where B is an arbitrary quantity.
A þ B
ð
Þ ¼ A þ B
ð3:6Þ
dA
dt
¼
dA
dt
ð3:7Þ
Applying these rules to the variables composed of mean values and fluctuations
(Reynolds means) as in Eq. (3.1):
A
À Á ¼ A þ A 0
À
Á ¼ A
À Á þ a 0 ¼ A
À Á
ð3:8Þ
as the mean fluctuations:
a
0
¼ A À A
ð3:9Þ
is zero.
It follows then that
A B
ð Þ ¼ A þ a 0
À
Á
B þ b 0
À
Á ¼ AB þ a 0 B þ Ab 0 þ a 0 b 0
À
Á ¼ AB
À Á þ a 0 B
À
Á þ B b 0
À
Á þ a 0 b 0
ð
Þ¼
AB
À Á þ a 0 b 0
ð
Þ¼AB þ a 0 b 0
ð
Þ
ð3:10Þ
The mean of the fluctuation’s product is a non-linear variable, in principle
non-zero.
Other non-linear variables are, for example, a 02
ð Þ, a 0 b 02
ð
Þ, a 02 b 0
ð
Þ, or a 02 b 02
ð
Þ.
These variables are important for the characterization of atmospheric turbulence. It
is necessary to consider the definitions of variance for any scalar or vector quantity
where for large data sets 1/N % 1/(N−1):
r
2
A ¼
1
N
X NÀ1
i¼0
ðA i À AÞ
2 ¼
1
N
X NÀ1
i¼0
ða
0
i Þ
2 ¼ a 02
ð Þ
ð3:11Þ
The standard deviation is the square root of the variance:
r A ¼ a 02
1=2
ð3:12Þ
The standard deviation of velocity is a measure of the magnitude of the deviation
or dispersion of the measured values around the mean. Thus, the intensity of
turbulence I can be defined as (Stull 1994)
36
3 Characterization of Turbulent Flow in the Surface Boundary Layer