By Dave Burton, Oct. 11, 2020. (Adapted, in part, from a blog comment.)
The formula for estimating radiative forcing from a change in atmospheric CO2 concentration is usually given as:
ΔF = 𝞪·ln(C/C₀) W/m²
C/C₀ is the ratio of new to old CO2 concentrations
and, the most common estimate is:
𝞪 = 5.35 ±0.58
Those figures are from Myhre 1998, who also reported an uncertainty “of order 10%,” which would be about ±0.54.
Prof. Will Happer (2013) reports calculating (and see also Happer 2015), based on corrected modeling of CO2 lineshapes, that the “5.35” coefficient is about 40% too high, which makes:
𝞪 ≈ 3.8 ±0.5 (which is 2.6 ±0.5 W/m² per doubling)
Prof. Joshua Halpern reports:
𝞪 = 4.35 (but I don't know what his source is):
Feldman et al 2015 measured downwelling longwave IR “back radiation” from CO2, at ground level, under clear sky conditions, for a decade, to determine the effect of increasing atmospheric CO2 concentration. They reported that a 22 ppmv increase in atmospheric CO2 level (+5.953%, starting from 369.55 ppmv in 2000) resulted in a 0.2 ±0.06 W/m² increase in downwelling LW IR from CO2. 5.953% happens to be almost exactly 1/12 of a doubling, which makes their measured result +2.40 ±0.72 W/m² per doubling of CO2.
However, radiative forcing is customarily defined at TOA, not at the surface. That is, it's specified in terms of an equivalent change in solar radiation. The effect of GHGs on radiation balance at the surface is similar, but not identical, to their effect at TOA.
The only way that energy leaves the Earth at TOA is by radiation, but at the Earth's surface the energy fluxes are much more complicated, including important mechanisms like convection and evaporative cooling. Warming the Earth causes its rate of cooling to increase by all of those mechanisms, making them negative feedbacks. Those negative feedbacks limit the temperature increase which results from an increase in any forcing.
GHGs in the atmosphere have their effect on temperature by absorbing outbound longwave IR radiation from below, which otherwise would have escaped to space. Absorbing that radiation warms the atmosphere. That, in turn, increases radiation from GHGs in the atmosphere, about half of which goes back down toward the surface, as “downwelling” or “back radiation,” which warms the surface. Hence, the surface and the lower troposphere warm or cool together.
Here's NASA's 2014 (latest?) version of the famous Trenberth energy flow diagram (click to enlarge it):
The diagram was discussed on WUWT, here:
The diagram shows:
1. Incoming solar radiation at TOA = 340.4 W/m². That's 1/4 of the 1360 to 1370 W/m² which is typically estimated as the value of the “solar constant,” because the 136x W/m² figure is radiation at TOA where the Earth directly faces the Sun, and the 340.4 figure is averaged over the entire globe, including the back side. (The surface area of a sphere is exactly four times the area of a circle of same radius.)
2. Estimated outgoing radiation at TOA is 239.9 W/m² IR + 77.0 + 22.9 = 339.8 W/m², which is 0.6 W/m² less than the incoming solar radiation. (That figure is poorly constrained, and not directly measurable; I’ve also seen estimates of 0.7 or 0.8 W/m².) That is their estimate of the “radiation imbalance,” which represents warming “in the pipe,” but not yet realized (roughly 0.2°C to 0.4°C, depending on sensitivity estimates and timeframe).
3. Downwelling “back radiation” at the surface, from GHGs in the atmosphere, is estimated as averaging 340.3 W/m². That's about the same as the solar irradiance at TOA, but ≈29% greater than the average amount of solar radiation which makes it to the surface.
4. About 22.6% (77 W/m²) of the incoming solar radiation is reflected back into space, without either reaching the surface or being absorbed.
Thus, an increase of 1 W/m² in LW IR at the surface from GHGs in the atmosphere has an effect on surface temperatures which is similar to a
(That 1.29 ratio is approximate, because the effect of sunlight is not identical to the effect of LW IR from CO2. On one hand, there's considerable overlap between the spectrums of CO2 and water vapor, which reduces the warming effect from CO2, especially in the humid tropics. On the other hand, an estimated 22.9 W/m² (6.7%) of sunlight is reflected from the surface back into space, and that does not happen for LW IR.)
Adjusting the Feldman values for having measured at the surface, rather than TOA, yields about 1.29 × (2.40 ±0.72) per doubling at TOA, and dividing by ln(2), yields the coefficient:
𝞪 = 4.47 ±1.34 (which is 3.10 ±0.93 W/m² per doubling)
That figure is very close to Halpern's “4.35”, and closer to Happer's “3.8” than to Myhre's “5.35,” but the uncertainty interval is wide enough to encompass all three estimates. It does preclude the SAR's “6.3” figure.
Rentsch 2020 (unpublished draft), analyzed orbital Atmospheric Infrared Sounder (AIRS) spectroscopy, and found that CO2-induced radiative forcing at TOA from a 37 ppmv CO2
concentration increase (13.65% of a doubling), under nighttime, cloud-clear conditions, caused +0.358 ±0.067 W/m² radiative forcing increase.
Dividing by 0.1365 × ln(2), yields the coefficient:
𝞪 = 3.79 ±0.71 (which is 2.62 ±0.49 W/m² per doubling)
That's about 70% of the Myhre 1998 / IPCC prediction, and very close to Happer's result.