Appendix B. Sensible Heat Flux without a FW05
B-2
(
)
(
)
(
)
c
C
C
q
q
R T
q
p
v
d
d
d
2
1 0 84
1 0 94
1 0 61
=
+
+
+
.
.
.
(B-6)
Expand and collect terms.
(
)
(
)
c
C
C
R T
q
q
q
p
v
d
d
d
2
2
1 145
0 5124
1 0 94
=
+
+
+
.
.
.
(B-7)
Multiply the above by
(
)
(
)
1 0 94
1 0 94
−
−
.
.
q
q
and ignore the second and third order
terms.
(
)
c
C
C
R T
q
R T
p
v
d
d
d s
d
d
2
1 0 51
=
+
=
.
γ
(B-8a)
(
)
T
T
q
s
=
+
1 0 51
.
(B-8b)
γ
d
p
v
C
C
d
d
=
=
14
.
(B-8c)
B.2 Sonic Temperature, Temperature, and Humidity
In the following section define
T
,
q
,
T
s
, and
w
in terms of an instantaneous
value, a mean, and a deviation from the mean, where
w
is the vertical wind in
the natural wind coordinates.
Recall the rules of Reynolds averaging.
(
) ( )
A
A A
A
A
A A
=
+ ′ =
+ ′ = + ′
(B-9)
The only way the Equation (B-9) is true is if
′ =
A
0
.
Write the instantaneous
T
,
q
,
w
and
T
s
in terms of a mean and a fluctuation.
T T T
= + ′
(B-10)
q q q
= + ′
(B-11)
w w w
= + ′
(B-12)
T
T
T
s
s
s
=
+ ′
(B-13)
Substitute Equations (B-10), (B-11), and (B-12) into (B-8b) to write the sonic
temperature in terms of temperature and specific humidity.
(
)
(
)
(
)
T
T
T
T T
q q
s
s
s
=
+ ′ =
+ ′
+
+ ′
1 0 51
.
(B-14)