Published on *METEO 300: Fundamentals of Atmospheric Science* (https://www.e-education.psu.edu/meteo300)

Meteorologists are often so busy looking at weather maps for 850 mb and above that they don’t pay attention to what is happening near Earth’s surface. Yet we walk on Earth’s surface and we live in the air near Earth’s surface, which is called the **planetary boundary layer (PBL)**. The PBL has repeatable cycles from day-to-day, but it also has chaotic, turbulent behavior that cannot easily be described by mathematics and certainly cannot be predicted. These repeatable cycles and chaotic behavior have a huge impact on the quality of the air that we breathe and on the sources of sensible heat and water vapor that are essential for the weather above the PBL. In this lesson, we will show how the PBL rises during the day and drops down at night. This behavior is driven by turbulent motion, so we will see how all motion can be separated into the mean flow and the turbulent flow using some simple concepts, including Reynolds averaging. We will see that while the mean wind is most important for moving water vapor and momentum horizontally, turbulent transport is most important for moving these quantities and others vertically throughout the PBL to the base of the free atmosphere above it. We will see how turbulent eddies of different sizes are related and which ones have the most energy. Finally, we will take a look at the energy budget for different types of locations and different times of day. Hopefully you will understand that while weather forecast models treat the planetary boundary layer as a boundary condition, the atmosphere treats it as an integral part of weather.

By the end of this lesson, you should be able to:

- draw the PBL and its diurnal variation
- perform Reynolds averaging on an equation and derive an equation for the turbulent parts
- explain kinematic fluxes
- show vertical motion using eddy fluxes
- explain turbulent kinetic energy (TKE) and its behavior
- sketch the surface energy budget for different conditions

Please see Canvas for a list of required assignments, due dates, and submission instructions.

If you have any questions, please post them to the Course Questions discussion forum, located in Canvas. I will check that discussion forum daily to respond. While you are there, feel free to post your own responses if you, too, are able to help out a classmate.

The **atmospheric boundary layer (ABL)** is the tropospheric layer that is directly influenced by the presence of Earth’s surface and responds to surface forcings in an hour or less. It is also called the *planetary boundary layer* or just the *boundary layer*. The atmospheric boundary layer is typically 1 km deep during the day and ~100 m deep during the night. Above the boundary layer is the free atmosphere. We live in the atmospheric boundary layer.

Credit: NOAA Earth System Research Laboratory

The surface forms a boundary for the atmosphere and is actually responsible for the existence of the planetary boundary layer.

The surface influences the atmosphere in three main ways:

- sensible heat flux
- moisture flux from the surface to the atmosphere
- radiation

The water vapor that is in the air came from the evaporation of liquid water or sublimation of ice in or on the surface. Once that water vapor is in the atmosphere, it can condense, thus providing energy that heats the air and creates buoyant energy, as you learned in Lesson 3.

The picture below was taken on a sunny summer day over Maryland. What do you see? Starting at the bottom, you can barely make out the buildings and the roads because the air is so uniformly hazy. The haze extends all the way up to the lower parts of the fair weather cumulus clouds, which seem to be bobbing on top of the haze, like Styrofoam peanuts bobbing on top of a pool of water. Above the clouds, it’s blue sky with some thin cirrostratus clouds. This haze layer is the atmospheric boundary layer and the fair weather cumulus clouds mark its top.

Credit: W. Brune

Where did the haze come from? How did the clouds form? How did the haze get to be so uniform? Why are the fair weather cumulus clouds bobbing on top with blue sky above? You already know answers to the first two questions. The haze is small particles that came from the surface or were made in the atmosphere by gas-to-particle conversion and then swelled in the hot humid air. The clouds come from moist air containing aerosol that rises and cools until its supersaturation is sufficient to form cloud drops. The condensation releases energy, creating buoyancy, and the clouds rise until they reach their **level of neutral buoyancy (LNB)**. You will learn in this lesson why the haze is so uniform (turbulent mixing) and why the clouds bob on top (a stable layer with increasing potential temperature above the boundary layer).

There are three basic ways that air can be moved: the mean wind, waves, and turbulence. You have already learned about the mean wind and its causes, but mostly we have dealt with the wind in the free troposphere. In the boundary layer, transport of moisture, heat, and momentum are dominated by the mean wind in the horizontal and by turbulence in the vertical. Turbulence is a much more persistent part of boundary layer flow than it is of flow in the free atmosphere.

For any variable over time or space, we can split that variable into two values—the *mean* and the *perturbation*. We will see more about this later.

Turbulent transport consists of swirling motions called eddies. These eddies occur in a range of sizes, and they can have different intensities, meaning wind velocities. We can plot the turbulent intensity as a function of size of the eddies to get a turbulence spectrum. To get an idea about the behavior of eddies and clouds, watch the following short video of cloud formation.

Here is your **last chance** to earn **one point of extra credit via** **Picture of the Week**!

- You take a picture of some atmospheric phenomenon—a cloud, wind-blown dust, precipitation, haze, winds blowing different directions—anything that strikes you as interesting.
- Add a short description of the processes that you think are causing your observation. A Word file is a good format for submission.
- Use your name as the name of the file. Upload it to the
**Picture of the Week Dropbox**in this week's lesson module. To be eligible for the week, your picture must be submitted by 23:59 UT on Sunday of this week. - I will be the sole judge of the weekly winners. A student can win up to three times.
- This is your last chance to enter!

The boundary layer is not frozen in time but instead changes dramatically during the course of the day. Let’s start with the midday when the boundary looks like the hazy scene over Maryland (figure in 11.1). The boundary layer consists of a mixed layer that is stirred by solar heating of the surface and convection of warm moist air that pops up sporadically from place-to-place and time-to-time, and, as a result, mixes the air within the boundary layer. This convective stirring takes about ten to twenty minutes to go from bottom to top. As the air bubbles up, it mixes with the air surrounding it and with the air from the free troposphere at the top, thus creating an entrainment zone, which is where the clouds are.

Does ten to twenty minutes for boundary layer vertical stirring make sense?

You learned in Lesson 2 that buoyant acceleration equaled the gravity times the difference between the air parcel virtual temperature and the virtual temperature of its surroundings divided by the virtual temperature of its surroundings. Let's assume that temperature difference between an air parcel above a heated surface and its surroundings is 0.1 K, which seems pretty reasonable, and that the temperature is 300 K. The buoyant acceleration, *B*, is just 9.8 m s^{–2} times 0.1/300, or 0.0033 m s^{–2}. So, if the initial air parcel velocity is 0 m s^{–1} and the top of the PBL, *z*_{0}, = 1 km, then since *z*_{0 }= 1/2 *Bt*^{2}, where *t* is time, then *t* is the square root of 2*z*_{0}/*B* ~ 13 minutes. So now you can see that it takes a very small virtual temperature difference to stir the planetary boundary layer.

As the sun sets, the solar heating of the surface and the convection and associated turbulent eddies cease. Air from the surface no longer mixes with air throughout the convective boundary layer, and the air that was mixed during the day stays above the much lower nighttime stable boundary layer in a layer called the residual layer. Any gaseous or particle emissions from the surface are mixed within this nocturnal boundary layer. Because convection ceases at night, the winds in the residual layer are no longer affected by the friction caused by convection and they accelerate in the presence of a horizontal pressure gradient. So, the residual layer winds accelerate, blowing harder across the top of the more stagnant nocturnal boundary layer and a shear develops. This shearing is unstable and creates turbulence that mixes the boundary layer air and the residual layer air near the interface, so the nocturnal boundary layer grows a little during the night.

In the morning, the sun returns to heat the surface and to start driving convection and mixing again. This convection bubbles up, bumping into and entraining air from the residual layer. As the solar heating increases, the convection has more energy and can rise higher and entrain more air from the residual layer. Eventually, the air driven by convection reaches its maximum energy level and this maximum energy limits how high the boundary layer will grow into the stable free troposphere above it.

