Simple greenhouse gas layer model

Since carbon dioxide in the atmosphere is still increasing and it seems as if it will not stop increasing in the near future, it is important to know about the greenhouse gas effect. On earth is it neither as cold as on Mars (-55°C) nor as hot as on Venus (>400°C). This has not only to do with the distance from sun, but also with the greenhouse gases. On Mars there is hardly any atmosphere, leading to warm temperatures during day and very cold temperatures during night. You might have experienced clear nights with a much larger temperature difference to daytime temperatures, whereas with a cloudy sky the temperature difference between day and night is not as large. This is because water is a very effective greenhouse gas. However, there are hardly any possibilities to alter the water content in the atmosphere, but we could reduce our CO2 emissions. On Venus on the other side the atmosphere contains a lot of CO2 and other gases leading to a very strong greenhouse gas effect.

Quiet a while ago I’ve written a simple program to calculate the greenhouse gas effect, and since I’m not really good in documenting my code the following text is an extended documentation of a simple greenhouse gas layer model written in R, which is based on chapter 3 “The layer model” of David Archer’s book “Global Warming: Understanding the Forecast” (2008) (web page, publisher). The book is easy to read and the formulas are even understandable by a non-physicist like me. In the following I will briefly explain the simple radiation budget of the earth.

Solar radiation

If the earth is in equilibrium, with respect to its radiation budget, the amount of energy reaching the earth must be equal the amount of energy leaving the earth for a given time period. Energy is measured in Joule and energy per time is called power and is measured in Watts, where Watts = Joule/second or in scientific notation W = J s-1. The energy source for the earth is the sun and it is possible, to calculate suns mean energy reaching the earth, the so called solar constant Isolar (it is not really a constant, but for simplicity it is assumed to be constant).

According to the Stefan-Boltzman law the power emitted by a so called “black body” per area in Watts per square meter is equal to the surface temperature to the power of 4 times the Stefan-Boltzman constant σ. With the average surface temperature of the sun in Kelvin Tsun = 5778 K and σ = 5.671*10-8W m-2 K-4 the sun emits σ*Tsun4 = 63.21 MW m-2. That’s really much. The sun has a radius of rsun = 696*106 m and therefore a surface of 4*π*r2 = 6.09*1018 m2. By multiplying these two large numbers, the power emitted per area and suns surface area, we end up with the total power emitted by the sun in all directions, which is 3.85*1026 W. In Wikipedia’s list of the largest power stations in the world the Three Gorges Dam with 2.5*109 W is the largest power station to date, which is still nothing compared to the sun.

Of course, not all this power is reaching the earth, the total power emitted by the sun must be divided by the surface of a sphere with the distance of the earth from the sun dsun-earth The distance is on average dsun-earth = 149.6*109 m. This is so far, that the radiuses of earth and sun can be neglected. According to the above already used formula to calculate the surface of a sphere, the solar constant calculated here is Isolar = 3.85*1026 W / (4*π*dsun-earth2) = 1368.123 W m-2.


Radiation budget without atmosphere

Everybody might have seen pictures of the earth and realized that some parts are bright and others are dark. This means a part of the light and therefore the energy from the sun is directly reflected back to space, without being taken up by the earth surface and converted to “heat” or infrared radiation. The fraction of back scattered light or energy is called albedo with the symbol α and is on average 33% of incoming solar radiation. As mentioned above, the average incoming power must be equal the outgoing power. Because the sun only shines on one side of the earth the solar constant must be multiplied by the surface of a circle, the cross section of the earth (Isolar*(1 – α)*π*rearth2), but the earth emits in all directions, therefore the outgoing power must be multiplied by the surface of a sphere and we assume that the earth also behaves like a “black body” (4*π*rearth2*σ*Tearth4) to calculate earth surface temperature without atmosphere. As already mentioned the incoming and outgoing radiation must be equal and by solving Isolar*(1 – α)*π*r2 = 4*π*rearth2*σ*Tearth4 it is possible to calculate earth surface temperature if it had no atmosphere: Tearth = [(1-α)*Isolar / (4*σ)]0.25 (where the power to 0.25 stand for the forth root). By filling in the above values the earth surface would have an average temperature of 252.13 K equal to -20°C, pretty cold. So luckily we have an atmosphere on earth not only to breath, but also to raise the temperature.

Greenhouse layer

If now the atmosphere comes in our model, the above radiation budget is valid for the top of the atmosphere. So incoming solar power must equal outgoing infrared radiation at the top of the atmosphere. We assume a very simple atmosphere, which is transparent to the incoming radiation, but absorbing and emitting all infrared radiation. However, it takes up infrared radiation from the earth but emits it on both sides, back to earth and to space. By solving a set of formulas the surface temperature below the greenhouse layer equal 20.25*252.13 K = 299.8 K, a much more comfortable temperature now. We could now add more and more greenhouse layers, each of them would increase the surface temperature by the factor of 20.25 = 1.189.

Dust layer

One could also assume a dusty atmosphere layer, which absorbs a fraction of sun’s radiation, and emits infrared radiation to all direction. Guess what happens. Of course the earth surface would be cooler. I don’t want to go into details here, if you want to know more, look at the code, read David Archer’s or another atmospheric physics book. I was not yet able to combine the two models the greenhouse and the dust layer. And I will not finish it.

Final remarks

In the real world things are not so simple. The solar constant is not equal through out the year and over geological time periods, since the distance to earth changes and there are times with more or less sun spots. The albedo is not the same everywhere, some clouds and ice are much more reflective than forests or the oceans. A fraction of the solar radiation energy is not converted to infrared radiation and reemitted, but used for evaporation and photosynthesis. The energy in the evaporated and transpired water vapor is transported elsewhere and released, when the water condenses and precipitates. We don’t have a closed greenhouse gas layer and no equally distributed dust layer in the atmosphere.

Most of these processes are implemented in global climate models in a simplified way, which we call parameterization. These models divide the earth surface in small squares and the atmosphere in boxes above these squares to calculate all known processes. The models are parameterized to our best knowledge to represent reality as good as possible. And these global climate models do not run on a single PC they need several tens to hundreds, so called clusters. However, the above described model is already good enough to yield an approximation of the greenhouse gas effect and reality (observed mean earth temperature: 295 K or 22°C) is in between the two calculated temperatures of 252 K and 300 K, somewhat closer to the single greenhouse layer model.