Geoscience Reference
In-Depth Information
5.6 EXERCISES
5.1.
(a) Derive Eq. 5.3 from Eq. 5.2. Start by taking the total time derivative
of the idea gas law, and then use Eq. 5.2 to eliminate a. Note that
ccR
p
=+.
(b) Derive Eq. 5.5.
5.2. Show that potential temperature is conserved under adiabatic conditions.
Hint
: Show that Eq. 5.7 is equivalent to Eq. 5.3 with
Q
0.
5.3. Is the planetary albedo equal to the sum of the surface albedo and the
atmospheric albedo? Explain your answer, perhaps with a diagram.
5.4. Instantaneous measurements over a layer of water 5 cm thick are as
follows:
incident solar radiation 250 W/m
2
water temperature 287 K
atmospheric back radiation 275 W/m
2
sensible heat flux 30 W/m
2
latent heat flux 70 W/m
2
The sun is overhead. Is the water heating up or cooling down? At what rate
is its temperature changing? After 10 minutes, do you expect that the rate
of change of temperature of the water will be larger or smaller? Explain.
5.5. Comparison of thermal emission and the sensible heat flux at the earth's
surface.
(a) Use a Taylor series expansion to show that the blackbody emission
from a surface can be written
FT T
==+
σ
4
σ
4
4(
σ
T TT
3
−
) ..,
+
s
0
0
s
0
where
T
0
is some reference temperature. (Write out at least four terms
on the right-hand side of the above equation.)
(b) Show that if
T
0
and
T
S
are only a few degrees apart, that is, if you
choose the reference temperature
T
0
to be close to the surface
temperature
T
S
, then retaining only two terms in the Taylor expansion
is a reasonable approximation.
(c) Write the bulk aerodynamic formula for the sensible heat flux. Compare
the form of this equation with your approximation for blackbody
emission. (For example, what if you choose
TT
0
in the blackbody
emission temperature formula?)
(d) Derive equations that express the sensitivity of longwave and sensible
heat fluxes to changes in surface temperature. [
Hint
: The sensitivity of a
flux
F
to
T
S
is expressed mathematically as
22
]
(e) Which is more sensitive to changes in surface temperature—longwave
or sensible heat fluxes?
FT
S
.
