# The Azimuth Project Blog - Doubling CO2... then what?

We want to investigate the direct effect of a rise in carbon dioxide concentration on an average earth’s temperature. By direct it is meant that we only consider the feedback from the temperature change itself on the atmospheric thermal emission characteristics; we neglect any other secondary feedback effects such as, e.g., a change in water vapour concentration, which actually affects the absorption characteristics of the atmosphere considerably. The current understanding is that these secondary feedback effects are important because they add considerably to the direct effect, but they are difficult to study because they involve many earth system processes, whereas the direct effect is relatively straightforward to understand.

The direct effect of a rise in carbon dioxide concentration on the earth’s temperature is positive and logarithmic. When we will write logarihtmic forcing/response below, we mean the linear response of the earth’s surface temperature on a doubling of the $CO_2$ concentration. Note that in the previous paragraph we wrote an average temperature: to keep matters simple we assume that we can consider a 0D-earth, and a 1D-atmosphere, and that the forcing derived from this case can actually still tell us something about our earth as it is, where the incoming radiation is inhomogeneous, which leads to heat transport through the atmosphere and oceans, etc.

It is instructive to consider first an extremely simplified toy model for the greenhouse effect, which will allow us to highlight the important ingredients necessary to derive the logarithmic forcing. Let us consider one effective emission level in the atmosphere, with a greenhouse gas that is transparent for the incoming radiation, but absorbs part of the thermal radiation emitted by the earth. The earth’s surface temperature is called $T_s$, and an energy balance at the surface yields:

$F_{in;short} + F_{gh_down;long}= \epsilon T_s^4$

i.e. the energy of the radiation emitted by the earth’s surface in the right hand side is supplied by the incoming shortwave radiation plus a longwave contribution from the greenhouse gases (simplistically assumed to be present at only one level). We assume the earth’s surface radiates as a greybody, with $\epsilon$ the greybody factor. Considering the energy balance at the effective emission height we get:

$\alpha \epsilon T_s^4 = F_{gh_up;long} + F_{gh_down;long}$

i.e. a certain fraction $\alpha$ of the longwave radiation emitted by the earth’s surface is absorbed by the greenhouse gases, then re-emitted as longwave radiation. Because our atmosphere is one-dimensional, half goes upward, half goes downward (of course in reality the radiation will be emitted diffusively in all directions). Therefore we get:

$F_{in;short} = (1-\alpha/2) \epsilon T_s^4$

and we see that the surface temperature depends on the absorption characteristics of the atmosphere, and for positive $\alpha$ the temperature is larger than it would be without. In addition, we see that even for perfect absorption in the atmosphere ($\alpha=1$ it is impossible for the earth not to radiate (which it should to balance the incoming radiation!) because the greenhouse gases emit thermal radiation too.

While the above example is perhaps instructive as a very simple toy model for the greenhouse effect, it does not allow us to derive the logarithmic response of surface temperature to changes in $C0_2$ concentration. To this end, we need several corrections:

• radiation is absorbed at specific wavelengths, we have to break down the energy balance equation to Schwarzschild’s equation for radiative transfer
• radiation is not absorbed and emitted at a single level, but continuously through the atmosphere
• the radiation emitted at a certain wavelength is temperature dependent, hence the temperature profile of the atmosphere is important
• radiation is absorbed proportionally to the density of greenhouse gases, hence their density profile is important

We will now go through these corrections two by two.

## Radiative transfer is a function of optical depth

Radiative transfer through the atmosphere is expressed by Schwarzschild’s equation. When there is no emission, Schwarzschild’s equation can be readily nondimensionalised in terms of optical depth, a quantity that combines both the distance and the absorption strength. This means that in the atmosphere the height and the absorption characteristics cannot be considered separately. When optical depth is near one, we call that location (the height could be anywhere, depending on the absorption strength) optically thick. When it is near zero, we call it optically thin. Moving upward from the surface, a long as the atmosphere remains optically thick for a certain wavelength, radiation effectively cannot escape. The radiation that effectively escapes from the earth with height is of course dependent on wavelength. If a camera in a satellite looks at the earth, the details that are seen are at a height dependent on the wavelength, for visible light the surface can be distinguished, but not true for all wavelengths in the infrared.

