Because of a typo and dropping the exponent, the numbers above are off. I'll revise the numbers in the morning. That's what I get for posting without letting it set before final proof.
Looking at the NASA cartoon a couple things stick out, the 6% radiated directly to space and the 23% heating by latent or evaporation to water vapor. The average temperature of Earth is 288 degrees K which is 15 degrees above 273K which is the freezing point of fresh water. Using another simple ratio, 23%/15 equals 1.5% per degree. So with a three degree warmer atmosphere, the latent heat would increase by 4.5% to 27.5%. Since the original 23% is 78 Watts/m^2, the increase due to the 3K is about 15 Watts/m^2 or 5 Watts/m^2 per degree.
The 6% radiated to space is about 20.5 Watts/m^2. CO2 doubling will close some of that atmospheric window. Just for an estimate, let's say it closes it all. So instead of the ~4 Watts/m^2 estimated by the IPCC we have 20.5.
While this is essentially mixing apples and oranges, the 20.5 Watt/m^2 by closing the window plus the 15 Watts/m^2 assuming increased water vapor due to 3 C warming is about 36 Watts/M^2.
We have a lot of apples and oranges plus maybe a mango. The Mango is the Sun. It puts out 342 Watt/m^2 roughly and that is not changing, much anyway. If we close the window and assume 3 degrees K worth more latent heat, we only have 36 Watts/m^2. The S-B law says that radiation will increase by
Just for shits and grins, let's say that a doubling of CO2 with water vapor feedback leads to 1.3 degrees K of warming. If you look at the NASA cartoon you will see that there are three layers of different color blue. In the bottom darker layer, no radiation from the Earth is absorbed. There is some of course, but it is dominated by latent heat and conduction/rising air. The absorbed radiation from the Earth's surface dominates in the middle blue layer even though there is some latent and conduction/rising air in that layer due to major convective storms. The change in CO2 impact should dominate in that layer. The doubling of CO2 may very well increase the absorbed radiation budget of that layer by 4 Watt/m^2. That does not mean the impact of the 4 Watts/m^2 in that layer will manifest itself as 3 K warming at the surface. It will have some impact at the surface and that impact will increase evaporation. Just as a guess, let's say the impact by CO2 alone is 1 K at the surface with the other 0.3 K due to water vapor.
Based on that assumption, the window is shut a little so now it is 16.5 Watts/m^2 The latent heat increases from 78Watt/m^2 to 79.5 Watt/m^2. So why would I think something so silly? I mean, CO2 doubling increases the radiative forcing by about 4 Watts/m^2, but only increases the surface by 1 degree with water feedback that kicks the impact up to 1.3 degrees. The reason is where the 4Watts/m^2 main impact dominates, is a low temperature region of the atmosphere. Energy in is going to equal energy out, with minor fluctuations. The apparent temperature from space is the thermosphere where the temperature is warmer than the surface of the Earth. The top of the troposphere (on average) into the lower stratosphere is the lowest temperature of the atmosphere in relation to the major layer where CO2 increased forcing will have an impact. Just like the apparent temperature of the ocean is the surface. The change in the radiation budget will not be at the top of the thermosphere, it will be in the upper troposphere into the lower stratosphere and less so at the surface. While the increased CO2 will decrease the lapse rate, heat still flows from warm to cold, which implies that most of the heat from CO2 doubling will flow to the nearest cold region of the atmosphere. Major warming of the surface will not occur until the nearest cold region of the atmosphere, the region around the tropospause and 10 kilometers into the stratosphere, warms uniformly. So the lapse rate would have to decrease by over 50%, for the majority of the CO2 forcing to be of impact at the surface.
Since I can't find an English copy of Arrhenius' second paper where his impact of CO2 doubling reduced from 5 C to 1.6 C, I can't be sure. I doubt he just caved. I suspect he reconsidered the Tropopause sink, which lead to his re-evaluation. If I were to redo the Energy Budget cartoon's, I would be more specific about the layers of the atmosphere where the dominate impacts occur. That would simply the cartoons without over simplifying them.
So my answer is there will be no 3 degrees warming by 2100 at the Earth's surface due to CO2 doubling, but the window will shut a little and there will be more water vapor. Then Stefan's law is not violated, Arrhenius' revised law is not violated nor are the laws of thermodynamics. But then, I am just a fisherman, that applied for a job at a Holiday Inn Express.
My Epiphany may be shot because of the S-B mistake, I will need to redo my spread sheet, because of the garbage. The flow from the CO2 heating "Zone" should still have a significant sink component.