Numeracy in Climate Discussions – how long will it take to get a 6C rise in temperature?

The answer may surprise you

Note: This essay is a result of an email discussion this morning, I asked Dr. Happer to condense and complete that discussion for the benefit of WUWT readers. This is one of the most enlightening calculations I’ve seen in awhile, and it is worth your time to understand it because it speaks clearly to debunk many of the claims of temperature rise in the next 100 years made by activists, such as the 6c by 2050 Joe Romm claims, when parroting Fatih Birol in Reuters:

“When I look at this data, the trend is perfectly in line with a temperature increase of 6 degrees Celsius (by 2050), which would have devastating consequences for the planet,” Fatih Birol, IEA’s chief economist told Reuters.

Graph source: IEA.org scenarios and projections

– Anthony

Guest post by Dr. William Happer

For any rational discussion of the effects of CO2 on climate, numbers are important. An average temperature increase of 1 C will be a benefit to the planet, as every past warming has been in human history. And the added CO2 will certainly increase agricultural yields substantially and make crops more resistant to drought. But in articles like “Scant Gains Made on CO2 Emissions, Energy Agency Says” by Sarah Kent in the Wall Street Journal on April 18, 2013, we see a graph with a 6 C temperature rise by 2050 – if we don’t reduce “carbon intensity.” Indeed, a 6 C temperature rise may well be cause for concern. But anyone with a little background in mathematics and physics should be able to understand how ridiculous a number like 6 C is.

The temperature change, ∆T , from the mean temperature of the present (the year 2013), if the concentration N of CO2 is not equal to the present value, N = 400 ppm, is given by the simple equation

(1)

Here ∆T2 is the temperature rise that would be produced by doubling the CO2 concentration from its present value, and ln x denotes the natural logarithm of the number x.

The proportionality of the temperature increment ∆T to ln N is widely accepted. But few know that this is a bit of a “miracle.” The logarithmic law, Eq. (1) comes from the odd fact that the average absorption cross section of infrared light by CO2 molecules decreasesvery nearly exponentially with the detuning of the infrared frequency from the 667 cm−1 center frequency of the absorption band. More details can be found in a nice recent paper by Wilson and Gea-Banacloche, Am. J. Phys. 80 306 (2012). Eq. (1) exaggerates the warming from more CO2 because it does not account of the overlapping absorption bands of water vapor and ozone, but we will use it for a “worst case” analysis.

Recalling the identity for natural logarithms,   , we write Eq. (1) as

   (2)

The only solution of the equation ln x = ln y is x = y, so (2) implies that

   (3)

Recent IPCC reports claim that the most probable value of the temperature rise for doubling is ∆T2 = 3 C. Substituting this value and a warming of ∆T = 6 C into Eq. (3) we find

   (4)

But the rate of increase of CO2 has been pretty close to 2 ppm/year, which implies that by the year 2050 the CO2 concentration will be larger by about (50−13) years×2 ppm/ year = 74 ppm to give a total concentration of N = 474 ppm, much less than the 1600 ppm needed.

The most obvious explanation for the striking failure of most climate models to account for the pause in warming over the past decade is that the value of ∆T2 is much smaller than the IPCC value. In fact, the basic physics of the CO2 molecule makes it hard to justify a number much larger than ∆T2 = 1 C – with no feedbacks. The number 3 C comes from various positive feedback mechanisms from water vapor and clouds that were invented to make the effects of more CO2 look more frightening. But observations suggest that the feedbacks are small and may even be negative. With a more plausible value, ∆T2 = 1 C , in Eq. (3) we find that the CO2 concentration needed to raise the temperature by ∆T = 6 C is

   (5)

This amount of CO2 would be more than a warming hazard. It would be a health hazard. The US upper limit for long term exposure for people in submarines or space craft is about 5000 ppm CO2 (at atmospheric pressure). To order of magnitude, it would take a time

t = 25, 600 ppm/(2 ppm/year) = 12,800 years.

(6)

to get 6 C warming, even if we had enough fossil fuel to release this much CO2.

A 6 C warming from CO2 emissions by 2050 is absurd. It is a religious slogan, a sort of “Deus vult” of the crusade to demonize CO2, but it is not science.

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Dr. William Happer is the Cyrus Fogg Brackett  Professor of Physics at Princeton University, and a long-term member of the JASON advisory group,where he pioneered the development of adaptive optics. From 1991-93, Happer served as director of the Department of Energy’s Office of Science.

UPDATE: Dr. Happer has contacted the author of the paper cited, and he has graciously setup a free link to it: http://comp.uark.edu/~jgeabana/gw.html