Problem Set 1: Build and Analyze a Simple Climate Model in EXCEL

Models are ubiquitous in the analysis of environmental policy. Understanding what they are and how to evaluate them is critical. In this problem set you will use Excel to build the climate model used in the DICE economic model created by Bill Nordhaus.1 This is a real model used by researchers and by the U.S. government to analyze climate and energy policy.

If you wish, you can work in pairs to problem set.

Due Date: Tuesday, October 10th. Upload an Excel and Word document to Canvas. Make sure you answer all the questions except question 2 in the Word document, including all graphs. If you work in pairs, you should both upload the Word and Excel files, but clearly put both your names on the assignment.

A Simple Model of the Climate System

Understanding how human emissions of greenhouse gases affect global temperature involves several steps.

CO2 1 Atmospheric 2 Radiative 3 Global Emissions CO2 Forcing Temperature

Step 1

This describes how the CO2 that we emit causes changes in the mass of CO2 in the atmosphere. Not all our CO2 emissions stay in the atmosphere, some are taken up by the ocean. We keep track of CO2 in three different parts of the Earth system: the atmosphere, the upper ocean, and the lower ocean.

Where are CO2 emissions at time period t, is the mass of carbon in reservoir i at time period t, is the mass of carbon in reservoir i in the previous time period1, and are a set of constants that describe how quickly carbon moves between different parts of the earth system.

1 Time periods in the model are 5 years so t-1 refers to the value 5 years previously

Step 2

The second step describes how increased CO2 concentrations disrupt the planet's energy balance by trapping infra-red radiation. This is a quantity called radiative forcing, which is an energy flux measured in W/m2.

2 (1750)

Where is the radiative forcing at time period t, is the same as in Step 1, (1750) is the mass of carbon in the atmosphere in 1750 and is a constant that describes how effective CO2 is at trapping energy.

Step 3

The final step describes how the disruption to the Earth's energy imbalance by the addition of greenhouse gases causes the global temperature to change. We keep track of two temperatures, the surface atmospheric temperature, which is primarily what we care about, and the ocean temperature.

Where is the temperature at time period t, is the temperature at the previous time

period, is the same as in Step 2, and are constants describing how quickly the temperature of

the atmosphere and ocean adjusts.

Questions (Total - 15 Points)

1. Draw a full system diagram that includes all the relationships given in steps 1, 2, and 3 above. Label the direction of all relationships and identify any feedback loops and determine if they are positive or negative. What variables are exogenous? What variables are endogenous? (2.5 points)
2. The Excel file on Canvas includes all parameters and initial values needed for the model, as well as CO2 emissions for 1990-2100. Use the formulas above to complete the Climate Model worksheet by adding in the equations above for (0.5 points each):
3. Mass of carbon in the atmosphere (1995 - 2100)
4. Mas of carbon in the upper ocean (1995 - 2100)
5. Mass of carbon in the lower ocean (1995 - 2100)
6. Radiative forcing from CO2 (1990 - 2100) - HINT: Make sure to specify in the logarithm
7. formula that the base is 2, not the default value.
8. Atmospheric temperature change (1995 - 2100)
9. Ocean temperature change (1995 - 2100)
10. a. Create a plot showing atmospheric temperature change and ocean temperature change between 1990 and 2100. Make sure the axes are labelled correctly and that you include a legend and a title. (2 points)
11. b. 1.5 of warming and 2 of warming above pre-industrial are both discussed as limits that
12. global climate policy should aim to avoid. When does atmospheric temperature pass 1.5?
13. What about 2? (0.5 points)
14. c. What do you notice about the difference between warming in the atmosphere and ocean?
15. What do you think is going on? (0.5 points)
16. We don't know what carbon emissions will be between 2015 and 2100, which means we should explore a range of possibilities. The worksheet "Emissions Scenarios" in the Excel file has three possible sets of emissions for 2015 to 2100. Right now the "Medium Emissions" scenario is used in the Climate Model worksheet and the actual observed emissions are given from 1990-2010. Make a single plot showing how atmospheric temperature changes between 1990 and 2100 with the three different emissions scenarios. (2.5 points)
17. It is important to test models against observations to evaluate performance. In the "Observed Temperatures" worksheet you will find actual global temperature data for 1990-2015.
18. Make a plot of the model error (difference between model output and observations). (1 point)
19. Calculate the root mean squared error (RMSE) of the model. Make sure to give the correct units. (1 point)
20. One reason that the model output is different from observations is that emissions of aerosols have had a cooling effect on the climate over this time period, off-setting some of the warming from CO2. A model of forcing that includes this effect is given by:

