Previously I did a three disk model with 2T as the source and T as the sink, then inserted another disk. I was going to add another disk representing CO2, but my spreadsheet to calculate the CO2 spectrum at lower temperatures is being fought by my urge to nap frequently following typical seasonal gluttony.
Since I forced myself to stay awake for a football game that deserves a nap, I figured I would lay out a slightly different model using the average tropopause versus surface.
So T this time is 288K @ 390Wm-2 and the tropopause will be approximately 213K or - 60C at 117Wm-2, so that will be 213/288=.74T @ 0.3F. As before, this would be a night thing we don't have to deal with solar absorption in the atmosphere.
The surface disc in this case is massive compared to the Tropopause disc and the CO2 disc. If I insert the CO2 disc at some temperature between the source and sink, it will stabilize at a temperature and flux that should represent the atmospheric effect of CO2 to some degree. Note that the Tropopause that does exist is stabilized at 0.3F.
When I insert the CO2 disc, it absorbs from the source and returns to both source and sink equally. If the CO2 disc absorbs 117Wm-2 plus 3.7Wm-2, then its initial effective temperature would be 214.8K @ 120.7Wm-2, it would return half or 60.3Wm-2 which since source is massive would return the same amount. However, since the CO2 disc can absorb at a maximum, 120.7/390 or 30.9% of the broad source emission due to the returned flux, it will absorb 18.8Wm-2 returning 9.4Wm-2 the second time around. So, 120.7+60.7+18.8+2.9... so we will say it would approach 210Wm-2 for a round number. The flux from the source not absorbed by the CO2 spectrum would pass through to the Tropopause so there may be some slight change to the Tropopause temperature. The Tropopause is fairly stable, so just from grins we will assume its change is negligible for the moment.
If this happens to be the stable temperature of the CO2 disc, then it is at an equivalent temperature of 246.9K @ 210Wm-2 which is -26C. So the addition of 3.7Wm-2 worth of CO2 forcing would change the effective emissivity of some layer of the atmosphere from some temperature to approximately 246.9K degrees.
This is where the monster be! It is assumed that the 3.7Wm-2 additional forcing would produce approximately 1 to 1.2 degrees of surface warming. If this layer of the atmosphere were initially at 206.3Wm-2 (245.6K), the the add 3.7Wm-2 produced a 1.3K increase in the temperature of this layer. Since the source initially saw a sink at 117Wm-2 with a layer at 206.3Wm-2, and now sees that intermediate layer shifted to 210Wm-2, it is seeing 3.7Wm-2 which it politely returns in full from a different location. The location shifted from 245.6K to 246.9K, so what would really be the impact on the source? That would totally depend on the layers between this new CO2 effective radiant layer and the source.
The reason I use source is because CO2 can only absorb in its spectrum based on its temperature. The higher the temperature of this layer the greater the potential broadening of the spectrum of the CO2 layer. Layers below this temperature with spectra compatible with CO2, as in other CO2, overlapping water vapor and water in liquid or solid form, can return radiation from the new CO2 layer without as much impact on surface temperature.
So we end up with a variety of local sensitivities to a doubling of CO2 dependent on the radiant spectra of the atmospheric layers between the CO2 and the ultimate source of energy for the CO2 layer to return.
In the Antarctic, where there is little water of any phase in the air, CO2 is the main source of the energy to be returned. With the average temperature of both the source and the CO2 layer well below -26C, the energy available cannot produce the full 3.7Wm-2 anywhere near the surface. Also with CO2 being the primary source and return, the available spectrum is effectively filtered to the
co2 spectrum for any radiant interaction. This effect is noted by the lack of Antarctic temperature increase with increased CO2.
In the Tropics with the preponderance of water in all phases, the average altitude and therefore temperature of the radiant layers would be more greatly separated from the initial source at the surface. The additional CO2 would interact with clouds and water vapor near saturation producing upper level convection or acceleration of the rate of convection, which I believe has been noted.
Finally, in the Arctic region where surface energy is available, the increase in CO2 can interact with water and water vapor closer to the surface to have a noticeable impact on surface temperatures as advertised. Since available energy is required for CO2 enhanced return, changes in snow and ice cover greatly impact the CO2 enhancement of the "Greenhouse Effect". Which I believe is evident in the ice core and other paleo data in the northern hemisphere. This impact is highly dependent on internal variability of energy and moisture associated with natural weather circulations, increasing the amplitude of the temperature variability in these Arctic and near Arctic regions.
It should be noted that the much greater land use changes made in the past 500 to 1000 years, may have enhanced the atmospheric effect sans additional fossil fuel CO2, by reducing the expanse of snow fields which tend to sequester biological carbon. Draining swamps and marshes, building large urban areas, and in general doing the things civilized man does, would also impact the available energy and radiant gases to enhance the "Greenhouse Effect".
There are a wealth of sources that deserve mentioning for gleaned information used to write this little post. Any and everyone that ever published or thought of publishing anything that may have contributed to this post can request mention, complain that they were not mentioned or attempt to sue the crap out of me. I will try to eventually get around to providing sources. For now, I think I will just pester someone on some blog some where :)