Credit: NikNaks (Own work, based on [1]) [CC BY-SA 3.0], via Wikimedia Commons

The following video explains the variation of the planetary boundary layer over the course of a typical day:

Let’s summarize the diurnal behavior of the boundary layer with a bulleted list of technical terms:

**Mixed Layer (Convective Boundary Layer):**

- turbulence driven by convection (large eddies or thermals)
- heat transfer from solar heating of the ground to the atmosphere
- mixed layer grows by entrainment of air from above it
- virtual temperature nearly adiabatic in middle; superadiabatic (i.e., potential temperature decreases with height) near surface; subadiabatic (i.e., potential temperature increases with height) at top, where exchange of air between the ABL and the free troposphere occurs
- wind speeds are sub-geostrophic in mixed layer, crossing isobars because of turbulent drag

**Surface Layer**

- directly in contact with Earth’s surface
- usually has vertical gradients in potential temperature, water vapor, and other quantities
- logarithmic wind speed profile with height, with low wind speed near ground
- typically is ~10% of the mixed layer

**Residual Layer**

- disconnected from boundary layer and Earth’s surface
- neutrally stratified, with small but near-equal turbulence in all directions
- contains moisture and trace atmospheric constituents from the day before

**Stable Boundary Layer**

- statically stable with weaker turbulence that occurs sporadically
- winds aloft may increase to supergeostrophic speeds (low-level jet or nocturnal jet)
- stability tends to suppress turbulence, except for occasional shear-generated turbulence caused by the low-level jet

Recall the concept of virtual potential temperature, which was introduced in Lesson 2. The virtual potential temperature is found by replacing the temperature in the formula for virtual temperature with the potential temperature:

${T}_{v}=T\left(1+0.61q\right)$[1]@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWaqaaaaaaaaaWdbiaa=rfapaWaaSbaaSqaa8qacaWF2baapaqabaGcpeGaeyypa0Jaa8hvamaabmaapaqaa8qacaaIXaGaey4kaSIaaGimaiaac6cacaaI2aGaaGymaiaa=fhaaiaawIcacaGLPaaaaaa@3E25@

$\theta =T{\left(\frac{{p}_{o}}{p}\right)}^{0.286}$[1]@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbiabeI7aXjabg2da9Gqadiaa=rfadaqadaWdaeaapeWaaSaaa8aabaWdbiaa=bhapaWaaSbaaSqaa8qacaWFVbaapaqabaaakeaapeGaa8hCaaaaaiaawIcacaGLPaaapaWaaWbaaSqabeaapeGaaGimaiaac6cacaaIYaGaaGioaiaaiAdaaaaaaa@3F90@

${\theta}_{v}=\theta \left(1+0.61q\right),forunsaturatedair$[1]@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH4oqCdaWgaaWcbaacbmaeaaaaaaaaa8qacaWF2baapaqabaGcpeGaeyypa0JaeqiUde3aaeWaa8aabaWdbiaaigdacqGHRaWkcaaIWaGaaiOlaiaaiAdacaaIXaGaa8xCaaGaayjkaiaawMcaaiaacYcacaWFGcGaa8Nzaiaa=9gacaWFYbGaa8hOaiaa=vhacaWFUbGaa83Caiaa=fgacaWF0bGaa8xDaiaa=jhacaWFHbGaa8hDaiaa=vgacaWFKbGaa8hOaiaa=fgacaWFPbGaa8NCaaaa@539F@

*θ _{v}* is the virtual potential temperature. It is a useful quantity because it takes the moisture into account as well as temperature when considering buoyancy and stability. Thus, adiabatic ascent or descent in moist air follows the constant virtual potential temperature line all the way up to the lifting condensation level (LCL), where the potential temperature increases. So let’s look at the evolution of the profile of virtual potential temperature in a cloud-free boundary layer.

Start with the late afternoon (S1 above). Surface heating causes the air near the surface to have a higher virtual potential temperature than the air just above it, so that the air is superadiabatic. Thus, air parcels of this warm moist air rise all the way up to the point in the free atmosphere where the virtual potential temperature is as great or greater than the air parcel’s value. At this point the air parcel likely mixes with the surrounding air and therefore contributes to raising the mixed layer’s temperature. As convection of air parcels continues, the slightly cooler air from the upper boundary layer sinks around the rising air parcels and continues the mixing process as the air parcels going up encounter air coming back down. The result is a well-mixed boundary layer.

Just after sunset (S2 above), the surface cools by infrared radiation [2]and the air near the surface becomes stable with a virtual potential temperature (and thus temperature) inversion that prevents air from the surface to rise. As a result, the nocturnal boundary layer becomes quite stable, and as the surface continues to cool, the boundary layer becomes even more stable during the night (S3 above). Note also that the stable boundary layer grows, not by convective mixing, but by shear mixing.

Just after sunrise (S4 above), the surface is warmed by the sun, and as it continues to get warmed throughout the morning (S5 above), convection begins to mix warm air up first throughout the stable boundary layer and then into the mixed layer. Any trace atmospheric constituents left in the residual layer from the day before are now mixed back into the boundary layer as it grows higher. Finally, the residual layer is entirely mixed into the growing boundary layer (S6 above), and the boundary layer returns to its condition of the previous afternoon (S1 above).

Credit: Nina Hale [3] via flickr

Think about morning and evening rush hours. During morning rush hour, which is near sunrise, the vehicle emissions mix into a shallow boundary layer and hence the mixing ratios of the pollutants can be quite substantial. This situation leads to the photochemistry that makes pollution, including ozone. In the evening, the rush hour traffic also emits the same amounts of pollutants into the boundary layer, but because the boundary layer height is so much greater in the early evening than in the morning, the pollutant mixing ratios are less because the same level of emissions are being mixed into a larger volume of air. Thus the effects of evening emissions are not as severe as the effects of morning emissions. In addition, modeling the planetary boundary layer height correctly is essential for accurate air quality modeling.

The diurnal variation of the planetary boundary layer height is more than just a curiosity--it has influence on our daily lives and health because we live, work, and breathe mostly in the atmospheric planetary boundary layer. Thus it is important that meteorologists and atmospheric scientists gain a better understanding of the atmospheric motions and energy budget of the planetary boundary layer. Gaining this understanding means learning something about atmospheric turbulence, which is the small-scale chaotic winds that are a significant factor in the planetary boundary layer.

We would like to be able to take snapshots of the eddies in three dimensions and measure all their sizes each instant. Unfortunately, we do not have a good way to do this. Instead, we can simply measure the fluctuations of a variable such as wind speed, specific humidity, or temperature with a sensor at one location for a period of time. In this way, we watch the eddies drift by the sensor. But the eddies could be changing size and shape as they drift by the sensor. Let’s put this physical concept in the context of the total derivative.

Take a variable like temperature, *T*. We know that the change in *T* with time at any location (such as where a sensor might be placed) is the sum of the total derivative and the temperature advection:

$\frac{\partial T}{\partial t}=\frac{DT}{Dt}-\overrightarrow{U}\xb7\overrightarrow{\nabla}T$[1]@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacqGHciITieWacaWFubaapaqaa8qacqGHciITcaWF0baaaiabg2da9maalaaapaqaa8qacaWFebGaa8hvaaWdaeaapeGaa8hraiaa=rhaaaGaeyOeI0Iab8NDa8aagaWca8qacqWIpM+zcuGHhis0paGbaSaapeGaa8hvaaaa@4389@

The temperature advection is the change in temperature at the sensor due to the advection of warmer or colder air past the sensor. The total derivative is the change in temperature of an air parcel moving past the sensor. Such a temperature change may be caused by any number of processes, such as the absorption or emission of radiation, condensation or evaporation (latent heating or cooling), or compression and expansion. **Taylor’s hypothesis** says that we can assume that the turbulent eddies (which we can think of as large air parcels) are frozen as they advect past the sensor and thus the change in temperature within each eddy is negligible:

$\frac{DT}{Dt}~0$[1]@5@5@+=faaahmart1ev3aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacaWGebGaamivaaWdaeaapeGaamiraiaadshaaaGaaiOFaiaaicdaaaa@384C@

so that:

$\frac{\partial T}{\partial t}=-\overrightarrow{U}\xb7\overrightarrow{\nabla}T$[1]@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacqGHciITieWacaWFubaapaqaa8qacqGHciITcaWF0baaaiabg2da9iabgkHiTiqa=zhapaGbaSaapeGaeS4JPFMafy4bIe9dayaalaWdbiaa=rfaaaa@3FE7@

Local temperature gradients, which might be present from one side of an eddy to another, are advected across the sensor by the mean wind without the eddy changing.

When is this condition valid? Experiments suggest that this hypothesis is valid when the variation of the wind speed due to turbulence is less than ½ of the mean wind speed.

We start this study with methods to separate wind motion driven by larger scale processes, such as gradient flow or geostrophic flow, from turbulence.

Credit: Department of Energy Atmospheric Radiation Measurement Carbon Program

- Please note:
**there is no practice quiz**for Quiz 11-1. - When you feel you are ready, take
**Quiz 11-1**in Canvas. You will be allowed to take this quiz only**once**. Good luck!

For any variable, the observed value can be written as a sum of the mean value and a turbulent value:

$$u=\overline{u}+{u}^{\prime}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaacckapaGaamyDaiabg2da9iqadwhagaqeaiabgUcaRiqadwhagaqbaaaa@3B0B@

where $\overline{u}$[1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaanaaabaGaamyDaaaaaaa@33D5@ is the mean, or average value, and ${u}^{\prime}$[1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadwhagaqbaaaa@33D0@ is the turbulent part.

The average can be obtained by averaging *u* over time or over space or even by doing a number of samples and averaging over the samples.

$$\text{temporalaveraging:}\overline{u}=\frac{{\displaystyle \int u(t)dt}}{{\displaystyle \int dt}}=\frac{{\displaystyle \sum _{i=0}^{N-1}u({t}_{i})}}{N};\text{\hspace{1em}}\text{spatialaveraging}\text{\hspace{1em}}\overline{u}=\frac{{\displaystyle \int u(x)dx}}{{\displaystyle \int dx}}=\frac{{\displaystyle \sum _{i=0}^{N-1}u({x}_{i})}}{N}$$[1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@8EBB@

If the turbulence does not change with time and is homogeneous (i.e., the same in all directions and for all time), then these averages equal each other.

