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
where:
    C/C₀ is the ratio of new to old CO₂ concentrations

Myhre 1998 and 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 ERF at the mesopause (similar to TOA) from doubling CO₂ 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.

Kramer, RJ et al. 2021. Observational evidence of increasing global radiative forcing. Geophysical Research Letters, 48, e2020GL091585. doi:10.1029/2020GL091585. (See discussion.)