Where is the aerosol forcing in time t. Aerosol forcing for time t is given in the "Aerosol Forcing" sheet of the Excel workbook. Incorporate this into the model and recalculate the RMSE of the model temperature. Has this change improved or worsened the model? (2 points)

Problem Set 1: Build and Analyze a Simple Climate Model in EXCEL Models are ubiquitous in the analysis of environmental policy. Understanding what they are and how to evaluate them is critical. In this problem set you will use Excel to build the climate model used in the DICE economic model created by Bill Nordhaus.1 This is a real model used by researchers and by the U.S. government to analyze climate and energy policy. If you wish, you can work in pairs to complete this problem set. Due Date: Tuesday, October 10th. Upload an Excel and Word document to Canvas. Make sure you answer all the questions except question 2 in the Word document, including all graphs. If you work in pairs, you should both upload the Word and Excel files, but clearly put both your names on the assignment. A Simple Model of the Climate System Understanding how human emissions of greenhouse gases affect global temperature involves several steps. CO2 Emissions 1 Atmospheric CO2 2 Radiative Forcing 3 Global Temperature Step 1 This describes how the CO2 that we emit causes changes in the mass of CO2 in the atmosphere. Not all our CO2 emissions stay in the atmosphere, some are taken up by the ocean. We keep track of CO2 in three different parts of the Earth system: the atmosphere, the upper ocean, and the lower ocean. () = () + 11 ( 1) + 12 ( 1) () = 21 ( 1) + 22 ( 1) + 23 ( 1) () = 32 ( 1) + 33 ( 1) Where () are CO2 emissions at time period t, () is the mass of carbon in reservoir i at time period t, ( 1) is the mass of carbon in reservoir i in the previous time period1, and are a set of constants that describe how quickly carbon moves between different parts of the earth system. 1 Time periods in the model are 5 years so t-1 refers to the value 5 years previously Step 2 The second step describes how increased CO2 concentrations disrupt the planet's energy balance by trapping infra-red radiation. This is a quantity called radiative forcing, which is an energy flux measured in W/m2. () = 2 () (1750) Where () is the radiative forcing at time period t, () is the same as in Step 1, (1750) is the mass of carbon in the atmosphere in 1750 and is a constant that describes how effective CO2 is at trapping energy. Step 3 The final step describes how the disruption to the Earth's energy imbalance by the addition of greenhouse gases causes the global temperature to change. We keep track of two temperatures, the surface atmospheric temperature, which is primarily what we care about, and the ocean temperature. () = ( 1) + 1 [() 2 ( 1) 3 ( ( 1) ( 1))] () = ( 1) + 4 [ ( 1) ( 1)] Where () is the temperature at time period t, ( 1) is the temperature at the previous time period, () is the same as in Step 2, and are constants describing how quickly the temperature of the atmosphere and ocean adjusts. Questions (Total - 15 Points) 1. Draw a full system diagram that includes all the relationships given in steps 1, 2, and 3 above. Label the direction of all relationships and identify any feedback loops and determine if they are positive or negative. What variables are exogenous? What variables are endogenous? (2.5 points) 2. The Excel file on Canvas includes all parameters and initial values needed for the model, as well as CO2 emissions for 1990-2100. Use the formulas above to complete the Climate Model worksheet by adding in the equations above for (0.5 points each): a. Mass of carbon in the atmosphere (1995 - 2100) b. Mas of carbon in the upper ocean (1995 - 2100) c. Mass of carbon in the lower ocean (1995 - 2100) d. Radiative