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available for conversion to kinetic energy because no adiabatic process can further
reduce E
I
.
The
available potential energy
(APE) can now be defined as the difference
between the total potential energy of a closed system and the minimum total
potential energy that could result from an adiabatic redistribution of mass. Thus, for
the idealized model given earlier, the APE, which is designated by the symbol P ,is
=
c
p
/c
v
E
I
−
E
I
P
(8.33)
which is equivalent to the maximum kinetic energy that can be realized by an
adiabatic process.
Lorenz (1960) showed that available potential energy is given approximately
by the volume integral over the entire atmosp
h
ere of the variance of potential
temperature on isobaric surfaces. Thus, letting θ designate the average potential
temperature for a given pressure surface and θ
the local deviation from the average,
the average available potential energy per unit volume satisfies the proportionality
V
−
1
θ
2
/ θ
2
dV
P
∝
where V designates the total volume. For the quasi-geostrophic model, this pro-
portionality is an exact measure of the available potential energy, as shown in the
following subsection.
Observations indicate that for the atmosphere as a whole
P/
c
p
/c
v
E
I
∼
10
−
3
,
10
−
1
5
×
K/P
∼
Thus only about 0.5% of the total potential energy of the atmosphere is available,
and of the available portion only about 10% is actually converted to kinetic energy.
From this point of view the atmosphere is a rather inefficient heat engine.
8.3.2
Energy Equations for the Two-Layer Model
In the two-layer model of Section 8.2, the perturbation temperature field is propor-
tional to ψ
1
−
ψ
3
, the 250- to 750-hPa thickness. Thus, in view of the discussion in
the previous section, we anticipate that the available potential energy in this case
will be proportional to (ψ
1
−
ψ
3
)
2
. To show that this in fact must be the case, we
derive the energy equations for the system in the following manner: We first mul-
tiply (8.9) by
ψ
3
). We then integrate the
resulting equations over one wavelength of the perturbation in the zonal direction.
ψ
3
, and (8.11) by (ψ
1
−
−
ψ
1
, (8.10) by
−
