where j h is the molecular thermal diffusivity, LE is the latent heat associated with
the change in phase flux of water vapor; and R nj is the component of the radiative
budget in j direction.
In Eq. (3.78) I, II, and III represent the terms for storage, advection, and
molecular diffusion and terms IV and V represent the sources (or sinks) of heat
from radiation and change in the water vapor phase.
Averaging for converting variables into the mean and turbulent components,
gives
@ h
@t þ
@ h0
@t þ u j
@h
@x j
þ u j
@ h 0
@x j
þ u
0
j
@h
@x j
þ u
0
j
@ h 0
@x j
¼ j h
@
2 h
@x 2
j
þ j h
@
2 h 0
@x 2
j
À
À
1
qc p
@R nj
@x j
À
1
qc p
@R
0
nj
@x j
À
LE
qc p
ð3:79Þ
Calculating again the means for Eq. (3.79), we have
@h
@t
þ u j
@h
@x j
¼ j h
@
2 h
@x 2
j
À
@ u 0
j h
0
@x j
À
1
qc p
@R nj
@x j
À
LE
qc p
I
II
III
IV
V
VI
ð3:80Þ
The new term IV (nonexistent in Eq. 3.79) is the vertical flow of sensible heat.
Subtracting now Eq. (3.80) from Eq. (3.79) yields a budget equation for the
temperature fluctuations:
@ h
@t
þ u j
@ h 0
@x j
þ u
0
j
@h
@x j
þ u
0
j
@ h 0
@x j
¼ j h
@
2 h
0
@x 2
j
þ
@ u 0
j h
0
@x j
À
1
qc p
@R
0
yj
@x j
ð3:81Þ
and multiplying the terms of Eq. (3.81) by fluctuations u i ' and applying the means
u 0
i
@ h0
@t þ u j u 0
i
@ h 0
@x j
þ u 0
j u 0
i
@h
@x j
þ u 0
j u 0
i
@ h 0
@x j
¼ j h u 0
i
@ 2 h 0
@x 2
j
þ
þ u 0
i
@ h
0 u 0
j
À Á
@x j
À
1
qc p
u 0
i
@R 0
nj
@x j
ð3:82Þ
Taking now Eq. (3.60) relative to the balance of fluctuations u i ', multiplying by
fluctuation h i
' and applying averages, it will come
h
0 @u 0
i
@t þ h
0 u j
@u 0
i
@x j
þ h
0 u 0
j
@U i
@x j
þ h
0 u 0
j
@u 0
i
@x j
¼ d i3 h
0 h
0
v
h
v
v
!
g þ f c e ij3 u 0
j h
0
À
À
h
0 @p 0
q @x i
þ vh
0 @ 2 u 0
i
@x 2
i
ð3:83Þ
52
3 Characterization of Turbulent Flow in the Surface Boundary Layer