A Calculation for Equilibrium Climate Sensitivity (ECS)

Tree Overhanging the Econ River, Little Big Econ State Forest, May 9, 2021

My interest in "wood romances" spans the gamut from nature photography to hiking, to conservation, to understanding the future impacts of climate change. I frequently participate in on-line discussions on these matters, and many of the same issues get recycled in numerous contexts. I've taken to addressing these in relatively short responses so that I don't have to rewrite the same arguments, evidence and calculations. Here is a basic calculation of equilibrium sensitivity (ECS) that I did recently. ECS is the amount of increase in global mean surface temperature (GMST) after reaching equilibrium with a doubling of CO2 concentrations. There are many ways to do this, and this is a simple energy balance equation that uses empirical data for various forcings to arrive at the estimate. This of course is not the end of discussion on the matter. Many scientific studies have been written estimating the value for ECS and attempting to constrain the range of plausible estimates. What this calculation does is show that the empirical evidence we have shows that ECS values of around 3 C are consistent with observational evidence.

The earth’s mean temperature is a function of energy entering vs escaping the climate system. If more energy enters the climate system than escapes into space, the planet warms. If more escapes than enters, the planet cools.  An increase in  radiative forcing will create an energy imbalance, and the earth’s surface temperature will increase in response. As temperatures rise, this imbalance will decrease  until a new equilibrium is reached (energy entering the climate system matches energy escaping into space). This concept can be expressed in terms of a simple formula:

ΔF = ΔT/λ + EEI, where

ΔF is the total change in radiative forcing,

ΔT is the observed change in temperature from that change in forcing,

EEI is the Earth’s Energy Imbalance

λ is sensitivity

Rearranging terms we get:

λ= ΔT/(ΔF - EEI)

The error limits for each will determine the range of reasonable values for sensitivity. A recent paper[1] demonstrates that changes in GHGs and aerosols account for practically all the increase in temperatures above the 1850-1900 mean, with natural forcings being pretty much negligible over that time frame. We can thus estimate sensitivity using reasonable values for the above. I used estimates of forcings from both the increase in GHGs (ΔFghg) and aerosols (ΔFaer) separately.

1. For ΔT, I used the estimate from Gillett 2021[1] that 2010-2019 was 1.1 ± 0.2 C above the 1850 - 1900 mean. 

2. For EEI, I used the value of 0.87 ± 0.12 W/m^2 from von Schuckmann et al 2020[2]

3. For ΔFghg, I used values from NOAA’s website and averaged forcings from 2010-2018 to match the dates for the EEI estimate. That came out to 2.94 W/m^2.

4. For ΔFaer, I used the value from Forest 2018[4] of “− 0.77 Wm−2 with a 5–95% range of − 1.15 to − 0.31 Wm−2.”

Using these values, we can estimate λ and therefore ECS as:

ΔF = ΔFghg + ΔFaer = 2.94 - 0.77 = 2.17 W/m^2

λ = T/(F - EEI) = 1.1/(2.17-0.87) = 0.85 C/W/m^2

To calculate ECS, you need to multiply this number by the radiative forcing for a doubling of CO2:

ΔF = 5.35*ln(2) = 3.71 W/m^2, so

ECS = 0.85*3.71 = 3.1 C

We can determine the portion of the ΔFghg (above) from CO2 using the following formula:

ΔFco2 = 5.35*ln(415/280) = 2.11 W/m^2

That means the ΔF from non-CO2 GHGs is 0.83 W/m^2. That’s very close to the estimated value for ΔFaer above (0.77 W/m^2). So forcings from non-CO2 GHGs and aerosols roughly cancel each other out. If we were to estimate ECS from only the increase in CO2, you would get this:

λ = T/(F - EEI) = 1.1/(2.11-0.87) = 0.89 C/W/m^2

ECS = 0.89*3.71 = 3.3 C

If you look closely at the error limits for each of the estimates used in this calculation, the range of reasonably possible values is fairly wide. But this calculation puts ECS square in the middle of the best estimates for ECS in peer-reviewed literature.

References:

[1] Gillett, N.P., Kirchmeier-Young, M., Ribes, A. et al. Constraining human contributions to observed warming since the pre-industrial period. Nat. Clim. Chang. 11, 207–212 (2021). https://doi.org/10.1038/s41558-020-00965-9

[2] von Schuckmann, K., Cheng, L., Palmer, M. D., Hansen, J., Tassone, C., Aich, V., Adusumilli, S., Beltrami, H., Boyer, T., Cuesta-Valero, F. J., Desbruyères, D., Domingues, C., García-García, A., Gentine, P., Gilson, J., Gorfer, M., Haimberger, L., Ishii, M., Johnson, G. C., Killick, R., King, B. A., Kirchengast, G., Kolodziejczyk, N., Lyman, J., Marzeion, B., Mayer, M., Monier, M., Monselesan, D. P., Purkey, S., Roemmich, D., Schweiger, A., Seneviratne, S. I., Shepherd, A., Slater, D. A., Steiner, A. K., Straneo, F., Timmermans, M.-L., and Wijffels, S. E.: Heat stored in the Earth system: where does the energy go?, Earth Syst. Sci. Data, 12, 2013–2041, https://doi.org/10.5194/essd-12-2013-2020, 2020.

https://essd.copernicus.org/articles/12/2013/2020/

[3] NOAA “Annual Greenhouse Gas Index (AGGI).” https://www.esrl.noaa.gov/gmd/aggi/aggi.html#:~:text=The%20AGGI%20is%20a%20measure,onset%20of%20the%20industrial%20revolution

[4] Forest, C.E. Inferred Net Aerosol Forcing Based on Historical Climate Changes: a Review. Curr Clim Change Rep 4, 11–22 (2018). https://doi.org/10.1007/s40641-018-0085-2