Geography Reference
In-Depth Information
For a saturated parcel undergoing pseudoadiabatic ascent the rate of change in q
s
following the motion is much larger than the rate of change in T or L
c
. Therefore,
d
L
c
q
s
c
p
T
d ln θ
≈−
(9.39)
Integrating (9.39) from the initial state (θ,q
s
,T)to a state where q
s
≈
0 we obtain
ln
θ
θ
e
≈−
L
c
q
s
c
p
T
where θ
e
, the potential temperature in the final state, is approximately the
equivalent potential temperature defined earlier. Thus, θ
e
, for a saturated parcel
is given by
θ exp
L
c
q
s
c
p
T
θ
e
≈
(9.40)
The expression in (9.40) may also be used to compute θ
e
for an unsaturated parcel
provided that the temperature used in the formula is the temperature that the parcel
would have if expanded adiabatically to saturation (i.e., T
LCL
) and the saturation
mixing ratio is replaced by the
actual
mixing ratio of the initial state. Thus, equiv-
alent potential temperature is conserved for a parcel during both dry adiabatic and
pseudoadiabatic displacements.
An alternative to θ
e
, which is sometimes used in studies of convection, is the
moist static energy
, defined as h
≡
s
+
L
c
q, where s
≡
c
p
T
+
gz is the
dry static
energy
. It can be shown (Problem 9.7) that
c
p
Tdln θ
e
≈
dh
(9.41)
Hence, moist static energy is approximately conserved when θ
e
is conserved.
9.5.2
The Pseudoadiabatic Lapse Rate
The first law of thermodynamics (9.38) can be used to derive a formula for the rate
of change of temperature with respect to height for a saturated parcel undergoing
pseudoadiabatic ascent. Using the definition of θ (2.44), we can rewrite (9.38) for
vertical ascent as
d ln T
dz
R
c
p
d ln p
dz
L
c
c
p
T
dq
s
dz
−
=−
which upon noting that q
s
≡
q
s
(T,p) and applying the hydrostatic equation and
equation of state can be expressed as
∂q
s
∂T
ρg
∂q
s
∂p
dT
dz
+
g
c
p
=−
L
c
c
p
dT
dz
−
p
T
