Simple Earth Climate Model - Single-Layer Perfect Greenhouse Atmosphere

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For the purposes of this model, let's assume that there is only one layer of Earth's atmosphere, which can be modelled as a slab. The atmosphere allows most of the incident solar radiation through, but absorbs radiation emitted by Earth. The atmosphere then radiates equally from both its topside and underside (Fig. 1). Let's assume that Earth's temperature is constant, which of course it isn't. We can assume this balance for now as the imbalance is about 1.5 W/m2 in 240 W/m2, which is less than 1%1. As a result, our balanced equation for the conservation of energy on Earth's surface is

<br />
\begin{eqnarray}<br />
   I_{in} &=& I_{out} \nonumber \\<br />
   \dfrac{S}{4}(1-A) + \sigma T_a^4 &=& \sigma T_e^4 \nonumber<br />
   \end{eqnarray}<br />

while the balanced equation for the conservation of energy of Earth's atmosphere becomes

<br />
\begin{eqnarray}<br />
   I_{in} &=& I_{out} \nonumber \\<br />
   \sigma T_e^4 &=& 2 \sigma T_a^4 \nonumber<br />
   \end{eqnarray}<br />

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). Combining the equations for surface and atmosphere yields an expression for the atmosphere temperature

<br />
\dfrac{S}{4}(1-A) & = & \sigma T_a^4 \nonumber<br />

 

and a relation between surface and atmosphere temperature

<br />
T_e & =&  1.19  T_a \nonumber<br />

 

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. The green arrows represent the incident solar intensity, which is not absorbed by Earth's atmosphere. 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 mean solar intensity, Iin or Iout, where ε = 1.

Using our model, we obtain the following results:

<br />
\begin{eqnarray}<br />
   \textnormal{\underline{Earth's Atmosphere}} & & \textnormal{\underline{Earth's Surface}} \nonumber \\<br />
   I_{in} \approx 240 \textnormal{ W/m}^2 & & I_{in} \approx 480 \textnormal{ W/m}^2\nonumber \\<br />
   I_{in} = I_{out} & & I_{in} = I_{out} \nonumber \\<br />
   I_{in} \approx \sigma T_a^4 & & I_{in} \approx \sigma T_e^4 \nonumber \\<br />
   I_{a} = 255 \textnormal{ K} & & I_{a} = 303 \textnormal{ K} \nonumber \\<br />
   I_{a} = -18^{\circ}\rm{C} & & I_{a} = 30^{\circ}\rm{C} \nonumber<br />
   \end{eqnarray}<br />

The temperature of Earth's surface is much too hot, as its average temperature is recorded as a mean of 14.5°C2. Remember, however, that this assumes a single but perfect greenhouse layer, which we can deduce is not the case. Nevertheless, the difference in temperatures between Earth's surface and Earth's atmosphere surface indicate that life would be a lot colder if Earth lacked an atmosphere. Earth's atmosphere makes Earth habitable for this reason, amongst others. 

The consequences are summarized below:

The calculated surface temperature on Earth is 30°C assuming that Earth has a single, perfect greenhouse atmospheric layer.

GOOD: Earth's atmosphere is not a perfect absorber of IR radiation, which accounts for Earth's measured mean surface temperature of 14.5°C.

BAD: It gets closer to perfection each year as billions of tonnes of greenhouse gases (IR absorbers like CO are added to it3.

GOOD: The T4 factor means that a small rise in temperature means a great increase in Pout. This is an example of negative feedback.

BAD: The warmer Earth gets, the lower our albedo becomes as the white, sunlight-reflecting ice in the Arctic disappears. This is an example of positive feedback.

BAD: There is no reason to stop at one atmosphere in our model. For example, Venus' atmosphere absorbs and re-emits IR radiation many times over. Venus' surface absorbs the same direct solar power as does Earth (it is closer to the Sun but has a much higher albedo), yet its temperature is 735 K4. This is an example of a runaway greenhouse effect.

CONCLUSION: There is nothing guaranteed about our current mean surface temperature of 14.5°C; with plausible changes to our albedo and atmosphere, we can dial up pretty much any temperature we like, from freezing to boiling.

For a more extensive study on the amount of CO2 produced and its effect on Earth's climate, check out the Carbon_Dioxide_Emissions_per_Capita_Spreadsheet.xls.

 

  • 1. Intergovernmental Panel on Climate Change. Graphics; IPCC AR4 Synthesis Report; Figure 2.4 (online). http://www.ipcc.ch/graphics/syr/fig2-4.jpg [12 June 2009].
  • 2. Çengel, Yunus A. Steady Heat Conduction. In: Heat Transfer a Practical Approach (2). New York: McGraw Hill Professional, 2003, p. 173.
  • 3. Forget, F. and Pierrehumbert, R.T. Warming early Mars with carbon dioxide clouds that scatter infrared radiation. Science vol. 278, iss. 5341: 1997, p. 1273.
  • 4. Aubrecht GJ. Solar Energy: Wind, Photovoltaics, and Large-Scale Installatons. In: Energy - Physical, Environmental, and Social Impact (3), edited by Erik Fahlgren. Upper Saddle River, NJ: Pearson Education Inc., 2006, chapt. 16, pp. 334,336.

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