Simple Climate Model

Overview

Simple climate models can be used to teach a variety of topics, including the utility of models, uncertainty in climate projections, and how past temperature changes can help estimate future warming. This lab uses a basic climate model that can be run with Excel. For this lesson plan, we focus on providing students experience testing climate models, making projections of 21st century warming, and comparing their projections with that of their peers’. Students are divided into groups of two or three, each group representing a distinct modeling group. Each group will provide a 21st century temperature projection based on their model that is first tuned to 20th century warming. The end product will be a graph of projected 21st century warming based on their models, much like the graphs found in Intergovernmental Panel on Climate Change (IPCC) reports. This will give the students a firsthand experience making climate projections and show students how climate modelers can get uncertainty in projected temperatures of the future, even when they use the same historical observations to constrain the model parameters.

Grade level: 9-12

Lesson Time: 2 x 50 minute class periods

Focus Questions

  • What is a model? Why do we need global climate models?
  • What is one way to test a model’s performance? (How can 20th century temperature records be used to test a model’s behavior?)
  • Why do the wiggles exist in the temperature records and what determines how wiggly they are?
  • Where do uncertainties in climate projections come from?
  • Why has the temperature been rising in the last hundred years?
  • What is climate sensitivity and what aspect of the Earth’s mean temperature does it control?

Learning Goals

  • To look at Earths’ temperature record and understand the changes in the temperature in terms of simple forcing (anthropogenic and natural), climate sensitivity (feedback), and heat storage.
  • To understand why scientists would be interested in estimating the Earth’s climate sensitivity.  Students should see that depending on the value of the climate sensitivity, the warming we get in the next 100 years would substantially change.
  • Use the IPCC temperature projections for the 21st century and explain:
    • Why the SSP585 scenario projections show more warming in 2100 than the SSP126 scenario projections.
      • Explanation: The SSP585 projections are based on emission forecasts of more greenhouse gases emissions over the course of the 21st century. It is a scenario where greenhouse gas emissions are not curbed.
    • Why, for each emission scenario, there is a range of temperatures predicted for the year 2100.
      • Explanation: Multiple models are used to make temperature projections, and each model is slightly different from the other. Even if one uses the same model and tests the model on the same 20th century temperature record, one can still come up with a range of temperatures for 2100. Therefore, this represents an inherent uncertainty in our projections for the future. Even if we knew future emissions of greenhouse gases, there will still be some uncertainty to the amount of warming, given our simplified understanding of the climate in the Excel model.

Prior Knowledge and Learning Assets

  • Concept of energy budget and conservation
  • Knowledge of what models are
  • Intergovernmental Panel on Climate Change (IPCC), climate models, and how they are being used to inform policy or understand the climate system
  • Some familiarity with Excel spreadsheets

Anticipated Challenges

  • Learning to use the Excel spreadsheet
  • Understanding why we need models, why we need climate models
  • Understanding what the different components of the model represent
  • Understanding the difference between uncertainty in understanding of the physical processes that affect Earth’s climate and uncertainty in future anthropogenic emissions of greenhouse gases and aerosols
  •  Conveying what we know about future warming without getting bogged down with what we don’t know

Background Information for Teachers

Understanding the energy budget at the Earth’s surface can take us a long way to understand recent changes in global mean temperatures. The simple climate model is essentially an energy balance model that tries to account for the major components of the climate system (feedbacks, natural variability, and an ocean) to help estimate changes in global mean temperature.

Before working with the simple climate model, some background information about models is necessary. So why do we even need climate models? Unlike many physics experiments that we can perform inside a laboratory, doing experiments on the Earth’s climate is not as easy. Therefore, scientists must use computer models of the climate to represent the key processes and attributes that control Earth’s climate, some of which are represented in the picture below.

key processes and attributes that control Earth’s climate,

It is important to test whether models accurately represent the Earth’s climate. For this reason, climate models are compared with observations, such as global temperatures or global precipitation fields. If models can reproduce major features of the observed climate record, we have more confidence in their ability to project future changes in climate. Even state-of-the-art climate models are imperfect. In order to improve the accuracy of climate simulations, models are closely compared against observations to determine what model aspects need improvement.

