This post examines how radiative forcing depends on CO2 concentrations in the atmosphere. In CO2 greenhouse demystified, we calculated the effective emission height where “thermal” photons escape to space . This height depends on the lapse rate temperature and defines the outgoing radiative flux for a given wavelength. By integrating over all lines in the CO2 15 micron band we were able to derive by how much the radiative flux reduced for a doubling of CO2. This “instantaneous” energy imbalance is usually called radiative forcing. Now we study how in detail how this radiative forcing depends on CO2 concentration. Figure 1 shows CO2 induced radiative forcing varies as a function of fractional concentration (C/C0), assuming a constant surface temperature of 288 K and a constant lapse rate.

The increase shows an approximately logarithmic dependency. A best fit to the data for CO2 concentrations from 100ppm(as reference) up to 1000 ppm(C/C0=10), assuming a fixed surface temperature and stable atmosphere gives

**R.F = 6.6 log (C/C0) , where C is CO2 .**

We have derived the often quoted formula in climate science for the radiative forcing for a CO2 increase from concentration Co to C. This canonical equation is given by ~5.4 log (C/C0)[1] ! OK so our result is about 20% higher – but that is pretty good IMHO, since the formula is never explained without reference to results from various GCM black boxes. Now we see here how it can be approximately derived just from changes to the emission height with increasing CO2.

Figure 2 shows the Planck spectra for a range of concentrations showing how OLR reduces with increasing CO2 to produce this dependence.

Finally we can also make an estimate for the net change in surface temperature due to CO2.

Averaging over clouds(), oceans and land() we then get a global averaged

If we assume that each increment in forcing DS is offset by the same increase in Black Body radiation due to a small surface temperature rise DT, then we can iterate through long term increases in CO2 concentrations. The result is shown in Figure 3.

**Summary:** By calculating effective emission heights for Co2 we have shown that:

1. Modern levels of 300 ppm of CO2 have resulted in an average surface temperature ~ 4 deg.C higher than an atmosphere free of CO2.

2. A doubling of CO2 from 300 – 600 ppm results in a further ~1.5C increase in surface temperatures.

Published results from more sophisticated GCMs reduce these figures by ~ 20%.

**References**

[1] Myhre et al. New estimates of radiative forcing due to well mixed greenhouse gasses Phys.Rev.Lett., 25, 2715-2718,1998

CB, I appreciate what you are doing with this series of posts, in providing an outsider’s sanity check.

To me it seems much of the uncertainty is partitioning the marginal effects between radiative transfer for an atmospheric profile and how that atmospheric profile is modified by convective properties of the variable gases. These pieces don’t lock together tightly, and nothing appears to dominate. Yet they have to work together to balance the energy flows and approach some sort of steady state.

Whenever this kind of coupling occurs, I have noticed that physicists like to use variational approaches ala Lagrange multipliers as a solution strategy.

This paper (On Maximum Entropy Profiles) by Verkley and Gerkema use a similar clean exposition of thermodynamic relationships that you are applying and adds in a variational strategy:

http://journals.ametsoc.org/doi/pdf/10.1175/1520-0469%282004%29061%3C0931:OMEP%3E2.0.CO%3B2

Attaching your radiative profile to their variational approach may be the missing link in narrowing down the uncertainty. We still may not know what the multipliers mean but it is something to consider.

Thanks for the suggestion. I am going to study their paper.

One interesting coincidence I found was that the CO2 radiative energy flux through the atmosphere seems to be maximised at exactly 300ppm. Too little and surface radiation takes over, too much and there is less radiative transfer upwards. It is almost as if the CO2 concentrations are driven to this value by thermodynamics.

Hi Clive. The ‘modern level is 400ppm not 300ppm isn’t it?

Very true !

I think I was meaning modern as in mid 20th century ! The exercise was to understand the formula DS = 5.3ln(C/C0). I understand now how it arises. Of course there is also the assumption that nothing else changes. So for example that the lapse rate remains constant. If the lapse rate decreases the GHE also decreases.

Thanks Clive. The assumption that ‘all other things remain equal’ is indeed a dangerous one. Miskolczi’s analysis of radiosonde data revealed that atmopheric opacity hardly varied over the period of record, despite the increase in airorne co2.