Statistical Tools - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 160

increments (e.g., wind speed changes in 1 or 3 s intervals) are much more frequent
than could be expected from Gaussian statistics (see, e.g., Böttcher et al. 2007 ).
Morales et al. ( 2010 ) show that only u 0 values from single 10 min intervals which
have been detrended show a distribution close to a normal distribution (excess
kurtosis slightly less than zero). u 0 values from longer time series over many
10 min intervals exhibit an excess kurtosis in the order of 3.3, i.e., large deviations
from the mean are much more frequent than it could be expected from a normal
distribution. Only normalizing the wind speed deviations u 0 by the corresponding
standard deviation of the respective 10 min interval produces a normal
distribution. Motivated by the non-stationarity of atmospheric winds Böttcher
et al. ( 2007 ) suggest understanding the intermittent distributions for small-scale
wind fluctuations as a superposition of different subsets of isotropic turbulence.
Therefore, a different statistical approach is necessary. Often, wind fluctuation and
gust statistics are described by a Gumbel distribution (Gumbel 1958 ) which has
proven to be especially suitable for extreme value statistics (see Sect. A.3 below).
The fluctuations of wind speed parallel to the mean wind direction (longitudinal
component) u 0 , normal to the mean wind direction (transverse component) v 0 and
the vertical component w 0 are not independent of each other, i.e., they have non-
zero correlation products. The most important of these products is:

Z

T

u 0 w 0 ¼ 1
T

u 0 ð t Þ w 0 ð t Þ dt ð A : 12 Þ

0

This product is usually negative, because the mean wind speed increases with
height and negative (downward) fluctuations of the vertical velocity component
bring down positive (higher) longitudinal wind fluctuations from upper layers
while positive (upward) fluctuations of the vertical velocity are connected with
negative (lower) longitudinal wind fluctuations from lower layers. The square root
of the negative value of this correlation product is usually called friction velocity
which often serves as a suitable velocity scale in the (mechanically) turbulent
atmospheric boundary layer:

p

u 0 w 0 u ¼ ð A : 13 Þ

It is a measure how fast horizontal momentum is transported downward by
turbulent motions in the atmospheric boundary layer.
Often, one-point statistics are not sufficient to describe the characteristics of
atmospheric turbulence. The next step is therefore to look at two-point statistics.
A simple example for a two-point statistics in the time domain is the

autocorrelation function:

R u 0 u 0 ð s Þ¼ 1

r u 0 u 0 ð t þ s Þ u 0 ð t Þ

ð A : 14 Þ