In Lesson 10, we developed the equation of motion without really considering the short-term and small-scale variations, except to say that they led to turbulent drag, which acts to resist the mean flow in the upper boundary layer. Now we want to think about how to correctly capture the dynamic effects of turbulent motion. What we want to do is to write down the equations of motion that you learned in Lesson 10; substitute mean and turbulent parts for the variables such as *u*, *v*, and *w*; average over all the terms; and then see if we can sort out the terms to create an equation for the mean wind and an equation for the turbulent wind. This type of averaging is called **Reynolds averaging**.

But first we need to learn the rules for averaging.

$$\begin{array}{l}c\text{isconstant;}u\text{and}v\text{arevariables}\hfill \\ \overline{{u}^{\prime}}=0\hfill \\ \overline{c\text{\hspace{0.17em}}u}=c\text{\hspace{0.17em}}\overline{u}\hfill \\ \overline{u+v}=\overline{u}+\overline{v}\hfill \\ \overline{\left(\overline{u}\text{\hspace{0.17em}}v\right)}=\overline{u}\text{\hspace{0.17em}}\overline{v}\hfill \\ \overline{\left(\frac{\partial u}{\partial t}\right)}=\frac{\partial \overline{u}}{\partial t}\hfill \end{array}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@807D@

Now let’s apply these rules to a variable with a mean and a turbulent part. For example, consider the product *uv*, which is just the advection of horizontal wind in one direction by the horizontal wind in the other direction.

So, using the rules:

$$\begin{array}{l}\overline{u\text{\hspace{0.17em}}v}=\overline{(\overline{u}+u\text{'})(\overline{v}+v\text{'})}\hfill \\ \text{}=\overline{\overline{u}\text{}\overline{v}}\text{\hspace{0.17em}}+\overline{\overline{u}v\text{'}}+\overline{u\text{'}\overline{v}}+\overline{u\text{'}v\text{'}}\hfill \\ \text{and}\hfill \\ \overline{\overline{u}\overline{v}}=\overline{u}\text{}\overline{v}\hfill \\ \overline{\overline{u}\text{\hspace{0.17em}}v\text{'}}=\overline{u}\text{\hspace{0.17em}}\overline{v\text{'}}=\overline{u}\cdot 0=0\hfill \\ \overline{u\text{'}\overline{v}}=\overline{u\text{'}}\overline{\overline{v}}=0\cdot \overline{v}=0\hfill \\ \text{so}\hfill \\ \overline{u\text{\hspace{0.17em}}v}=\overline{u}\text{}\overline{v}\text{\hspace{0.17em}}+\overline{u\text{'}v\text{'}}\hfill \end{array}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=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aadaqdaaqaa8qacaWG2bGaai4jaaaacqGH9aqpceWG1bWdayaaraWdbiabgwSixlaaicdacqGH9aqpcaaIWaaapaqaamaanaaabaWdbiaadwhacaGGNaGabmODa8aagaqeaaaapeGaeyypa0ZdamaanaaabaWdbiaadwhacaGGNaaaa8aadaqdaaqaa8qaceWG2bWdayaaraaaa8qacqGH9aqpcaaIWaGaeyyXICTabmODa8aagaqea8qacqGH9aqpcaaIWaaapaqaa8qacaqGZbGaae4BaaWdaeaadaqdaaqaa8qacaWG1bGaaGPaVlaadAhaaaGaeyypa0JabmyDa8aagaqea8qacaqGGaGabmODa8aagaqea8qacaaMc8Uaey4kaSYdamaanaaabaWdbiaadwhacaGGNaGaamODaiaacEcaaaaaaaaa@9658@

This second term, a product of two turbulent terms is *not necessarily zero*! Whereas the average of one turbulent term is zero by definition, the average of two turbulent terms is not necessarily zero.

The following video (3.11) further describes Reynolds averaging:

Consider two random numbers varying between –0.5 and 0.5, called *u* and *v*. The figure below shows *u*, *v*, the mean of *uv* and the mean of *u’v’*. Of course, the mean of *u’v’* might be zero, but is it not necessarily zero, as shown in this figure.

Credit: W. Brune

The same thinking applies to *u*^{2}.

$$\overline{u{\text{\hspace{0.17em}}}^{2}}=\overline{(\overline{u}+u\text{'})(\overline{u}+u\text{'})}=...=\overline{u}{\text{\hspace{0.05em}}}^{2}\text{\hspace{0.17em}}+\overline{u{\text{'}}^{2}}$$[1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaanaaabaGaamyDaiaaykW7daahaaWcbeqaaiaaikdaaaaaaOGaeyypa0Zaa0aaaeaacaGGOaGabmyDayaaraGaey4kaSIaamyDaiaacEcacaGGPaGaaiikaiqadwhagaqeaiabgUcaRiaadwhacaGGNaGaaiykaaaacqGH9aqpcaGGUaGaaiOlaiaac6cacqGH9aqpceWG1bGbaebacaaMi8+aaWbaaSqabeaacaaIYaaaaOGaaGPaVlabgUcaRmaanaaabaGaamyDaiaacEcadaahaaWcbeqaaiaaikdaaaaaaaaa@4E12@

Remember your statistics and the concept of variance:

$${\sigma}_{u}{}^{2}=\frac{1}{N-1}{{\displaystyle \sum _{i=0}^{N-1}\left({u}_{i}-\overline{u}\right)}}^{2}\approx \frac{1}{N}{{\displaystyle \sum _{i=0}^{N-1}\left({u}_{i}-\overline{u}\right)}}^{2}=\frac{1}{N}{{\displaystyle \sum _{i=0}^{N-1}\left({u}_{i}\text{'}\right)}}^{2}=\overline{u{\text{'}}^{2}}$$[1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@6D6A@

So, the variance is the same as the mean value for the square of the turbulent part of the variable.

The covariance of *u* and *v* is given by the equation:

$$\text{covariance}(u,v)=\overline{u\text{'}v\text{'}}$$[1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabogacaqGVbGaaeODaiaabggacaqGYbGaaeyAaiaabggacaqGUbGaae4yaiaabwgacaGGOaGaamyDaiaacYcacaWG2bGaaiykaiabg2da9maanaaabaGaamyDaiaacEcacaWG2bGaai4jaaaaaaa@4463@

The standard deviation is just:

$${\sigma}_{u}={\left(\overline{u{\text{'}}^{2}}\right)}^{1/2}$$[1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBaaaleaacaWG1baabeaakiabg2da9maabmaabaWaa0aaaeaacaWG1bGaai4jamaaCaaaleqabaGaaGOmaaaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaigdacaGGVaGaaGOmaaaaaaa@3D4C@

We can get a better sense of how large this variance is by dividing by the mean value:

$$\text{I=}\frac{{\sigma}_{u}}{\overline{u}}$$[1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabMeacaqGGaGaaeypaiaabccadaWcaaqaaiabeo8aZnaaBaaaleaacaWG1baabeaaaOqaaiqadwhagaqeaaaaaaa@39B1@

Now that we know how to separate out the mean and turbulent parts, we can look at the concept of **kinematic flux**.

A flux is the transfer of some variable per unit area per unit time. Generally in meteorology, we care about variables like mass, heat (i.e., temperature), kinetic energy, moisture, momentum. Those who study the atmosphere's composition are also interested in the flux of chemicals emitted into the atmosphere from the surface and the flux of atmospheric pollutants, such as ozone, back to Earth's surface. For this discussion, let's consider only the fluxes directly relevant to meteorology.

So the SI units of the following fluxes would be ${\text{kgm}}^{-2}{\text{s}}^{-1}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaabUgacaqGNbGaaeiOaiaab2gapaWaaWbaaSqabeaapeGaeyOeI0IaaGOmaaaakiaabckacaqGZbWdamaaCaaaleqabaWdbiabgkHiTiaaigdaaaaaaa@3ED5@ for mass (*m*); ${\text{kg}}_{\text{water}}{\text{m}}^{-2}{\text{s}}^{-1}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaabUgacaqGNbWdamaaBaaaleaapeGaae4DaiaabggacaqG0bGaaeyzaiaabkhaa8aabeaak8qacaqGGcGaaeyBa8aadaahaaWcbeqaa8qacqGHsislcaaIYaaaaOGaaeiOaiaabohapaWaaWbaaSqabeaapeGaeyOeI0IaaGymaaaaaaa@43FB@ for moisture (*m _{v}*); $({\text{kgms}}^{-1}){\text{m}}^{-2}{\text{s}}^{-1}={\text{kgm}}^{-1}{\text{s}}^{-2}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaacIcacaqGRbGaae4zaiaabckacaqGTbGaaeiOaiaabohapaWaaWbaaSqabeaapeGaeyOeI0IaaGymaaaakiaacMcacaqGGcGaaeyBa8aadaahaaWcbeqaa8qacqGHsislcaaIYaaaaOGaaeiOaiaabohapaWaaWbaaSqabeaapeGaeyOeI0IaaGymaaaakiaacckacqGH9aqpcaqGGcGaae4AaiaabEgacaqGGcGaaeyBa8aadaahaaWcbeqaa8qacqGHsislcaaIXaaaaOGaaeiOaiaabohapaWaaWbaaSqabeaapeGaeyOeI0IaaGOmaaaaaaa@53A6@ for momentum (

