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The latest IPCC Scientific Basis report includes a well-known diagram (see http://www.grida.no/climate/ipcc_tar/wg1/pdf/TAR-01.PDF, page 90, Figure 1.2). This diagram appears just about everywhere. The source is a paper (Kiehl and Trenberth, 1997: Earth’s Annual Global Mean Energy Budget, Bull. Am. Met. Soc. 78, 197-208) published in 1997. The diagram has appeared in every IPCC report since 1997 (probably because Trenberth is an IPCC contributor).

I was hoping someone here could check my math and tell me where I've gone wrong. The Trenberth paper (see http://www.cgd.ucar.edu/cas/trenberth.papers/KiehlTrenbBAMS97.pdf, pages 205-206) makes the following statement:

"The estimate of the amount leaving via the atmospheric window is somewhat ad hoc. In the clear sky case, the radiation in the window amounts to 99 W/m^2, while in the cloudy case the amount decreases to 80 W/m^2, showing that there is considerable absorption and re-emission at wavelengths in the so-called window by clouds."

If I understand these sentences correctly--when the sky is clear (IOW cloudless) the atmospheric window lets 99 watts per square meter escape. When it's cloudy, that amount drops to 80 watts per square meter. The paper continues:

"The value assigned in Fig. 7 of 40 W/m^2 is simply 38% of the clear sky case, corresponding to the observed cloudiness of about 62%. This emphasizes that very little radiation is actually transmitted directly to space as though the atmosphere were transparent."

The first problem is the 40 W/m^2 value. Thirty-eight percent of 99 W/m^2 is 37.62 W/m^2. Why did the authors round to 40 instead of 38?

The second problem is also the 40 W/m^2 value. If 0% cloudiness allows 99 W/m^2 to escape, and 100% cloudiness drops that value to 80 W/m^2, how did we get 40 W/m^2? Shouldn’t it be 38% of the difference between 99 and 80 or 87.22 W/m^2?

This model has many other problems too. You could use it to disprove AGW.

Jim