Geoscience Reference
In-Depth Information
tke
¼
q
2
u
0
2
þ
v
0
2
þ
w
0
2
ð
A
:
7
Þ
is called turbulent kinetic energy and is a prognostic variable in many numerical
flow simulation models.
If wind speed values were distributed totally random, then the probability
density function f(u,T) of a stationary time series u(t) for a given averaging period
T would follow a normal distribution or Gaussian distribution, which is fully
determined by the mean value and the standard deviation:
0
@
1
A
!
2
u
0
u
ð
T
Þ
r
ð
T
Þ
1
r
ð
T
Þ
p
exp
1
2
f
ð
u
;
T
Þ¼
ð
A
:
8
Þ
2p
Higher-order moments can be used to check whether a time series is normally
distributed or not. The next two higher moments are skewness:
skew
u
ð
t
;
T
Þ¼
u
0
3
ð
t
;
T
Þ
r
u
ð
t
;
T
Þ
ð
A
:
9
Þ
and flatness or kurtosis:
Fl
u
ð
t
;
T
Þ¼
u
0
4
ð
t
;
T
Þ
r
u
ð
t
;
T
Þ
ð
A
:
10
Þ
The latter is often calculated as excess kurtosis in order to highlight the
deviation from normally distributed values which have a flatness of 3:
kur
u
ð
t
;
T
Þ¼
u
0
4
ð
t
;
T
Þ
r
u
ð
t
;
T
Þ
3
ð
A
:
11
Þ
For a normal distribution the skewness is zero because this distribution is
symmetric. Thus, skewness is a measure for asymmetry of a distribution.
Likewise, the excess kurtosis of the normal distribution is zero, because its flatness
is 3. Distributions with a negative excess kurtosis have a high central peak and low
tails while distributions with a positive excess kurtosis have a lower peak and
higher tails.
10 min mean wind speed distributions usually have a positive skewness, i.e.,
they have a long right tail. This means that large positive deviations from the mean
wind speed are more frequent than negative deviations of the same magnitude.
This is, because wind speed values are one-sidedly bounded. Negative wind speeds
are not possible. Therefore, they cannot be normally distributed and have to be
described by other more suitable distributions than the Gaussian distribution, e.g.,
the two-parametric Weibull distribution (see
Sect. A.2
below).
Occurrences of wind speed fluctuations even deviate more from Gaussian
statistics. Although the distribution of fluctuations is more or less symmetric
around the mean wind speed, analyses have shown that larger
wind speed
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