Geography Reference
In-Depth Information
Substituting for pDα/Dt in (2.41) and using c
p
=
1004 J
kg
−
1
K
−
1
) is the specific heat at constant pressure, we can rewrite the first law of
thermodynamics as
c
v
+
R, where c
p
(
=
c
p
DT
α
Dp
Dt
−
Dt
=
J
(2.42)
Dividing through by T and again using the equation of state, we obtain the entropy
form of the first law of thermodynamics:
D ln T
Dt
R
D ln p
Dt
J
T
≡
Ds
Dt
c
p
−
=
(2.43)
Equation (2.43) gives the rate of change of entropy per unit mass following the
motion for a thermodynamically
reversible
process. A reversible process is one in
which a system changes its thermodynamic state and then returns to the original
state without changing its surroundings. For such a process the entropy s defined
by (2.43) is a field variable that depends only on the state of the fluid. Thus
Ds
is
a perfect differential, and Ds/Dt is to be regarded as a total derivative. However,
“heat” is not a field variable, so that the heating rate J is not a total derivative.
2
2.7.1
Potential Temperature
For an ideal gas undergoing an
adiabatic
process (i.e., a reversible process in which
no heat is exchanged with the surroundings), the first law of thermodynamics can
be written in differential form as
D
c
p
ln T
R ln p
=
c
p
D ln T
−
RD ln p
=
−
0
Integrating this expression from a state at pressure p and temperature T to a state
in which the pressure is p
s
and the temperature is θ , we obtain after taking the
antilogarithm
T (p
s
/p)
R/c
p
θ
=
(2.44)
This relationship is referred to as Poisson's equation, and the temperature θ defined
by (2.44) is called the
potential temperature
. θ is simply the temperature that a
parcel of dry air at pressure p and temperature T would have if it were expanded or
compressed adiabatically to a standard pressure p
s
(usually taken to be 1000 hPa).
Thus, every air parcel has a unique value of potential temperature, and this value
is conserved for dry adiabatic motion. Because synoptic scale motions are approx-
imately adiabatic outside regions of active precipitation, θ is a quasi-conserved
quantity for such motions.
2
For a discussion of entropy and its role in the second law of thermodynamics, see Curry and
Webster (1999), for example.
