What is the Gain of the CO2 Control Knob?

When I was playing the the Kimoto equation, the perturbation produced a maximum Greenhouse response of -271 W/m-2 at the surface. When Steven Mosher asked on Dr. Curry's blog about what would be the gain of the CO2 control knob, that got me thinking.

A value that I came up with the approximate Effective emissivity of 0.71 was interesting. I was hoping to get some up/down emissivity data online, but haven't had much luck, so I just quit. The looking down emissivity at the TOA is about 0.61, so I thought of doing some estimating.

0.71*390=276W/m-2 This is a higher value than my estimated atmospheric effect of 220. So that value makes sense, because that would be all radiative at some altitude.

0.61*390=238W/m which should be the forcing at a lower altitude. Don't get crazy, these are just guesstimates for fun.

If, the 276Wm-2 were at the same altitude as with water vapor, the 0.7*276=193, which would indicate warming if the same downward emissivity is assumed.

0.7*238=167 at the surface which would also be some warming as the value at the would be 155Wm-2 in a line drawing, for no temperature change. The difference, 193-167=26 is the for some odd reason the value that Trenberth has for atmospheric absorption of OLR. Which would be reasonable as his OLR from the surface to space is the reason for his miscalculation. The free to space radiation interacts with the atmosphere as shown in the NASA drawings. This is what I consider part of the key issue with the choice of frame of reference. It is easy to miss something swapping your reference around. So if his drawing showed the 321Wm-2 in the tropopause, corrected for the missing ~24Wm-2 his cartoon would have been dead on. 321+24=345Wm-2 tropo Times the 0.71 Emissivity at the tropopause would equal 244 which is close to the maximum value and within the rather large margin of error of my calculations. Or the 321Wm-2 at the tropopause times the Effective emissivity would be 0.71*321=228 which is well within my margin of error.

Since in my opinion, energy must be conserved, the triangles are showing the impact of the down welling opposed by the down welling. He perhaps was inverting the triangles to show 321Wm-2 at the tropopause opposed by 228 at the surface. An alternate description that would be correct, but not illustrative of conservation of energy. In either case, 321 at the surface is incorrect, but there are options with the selection of frame of reference. And trust me, I am looking to find my error, if there is a large one.

Note: While correct, the 216Wm-2 is a number with a value and 321Wm-2 requires manipulation to sense its value. More accurate communication is important for the guys getting paid to do science. I just fish, so who cares?

As I said, don't get crazy, this was just a guesstimate, but I am interested in exactly how he made is small error that gets magnified.

If he would address the issue, it would be interesting. Oh, no water vapor interaction with CO2 would produce more warming in the upper troposphere which would produce more surface warming as the optical window to the surface would be clearer. How much, don't know, but it is the water vapor barrier to down welling long wave that is an important consideration.

Just so people don't have to run around, the initial estimate of GHG forcing, Which is really 155Wm-2, would be felt as ~345Wm-2 at the surface if you assume there was no tropopause. Regardless, of the height of a no GHG tropopause, it would exist, so the initial conditions are assuming a tropopause, if the 30% albedo was included for initial temperature and OLR. The reason my numbers balance better is I adjust for the change the height of the tropopause to maintain a consistent surface frame of reference.