Geoscience Reference
In-Depth Information
buoyant production is generally smaller, and the loss terms are turbulent trans-
port and pressure destruction. In the bottom-up budget buoyant production, not
mean-gradient production, is the principal source; mean-gradient production and
turbulent transport are generally smaller, and the principal loss term is pressure
destruction.
11.3.1.2 Scalar variances
With the top-down, bottom-up decomposition
(11.21)
we can write the variance of
a passive, conserved scalar
c
as
c
t
c
b
.
c
2
=
+
2
c
t
c
b
+
(11.27)
The simplest scaling hypothesis that accounts for the scalar flux a
t m
ixed-layer top
takes the governing param
ete
rs for
c
t
statistics to be
w
∗
,
z
i
,and
cw
1
; those f
or
c
b
stati
stic
s to be
w
∗
,
z
i
,and
cw
0
; and those for their joint statistics to be
w
∗
,
z
i
,
cw
0
,
and
cw
1
. Thus from
Eq. (11.27) Moeng and Wyngaard
(
1984
)wrote
cw
1
w
∗
2
2
cw
0
cw
1
w
2
f
tb
(z/z
i
)
cw
0
w
∗
2
c
2
=
f
t
(z/z
i
)
+
+
f
b
(z/z
i
).
∗
(11.28)
They found that in midlayer
f
b
∼
6
.
The correlation coefficient of the top-down and bottom-up scalar fields is
1
,f
tb
∼
1
,f
t
∼
c
t
c
b
c
t
c
b
1
/
2
f
tb
r
tb
=
=
(f
t
f
b
)
1
/
2
sgn
(cw
0
cw
1
),
(11.29)
where sgn means “the sign of.”
Moeng and Wyngaard
(
1984
) found that
0
.
5
in mid-CBL. The positive sign of
f
tb
has interesting implications. For example, if
two
spec
ies are diffu
sing
into the mixed layer, one from above and one from below,
then
cw
0
is positive,
cw
1
is negative, and in midlayer
r
tb
−
|
r
tb
|∼
0
.
5. If the two species
undergo a binary reaction, the mean reaction rate is
mean reaction rate
∝ ˜
c
1
˜
c
2
=
C
1
C
2
+
c
1
c
2
.
(11.30)
Because
Eq. (11.29)
says that
c
1
c
2
is negative here, the mean reaction rate is less
than the mean concentrations would indicate.
We saw in
Figure 11.13
that the top-dow
n an
d bo
tto
m-up scalar flux budgets have
sig
ni
ficant
bu
oyant-production terms. For
wθ
and
wc
these terms are proportional
to
θ
2
and
θc
, respectively. By using the top-down and bottom-up decomposition