We can write these in the kinematic form by dividing by the density of air:

$\text{kinematicmassflux}=({\text{kgm}}^{-2}{\text{s}}^{-1})/({\text{kgm}}^{-3})={\text{ms}}^{-1}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=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@6159@

$\text{kinematicmoistureflux=}({\text{kg}}_{\text{water}}{\text{m}}^{-2}{\text{s}}^{-1})/({\text{kg}}_{\text{air}}{\text{m}}^{-3}){\text{=kg}}_{\text{water}}{\text{kg}}_{\text{air}}{}^{-1}{\text{ms}}^{-1}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@8061@

${\text{kinematicmomentumflux=kgm}}^{-1}{\text{s}}^{-2}/(\text{kg}{\text{m}}^{-3}){\text{=m}}^{2}{\text{s}}^{-2}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@663E@

For **kinematic heat flux**, the heat flux is usually divided by the air density times the specific heat of air to give units of $({\text{Jm}}^{-2}{\text{s}}^{-1})/({\text{kgm}}^{-3}{\text{Jkg}}^{-1}{\text{K}}^{-1})={\text{Kms}}^{-1}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaacIcacaqGkbGaaeiOaiaab2gapaWaaWbaaSqabeaapeGaeyOeI0IaaGOmaaaakiaabckacaqGZbWdamaaCaaaleqabaWdbiabgkHiTiaaigdaaaGccaGGPaGaai4laiaacIcacaqGRbGaae4zaiaabckacaqGTbWdamaaCaaaleqabaWdbiabgkHiTiaaiodaaaGccaqGGcGaaeOsaiaabckacaqGRbGaae4za8aadaahaaWcbeqaa8qacqGHsislcaaIXaaaaOGaaeiOaiaabUeapaWaaWbaaSqabeaapeGaeyOeI0IaaGymaaaakiaacMcacqGH9aqpcaqGlbGaaeiOaiaab2gacaqGGcGaae4Ca8aadaahaaWcbeqaa8qacqGHsislcaaIXaaaaaaa@59F0@. Note that this is just a temperature times a velocity.

*The word “kinematic” is often dropped and assumed.*

Let’s look at the kinematic heat flux in the vertical direction. Also, we are interested in the vertical transport of thermal energy and so we do not want to think about the adiabatic variation in temperature due to vertical motion, since an air parcel's temperature decreases with height even in adiabatic ascent. We can accomplish this by using the potential temperature and not the virtual temperature, which falls off with height adiabatically while the potential temperature is constant with height. *θ* does not equal *T* except at 1000 hPa, so we need to keep that in mind.

Write down the expression for the vertical flux of potential temperature, which is just equal to the vertical wind times the potential temperature, and then take the Reynolds average of that flux:

$$\overline{w\theta}=\overline{w}\overline{\theta}+\overline{{w}^{\prime}{\theta}^{\prime}}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaabbaaaaaaaaIXwyJTgapeGaam4DaiabeI7aXbaacqGH9aqpceWG3bWdayaaraWdbiqbeI7aX9aagaqea8qacqGHRaWkpaWaa0aaaeaapeGabm4DayaafaGafqiUdeNbauaaaaaaaa@3E64@

The first term on the right is the vertical flux of potential temperature due to the average vertical motion, but *w* ~ 0 on average in the boundary layer, so we can usually drop the first term on the right.

Let’s consider how this flux works at moving thermal energy. A typical condition for the mixed boundary layer is a superadiabatic layer near the heated surface (i.e., *θ *decreases with height).

Credit: W. Brune

Let’s look at the different cases.

Daytime case with surface heating such that $\partial \theta /\partial z\text{}\text{}0$ (as in figure above). Eddies can make air trade places. They move some air parcels down,

for $w\text{'}<0\text{and}\theta \text{'}0:\text{}\overline{w\text{'}\theta \text{'}}0$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaadEhacaGGNaGaeyipaWJaaGimaiaabckacaqGHbGaaeOBaiaabsgacaqGGcGaeqiUdeNaai4jaiabgYda8iaaicdacaGG6aGaaeiiaiaabccacaqGGaGaaeiia8aadaqdaaqaa8qacaWG3bGaai4jaiabeI7aXjaacEcaaaWdaiabg6da+iaaicdaaaa@4AB7@

and at the same time move some up,

for $w\text{'}>0\text{and}\theta \text{'}0:\text{}\overline{w\text{'}\theta \text{'}}0$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaadEhacaGGNaGaeyOpa4JaaGimaiaabckacaqGHbGaaeOBaiaabsgacaqGGcGaeqiUdeNaai4jaiabg6da+iaaicdacaGG6aGaaeiiaiaabccacaqGGaGaaeiia8aadaqdaaqaa8qacaWG3bGaai4jaiabeI7aXjaacEcaaaWdaiabg6da+iaaicdaaaa@4ABF@

Both of these processes are happening simultaneously. If the air parcels mix at their new heights, they move cooler air lower (left panel above) or warmer air higher (right panel above) and in both cases, they move heat up. Thus, when the potential temperature decreases with height (i.e., is superadiabatic), the turbulent eddies move warmer air up and the heat flux is positive.

Credit: W. Brune

Nighttime case with surface cooling such that $\partial \theta /\partial z\text{}\text{}0$ (see figure above). Eddies make air parcels trade places vertically. They move some air parcels down,

for $w\text{'}<0\text{and}\theta \text{'}0:\text{}\overline{w\text{'}\theta \text{'}}0$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaadEhacaGGNaGaeyipaWJaaGimaiaabckacaqGHbGaaeOBaiaabsgacaqGGcGaeqiUdeNaai4jaiabg6da+iaaicdacaGG6aGaaeiiaiaabccacaqGGaGaaeiia8aadaqdaaqaa8qacaWG3bGaai4jaiabeI7aXjaacEcaaaWdaiabgYda8iaaicdaaaa@4AB7@

and some up,

for $w\text{'}>0\text{and}\theta \text{'}0:\text{}\overline{w\text{'}\theta \text{'}}0$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaadEhacaGGNaGaeyOpa4JaaGimaiaabckacaqGHbGaaeOBaiaabsgacaqGGcGaeqiUdeNaai4jaiabgYda8iaaicdacaGG6aGaaeiiaiaabccacaqGGaGaaeiia8aadaqdaaqaa8qacaWG3bGaai4jaiabeI7aXjaacEcaaaWdaiabgYda8iaaicdaaaa@4AB7@

Both of these processes are happening simultaneously. If the air parcels mix at their new heights, they either move warmer air lower (left panel above) or cooler air higher (right panel above), and in both cases, they move heat down. Thus, when the potential temperature increases with height (i.e., is subadiabatic), the turbulent eddies move warmer air down and the heat flux is negative.

We see that by turbulent motions, the thermal energy tends to flow from higher mean temperature to lower mean temperature, or to put it another way, down the gradient of the mean value.

So, the heat flux $\overline{w\text{'}\theta \text{'}}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaabbaaaaaaaaIXwyJTgapeGaam4DaiaacEcacqaH4oqCcaGGNaaaaaaa@38F7@ tells us about the transport of thermal energy by eddies. Note that the transport is in the direction from the higher values to the lower values for the mean potential temperature.** **The heat flux is the way that sensible heat gets transferred to the free troposphere in the atmosphere's energy budget [2]. The heat flux combined with the moisture flux create conditions that can lead to thunderstorms and severe weather.

Watch the following video (1:49) for further discussion of the eddy heat flux:

You have already encountered the idea of molecular fluxes in cloud physics, in the section on vapor deposition [4]. Water vapor molecules are moving randomly all the time, but when water vapor molecules hit a liquid or ice surface and stick, the water vapor concentration near the drop is less than it is farther away. As a result, molecular diffusion tends to move water vapor from regions of higher water vapor concentration far from the drop towards regions of lower water vapor concentration near the drop. The greater the difference between the concentration far away from the drop and the concentration near the drop, the greater will be the flux of water molecules to the drop.

$${F}_{molecules}=-{D}_{v}\frac{\partial n}{\partial r}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaadAeapaWaaSbaaSqaa8qacaWGTbGaam4BaiaadYgacaWGLbGaam4yaiaadwhacaWGSbGaamyzaiaadohaa8aabeaak8qacqGH9aqpcqGHsislcaWGebWdamaaBaaaleaapeGaamODaaWdaeqaaOWdbmaalaaapaqaa8qacqGHciITcaWGUbaapaqaa8qacqGHciITcaWGYbaaaaaa@4683@

where ${F}_{molecules}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaadAeapaWaaSbaaSqaa8qacaWGTbGaam4BaiaadYgacaWGLbGaam4yaiaadwhacaWGSbGaamyzaiaadohaa8aabeaaaaa@3D3A@ is the molecular flux (SI units of molecules m^{–2} s^{–1}), $\partial n/\partial r$[1]@5@5@+=faaahmart1ev3aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiabgkGi2kaad6gacaGGVaGaeyOaIyRaamOCaaaa@3909@ is the change in concentration $n$ (SI units of molecules m^{–3}) as a function of radial distance $r$ from the drop (SI units of m), and ${D}_{v}$ is the molecular diffusion coefficient (SI units of m^{2} s^{–1}). When $n$ increases with $r$, then the flux is negative, which means that the flux is toward the drop, in the negative $r$ direction.

