Vertical Profiles Over Flat Terrain - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 27

3.1.1 Logarithmic Wind Profile

The most important atmospheric feature which influences the generation of energy
from the wind is the vertical increase of wind speed with height. This increase is
described by the laws for the vertical wind profile. Different descriptions of the
vertical wind profile exist. We will have a look at the classical logarithmic wind
profile first, which can be derived from simple physical considerations valid for the
surface layer. The empirical power law, which is often used instead of the loga-
rithmic law, will be presented in the subsequent Sect. 3.1.2 .

3.1.1.1 Neutral Stratification

We start the derivation of the logarithmic wind profile with dynamical consider-
ations, which suggest formulating the vertical momentum exchange coefficient K M
in ( 3.1 ) as being proportional to the mixing length l = jz, which in turn is pro-
portional to the distance to the ground and the friction velocity (K M = ju * z). This
leads to the following equation for the vertical wind speed gradient (or wind shear)
in the Prandtl layer derived from Eq. ( 3.1 ) (with the van Kármán constant
j = 0.4):

o u

oz ¼ u l ¼ u

ð 3 : 5 Þ

jz

Integration of the wind shear Eq. ( 3.5 ) from a lower height z 0 where the wind
speed is assumed to vanish near the ground up to a height z within the Prandtl layer
then yields the well-known logarithmic wind profile for this layer with the
roughness length z 0 :

u ð z Þ¼ u

j ln z d

ð 3 : 6 Þ

z 0

where d is called the displacement height and is relevant for flows over forests and
cities (see Sects. 3.6 and 3.7 ). The displacement height gives the vertical dis-
placement of the entire flow regime over areas which are densely covered with
obstacles such as trees or buildings. Otherwise, we will disregard this parameter in
the following considerations. If the displacement height is a relevant parameter,
then in the following equations all dimensionless ratios z/z 0 and z/L * [see ( 3.10 ) for
the definition of L * ] have to be replaced by (z-d)/z 0 and (z-d)/L * respectively.
The roughness length, z 0 and the displacement height, d are not purely local
values. They depend in a non-linear way from the surface properties upstream of
the place where the wind profile has to be computed from ( 3.6 ). The size of this
influencing upstream area, which is called fetch or footprint, is increasing with
increasing height z in the wind profile. Thus, the determination of these two values
is not an easy task, but requires the operation of footprint models (Schmid 1994 ;
Foken 2012 ). The footprint increases with wind speed, decreases with increasing