Radiative forcing from a change in atmospheric CO2 concentration (synopsis)

Calculated estimates of CO2 forcing

The formula for estimating radiative forcing (RF or ΔF) 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
Myhre 1998 & the IPCC (TAR & later) estimate:
    𝞪 = 5.35 ±0.58  (which is 3.7 ±0.4 W/m² per doubling of CO2)
Happer 2013[2] (and 2015) reports calculating, based on corrected modeling of CO2 line­shapes, that that's ≈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) (see also 2020 & 2022) report calculating CO2's ERF at the mesopause (similar to TOA) to be:
    𝞪 = 4.28  (which is 2.97 W/m² per doubling)

Myhre's estimate is about 15% lower than the previous IPCC estimate of:
    𝞪 = 6.3  (which is 4.4 W/m² per doubling; see FAR p.52 & SAR p.320)
AR5 reports that RF estimates for a doubling of CO2 assumed in 23 CMIP5 GCMs vary from 2.6 to 4.3 W/m² per doubling, so:
    𝞪 ≈ 3.7 to 6.2 
Prof. Joshua Halpern reports:
    𝞪 = 4.35  (https://twitter.com/EthonRaptor/status/1254176110507626499)

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. They reported that a 22 ppmv (= +5.946%) increase in atmospheric CO2 level (+8.333% forcing) resulted in a 0.2 ±0.06 W/m² increase in downwelling LW IR from CO2, which is +2.40 ±0.72 W/m² per CO2 doubling.
However, ≈22.6% of incoming solar radiation is reflected back into space, without either reaching the surface or being absorbed in the atmosphere. So, adjusting for having measured at the surface, rather than TOA, gives ≈1.29 × (2.40 ±0.72) per doubling at TOA, and dividing by ln(2), yields:
    𝞪 ≈ 4.47 ±1.34  (which is 3.10 ±0.93 W/m² per doubling)
That's 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 (draft), analyzed AIRS satellite spectroscopy, and found that under nighttime, cloud-clear conditions, a 37 ppmv CO2 increase caused +0.358 ±0.067 W/m² radiative forcing increase at TOA, which is:
    𝞪 = 3.79 ±0.71  (which is 2.62 ±0.49 W/m² per doubling)
That's about 70% of Myhre 1998, and very close to Happer's result.
NEW: Kramer, RJ et al. 2021. Observational evidence of increasing global radiative forcing. Geophysical Research Letters, 48, e2020GL091585. doi:10.1029/2020GL091585. (See discussion.)

For full details see:  https://sealevel.info/Radiative_Forcing.html
For effect on temperatures see:  https://sealevel.info/sensitivity.html
For similar calculations for CH4 see: https://sealevel.info/methane.html
For graphs of atmospheric CO2 concentration see: https://sealevel.info/co2.html
For CO2 emissions data see: https://sealevel.info/carbon/
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