A Curious Paleo Puzzle

Guest Post by Willis Eschenbach

I was wandering through the fabled land of X yesterday and came across the following post:

Figure 1. Post on X showing a shortened version of the original Figure 6 from the paper linked below.

Hmmm, sez I … looks like an interesting study. Paleo CO₂ levels back to about 65 million years ago, which peaked at about 2000 ppmv.

I was reminded of an earlier graphic I’d done, showing paleo CO₂ levels over a much longer time span. Figure 1 above only covers the Tertiary and Quaternary ages, the far right two boxes on Figure 2 below.

Figure2. Full history of CO2 since the Cambrian Explosion of life.

So I went to the source listed in Figure 1, a study yclept “Atmospheric CO2 over the Past 66 Million Years from Marine Archives” by Rae et al. The graphic below is a shortened version of Figure 6 of the Rae et al. study.

FIgure 3. Panels a) and c) of Rae et al., Figure 6

The authors have fully bought into the “CO₂ Roolz Climate” theory, saying inter alia:

Changing levels of atmospheric CO₂ have long been implicated in the well-documented cooling of the climate through the Cenozoic; however, outside of a handful of well-studied climate transitions, it has been hard to make a close link between CO₂ and climate. Our new combined marine-based CO₂ compilation shows, more clearly than in previous studies, a close correlation between CO₂ and records of global temperature (based on either geochemical reconstructions and/or the state of the cryosphere) through the entire Cenozoic (Figure 6).

…

Nonetheless, it is clear, even with these caveats, that atmospheric CO₂ and temperature are closely coupled, both across the data set as a whole and within shorter time windows. While the data set as a whole suggests a relatively high climate sensitivity, much of this temperature change is apparently accomplished by jumps between different climate states.

Hmmm, sez I … I wondered why they didn’t mention the value of what they claim is “a relatively high climate sensitivity” for the data set as a whole. So I figured I’d take a look at their data.

For the CO₂ data, that was easy. They provide it in an Excel file in their Supplemental Material.

For the temperature data, the total opposite. They say “Surface temperature estimated from the benthic δ18O stack of Westerhold et al. (2020), using the algorithm of Hansen et al. (2013)” … except they don’t give you the relevant links to the data or the algorithm.

Grrr. I did that, dug out Westerhold’s data, found and applied the Hansen algorithm, links above, and dang, it took a while.

In any case, here’s their CO₂ data.

Figure 4. Paleo CO₂ levels to 66 million years ago

Our current levels are far from the highest even in just the last 5 million years, let alone 60 million years. (Note that I haven’t added a LOWESS smooth of the data as they did, because that effectively invents tens of millions of years of data.)

So my next question, of course, was how well the temperature corresponds with the log of the CO₂ levels. Figure 5 shows that relationship.

Figure 5. Paleo temperatures, and linear fit of the base 2 log of the CO₂ levels.

I have to say that is a very good agreement by climate and paleo standards, where measurements are always uncertain.

Next I looked in detail at the cluttered area at the lower right. That’s the time of the “ice ages”, periods of glaciation followed by periods of warmth. Figure 6 shows the ice ages.

Figure 6. Paleo temperatures, and linear fit of the base 2 log of the CO₂ levels. I’ve added lines between the temperature and fitted CO₂ values for each measurement time.

Once again, by the standards of paleo and climate, these are good fits.

So … what’s not to like? Does this actually show that CO₂ at 0.04% of the atmosphere is really the secret global temperature control knob as the authors claim?

Perhaps not.

Let’s start by looking at the math. I’ll divide the math off so as not to bother folks who are math-averse … for you, just jump over this section. Here is the summary of the linear fit of log2(CO₂) and temperature.

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Coefficients:

             Estimate   Std. Error   t value    P-value
(Intercept)    7.4810     0.1954       38.3      <2e-16
log2_CO2       6.3137     0.1087       58.1      <2e-16

Residual standard error: 2.468 on 644 degrees of freedom
Multiple R-squared: 0.8398, Adjusted R-squared: 0.8395
F-statistic: 3375 on 1 and 644 DF, p-value: < 2.2e-16

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“log2_CO2” is the base 2 logarithm of the change in CO₂. The “Estimate” of 6.3 in bold above is the climate sensitivity, the estimate of the temperature change corresponding to a doubling of CO₂.

