The disputed area is the yellow zone in this image:
This image shows the absorption spectrum for the atmosphere. Only a slice in the center is shown, because the concern is the absorption peaks for CO2 and water vapor. Not shown are nearby peaks for methane, ozone, sulfur dioxide, nitrous oxide, more water vapor and more CO2. The complete spectrum is shown here.
The zone has a low density of CO2 molecules
The most significant fact is that CO2 absorbs all available radiation at it's peak wavelength of 15µ in a distance of about 10 meters near the surface of the earth. Doubling the amount of CO2 in the atmosphere would shorten the distance to 5 meters. Shortening the distance is not increasing the heat. This area is the dark red color. Water vapor does about the same thing, and it is shown in blue.
Global warming cannot occur in the red or blue areas, because all available radiation is already being absorbed there. Radiation is the limiting factor, not the amount of greenhouse gas.
So the rationalizers said that on the edge of the absorption peaks there are fewer molecules, so they are not absorbing all of the available radiation at this time, and when CO2 in the atmosphere is doubled, they will absorb more radiation.
One problem is that overlap of the curve for water vapor results in all of the radiation being absorbed in the area between the curves for CO2 and water vapor. However, higher in the atmosphere, the peaks get sharper, and the overlap disappears.
Not the least problem is that there is no explanation of how the supposed temperature increase at the top of the troposphere, which is very cold, can produce heat at ground level.
The yellow zone is that area between the curves for CO2 and water vapor where there are a few CO2 molecules which are not mixed with water vapor molecules, so that an increase in the amount of CO2 will result in more radiation being absorbed which would otherwise go into outer space (or the stratosphere, which does not mix with the troposphere).
In closeup, that zone looks like this:
There are a small number of CO2 molecules in the zone, and they separate from water vapor molecules as height increases.
Humans supposedly put 30% of the CO2 in the air (116 ppm), and it increased the temperature by 0.2°C (with water vapor multiplying it to 0.6°C). If doubling the CO2 adds another 385 ppm at exactly the same rate, it would increase the temperature 0.66°C. Yet the claimed result of doubling CO2 in the air is said to be 11.5°C—about twice as much heat per CO2 molecule, even though almost all radiation is absorbed already. That's not honest science; that's fraud.

Calculating heat in the yellow zone
So how much radiation will those thin CO2 molecules absorb which would otherwise escape upward? There's not the slightest chance of anyone knowing. But there are knowable influences which set limits on the result. The unknowable variables include humidity, temperature and pressure and how they separate molecules at various heights.
The unknowns are not knowable, because they involve large amounts of randomness; and each complexity multiplies the randomness. Fake gods cannot turn the randomness into knowability.
Also variable beyond analysis is how often does radiation get reemitted as blackbody radiation after it is absorbed under such illusive conditions. Physicists assume that so many watts per square meter are emitted at a given temperature, but that analysis is only relevant to a two dimensional surface. Air does not have a two dimensional surface, yet climatologists apply that formula (called the StefanBoltzmann constant) to the atmosphere.
Out of that mess, fakes claim the globe will be 1°C or more hotter when the amount of CO2 in the atmosphere doubles. Then water vapor will increase due to the increased temperature, and since water vapor is a stronger greenhouse gas than CO2, it will supposedly multiply the result by a factor of 36 causing a total temperature increase of 36°C.
One major problem is that thermal runaway would occur, if a multiplier really existed. In other words, the 36°C heating the atmosphere should also cause more water vapor to enter the air and multiply the result by another factor of 36 getting 936°C increase, and then that result would be repeated, etc. The fact that there is no such thing happening shows that there is no such multiplier.
They didn't get the 1°C by plugging equations into a computer model. They started with the end result of 1°C and tweaked the computer model until it showed what they wanted. This can be known because there is no theory to go by. There are hundreds of complex interacting factors, and no one can reduce it to simplicity or human comprehension. To get an unbiased prediction would be like balancing five balls on top of each other and predicting which way they will fall. There are too many unknowns, too much complexity and too much fluidity in natural conditions. No one knows what the average, global humidity should be, let alone how CO2 molecules will mix with water vapor molecules 510 kilometers up in the atmosphere.
In fact, using averages for everything is a misrepresentation. There are no averages in nature; averaging is a human construct. The extremes will do different things than an average will do. It's like hitting someone in the head with a hammer and following it with five soft punches. It isn't the average that determines the result. Or it's like Bill Gates walking into a bar. The average income increases by a factor of a million, but no one is any richer.
The Heinz Hug Test
Heinz Hug did a rudimentary test by doubling the CO2 under laboratory conditions and comparing absorption spectra to see how much change occurred on the outer edges of the curve. He said that 0.17% of the curve was involved in absorbing additional radiation. He said it was 80 times less than assumed in climate models by the IPCC. Of course, opposition is not allowed in modern science, so the fakes said his measurement did not account for the colder temperature and lower pressure at high altitudes. But colder temperature will make the absorption curves narrower, which reduces the effective area even more. Lower pressure would narrow the curves some more. But notice how impossible it is to account for so many phantom variables.
Reducing the complexities to rudimentary components doesn't work for predicting, because they all interact with too many unknowns which include too much randomness. But there are limitations which can be evaluated, and they show that the quantities are absurdly miniscule.