Molecular diffusion, by the way, is very slow at transferring molecules from one place to another in the troposphere. By solving the equations of motion for a simple case, we find that the characteristic time to travel a distance *L* by molecular diffusion is:

$$\tau =\frac{{L}^{2}}{{D}_{v}}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiabes8a0jabg2da9maalaaapaqaa8qacaWGmbWdamaaCaaaleqabaWdbiaaikdaaaaak8aabaWdbiaadseapaWaaSbaaSqaa8qacaWG2baapaqabaaaaaaa@3AB9@

By molecular diffusion, how long would it take water vapor molecules to move from Earth's surface to the top of the planetary boundary layer, 1 km away? A typical value for *D*_{v} is 2 x 10^{–5} m^{2} s^{–1}.

Put 1000 m for the *L* in Equation [11.8] along with the value for *D _{v}*. The resulting characteristic time is about 1600 years. So, it is clear that molecular diffusion cannot be responsible for water vapor (and, as it turns out, anything else, such as ozone, heat, and momentum) over distances on the scale of even the planetary boundary layer.

Molecular diffusion cannot transport anything fast enough for the atmosphere except on small scales of a centimeter or less. However, on the spatial and temporal scales of the planetary boundary layer, eddies are quite effective at moving heat, molecules, and momentum. In the last section, we saw that turbulence tends to move heat from heights where the air is warmer to heights where the air is cooler. Eddy "diffusion" shares this characteristic with molecular diffusion.

We can write the heat flux in the same way that we write the molecular flux:

$${F}_{heat}=\overline{{w}^{\prime}{\theta}^{\prime}}=-K\frac{\partial \overline{\theta}}{\partial z}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaadAeapaWaaSbaaSqaa8qacaWGObGaamyzaiaadggacaWG0baapaqabaGcpeGaeyypa0ZdamaanaaabaWdbiqadEhapaGbauaapeGafqiUde3dayaafaaaa8qacqGH9aqpcqGHsislcaWGlbWaaSaaa8aabaWdbiabgkGi2kqbeI7aX9aagaqeaaqaa8qacqGHciITcaWG6baaaaaa@457C@

where *K* is the eddy diffusion coefficient. Since *K* is always positive, this equation makes it clear that the flux of any quantity goes from where there is more of that quantity to where there is less of that quantity.

Let’s go all the way back to the idea of advection [5] in Lesson 8.7. We will replace the scalar *R* with potential temperature:

$$\begin{array}{l}\frac{\partial \theta}{\partial t}=\frac{D\theta}{Dt}-\overrightarrow{U}\u2022\overrightarrow{\nabla}\theta \hfill \\ \text{oronlyinthevertical}\hfill \\ \frac{\partial \theta}{\partial t}=\frac{D\theta}{Dt}-w\frac{\partial \theta}{\partial z}\hfill \\ \text{lookatthemeanandperturbedparts:}\hfill \\ \frac{\partial \left(\overline{\theta}+{\theta}^{\prime}\right)}{\partial t}=\frac{D\left(\overline{\theta}+{\theta}^{\prime}\right)}{Dt}-\left(\overline{w}+{w}^{\prime}\right)\frac{\partial \left(\overline{\theta}+{\theta}^{\prime}\right)}{\partial z}\hfill \\ ...\text{taketheReynoldsaverage.}\hfill \end{array}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@D411@

After taking the Reynolds average, we achieve the equation:

$$\frac{\partial \overline{\theta}}{\partial t}=\frac{D\overline{\theta}}{Dt}-\overline{w}\frac{\partial \overline{\theta}}{\partial z}-\overline{{w}^{\prime}\frac{\partial {\theta}^{\prime}}{\partial z}}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbmaalaaapaqaa8qacqGHciITpaGafqiUdeNbaebaaeaapeGaeyOaIyRaamiDaaaacqGH9aqpdaWcaaWdaeaapeGaamira8aacuaH4oqCgaqeaaqaa8qacaWGebGaamiDaaaacqGHsislceWG3bWdayaaraWdbmaalaaapaqaa8qacqGHciITpaGafqiUdeNbaebaaeaapeGaeyOaIyRaamOEaaaacqGHsislpaWaa0aaaeaapeGabm4DayaafaWaaSaaa8aabaWdbiabgkGi2kqbeI7aXzaafaaapaqaa8qacqGHciITcaWG6baaaaaaaaa@4F34@

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 [6]. 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:

$$\frac{\partial \overline{\theta}}{\partial t}=-\frac{\partial \left(\overline{{w}^{\prime}{\theta}^{\prime}}\right)}{\partial z}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbmaalaaapaqaa8qacqGHciITcuaH4oqCpaGbaebaaeaapeGaeyOaIyRaamiDaaaacqGH9aqpcqGHsisldaWcaaWdaeaapeGaeyOaIy7aaeWaa8aabaWaa0aaaeaapeGabm4Da8aagaqba8qacuaH4oqCpaGbauaaaaaapeGaayjkaiaawMcaaaWdaeaapeGaeyOaIyRaamOEaaaaaaa@444D@

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 $\frac{\partial \left(\overline{{w}^{\prime}{\theta}^{\prime}}\right)}{\partial z}<0$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbmaalaaapaqaa8qacqGHciITdaqadaWdaeaadaqdaaqaa8qaceWG3bWdayaafaWdbiqbeI7aX9aagaqbaaaaa8qacaGLOaGaayzkaaaapaqaa8qacqGHciITcaWG6baaaiabgYda8iaaicdaaaa@3E37@ , which means that the mean potential temperature increases with time ($\frac{\partial \overline{\theta}}{\partial t}>0$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbmaalaaapaqaa8qacqGHciITpaGafqiUdeNbaebaaeaapeGaeyOaIyRaamiDaaaacqGH+aGpcaaIWaaaaa@3B42@). At night, the opposite is generally true.

Consider the sensible heat flux, *F _{SH}* (SI units of W m

$${F}_{SH}={\rho}_{air}{c}_{p}\overline{{w}^{\prime}{\theta}^{\prime}}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaadAeapaWaaSbaaSqaa8qacaWGtbGaamisaaWdaeqaaOWdbiabg2da9iabeg8aY9aadaWgaaWcbaWdbiaadggacaWGPbGaamOCaaWdaeqaaOWdbiaadogapaWaaSbaaSqaa8qacaWGWbaapaqabaGcdaqdaaqaa8qaceWG3bWdayaafaWdbiqbeI7aX9aagaqbaaaaaaa@41E2@

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:

