Geography Reference
In-Depth Information
Taking the logarithm of (2.44) and differentiating, we find that
D ln θ
Dt
D ln T
Dt
R
D ln p
Dt
c
p
=
c
p
−
(2.45)
Comparing (2.43) and (2.45), we obtain
D ln θ
Dt
J
T
=
Ds
Dt
c
p
=
(2.46)
Thus, for reversible processes, fractional potential temperature changes are indeed
proportional to entropy changes. A parcel that conserves entropy following the
motion must move along an isentropic (constant θ ) surface.
2.7.2 The Adiabatic Lapse Rate
A relationship between the
lapse rate
of temperature (i.e., the rate of
decrease
of
temperature with respect to height) and the rate of change of potential tempera-
ture with respect to height can be obtained by taking the logarithm of (2.44) and
differentiating with respect to height. Using the hydrostatic equation and the ideal
gas law to simplify the result gives
T
θ
∂θ
∂z
=
∂T
∂z
+
g
c
p
(2.47)
For an atmosphere in which the potential temperature is constant with respect to
height, the lapse rate is thus
dT
dz
=
g
c
p
≡
−
d
(2.48)
Hence, the dry adiabatic lapse rate is approximately constant throughout the lower
atmosphere.
2.7.3
Static Stability
If potential temperature is a function of height, the atmospheric lapse rate,
≡
−
∂T/∂z, will differ from the adiabatic lapse rate and
T
θ
∂θ
∂z
=
d
−
(2.49)
If <
d
so that θ increases with height, an air parcel that undergoes an adiabatic
displacement from its equilibrium level will be positively buoyant when displaced