Calculating the limitations
The atmosphere is said to be responsible for 33°C of heat. Of that, 23°C gets into the atmosphere through conduction, convection and evaporation. That leaves 10°C due to radiation. If the thin CO2 molecules were responsible for 0.17%, as Heinz Hug claimed (in an oversimplified way), then 0.17% x 10°C = 0.017°C. That's not the 1°C claimed by the fakes.
CO2 is assumed to absorb 8% of blackbody radiation due to bandwidth. In other words, the width of the CO2 absorption peaks is about 8% as wide as the total blackbody spectrum. If then, radiation is responsible for 10°C of atmospheric heat, and CO2 is responsible for absorbing 8% of the radiation, then 8% x 10°C = 0.8°C.
Notice how these numbers compare. All CO2 at this time is responsible for 0.8°C, but doubling it will supposedly be responsible for an additional 1°C or more, while 99.9% of the CO2 is already absorbing all radiation available to it, and only the thinnest edges on the shoulders of the peaks are able to absorb more. The edges are due to unusual energy states for CO2 molecules causing them to absorb slightly different wavelengths.
At the center of the 15µ absorption peak for CO2, all available radiation is absorbed in 10 meters. This means that where the CO2 is 1% as thin, radiation must travel 100 times as far to be totally absorbed, which is 1,000 meters. All available radiation is still being absorbed in 1 kilometer in that area. Doubling the CO2 in the atmosphere would do the same thing in 500 meters. Is that increasing the heat? Not enough to say so.
Where there are one thousandth as many CO2 molecules as at the peak, the distance for absorption increases to 10 kilometers, and doubling the CO2 shortens it to 5 kilometers. The shorter distance could result in some of the radiation creating heat within the troposphere, where it would otherwise go into the stratosphere.
The whole global warming effect of carbon dioxide is supposedly caused by about 0.1% of the CO2 molecules, and they are absorbing a few fleeting rays before they enter the stratosphere. If the wavelength here is 14.7µ, not all radiation at that wavelength is adding to the heat of the troposphere, because most of it is being absorbed before getting to the stratosphere; and this is occuring before the doubling of CO2 which is supposed to create global warming.
So there is no way that doubling the CO2 is going to more than double the amount of heat CO2 creates. It's more like 1/4 of 1/1,000 of 8% of 10°C, which is 0.0002°C.
The Other Side of the Story by Alarmists
This subject is described on the Real Climate web site— the usual reference for alarmists—exactly as I describe it here, except with the conclusion that the result is significant rather than irrelevant. Their conclusion is explained with this image:
(Image at Real Climate)
Their explanation is a total fraud, because it says the bandwidth widens with increased CO2 in the atmosphere. It does not. Only energy state of the molecules determines bandwidth, and it does not change with more being added to the atmosphere.
Heinz Hug measured absorption with doubling of CO2 and showed that the bandwidth does not increase.
Notice that the slope of the absorption spectrum used by alarmists is about 45 degrees of angle. This leaves a lot of space for change. But the actual absorption spectrum for CO2 is nothing resembling this. The edges are sharp. The actual absorption spectrum for CO2 is bounded by the wavelengths 14 to 16 microns. In the above image, the relevant band is shown as 13 to 17.5 microns. CO2 cannot absorb at such wavelengths. The wavelength of absorption is determined by the energy state of the molecules, while increasing the amount of CO2 does not change the energy state.
So where do these 45 degree angles come from? This graph is based upon measurements high in the atmosphere by the Air Force during WWII. The atmosphere doesn't represent anything close to an absorption spectrum for a single gas such as CO2. After radiation goes through several kilometers of atmosphere it is influence by a large number of unknown factors including scattering and absorption by any number of unknown substances. This means that the angled slopes on the curve are not caused by CO2 but other things. So the entire rationale for absorption on the edges of the absorption spectrum is not valid.
This graph is not a properly scaled representation of anything. It's designed to catch the eye of the unwary, much like the fake hockey stick graph. Oddly, they multiplied the CO2 level by a factor of four, where the usual question is what happens upon doubling of CO2. Then the lines across the graph for 1X and 4X have no meaning. They don't intersect with anything relevant on the graph.
The 1X line is arbitrarily placed at one absorption unit, and the 4X line is placed at the 0.5 absorption units, while the peak is at 10,000 absorption units. So how do they know that the increase will widen the effective absorption by almost one micron of wavelength? There is nothing on the graph or anywhere in science which will tell them that.
As the CO2 concentration increases, the slopes on the absorption curves absorb more infrared. This would be irrelevant at ground level, but it supposedly prevents more radiation from going into space at the top of the troposphere. But is the amount a lot of very little? I explain why it is very little, while alarmists say it is a lot. Is there a science to resolve this dispute? None whatsoever. In fact, the only resolution of this whole subject is to measure the claimed temperature increase of the past and assume something similar will happen in the future. (See above)
I give explanations and logic for a miniscule effect (see Crunching the Numbers), while alarmists do not explain why the heat would be significant. But regardless of the numbers, there is no explanation of how the supposed temperature increase at the top of the troposphere, which is very cold, can produce heat at ground level.
The primary effect of increasing CO2 in the atmosphere is to shorten the distance by which it does exactly the same thing it was doing. Shortening the distance is not increasing the temperature.
Over long distances, reducing the distance to one half might seem like a lot, but a few watts of heat spread over a few kilometers of height is an irrelevant temperature increase&$151like 16 millionths of a degree centigrade if allowed to accumulate for ten seconds, but will equilibrate with heat leaving on an instantaneous basis. And this is supposed to create 3°C temperature increase at the surface upon doubling of CO2 in the atmosphere.
