In this equation, the various terms are equivalent to those corresponding to
Eq. (3.71). Term VI is the redistributive term of return to isotropy. This term must
be inserted in the equations equivalent to Eq. (3.72), relative to the variance of each
fluctuation of the wind speed components. It should be noted that if Eq. (3.72) is
applied to u 3 , then term III of Eq. (3.71) for quantification of vertical buoyancy
effects should also be considered.
3.5.4 Equations for the Vertical Momentum Flux
The equations for turbulent flow of momentum, sensible heat, water vapor, and
other gases, such as carbon dioxide, (u 0
i u 0
k , u 0
i T 0 ; u 0
i E 0 ; or u 0
i c 0 ) are obtained by adding
equations for instantaneous fluctuations. By taking Eq. (3.60), multiplying each
term by u k ′ and calculating the means gives
u 0
k
@ u 0
i
@t þ u 0
k u j
@ u 0
i
@x j
þ u 0
k u 0
j
@u i
@x j
þ u 0
k u 0
j
@ u 0
i
@x j
¼ d i3 u 0
k
h
0
v
h v
g þ
þ f c e ij3 u 0
k u 0
j À
u 0
k
q
@p
@x i
þ þ vu
0
k
@
2 u
0
i
@x 2
j
ð3:73Þ
then, by exchanging the indices i and k in Eq. (3.73) gives:
u 0
i
@ u 0
k
@t þ u 0
i u j
@ u 0
k
@x j
þ u 0
i u 0
j
@u k
@x j
þ u 0
i u 0
j
@ u 0
k
@x j
¼ d k3 u 0
i
h
0
v
h v
g þ
þ f c e kj3 u 0
i u 0
j À
u 0
i
q
@p 0
@x k
þ vu
0
i
@
2 u
0
k
@x 2
j
ð3:74Þ
From the addition of Eqs. (3.73) and (3.74) and applying the product derivative
rule to expressions of the type u 0
i @u 0
k =@t þ u 0
k @u 0
i =@t ¼ @ðu 0
i u 0
k Þ=@t, we have
@u 0
i u 0
k
@t þ u j
@u 0
i u 0
k
@x j
þ u 0
i u 0
j
@u k
@x j
þ u 0
k u 0
j
@u i
@x j
þ
u 0
j @u 0
i u 0
k
@x j
¼
¼ d i3 u 0
k
h
0
v
h v
g þ d k3 u 0
i
h
0
v
h v
g þ f c e ij3 u 0
k u 0
j þ f c e kj3 u 0
i u 0
j À
À
u 0
i
q
@p 0
@x k
À
u 0
k
q
@p 0
@x i
þ vu
0
k
@ 2 u 0
i
@x 2
j
þ vu
0
i
@
2 u
0
k
@x 2
j
ð3:75Þ
The most common simplification of Eq. (3.75) involves vertical transport of
momentum (i = 1, k = 3). For this application, it is assumed that the mean vertical
component is zero, the horizontal plane is homogeneous, and that the coordinate system
is aligned with the average wind. This gives (Stull 1994; Tennekes and Lumley 1980)
@ u 0
1 u 0
3
À
Á
@t
¼ Àu 02
3
@u 1
@x 3
À
@ð u 0
1 u 02
3 Þ
@x 3
þ
g
h m
u 0
1 h
0
v þ p0
@u 0
1
@x 3
þ
@u 0
3
@x 1
"
#
1
q
I
II
III
IV
V
50
3 Characterization of Turbulent Flow in the Surface Boundary Layer
Eq. (3.71). Term VI is the redistributive term of return to isotropy. This term must
be inserted in the equations equivalent to Eq. (3.72), relative to the variance of each
fluctuation of the wind speed components. It should be noted that if Eq. (3.72) is
applied to u 3 , then term III of Eq. (3.71) for quantification of vertical buoyancy
effects should also be considered.
3.5.4 Equations for the Vertical Momentum Flux
The equations for turbulent flow of momentum, sensible heat, water vapor, and
other gases, such as carbon dioxide, (u 0
i u 0
k , u 0
i T 0 ; u 0
i E 0 ; or u 0
i c 0 ) are obtained by adding
equations for instantaneous fluctuations. By taking Eq. (3.60), multiplying each
term by u k ′ and calculating the means gives
u 0
k
@ u 0
i
@t þ u 0
k u j
@ u 0
i
@x j
þ u 0
k u 0
j
@u i
@x j
þ u 0
k u 0
j
@ u 0
i
@x j
¼ d i3 u 0
k
h
0
v
h v
g þ
þ f c e ij3 u 0
k u 0
j À
u 0
k
q
@p
@x i
þ þ vu
0
k
@
2 u
0
i
@x 2
j
ð3:73Þ
then, by exchanging the indices i and k in Eq. (3.73) gives:
u 0
i
@ u 0
k
@t þ u 0
i u j
@ u 0
k
@x j
þ u 0
i u 0
j
@u k
@x j
þ u 0
i u 0
j
@ u 0
k
@x j
¼ d k3 u 0
i
h
0
v
h v
g þ
þ f c e kj3 u 0
i u 0
j À
u 0
i
q
@p 0
@x k
þ vu
0
i
@
2 u
0
k
@x 2
j
ð3:74Þ
From the addition of Eqs. (3.73) and (3.74) and applying the product derivative
rule to expressions of the type u 0
i @u 0
k =@t þ u 0
k @u 0
i =@t ¼ @ðu 0
i u 0
k Þ=@t, we have
@u 0
i u 0
k
@t þ u j
@u 0
i u 0
k
@x j
þ u 0
i u 0
j
@u k
@x j
þ u 0
k u 0
j
@u i
@x j
þ
u 0
j @u 0
i u 0
k
@x j
¼
¼ d i3 u 0
k
h
0
v
h v
g þ d k3 u 0
i
h
0
v
h v
g þ f c e ij3 u 0
k u 0
j þ f c e kj3 u 0
i u 0
j À
À
u 0
i
q
@p 0
@x k
À
u 0
k
q
@p 0
@x i
þ vu
0
k
@ 2 u 0
i
@x 2
j
þ vu
0
i
@
2 u
0
k
@x 2
j
ð3:75Þ
The most common simplification of Eq. (3.75) involves vertical transport of
momentum (i = 1, k = 3). For this application, it is assumed that the mean vertical
component is zero, the horizontal plane is homogeneous, and that the coordinate system
is aligned with the average wind. This gives (Stull 1994; Tennekes and Lumley 1980)
@ u 0
1 u 0
3
À
Á
@t
¼ Àu 02
3
@u 1
@x 3
À
@ð u 0
1 u 02
3 Þ
@x 3
þ
g
h m
u 0
1 h
0
v þ p0
@u 0
1
@x 3
þ
@u 0
3
@x 1
"
#
1
q
I
II
III
IV
V
50
3 Characterization of Turbulent Flow in the Surface Boundary Layer