$$\begin{array}{l}\frac{1}{h}\underset{0}{\overset{h}{{{{\displaystyle \int}}^{\text{}}}^{\text{}}{{}^{\text{}}}^{\text{}}}}\frac{\partial \overline{\theta}}{\partial t}\partial z=-\frac{1}{h}\underset{0}{\overset{h}{{{{\displaystyle \int}}^{\text{}}}^{\text{}}{{}^{\text{}}}^{\text{}}}}\frac{\partial \left(\overline{{w}^{\prime}{\theta}^{\prime}}\right)}{\partial z}\partial z\hfill \\ h\text{=boundarylayerheight;}z=0\text{isthesurface}\hfill \\ \text{\hspace{1em}}\text{\hspace{1em}}\frac{\partial \overline{\theta}}{\partial t}=\frac{1}{h}\left[{\left(\overline{{w}^{\prime}{\theta}^{\prime}}\right)}_{\text{0}}-{\left(\overline{{w}^{\prime}{\theta}^{\prime}}\right)}_{h}\right]\hfill \\ \text{}\cong \frac{1}{h}{\left(\overline{{w}^{\prime}{\theta}^{\prime}}\right)}_{\text{0}}\hfill \end{array}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaafaWaaeabbaaaaeaaqqaaaaaaaaGySf2yRbWdbmaalaaapaqaa8qacaaIXaaapaqaa8qacaWGObaaa8aadaWfWaqaamaavacabeWcbeqaaiaaygW7a0qaa8qacqGHRiI8aaGcpaWaaWbaaSqabeaapeGaaGzaVdaak8aadaahaaWcbeqaa8qacaaMb8oaaOWdamaaCaaaleqabaWdbiaaygW7aaaapaqaa8qacaaIWaaapaqaa8qacaWGObaaaOWaaSaaa8aabaWdbiabgkGi2kqbeI7aX9aagaqeaaqaa8qacqGHciITcaWG0baaaiabgkGi2kaadQhacqGH9aqpcqGHsisldaWcaaWdaeaapeGaaGymaaWdaeaapeGaamiAaaaapaWaaCbmaeaadaqfGaqabSqabeaacaaMb8oaneaapeGaey4kIipaaOWdamaaCaaaleqabaWdbiaaygW7aaGcpaWaaWbaaSqabeaapeGaaGzaVdaak8aadaahaaWcbeqaa8qacaaMb8oaaaWdaeaapeGaaGimaaWdaeaapeGaamiAaaaakmaalaaapaqaa8qacqGHciITdaqadaWdaeaadaqdaaqaa8qaceWG3bWdayaafaWdbiqbeI7aX9aagaqbaaaaa8qacaGLOaGaayzkaaaapaqaa8qacqGHciITcaWG6baaaiabgkGi2kaadQhaa8aabaWdbiaadIgacaqGGcGaaeypaiaabckacaqGIbGaae4BaiaabwhacaqGUbGaaeizaiaabggacaqGYbGaaeyEaiaabckacaqGSbGaaeyyaiaabMhacaqGLbGaaeOCaiaabckacaqGObGaaeyzaiaabMgacaqGNbGaaeiAaiaabshacaqG7aGaaeiOaiaadQhacqGH9aqpcaaIWaGaaeiOaiaabMgacaqGZbGaaeiOaiaabshacaqGObGaaeyzaiaabckacaqGZbGaaeyDaiaabkhacaqGMbGaaeyyaiaabogacaqGLbaapaqaa8qacaaMf8UaaGzbVpaalaaapaqaa8qacqGHciITcuaH4oqCpaGbaebaaeaapeGaeyOaIyRaamiDaaaacqGH9aqpdaWcaaWdaeaapeGaaGymaaWdaeaapeGaamiAaaaadaWadaWdaeaapeWaaeWaa8aabaWaa0aaaeaapeGabm4Da8aagaqba8qacuaH4oqCpaGbauaaaaaapeGaayjkaiaawMcaa8aadaWgaaWcbaWdbiaabcdaa8aabeaak8qacqGHsisldaqadaWdaeaadaqdaaqaa8qaceWG3bWdayaafaWdbiqbeI7aX9aagaqbaaaaa8qacaGLOaGaayzkaaWdamaaBaaaleaapeGaamiAaaWdaeqaaaGcpeGaay5waiaaw2faaaWdaeaapeGaaeiOaiaabckacaqGGcGaaeiOaiaabckacaqGGcGaaeiOaiaabckacaqGGcGaaeiOaiaabckacaqGGcGaaeiOaiaabckacaqGGcGaeyyrIa0aaSaaa8aabaWdbiaaigdaa8aabaWdbiaadIgaaaWaaeWaa8aabaWaa0aaaeaapeGabm4Da8aagaqba8qacuaH4oqCpaGbauaaaaaapeGaayjkaiaawMcaa8aadaWgaaWcbaWdbiaabcdaa8aabeaaaaaaaa@C4E2@

Note that in the last line of the above, we have assumed ${\left(\overline{{w}^{\prime}{\theta}^{\prime}}\right)}_{h}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbmaabmaapaqaamaanaaabaWdbiqadEhapaGbauaapeGafqiUde3dayaafaaaaaWdbiaawIcacaGLPaaapaWaaSbaaSqaa8qacaWGObaapaqabaaaaa@39A7@ is small because it turns out that it is driven by ${\left(\overline{{w}^{\prime}{\theta}^{\prime}}\right)}_{0}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbmaabmaapaqaamaanaaabaWdbiqadEhapaGbauaapeGafqiUde3dayaafaaaaaWdbiaawIcacaGLPaaapaWaaSbaaSqaa8qacaaIWaaapaqabaaaaa@3974@ and is a negative but small fraction of ${\left(\overline{{w}^{\prime}{\theta}^{\prime}}\right)}_{0}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbmaabmaapaqaamaanaaabaWdbiqadEhapaGbauaapeGafqiUde3dayaafaaaaaWdbiaawIcacaGLPaaapaWaaSbaaSqaa8qacaaIWaaapaqabaaaaa@3974@ .

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,

$$\frac{\partial \overline{\theta}}{\partial t}\approx \frac{0.2{\text{Kms}}^{-1}}{1000\text{m}}\times 3600{\text{shr}}^{-1}=0.7{\text{Khr}}^{-1}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@60FB@

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.

We have focused on the sensible heat flux up to now, but turbulence creates other vertical fluxes. There are many vertical turbulent fluxes, but two important ones are the **latent heat flux**, which involves the vertical transport of water vapor, and the** horizontal momentum flux**, which involves the vertical transport of horizontal wind.

For the purpose of this discussion, use the specific humidity, *q*. There is the mean value for *q* at different heights, and then there is the kinematic eddy flux. Using the same methods as before, we find that the kinematic water vapor flux (or specific humidity flux) is given by:

$${F}_{v}=\overline{w\text{'}q\text{'}}$$

This flux has SI units of ${\text{kg}}_{\text{water}}{\text{kg}}_{\text{air}}{}^{-1}{\text{ms}}^{-1}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaabUgacaqGNbWaaSbaaSqaaiaabEhacaqGHbGaaeiDaiaabwgacaqGYbaabeaakiaabckacaqGRbGaae4zamaaBaaaleaacaqGHbGaaeyAaiaabkhaaeqaaOWdamaaCaaaleqabaWdbiabgkHiTiaaigdaaaGccaqGGcGaaeyBaiaabckacaqGZbWdamaaCaaaleqabaWdbiabgkHiTiaaigdaaaaaaa@4873@. Usually the specific humidity is greatest near Earth's surface and decreases with height. Using the same logic as for the sensible heat flux, we expect a flux of water vapor from the surface, where the specific humidity is greater, to the free troposphere, where the specific humidity is less.

However, we usually want to compare energy fluxes caused by different processes as in Lesson 7.3 [7], so we multiply the specific humidity flux by the terms necessary to convert it into an energy flux that would result from the condensation of that water vapor. We end up with a latent heat flux:

$${F}_{LH}={\rho}_{air}{l}_{v}\overline{w\text{'}q\text{'}}$$

with SI units of $({\text{kg}}_{\text{air}}{\text{m}}^{-3})({\text{Jkg}}_{\text{water}}{}^{-1})({\text{kg}}_{\text{water}}{\text{kg}}_{\text{air}}{}^{-1}{\text{ms}}^{-1})={\text{Jm}}^{-2}{\text{s}}^{-1}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@6B06@.

Note that we have multiplied the specific humidity flux by the density of air and the latent heat of vaporization to put the specific humidity flux in terms of an energy flux, which we see is comparable to the sensible heat flux and is a significant fraction of the global energy balance at Earth’s surface. In fact, on a global scale, the latent heat flux is about five times larger than the sensible heat flux and is about half the total absorbed solar irradiance.

Latent heat flux is the primary way that water vapor gets into the atmosphere and is thus the primary source of water vapor for convection and clouds. Predicting convection and precipitation depends on knowing the latent heat flux.

The mean horizontal wind velocity is the vector sum of the wind components in the *x*-direction and the *y*-direction. The magnitude of the mean horizontal wind velocity is given by:

$$\overline{{V}_{}}=\sqrt{{\overline{u}}^{2}+{\overline{v}}^{2}}$$

The horizontal momentum flux is basically vertical turbulent eddies bringing high-wind-velocity air down from above. You all have experienced this phenomenon if you have ever been out early in the morning, just as the solar heating of the surface has begun to create convection and mix calm near-surface air up and windier residual layer air down.

The equations for the (kinematic) vertical fluxes of *x*-momentum and *y*-momentum air are, respectively:

$${F}_{mx}=\overline{{u}^{\prime}{w}^{\prime}}\text{and}{F}_{my}=\overline{{v}^{\prime}{w}^{\prime}}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbiaadAeapaWaaSbaaSqaa8qacaWGTbGaamiEaaWdaeqaaOWdbiabg2da98aadaqdaaqaa8qaceWG1bGbauaaceWG3bGbauaaaaGaaeiOaiaabckacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaabckacaqGGcGaamOra8aadaWgaaWcbaWdbiaad2gacaWG5baapaqabaGcpeGaeyypa0ZdamaanaaabaWdbiqadAhagaqbaiqadEhagaqbaaaaaaa@4913@

where the SI units are m^{2} s^{–2} and where *u'* and *v'* are wind speed perturbations in the *x* and *y* directions, respectively.

Note that the horizontal wind speed, *V*, is zero at Earth's surface (because of molecular friction) and increases with height. Just as the turbulent heat flux moves air with a higher potential temperature to heights where the potential temperature is lower, the turbulent momentum flux moves air with higher horizontal momentum (i.e., horizontal velocity) to heights where the mean horizontal momentum is lower. That is, the horizontal momentum is moved downward through the boundary layer to the Earth's surface, where it is dissipated by molecular friction.

Just as the heat flux is equal to a constant times the vertical gradient of the mean potential temperature (Equation [11.9]), the *x*-momentum flux is equal to a constant times the vertical gradient of the mean *x*-wind:

$$\overline{{u}^{\prime}{w}^{\prime}}=-K\frac{\partial \overline{u}}{\partial z}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaabbaaaaaaaaIXwyJTgapeGabmyDayaafaGabm4Dayaafaaaaiabg2da9iabgkHiTiaadUeadaWcaaWdaeaapeGaeyOaIyRabmyDa8aagaqeaaqaa8qacqGHciITcaWG6baaaaaa@3DAC@

where *K* is the eddy diffusivity.

