Illustrative model of greenhouse effect on climate change

There is a strong scientific consensus that greenhouse effect due to carbon dioxide is a main driver of climate change.

Following is an illustrative model meant for a pedagogical purpose, showing the main physical determinants of the effect.

Under this understanding, global warming is determined by a simple energy budget: In the long run, Earth emits radiation in the same amount as it receives from the sun.

In most of the electromagnetic spectrum, atmospheric carbon dioxide either blocks the radiation emitted from the ground almost completely, or is almost transparent, so that increasing the amount of carbon dioxide in the atmosphere, e.g. doubling the amount, will have negligible effects.

Since the upper layers are colder, the amount emitted would be lower, leading to warming of Earth until the reduction in emission is compensated by the rise in temperature.

The troposphere is thicker in the equator and thinner at the poles, but the global mean of its thickness is around 11 km.

At higher altitudes, up to 20 km, the temperature is approximately constant; this layer is called the tropopause.

Higher through the tropopause, density continues dropping exponentially, albeit faster, with a typical length of 4.2 km.

Earth constantly absorbs energy from sunlight and emits thermal radiation as infrared light.

Atmospheric CO2 absorbs some of the energy radiated by the ground, but it emits itself thermal radiation: For example, in some wavelengths the atmosphere is totally opaque due to absorption by CO2; at these wavelengths, looking at Earth from outer space one would not see the ground, but the atmospheric CO2, and hence its thermal radiation—rather than the ground's thermal radiation.

The amount of ground radiation that is transmitted through the atmosphere in each wavelength is related to the optical depth of the atmosphere at this wavelength, OD, by: The optical depth itself is given by Beer–Lambert law: where σ is the absorption cross section of a single CO2 molecule, and n(y) is the number density of these molecules at altitude y.

Note that the OD depends on the total number of molecules per unit area in the atmosphere, and therefore rises linearly with its CO2 content.

When it reaches the tropopause, any further increase in CO2 levels will have no noticeable effect, since the temperature no longer depends there on the altitude.

Therefore, at these wavelengths Earth radiates mainly in the tropopause temperature, and addition of CO2 does not change this.

Thus, the weight of the troposphere in determining the radiation that is emitted to outer space is: A relative increase in the CO2 concentration means an equal relative increase in the total CO2 content of the atmosphere, dN/N where N is the number of CO2 molecules.

Adding a minute number of such molecules dN will increase the troposphere's weight in determining the radiation for the relevant wavelengths, approximately by the relative amount dN/N, and thus by:

Since CO2 hardly influences sunlight absorption by Earth, the radiative forcing due to an increase in CO2 content is equal to the difference in the flux radiated by Earth due to such an increase.

Thus, a 2-fold increase in CO2 content will reduce the radiation emitted by Earth by approximately: More generally, an increase by a factor c/c0 gives: These results are close to the approximation of a more elaborate yet simplified model giving We may make a more elaborate calculation by treating the atmosphere as compounded of many thin layers.

Therefore, OD(y) is proportional to the pressure p(y), which within the troposphere (height 0 to U) falls exponentially with decay constant 1/Hp (Hp~5.6 km for CO2), thus: Since

with T1 taken at Hp, so that totally: giving the same result as in the one-layer model presented above, as well as the logarithmic dependence on N, except that now we see T1 is taken at 5.6 km (the pressure drop height scale), rather than 6.3 km (the density drop height scale).

The total average energy per unit time radiated by Earth is equal to the average energy flux j times the surface area 4πR2, where R is Earth's radius.

An exact calculation using the MODTRAN model, over all wavelengths and including methane and ozone greenhouse gasses, as shown in the plot above, gives, for tropical latitudes, an outgoing flux

295.286 W/m2 after CO2 doubling, i.e. a radiative forcing of 1.1%, under clear sky conditions, as well as a ground temperature of 299.7o K (26.6o Celsius).

As CO2 levels rise, the emitted radiation can maintain this equilibrium only if the temperature increases, so that the total emitted radiation is unchanged (averaged over enough time, in the order of few years so that diurnal and annual periods are averaged upon).

According to Stefan–Boltzmann law, the total emitted power by Earth per unit area is: where σB is Stefan–Boltzmann constant and ε is the emissivity in the relevant wavelengths.

The relative change in the total radiated energy flux due to changes in emissivity and temperature is: Thus, if the total emitted power is to remain unchanged, a radiative forcing relative to the total energy flux radiated by Earth, causes a 1/4-fold relative change in temperature.

As a rough estimate, we note that the average temperature on most of Earth are between -20 and +30 Celsius degree, a good guess will be that 2% of its surface are between -1 and 0 °C, and thus an equivalent area of its surface will be changed from ice-covered (or snow-covered) to either ocean or forest.

For comparison, in the northern hemisphere, the arctic sea ice has shrunk between 1979 and 2015 by 1.43x1012 m2 at maxima and 2.52x1012 m2 at minima, for an average of almost 2x1012 m2,[6] which is 0.4% of Earth's total surface of 510x1012 m2.

The antarctic ice cap size oscillates,[7] and it is hard to predict its future course,[9][10] with factors such as relative thermal insulated and constraints due to the Antarctic Circumpolar Current probably playing a part.

However this will mainly happen in northern and southern latitudes, around 60 degrees off the equator, and so the effective area is actually 2% * cos(60o) = 1%, and the global albedo drop would be 2/3%.