It is claimed that a doubling of carbon dioxide in the atmosphere will cause a temperature increase of 1.2°C and the only question is how much increase will be due to forcing effects, mostly by water vapor. Computer models are used to evaluate the forcing.
There is an equation for calculating the 1.2°C temperature increase; and it is such a simple equation that it seems to never be questioned. The problem is, it is too simple. It is a three component equation.
The equation is this: Heat increase = 5.35 ln C/C0. Temperature increase = 0.75 times heat increase. C sub zero is the starting concentration of CO2 and C is the final amount. Since the question is doubling, this ratio is two. The natural log of two is multiplied times the constant 5.35. Then the heat increase is multiplied times 0.75 to get the temperature increase. If only seventh grade general science were so simple.
Determining where this equation came from is no easy task. Steve McIntyre tried to trace down the citations for it in the IPCC documents and failed (1). All of the references led to no real explanation. The citation problem wasn't just glaciergate, Amazongate ad infinitum; it was also in the claimed, unquestionable physics.
Norm Kalmanovitch (2) determined that the first appearance of the equation was in 1988 paper by Hansen et al (3) and drew the conclusion that the equation would have been based on the claimed increase in CO2 of 100 ppm since the industrial revolution combined with a supposed temperature increase of 0.6°C.
As the equation first appeared, with the constant 5.35, it would show an atmospheric temperature increase of 3°C upon doubling of CO2. But that amount was not credible, so somewhere along the line it was changed to 1.2°C. To get that amount, the constant would have to be 2.31, not 5.35. Lately, the expected temperature increase is being stated as 1°C, which would require the constant to be further reduced to 1.92. Then 15% was removed for overlap of the absorption peak with water vapor, which would require an increase in the constant to get the desired 1°C.
This isn't how laws of physics are supposed to work. It's how fudge factors work. If the equation were a simplified expression of some complex analysis, it wouldn't be quite so fluid or unquestionable. Supposedly, this type of analysis involves "radiative transfer equations" based on Beers-Lambert laws with some Planck's equations added. This requires some of the largest computers models in existence. There is no indication that Hansen et al went to that much trouble. They appear to simply have written a fudge factor to meet their needs.
One of the problems with applying the radiative transfer equations to the atmosphere is that they do not apply to saturation. Most of the greenhouse gasses totally saturate in the atmosphere. Heinz Hug (4) indicated that carbon dioxide saturates at the center of its main absorption peak in ten meters. Rationalizers claim that the shoulders do not saturate; however, it would take very thin shoulders to go from 10 meters to outer space. Not only would such etheric effects be inconsequential, any number of other minute effects would overlap the absorption.
The claim is made on Real Climate (5) that the absorption band gets wider as the amount of CO2 increases in the atmosphere, but that's not true. The bandwidth is determined by the energy state of the molecules which is not influenced by concentration.
Then there is a major contradiction inherent in the whole process of using the equation/fudge factor. If the heat created by CO2 is multiplied by a factor of three due to increased water vapor, the same thing would have been true for the first 0.6°C which humans supposedly caused since the industrial revolution. This means that only 0.2°C can be attributed to CO2, while the other 0.4°C was due to water vapor. To get 0.2°C out of the fudge factor, the constant needs to be 0.87.
In other words, the multiplication by water vapor would have occurred twice--once to get to the fudge factor and then again after applying the fudge factor. Otherwise, the fudge factor should show much less than the 1 or 1.2°C that is claimed as the unquestionable physics which creates the settled science of climate change.
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3. Hansen j. et al., J. Geophys. Res., 1988, 93-D8, 9341-9364.