### Estimating TCR and ECS from the Logarithmic Relationship Between CO2 and GMST

In a previous post, I calculated ECS (accounting for increases in GHGs and aerosols) to be about 3.3 C.  The calculation was based on CO2 causing 2.11 W/m^2 increase in radiative forcing with a total increase, after accounting for GHGs other than CO2 and aerosols, of 2.17 W/m^2 (aerosols cancel out most of the effects of GHGs outside of CO2). One weakness of that approach is that it used a value for EEI that was an average for 2011-2018 with forcings that were current through 2020. I've been thinking about a way to improve this, and here's what I came up with.

#### Transient Climate Response (TCR)

Since the relationship between CO2 and temperature is logarithmic, I decided to plot the relationship between temperature and ln(rCO2) to see what that might be able to tell us about sensitivity from empirical data. So in the above graph, on the y-axis I plotted GMST from HadCRUT5 using a 1850-1900 baseline to match the IPCC's approximation of preindustrial levels. On the x-axis, I plotted 5.35*ln(CO2/280). The relationship between these two should be linear, and the slope of this relationship should let us calculate TCR. The best fit line through this data is 0.625x - 0.158 with an r^2 of 0.87.

The relationship between changes in radiative forcing (Î”F) and temperature (Î”T) is linear, and constant of proportionality between them is referred to as "sensitivity," often expressed as Î», and this is the slope of the above graph. So assuming all the increase in radiative forcing is due to CO2, this would imply a TCR value of 0.625 C/W/m^2 x 3.71 W/m^2 = 2.3 C.  However, we've already seen above that CO2 accounts for roughly 2.11 of 2.17 W/m^2, or 97.24% of the increase in radiative forcings. So while using CO2 alone would produce a good estimate for TCR, it will be slightly too high. For the sake of this post, I think it's fair to express the relationship between Î”Fc and Î”Ft as Î”Fc = 0.9724 x Î”Ft, where Î”Ft is total forcings and Î”Fc is the forcings from CO2 changes alone. Obviously, this is not precisely true every year, but I think it's fair that this will be approximately true over the last couple decades or so. The adjusted Î» for TCR would thus become 0.625*0.9724 = 0.608 C/W/m^2. Since doubling CO2 causes 3.71 W/m^2 increase in radiative forcing, we can calculate that TCR as 0.608 x 3.71 = 2.25 C for doubling CO2.

#### Equilibrium Climate Sensitivity (ECS)

This above value for TCR is lower than ECS because the graph doesn't take into consideration the Earth's energy imbalance (EEI). So what I decided to do is to estimate ECS using three values for EEI. I used
1. Hansen's estimate of 0.58 W/m^2 from 2005-2010[1]
2. von Schuckmann's estimate of 0.87 W/m^2 from 2011-2018.[2]
3. Loeb's estimate of 0.77 W/m^2 from 2005-2019[3]
The first two estimates indicate an increase in EEI of about  10*(0.87-0.58)/7 = 0.41 W/m^2/decade., and this agrees with Loeb, who estimated the trend in the increase of EEI from both in situ and satellite measurements to be between 0.43 ± 0.40 (in situ) and 0.50 ± 0.47 (CERES) W/m^2/decade. So these values are all consistent with each other. You can read more about trends in EEI since 2005 here. I averaged the Î”Ft values for each of these time frames and then used the line equation from the above graph to calculate the corresponding Î”T:

Î”T = 0.608*Î”Ft - 0.158
(y = Î»x + b)

This lets me calculate the expected average GMST (the average y-value on the best fit line in the above graph given a range of dates, Î”T) for each time frame correlating with the three EEI estimates. Incidentally, if I use Î”Fc values with the slope calculated from those (0.625*Î”Fc - 0.158), I predictably get the exact same Î”T values. Using values for EEI for each of the three time frames, I can calculate the Î» for ECS using the following equation.[1]

Î»= Î”T/(Î”Ft - EEI)

This is just the energy balance equation that is solved for Î», allowing me to use empirical values for the other terms to discover the ECS value that best summarizes the data we have.  Here are the values for Î”Fc for each of the time frames used in the three studies, where Î”Fc is the Î”F from CO2 alone:

2005-2010, Î”Fc = 1.703 W/m^2 for Hansen[1]
2011-2018, Î”Fc = 1.912 W/m^2 for von Schuckmann[2]
2005-2019, Î”Fc = 1.838 W/m^2 for Loeb[3]

However, since we estimated that Î”Fc = 0.9724 x Î”Ft, using these would slightly overestimate sensitivity, so I'm going to use my estimated Î”Ft values for these ECS calculations. With this adjustment, we get the following:

2005-2010 (Hansen)
Î”Fc = 1.703 W/m^2
Î”Ft = 1.751 W/m^2
EEI = 0.58 W/m^2[1]
Î”T = 0.907 C
Î» = 0.774 C/W/m^2

2011-2018 (von Schuckmann)
Î”Fc = 1.912 W/m^2
Î”Ft = 1.966 W/m^2
EEI = 0.87 W/m^2[2]
Î”T = 1.037 C
Î» = 0.947 C/W/m^2

2005 - 2019 (Loeb)
Î”Fc = 1.838 W/m^2
Î”Ft = 1.890 W/m^2
EEI = 0.77 W/m^2[3]
Î”T = 0.991 C
Î» = 0.885 C/W/m^2

Since Loeb's EEI value covers the longest time frame, this is probably the most accurate estimate for EEI, and the sensitivity (Î») value calculated from it is in the middle between the other two. The average of all these values for sensitivity (Î») is 0.869 C/W/m^2, meaning that, according to this data, GMST increases by 0.869 C for any 1 W/m^2 increase in Î”F, regardless of the causes of Î”F. So we can convert this value to ECS by multiplying 0.869 C/W/m^2 by the Î”F for doubling CO2.

Average
Î» = 0.869 C/W/m^2
Î”F2x = 3.71 W/m^2
ECS = Î»*Î”F2x = 3.22 C

This of course is a very simple estimate, and the accuracy of the estimate depends on the accuracy of the Î”F values, the Î”T values, and EEI estimates. Of these, Î”F from aerosols have the largest uncertainty. Internal variability affecting the increase in EEI over basically 15 years should also be acknowledged. Further, in the y = Î»x + b equation above b should technically equal zero (since ln(1) = 0) but since CO2 was already 285 ppm in 1850 (instead of 280 ppm), the y-intercept is negative. I could fix that by changing 280 ppm to the 1850-1900 mean (290 ppm), but that would increase sensitivity, and I want to err on the side of being conservative. The main point I take from this is that this is another look at empirical data that shows that ECS estimates of ~3 C remain very consistent with observational data.

References

[1] Hansen, J., Mki. Sato, P. Kharecha, and K. von Schuckmann, 2011: Earth's energy imbalance and implications. Atmos. Chem. Phys., 11, 13421-13449, doi:10.5194/acp-11-13421-2011.
https://pubs.giss.nasa.gov/abs/ha06510a.html

[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., DesbruyÃ¨res, D., Domingues, C., GarcÃ­a-GarcÃ­a, 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] Loeb, N. G., Johnson, G. C., Thorsen, T. J., Lyman, J. M., Rose, F. G., & Kato, S. (2021). Satellite and ocean data reveal marked increase in Earth’s heating rate. Geophysical Research Letters, 48, e2021GL093047. https://doi.org/10.1029/2021GL093047