Homework #1
EPS 208: Physics of Climate
Due, Sep 30
1. Planetary temperature (10 points). Based on the following table, compute the
emission temperatures for Venus, Earth, and Mars. Compare the results with the actual
mean surface temperatures of these planets. How large are the greenhouse effects on
these planets?
Distance to the Sun Albedo
Mean surface
(AU)
temperature
Venus
0.72
0.77
750
Earth
1.00
0.30
288
Mars
1.52
0.15
218
2. Ice-albedo feedback (10 points). Take the leaky Greenhouse model talked about in
class, and use an emissivity of 0.75 for the atmosphere. Compute the surface temperature.
Now if we consider the planetary albedo as a function of mean surface temperature:
0.3
Ts > 0C
⎧
⎪
0C − Ts ⎞
⎪ ⎛
α = ⎨0.3 ⎜ 1 +
⎟⎠ −10C ≤ Ts ≤ 0C
⎝
10C
⎪
Ts < −10C
⎪⎩0.6
This is meant to crudely represent the transition from an ice-free to an ice-covered planet
as temperature decreases. Now gradually decrease the solar constant from its current
value towards 70% of its current value (thought to be the value early in the solar system
history) and compute Ts. Then gradually increase the solar constant towards 130% of its
current value and compute Ts. What do you see?
3. The carbon dioxide greenhouse (10 points). The following figure shows the outgoing
radiance from Earth as measured from an orbiting satellite. The energy distribution has
some resemblance to a black body at the surface temperature of the earth, Tg, but for the
strong band of carbon dioxide centered on 667cm-1 (for a rough approximation we may
ignore the ozone band at 1042cm-1). Emission in the center of the carbon dioxide band is
only about a quarter that would occur without the gas. Estimate the greenhouse effect
caused by carbon dioxide with the following simplifying assumptions. Assume that the
only atmospheric absorber is the 667cm-1 band and that it occupies the frequency range
600 to 725 cm-1. Assume that this band is represented by a rectangle and that the two
sections of the spectrum on either side can be represented by triangles with bases 50 to
600 cm-1 and 725 to 1500 cm-1. The object is to calculate the present ground temperature
and the temperature that would occur if the carbon dioxide were removed but if all else
were the same.
4. Radiative equilibrium (30 points). Use the radiative transfer code package attached
to this homework to examine radiative equilibrium. I have provided an example of how
to compute radiative equilibrium in radiative_equilibrium.m. You may need to modify
the makefile to use the correct Fortran compiler on your machine. Other than this, you
should be able to run the matlab code straight out of the box.
(a) Use the Galapagos Island sounding, and call the rad_bugs function, which returns the
longwave and shortwave heating rates in K/day and fluxes in W/m2. How much does the
surface receive in terms of longwave radiation from the atmosphere and absorbed
shortwave radiation?
(b) Compute the radiative heat rates due to CO2, H2O, O3 separately by setting all other
gases to very small amounts (but not zero, otherwise the code will not work…), and plot
them. Is the sum of the individual contributions the same as the results when you include
all the gases in the calculation? Why?
(c) Plot the O3 volume mixing ratio and partial pressure as a function of pressure and
verify that the shortwave heating due to ozone peaks above the ozone volume mixing
ratio peak. Estimate the scale height for Earth’s atmosphere, assuming a constant
temperature of 300K. For a surface pressure of 980hPa, estimate the heights of the ozone
volume mixing ratio peak and the ozone partial pressure peak.
(d) Now take away CO2, H2O, O3 one at a time and look at changes in the net radiative
flux at the tropopause level (about 100hPa). This gives us the radiative forcing1. Rank the
three gases in terms of their radiative forcing. Now, keep H2O, O3 the same as from the
Galapagos Island sounding, but double the CO2, and compute the radiative forcing of
doubling the amount of CO2.
(e) Compute the radiative-equilibrium profile (including the sea surface temperature)
using the example code (it might take a little while depending on your computer).
(f) Re-compute the radiative-equilibrium profile with doubled CO2 mixing ratio.
Compare this to what you get from (e). Interpret the main features that you see (hopefully
without consulting the lecture notes…).
(g) Try other climate changes that you might think of (e.g. changes in O3, solar constant)
and have fun.
Note that in reality, water vapor concentration cannot be viewed as given and the
atmosphere, in particular the troposphere, is not in radiative equilibrium. Nonetheless, the
above excise is a good first step… The radiation code is taken from a general circulation
model called BUGS developed at Colorado State University, but with a bug fix (trust no
one’s code!). The Matlab wrapper is based on one written by Dr. Peter Caldwell.
1
The reason that radiative forcing is defined at the tropopause level is because the
troposphere is assumed to adjust as a whole so changes in the net flux at this level affects
the whole troposphere.