Forcings for Doubling CO2

At least as far back as Myhre et al 1998[1], scientists have understood that it's possible to approximate the effective radiative forcing (ERF, the amount of change in the outgoing energy flux near the tropopause) by a simple logarithmic equation:

ΔERF ≈ α*ln(C/C0)

I say "approximate" because the actual calculations for the relationship between ΔF and CO2 from line by line radiative transfer models are a bit more complex than this. The above equation is simply the result of curve fitting that matches those calculations over the range of CO2 concentrations that we're mostly concerned with. The value for α scales the radiative forcing change for the log change in CO2 concentrations. Myhre's value for α was 5.35, and this was used in the IPCC's TAR and AR4 reports. More recent IPCC reports, though, have improved the ΔF2xco2 estimates, and we can solve for the α values implied by these changes in ΔF2xco2 with α = ΔF2xco2/ln(2)

ΔF2xco2 ≈ α*ln(2)

AR3: 3.71 ± 0.4 W/m^2, so  α = 5.35

AR5: 3.80 ± 0.38 W/m^2, so α = 5.48

AR6: 3.93 ± 0.47 W/m^2, so α = 5.67

All of these values fit within the uncertainty of AR3, so I typically use Myhre's value in the back-of-the-envelope calculations on this site, unless the context has to do with AR5 or AR6. However, while this is basically just improving estimates for the value of ERF, there's some evidence that this value may not be stationary; it may actually increase with warming.

From He et al 2023

A recent study from He et al 2023,[2] has shown that the instantaneous radiative forcing (IRF) for CO2 has actually increased by about 10% since 1850 and is likely to increase by ~25% for 2xCO2. Now IRF is a portion of ERF. ERF is made up of two components, IRF and the sum of all rapid adjustments (RAs), which consists of radiative forcings caused by the atmospheric response to IRF not at the surface. As a general rule of thumb, IRF makes up ~60-70% of ERF[2]. He writes,

When evaluating the effect of carbon dioxide (CO2) changes on Earth’s climate, it is widely assumed that instantaneous radiative forcing from a doubling of a given CO2 concentration (IRF2×CO2) is constant and that variances in climate sensitivity arise from differences in radiative feedbacks or dependence of these feedbacks on the climatological base state. Here, we show that the IRF2×CO2 is not constant, but rather depends on the climatological base state, increasing by about 25% for every doubling of CO2, and has increased by about 10% since the preindustrial era primarily due to the cooling within the upper stratosphere, implying a proportionate increase in climate sensitivity. This base-state dependence also explains about half of the intermodel spread in IRF2×CO2, a problem that has persisted among climate models for nearly three decades.

This finding gives support to other studies that have concluded that sensitivity during glacial periods is less than during interglacial periods (Friedrich et al 2016)[3] and even higher during hot periods like the PETM  (Tierney et al 2022)[4]. Perhaps also it's partial explanation for why some high sensitivity models seem to be performing better than scientists expected (Myhre et al 2025)[5]? I'm only guessing on that point. But perhaps it is better to use the AR6 estimate for ΔERF2xco2 over earlier estimates for projections of future warming, and maybe this is yet one more indicator that the IPCC's central estimate of 3 C for ECS is a conservative estimate, or maybe better stated, a temporary estimate.

References:

[1] Myhre, G., E.J. Highwood, K.P. Shine, and F. Stordal, “New estimates of radiative forcing due to well mixed greenhouse gases.” Geophys. Res. Lett., 25.14 (1998): 2715-2718.
http://go.owu.edu/~chjackso/Climate/papers/Myhre_1998_New%20eatimates%20of%20radiative%20forcing%20due%20to%20well%20mixed%20greenhouse%20gasses.pdf

[2] Haozhe He et al. ,State dependence of CO2 forcing and its implications for climate sensitivity.Science382,1051-1056(2023).DOI:10.1126/science.abq6872

[3] Tobias Friedrich et al. Nonlinear climate sensitivity and its implications for future greenhouse warming. Sci. Adv.2,e1501923(2016).DOI:10.1126/sciadv.1501923

[4] J.E. Tierney, J. Zhu, M. Li, A. Ridgwell, G.J. Hakim, C.J. Poulsen, R.D.M. Whiteford, J.W.B. Rae, L.R. Kump, (2022) Spatial patterns of climate change across the Paleocene–Eocene Thermal Maximum, Proc. Natl. Acad. Sci. U.S.A. 119 (42) e2205326119,
https://doi.org/10.1073/pnas.2205326119.

[5] Gunnar Myhre et al., Observed trend in Earth energy imbalance may provide a constraint for low climate sensitivity models.Science388,1210-1213(2025).DOI:10.1126/science.adt0647