Why does the emission of carbon dioxide influence our climate?

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 of the atmosphere $ε$ 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), as indeed Earth-sensing satellites do. Our balanced equation for the conservation of energy on Earth’s surface is

$$\begin{eqnarray}

I_{in} &=& I_{out} \nonumber \tag{1}\\

\dfrac{S}{4}(1-A) + ϵ σ T_a^4 &=& σ T_e^4 \nonumber\tag{2}

\end{eqnarray}$$

As before, $S$ is the solar constant ($S = 1367\text{ W}\text{ m}^{\text{-}2}$), $A$ is the albedo ($A = 0.3$), and $σ$ is the Stefan-Boltzmann constant ($σ = 5.67 \cdot 10^{\text{-}8}\text{ W}\text{ m}^{\text{-}2}\text{ K}^{\text{-}1}$). 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

$$\begin{eqnarray}

I_{in} &=& I_{out} \nonumber \tag{3}\\

ϵ σ T_e^4 &=& 2 ϵ σ T_a^4 \nonumber\tag{4}\\

T_e &=& 1.19 T_a \nonumber\tag{5}

\end{eqnarray}$$

We can solve for either the surface temperature or the atmosphere temperature by combining the equations for the surface and the atmosphere.

So what is a reasonable value for the emissivity of the atmosphere? Based on measured spectra[note] Science Briefs – Taking the Measure of the Greenhouse Effect, Gavin Schmidt, http://www.giss.nasa.gov/research/briefs/schmidt_05 [2019-10-11].[/note], we know that the atmosphere is transparent for some wavelength, even in the thermal IR. An example is shown below.

This is compared to the radiation of a perfect blackbody corresponding to a temperature of $294\text{ K}$ (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 vapour, 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

$$I_{measured} = (1 – \epsilon) \sigma T_e^4 + \epsilon \sigma T_a^4 \tag{6}$$

and also

$$I_{measured} = 0.65 \sigma T_e^4\tag{7}$$

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 $T_e = 285\text{ K}$, close to the current measured value of $288\text{ K}$.

A more refined analysis[note] http://en.wikipedia.org/wiki/Idealized_greenhouse_model [2019-10-11].[/note] yields $ε = 0.78$ and a temperature of $288\text{ K}$.

**Influence of CO _{2} 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 CO_{2} in the atmosphere, a simple model predicts that the emissivity increases from $ε = 0.78$ to $ε = 0.80$[note] http://en.wikipedia.org/wiki/Radiative_forcing [2019-10-11].[/note].

Using our spreadsheet again, we see that the surface temperature would increase by $1.2\text{ K}$. Additional effects such as ‘positive feedback’ due to increased water vapour lead to an increase in emissivity by another increment of $0.02$. In a static model, this would raise the Earth’s temperature to $292\text{ K}$. However, more sophisticated climate models indicate that further positive feedbacks would cause the temperature to go on rising for many centuries.

**Updated (CEW) 2019-10-15**