In this lab, the students will have the chance to use their simple models to make projections of 21st century warming, given a few possible emission scenarios. In the first part of the lab, the students will tune their models so that they produce realistic simulations of the 19th and 20th century. The students will use the same model parameters to make temperature projections for the future.

What follows is a mathematical description of the climate model. Some basic calculus is needed to understand the equations. It is up to the instructor to decide how much detail to convey to the students.

This simple climate model is an energy balance model. When a system (like the Earth) is in energy balance, the energy entering the system equals the energy leaving the system and the temperature of the system will stay the same. If the climate system experiences a positive forcing (e.g., from increased solar intensity or the effects of increased atmospheric greenhouse gas) the Earth will absorb heat and warm. A warmer Earth also emits more energy to space via “longwave cooling.” Over time the system reaches a new equilibrium such that the initial forcing is balanced by longwave cooling.

We can write this balance in terms of an equation:

Heat storage rate = longwave cooling + radiative forcing [1]

If we want to include effects from year to year differences in the weather, then we add weather noise such that:

Heat storage rate = longwave cooling + radiative forcing + weather noise [2]

When the system is in equilibrium, the heat storage rate will be on average zero, which means that longwave cooling + radiative forcing + weather noise will be zero. Since weather noise is zero on average, the system will be in equilibrium if longwave cooling + radiative forcing is zero.

Now, Eq. 2 can be written in the following form:

\rho c_p H \frac{d T(t)}{dt}=-bT(t)+R(t)+W(t) [3]

where
\rho is the density of seawater (1025 kg m-3),
cp is the specific heat of seawater (3985 Joules kg-1 ºC-1)
H is the depth of the upper ocean where heat is absorbed (m)
T is the temperature change from preindustrial conditions (ºC)
t is time (seconds)
b is the climate sensitivity (Watts m-2 ºC-1)
R is the radiative forcing from natural and anthropogenic effects (Watts m-2)
W is the year to year random weather forcing (Watts m-2)

Note that bT(t) in Eq. 3 does not just represent longwave cooling. As discussed earlier, a warmer Earth will emit more energy to space in the form of longwave energy. But a warmer Earth also results in other changes that can modify the amount of energy going to space. For example, sea-ice extent, snow cover, and some types of low clouds are diminished when the Earth warms. Since ice, snow, and clouds all reflect sunlight, their loss means that less sunlight will be reflected when the Earth warms. These temperature-dependent changes in outgoing radiation are known as climate feedback processes. These processes are also included as part of b, which is the climate sensitivity.

We can now track changes in temperature by taking the finite-difference form of Eq. 3 to look at the changes in temperature from year to year.

\rho c_p H \frac{(T^{(n+1)}-T^{n})}{\Delta t}=-bT^{n}+R^{(n+1)}+W^{(n+1)} [4]

Where the n superscript is the value at time n where n goes from 1850 through 2019 in the first tab of the excel spreadsheet file (climatereal3.xlsx).

We can rearrange Eq. 4 to get

T^{(n+1)}=T^{n}+\frac{-bT^{n}+R^{(n+1)}+W^{(n+1)}}{\rho c_p H} \Delta t [5]

This is the equation that is in the spreadsheet .

Graph of Observed global mean temperature variability and change since 1880.

Figure 1. Observed global mean temperature variability and change since 1880.

Now if we look at the global mean temperature record in this context, the temperature has been increasing, which means that the Earth has been storing energy (in the form of a warmer atmosphere and ocean). This also means that the radiative forcing has been larger than the longwave cooling (including effects from other feedbacks). The rate of the temperature increase then is set by both the ocean heat storage capacity and the climate sensitivity. The wiggles in the temperature can be either understood as either the effect of radiative forcing or weather noise.  By doing this experiment, we want the students to look at the temperature series and understand what controls the general trend, the rate of the trend (slope of the line), and the wiggles in the temperature.

