## Calculated estimates of CO_{2} forcing

The formula for estimating radiative forcing (RF or Δ*F*) from a change in atmospheric CO_{2} concentration is usually given as:

**Δ***F* = 𝞪 · ln(*C*/*C*₀) W/m²

where:

*C*/*C*₀ is the ratio of new to old CO_{2} concentrations

Myhre 1998 & the IPCC
(TAR & later) estimate:

𝞪 =** 5.35 ±0.58 ** (which is 3.7 ±0.4 W/m² per doubling of CO_{2})

Happer 2013[2]
(and 2015) reports calculating, based on corrected modeling of
CO_{2} 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 CO_{2}'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

SAR §6.3.2,

p.320)

AR5 reports that RF estimates for a doubling of CO

_{2} 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 CO_{2} forcing

Feldman *et al* 2015 measured downwelling longwave IR “back radiation” from CO_{2},
at ground level, under clear sky conditions, for a decade. They reported that a 22 ppmv (+5.953%) increase in atmospheric CO_{2} level resulted in a 0.2 ±0.06 W/m² increase in downwelling LW IR
from CO_{2}, which is +2.40 ±0.72 W/m² per doubling of CO_{2}.

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 CO_{2} 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.

**UPDATE:** There's a new paper on this topic:

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