unicode[STIX]{x1D705}}}\ln z_{io}+C={\displaystyle \frac{1}{\unicode[STIX]{x1D705}}}\ln \left(-{\displaystyle \frac{z}{L}}\right)+C.\end{eqnarray}$$

Hence,

(2.92)$$\begin{eqnarray}U=U_{m}+u_{\ast }\left({\displaystyle \frac{1}{\unicode[STIX]{x1D705}}}\ln \left(-{\displaystyle \frac{z}{L}}\right)+C\right).\end{eqnarray}$$

Equation (2.90) provides a physical interpretation of the (inverse of) the von Kármán constant

(2.93)$$\begin{eqnarray}{\displaystyle \frac{1}{\unicode[STIX]{x1D705}}}={\displaystyle \frac{\text{d}U_{m}/u_{\ast }}{\text{d}\ln (-L/h_{0})}}.\end{eqnarray}$$

It is the rate of change of the friction velocity ratio $U_{m}/u_{\ast }$ with respect to the logarithm of the ratio of the height of the shear-production-dominant layer to the roughness height. In a neutral boundary layer, it becomes

(2.94)$$\begin{eqnarray}{\displaystyle \frac{1}{\unicode[STIX]{x1D705}}}={\displaystyle \frac{\text{d}U_{m}/u_{\ast }}{\text{d}\ln z_{i}/h_{0}}},\end{eqnarray}$$

which also has a clear physical interpretation. Equations (2.90) and (2.91) are the log law and the surface-layer velocity-defect law in the matching layer. Combining the two equations results in

(2.95)$$\begin{eqnarray}{\displaystyle \frac{U_{m}}{u_{\ast }}}={\displaystyle \frac{1}{\unicode[STIX]{x1D705}}}\ln \left(-{\displaystyle \frac{L}{h_{0}}}\right)-C.\end{eqnarray}$$

Differentiating this equation with respect to $-L/h_{0}$ and considering (2.93), we find that $C$ also does not depend on $-L/h_{0}$, and therefore is a constant. Equation (2.95) is the logarithmic friction law for the convective atmospheric boundary layer, which we term the convective logarithmic friction law. Thus, for the same $U_{m}$, the surface stress increases with the surface temperature flux. Compared with the logarithmic friction law for neutral boundary layers, the Obukhov length replaces the boundary layer (inversion) height, because the convective eddies result in an asymptotically small mean velocity gradient in the mixed layer, pushing $U_{m}$ down to the height $-L$.

The discovery of the convective logarithmic friction law allows us to make several comments on the previous geostrophic drag laws (or resistance laws) (Lettau Reference Lettau1959; Kazanski & Monin Reference Kazanski and Monin1960; Csanady Reference Csanady1967; Blackadar & Tennekes Reference Blackadar and Tennekes1968; Brown Reference Brown1973; Zilitinkevich & Deardorff Reference Zilitinkevich and Deardorff1974; Zilitinkevich Reference Zilitinkevich1975; Zilitinkevich et al. Reference Zilitinkevich, Fedorovich and Shabalova1992). First, the convective logarithmic friction law is derived analytically from first principles whereas the geostrophic drag law was largely based on phenomenology. Second, the convective logarithmic friction law is derived from the law of the wall and the surface-layer velocity-defect law and is an exact leading-order result relating $u_{\ast }$ to the velocity scale $U_{m}$, not an extrapolation of the log law. No approximate velocity is used. By contrast, some previous drag laws (e.g. Zilitinkevich et al. Reference Zilitinkevich, Fedorovich and Shabalova1992) match the geostrophic wind with an approximate velocity beyond the log layer (at a height $z\sim z_{i}$), and therefore are inherently approximate and empirical. Third, as mentioned above, the convective logarithmic friction law is derived using the velocity defect $U-U_{m}$ which scales with $u_{\ast }$ in the surface layer, whereas some other previous drag laws (e.g. Zilitinkevich & Deardorff Reference Zilitinkevich and Deardorff1974; Zilitinkevich Reference Zilitinkevich1975) used the velocity defect $U-U_{g}$, part of which ($U_{m}-U_{g}$) does not scale with $u_{\ast }$ (2.26). Since $u_{\ast }$ is completely determined by $U_{m}$ and $L/h_{0}$, $U_{m}-U_{g}$ does not play a role in determining the friction, and therefore should not be an inherent part of the drag law. Garratt et al. (Reference Garratt, Wyngaard and Francey1982) proposed a geostrophic drag law assuming that the mean velocity at the top of the surface layer equals that in the mixed layer. However, it uses the empirical stability function $\unicode[STIX]{x1D713}_{m}$, which may be inaccurate for large values of $-z/L$, whereas the convective logarithmic friction law does not involve any empirical functions.

