Math 5421 Spring 2025
Introduction to Climate Models
Assignment 3
due January 30
Carbon Budget
The concept of a global carbon budget has been around for all of the 21st century, but it became a policy concept after the 2015 Paris Agreement, where the goal of the participating countries was to hold the global mean temperature (GMT) "to well below 2°C above pre-industrial levels" and to pursue "efforts to limit the temperature increase to 1.5°C above pre-industrial levels." Since the data indicate a nearly linear relation between the GMT and the cumulative worldwide carbon dioxide emissions, the goal of keeping the GMT increase to 1.5°C or 2.0°C translates to holding the total amount of emissions below a computable amount. That amount is called the carbon budget, and, unfortunately, each year that we continue to add carbon dioxide to the atmosphere through the use of fossil fuels, the remaining carbon budget decreases accordingly.
Another term that you may have heard is net zero. When applied to the planet, it means that, whatever carbon is emitted by fossil fuel burning, there is an equal amount being removed from the atmosphere. Presumably, if the planet reaches net zero emissions, the atmospheric CO2 will remain constant.
In the exercises below, we determine the carbon budget for 1.5°C, well-below 2.0°C, and 2.0°C.
Fossil Fuel Emissions
Emissions can be measured in two units: gigatonnes of carbon (GtC) or gigatonnes of carbon dioxide (GtCO2).
A tonne, also called a metric ton, is one thousand kilograms. A ton usually refers to an imperial ton, which is two thousand pounds. A tonne is about 1.1 tons.
A gigatonne is 109 tonnes, or 1012 kilograms, or 1015 grams, called a petagram.
A carbon dioxide molecule consists of one carbon atom and two oxygen atoms. Carbon has an atomic weight of 12, while oxygen has an atomic weight of 16. Therefore carbon dioxide has a molecular weight of 44. A tonne of carbon dioxide contains 12/44 ≈ 0.273 tonnes of carbon. Conversely, if one measures the emissions by GtC, then one can compute the weight of the carbon dioxide by multiplying by 44/12 ≈ 3.67.
Emitting carbon dioxide into the atmosphere results in an increase in the concentration of atmospheric carbon dioxide. The conversion factor is about 0.469, so emitting a GtC of carbon dioxide results in an increase in the atmospheric concentration of CO2 of 0.459 ppm. This can be written as 1 GtC = 0.459 ppm or 1 ppm = 2.13 GtC. Therefore, the mass of all the carbon in the all the carbon dioxide in the atmosphere if the concentration is 420 ppm is 2.13×420 = 895 gigatonnes. If you want the mass of all the carbon dioxide in the atmosphere, you have to multiply by the conversion factor 44/12, so we see that the total mass of carbon dioxide in the atmosphere if the concentration is 420 ppm is 3.67×895 = 3285 gigatonnes or 3.285 teratonnes.
The Global Carbon Project is an international organization devoted to solving the greenhouse gas problem. A part of the Global Carbon Project is the Global Carbon Budget, an organization with an office at the University of Exeter, housing the data hub containing the latest greenhouse gas data.
The latest data in Excel format can be downloaded from the page https://globalcarbonbudgetdata.org/latest-data.html by clicking on the link "Global Carbon Budget". You might want to do that for yourself in the future, but, to save time, I have downloaded the data and put the part of relevance to us into our familiar format.
https://www-users.cse.umn.edu/~mcgehee/Course/Math5421/data/Emissions_1959-2023.csv
The first column is the year, while the second column is the amount of carbon dioxide (measured in GtC) emitted globally by all the fossil fuel burning during that year.
Exercise 1
Using the data in the file linked above, draw two figures. The first should show the emissions for each year starting in 1959 and ending in 2023. The second should show the cumulative emissions for each year during that period. For example, the cumulative emissions in 2020 is the sum of the emissions for the years starting in 1959 and ending in 2000.
(a) For the first figure, use a linear trend and extend the axes out to 2050. If the trend continues, what do you expect the annual emissions to be in 2030? In 2050?
(b) For the second figure, used a quadratic trend and extend the axes out to 2050. If the trend continues, what do you expect the cumulative emissions to be in 2030? In 2050?
(c) Using your knowledge of the Fundamental Theorem of Calculus, justify the use of the quadratic trend in the second figure.
Atmospheric CO2 vs Emissions
Why do we think that the increase in atmospheric CO2 is caused by human activities, in particular, the burning of fossil fuels? As we see in the next exercise, the circumstantial evidence is pretty strong.
You could combine some of the data sets you have already downloaded in the previous exercises to produce an appropriate data set to see the evidence, but, to save time, I have constructed a csv file for you. Column 1 contains the year, column 2 the month, column 3 the digital date, column 4 the atmospheric CO2 for that month, and column 5 the cumulative emissions up to and including that month. Here is the file:
https://www-users.cse.umn.edu/~mcgehee/Course/Math5421/data/MLOCO2vsCumulEmiss.csv
Exercise 2
Using the data in the file linked above, produce a scatter plot with the cumulative emissions on the horizontal axis and the atmospheric CO2 on the vertical axis. Include in your figure the linear trend.
(a) Ask your favorite AI whether the increase in atmospheric CO2 is caused by human activity. Report the answer, and use your figure to either support or dispute the AI's answer.
(b) Using the slope of your trend line compute the percentage of emitted CO2 that has remained in the atmosphere since 1959.
(c) Assuming that the trend in your figure continues into the future, what will be the atmospheric CO2 if we emit another 100 GtC?
Global Mean Temperature vs Emissions
So far, we have seen that, when the cumulative emission go up, the atmospheric concentration of CO2 goes up, more or less linearly. In previous exercises, we have seen a linear relation between increasing atmospheric CO2 and global mean temperature. We should therefore expect to see a linear relation between between the global mean temperature and the cumulative emissions.
I have prepared a csv file with all three quantities that we have been discussing: cumulative emissions (column 4), atmospheric CO2 concentration (column 5), and global mean temperature anomaly (column 6). The first three columns are year, month, and digital date. I have included only the twenty-first century data up to and including 2023..
https://www-users.cse.umn.edu/~mcgehee/Course/Math5421/data/AllTogetherNow.csv
Exercise 3
Using the data in the file just described, produce a scatter plot with the cumulative emissions on the horizontal axis and the GMT anomaly on the vertical axis. Include in your figure the linear trend.
(a) Using the slope of the trendline, estimate how much more CO2 can be emitted before the temperature anomaly reaches 1.5°C. Same estimates for 1.7°C and 2.0°C.
For the following questions, assume that, starting in 2024, you owed all the carbon emitting industries on the planet and that you have complete control of what they do.
(b) If, starting in 2024 you held all carbon emissions to 10 GtC/year, when would the GMT anomaly reach 1.5°C? When would it reach 2.0°C?
(c) If, starting in 2024, you held emissions to 10 GtC, then reduced the emissions by 2 GtC each year until you reached net zero in 2029. What would the temperature anomaly be in 2029?
(d) Same as (c), except that you reduce the emissions by 1 GtC each year until you reach net zero in 2034.
(e) Same scenario as (c) and (d), except that you reduce the emissions by x GtC each year. What value of x will produce a temperature anomaly of 1.5°C when net zero is reached? Same question for an anomaly of 2.0°C.
Instructions for all Three Exercises
Transfer your figures to a Google Docs document. Locate the answers to your questions near the corresponding graph. Share your document with mcgehee@umn.edu.