Without emission into play, we see that a doubling of carbon dioxide concentration would not have any effect, since for each wavelength the effective emission height would simply shift upward, without further consequence. When we now include emission in Schwarzschild’s equation, it can still be nondimensionalized, but we have to take into account that the emission depends on the local temperature, which, through the local height, will depend on the local optical depth. So now a change in carbon dioxide concentration will affect the radiative transfer, because at this new effective emission height the temperature will be different, and the local emission will be different too. This goes at the heart of the logarithmic forcing.

The absorption characteristics of carbon dioxide are of course wavelength dependent, with some strongly absorbing bands around the peak of the emission spectrum of the earth and no significant absorption of incoming solar wavelengths at the shortwave infrared (this is what makes it a greenhouse gas). Carbon dioxide is a well-mixed trace gas, which means that we can consider its density proportional to the atmospheric density, and its density affects its absorption strength. (Pressure affects the absorption characteristics too, but it does not affect the conclusion of this qualitative discussion.)

## In the troposphere the temperature drops linearly

In the lowest layer of the atmosphere, called the troposphere, the profiles for density, pressure and temperature are approximately isentropic. They can be derived by also considering the hydrostatic equation (linking pressure and density) and the ideal gas law (linking all three). The resulting profiles for density and pressure drop approximately exponentially and the temperature drops linearly with respect to the value at the earth’s surface.

Above the troposphere lies the stratosphere, where pressure and density still follow an exponential profile (half of the atmosphere’s mass is in the troposphere), but the temperature is approximately constant in the lower stratosphere. The reason why the temperature does not drop further is because at the top of the stratosphere the temperature rises again due to the absorption of UV shortwave radiation in the atmosphere, which acts as a local heat source due to the emission of thermal radiation. Because for the logarithmic forcing response we primarily have to consider the longwave radiation of wavelengths corresponding to the earth’s surface temperature, so the UV absorption does not complicate our analysis too much. The absorption takes place at much shorter wavelengths, and the emission is at longer wavelengths (the subsequent thermal radiation is emitted at lower temperatures) or at wavelengths that are already saturated (optically thick).

## Only the Goldilocks wavelengths lead to a logarithmic response

From the optical depth we have already concluded that length scales and absorption strength act together, and that a shift in height is similar to a shift in absorption strength. Analogously, for the absorption strength not only the carbon dioxide concentration matters, also the wavelength which is affected comes into play: for high concentrations also weakly absorbing wavelengths become important, for low concentrations only strongly absorbing wavelengths are affected.

• Wavelengths that are strongly to very strongly absorbed (optically thick) are hardly affected by a change in carbon dioxide concentration. Because they are optically thick in the whole troposphere, they are either already fully saturated or their effective emission height lies in the stratosphere, where their emission characteristics remain unchanged due to the isothermal profile there.

• Wavelengths that are very weakly absorbed (optically thin) respond linearly to a change in carbon dioxide concentration, which can be deduced from analyzing Schwarzschild’s equation. This is important for greenhouse gases with very low concentrations, or for wavelengths outside the main absorption bands. (Though a linear response seems stronger than a logarithmic response, the absolute absorption of the radiation around these wavelengths is low, so they don’t contribute much to the total energy balance.)

• The wavelengths that are most affected by a change in carbon dioxide concentration are those at the fringes of the major absorption bands: they are optically thick near the earth’s surface, but they already become optically thin higher up in the troposphere, where the temperature still drops linearly with height. This means that when the $CO_2$ concentration changes, their effective emission height will shift upward. On the other hand, the temperature is lower higher up, which would lead to less emission (and the emission will shift toward lower wavelengths). To restore the energy balance, the temperature at this level has to rise to equal the original temperature at the previous effective emission height, which can only happen if the earth’s surface temperature rises. Because the density profile is approximately logarithmic, a doubling of carbon dioxide therefore implies a linear rise of the effective emission height, so it has to be accompanied by a linear rise of the earth’s surface temperature.