2.6 Overall mean velocity profile

The different functional forms of mean velocity in the three layers discussed in the preceding subsections form an overall mean velocity profile. A schematic including the relationships among the different layers and scaling ranges is shown in figure 1. In a neutral ABL, the inner layer, where the law of the wall is valid, covers the entire surface layer. The outer layer, where the velocity-defect law holds, extends into the surface layer, down to a height $z\gg h_{0}$. By contrast, in a CBL, the inner-inner layer, where the law of the wall (for rough wall) is valid, only extends to a height $z<-L$. The inner-outer layer covers the rest of the surface layer ($h_{0}\ll z\ll z_{i}$), and follows the surface-layer velocity-defect law, with the velocity scale $u_{\ast }$. The outer layer extends into the surface layer, but only down to $z\gg -L$, and follows the mixed-layer velocity-defect law, with the velocity scale $u_{\ast }^{2}/w_{\ast }$. There are two overlapping layers in which the mean velocity profile conforms to the log law and the local-free-convection scaling, respectively.

The surface-layer velocity-defect law provides an important interpretation of the velocity profile given in Monin & Obukhov (Reference Monin and Obukhov1954),

(2.96)$$\begin{eqnarray}U(z)={\displaystyle \frac{u_{\ast }}{\unicode[STIX]{x1D705}}}\left[f\left({\displaystyle \frac{z}{L}}\right)-f\left({\displaystyle \frac{h_{0}}{L}}\right)\right],\end{eqnarray}$$

where

(2.97)$$\begin{eqnarray}f\left({\displaystyle \frac{z}{L}}\right)=\int {\displaystyle \frac{\unicode[STIX]{x1D719}_{m}(z/L)}{z/L}}d\left({\displaystyle \frac{z}{L}}\right).\end{eqnarray}$$

For $h_{0}/L\ll 1$, $f(h_{0}/L)\approx \ln (L/h_{0})$ since $\unicode[STIX]{x1D719}_{m}(0)=1$. Thus,

(2.98)$$\begin{eqnarray}U(z)\approx {\displaystyle \frac{u_{\ast }}{\unicode[STIX]{x1D705}}}\left[f\left({\displaystyle \frac{z}{L}}\right)+\ln \left({\displaystyle \frac{L}{h_{0}}}\right)\right].\end{eqnarray}$$

Using the convective logarithmic friction law (2.95), we can write

(2.99)$$\begin{eqnarray}U(z)\approx {\displaystyle \frac{u_{\ast }}{\unicode[STIX]{x1D705}}}\left[f\left({\displaystyle \frac{z}{L}}\right)+{\displaystyle \frac{U_{m}\unicode[STIX]{x1D705}}{u_{\ast }}}+\unicode[STIX]{x1D705}C\right],\end{eqnarray}$$

and

(2.100)$$\begin{eqnarray}U-U_{m}\approx {\displaystyle \frac{u_{\ast }}{\unicode[STIX]{x1D705}}}\left[f\left({\displaystyle \frac{z}{L}}\right)+\unicode[STIX]{x1D705}C\right].\end{eqnarray}$$

Thus $f(z/L)$ is essentially part of the velocity defect. However, this relationship only becomes clear with the derivation of the surface-layer defect-law.

