Radiative Forcing from a change in atmospheric CO2 concentration

By Dave Burton, Oct. 11, 2020.  (Adapted, in part, from a blog comment.)
Last updated: Feb. 22, 2021.

Synopsis:  https://sealevel.info/Radiative_Forcing_synopsis.html
Effect on temperatures:  https://sealevel.info/sensitivity.html
Similar calculations for CH4: https://sealevel.info/methane.html


1. Calculated estimates of CO2 forcing
2. Measurements of CO2 forcing
    2.1 Feldman, 2015
    2.2 Rentsch, 2020
    2.3 Kramer, 2021

Calculated estimates of CO2 forcing

The formula for estimating radiative forcing (RF) from a change in atmospheric CO2 concentration is usually given as:
    ΔF = 𝞪·ln(C/C₀) W/m²
    ΔF = ERF·log2(C/C₀) W/m²
    C/C₀ is the ratio of new to old CO2 concentrations, and
    ERF = 𝞪·ln(2) = “Effective Radiative Forcing” per doubling of CO2 level
and the most common estimate is:
    𝞪 = 5.35 ±0.58

For a doubling of CO2 concentration that yields:
    ERF = 𝞪·ln(2) = 3.7 ±0.4 W/m² per doubling of CO2 concentration

Those are the figures used by the IPCC (TAR & later). They are from Myhre 1998 (pdf), who also reported an uncertainty “of order 10%,” which would be about ±0.54.

That represented about a 15% reduction from an earlier IPCC estimate of
    𝞪 = 6.3  (which is ERF=4.4 W/m² per doubling; see SAR §6.3.2, p.320)

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)

van Wijngaarden & Happer 2021 (preprint) (and 2020) report calculating CO2's ERF at the mesopause (which should be similar to TOA) to be 2.97 W/m² per doubling (see their Table 2 [or 2020 version], rightmost column). That makes:
    𝞪 = 4.28

The IPCC's AR5 reported that the radiative forcing estimates for a doubling of CO2 assumed in 23 CMIP5 GCMs vary from 2.6 to 4.3 W/m² per doubling, so that:
    𝞪 ≈ 3.7 to 6.2

Prof. Joshua Halpern reports:
    𝞪 = 4.35  (I don't know his source; [I asked, but he didn't answer):

Joshua Halpern on Twitter: 'CO2 forcing is approximated by 4.35 ln ([CO2(current)]/[CO2(1880)])''

Measurements of CO2 forcing

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):

NASA updated energy budget diagram (2014)

The diagram was discussed on WUWT, here:

The diagram shows:

1. Incoming solar radiation at TOA = 340.4 W/m². That's one-fourth of 1361.6 W/m², which is their estimate for the average value of the “Solar Constant.” (The so-called “Solar Constant” actually varies somewhat, and other estimates for its average vary from 1360 to 1370 W/m².) 

It's one-fourth because the 1361.6 W/m² figure is solar radiation intensity 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 the 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 0.6 × 0.3±0.1°C ≈ 0.2 °C, depending on sensitivity estimates). 

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 (1 / 0.774) = 1.29 W/m² increase in average solar irradiance at TOA (like a 5.17 W/m² increase in the Solar Constant).

That 1.29:1 ratio is approximate, because the effect of sunlight at the surface is not identical to the effect of downwelling LW IR from atmospheric 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, which does not happen for LW IR. If that 6.7% were included in the calculation above, it would make an an increase of 1 W/m² in downwelling LW IR at the surface similar in effect to a (1 / 0.706) = 1.42 W/m² (rather than 1.29 W/m²) increase in average solar irradiance at TOA, like a 5.66 W/m² increase in the Solar Constant.

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.

UPDATE: There's a new paper out on this topic:
Kramer, RJ et al. 2021. Observational evidence of increasing global radiative forcing. Geophysical Research Letters, 48, e2020GL091585. doi:10.1029/2020GL091585.
The abstract reports a TOA radiative forcing change of 0.53±0.11 W/m² from 2003 to 2018.
Per Mauna Loa measurements, average atmospheric CO2 concentration rose from 375.80 ppmv in 2003 to 408.52 ppmv in 2018. log2(408.52/375.80) = 0.12044, so that increase was about 12% of a doubling.
CH4 concentration rose from 1.777 ppmv in 2003 to 1.857 in 2018. According to MODTRAN, the CH4 increase should have accounted for about 298.488 - 298.426 = 0.062 W/m². If CO2 and consequent feedbacks accounted for the rest, that leaves 0.53±0.11 - 0.06 = 0.47±0.11 W/m², from a 12% CO2 forcing increase (due to an 8.7% CO2 concentration increase).
So ERF at TOA from a doubling of CO2 + feedbacks should be (0.47±0.11) / 0.12044 = 3.90±0.92 W/m². That's very close to the Myhre 1998 and IPCC (TAR & later) estimates for radiative forcing from CO2 alone, without feedbacks (3.7±0.4 W/m² per doubling).
Kramer's result isn't directly comparable to their figures, because his result includes the effects of feedbacks, but if feedbacks had no net effect then his result would imply 𝞪 is 5.63 ±1.33.

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