So yes, as the authors say above, this does show a “relatively high climate sensitivity”. The climate sensitivity of 6.3 is the third highest of 172 various past estimates of climate sensitivity. Here’s a look at previous estimates.

Figure 7. Estimates of climate sensitivity from theory and reviews, observations, paleoclimate, climatology, and GCMs.

Another problem, beyond the climate sensitivity being one of the highest of the 173 estimates, is the much higher CO₂ values earlier in the past shown in Figure 2. If climate sensitivity is 6.3°C per doubling, that would put the global average surface temperature at around 40°C in the Cambrian and around 36°C in the Devonian … seems unlikely.

However, there’s a larger problem. The CO₂ theory is that a rise in CO₂ absorbs more upwelling longwave radiation. This causes an imbalance in the net radiation at the top of the atmosphere by redirecting some of the upwelling longwave back to the surface. The amount of this increased downwelling radiation is called the CO₂ forcing.

The surface temperature then warms, increasing the upwelling surface longwave radiation to restore the balance.

Figure 8 below shows the difficulty factor. When the earth’s surface warms, it emits more radiation following something called the Stefan-Boltzmann equation. Figure 8 below shows the increased forcing due to increased CO₂ levels in the past, along with the corresponding increase in surface upwelling longwave radiation from the temperature increases.

Figure 8. Changes in upwelling surface longwave radiation, and downwelling atmospheric longwave radiation due to CO₂.

Yikes. So there’s the perplexitude—we’re told to believe that a change in CO₂ forcing of 13 W/m2 causes a corresponding 128 W/m2 in upwelling surface longwave radiation.

But where does the extra energy come from? Seems like the tail is wagging the dog. After allowing for the 13 W/m2 of CO₂ forcing, there’s another 115 W/m2 of extra energy leaving the surface … but where did it all come from?

By comparison, solar energy absorbed by the surface is 164 W/m2. So the surface needs to be getting another three-quarters of a sun’s worth of energy from … somewhere …

It would have to be some extraordinarily large feedback to the warming if that’s what caused more warming. The feedback factor would have to be ~ 0.9 … and if the feedback factor is greater than 1.0, it grows without end. And that would suggest that at some point in the past, the natural fluctuations in this feedback would have led to endless growth.

And it’s difficult to think of a physical process that would supply that 115 W/m2 to the surface. For example, total cloud albedo reflects about 75 W/m2 back to space. So if the positive cloud feedback led to the complete disappearance of the clouds, that would only increase the surface absorbed solar by 66 W/m2 after adjusting for increased surface reflection … and we’re looking for 115 W/m2.

The same is true for the positive water feedback. The numbers aren’t big enough. The IPCC AR6 WG1 Chapter 7 Section 7.4.2.2 estimates a combined water vapor and lapse rate feedback at 1.12 W/m2 per °C of warming. The warming back to 60 million years ago is about 15°C. So water vapor+lapse rate feedback would be on the order of 15°C * 1.12 W/m2 per °C = 17 W/m2 … and we’re looking for 115 W/m2.

What else … the paleo temperature or paleo CO₂ might be incorrectly calculated, in which case none of this shows anything.

A final possibility, of course, is that the warming has little to do with CO₂ and that the CO₂ levels are a function of temperature and not the other way around …

I titled this post “A Curious Paleo Puzzle”. That’s the puzzle. How can an increase of 13 W/m2 in CO₂ forcing cause an increase of 115 W/m2 of upwelling surface longwave radiation?

All suggestions welcome.

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Here on our redwood forested northern California hillside, there’s a small triangle of the Pacific Ocean visible between the far hills on a warm sunny day after a long string of storms. The wind brings us the sound of the surf gnashing and gnawing on the coast six miles (ten km) in the distance. The grandkids, two and four years old, play and laugh in the next room.

With best wishes for sun, rain, and the joys of family in your life,

w.

You May Have Heard This Before: When you comment, I ask that you quote the exact words you are discussing. It avoids endless misunderstandings.

Data: I’ve made two CSV files containing the data used in this analysis so they can be used in Excel or the computer language of your choice. One contains the temperatures, 23,722 different paleo measurements.

The other contains the paleo CO₂ data, 646 measurements. That one also contains the temperatures from the other dataset, interpolated at the dates of the CO2 measurements. This allows for the calculation and graphing of the relationship between the datasets.

The datasets are “Rae CO2 and Interpolated Temps.csv” and “Rae Temperatures.csv“, available at the links from my Dropbox.