TUHO: People who write about climate change are accustomed to getting emails explaining why they are mistaken. The writer, often a retired engineer, sends a couple of pages of equations “proving” that adding carbon dioxide gas (CO2) to the atmosphere cannot cause global warming. Is there a simple physics model that shows in a transparent way how humanity’s emissions of gases do heat the planet? History offers an instructive approach to this question. When scientists attacked the problem, what mental obstacles did they encounter, and how were those overcome? Two centuries of effort, summarized below, concluded that greenhouse calculations require computer models far too complex to be understood intuitively—but simple, readily grasped observations show that the models’ conclusions are plausible.
Intuitive models
The struggle began in 1824 when Joseph Fourier, as a minor aside from his landmark contributions to the physics and mathematics of heat flow, published a speculation. He proposed (wrongly) that interplanetary space is inherently very cold, and he wondered why our Earth is not frozen. Perhaps our atmosphere retains heat like a blanket? He compared the air to a pane of glass covering a box: the glass lets sunlight in but stops heat (infrared) radiation from leaving. This would later be called the “greenhouse effect.” Not until 1909 did a physicist, Robert W. Wood, point out that the phrase is misleading; the main work of the glass in an actual greenhouse is to separate the warm air inside from the cold winds outside. Still, Fourier’s rudimentary model of the atmosphere raising Earth’s temperature by blocking outgoing infrared radiation sounded plausible.
The idea got little traction. There was no actual evidence that Earth needed help in keeping warm, and anyway air seemed to be entirely transparent to radiation. But then geologists discovered the ice ages: a constant global temperature could no longer be taken for granted. Could an ice age be caused by a change in the composition of the atmosphere? John Tyndall decided to check that by devising an apparatus to measure the passage of infrared rays through gases. In 1859, he found that the main constituents of the atmosphere, nitrogen and oxygen, are indeed transparent—but water vapor, CO2, methane, and some other gases absorb infrared rays.
How does that affect Earth’s climate? Tyndall, a superb science popularizer, came up with a simple model of the process that has never been bettered: “As a dam built across a river causes a local deepening of the stream, so our atmosphere, thrown as a barrier across the terrestrial [heat] rays, produces a local heightening of the temperature at the Earth’s surface.” A fine analogy—but understanding a process doesn’t signify much until you get numbers. How much would global temperature change if the amount of CO2 in the atmosphere changed?
Calculating a number
In 1896, after half a century of advances in infrared measurements, Svante Arrhenius attempted to quantify the greenhouse effect. He began with a short list of equations, the first real physics model. There was much to calculate. Adding CO2 at a given height in the atmosphere would absorb a certain amount of radiation and warm that level. But then the warmer air would hold more water vapor, itself a potent greenhouse gas. So that had to be calculated too. Arrhenius made a separate calculation for each band of latitude, noting that when the surface in northern latitudes grew warmer, it would retain less ice and snow, uncovering dark ocean and soil that would absorb additional heat. In the end, he spent a full year on pencil-and-paper computations. Yet it was a simple model; one modern microchip could do the calculation in a fraction of a second.
Arrhenius announced that doubling the amount of CO2 in the atmosphere should warm the planet something like 4 °C. That was obviously only a rough estimate, but the exact number did not seem to matter much. At the rate that humanity was burning coal, Arrhenius figured it would take thousands of years to double the CO2.
Other scientists soon decided that Arrhenius’s estimate was worthless. They were right, for as we will see, he left out factors that are crucial for climate. But their main argument was a simple one that apparently refuted the greenhouse effect altogether. A basic laboratory measurement indicated that doubling the CO2 in the atmosphere could make no difference at all. For in the broad bands of the infrared spectrum where CO2 acts to absorb radiation, there was already enough of the gas in the atmosphere to make the air utterly opaque: that part of the infrared spectrum was “saturated.”
