Debunking the Most Ridiculous Climate Paper I've Seen Yet

Even though there is no theoretical basis for the Beer-Lambert formula, ∆RF = αln(C/Co), it has been accepted by the scientific community as a reasonable approximation. In this paper we propose an improved mathematical approximation that... has no theoretical basis.

~ H. Douglas Lightfoot

A paper was published in Energy & Environment by Lightfoot & Mamer back in 2014 (LM14)[1] arguing that we should toss out decades of research establishing a theoretical basis for and quantifying the logarithmic relationship between changes in CO2 (∆CO2) and radiative forcing (∆RF) for no good reason. In it's place, they sought to replace it with the results of a curve fitting exercise for no good reason except to generate an equation that would conform with a climate myth that was conclusively refuted in the 1950s.

A theoretical basis for the near-logarithmic relationship between ∆CO2 and ∆RF was established beginning in 1896 with Arrhenius' landmark paper. Arrhenius quantified the logarithmic relationship between ∆CO2 and GMST to be 4-6°C for 2xCO2. This conclusion has been confirmed ever since, though the best ECS estimates are lower than his at about 3°C. The near log relationship is not seriously challenged in studies published in reputable journals. As research has progressed more sophisticated radiative transfer calculations, with results confirmed be experimental data and observations, have allowed scientists to conclude that the relationship between ∆CO2 and ∆RF is approximately logarithmic and can be approximated with the equation: 

∆RF = α*ln(C/Co), where

        Co is an initial concentration of CO2 (usually preindustrial or 280 ppm),
        Cis concentration at a later time, and
        α is a constant, calculated to be 5.35 in 1998.

This relationship is clearly an approximation, and this is acknowledged in the literature. In fact, multiple studies have noted that the actual ∆RF for 2xCO2 may be larger at high CO2 concentrations. "For high CO2 cases (2000 ppm...) the new calculations are typically 10% higher than the old fits, indicating that the CO2 forcing increases more rapidly than expected from a purely logarithmic dependence."[2]

LM14 apparently decided to capitalize on the fact that this relationship is approximate to justify tossing out the logarithmic relationship altogether. After wrongly claiming that decades of research on the near logarithmic relationship between ∆CO2 and ∆RF had no theoretical basis, LM14 engaged in a curve-fitting exercise. They just calculated a quadratic equation to approximate the radiative forcing changes from 275 ppm to 378 ppm. This is the equation they came up with:

RF = -0.0000248*C^2+0.03231*C

Almost certainly they just put data into a spreadsheet and let it calculate the best-fit quadratic curve through that data, and this is the equation the spreadsheet gave them. And because they just had a spreadsheet come up with a best fit quadratic, it does actually match changes in GMST with a good correlation that are near the range where the curve fit was performed. I plotted the hypothetical LM14 forcings with HadCRUT5 and got the expected good correlation. 

But one way to show that this equation is absolute nonsense see how well it predicts ∆RF above 378 ppm CO2. What happens to ∆RF as CO2 increases? Well, this can be plotted, and things quickly go bonkers. According to this equation, ∆RF maxes out at about 655 ppm, and then it descends into an abyss. According to this quadratic model, as CO2 approaches 2200 ppm, the Earth should plunge into a snowball earth, with a massive cooling impact of -48 W/m^2. Below you can see a comparison of the two models with LM14 plotted as what they consider absolute RF and conventional estimates as ∆RF (where RF at 280 ppm is 0 W/m^2).

Pretty much by admission, LM14 chose to use this logarithmic model because of the maximum RF at 655 ppm. According to LM14, "Thus, at 378 ppmv, RF = 8.67 W m-2. At some point above this, CO2 has absorbed all of the IR that it can absorb and it is impossible for additional CO2 to increase RF further. This is the concentration where the model plateaus." Except it doesn't plateau above 655 ppm. It takes a nose dive. RF decreases with CO2^2, so as CO2 increases, its cooling influence will only increase. In other words, because LM14 have a prior commitment to the myth that CO2 saturates at some point and will permit no further warming, they chose to use a quadratic model, but they hid the impact of the model above 655 ppm. And saturation is not a valid reason to propose the quadratic alternative. The saturation myth is debunked here, and it has been shown to be a myth since Plass' work in the 1950s.[4] 

I think we can demonstrate how ridiculous this paper is by comparing their quadratic model to paleoclimate observations. Afterall, LM14's argument is that the quadratic model works just as well as the logarithmic model between 275 and 378 ppm, and it works better than the logarithmic model above 378 ppm: "Outside of this range, the quadratic model is a better fit to the origin point and to the plateau where CO2 has absorbed all possible back-emitted IR." In other words, they're claiming the quadratic model makes better predictions than decades of climate research. Let's see if that holds up. 