The surface-layer velocity-defect law also provides an understanding of another form of the velocity profile (Panofsky & Dutton Reference Panofsky and Dutton1984),

(2.101)$$\begin{eqnarray}{\displaystyle \frac{U}{u_{\ast }}}={\displaystyle \frac{1}{\unicode[STIX]{x1D705}}}\left[\ln {\displaystyle \frac{z}{h_{0}}}-\unicode[STIX]{x1D713}_{m}\left({\displaystyle \frac{z}{L}}\right)\right],\end{eqnarray}$$

where

(2.102)$$\begin{eqnarray}\unicode[STIX]{x1D713}_{m}\left({\displaystyle \frac{z}{L}}\right)=\int _{0}^{z/L}[1-\unicode[STIX]{x1D719}_{m}(\unicode[STIX]{x1D701})]{\displaystyle \frac{\text{d}\unicode[STIX]{x1D701}}{\unicode[STIX]{x1D701}}}.\end{eqnarray}$$

We note that (2.102) is only valid for $z/h_{0}\gg 1$, i.e. in the matching (log) layer and above, because it is obtained using the non-dimensional mean shear $\unicode[STIX]{x1D719}_{m}(z/L)$, which has the same region of validity. Furthermore, although (2.101) might give the impression of a modified law of the wall (with an added dependence on $-z/L$), in which a wall velocity scale ($u_{\ast }$) and a wall length scale ($h_{0}$) are involved, it is not one. This can be seen from the surface-layer velocity-defect law (2.46), which shows that the dependence of $U$ on $-z/L$ comes entirely in the form of the velocity defect and does not depend on $h_{0}$ (part of $\unicode[STIX]{x1D713}_{m}$ cancels the logarithmic term in (2.101)).

To further examine (2.101), we derive it by inserting $U_{m}$ (2.95) into the surface-layer defect-law (2.46),

(2.103)$$\begin{eqnarray}\displaystyle U & = & \displaystyle {\displaystyle \frac{u_{\ast }}{\unicode[STIX]{x1D705}}}\ln \left(-{\displaystyle \frac{L}{h_{0}}}\right)-u_{\ast }C+u_{\ast }U_{io,2a}\nonumber\\ \displaystyle & = & \displaystyle {\displaystyle \frac{u_{\ast }}{\unicode[STIX]{x1D705}}}\left[\ln {\displaystyle \frac{z}{h_{0}}}+\ln \left(-{\displaystyle \frac{L}{z}}\right)\right]-u_{\ast }C+u_{\ast }U_{io,2a}\nonumber\\ \displaystyle & = & \displaystyle {\displaystyle \frac{u_{\ast }}{\unicode[STIX]{x1D705}}}\ln {\displaystyle \frac{z}{h_{0}}}-{\displaystyle \frac{u_{\ast }}{\unicode[STIX]{x1D705}}}\ln \left(-{\displaystyle \frac{z}{L}}\right)-u_{\ast }C+u_{\ast }U_{io,2a}.\end{eqnarray}$$

It can be seen that $\unicode[STIX]{x1D713}_{m}(z/L)$ contains the velocity defect which, however, also only becomes clear with the derivation of (2.46). Since (2.101) was obtained without the knowledge of the surface-layer velocity-defect law, it could only be obtained empirically.

The defect law (2.46) also shows that for $z\sim -L$, the mean velocity $U$ is of order $U_{m}$ because $U_{m}-U\sim u_{\ast }\ll U_{m}$. This behaviour, however, cannot be inferred from (2.96) and (2.101) as they do not contain $U_{m}$. Thus, while numerically the empirical velocity profile (2.101) has the correct behaviour and is useful for practical applications, its origin and interpretation only become clear with the discovery of the surface-layer velocity-defect law. From a practical point of view, it may be more accurate to use the surface-layer velocity-defect law rather than (2.101) for $z/L\gtrsim 1$, as it is the difference of two large terms (the term $\ln (z/h_{0})$ is actually cancelled).

The surface-layer velocity-defect law indicates that buoyancy effects reduce the extent of the law of the wall. In a neutral boundary layer, the law of the wall is valid in the entire surface layer. By contrast, in a convective boundary layer, it is only valid up to $z<-L$, i.e. it ceases to be valid as $z$ approaches $-L$. Thus buoyancy disrupts the law of the wall rather than modifying it. A key factor for the law of the wall is the no-slip boundary condition (at $z=h_{0}$). For $z\geqslant -L$ the larger vertical velocity variance due to buoyancy production reduces the outer-layer velocity defect, imposing $U_{m}$ as the (upper) boundary condition on the leading-order inner-outer layer velocity, thereby disrupting the law of the wall and resulting in the surface-layer velocity-defect law. This defect law suggests that $-L$ can also be interpreted as the height at which the mean velocity ceases to be coupled to the surface velocity, and instead becomes coupled to the mixed-layer velocity scale $U_{m}$.