Note: I have started a supplemental post to discuss the potential error range of the approximation.
Update: Well, Trenberth's cartoon for one thing.
I was thinking my method sucked, but I am getting the idea most of the issue is in the cartoon. In any case, the NASA budget and the K&T budget have some inconsistencies.
First, seeing that whopping 324 Wm-2 coming out of that huge greenhouse cloud, I can't take it anymore. The Earth is warmer by about 33 degrees thanks to the greenhouse effect and that is because it re-radiates 155 W/m-2. The sky is still warmed by the sun, so the 169 in red is its contribution.
In trying to figure out what Christopher Monckton was on about, I started looking at the cartoon again. To me there is a pretty obvious relationship at the surface with three heat flows or fluxes, Thermal, Latent and Radiative,All three depend on the Earth for their energy, heat flowing from a source at 288K and each have different factors that effect their rates of flow. At the surface the total outward heat flow is 390Wm-2 based on the black body temperature. At the top of the atmosphere, the flow rate is 235 Wm-2, or equivalent to about 255K degrees (actually, 240Wm-2 is the more typical value used for 255K). The difference is due to the atmospheric effect or greenhouse effect.
So for my explanation I assume that layers of the atmosphere and the surface want to be in energy equilibrium, Ein=Eout. That is the way it is at the top of the atmosphere, that should be the way it is in the atmosphere and that is the way it should be at the surface, though it will never be in true equilibrium. Over some length of time, the approximation of equilibrium would be valid. This is a pretty common assumption and no one seems to be bent out of shape over that.
Since we are mainly concerned with a change in the radiative flux due to more CO2, CO2 change mainly impacts the outgoing radiation, I am looking for what impact CO2 change would have on surface radiative flux. An addition 3.7Wm-2 of flux would be a 14% change in the portion of the radiative flux impacted by the atmosphere. Since the total flux at the surface is 168, that would be a 0.022% change in the total flux. As the radiative impact of the atmosphere is 33 degree (288K-255K), that would cause a 0.73 degree change in surface temperature. The estimated value of a change of 3.7 Wm-2 is 1.2 - 1.5 Wm-2. If I take the difference between the total flux at the surface for a black body at 288K = 390Wm-2 and subtract the TOA flux of 235 there is 155 Wm-2 difference. 3.7Wm-2 divided by 155 would be 0.024% change or 0.78 degrees change. So there are two quick estimates of climate sensitivity based on the flux averages listed on the K&T cartoon, both are in the ballpark, but neither are correct.
Also if I divided 3.7 by 390Wm-2, the surface flux, I get 0.0095, multiply that by 288K and I get 2.73 and at the TOA 3.7/235Wm-2 = 0.0157, multiply that by 255K, I get 4.0K. Why? Because the atmosphere only adds approximately 33 degrees to the surface temperature. A simple ratio will only work for extrapolating the actual range impacted for small changes. I can use the S-B equation for the change in surface radiation from 390 to 393.7 Wm-2 and get a 0.68 change in surface temperature. That is still not 1.2 - 1.5 degrees, but using the more standard method of calculating temperature of a black body, there is some agreement with the ratio extrapolation. S-B does require an estimate of emissivity, since the Earth is not a true black body.
Assumption for the use of ratio and proportion to estimate change in surface flux: In purely radiative heat flow, if one portion of the spectrum is restricted, the unrestricted portion will uniformly increase to regain equilibrium. As energy is fungible, in a mixed heat flux environment, if one portion of any flux is restricted, the unrestricted fluxes will uniformly increase to regain equilibrium within the limits of their physical heat flow characteristics.
The way I chose to handle the problem was with a simple ratio of the change in radiative flux at the surface. The fluxes are, Et(thermals or sensible heat), El (latent heat), Era (radiant energy interacting with the atmosphere)and Ers(radiant energy lost directly to space). Using the above assumption, the Eout = Et+El+Era+Ers, a change in Era would cause a change in the remaining heat flux to maintain equilibrium. I had to calculate the Era by subtracting the 324 down welling from the 350 upwelling after the Ers, or radiation directly to space was subtracted. That gives us 24+78+26+40=168 Wm-2 leaving the surface. CO2 will only impact the Era. Feedbacks from that will have some impact on the rest, but they are not directly impacted. Using a ratio of the change in Era by 3.7Wm-2 to total upward energy fluxes, I came up with 29.7/171.7=0.179 or 17.9 percent change. I could have calculated a reduction, 22.3/164.3=0.135, but it is just an estimate, so why bother?
