Geography Reference
In-Depth Information
mixture of air and vapor by an amount of heat that goes into the liquid water, the
saturated parcel must satisfy the energy equation
R
∗
m
v
d (p
−
e
s
)
de
s
e
s
c
p
dT
+
q
s
c
pv
dT
−
RT
−
q
s
T
=−
L
c
dq
s
(D.3)
p
−
e
s
where q
s
and e
s
are the saturation mixing ratio and vapor pressure, respectively.
The quantity de
s
/e
s
may be expressed in terms of temperature using the Clausius-
Clapeyron equation
1
de
s
dT
=
m
v
L
c
e
s
R
∗
T
2
(D.4)
Substituting from (D.4) into (D.3) and rearranging terms we obtain
L
c
d
q
s
T
dT
T
Rd (p
−
e
s
)
dT
T
−
=
c
p
−
+
q
s
c
pv
(D.5)
p
−
e
s
If we now define the potential temperature of the dry air θ
d
, according to
c
p
d ln θ
d
=
c
p
d ln T
−
Rd ln (p
−
e
s
)
we can rewrite (D.5) as
L
c
d
q
s
T
−
=
c
p
d ln θ
d
+
q
s
c
pv
d ln T
(D.6)
However, it may be shown that
dL
c
/dT
=
c
pv
−
c
w
(D.7)
where c
w
is the specific heat of liquid water. Combining (D.7) and (D.6) to eliminate
c
pv
yields
d
L
c
q
s
T
−
=
c
p
d ln θ
d
+
q
s
c
w
d ln T
(D.8)
Neglecting the last term in (D.8) we may integrate from the originial state (p, T ,
q
s
, e
s
, θ
d
) to a state where q
s
→
0. Therefore, the equivalent potential temperature
of a saturated parcel is given by
θ
d
exp
L
c
q
s
/c
p
T
≈
θ exp
L
c
q
s
/c
p
T
θ
e
=
(D.9)
Equation (D. 9) may also be applied to an unsaturated parcel provided that the tem-
perature used is the temperature that the parcel would have if brought to saturation
by an adiabatic expansion.
1
For a derivation, see Curry and Webster (1999, p. 108)
