V1003 Climate & Society
Due Nov. 15 in class.
Homework #4
Understanding the Greenhouse effect
Due date: Nov. 6th, in class. Show all calculations and do this on your own.
1. Effective Temperature. The temperature of a planet is determined by a balance between the
energy received from the Sun and the energy radiated back out to space (Ein = Eout). The
Earth’s effective temperature, the temperature the Earth should be without an atmosphere,
can be calculated knowing the solar energy input to the and the planet’s albedo. (Ch. 3 of
Archer’s book).
a) Calculate the Earth’s effective temperature (Tearth) in Kelvins based on the following
constants and assumptions:
Albedo () = 0.3
Isolar = 1365 Wm-2
Emissivity () = 1 (blackbody)
Stefan-Boltzmann constant (s) = 5.67 x 10-8 W m-2K-4
To solve this they need to set incoming = outgoing equations in figure below, and then rearrange
for Tearth, then put in the numbers above.
ANS = 254.8 K
2. Solar Variability. From the “Climate of the last Millennium” lecture we learned that the sun
is a variable star, with regular sunspot cycles roughly every 11 years and longer term cycles
that were (partly) responsible for the Little Ice Age cool period between 1500-1900 AD.
Using satellites, we’ve been able to measure solar variability for the last three solar cycles.
a) If we consider the just the last few decades when we have actual measurements of
solar irradiance changes, the solar cycle amplitude is roughly 1365 to 1367 Wm-2
(below) or 0.15% - quite small. What is the associated range in Earth effective
temperature from this solar variability in K?
To solve this they need to set incoming = outgoing equations in figure above, and then rearrange
for Tearth, then put in numbers for Isolar (1365 and 1367 W/m2), the difference in ∆Tearth sets the
“variation” = 0.1°C.
ANS = 254.9K, or 0.1°C variation
b) If we consider the whole ~400 year sunspot record, solar irradiance has probably
varied by a maximum of 0.5 % from the average of 1365 Wm-2. What is the associated
range in Earth effective temperature from this solar variability in K?
ANS = 255.0 K, or about 0.2°C variation
c) How do these estimates of solar irradiance changes on Earth surface temperature
compare to the observed magnitude of global warming from 1900 to the present?
What percent of total global warming can be explained by Solar variability alone?
ANS = global warming is +1.0°C from 1900-2007 based on figure above, solar variability can
only account for 0.1 to 0.2°C of this, or 10-20%.
d) Why do we think the global warming trend is NOT due to solar variability?
ANS = Solar irradiance goes up and down, global warming has a trend upward.
3. Volcanic eruptions. When Mt. Punatubo erupted in 1991 in the Philippines, the volcanic
dust and aerosols were injected into the stratosphere and were quickly mixed around the
globe. The increase in atmospheric aerosols increased the Earth’s reflectivity (albedo) from
0.30 to 0.31.
a) What impact would this have on The Earth’s effective temperature?
Here they need to calculate Tearth using the (1-) term, setting  to 0.30 initially and then setting
to 0.31. The ∆Tearth is the effect on albedo due to volcanic eruptions.
ANS = 253.9 K or -0.95 K – a big cooling!
b) Some geophysicists predict a big volcanic eruption in the next 10 years and that this
will reverse global warming. What do you say?
ANS = Eruptions cool climate but only briefly, for 1-3 years.
4. The Greenhouse Effect. To understand the impact of changes in the greenhouse gas
concentrations we have to makes things just a bit more complicated. Up to now, we’ve only
considered the Earth as a one-dimensional model with no atmosphere. Now we have to add in
an atmosphere to calculate the greenhouse effect.
This can get pretty complicated, but as the Archer book points out in chapter 3, the principles are
pretty easy to understand. Adding an atmosphere with CO2 and other greenhouse gasses (that
absorb and reradiate back to Earth some of the otherwise spacebound "Earthlight" (Iup, ground)) means that the surface will warm
if you ad more greenhouse gasses to the atmosphere.
Calculating just how much warmer is the complicated part.
Luckily, Archer has provided a (pretty sophisticated) online model
that will calculate this for you that is very easy to run. The model
determines the "Equilibrium near-surface air temperature" (i.e.
ground temperature) based on a number of parameters that can be
adjusted.
You can locate the model here:
http://forecast.uchicago.edu/Projects/full_spectrum.html
For this question, you'll just be changing the CO2 content, called the "CO2 mixing ratio", located
on the left hand panel of the model. The "Equilibrium near-surface air temperature" that is
calculated for a given CO2 value is calculated and displayed on the upper right hand panel (after
you press the "Do it!" button; you'll get a lot of plots and information here but just pay attention
to the "Equilibrium near-surface air temperature").
The purpose of this exercise is to determine how much the ground surface would warm with
changing CO2 (and CH4), and to compare this to the results you've calculated so far.
a) Use the model to calculate the ground temperature (in K) for 280 ppm CO2 and 0.6 ppm CH4
– these are the "pre-Industrial" concentrations. Now calculate the ground temperatures for any
three other CO2 levels up to about 500 ppm, leaving CH4 fixed. Plot up your results on the graph
paper below, labeling the axes.
295.5
y = 0.0114x +
289.17
295
CO2 (ppm)
180
250
280
350
385
Ground T (K)
294.5
291
292.1
292.5
293.3
293.7
R2 = 0.985
294
293.5
293
292.5
Ground
Temp.
(°C)
292
291.5
291
290.5
100
200
300
400
500
CO2 (ppm)
b) The current CO2 level is about 385 ppm. How much has the surface warmed according to this
model with these parameters since the Pre-Industrial period, and how does this compare with
observations?
ANS = (293.7-292.5 = +1.2K), warming since preindustrial period has been ~1.0°C (pretty close)
c) We haven't taken into account the current high methane (CH4) levels, which are now about 1.7
ppm. Enter this number now into the model and calculate the change in ground temp from
preindustrial conditions (CO2=285, CH4=0.6).
ANS = (294.3-292.5 = +1.8K warmer than pre-Industrial)
d) How much might climate warm in addition to your answer (c) 50 years into the future if CO2
emissions continue at the current 1.5 ppm/year increase (fixed 1.7 ppm CH4)?
ANS = (295.0 – 294.3 = +1.8 °C (for CO2 = 385+50*1.5=460 ppm)
e) Add answers c and d together and this would be the total temperature change from the
preindustrial period to a time 50 years into the future – this is one estimate for the future
warming. Do you think this is an underestimate, or an overestimate of the future warming, why?
Underestimate, because it doesn't take into account the positive feedbacks such as sea ice melting
which would lower albedo, raise temps.