It is often said that the science behind CO2 enhanced global warming is simple and settled. However, calculating the top of the atmosphere ‘radiative forcing’ of CO2 is really not that straightforward at all. First you need to assume an initial temperature profile in the atmosphere and then calculate the net upward transfer of IR photons to space from the surface. This procedure is called line-by line radiative transfer. The term ‘line’ refers to hundreds of quantum energy transition levels for vibration and rotation states of CO2 molecules.
CO2 absorbs and emits photons mainly in the 15 micron band. Chemists have measured the absorption coefficients (cross-sections) for each of the individual lines, and these are available in the Hitran database. One must assume a standard atmospheric temperature and density profile and then loop over all lines to calculate the net outgoing radiation flux from the top of the atmosphere. Next you repeat the calculation many times for small increases of CO2 in the atmosphere. The radiative forcing DS is then equal to the reduction in outgoing IR for a given CO2 concentration. This energy imbalance is the difference between the incoming solar radiation and outgoing IR radiation. This imbalance is assumed to be offset mainly by the surface warming up a little (DT) so as to increase IR flux to compensate. So it is really the sun that causes global warming not CO2 at all. All that increasing CO2 does is to adjust the radiative profile with altitude.
Once you have done all that, you find that there is an approximately logarithmic fall-off of forcing DS with CO2 concentrations. The main reason for this is that most of the strongest lines are already saturated up to the tropopause where any increased concentration (height) makes no change to the outgoing flux because temperature remains constant with height. In fact the central line is saturated so far up into the stratosphere that it actually cools the planet with increasing concentrations because temperatures rise higher in the stratosphere. You can see that in both the calculated and measured spectra as shown below.
About a year ago I wrote a program to calculate the effective height at which IR photons escape to space for each wavelength in the 15 micron band while CO2 levels increase. I indeed found a clear logarithmic fall off in radiative forcing (DS) as CO2 concentration increases. This can then be fitted this to a simple formula. The accepted fit by IPCC follows that made by Mehl et al. This is
where is the initial value (300ppm) and C is the new value (say 600ppm). (I actually got a bit higher than that)
To derive the temperature response (DT) to that forcing (DS) you need to consider Stefan Boltzmann. To cancel out the extra ‘forcing’ and rebalance energy at the top of the atmosphere, the surface will heat up so as to radiate to space exactly the same amount as before the perturbation. Crudely speaking.
Climate sensitivity (CS) is usually defined as the change in temperature after CO2 levels have doubled. So ignoring feedbacks for the moment
and putting the numbers in gives you then get
Equilibrium Climate Sensitivity (ECS) = 1.1C
To get the more scary scenarios as reflected in IPCC climate models various assumptions need to be made for feedbacks (H2O, clouds, lapse rate, ice albedo ….). Some feedbacks like lapse rate, and maybe clouds can also be negative, but overall the net feedback is mostly from moderately to strongly positive. Let’s call this net value of all feedbacks f. Due to the iterative nature of feedbacks (feedback acting on feedback), it turns out that the resulting temperature change DT becomes
f = 1 gives a runaway warming. This is a bit like guitar feedback through an amplifier where the loud speaker sound gets ampilied iteratively by the pickup.
So what is the value of f? Well if you fit the Hadcrut4 data to a logarithmic dependency plus a natural variation then you get a value
which gives f ~ 0.3 and the Transient Climate Response (TCR) = 1.5C. However it is quite possible that f is not actually a constant with temperature and may even be tuned on earth to some rough equilibrium temperature. It is remarkable that the earth’s temperature has remained favorable to life for 4.5 billion years, during which time the sun’s output has increased by 30%. The obvious candidate for such a thermostat are the oceans. They cover 70% of the earth’s surface and can both warm and cool the surface depending on temperature. The largest uncertainties in climate models are clouds and aerosols. These are the very mechanisms that could provide a natural thermostat more powerful than man’s turning up of the CO2 control knob.
You hear a lot about the debate being over and the science being settled, but that isn’t really true. There is still a large uncertainty about the future climate response to inceasing CO2 levels because the value of f remains unknown. We don’t even know whether it is a constant. You only have to look at the spread in climate models to see how uncertain the future is.
So the consensus among climate scientists is really only about the first order greenhouse effect. There are still large uncertainties about how the rest of the climate system reacts thereafter.