Simple Earth Climate Model: Single-Layer Imperfect Greenhouse Atmosphere
Why does the emission of carbon dioxide influence our climate?
Whether radiation is absorbed or transmitted depends on the wavelength.
The absorption of the IR radiation in the atmosphere depends mainly on the presence of water vapor, ozone, and carbon dioxide.
Increasing the concentration of carbon dioxide will increase the emissivity and the surface temperature.
The atmosphere is not a perfect absorber for all radiation. We know already that the atmosphere is transparent for sunlight but it is also transparent for some of the thermal infrared radiation emitted from the Earth's surface. Consequently, only a fraction of the thermal IR radiation is absorbed by the atmosphere, which means that the emissivity ε is not equal to one. Therefore, an observer in space would detect IR radiation emitted from the surface as well as from the atmosphere, rather than just Earth's atmosphere (Fig. 1). Our balanced equation for the conservation of energy on Earth's surface is
As before (see main article), S is the solar constant (S = 1367 W/m2), A is the albedo (A = 0.3), and σ is the Stefan-Boltzmann constant (σ = 5.67 x 10-8 W/m2 K). Notice that the emissivity is still equal to 1 for the surface (so we did not write it explicitly in the second equation) but is less than 1 for the atmosphere now. The balanced equation for the conservation of energy of Earth's atmosphere becomes
We can solve for either the surface temperature or the atmosphere temperature by combining the equations for the surface and the atmosphere. We leave this up to the reader as an exercise. You can study influence of the emissivity on the surface temperature with our spreadsheet Earths_Surface_Temperature_Spreadsheet.xls.
Figure 1. A diagram of the exchange of EM radiation between the Sun, Earth, and Earth's atmosphere. All three objects are assumed to be black bodies, and so energy is conserved. The green arrows represent the incident solar intensity, which is not absorbed by Earth's atmosphere as the solar EM radiation spectrum consists of 37% visible, 51% near IR, and 12% UV radiation. The red arrows represent IR radiation, which is emitted by both Earth and Earth's atmosphere. The difference in the wavelength of EM radiation is due to the temperature of the radiating object. The red equations represent the intensities that are either emitted (outgoing arrows) or absorbed (incoming arrows).
So what is a reasonable value for the emissivity of the atmosphere? Based on measured spectra 1, we know that the atmosphere is transparent for some wavelength, even in the thermal IR. An example is shown below.
Figure 2. The graph 1shows the total outgoing flux measured at the top of Earth's atmosphere (blue curve). This is compared to the radiation of a perfect blackbody corresponding to a temperature of 294 K (red curve). The difference between the red and the blue curve is due to absorption. Most of the absorption is due to the presence of water vapor, ozone, and carbon dioxide.
An estimate of the difference between the measured flux and the flux of an ideal blackbody from figure 2 yields roughly 35%. (For this you compare the areas under the two curves.) This is the fraction of the Earth’s thermal radiation that is not absorbed by the Earth atmosphere: So the measured flux is 65% of the flux we expect from a perfect blackbody. Looking at the flux diagram, the measured flux should be
Combining these equations and using the relationship between surface and atmosphere temperature yields ε = 0.7. Entering the data into our spreadsheet yields a surface temperature of Te = 285 K, close to the current measured value of 288 K.
A more refined analysis 2 yields ε = 0.78 and a temperature of 288 K.
Influence of CO2 and other greenhouse gases
The concepts developed above allow us now to understand the influence of carbon dioxide and other greenhouse gases on our climate. The absorption of radiation is due to the molecules in our atmosphere. Most of the absorption is due to water, ozone, and carbon dioxide, as shown in the spectrum above. If we double the concentration of CO2 in the atmosphere, a simple model predicts that the emissivity increases from ε = 0.78 to ε = 0.80 2 3.
Using our spreadsheet again, we see that the surface temperature would increase by 1.2 K. Additional effects such as ‘positive feedback’ due to increased water vapor lead to an increase in emissivity by another increment of 0.02 3. In a static model this would raise the Earth's temperature to 292K. However, more sophisticated climate models indicate that further positive feedbacks would cause the temperature to go on rising for many centuries. [IPC ref.]
To take a more extensive look at how changing different variables effects Earth's surface temperature, check out Earths_Surface_Temperature_Spreadsheet.xls.
- 1. a. b. http://www.giss.nasa.gov/research/briefs/schmidt_05
- 2. a. b. http://en.wikipedia.org/wiki/Idealized_greenhouse_model
- 3. a. b. http://en.wikipedia.org/wiki/Radiative_forcing
© Physics and Astronomy Outreach Program at the University of British Columbia (Claire Wheeler and Georg Rieger Dec 3, 2010)