The Greenhouse Effect

Basic dynamics

On the level of the radiative energy balance, the Greenhouse effect is fairly easy to understand and parametrize. Earth's temperature in radiative equilibrium is determined by the balance the incoming solar radiation flux (predominantly in the optical wavelengths) from which a fraction determined by the albedo is reflected and the outgoing infra-red (IR) radiation flux of a warm planet. The solar flux given Earth's orbital position can be computed, and Earth's radiative temperature follows from setting incoming and outgoing flux equal.

The average temperature thus determined is about 254 K, i.e. much too chilly. The missing piece is that the the atmosphere in fact interacts with the radiation, and in the event not all of the IR radiation emitted by the surface of Earth reaches space, but part of it is radiated back to the surface. Defining a retention coefficient c allows to parametrize this effect of the atmosphere, and it turns out c ~ 0.38 provides a reasonable global average temperature of about 291.3 K.

Note that this temperature varies - the albedo of Earth is not constant everywhere, it matters whether the northern hemisphere with more landmass or the southern hemisphere with more ocean is tilted towards Sun, snowfall and cloud developments change the albedo,...) - all these effects can in fact be looked at with the worldbuilder code.

Atmosphere-radiation interactions

The first major challenge is to compute the retention coefficient. This is done as follows: The interaction between IR radiation and any gas depends on whether the molecules have suitable modes which can be excited at a given energy. Measurements for a large number of gases are available e.g. from the HITRAN data base.

IR line strength of CO2 - HITRAN data

IR line strength of water vapour - HITRAN data

Since the raw HITRAN data is a bit unwieldy (there's about half a million of individual lines for CO2 alone) a coarse-grained version of this data has been used in the worldbuilder code.

The absorption of IR radiation of a given wavelength by the atmosphere is now given by the line strength at that wavelength times the amount of molecules of the gas along the ray. Given the gas mixture of an atmosphere, the code can evaluate the product to result in the following wavelength-dependent absorption curve in the IR:

Combined absorption due to CO2, H2O, CH4 and O3 in Earth atmosphere.

The absorption curve is often either near zero or near unity - that's because the line strength often varies over five or six orders of magnitude, so the expected number of interactions alone the ray is rarely between 0.1 and 1 - either it is very small, or very large, in which case the lines just saturate. Note also the fact that IR radiation has been absorbed does not mean it is radiated back to the surface - since the re-emission is random, only half of absorbed radiation is actually radiated back to Earth (however, the fraction that is emitted skyward can be absorbed again). The code takes this effect into account with a simple heuristic formula.

The absorption curve now must be applied to the blackbody radiation curve of a warm Earth (which peaks around 1e-5), and from the ratio of the primary spectrum to the spectrum after absorption corrections, the IR retention factor can be computed. Using the so-obtained factor of 0.377 gives a mean temperature of Earth of 291 K at periapsis - not so much off from reality.


Having the model specified, one can investigate the more interesting questions. For instance, isolate the role of individual greenhouse gases. This is the absorption contribution of CO2 only:

Absorption due to CO2 alone in Earth's atmosphere.

It's in fact not a fearsome absorber. Most of its effects results from the coincidence that the main 15 micron line cuts right into the left flank of the blackbody radiation peak. But in itself, it would not do much. The most significant atmospheric absorber is in fact water vapour. The role of CO2 and other trace gases primarily is to drive water vapour. If Earth heats even modestly due to a higher CO2 concentration, the air can hold more water vapour - heating the air further, so that it can hold yet more water vapour.

So - how does the worldbuilder code do? Doubling the CO2 concentration in the atmosphere increases the perhelion temperature by 0.7 K.

According to much more sophisticated climate models, the radiative forcing due to CO2 is given by Delta F = 5.35 * log(C/C0) W/m^2 of additional heat flux, which translates into about 3.7 W/m^2 for a doubling of the CO2 content of the atmosphere (cf. the Modtran code). That number results in an 1.05 K increase of the temperature.

That's actually suspiciously good for the worldbuilder code which was not really designed to get that fine effects. So part of it probably is accidential.

Another interesting question is - what would happen if Earth atmosphere were 100% CO2? Note that while the linestrength away from the peaks is often several orders of magnitude down, assuming a pure CO2 atmosphere increases the amount by good factor thousand as well. What then happens is that we get to see the minor lines also in the absorption spectrum, while the major lines get somewhat broaden at the edges.

Absorption assuming 100% of Earth atmosphere would be CO2.

In this scenario, Earth would heat to an uncomfortable average 311 K. Note that Venus has an atmosphere that is another factor hundred thicker, dragging yet more minor lines into the absorption. This goes some way in understanding why the Greenhouse effect on Venus is so extremely strong. However, in such a thick atmosphere, the heat transport is more complex, and there's no reason to expect the worldbuilder code in its current form to handle this fully correctly.

Definition file

The basic worldbuilder definition file which has been used to create the above simulation is as follows:

materials materials.cfg
name Sun
mass 1.0
T_surf 5778.0
name Earth
mass 1.0
radius 1.0
semimajor_au 1.0
eccentricity 0.03
inclination 22.0
dec_offset 270.0
rot_period_h 24.0
mean_albedo 0.3
materials_file material_map_earth.cfg
internal_heat 0.86
elements_lat 12
elements_lon 16
infrared_blocking 0.377
transport_coeff 1.0
h2o 0.5
ch4 0.00018
o3 0.00006
n2 78.08
o2 20.95
ar 0.934
co2 0.04
surface_pressure 1.0

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Created by Thorsten Renk 2016 - see the disclaimer and contact information.