# 11.11: Let’s see how vertical turbulent transport can be quantified.

$\frac{\partial \theta}{\partial t}=\frac{D \theta}{D t}-\vec{U} \bullet \vec{\nabla} \theta$

or only in the vertical

$\frac{\partial \theta}{\partial t}=\frac{D \theta}{D t}-w \frac{\partial \theta}{\partial z}$

look at the mean and perturbed parts:

$$\frac{\partial\left(\bar{\theta}+\theta^{\prime}\right)}{\partial t}=\frac{D\left(\bar{\theta}+\theta^{\prime}\right)}{D t}-\left(\bar{w}+w^{\prime}\right) \frac{\partial\left(\bar{\theta}+\theta^{\prime}\right)}{\partial z}$$

... take the Reynolds average

After taking the Reynolds average, we achieve the equation:

$\frac{\partial \bar{\theta}}{\partial t}=\frac{D \bar{\theta}}{D t}-\bar{w} \frac{\partial \bar{\theta}}{\partial z}-\overline{w^{\prime} \frac{\partial \theta^{\prime}}{\partial z}}$

The term on the left is the rate of change of the mean potential temperature at a given height, although it applies to any height. The first term on the right is the local heating from the divergence of the radiant energy and from phase changes. This term is generally small, except in clouds, so we can ignore it in the typical convective boundary layer. The second term on the right is the mean advection, but can typically be ignored in the fair-weather boundary layer. Scale analysis shows that the mean horizontal advection (ignored here for the moment) is usually quite large, and must be kept in the heat conservation equation.

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, turbulence is fairly homogeneous, meaning that the velocity perturbations do not vary much in space (i.e., are about equal in the x, y,and z directions). Thus, w' is independent of z, which means that w' can be taken inside of the derivative in the third term on the right of Equation [11.10]. With the assumptions described above for a fair-weather convective boundary layer, we now have:

$\frac{\partial \bar{\theta}}{\partial t}=-\frac{\partial(\overline{w^{\prime} \theta^{\prime}})}{\partial z}$

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 $$\frac{\partial(\overline{w^{\prime} \theta^{\prime}})}{\partial z}<0$$, which means that the mean potential temperature increases with time ($$\frac{\partial \bar{\theta}}{\partial t}>$$). At night, the opposite is generally true.

Consider the sensible heat flux, FSH (SI units of W m–2). As we saw in the average atmospheric energy budget, the sensible heat flux plays an important role.

$F_{S H}=\rho_{\operatorname{air}} c_{p} \overline{w^{\prime} \theta^{\prime}}$

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:

$\frac{1}{h} \int_{0}^{h} \frac{\partial \bar{\theta}}{\partial t} \partial z=-\frac{1}{h} \int_{0}^{h} \frac{\partial(\overline{w^{\prime} \theta^{\prime}})}{\partial z} \partial z$

h = boundary layer height; z=0 is the surface

$\frac{\partial \bar{\theta}}{\partial t}=\frac{1}{h}\left[(\overline{w^{\prime} \theta^{\prime}})_{0}-(\overline{w^{\prime} \theta^{\prime}})_{h}\right]$

$\cong \frac{1}{h}(\overline{w^{\prime} \theta^{\prime}})_{0}$

Note that in the last line of the above, we have assumed $$(\overline{w^{\prime} \theta^{\prime}})_{h}$$ is small because it turns out that it is driven by $$(\overline{w^{\prime} \theta^{\prime}})_{0}$$ and is a negative but small fraction of $$(\overline{w^{\prime} \theta^{\prime}})_{0}$$.

### Example

We can put some numbers to these values to show how temperature changes from turbulent eddy fluxes alone. Reasonable values for the boundary layer depth and daytime surface kinematic heat flux are 1000 m and 0.2 K m s–1, respectively. Thus,

$\frac{\partial \bar{\theta}}{\partial t} \approx \frac{0.2 \mathrm{K} \mathrm{m} \mathrm{s}^{-1}}{1000 \mathrm{m}} \times 3600 \mathrm{s} \mathrm{hr}^{-1}=0.7 \mathrm{K} \mathrm{hr}^{-1}$

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

Thus, the heating 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.