Climate scientists use models to understand how the Earth is changing. Models of Earth can be experimented upon to assess the impact of various perturbations on the planet. Climate models describe our planet with mathematical equations. Because Earth is complex, it takes hundreds of very complex equations to model the atmosphere, oceans, and land surface. Because of the complexity, climate models are usually run on powerful computers.
The Very, Very Simple Climate Model is, as the name implies, very simple. In this model, average global temperature is determined entirely by the atmospheric carbon dioxide concentration via greenhouse warming of the atmosphere. The impacts of the Earth's biosphere, changes in land use, wind and precipitation patterns, other greenhouse gases, uptake of carbon dioxide by the oceans and other factors are not considered by this very simple model. Many of these factors decrease the amount of carbon dioxide in the atmosphere over time.
George E.P. Box once said, "all models are wrong, but some models are useful." The more a complex system like Earth is simplified in a model, the more wrong the model is. However, in simplifying this model to temperature and carbon dioxide, The Very, Very Simple Climate Model allows students to focus on the cause and effect relationships of greenhouse gases and climate change. A major educational point embodied in this model is that temperatures depend on concentration, which rises whenever emissions are greater than zero. When you hear world leaders saying that they are working hard to reduce the rate of growth of greenhouse gas emissions, remember that reducing the rate of growth does not lead to reduced temperatures. Instead, the amount of greenhouse gases continues to grow in the atmosphere whenever emissions are greater than zero.
The assumptions behind this model, though rather limited, are valid as far as they go. The starting values for concentration, emission rate, and temperature are right around actual values for the year 2000. The ranges for emission rate choices are in line with predictions scientists think we are likely to see in this Century. The relationship between atmospheric carbon dioxide concentration and temperature is well-established; basically, temperature rises about 3° C for each doubling of carbon dioxide concentration. So, for example, if the concentration goes from 400 ppmv to 800 ppmv, we expect to see temperature go up by 3° C.
According to the Intergovernmental Panel on Climate Change and their 2007 Assessment Report, Earth’s average temperature rose 0.6° Celsius (1.1°F) during the 20th Century. Based on the results from about a dozen computer models, the IPCC projects that global warming will continue. Model results project that Earth's average global temperature will rise between 1.8° and 4.0° Celsius (3.2° and 7.2° F) depending largely on how humans change the ways they live on the planet.
Details of the Math Behind the Model
Based on a mix of theory and observations, scientists now know approximately how much Earth's surface temperature as CO2 levels increase... everything else being equal, which is, of course, never the case. Stated simply, a doubling of the atmospheric concentration of CO2 is expected to produce about a 3° C (5.4° F) rise in average global surface temperatures.
An example: the CO2 level in the year 2000 was 368 parts per million (ppm), while the average global temperature was 14.3° C (57.7° F). If, in the future, the CO2 concentration were to rise to 736 ppm ( = 2 x 368 ppm) we would expect global temperatures to rise roughly 3° C (5.4° F) to 17.3° C (63.1° F). CO2 concentration would have to double again to 1,472 ppm ( = 2 x 736 ppm) to cause another 3° C (5.4° F) temperature rise to 20.3° C (68.5° F).
Mathematically, here's the formula which expresses this relationship:
|T = T0 + S log2 (C / C0)
- T is the new/current temperature
- T0 is the know temperature at some reference time (for example, 14.3° C in the year 2000)
- S is the "climate sensitivity" factor; we've been using 3° C (more on that below); the temperature rise as a result of CO2 doubling
- C is the new/current atmospheric CO2 concentration
- C0 is the known atmospheric CO2 concentration at some reference time (must be the same time as T0; 368 ppm in 2000 would match the T0 example mentioned above)
Let's look at an example calculation. What would we expect the average temperature (T) to be if the CO2 concentration rose to 600 ppm? Let's use 14.3° C for T0 (in 2000) and 368 ppm for C0.
T = T0 + S log2 (C / C0) = 14.3° C + [ 3° C x log2 (600 ppm/ 368 ppm)]
= 14.3° C + [ 3° C x log2 (1.63)] = 14.3° C + [ 3° C x 0.705] = 14.3° C + 2.1° C = 16.4° C
The "climate sensitivity" factor, S, is actually an estimate. According to the Fourth Assessment Report by the IPCC (Intergovernmental Panel on Climate Change), the climate sensitivity value is "likely to be in the range 2 to 4.5° C with a best estimate of about 3° C, and is very unlikely to be less than 1.5° C. Values substantially higher than 4.5° C cannot be excluded, but agreement of models with observations is not as good for those values." We have used 3° C for climate sensitivity in this simple model.
How does the CO2 emission rate affect the CO2 concentration level? Based on estimates of the total quantity of CO2 in the atmosphere (in gigatons, abbreviated GtC) and of the CO2 concentration, every 2.3 GtC of emissions would be expected to raise atmospheric CO2 concentration by 1 ppm. So, if emissions in a given year were about 7 GtC, we would expect CO2 concentration to rise by almost 3 ppm ( = 7 GtC ÷ 2.3 GtC) that year. Total global carbon emissions were around 8.2 GtC in 2000, and had climbed to 9.2 GtC by 2006.
To keep this model simple, a couple of major factors were left out. We intend to add them as optional elements in the simulation in the future.
- Ocean Absorption of CO2: Sea water absorbs carbon dioxide from the air, creating carbonic acid in the water. Scientists estimate that roughly a third of the anthropogenic CO2 released into the atmosphere is (currently) absorbed by Earth's oceans. Thus, only about 2/3rds of emissions actually go towards increasing the atmospheric concentration of CO2. Over time, as more carbon is sequestered into the oceans and the oceans therefore become more acidic (as a result of the carbonic acid produced), scientists expect that the oceans will be able to absorb less and less CO2 from the atmosphere. Thus, in the future, we expect a higher percentage (than the current 67%) of emitted CO2 to remain in the atmosphere... thus promoting greenhouse warming.
- Residence Time of CO2 in Earth's Atmosphere: Carbon dioxide is constantly cycled between the atmosphere, oceans, soil, living creatures, rocks, and so on. Every year, some of the CO2 humans have emitted into the atmosphere is naturally absorbed by other elements of the carbon cycle, reducing atmospheric CO2 concentration and thus greenhouse forcing. However, it takes a very long time for excess CO2 in the atmosphere to naturally migrate to other carbon reservoirs. As an example, if we had completely eliminated all anthropogenic carbon dioxide emissions in 2000 (and every year hence), scientists estimate that by the year 2100 CO2 concentration in the atmosphere would drop to about 12% below year 2000 levels. Because of this long residence time of CO2 in the atmosphere, the omission of this factor from our simple model is a relatively minor source of inaccuracy.