Graphing CO2 and Temperature for the Phanerozoic

My last post was a bit of a rant. I apologize for that (sort of), though I get bombarded with these obviously erroneous graphs, and I wanted to put down in one place all the problems I have with that family of contrarian graphs. But it occurred to me today that it would be more helpful (and less of a rant) if I demonstrate how the data used in those graphs would look different if the problems I described were fixed. So to do this, I went on an internet search and found a wonderful set of data from a paper I described in an earlier post, Royer 2004. It's a spreadsheet with the data from the GEOCARB III model (which originates in Berner 2001) and global temperatures corrected from Vezier.[1] This only goes back 520 million years, but I found the full Berner CO2 data on another site.[2]

The CO2 data is reported in RCO2, which is the ratio of CO2 at a time in the past and preindustrial CO2 (280 ppm). So I converted these values to a change in radiative forcing (RF) from preindustrial levels using the well-known equation RF = 5.35*ln (RCO2). I then needed to calculate the effect of radiative forcing changes from solar evolution, since the Sun was only about 70% as bright at the Earth's formation as it is today. I found this formula in Foster et al 2017[3]:

F = Fs/[1+0.4(1-t1/t0)], where

F = the solar constant at t1
Fs = the solar constant at present (1368 W/m^2)
t1 = time in the past from the age of the earth
t0 = the age of the earth (4567 million years)

This calculation put the solar constant at 1303 W/m^2 570 million years ago (about 95% of today's solar luminosity). I then needed to calculate the change in RF from this change in the solar constant. For this I used the following formula:

RF = dSo*(1-α)/4 , where

RF = the change in radiative forcing
dSo = change in solar constant
α = albedo

For α, I used a constant 0.3 (the contemporary value), which is certainly wrong for much of the Earth's history. During the warmest portions of the earth's history there was no glaciation, so the Earth's albedo was certainly lower. The change in albedo should amplify the signal of the graph, but this is a simple calculation, so I think it's fine to keep albedo constant. After this calculation I added the radiative forcings from CO2 and solar evolution and plotted the sum with global temperatures from 2004. Here's the effect of that adjustment on CO2 plotted as radiative forcing data.

The resulting correlation between CO2+Solar radiative forcing and temperature is at the top of the post. You can see the remarkable difference this makes to the correlation between CO2 and temperatures across the Phanerozoic. To show you the difference between CO2 plotted as concentration in the atmosphere and CO2 plotted as W/m^2 and adjusted for solar evolution, see below. Below I plotted the GEOCARB III values for CO2 expressed both in ppm and in W/m^2 corrected for solar evolution.

One more note; the resolution of both of these datasets is 10 million years, and so you can see that the short but significant glaciation at the end of the Ordovician does not show up in either time series - this is simply a function of the lack of resolution in the data.  Also, I haven't found values for the uncertainty envelope for this graph, so I looked at the graph published in Berner 2001 and estimated them. It looks like Berner drew straight lines for these, so I put in value that should be very close to what is in the paper. You can see what I did here.

Then I made the same transformation to radiative forcing and the same adjustment for solar evolution to both, and I plotted those with with the temperature time series. The results are basically the same as the graph at the top of this post, but does give an indication of the kind of agreement that exists between Berner's model and the temperature data.

I think more work can be done here (and plenty more has been done), but for now I think it's good just to see what a proper graph of peer-reviewed data regarding temperature and the GEOCARB III model plotted as CO2+Solar radiative forcing would look like.

References:


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