The second part of the lab is for the students to make temperature projections based on different scenarios, called Shared Socioeconomic Pathways (SSPs). The SSPs are meant to serve as inputs for climate models that capture different scenarios of changing emissions, concentrations, and land-cover change projections. In this lab, the SSPs provide a range of plausible estimates of future forcing for the simple climate model.

Materials

Conducting the Lesson

Teacher Note:

It is important to test the model on the computers that the students will be using. In the past, there have been issues with different versions of Excel not being able to correctly run the scripts that are embedded in the provided Excel spreadsheets.

If the students are not familiar with Excel, the instructor may need to spend a class period getting the students familiar with Excel.

Some of the necessary spreadsheet skills : entering data, creating new spreadsheets, copying and pasting data in excel, knowing the paste options so that the students can just paste the values, making simple scatter graphs on excel.

 

 

Preparation

Read the materials above and acquaint yourself with the student worksheet and the Excel climate model. This will help in gaining intuition for how the model responds to different parameter choices and sets of parameters that produce reasonable agreement with the observations of global mean temperature.

Preparation before class

  • Anticipate how many modeling groups (groups of twos or threes are suggested) you will have in your class and make sure that climatemodel_Combined.xlsx is organized to accommodate that number of groups.
  • Make sure the Excel model runs on every computer that will be using it. A recent version of Excel that can read .xlsx will be needed.

In class

INTRODUCTION TO THE LAB

15 min: Introduce the students to models and major components of the Earth’s climate with the presentation (ClimateModel_presentation.pptx) that comes with this lesson plan. The presentation may be tailored to fit the students’ level of understanding of models, energy budget, and major constituents of the Earth’s climate.

It may be helpful to ask the students whether they know of any models that are used to simulate different things (model airplane, model robot, flight simulator, weather forecast model, etc.) Then ask them who might use the models and why. For example, why would you even need a model of an airplane or a weather forecast model? Then transition to climate models.

15 min: Introduce the IPCC (Intergovernmental Panel on Climate Change) report and the projections of future warming provided by various international climate modeling groups (graph can be found in Powerpoint presentation). Introduce the graph with temperature projections for the 21st century. Ask them what they think each line represents and the meaning of the shading associated with each line. You will want to circle back to this during the post-lab discussion.

10 min: (optional) Talk about how these projections of future warming may be use to used to inform government official’s plans to mitigate or adapt to climate change.

10 min: (optional) If the students have not covered energy budgets, then introduce them to the concept of energy budget. Some sample slides from the presentation may be used for this purpose.

20 min: Introduce the students to the Excel climate model. Here, varying degrees of detail may be presented. In this lesson plan, we will try to present the most basic detail that is needed to do the experiment.

GETTING STARTED

5 min: Have the students break up into groups (2-3 students per group). Each group will represent a climate modeling center. They may choose their own group name if they wish. They may also come up with their own logos. For our examples, we’ve come up with the creative group names of A, B, & C. The Excel worksheet that combines everyone’s results allows for up to 5 groups.  More can be added if necessary.

Handout the worksheet to everyone.

30-40 min: By the end of the lab, the students will ultimately need to come up with 3 temperature projections for the 1850 to 2100 time period. The groups’ projections will then be combined using the Excel spreadsheet climatemodel_Combined.xlsx.

POST-LAB DISCUSSION

10 min: Combine the projections from all the modeling groups and graph them in climatemodel_Combined.xlsx. Then on the blackboard make a table with each group’s name and their selected parameter values (weather noise (W), ocean depth (H), and climate sensitivity (b)). This allows the students to see what other groups came up for their parameter values. It may be good to point out that we do not know the true values for these parameters in our simplified model. Ask the students what happened when they changed the a) weather noise, b) ocean depth, and c) climate sensitivity parameters.

