Turbulent Transport in the Atmospheric Surface Layer - Transport in the Atmosphere-Vegetation-Soil Continuum - page 98

3.5 Similarity Theory
In the context of this topic we are interested in the surface luxes of, for example,
heat and water vapour. In Section 3.4 it was shown that the turbulent luxes in the sur-
face layer are closely related to those surface luxes and it was shown how they can
be measured. However, in some cases the direct determination of turbulent luxes is
not feasible: for some gases fast response sensors are not available, and in the case
of modelling, luxes need to be modelled in terms of variables that are available in a
model. Hence we need to ind the relationship between the turbulent luxes and quan-
tities that are accessible for measurement or modelling (e.g., vertical gradients).
In Section 3.1 we saw that the turbulent diffusivity varies with height and meteoro-
logical conditions. In Section 3.3 we concluded that the turbulent kinetic energy may
play an important role in that variation. But the equation that describes the evolution
of the turbulent kinetic energy cannot be solved easily. 13 So, the link between luxes
and gradients (i.e., the derivation of K h in Eq. ( 3.1 )) cannot be solved from irst prin-
ciples. Therefore, one of the tools often used in luid mechanics is similarity theory,
which assumes that two lows are similar if certain dimensionless characteristics are
identical for the two situations.
A real-life example would be a bicycle shop that can sell bicycles in a range of
sizes, so the bikes are not identical . But if one would take the ratio of the height of the
bike and the height of the person who wants to buy it, all bikes would be similar , for
example, their height may be half the height of the buyer. This ratio can be considered
as a similarity law: the dimensionless ratio of bike height and a rider's height is a con-
stant. The bike seller then could use this similarity law by irst asking the customer
his length, before offering him a bike.
The formal method to determine which are the relevant dimensionless groups in
a physical problem, and how they are related, is called dimensional analysis. The
details of the method are presented in Appendix C . The main steps in dimensional
analysis are important to consider:
1. Find the relevant physical quantities that (may) determine the quantity of interest. For
example, one can guess that the temperature at 2 m height depends on the surface sensible
heat lux, the wind speed and the height above the surface.
2. Make dimensionless groups out of the quantities selected in step 1.
3. Do an experiment in which all quantities selected in step 1 are measured or obtain exist-
ing data.
4. Calculate, from the data obtained in step 3, for each measurement interval the values of
the dimensionless groups that were constructed in step 2. If all goes well, the dimension-
less groups show a universal relationship (or similarity relationship ) that can also be used
13 Differential equations, similar to Eq. ( 3.10 ) can be derived that describe the evolution of turbulent luxes, rather
than TKE. One of the omitted terms contains a third-order covariance. We could of course write down an equation
for the evolution of that term, but that would include a fourth-order covariance. This shifting of problems is called
the 'closure problem': at a certain order, we need to make a model for that higher-order term.