2.4 Adiabatic Processes 71
and isochoric processes describe a family of
isochores
(v = const), which are
superposed in Fig. 2.8. Just as each of the foregoing paths is characterized by
invariance of a certain state variable, so too are adiabats.
2.4.1 Potential Temperature
Poisson's relation between pressure and temperature motivates the introduc-
tion of a new state variable that is preserved during an adiabatic process. The
potential temperature 0
is defined as that temperature assumed by the system
when compressed or expanded adiabatically to a reference pressure of P0 =
1000 mb. According to (2.30.2), an adiabatic process from the state (p, T) to
the reference state (P0, 0) satisfies
Opo~= rp-~.
Hence, the potential temperature is described by
0
~= P . (2.31)
A function of pressure and temperature, 0 is a state variable. According to
(2.31) and Poisson's relation (2.30.2), 0 is invariant along an adiabatic path in
state space.
Adiabatic behavior of individual air parcels is a good approximation for
many atmospheric applications. Above the boundary layer and outside of
clouds, the timescale of heat transfer is of order two weeks and thus long
compared to the characteristic timescale of displacements, which influence an
air parcel through changes of pressure and expansion work. For instance, ver-
tical displacements of air and accompanying changes of pressure and volume
occur in cumulus convection on a timescale of minutes to hours. Even in mo-
tions of large horizontal dimension (e.g., in sloping convection associated with
synoptic disturbances), air displacements occur on a characteristic timescale
of order one day. Thus, over a fairly wide range of motions, the timescale for
an air parcel to adjust to changes of pressure and to perform expansion work
is short compared to the characteristic timescale of heat transfer.
Under these circumstances, the potential temperature of individual air
parcels is approximately conserved. An air parcel descending to greater pres-
sure experiences an increase of temperature according to (2.30.2) due to com-
pression work performed on it, but in such proportion to its increase of pres-
sure as to preserve the parcel's potential temperature through (2.31). Similar
considerations apply to an air parcel that is ascending. It follows that, un-
der adiabatic conditions, 0 is a conserved quantity and therefore behaves as
a tracer of air motion. On timescales for which individual parcels can be re-
garded as adiabatic, particular values of 0 track the movement of those bodies
of air. Conversely, a collection of air parcels that has a particular value of 0