15 min: Plot the temperature projection from one of the scenarios. An example figure with three temperature estimates (A, B, & C) for RCP 45 is shown below.

Example plot from SSP370 run using 3 models (A, B, & C).

Figure 2 : Example plot from SSP370 run using 3 models (A, B, & C). The black line indicates the multi-model mean.

Here, ask the students why certain models predict more warming than others. Then ask them about the wigglier graphs (if this is apparent). What appears to control the wiggles? This part is to reinforce the concept of how the feedback parameter and weather noise control the behavior of the temperature change. You can also ask about the line that represents the mean (average) here.

Next, show the multi-model mean plot, as shown below.

 

Example plot from the multi-model mean

Figure 3: Example plot from the multi-model mean. This figure can be found under the “All” worksheet in the spreadsheet named climatemodel_Combined.xlsx.

Here, only the multi-model means are plotted. The Excel spreadsheet may be modified so that the inter-model spread is also shown. Ask the students why they think each line ends up at different temperatures. Then if you look at the behavior of each graph, you will notice that SSP126 has actually leveled off. You can ask them why they think it’s flattened, compared to say SSP370. Finish the discussion with some of the following suggested questions to get the students to start thinking about policy implications, limitations of our simple climate model, and speculation about what temperatures would look like if we ran beyond 2100.

Discussion questions:

  1. What are the numbers in SSP YYY related to? Answer: The YYY are related to the radiative forcing values at the end of the 21st century (2100 values).
  2. If we run the model to 2200, what do you think will happen to the temperature change in SSP126 and SSP370? Answer:  SSP126 is starting to flatten out, while SSP370 is increasing. Therefore, the temperatures at 2200 will be higher for SSP370, while they would stay close to 2100 values for SSP126. The follow up question to this would be, “is there a way we can test this?” And of course, the original spreadsheet ran the models through 2500, instead of 2100, so the students can actually look back at their model’s projections.
  3. Given the way the models are constructed, can we ever get rid of the uncertainty (temperature range) that we find in each emission scenario? Answer: No, we will not be able to get rid of the uncertainty, because each model has a unique evolution of weather noise (assuming weather isn’t turned off!). Therefore, even if two models are run with the same climate sensitivity and the same size of weather noise, they will give you slightly different temperature projections, which will manifest as uncertainty in the temperature projections at 2100.
  4. Why do the SSP585 scenario projections show more warming in 2100 than the SSP126 scenario projections? Answer: The SSP585 projections are based on emission forecasts of more greenhouse gases emissions over the course of the 21st Simplistically put, it is a scenario where greenhouse gas emissions are not curbed.
  5. We can talk about uncertainty in our temperature estimates. But what can we say from our modeling exercise? Are the 2100 temperatures for SSP126 and SSP585 different enough that we can say that the two alternate ‘futures’ would give us very different temperatures? Answer: From the modeling exercise, the SSP126 projections should almost all lie below the SSP585 scenarios, even if one takes into account the spread in 2100 temperatures. This can show us that if we continue to emit, as projected by SSP585, we will have significantly different temperatures at 2100, compared to if we start reducing emissions, as in SSP126.
  6. For each emission scenario, why is there a range of temperatures predicted for the year 2100? Answer: Multiple models are used to make temperature projections, and each model is slightly different from another. The students should have found by now that even if one uses the same model and tests the model on the same 20th century temperature record, one can still come up with a range of temperatures for 2100. Therefore, this represents an inherent uncertainty in our projections for the future. Even if we knew how much emissions will exist in the future, there will still be some uncertainty to the amount of warming, given our simplified understanding of the climate in the Excel model.

Additional Resources:

 

 

Attribution:  Thompson, LuAnne; Po-Chedley, Stephen and Terai, Chris. “Simple Climate Model” Climate Science for the Classroom edited by Po-Chedley, Bertram and Herzfeld, 2025. https://uw.pressbooks.pub/climate/chapter/simple-climate-model/ Date of Access.

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