Geoscience Reference
In-Depth Information
and the potential temperature conservation
equation (8.26)
becomes
D θ
Dt
=
θ
ρc
p
T
∂ R
i
∂x
i
.
2
θ
α
∇
−
(8.28)
θ
θ
0
+
θ
,
If we define a base state and a deviation for potential temperature,
=
then from
(8.22)
we can write
T
)
˜
T
)
p
0
(
0
)
p
0
(z)
R
c
p
R
c
p
p(
0
)
˜
θ
0
+
θ
=
(T
0
+
(T
0
+
.
(8.29)
p(z)
The base states and the deviations are then related by
T
0
p(
0
)
˜
θ
=
T
p(
0
)
˜
R
c
p
R
c
p
θ
0
=
,
.
(8.30)
p(z)
p(z)
Equation (8.30)
indicates that the deviations of temperature and of potential
temperature, and therefore their vertical gradients, differ in general.
8.2.4 Scalar-constituent conservation
The density
ρ
c
of an advected constituent that has no sources or sinks (i.e., is
mass-conserving) and diffuses molecularly within the fluid follows
˜
∂
2
∂ ρ
c
ρ
c
u
i
∂x
i
˜
ρ
c
∂
∂t
+
=
γ
∂x
i
∂x
i
.
(1.29)
We can rewrite this as
∂
2
D
ρ
c
Dt
=−˜
˜
ρ
c
∂
u
i
∂x
i
+
˜
ρ
c
∂x
i
∂x
i
,
˜
γ
(8.31)
and using
Eq. (8.12)
to express the velocity divergence yields
∂
2
D
ρ
c
Dt
=−˜
˜
ρ
c
˜
u
3
H
ρ
+
ρ
c
∂x
i
∂x
i
.
˜
γ
(8.32)
This says that only in the limit of large density scale height is the density of an
advected, mass-conserving constituent conserved. In general it tends to decrease
in upward motion and increase in downward motion because of the adiabatic
expansion and compression, respectively, accompanying these motions. For that
reason turbulent mixing cannot produce vertically uniform mean density of a
mass-conserving trace constituent.
