Geography Reference
In-Depth Information
Integrating over the entire volume we get
ρ
0
u
2
2
ρ
0
∂v
∂y
ρ
0
u
v
+
ρ
0
uX
d
dt
∂u
∂y
=+
+
(10.55)
=
0 and u
v
=
=±
where we have assumed that v
D. Terms on the
right-hand side in (10.55) can be interpreted as the work done by the zonal-mean
pressure force, the conversion of eddy kinetic energy to zonal-mean kinetic energy,
and dissipation by the zonal-mean eddy stress. Alternatively, the first term on the
right can be rewritten with the aid of the continuity equation to yield
0 for y
ρ
0
w
∂
∂z
ρ
0
∂v
∂y
∂ρ
0
w
∂z
ρ
0
wT
R
H
=−
=
=
where we have assumed that ρ
0
w
. Thus, averaged
over the whole domain the pressure work ter
m
is proportional to the correlation
between the zonal-mean vertical mass flux ρ
0
w and the zonal-mean temperature
(or thickness). This term will be positive if on the average warm air is rising and
cold air sinking, that is, if there is a conversion from potential to kinetic energy.
Section 8.3.1 showed that in the quasi-geostrophic system the available potential
energy is proportional to the square of the deviation of temperature from a standard
atmosphere profile divided by the static stability. In terms of differential thickness
the zonal-mean available potential energy is defined as
=
0atz
=
0, and z
→∞
ρ
0
N
2
∂
∂z
2
1
2
P
≡
Multiplying (10.49) through by ρ
0
∂
∂z
/N
2
and averaging over space gives
ρ
0
2N
2
∂
∂z
2
ρ
0
w
∂
∂z
ρ
0
κ J
N
2
H
∂
∂z
d
dt
=−
+
(10.56)
ρ
0
N
2
v
∂
∂z
∂
∂y
∂
∂z
−
The first term on the right is just equal and opposite to the first term on the right
in (10.55), which confirms that this term represents a conversion between zonal-
mean kinetic and potential energies. The second term involves the correlation
between temperature and diabatic heating; it expresses the generation of zonal-
mean potential energy by diabatic processes. The final term, which involves the
