I ¼ r V = À
V
ð3:13Þ
The covariance of two random variables A and B is defined as
cov A; B
ð
Þ¼
1
N
X NÀ1
i¼0
ðA i À AÞðB À BÞ ¼
1
N
X NÀ1
i¼0
a
0
i b
0
i ¼ a 0 b 0
ð3:14Þ
The concept of linear correlation coefficient r AB between two random variables is
closely associated with covariance:
r AB ¼
a 0 b 0
r A r B
ð3:15Þ
This statistic varies between −1, when the variables vary oppositely, and 1, when
the variables vary in the same way. Variables without any common pattern of
variation have a zero r AB value.
The turbulent kinetic energy (TKE) is another fundamental variable for the
characterization of turbulence in terms of energy transfer among eddies of different
dimensional scales. The TKE budget allows comparison between the terms for
production either through thermal buoyancy or shear stresses, with the terms for
heat dissipation due to viscosity. TKE per unit air mass can be defined in two ways
depending on whether the TKE average flow component is related with:
TKE=m ¼
1
2
À
U
2
þ À
V
2
þ À
W
2
ð3:16Þ
or fluctuations:
e ¼
1
2
u 02 þ v 02 þ w 02
ð3:17Þ
Turbulent eddies carry scalar and vector quantities, associated with different flows,
such as density q, vertical velocity W, and volumetric content of the scalar quantity k to
which the flow refers. The average value of W is zero, because in a flat location that is
sufficiently large and geometrically uniform, there is no preferential vertical flow, and
because the rising air mass equals the downward air mass during a reasonable period
(at least 10 min). The mean wind velocity in the constant flux layer can therefore be
considered horizontal. The fluctuation value of W and w' will be positive in the case of
upward movement, and negative if in the opposite direction.
The mean flow of the flux quantity F then becomes:
F ¼ ðq þ q 0 Þðw þ w 0 Þðk þ k 0 Þ
ð 3:18Þ
3.2 Mean Components and Turbulence for Turbulent Flows
37
V
ð3:13Þ
The covariance of two random variables A and B is defined as
cov A; B
ð
Þ¼
1
N
X NÀ1
i¼0
ðA i À AÞðB À BÞ ¼
1
N
X NÀ1
i¼0
a
0
i b
0
i ¼ a 0 b 0
ð3:14Þ
The concept of linear correlation coefficient r AB between two random variables is
closely associated with covariance:
r AB ¼
a 0 b 0
r A r B
ð3:15Þ
This statistic varies between −1, when the variables vary oppositely, and 1, when
the variables vary in the same way. Variables without any common pattern of
variation have a zero r AB value.
The turbulent kinetic energy (TKE) is another fundamental variable for the
characterization of turbulence in terms of energy transfer among eddies of different
dimensional scales. The TKE budget allows comparison between the terms for
production either through thermal buoyancy or shear stresses, with the terms for
heat dissipation due to viscosity. TKE per unit air mass can be defined in two ways
depending on whether the TKE average flow component is related with:
TKE=m ¼
1
2
À
U
2
þ À
V
2
þ À
W
2
ð3:16Þ
or fluctuations:
e ¼
1
2
u 02 þ v 02 þ w 02
ð3:17Þ
Turbulent eddies carry scalar and vector quantities, associated with different flows,
such as density q, vertical velocity W, and volumetric content of the scalar quantity k to
which the flow refers. The average value of W is zero, because in a flat location that is
sufficiently large and geometrically uniform, there is no preferential vertical flow, and
because the rising air mass equals the downward air mass during a reasonable period
(at least 10 min). The mean wind velocity in the constant flux layer can therefore be
considered horizontal. The fluctuation value of W and w' will be positive in the case of
upward movement, and negative if in the opposite direction.
The mean flow of the flux quantity F then becomes:
F ¼ ðq þ q 0 Þðw þ w 0 Þðk þ k 0 Þ
ð 3:18Þ
3.2 Mean Components and Turbulence for Turbulent Flows
37