In the atmosphere we have Esun (the solar energy absorbed by the atmosphere)+Et+El+Era warming the atmosphere. Using the same ratio, I get 29.7/(67+24+78+29.7)=29.7/198.7=0.149 or 15%. For a reduction I could have calculated 22.3/191.3=0.116 or 12%. In both cases, the temperature that would be changed is 33C so the range of temperature change due to the atmospheric imbalance would be 3.96 to 4.96 from an initial value of 4.4 degrees due to radiative absorption in the atmosphere or -.48 to 0.56 degrees. At the surface, 4.4 to 5.9 with 5.2 the initial value for a range of -0.8 to 0.7 degrees.
If I were to derive a better method, Ein=Eout at the TOA, so from the top down, the integration would be from TOA to surface which is a temperature range of approximately 33 degrees. There is no need to start with 288K. If I did want to start with the 288K S-B gives me an estimate of 0.68K with an assumption of emissivity equaling one.
In the interpolation for the atmosphere, the total energy in initially is 195Wm-2. Using the same assumption of equilibrium, the effective temperature of the atmosphere is 242.2 degrees K. With increased absorbed Era, that temperature would increase by 0.56 to 242.76K. The surface would increase by 0.7 to 288.7K. The temperature differential between the surface and the atmosphere would increase from 288-242.2=45.8 degrees to 288.7-242.76=45.96K. That is 0.16K increase relative to 45.8K differential or a virtually negligible, 0.35% increase. That indicates minimal increase in the thermal and latent fluxes. Assuming that the reduced surface flux Era would proportionally divide between the other less effect fluxes, the increase would be 3.7/168=0.022 or 2.2 percent. Since temperature difference and its effect on pressure, is an important factor in the latent and thermal, the flux adjustment to regain equilibrium would of the shift to Ers, the radiation direct to space, as minimal change to radiation window is indicated, the path of least resistance. There is no indication of significant water vapor, differential temperature or radiative feedback to the small change in Era flux.
Since the K&T drawing provides 324W/m-2 value for down welling radiation, the effective temperature of the atmosphere would be 279.5 degrees. On the drawing above I split the down welling into 169W/m^2 due to Esun, Et and El, the majority of this energy is day time accumulated, and 155W/m^2 the outgoing long wave radiation greenhouse effect down welling radiation. The change in the diurnal resistance to heat loss is better illustrate with this modification.
As previously noted, the TOA flux indicated is 235Wm-2 equivalent to 253.8K instead of the more common estimate of 255K. This can cause a margin of error of 3.6 percent. Since the tropics are effective saturated for changes in radiative flux in the spectrum of CO2, most of the change in radiative balance would impact the higher latitudes. Then the approximate 0.8 increase in global temperature would be realized as greater increase in the mid to upper latitudes of approximately 1.6 degrees, predominately in the Northern Hemisphere where the land mass ratio would amplify impact. This imbalance increases potential feed back of water vapor to a significant value.
Update: On the question of is a linear approximation of any use? For a 10% change in forcing of Era, the error is approximately 6 percent due to the approximation. Cumulative errors would be pretty large. For small changes it could be useful, though a little more sophisticated method would be better. The inter-dependency of the three fluxes is interesting.
Musing: El, the latent flux is also dependent on available water vapor. With the large temperature difference between the poles, the north pole summers are near the freezing point of water while the south pole is well below freezing. Latent feedback at the south pole will be less than the north for a small change in Era. In this musing, it appears that a simple model would include three distinct regions, the Northern Extent, the Modified Tropics and the Southern Extent. With the modified tropics defined as the region where 50% of the incident sunlight is absorbed, the the thermal characteristics of the Northern Extent, more impacted by water vapor and albedo feedback could be contrasted with the Southern extent where radiative feedback would be less pronounce in individual fluxes.
As always this is a work in progress, feel free to comment.
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