Geography Reference
In-Depth Information
9.5.1
Equivalent Potential Temperature
We have previously applied the parcel method to discuss the vertical stability of a
dry atmosphere. We found that the stability of a dry parcel with respect to a vertical
displacement depends on the lapse rate of potential temperature in the environment
such that a parcel displacement is stable, provided that ∂θ/∂z > 0 (i.e., the actual
lapse rate is less than the adiabatic lapse rate). The same condition also applies to
parcels in a moist atmosphere when the relative humidity is less than 100%. If,
however, a parcel of moist air is forced to rise, it will eventually become saturated
at a level called the lifting condensation level (LCL). A further forced rise will
then cause condensation and latent heat release, and the parcel will then cool at
the saturated adiabatic lapse rate. If the environmental lapse rate is greater than the
saturated adiabatic lapse rate, and the parcel is forced to continue to rise, it will
reach a level at which it becomes buoyant relative to its surroundings. It can then
freely accelerate upward. The level at which this occurs is called the level of free
convection (LFC).
Discussion of parcel dynamics in a moist atmosphere is facilitated by defining
a thermodynamic field called the
equivalent potential temperature
. Equivalent
potential temperature, designated by θ
e
, is the potential temperature that a parcel
of air would have if all its moisture were condensed and the resultant latent heat
used to warm the parcel. The temperature of an air parcel can be brought to its
equivalent potential value by raising the parcel from its original level until all
the water vapor in the parcel has condensed and fallen out and then compressing
the parcel adiabatically to a pressure of 1000 hPa. Because the condensed water
is assumed to fall out, the temperature increase during the compression will be
at the dry adiabatic rate and the parcel will arrive back at its original level with
a temperature that is higher than its original temperature. Thus, the process is
irreversible. Ascent of this type, in which all condensation products are assumed
to fall out, is called
pseudoadiabatic
ascent. (It is not a truly adiabatic process
because the liquid water that falls out carries a small amount of heat with it.)
A complete derivation of the mathematical expression relating θ
e
to the other
variables of state is rather involved and will be relegated to Appendix D. For most
purposes, however, it is sufficient to use an approximate expression for θ
e
that can
be immediately derived from the entropy form of the first law of thermodynamics
(2.46). If we let q
s
denote the mass of water vapor per unit mass of dry air in a
saturated parcel (q
s
is called the saturation mixing ratio), then the rate of diabatic
heating per unit mass is
L
c
Dq
s
Dt
where L
c
is the latent heat of condensation. Thus, from the first law of thermody-
namics
=−
J
D ln θ
Dt
L
c
T
Dq
s
Dt
c
p
=−
(9.38)
