Section 3
Convectively Available Potential Energy
Consider an air parcel initially at point O as shown in Figure 1. Since Γs < Γ < Γd the
atmosphere is conditionally unstable, that is stable under a small displacement from O and
unstable if the displacement is large, so that the parcel reaches its level of free convection. After
passing this level the parcel accelerates and moves upwards due to the large buoyancy force
(Γs < Γ). Its motion will stop as soon as the atmospheric temperature profile intersects the
saturated adiabat for the second time. This point is called the level of neutral buoyancy. Our
goal will be to calculate the maximum possible kinetic energy that this statically unstable parcel
can acquire during its upward ”free” motion that starts at the level of free convection.
So consider an air parcel of density ρ0 embedded in an environment of density ρ that is
initially at the level of free convection (where Γ > Γs ). The force exerted on the parcel when
displaced δz upwards is the buoyancy force (ρg) and its weight (−ρ0 g) :
ρ0
D2 (δz)
ρ − ρ0
D2 (δz)
0
=
−ρ
g
+
ρg
⇔
=
g
Dt2
Dt2
ρ0
Using the perfect gas law, along with the fact that for the pressure inside the parcel is equal to
the atmospheric pressure (p0 = p) we can write the above equation as :
D2 (δz)
p/(RT ) − p/(RT 0 )
T0 − T
=
g
=
g
Dt2
p/(RT 0 )
T
We can now rewrite
D2 (δz)
Dt2
using the chain rule :
D2 (δz)
Dt2
=
Dw
Dt
=
Dz Dw
Dt dz
(1)
=
D
(w2 /2)
Dz
and equation
(1) becomes :
T0 − T
D 2
(w /2) =
g
Dz
T
(2)
Integrating (2) from the level of free convection till the level of neutral buoyancy we get :
2
wmax
/2
=
Z zLN B 0
T −T
zLF C
1
T
gdz
(3)
This is the the Convectively Available Potential Energy (CAPE) and you can estimate it graphically by the size of the area enclosed by the saturated adiabat and the observed temperature
profile. Typical values for CAPE are of the order of 500m2 /sec2 yielding values of the order of
5 − 10m/sec. You should compare this to typical values for vertical velocity outside the cloud,
of the order of cm/sec.
2
20
18
LNB
16
14
Γs
Γ
z
12
Γd
10
LFC
8
6
LCL
4
O
2
0
−70
−60
−50
−40
−30
−20
−10
Temperature
0
10
20
30
40
Figure 1: Vertical profile of atmospheric temperature (dashed line). The dry and saturated
adiabat are also plotted (solid lines). The particle is initially forced to move from point O until
the level of free convection following the dry adiabat until its lifting condensation level and then
the saturated adiabat. After it reaches the lifting condensation level and can accelerate on its
own due to the large buoyancy force.
3