V1003 Climate & Society
Due Nov. 17 in class.
Understanding the Greenhouse effect
Show all calculations and do this on your own.
Homework #5
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 (E in
= E out
). 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 (T earth
) in Kelvins based on the following constants and assumptions:
Albedo (

) = 0.3
I solar
= 1365 Wm -2
Emissivity (

) = 1 (blackbody)
Stefan-Boltzmann constant (s) = 5.67 x 10 -8 W m -2 K -4
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?

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? 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? d) Why do we think the global warming trend is NOT due to solar variability?
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?

b) Some geophysicists predict a big volcanic eruption in the next 10 years and that this will reverse global warming. What do you say?
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" (I up
, 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 CO
2
content, called the "CO
2
mixing ratio", located on the left hand panel of the model. The "Equilibrium near-surface air temperature" that is calculated for a given CO
2
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 CO
2
(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.

CO2 (ppm) Ground T (K)
100 200 300 400 500 b) The current CO
2
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? 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). 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)? 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?