In this post I shall assume that rises in CO2 concentrations are responsible for all observed increases in global surface temperatures since before the Industrial Revolution. I shall ignore any other mechanism which has been proposed as a possible natural cause of the observed warming; for example recovery from the Little Ice Age, solar activity etc. Under this assumption we can then simply assume that changes in temperature deltaT = some function(CO2 concentration)
The atmosphere is opaque to IR radiation emitted from the Earth’s surface for the main CO2 absorption bands. Radiation to space in these bands then takes place in the upper troposphere where temperatures are much colder due to the Lapse Rate of 6-10 degrees/kilometer. As CO2 concentrations increase over time the fog gets a bit thicker, and the effective height where the fog clears moves to a higher level where the temperature is lower, thus reducing heat loss. As a result of all this the Earth’s surface warms slightly, increasing the outgoing IR radiation until the overall energy balance is maintained. This is my understanding of the CO2 greenhouse effect. Radiation transfer calculations based on this theory  result in a logarithmic dependence of the radiative forcing (extra heating effect), as follows.
deltaE = 5.3* Ln(C/C0) watts/m2
where C is the CO2 concentration at some time and C0 is a reference value for defining deltaE. So a doubling of CO2 concentrations will give an additional radiative forcing of 3.7 watts/m2 (5.3ln(2)), and a quadrupling of CO2 would give a radiative forcing of 7.3 watts/m2 above the baseline. The logarithmic dependence ensures a diminishing return for ever increasing CO2 concentrations for the radiative forcing.
The Earth responds to the forcing by warming up a little to radiate more energy thereby maintaining overall energy balance. So very simply the energy balance becomes
E + deltaE = e.sigma(T+deltaT)**4 , where E is total outgoing IR to space from the surface
E(1+deltaT/T)**4 = E +deltaE
binomial expansion gives:
deltaE/E = 4deltaT/T
deltaT = (5.3T/4E)ln(C/C0)
Putting in the numbers E=239 watts/m2, T=288K, deltaE=5.3ln(C/C0) watts/m2 gives
deltaT = 1.6ln(C/C0) in degreesC
So let’s look at the data. Here, I am using the full Hadcrut  data from 1850, overlayed with a smoothing curve as described in the previous post. For the CO2 data I am using the Mauna Loa data , and assuming an extrapolation back to a pre-industrial value of 280 ppm. The curve through the CO2 data is simply a least squares fit through all the points extrapolated to 1850 I then use these CO2 level for the model calculation, whereas The CO2 values used are the reference values as of 1990 (354 ppm)
The simple model underestimates the observed rise but the shape is reasonably good. A better fit is given by the red curve using 2.5ln(C/C0). It is also possible that there are other natural warming trends present particularly recovery from the little Ice Age, which lasted until the end of the 19th century.
We can then apply these formulas to make some predictions. Let’s assume that levels of CO2 concentrations continue indefinitely to increase at the same rate. The predicted temperature rises from current temperatures are:
Period delta(C/C0) deltaT(2.5) deltaT(1.6)
1990 ->2011 0.1 0.24 0.16 (see last post and graph below)
1990 ->2030 0.2 0.5 0.32 (see last post and graph below)
2011 ->2100 0.41 1.0 0.7
If this simple formula has validity, then it predicts that by 2100 average temperatures would have risen above current values by just 0.7 to 1.0 degrees C, even if no reduction in CO2 emissions occurs in the meantime.
To be fair considering what I wrote in the last post comparing IPCC 1990 predictions to data from 1990 to 2011, I enclose below a new comparison between the data and these two new simple predictions. The 1.6ln(C/C0) line also appears to be rather too low despite the limited range of data available.
Disclaimer: I fully admit that I am not an expert on climate science, with only a superficial knowledge of the detailed ocean/atmosphere interactions, or the various feedback mechanisms with water vapour, CO2 outgassing etc. that have been parameterised in various climate models. I accept ahead of time criticism on this point. Perhaps the increase of the constant from 1.6 to 2.5 required by the data parameterises the climate sensitivity response to past warming. Perhaps a natural recovery from the Little Ice age due to increased solar activity is mixed in as well.
An analysis of Radiative Equilibrium, Forcings and Feedback, Christopher M. Colose