How Do Cumulative Carbon Emissions Affect Warming?


The IPCC (and others) have observed that there has been a near linear increase in GMST with cumulative anthropogenic carbon emissions. "In the literature, units of °C per 1000 PgC (petagrams of carbon) are used, and the AR6 reports the TCRE likely range as 1.0°C to 2.3°C per 1000 PgC in the underlying report, with a best estimate of 1.65°C."[1] To be clear, 1000 PgC = 1000 GtC = 1 TtC. 

I decided to see if that has been observed in the empirical data. I took values from the 2021 global carbon budget[2] and HadCRUT5 set to a 1850-1900 baseline and plotted the relationship. The R^2 was 0.88, and the slope of the best fit line was 1.847 ± 0.104°C/TtC (2σ). So the 95% likely range is between 1.74 to 1.95 C/TtC from 1850-2021. If I start after we reach 200 GtC in 1951, the best fit line is 2.153 ± 0.176°C/TtC (2σ) and the R^2 increases to 0.90, but I think it's best to be conservative here. What this suggests to me is that the IPCC may again be a bit conservative their best estimate and in their likely range. This I believe is good practice when making projections for policy makers - projected likely ranges should be wider than those describing empirical data. 
But if the best estimate from empirical data holds true, then we can expect on average 0.1 C warming for every 54 GtC emissions. Since 2000, we have emitted on average 10 GtC/yr, so if this rate continues, we would expect about 0.1 C warming every 5.4 years. We're currently at about 1.25 C as of 2021, so if emissions continue at 10 GtC/yr, 

We hit 1.5 C warming in 0.25*54 = 13.5 years or 2035 
We hit 2.0 C warming in 0.75*54 = 40.5 years or 2062 

Obviously, these projections are subject to natural variability (like ENSO) and future emission rates, as well as the uncertainty of the estimate. But in 2021, total anthropogenic emissions eclipsed 700 GtC with very little evidence that our progress in transitioning away from fossil fuels is yet significantly lowering annual emission rates. Peak emissions was in 2019 with 11.06 GtC. That was lowered to 10.38 GtC in 2020 and 10.72 GtC in 2021, but that progress is way too small to have any hope of preventing 1.5 C warming, and it makes avoiding 2 C warming increasingly unlikely. 

References:

[1] IPCC AR6, Summary for Policy Makers. p. 28. https://www.ipcc.ch/report/ar6/wg1/downloads/report/IPCC_AR6_WGI_SPM_final.pdf 

[2] Global Carbon Budget 2021
https://www.icos-cp.eu/science-and-impact/global-carbon-budget/2021

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