Geoscience Reference
In-Depth Information
The equation now has a buoyancy term, the second from the last on the right side.
The earth's surface can be warmer than the air above (as over land on clear days,
due to surface absorption of solar radiation) or cooler (as on clear nights, through
surface emission). The surface temperature of a body of water also typically differs
from that of the air somewhat above. This causes heat transfer between the surface
and the air, which in turn causes the air temperature profile near the surface to
differ from
T
0
(z)
. If an air parcel at height
z
is warmer than
T
0
(z),
its temperature
deviation
T
is positive and according to
Eq. (8.18)
it feels an upward buoyancy
force - and so “warm air rises.” If a parcel is cooler than its local
T
0
,
T
is negative
and it feels a downward buoyancy force. A parcel with
T
=
0is
neutrally
buoyant.
It is convenient to use a temperature variable that is constant during isentropic dis-
placements in the atmosphere, like ordinary temperature in laboratory-scale flows.
For this purpose the
potential temperature θ
is defined through the specific entropy
equation (8.3)
rewritten with the ideal gas law,
c
p
T
c
p
θ
Ds
Dt
=
DT
Dt
−
R
d
p
Dp
Dt
=
Dθ
Dt
.
(8.19)
Thus
θ
is conserved in an isentropic displacement:
Ds
Dt
=
c
p
θ
Dθ
Dt
=
0
.
(8.20)
The solution of
Eq. (8.19)
is
−
R
d
c
p
θ(t)
=
C
[
p(t)
]
T(t),
(8.21)
with
C
a constant.
Scaling arguments indicate that the dominant pressure changes along a parcel
trajectory in the turbulent lower atmosphere are caused by vertical displacements in
the background vertical pressure gradient, rather than turbulent pressure fluctuations
(Problem 8.23)
. Thus we follow the convention of interpreting the independent
variable in
Eq. (8.21)
as
z
, distance from the surface, rather than time, and we
choose the constant
C
as
R
d
c
p
.Thisgives
[
p(
0
)
]
T(z)
p(
0
)
p(z)
R
d
c
p
θ(z)
=
.
(8.22)
In this convention the potential temperature at a height
z
is the temperature that a
parcel originating there would have after it traveled isentropically to the surface.
