Geoscience Reference
In-Depth Information
16.1.2 A scalar in a steady, horizontally homogeneous boundary layer
Next we'll consider a physical problem: a conserved scalar with mean vertical
gradient
C
,
3
in a steady, horizontally homogeneous boundary layer with a mean
horizontal velocity
U
1
. Here the equation for the fluctuating part of the scalar is
(Chapter 5)
c
,t
+
C
,
3
u
3
+
c
,
1
U
1
+
(cu
j
)
,j
−
(cu
3
)
,
3
=
γc
,jj
.
(16.25)
Multiplying by 2
c
, averaging, using horizontal homogeneity, and rewriting the
molecular term
(Chapter 5)
gives the scalar variance budget:
∂c
2
∂t
=−
(c
2
u
3
)
,
3
−
2
C
,
3
u
3
c
−
2
γ c
,j
c
,j
.
(16.26)
The terms on the right side represent, in order, the rates ofmean-gradient production,
turbulent transport, and molecular destruction.
Because of the vertical inhomogeneity in this problem we must restrict our
Fourier-Stieltjes representations to the horizontal plane. As in
Chapter 15
,we'll
use the notation
κ
h
=
(κ
1
,κ
2
),
x
h
=
(x
1
,x
2
)
:
e
i
κ
h
·
x
h
dZ(
κ
h
;
x
3
,t),u
3
(
x
h
;
x
3
,t)
=
e
i
κ
h
·
x
h
dZ
3
(
κ
h
;
x
3
,t)
;
c(
x
h
;
x
3
,t)
=
e
i
κ
h
·
x
h
dF
j
(
κ
h
;
cu
j
(
x
h
;
x
3
,t)
=
x
3
,t), j
=
1
,
2
;
(16.27)
e
i
κ
h
·
x
h
dV(
κ
h
;
(cu
3
−
cu
3
)
,
3
=
x
3
,t).
We shall not indicate the dependence on
x
3
and
t
hereafter. Substituting these into
Eq. (16.25)
yields
∂
∂t
dZ(
κ
h
)
=−
C
,
3
dZ
3
(
κ
h
)
−
iκ
1
U
1
dZ(
κ
h
)
−
iκ
j
dF
j
(
κ
h
)
−
dV(
κ
h
)
−
γκ
j
κ
j
dZ(
κ
h
)
+
γ(dZ)
,
33
(
κ
h
)
(j
summed on 1
,
2
).
(16.28)
We proceed as in
Subsection 16.1.1
, multiplying
Eq. (16.28)
by
dZ
∗
and averaging,
and adding to that the result of multiplying the complex conjugate of
Eq. (16.28)
by
dZ
and averaging. We define the spectral density of
c
in the horizontal plane,
φ
(
2
)
(
κ
h
)d
κ
h
=
dZ(
κ
h
)dZ
∗
(
κ
h
).
(16.29)
