AGRON/ENSCI/MTEOR 405/505
Lecture 8 - Chapter 5 - Turbulence
February 3, 2011
1
Introduction
Thought experiment: Think about what data these graphs could contain (hint: the x-axis can be
either a time or distance axis). Label the graphs reasonably with numbers and units.
Is taking an average of the first graph useful? What about the second graph? What can you do or
say about the deviations from the mean?
2
Mean and Turbulent Parts
To separate the slow, large-scale changes in some property x of the environment from fast, random,
small-scale changes, we often split the property into mean (x) and turbulent (x! ) parts:
x = x + x!
(1)
A justification we have for doing this is the existence of a spectral gap. (See: Figure 2.2 in Stull)
Large-scale changes tend to occur on the synoptic scale (1000 km/1 week), while small-scale changes
tend to occur on the micro scale (¡1 km/¡1 hour).
3
Eddies
When a fluid moves laterally over a rough surface (any surface with friction), random vertical motions develop (think about wind hitting a building). Similarly, when a surface is heated, pockets of
air next to the hottest locations on the surface will randomly warm and rise, forcing nearby air to
sink. We call these random vertical motions eddies.
Eddies are important because they transfer a fluid’s properties in the vertical direction, mixing
the air (draw figure). These properties include nearly everything we’ve talked about: temperature,
moisture, gas concentrations, and even wind speed.
Eddies can come in nearly any size, and it’s the large eddies that help build the atmospheric
boundary layer. During the day, the large eddies mix upward the very warm (and usually moist)
air near the ground and mix downward the very cold (and usually dry) air in the atmosphere.
4
Covariances
Previously, we have talked about heat and moisture fluxes being the product of conductances and
gradients, e.g.:
H = gH (Ts − Ta )
(2)
Another way we can model fluxes is by using the covariance of vertical wind speed with other
properties.
1
2
Definition: A covariance is the mean of the product of two turbulent terms. For example, the
vertical heat flux can be modeled as the covariance of vertical wind speed (w) and temperature
(T ):
H = ρcp w! T !
(3)
While this will likely end our discussion of covariances in this class, it is important to understand
the concept because covariances are used to measure fluxes in the real world. It is difficult and
very expensive to measure temperature, moisture, etc. gradients over a few m or cm, but it is
less difficult and less expensive (though not easy or cheap) to measure fluctuations in temperature,
wind speed, etc. at a single location.
5
Final Notes
While turbulence is often difficult and expensive to measure, there are many models that relate
turbulence to the mean flow. For example, on a windy day, there will often be “stronger turbulence” (i.e. larger eddies). Indeed, experiments have shown a strong link between wind speeds and
atmospheric boundary layer height!
The size of eddies has a strong influence on sensible heat and latent heat fluxes, but we have not yet
included their influence in our equations. In fact, our equations (like equation 2) currently act like
diffusion equations, but diffusion is not the main sensible heat/latent heat transport mechanism
between organisms and their environment; turbulent transport is! Thus, soon we will use some of
the simple models that relate mean wind speed to turbulence to better tune our equations, mainly
by adjusting the conductance terms to include wind speed.
3