So matters stood until 1956, when Gilbert Plass took a fresh look at the greenhouse question. The laboratory measurement of CO2 that supposedly refuted Arrhenius had been done at sea-level pressure. That seemed reasonable when everyone looked at the atmosphere from the bottom up, as if it indeed acted like a solid slab of glass. But if you looked down from space, you would see infrared radiation coming mostly from the thin air near the top of the atmosphere—air that was heated by absorbing radiation from below. Drawing on decades of progress in theory and spectroscopy, Plass knew that in this thin air, the bands of infrared absorption resolve into a thicket of individual lines. Adding CO2 would broaden the lines, and they would absorb more radiation. The place from which heat radiation finally escaped into space would migrate to a higher level. Everything below would get warmer, as in Tyndall’s analogy of a dam.
Even with the new digital computers, it was a huge job to calculate the effect, layer by layer through the atmosphere and point by point across the spectrum. Plass could model only a one-dimensional column of air, a simpler physical model than Arrhenius’s even as it required much more computation. Plass found that doubling the CO2 in his model did raise the temperature by a few degrees down to ground level: the greenhouse question was revived. However, he had left out so many things (water vapor, for one) that everyone knew the question was not answered. Indeed, when Fritz Möller tried the calculation including water vapor, he got an unreasonable surface temperature rise of 10 °C or more.
Complete calculations
Syukuro Manabe took up the challenge. His equations included a crucial process that almost everyone had overlooked: convection. Heat rises from Earth’s surface not only in radiation but in columns of air and moisture, carried skyward, for example, in thunderstorms. That is what prevents Möller’s runaway surface heating. Manabe’s model was in a sense still simple, equations that could be written down on a couple of pages. But he meticulously fed it the details of the actual infrared absorption and humidity at 18 levels of the atmosphere. Calculating it all just for a one-dimensional column of air still needed a state-of-the-art computer. In 1967, working with a collaborator, Manabe produced a simulated atmospheric profile that looked pretty much like the real one. Then, like Arrhenius and Plass, he doubled the CO2 level in his simulated atmosphere and calculated the change in surface temperature—a number that would be called the climate “sensitivity.” It was roughly 2 °C. The calculation was impressive, convincing many scientists that greenhouse warming was worth looking into. Yet Manabe’s model was clearly too simple. In particular, like everyone else, Manabe had left out a feature of climate that profoundly affects radiation: clouds.
Over the next decade, leaps in computer power enabled Manabe and his collaborators to clone their one-dimensional column thousands of times to wrap a globe in three dimensions, and to incorporate clouds and other essential climate features. To get the pattern of cloudiness, they had to calculate how the atmosphere exchanges moisture with simplified sea, land, and ice surfaces, and how rain or snow falls on the surfaces and evaporates or runs off in rivers, and more. Then there were the oceans, with their own circulation transporting vast amounts of heat from the tropics toward the poles. In the end, Manabe produced a simulated planet with trade winds, tropical rain bands, deserts, ice caps, and so forth in all the right places. Finally, a model complicated enough to look like the real world! Doubling the CO2 got, again, a sensitivity of roughly 2 °C.
Humanity was now burning fossil fuels an order of magnitude faster than in Arrhenius’s day. Measurements of the CO2 level in the atmosphere revealed it was rising fast. A doubling was not a thousand years off, but likely before the end of the 21st century. National policies for energy production might need to be reconsidered.
The U.S. President’s Science Adviser, geophysicist Frank Press, heard of the problem. In 1979, he turned to the nation’s traditional provider of trustworthy science advice: the National Academy of Sciences. The Academy duly convened a panel to conduct a study. The panel ploughed through publications on a variety of rudimentary models like Plass’s. They interviewed Manabe at length about his 2 °C finding. And they interviewed James Hansen, the author of the only other big climate model at that time, which computed a sensitivity of 4 °C. The panel found it very probable that doubling CO2 would seriously heat the planet. Splitting the difference between Manabe and Hansen, they estimated the sensitivity would be 3 °C give or take 50%, that is, 1.5–4.5 °C.
The Academy panel judged well. The scientific consensus today still puts the most likely sensitivity at 3 °C (a climate of severe global disruption). The range of uncertainty was not narrowed until 2021, when the Intergovernmental Panel on Climate Change put the likely lower bound at 2 °C and the upper at 4 °C, although they could not rule out 5 °C (an unimaginable catastrophe). So there persists a disturbing uncertainty. The most advanced models, embodying orders of magnitude more features than Manabe’s, disagree among themselves. Climate is inextricably complicated. That raises a different and urgent question: can these models, far too elaborate to be grasped intuitively, be trusted at all?