Just as the change with time of the mean potential temperature is related to the negative of the vertical gradient of the kinematic heat flux (Equation [11.11]), so is the change with time of the mean velocity related to the negative of the vertical gradient of the kinematic momentum flux. Thus, the *x*-component momentum equation in the boundary layer becomes (ignoring other terms for now, such as the pressure gradient force and the Coriolis force):

$$\frac{\partial \overline{u}}{\partial t}=-\frac{\partial \left(\overline{{u}^{\prime}{w}^{\prime}}\right)}{\partial z}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbmaalaaapaqaa8qacqGHciITceWG1bWdayaaraaabaWdbiabgkGi2kaadshaaaGaeyypa0JaeyOeI0YaaSaaa8aabaWdbiabgkGi2oaabmaapaqaamaanaaabaWdbiqadwhapaGbauaapeGabm4Da8aagaqbaaaaa8qacaGLOaGaayzkaaaapaqaa8qacqGHciITcaWG6baaaaaa@42D5@

Just as we assumed the mean potential temperature is constant with height in the boundary layer, we can assume that the mean *x*-momentum (i.e., zonal velocity, *u*) is constant with height in the boundary layer. We can then integrate the above equation from the surface (*z* = 0) to the top of the boundary layer (*z* = *h*) and make the same assumptions about the flux at the top being relatively small to get:

$$\frac{\partial \overline{u}}{\partial t}=-\frac{1}{h}{\left(\overline{{u}^{\prime}{w}^{\prime}}\right)}_{0}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbmaalaaapaqaa8qacqGHciITceWG1bWdayaaraaabaWdbiabgkGi2kaadshaaaGaeyypa0JaeyOeI0YaaSaaa8aabaWdbiaaigdaa8aabaWdbiaadIgaaaWaaeWaa8aabaWaa0aaaeaapeGabmyDa8aagaqba8qaceWG3bWdayaafaaaaaWdbiaawIcacaGLPaaapaWaaSbaaSqaa8qacaaIWaaapaqabaaaaa@41C6@

where ${\left(\overline{{u}^{\prime}{w}^{\prime}}\right)}_{0}$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbmaabmaapaqaamaanaaabaWdbiqadwhapaGbauaapeGabm4Da8aagaqbaaaaa8qacaGLOaGaayzkaaWdamaaBaaaleaapeGaaGimaaWdaeqaaaaa@38B8@ is the vertical flux of *x*-momentum at the surface. As Equation [11.16b] indicates, this flux depends on the local eddy diffusivity and the local vertical gradient of the mean *u* velocity. The eddy diffusivity near the surface will increase as the mean wind speed in the boundary layer increases because vertical shear, which is responsible for the mechanical generation of turbulence, will be greater as the mean wind speed increases. Also, the vertical gradient of the mean *u* velocity near the surface will increase as the mean *u* of the boundary layer increases. Thus, we expect

$${\left(\overline{{u}^{\prime}{w}^{\prime}}\right)}_{0}\propto \overline{V}\overline{u}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqqaaaaaaaaGySf2yRbWdbmaabmaapaqaamaanaaabaWdbiqadwhapaGbauaapeGabm4Da8aagaqbaaaaa8qacaGLOaGaayzkaaWdamaaBaaaleaapeGaaGimaaWdaeqaaOGaeyyhIuRabmOvayaaraGabmyDayaaraaaaa@3C47@

The coefficient of proportionality is called the drag coefficient, ${C}_{d}$, which depends on the roughness of the surface and the thermal stability at the surface. Hence, combining Equations [11.16d] and [11.16e], we have for both horizontal components:

$$\begin{array}{l}\frac{\partial \overline{u}}{\partial t}=-\frac{{C}_{d}\overline{V}}{h}\overline{u}\hfill \\ \frac{\partial \overline{v}}{\partial t}=-\frac{{C}_{d}\overline{V}}{h}\overline{v}\hfill \end{array}$$[1]@5@5@+=faaahmart1ev3aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=ribbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaafaWaaeGabaaabaaeeaaaaaaaaigBHn2Aa8qadaWcaaqaaiabgkGi2kqadwhapaGbaebaa8qabaGaeyOaIyRaamiDaaaacqGH9aqpcqGHsisldaWcaaWdaeaapeGaam4qa8aadaWgaaWcbaWdbiaadsgaa8aabeaakiqadAfagaqeaaqaa8qacaWGObaaaiqadwhapaGbaebaaeaapeWaaSaaaeaacqGHciITceWG2bWdayaaraaapeqaaiabgkGi2kaadshaaaGaeyypa0JaeyOeI0YaaSaaa8aabaWdbiaadoeapaWaaSbaaSqaa8qacaWGKbaapaqabaGcceWGwbGbaebaaeaapeGaamiAaaaaceWG2bWdayaaraaaaaaa@4C5B@

These are the turbulent resistance terms that we introduced as friction in Lesson 10. They come from the downward transfer of horizontal momentum to the surface, where it is dissipated by molecular friction at Earth's surface.

We have defined three mean variables, ${\overline{\theta}}_{v}$, $\overline{q}$, and $\overline{{V}_{}}$, and three eddy flux terms associated with these mean values, ${\overline{w\text{'}\theta \text{'}}}_{v}$, $\overline{w\text{'}q\text{'}}$, and $\overline{u\text{'}w\text{'}}$. Let’s examine their behavior as a function of height in the daytime boundary layer (see figure below).

Credit: W. Brune, after Deidonks and Tennekes, 1984

First, start with some general observations:

- All these height variations scale with the PBL height, whether it be 1 km or 3 km.
- Eddy fluxes move quantities from higher to lower values. If a mean quantity increases with height, then the eddy flux will be downward (negative); if a mean quantity decreases with height, then the eddy flux will be upward (positive).
- The entrainment zone is a layer of mixing between the boundary layer and the free troposphere.

*Before you read on, make sure that you believe these three concepts. *

The ${\overline{\theta}}_{v}$ profile is superadiabatic near the surface due to contact with the heated surface, neutral in the middle, and stable above. Eddies can rise from the surface to a height where ${\overline{\theta}}_{v}$ equals its surface value (assuming no cloud formation and condensation). The mixed layer will grow if surface heating or increased humidity by evaporation causes the surface ${\overline{\theta}}_{v}$ to increase, which means that air parcels can rise and be neutrally buoyant at greater ${\overline{\theta}}_{v}$ aloft and thus greater heights.

We see that the eddy flux ${\overline{w\text{'}\theta \text{'}}}_{v}$ is greatest at the surface, decreases nearly linearly with height, becomes slightly negative above the PBL height *h* because eddies are bringing warmer air down from above$\mathrm{}$. Remember, eddy fluxes carry a quantity like virtual potential temperature down the mean gradient. The eddy flux for virtual potential temperature (and all quantities) goes to very low values (essentially zero) above the entrainment zone even though a gradient is present because the eddy energy is much lower there.

Specific humidity is greatest at the surface, where moisture sources, such as water bodies and vegetation, are present. The specific humidity falls off slowly with height until it reaches the PBL height, and then falls off rapidly into the free atmosphere. Because $\overline{q}$ falls off with height, the humidity flux $\overline{w\text{'}q\text{'}}$ increases with height until $\overline{w\text{'}q\text{'}}$ becomes small because $\overline{q}$ becomes small.

$\overline{{V}_{}}$, the mean horizontal wind speed, increases logarithmically with height and then takes a jump above the PBL height as the wind no longer feels the friction from Earth’s surface and becomes geostrophic. The dashed line in the figure above indicates what the geostrophic wind would be without boundary layer friction. Because $\overline{{V}_{}}$is increasing with height, the horizontal momentum flux is negative throughout the boundary layer. It becomes less negative in the entrainment zone and approaches zero at heights above *h*.

So what does the daytime convective boundary layer look like? There are isolated convective updrafts surrounded by slower descending air, giving rise to the large-scale eddy circulation, as seen in the following video of the convective boundary layer, viewed from the top. Associated with the large eddies are smaller eddies that come about as the upward and downward air parcels move past each other. If you look closely, you will see eddies of all sizes in the video (:24), some quite large and some quite small, but the smaller ones seem to originate in the larger ones.

- Find
**Practice Quiz 11-2**in Canvas. You may complete this practice quiz as many times as you want. It is not graded, but it allows you to check your level of preparedness before taking the graded quiz. - When you feel you are ready, take
**Quiz 11-2**. You will be allowed to take this quiz only**once**. Good luck!

If an air parcel has a velocity, then it has kinetic energy. The air parcel's velocity can be broken down into average and perturbation parts and so can the kinetic energy. Now suppose we look at kinetic energy per volume of air: $\mathrm{\xbd}\text{}\rho {v}^{2}$. Then we find the kinematic kinetic energy by dividing by the density.