forcing from CO2 (1990 - 2100) - HINT: Make sure to specify in the logarithm formula that the base is 2, not the default value. e. Atmospheric temperature change (1995 - 2100) f. Ocean temperature change (1995 - 2100) 3. a. Create a plot showing atmospheric temperature change and ocean temperature change between 1990 and 2100. Make sure the axes are labelled correctly and that you include a legend and a title. (2 points) b. 1.5 of warming and 2 of warming above pre-industrial are both discussed as limits that global climate policy should aim to avoid. When does atmospheric temperature pass 1.5? What about 2? (0.5 points) c. What do you notice about the difference between warming in the atmosphere and ocean? What do you think is going on? (0.5 points) 4. We don't know what carbon emissions will be between 2015 and 2100, which means we should explore a range of possibilities. The worksheet \"Emissions Scenarios\" in the Excel file has three possible sets of emissions for 2015 to 2100. Right now the \"Medium Emissions\" scenario is used in the Climate Model worksheet and the actual observed emissions are given from 1990-2010. Make a single plot showing how atmospheric temperature changes between 1990 and 2100 with the three different emissions scenarios. (2.5 points) 5. It is important to test models against observations to evaluate performance. In the \"Observed Temperatures\" worksheet you will find actual global temperature data for 1990-2015. a. Make a plot of the model error (difference between model output and observations). (1 point) b. Calculate the root mean squared error (RMSE) of the model. Make sure to give the correct units. (1 point) c. One reason that the model output is different from observations is that emissions of aerosols have had a cooling effect on the climate over this time period, off-setting some of the warming from CO2. A model of forcing that includes this effect is given by: () () = 2 + () (1750) Where () is the aerosol forcing in time t. Aerosol forcing for time t is given in the \"Aerosol Forcing\" sheet of the Excel workbook. Incorporate this into the model and recalculate the RMSE of the model temperature. Has this change improved or worsened the model? (2 points) Year: 1990 1995 2000 Emissions (GtC per 5 years): 29.406 30.937 32.937 Carbon Cycle Mass C Atm (GtC): Mass C Upper Ocean (GtC): Mass C Lower Ocean (GtC): 775 1478 10010 Radiative Forcing Radiative Forcing from CO2 (W per m2): Climate Response Atmospheric Temperature Change (degrees C above pre-industrial): Ocean Temperature Change (degrees C above pre-industrial): Parameters 11 12 21 22 23 32 33 Matm(1750) 1 2 3 4 0.4 0.06 0.912 0.038 0.088 0.96 0.00034 0.0025 0.9997 3.8 588 0.104 0.84 0.088 0.025 2005 2010 2015 2020 2025 2030 2035 36.959 43.311 46.864 46.784 46.9865 47.190 50.6955 2040 2045 2050 54.202 58.55075 62.900 2055 67.865 2060 2065 2070 2075 72.830 77.60675 82.384 85.0035 2080 2085 87.624 80.20075 2090 2095 2100 72.778 71.22575 69.674 Aersol Forcing (Wm-2) 1990 -0.2 1995 -0.2 2000 -0.19 2005 -0.17 2010 -0.17 2015 -0.16 Observed Atmospheric Surface Temperature (degrees C above pre-industrial): 1990 0.44 1995 0.46 2000 0.42 2005 0.69 2010 0.72 2015 0.87 Low Emissions (GtC per 5 years): 2015 49.326 51.062 2025 53.455 55.848 2035 56.7655 57.683 Medium Emissions (GtC per 5 years): 46.864 46.784 46.9865 47.190 50.6955 54.202 56.0335 62.220 67.495 72.770 79.964 87.158 High Emissions (GtC per 5 years): 2020 2030 2040 2045 57.04125 2075 2080 56.400 52.16275 47.926 42.01775 36.110 28.52975 20.950 58.55075 62.900 67.865 72.830 77.60675 82.384 103.903 112.1945 120.486 126.1785 95.5305 2050 2055 2060 2065 2070 2085 21.024 21.098 2090 85.0035 87.624 80.20075 72.778 131.871 135.2233 138.576 140.6145 142.654 2095 21.17225 21.247 2100 71.22575 69.674 143.3695 144.086