Power Series or Opposing Forces?
There are dozens of ways to solve problems. The atmospheric effect is not different. It is actually a simple problem. Feedbacks in the dynamic system are the problem which should be a little easier with the correct solution of the atmospheric effect. Being a big fan of keep is simple, I choose a simple frame of reference, the take small steps from a solid foundation toward a better solution of the more complex issues.
The Stefan-Boltzman equation is required since radiant energy is involved, though in the lower atmosphere it is not. Different methods can arrive at the same answer.
Since the Average surface temperature is 288K and the average outgoing energy flux at the top of the atmosphere is 238Wm-2, the S-B relationship makes perfect sense for a beginning. Using F=5.67e-8(T^4) and T=(F/5.67e-8)^0.25 you simple get temperature @ flux values for the surface and TOA.
288K and 390Wm-2 Surface and 238Wm-2 and 254.5K at the TOA, is the simple relationship between the surface and the True TOA. This has nothing to do with a tropopause, it has nothing to do with a no greenhouse gas Earth. Thinking it is more is a problem,
So on one of the blog, a commenter mentioned using power series to determine what a change in surface flux would cause. There is nothing wrong with that approach. There may be something wrong with the hypothetical question of the change being felt at the surface.
If we take 238Wm-2 outgoing flux and divided that by the surface outgoing flux we have 238/390 or 0.61. This is the emissivity of the atmosphere of the Earth. So one can consider that 1-.61 or 0.39 is the atmospheric resistance, feedback, whatever term you like. I happen to call it the “Effective” Emissivity or the portion of the surface emissions that has an effect on or is affected by the atmosphere. The Effective emissivity includes all thermal flux interactions with the atmosphere, interactions I know and interactions I may not know.
Since an impact of change from the surface will impact the atmosphere which will in turn impact the surface, at some point stabilizing. This is good application for a power series as I understand it.
So if the surface were at 254.5K and 20K was suddenly added to the surface temperature, we would need to use the flux values, which are non-linear with respect to temperature, to determine the stable resulting temperature.
238Wm-2 @ 254.5 K is already known, with 20 more degrees the flux at 274.5 would need to be determined by S-B which is 263.8Wm-2 @ 274.5K
So, the difference in the flux would be 263.8-238=25.8 added to the atmospheric flux, of that 39% would be returned or 10.1Wm-2, requiring 0.39*10.1 or 3.94, with 0.39*3.94=1.54 etc. etc,
This solution is simply 25.8/(1-.61) or 25.8/.39 = 66.15 meaning the stable surface flux would be 238 + 66.15 = 304 Wm-2. The 66.15Wm-2 would be a portion of the atmospheric effect IF, the 0.39 return was linear. Just for grins, let us pretend it is not. Let’s say something crazy like the effective emissivity in the lower troposphere where higher, say 0.75, effective emissivity.
Then let’s say we added 20Kto the temperature at some point in the lower atmosphere. Just for a guess, the 274.5 but this time in the atmosphere, We have the same flux as before, 25.8Wm-2 only 0.25 instead of 0.39, yielding 25.8/(1-.75)=25.8/.25 = 103Wm-2 at some point in the lower atmosphere.
That 103Wm-2 plus the 66 Wm-2 =169Wm-2, which is pretty close to 390Wm-2 surface minus 238Wm-2 TOA or 152W atmospheric effect, not too bad for a guess, had I guess 0.7 instead of 0.75, the value , , 25.8/.3=86, would exactly equal 152W/m-2, the ideal atmospheric effect as seen at the TOA. That added 20K in the atmosphere is radiated in both up and down. Before we had the surface increase from 238 to 274.5K @ 304Wm-2, adding the 86Wm-2 from the atmosphere would yield 390Wm-2 at the surface.
This is my visualization of the Atmospheric Effects, with the 20 degrees at the surface being most of the surface temperature increase and radiant gases in the atmosphere creating a balancing effect in the atmosphere. It only requires one more element.
That element is the lapse rate, including latent heat, which transfers heat directly from the surface where condensation begins to release that hidden heat. This depresses the 274.5K atmospheric value by approximate 20K to 255K as appears to have been estimated by Arrhenius, with one small issue, the latent component of the lapse rate.
There are several ways to estimate the temperature of the average layer of radiant impact, of the source of down welling long wave radiation, The simplest being the average mass level of the atmosphere or the average energy level of the combined lapse rate, conductive, latent and radiant. That would where ½ of the energy transferred from the surface to the atmosphere can be approximated.
A simple estimate based on observation would be the average altitude of low cloud bottoms or the average layer where condensation begins. This altitude is approximately 4000 meters.
Of course, one would have to assume that the return value even after applying power series can never be greater than the input value or perpetual motion would be involved.