If we start with the equation:

$$\frac{KE}{\rho}=\frac{1}{2}\left({u}^{2}+{v}^{2}+{w}^{2}\right)$$

and we write each term as its mean and turbulent parts and then multiply all these terms out and take the Reynolds average, we can then apply the rules of averaging in Equation [11.1], and only two terms survive:

$$\begin{array}{l}\frac{\overline{MKE}}{\rho}=\frac{1}{2}\left({\overline{u}}^{2}+{\overline{v}}^{2}+{\overline{w}}^{2}\right)\\ \overline{e}=\frac{1}{2}\left(\overline{u{\text{'}}^{2}}+\overline{v{\text{'}}^{2}}+\overline{w{\text{'}}^{2}}\right)\end{array}$$

The first term is simply the kinetic energy associated with the mean wind. The second term is the kinetic energy associated with the turbulent wind and is called the **turbulent kinetic energy, or TKE**, for short.

Which size eddies have the most energy? We can look at the relative intensity of the different scales of wind by considering the energy associated with motions of different sizes. Remember that by the Taylor hypothesis, the size of the eddy and the period of the eddy are related so that large eddies have longer periods and smaller eddies have smaller periods.

So, relative spectral intensity is just the amount of kinetic energy associated with that size eddy and the eddy size is associated with a period required for the eddy to pass over a sensor (see figure below).

- The peak in energy at 100 hours is from fronts and weather systems as they pass over a location. The spatial scale of these phenomena is relatively large and is called the synoptic scale.
- The smaller peak at about 24 hours is the diurnal cycle of wind speed, which increases during the day and then decreases at night.
- There is often a mininum in energy (called a spectral gap) at a time scale of an hour, where the circulations or eddies are relatively weak.
- This smaller peak at about 0.1 to 0.01 hours is called the “turbulent scale." This peak is caused by production of turbulent kinetic energy by buoyancy production (i.e., convection) and shear production (i.e., viscous interaction of air masses with different velocities). These eddies have the time scales of minutes and the size of the PBL.
- As the period of the eddies decreases below about 0.01 hours (about a minute), the strength of the eddies decreases.
- Eventually, on the subsecond time scale, the eddies have very little energy indeed.

So what is happening? Energy is flowing from the larger-scale eddies to the smaller-scale eddies. Eventually, the energy is dissipated through viscosity, which is a molecular-scale process. So, the energy of the larger eddies is transferred into smaller eddies, and eventually that energy is lost to viscosity, which in turn generates heating.

Credit: W. Brune, after R. B. Stull's An Introduction to Boundary Layer Meteorology (1988), from a study of I. Van der Hoven (1957).

Lewis Richardson wrote a poem about this process for whorls (a.k.a. eddies) in 1922:

Big whorls have little whorls,

Which feed on their velocity;

And little whorls have lesser whorls,

And so on to viscosity

(in the molecular sense).

In Lesson 7, we looked at the energy budget of the average atmosphere. Now let’s look at the energy budget in the surface layer. We can think of this budget in terms of energy fluxes and energy storage. As you learned in Lesson 2, energy is conserved—it must go somewhere.

There are several basic terms that we can put together in an energy equation that is appropriate for different environments or times of day. Upward fluxes will be positive; downward fluxes will be negative. The units are W m^{–2}.

The main energy terms are:

- ${Q}_{s}$: net radiation = Earth’s IR (up) – (solar + atmospheric IR) (down)
- ${Q}_{H}$: sensible heat flux (up or down)
- ${Q}_{E}$: latent heat flux (up or down)
- ${Q}_{G}$: surface heating (up or down)
- $\Delta {Q}_{s}$: surface energy storage in trees, buildings, etc.

We will neglect the storage term, which is usually small compared to the fluxes.

The ratio of sensible heat flux to latent heat flux is called the **Bowen ratio**:

$$B=\frac{{Q}_{H}}{{Q}_{E}}$$[1]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkeacqGH9aqpdaWcaaqaaiaadgfadaWgaaWcbaGaamisaaqabaaakeaacaWGrbWaaSbaaSqaaiaadweaaeqaaaaaaaa@384C@

It can be either positive or negative:

- > 10 over deserts,
- 2 – 6 over semi-arid lands,
- 0.4 – 0.8 over grasslands and forests,
- 0.1 – 0.3 in tropical rainforests,
- < 0.1 over oceans,
- negative over oases.

Looking at this list, it is easy to see that the water vapor comes not only from the surface moisture but also from trees and other plants by a process called **evapotranspiration**.

Look at the figure above, which shows the relative values of energy fluxes for different environments and times of day:

**(a) daytime:**solar heating dominates; heating is going into upward latent and sensible heating and warming the ground. The Bowen ratio (ratio of the sensible to the latent heat fluxes) depends on the amount of vegetation and surface moisture. The heating of the ground’s surface causes a downward energy flux into the ground.**(b) nighttime:**no solar shortwave; radiation is all infrared and is less than during the day, which has a net loss of energy to space. Both latent and sensible heat fluxes are negative because the ground is cooling and water vapor is condensing, leading to a positive gradient in temperature and specific humidity. The subsurface ground is warmer than the surface, so there is an upward energy flux from the subsurface to the surface.**(c) daytime oasis with warm air advection over a moist surface:**solar heating, sensible heat flux downward from hot dry air, latent heat flux upward offsetting some of the heating from the solar flux and the downward sensible heat flux. The surface is still heating, so some energy flux goes into the ground. The oasis is cooler than the surrounding desert because of the large latent heat flux.**(d) daytime ocean:**sensible and latent heat fluxes are positive, but much of the downward surface flux carries significant energy down into the ocean away from the surface. The ocean has a lot of heat capacity, so that solar energy can be absorbed in the ocean with little temperature change. Most of the energy goes into the ocean, with fairly constant fluxes of latent heat and sensible heat.

- Please note:
**there is no practice quiz**for Quiz 11-3. - When you feel you are ready, take
**Quiz 11-3**in Canvas. You will be allowed to take this quiz only**once**. Good luck!

The atmospheric boundary layer has a strong diurnal cycle, which consists of a well-mixed layer that is ~1 km high during the day but collapses down to 10% of that at night, leaving behind a residual layer. The mixing is caused by turbulence, a chaotic mix of swirling eddies ranging in size from the height of the PBL to the size of your fist. Understanding turbulence is the key to understanding the motion of air in the PBL. Taylor’s Hypothesis shows that changes in eddy properties measured by a sensor in time can be used to describe the eddy properties in space.

Equations of motion by turbulent eddies arise from defining all quantities (e.g., wind speed, specific humidity, and potential temperature) in terms of their mean and turbulent parts, then substituting these two parts for each quantity into the appropriate conservation equations, expanding the equations, then taking the Reynolds averages. Terms with two turbulent components arise that are non-zero, and these give rise to turbulent fluxes of temperature, water vapor, and momentum. A key result is that eddies drive vertical transport of these quantities (and others) in the direction of where the mean quantities are the least from where they are the most. Furthermore, changes in the vertical profiles of mean quantities with time are caused by the vertical eddy fluxes of those quantities.

The energy in the PBL is a tale of the turbulent kinetic energy and the energy balance near the surface. Turbulence consists of eddies of all different sizes and lifetimes, ranging from sizes that are the height of the PBL to others as small as your fist. These eddies are related: the largest carrying the most energy, and the smaller ones fracturing off of the larger ones, thus cascading the energy from larger scales to smaller scales, ending eventually with molecular viscosity, which converts all that kinetic energy into heat. The driver of this turbulent kinetic energy is the exchange of energy at Earth’s surface and consists of net radiation, sensible heat flux, latent heat flux, and subsurface ground flux. Depending on the environment and the time of day, the relative values of these fluxes can be quite different, but in every case, the total energy budget must balance.

You have reached the end of Lesson 11! Double-check that you have completed all of the activities before you begin Lesson 12.

**Links**

[1] mailto:MathType@MTEF

[2] https://www.e-education.psu.edu/meteo300/node/649

[3] https://www.flickr.com/photos/94693506@N00/2419349835/in/photolist-4FMNqc-pPc62m-wS2w6-dvKPAm-wVoJm-4FRY7h-4FRYbS-pMxwKs-yWzUF-Sj99-3nff21-oedY-7XAjDH-5WwVkA-jiedz1-e7dzx1-4VhCaS-rykHgu-p1twH6-rx9ea-23ksSg-4FMNkz-5DFQT5-8rkZxX-s1V1Zg-e6PHj8-4EjNgb-dsb31d-s5Vvz-aDU11j-8pu59P-8pxghU-aFNN8T-8pUg3k-5sEV3X-8sPiea-bZo8D-5sJE9E-6SESpg-5sJmBG-8Hy5L2-8pu5pV-e6PGiX-8Hy72a-8pxfXC-8o99Zn-tEAtz-3CdVYd-9D52xH-dJ85ir

[4] https://www.e-education.psu.edu/meteo300/node/679

[5] https://www.e-education.psu.edu/meteo300/node/722

[6] https://www.e-education.psu.edu/meteo300/node/726

[7] https://www.e-education.psu.edu/meteo300/649