Judd et al 2024[3] developed a new reconstruction of GMST and CO2 for the last 485 million years, and the paper supports a logarithmic relationship of ~8°C for 2xCO2 on geologic time scales. The correlation is quite high in the Cenozoic, and data poverty causes some issues during the Mesozoic, but overall the relationship is pretty clear in the data. Especially accounting for solar and geographic forcings over geologic history, a logarithmic relationship between CO2 and GMST works (yes, I know correlation is not causation; that's settled with other evidence. My point here is only that the logarithmic model works as an explanation for GMST and CO2). Above I show the relationship for the Cenozoic from Judd's paper; below I show the long term GMST and CO2 changes for the last 485 million years (most of the Paleozoic). We can tell from this that GMST has increased to ~34°C and CO2 has been as high as 3200 ppm.

Since LM14 claim to calculate "atmospheric radiative forcing (warming effect) of carbon dioxide at any concentration," we ought to be able to use the LM14 quadratic model and make better explanations than the logarithmic model. But things go bonkers quickly. According to the quadratic model, CO2 at 3200 ppm would exert a cooling influence of 150 W/m^2, certainly plunging the planet into a snowball earth episode. And yet GMST during that time was ~15°C warmer than today. Let's call that a fail. More recently, during the PETM (56 mya), CO2 was ~1280 ppm and GMST was ~34°C or ~19°C warmer than today. This again is explained with a logarithmic model. But according to the quadratic model, RF would drop from 9.2 W/m^2 at 420 ppm today to 0.7 W/m^2 at 1280 ppm. So CO2 should be exerting a cooling influence of 8.5 W/m^2 relative to today, and yet the PETM was ~19°C warmer than today. In order to explain this, LM14 would have to imagine some set of forcings that would be massive enough to both overcome this 8.5 W/m^2 deficit and then warm the planet by an additional 19°C. In fact, there is no evidence of any "plateau" in the paleoclimate evidence, let alone high CO2 concentrations exerting a cooling influence on the climate. Once you get beyond the range of CO2 concentrations for which LM14 performed their curve fit, the model is bunk. And by the admission of the authors, they replaced the logarithmic model (which works and has explanatory power) with another model that they admit has "no theoretical basis." It has no predictive or explanatory value either. 

References:

[1] Lightfoot, H. D., & Mamer, O. A. (2014). Calculation of Atmospheric Radiative Forcing (Warming Effect) of Carbon Dioxide at Any Concentration. Energy & Environment, 25(8), 1439-1454. https://doi.org/10.1260/0958-305X.25.8.1439

[2] Etminan, M., G. Myhre, E. J. Highwood, and K. P. Shine (2016), Radiative forcing of carbon dioxide, methane, and nitrous oxide: A significant revision of the methane radiative forcing, Geophys. Res. Lett., 43, 12,614–12,623, doi:10.1002/2016GL071930.

[3] Emily J. Judd et al., A 485-million-year history of Earth’s surface temperature. Science 385,eadk3705 (2024).DOI:10.1126/science.adk3705

[4] James Rodger Fleming, Gavin Schmidt, Gilbert Plass. "Carbon Dioxide and the Climate: A 1956 American Scientist article explores climate change; two contemporary commentaries illuminate its relevance to the present." American Scientist. Vol. 98.1, page 58.
https://www.americanscientist.org/article/carbon-dioxide-and-the-climate
See also, Plass, G. N. 1956. The influence of the 15 micron ozone band on the atmospheric infra-red cooling rate. Quarterly Journal of the Royal Meteorological Society 82.