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turbulence, and increases with the measurement height. If no detailed information
is available, a rough first guess of the upstream extent of the footprint is one
hundred times the height z. The extent is modified by the thermal stability of the
surface layer. For unstable stratification the footprint is closer to the site of interest,
while for stable stratification it is further away. This means that surface features
such as hills and forests can influence the wind speed and profile at hub heights in
the order of 100 m even if they are several kilometres upstream. To guarantee a
good representativity of an estimated wind profile from ( 3.6 ) for a certain surface
type, the footprint should be horizontally as homogeneous as possible. The transfer
of the footprint concept to inhomogeneous terrain is discussed in Schmid ( 2002 ).
The wind speed increases with height without a turning of the wind direction in
the Prandtl layer. A scale analysis gives for the height of this layer, z p (Kraus
2008 ):
f
v g
z p 0 : 01 u
f 0 : 00064
ð 3 : 7 Þ
Putting in numbers (u * = 0.5 m/s, f = 0.0001 1/s, v g = 8 m/s) gives a typical
height of the Prandtl layer of 50 m.
In a well-mixed Prandtl layer the temperature, T decreases with height
according to the adiabatic lapse rate, g/c p (g is gravity acceleration, c p is specific
heat of the air at constant pressure). This yields a vertical temperature decrease of
roughly 1 K per 100 m in an unsaturated atmosphere, i.e., in an atmosphere in
which no moisture condensation or evaporation processes take place. Due to this
vertical decrease, the normal temperature is not appropriate to identify air masses.
For air mass identification, meteorologists and physicists have developed the
definition of an artificial temperature which stays constant during vertical dis-
placements without condensation processes. This artificial temperature is the
potential temperature. The potential temperature,
c p
p 0
p
H ¼ T
ð 3 : 8 Þ
is constant with height in a neutrally stratified Prandtl layer (R is the gas constant
for dry air). p 0 is the surface pressure.
Equations ( 3.5 ), ( 3.6 ) and ( 3.8 ) describe vertical profiles of mean variables in
the surface layer. We also have to specify the vertical distribution of turbulence.
The standard deviations [see Eq. (A.4) in Appendix A] of the 10 Hz turbulent
fluctuations of the three velocity components are assumed to be independent of
height in the surface layer and scale with the friction velocity u * as well (Stull
1988; Arya 1995 ). Usually the following relations are used:
r u
u 2 : 5;
r v
u 1 : 9;
r w
u 1 : 3;
ð 3 : 9 Þ
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