There is a recent post on Watts Up With That (WUWT) about the the effectiveness of CO_{2} as a greenhouse gas. It makes the following statement

According to well understood physical parameters, the effectiveness of CO2 as a greenhouse gas diminishes logarithmically with increasing concentration and from the current level of ~390 ppmv, (parts per million by volume). Accordingly only ~5% of the effectiveness of CO2 as a greenhouse gas remains beyond the current level,

and claims that this is well known in the scientific community and references Section 6.3.4 of the IPCC’s Third Assessment Report. The WUWT post then goes on to calculate how the temperature changes with changing CO_{2} concentration.

0-100 ppmv: according to David Archibald / Modtran data ~2.22°C ~2.22°C

100-200 ppmv: plants die below this level of CO2 +~0.29°C ~2.51°C

200-300 ppmv: noted as the preindustrial CO2 level +~0.14°C ~2.65°C

300-400 ppmv: current level IPCC attributes all as Man-made +~0.06°C ~2.71°C

400-600 ppmv: business as usual till 2100 +~0.08°C ~2.79°C

This post is basically claiming that the surface temperature will only rise by 0.57^{o}C if CO_{2} levels were to rise from 100 ppm to 600 ppm and by 0.14^{o}C if CO_{2} levels were to go from 300 ppm to 600 ppm.

This seemed a little lower than I was expecting, so I went to the IPCC report and found the following relationship between CO_{2} concentration and radiative forcing:

ΔF = α ln(C/C_{o}),

where ΔF is the change in forcing, α = 5.35, C is the new CO_{2} concentration, and C_{o} is a reference CO_{2} concentration. If you assume C = 600 ppm and C_{o} = 300 ppm then the above gives ΔF = 3.7 W m^{-2}. This change in forcing should produce a 1^{o}C change in surface temperature. There are two newer and more complicated equations (in the IPCC report) relating changes in CO_{2} to changes in radiative forcing, but they give the same kind of result. These are all calculations that do not include feedback and so they would be expected to be the minimum change in temperature, as feedback would tend to enhance the effect.

So, as far as I can tell, the IPCC does indeed acknowledge the logarithmic relationship between radiative forcing and CO_{2} concentration but suggests that – in the absence of feedback – a doubling of CO_{2} concentration would produce a 1^{o}C rise in surface temperature. For some reason, the author of this WUWT post gets a change of 0.14^{o}C for a doubling of CO_{2} concentration. Very odd.

I’m confused. Hasn’t the temperature already risen by 0.8C? So wouldn’t that suggest 0.57C is already too low? And 1.0C is also likely to be too low.

Well, yes. That would certainly be my conclusion. And, as you’re suggesting, the 0.8oC is from the end of what was a downward trend and so it is probably more than 0.8oC warmer than it would have been had CO2 levels not been rising since the mid-1800s.

In a good approximation the relationship is logarithmic. That is also why people always talk about the temperature increase due to a doubling of CO2. If the relation were linear, people would talk about the temperature increase due to a rise of, for example, 100 ppmv.

Exactly. I believe it is thought that the relationship might change at higher concentrations, but – I believe – that the expectation is that this would make it more sensitive, rather than less.

I think it comes from a

guest post by David Archibald.

The post My model used for deception at RealClimate explains were Archibald’s calculations goes wrong.

“But then Archibald multiplies the radiative forcing by an absurdly low value of the climate sensitivity parameter. In this case he is using the parameter in units of degrees C per Watt / m2. The two forms of the climate sensitivity parameter that we have discussed here are related by a factor of about 4 Watts / m2 for a doubling of CO2. The value Archibald uses is 0.1 degree C per Watt / m2 which was “demonstrated” in a paper entitled “CO2-induced global warming: a skeptic’s view of potential climate change” by Idso, 1998. Translated, Idso’s climate sensitivity winds up to be 0.4 degrees for doubling CO2. IPCC finds it essentially impossible (yeah, I know, highly unlikely or whatever) that the climate sensitivity could be less than 1.5 degrees C for doubling CO2, and 3 degrees C is a best-guess value.”More about Archibald at N3xus6.

That’s very interesting, thanks for the comments. Maybe you can answer something I was puzzling about today. It seems as though an increase in radiative forcing of 3.7 W m-2 is expected to produce a temperature increase of 1 K (in the absence of feedback). What I was puzzling about was that if I assume the surface temperature is 290K, then an increase of 1K produces an increase in (blackbody) flux of 5.5 W m-2. I would, therefore, naively have assumed that the radiative forcing would need to increase by 5.5 W m-2 to produce (without feedback) an increase of 1 K. My question is therefore, why does 3.7 W m-2 produce an increase of 1K?

If you use 255K, you get 3.7 W m-2 . 255 K which is about the temperature caused by the sun alone, without GHGs, This is also the temperature at about 6 km up in the atmosphere.

You can read more at this post at Science of Doom, including the comments. (You’ll find the issue much better explained there than I can do).

Thanks. I’d realised that 3.7 W m-2 was what one would expect for the temperature at something like 6km, but wasn’t sure why it was appropriate. I’ve read through the Science of Doom post and I’m starting to see why, but will probably have to read it again.