Let’s go all the way back to the idea of advection. We will replace the scalar R with potential temperature:

$$\begin{array}{l}\frac{\partial \Theta}{\partial t}=\frac{D\Theta}{Dt}-\overrightarrow{v}\xb7\overrightarrow{\nabla}\Theta \\ \text{orinthevertical}\\ \frac{\partial \Theta}{\partial t}=\frac{D\Theta}{Dt}-w\cdot \frac{\partial \Theta}{\partial z}\\ \text{lookatthemeanandperturbedparts:}\\ \frac{\partial \left(\overline{\Theta}+\Theta \text{'}\right)}{\partial t}=\frac{D\left(\overline{\Theta}+\Theta \text{'}\right)}{Dt}-\left(\overline{w}+w\text{'}\right)\cdot \frac{\partial \left(\overline{\Theta}+\Theta \text{'}\right)}{\partial z}\\ ...\text{taketheReynold'saverage.}\end{array}$$

After taking the Reynold’s average, we achieve the equation:

$$\frac{\partial \overline{\Theta}}{\partial t}=\frac{D\overline{\Theta}}{Dt}-\overline{w}\cdot \frac{\partial \overline{\Theta}}{\partial z}-\overline{w\text{'}\frac{\partial \Theta \text{'}}{\partial z}}$$

The first term is the rate of change of the mean potential temperature at a given height, although it applies to any height. The $\frac{D\Theta}{Dt}$ term is the local heating from the divergence of the radiant energy and from phase changes. The third term is the mean advection, but because it is small in the fair weather boundary layer, mean vertical advection is ~0. Note that the mean horizontal advection is usually quite large, and by scale analysis, must be kept in the equation of motion.

Generally, $\frac{D\overline{\Theta}}{Dt}$ is small for the entire atmosphere, except in clouds, so we can neglect that term for the typical convective boundary layer.

If we assume that the density does not change, then we can basically say the volume of air doesn’t change (i.e., incompressibility). We used this concept to show that horizontal convergence results in vertical divergence. For the typical convective boundary layer, the variations are fairly homogeneous (i.e. are about equal in the x, y ,and z directions) and thus we can say that:

$$\begin{array}{l}\frac{\partial w\text{'}}{\partial z}\approx 0\\ \text{andthus,wecanusethe product rule:}\end{array}$$

$$\begin{array}{l}\Theta \text{'}\frac{\partial w\text{'}}{\partial z}+w\text{'}\frac{\partial \Theta \text{'}}{\partial z}=\frac{\partial \left(w\text{'}\Theta \text{'}\right)}{\partial z}\end{array}$$

$$\frac{\partial \overline{\Theta}}{\partial \text{t}}=-\frac{\partial \overline{\left(w\text{'}\Theta \text{'}\right)}}{\partial z}$$

What does this mean? It means that the change in the boundary layer potential temperature in the daytime boundary layer is driven by the negative of the vertical gradient of eddy flux of thermal energy. During the day, the eddy heat flux is greatest at the surface and decreases with altitude. So $\frac{\partial \overline{\left(w\text{'}\Theta \text{'}\right)}}{\partial z}$ < 0, which means that the mean potential temperature increases with time ( $\frac{\partial \overline{\Theta}}{\partial \text{t}}$ > 0). At night, the opposite is generally true.

Consider the sensible heat flux, *F _{SH}*. As we saw in the average atmospheric energy budget, the sensible heat flux plays an important role.

$$H=\overline{{\rho}_{air}{c}_{p}w\text{'}\Theta \text{'}},\text{whereHiscalledthesensibleheatflux}$$

The average θ is often approximately constant over the height of the boundary layer. So, when we integrate both sides of Equation [11.11], we get the following:

$$\begin{array}{l}\frac{1}{{z}_{i}}{\displaystyle \underset{0}{\overset{{z}_{i}}{\int}}\frac{\partial \overline{\Theta}}{\partial \text{t}}}\partial z=-\frac{1}{{z}_{i}}{\displaystyle \underset{0}{\overset{{z}_{i}}{\int}}\frac{\partial \overline{\left(w\text{'}\Theta \text{'}\right)}}{\partial z}}\partial z\\ {\text{z}}_{\text{i}}\text{=boundarylayerheight;z=0isthesurface}\\ \text{\hspace{1em}}\text{\hspace{1em}}{\overline{\frac{\partial \overline{\Theta}}{\partial \text{t}}}}^{{z}_{i}}=\frac{1}{{z}_{i}}\left({\overline{\left(\text{w'}\Theta \text{'}\right)}}_{\text{0}}-{\overline{\left(w\text{'}\Theta \text{'}\right)}}_{{z}_{i}}\right)\\ \end{array}$$

### Example

We can put some numbers to these values to show how temperature changes from turbulent eddy fluxes alone. z_{i} is typically 1000 m.

$$\begin{array}{l}{\overline{\left(w\text{'}\Theta \text{'}\right)}}_{0}\approx 0.1-0.2{\text{Kms}}^{\text{-1}}\text{,and}{\overline{\left(w\text{'}\Theta \text{'}\right)}}_{{z}_{i}}\text{isdrivenby}{\overline{\left(w\text{'}\Theta \text{'}\right)}}_{0}\text{andisanegativebutsmallfractionof}{\overline{\left(w\text{'}\Theta \text{'}\right)}}_{0}.\\ {\overline{\frac{\partial \overline{\Theta}}{\partial \text{t}}}}^{{z}_{i}}\approx \frac{0.2Km{s}^{-1}}{1000m}=2x{10}^{-4}K{s}^{-1}\cdot 3600s\text{\hspace{0.05em}}h{r}^{-1}\approx 0.7\text{\hspace{0.05em}}K\text{\hspace{0.05em}}h{r}^{-1}\end{array}$$

So, at the top of the boundary layer, the potential temperature (and thus the temperature) is increasing during the day at a little less than 1 K hr^{-1}.

Thus, the heating at the top of the boundary layer is driven by the eddy heat flux from the surface. In this way, the heating at the surface due to the absorption of solar energy at Earth's surface is spread throughout the boundary layer.