How Will Happer computes temperature increase

Vincent J. Curtis

19 Oct 23

On the Tom Nelson podcast #158, guest Professor Will Happer explained (between 21:20 and 42:00) how he determines that a doubling of atmospheric CO₂ will produce only a 0.71℃ increase in global temperature.  I will try to present below concisely how he does it.

Dr. Happer, with co-author Dr. W. van Wijngaarden, released what I call the “Will & Bill” paper[1] in which they report their solutions to the “equation of transfer” to produce the Schwarzschild curve of the earth’s infra-red emission spectra as seen from space.  The equation of transfer was originally developed by astrophysicists to determine the radiation from the center of stars, and the solution of it for all relevant frequencies produces the Schwarzschild curve.  The Will & Bill computations reproduce with uncanny accuracy the actual IR spectrum of the earth observed from satellites, and therefore their solutions to the equation of transfer to produce the earth’s Schwarzschild curve can be deemed reliable.

The earth’s surface is at a average temperature of 288K (15℃), which means that if the earth were a perfect blackbody radiator, the radiation distribution function would follow the Planck curve, with the area under the curve being 394 Wm^-2.  The Planck curve is to the Schwarzschild curve as the ideal is to the actual.  Because of the absorption of some of this infra-red radiation in the atmosphere, notably by water vapor, ozone, and CO₂, only 277 Wm^-2 actually reaches space.  The area under the actual (Schwarzschild) curve Will & Bill computed to be 277 Wm^-2.  It is here that Will Happer introduces the Stefan-Boltzmann equation:

            F = ƐσT⁴;

where F will stand as the forcing power in Wm^-2, Ɛ is the emissivity, σ is the Stefan-Boltzmann constant (5.67 x 10^-8 Wm^-2 K^-4), and T is absolute temperature in degrees Kelvin.

Ɛ is the emissivity of the earth conceived as a blackbody radiator, and Happer computes it and ΔƐ as follows:

            Ɛ = (277/394) = 0.70

If the CO₂ content of the atmosphere were doubled from 400 to 800 ppm, then the area under the resulting Schwarzschild curve is reduced from 277 to 274 Wm^-2; hence, ΔƐ/Ɛ is computed to be:

            ΔƐ/Ɛ = [(274-277)/394](394/277) = -3/277 = -0.01

To maintain equilibrium, the heat radiated to space by the earth must equal the incoming radiation from the sun.  Hence, the F in the Stefan-Boltzmann equation above could stand for either the incoming solar flux, or the flux radiated from the earth back into space, as they must be the same value.  Since F of the sun is certainly constant for our purposes, Happer makes a first-order differentiation of the Stefan-Boltzmann equation as follows:

            ΔF_s = 0 = ΔƐσT⁴ + 4ƐσT³ΔT;

Rearranging:

            ΔT = - (ΔƐ/4Ɛ)T;

Substituting the values:

            ΔT = (0.01/4) (288) = 0.72K = 0.72℃

The result of reducing the flux to space by 3 Wm^-2 is to increase temperature by 0.72℃ to compensate.

Given the accuracy and robustness of the Will & Bill computations, this is a figure that can be relied on, for the only way the earth can shed heat is by radiation to space and that process is subject to the Stefan-Boltzmann equation.

On a personal note, the equation of transfer wasn’t pulled out of a hat for the purpose of climate science; it was developed much earlier and for completely different purposes.  That fact, combined with the uncanny results in the Will & Bill paper, convinces me of the correctness of Happer’s forecast.

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[1] W.A. van Wijngaarden, W. Happer, “”Dependence of the Earth’s Thermal Radiation on Five Most Abundant Greenhouse Gases” June 8, 2020.