# Why is dependence of radiation forcing on CO2 concentration logarithmic?

As a newbie on climate science and meteorology I'm trying to understand the almost logarithmic behavior of CO2 concentration on radiation forcing

$$F(c) = A \cdot \ln \frac{c}{c_0}$$

and why there is no "saturation" due to further increasing of CO2 even when the atmosphere is already totally absorbing as a whole.

So far I learned that the air layers which contribute most to outgoing radiation are placed where optical thickness, as measured from TOA, is about 1. Let's say in an undisturbed atmosphere this location is about a height of z=2 (arbitrary units), then a CO2 doubling leads to a shift of this layer to about z=3.

Since this higher layer has usually lower temperature, less energy is irradiated an, therefore, the planet will warm up.

On the other hand, other explanations, even mentioned in the cited publication, are based on spectral effects where the wings of the absorption band become more absorbing at higher CO2 concentrations:

Q1: Firstly, I don't understand why this last effect is, because CO2 is quite dilute in atmosphere and this effect cannot be due to pressure broadening. What is the reason for absorbing more CO2 at the wings when concentration rises up?

Q2: secondly, which of the two effects is more relevant to explain the observed log behavior? Are they related in any way or do they just happen to give the same overall result?

• Have a look at the 'curve of growth' of spectral line equivalent width: spiff.rit.edu/classes/phys440/lectures/curve/curve.html Even at high optical depths, 'saturation' doesn't mean that there is no increase in absorption anymore. Rather, absorption starts depending logarithmically, rather than linearly on the absorber density, as the line core is saturated, but the line doppler wings continue to contribute. The arguments for a single line then directly translate to the line forests of molecules, which is why the overall CO2 opacity also follows the log-behaviour. Nov 10, 2022 at 15:05
• Does it really mean that the absorption characteristics of a particular molecule, e.g. CO2 depend on the density of those molecules? Cannot imagine, because wit 300ppm CO2 molecules are very rare and they do not interact in any way. Most of the other molecules for a single CO2 are not CO2, but N2, O2, ... so if there is some modification because of collisions, most of the collisions come from other molecule types - not CO2. So I would assume some dependence on the total pressure, but not on partial pressure of CO2. Do we talk about the same thing? Nov 10, 2022 at 15:31
• The pressure broadening/collisional broadening does not matter on the collision partner when you are close to thermal equilibrium (we're far from the ionosphere here). Study the document I've linked carefully, it doesn't take a long time. The log(c) dependency of the total absorption depends on the number of excitations any single line gets hit with times the number of absorbers, which is just another way of encoding the optical depth and hence the partial pressure. Nov 10, 2022 at 17:45
• I would say that is correct. Nov 15, 2022 at 20:44

Your confusion, I think, comes from a subtlety, and I think I can only answer this from the perspective of an astrophysicist. We don't think in terms of "forcings", the vocabulary is still alien to me (I have yet to see a sensible, formal defition of it). But I will answer in terms of a certainly related concept.

In the optically thick part of an atmosphere (i.e. the one that potentially features a greenhouse effect), we know from analytic solutions that the Temperature $$T$$ as function of optical depth $$\tau$$ is approximately $$T^4=T^4_0(\frac{2}{3} + \tau)$$, where $$T_0$$ is a measure of the energy content of the optically thin radiation field.

The total optical depth $$\tau$$, itself is simply computed as a sum of all opacity carrier contributing species $$s$$, with number densities $$n_s$$, opacity function $$\kappa_s$$, and optical ray path $$ds$$ as $$\tau = \sum_s \int ds \, n_s \, \kappa_s.$$ This already answers the part of your question about the partial pressure dependency: $$\tau\propto n_s$$, no matter how the opacity is generated (I am using pressure and number density at constant temperature interchangeably, but to first order it is $$n_s$$ and not $$p_s$$ determining $$\tau$$).
Now collision-induced absorption, which gives rise to the broad Lorentzian line wings, which grow logarithmically at large optical depth, is dependent on all available collision partners, hence this is where the total pressure comes into play, via $$\kappa_{CO2} = \kappa_{CO2}(\sum_s n_s)$$. In the last sum, the $$CO_2$$ contribution would be negligible, but $$\kappa_{CO2}$$ still grows logarithmically.

So in my understanding we'd rather have something along the lines of $$\tau \propto n_s \times log(f(\sum_s n_s))$$

I would still like to better understand how the 'forcing' comes about from those quantities, but it does give you an answer as to why the temperature to which the system wants to get to (the above $$T^4$$) has the dependencies that you are confused about.

Just wondering about your statement above: "Since this higher layer has usually lower temperature, less energy is irradiated an, therefore, the planet will warm up". This doesn't sound right. For one thing, temperature at any layer is not a passive variable: if radiation (absorption and emission) is happening at a higher level, then the temperature at that level should rise too.