Verifying the number
The first convincing answer came in 1985 from Vostok, Antarctica, where the Soviet Union drilled a hole kilometers deep into the ice cap. Tiny bubbles in the ice preserved ancient air with its CO2. The ratio of oxygen isotopes (18O/16O) in the ice measured the temperature of the clouds at the time the snow had fallen, for the warmer the air, the more of the heavier isotope got into the ice crystals. Analysis showed that through the coming and going of entire ice ages, temperature and CO2 had soared and plunged in lockstep. And the sensitivity? Doubled CO2 meant a temperature rise of … wait for it … 3 °C give or take 50%.
In any field of science, when two utterly different approaches give you the same number, you can feel you are in touch with reality. Researchers took up the problem with other independent methods, working out ingenious ways to find temperature and CO2 in distant geological eras (for example, the density of pores in fossil leaves reflects the CO2 level of the air, as do carbon isotope ratios in carbonates precipitated in ancient soils, while oxygen isotope ratios in shells in seabed sediments vary with the ocean surface temperature, etc.). A variety of studies kept getting the same sensitivity. Meanwhile, other researchers used the actual warming of recent decades as a sort of natural experiment. They found that the patterns of heating measured deep in individual ocean basins neatly matched the patterns that computer models calculated for rising CO2. They found that the distribution of cloud types seen by satellites changed with warming much like the responses of computed clouds … and so forth.
The most impressive feature of the ongoing natural experiment is rudimentary. If you superimpose the rising curve of CO2 since the 1950s on the rising curve of observed global temperature, you find an ominous match (the match is particularly precise if you assume that an exponential rise of CO2 should cause a linear rise of temperature—Arrhenius, for one, found this intuitively plausible). Extrapolate to doubled CO2, and the temperature rise is, yes, near 3 °C.
In 1979, when the Academy panel made their estimate, the world was on track to reach doubled CO2 well before 2100. However, if nations adopt policies to fulfill the pledges they have made, we can arrest the rise a bit short of doubling—unless we have bad luck and, as some models find possible, the warming triggers a vicious cycle of additional greenhouse gas emissions.
Climate models today explore hundreds of interacting processes in computer runs lasting weeks at teraflop rates. Nature does not allow a simple, transparent model for global warming. But we have something perhaps better: simple, transparent ways to show that we must take the models seriously.
REFERENCES
1.Key papers by Fourier, Tyndall, Arrhenius, Plass, Manabe, the National Academy “Charney” panel, Vostok researchers, and more are reprinted with commentary in D. Archer and R. T. Pierrehumbert (editors), The Warming Papers: The Scientific Foundation for the Climate Change Forecast (Wiley-Blackwell, Hoboken, NJ, 2011).
2.For full history and references, see S. Weart, “Basic radiation calculations” and “Simple models of climate change” (American Institute of Physics, 2022)
S. Weart, The Discovery of Global Warming, 2nd ed. (Harvard University Press, Cambridge, MA, 2008).
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3.A short history from another viewpoint is H. Le Treut et al, “Historical overview of climate change science,” in S. Solomon, et al. (editors), Climate Change 2007:The Physical Basis of Climate Change. Contribution of Working Group I to the Fourth Assessment Report of the IPCC (Cambridge University Press, New York, 2007), pp. 93–127,
https://www.ipcc.ch/site/assets/uploads/2018/05/ar4_wg1_full_report-1.pdf.4.On matching CO2 and temperature curves, see J. Aber and S. V. Ollinger, “Simpler presentations of climate change,” Eos 103 (Sept. 13, 2022)
5.For a college-level “simple” but reasonably complete model, see R. E. Benestad, “A mental picture of the greenhouse effect,” Theor. Appl. Climatol. 128, 679–688 (2017). All websites accessed Oct. 1, 2022.
Spencer Weart published articles on solar physics in leading scientific journals and then turned to studying the history of science. From 1974 until his retirement in 2009, he was director of the Center for History of Physics at the American Institute of Physics. His publications include children’s science books, The Rise of Nuclear Fear, and The Discovery of Global Warming.