NZCLIMATE & ENVIRO TRUTH NO 155
Dr Vincent Gray
22 AUGUST 2007
I hesitate in even mentioning the word mathematics nowadays as hardly anybody seems to know any. I am always surprised when I get the correct change from a shop assistant, and nobody seems to want the receipt offered at a money machine.
It is therefore with some diffidence that I recount two major mathematical blunders perpetrated by the IPCC as a result of their use of non-linear equations to handle arithmetical averages.
The first involves a fundamental error in the "Earth's Annual Global Mean Energy Budget" as devised by J T Kiehl and Kevin L Trenberth 1997, Bulletin of the American Meteorological Society 78 (2), 197-208.(attached) which has been a major part of all the IPCC Reports and has been quoted very widely.
I converge on the figure given for the radiant energy emitted by the earth, which is given as 390 W/sqm.
Now the earth, like every other substance, emits radiation according to the Stefan-Boltzmann equation. If it is assumed to be "black body" the radiation intensity E, in W/sqm is related to the absolute temperature T in K by the equation
E = σ T to the fourth power
σ = 5.67x 10 to minus 8 W/sqm/K, is the Stefan-Boltzmann constant
The average temperature of the earth is generally agreed to be 288K.
If you put 288 into the Stefan-Boltzmann equation using a pocket calculator, you get 390 W/Sqm.
This means that Kiehl and Trenberth have assumed that the earth has a constant temperature of 288K (15oC).
Everyone knows that this is wrong. But they may not be aware that the average of the fourth power of the temperature is not the same as the fourth power of the average temperature. The distribution curve of T to the fourth power is skewed towards higher temperatures.
The radiation from the earth is very much greater from warmer regions than from cooler ones because of the fourth power dependency, so that temperature above average have a much greater influence than temperatures below average.
Let us assume, just as an example, that the earth is divided into four temperature zones, 313K (40C) 293K (20C), 283 (10K) and 263 (-10C) Average 288K (15C)
The energy of emission from each zone, by Stefan-Boltzmann is 544, 418, 363 and 271 W/sqm; average, 399W/sqm. 9W/sqm different from using the average
Since the effects of greenhouse gases are supposedly around 1.6W/sqm this sort of error cannot be considered negligible.
Another error from use of a non-linear equation comes from the calculation by the IPCC of "radiative forcing" (the extra radiation caused by increases in carbon dioxide from an average concentration, instead of from a properly weighted average).
The IPCC and the observers of atmospheric carbon dioxide have tried to conceal the fact that the concentration is variable. They only make measurements in a few extremely restricted circumstances and they conceal results from all other places and times
Beck 2007 Energy and Environment 18 "180 years of Atmospheric CO2 Gas Analysis by Chemical Methods" 259-280 shows that more than 90,000 measurements of atmospheric carbon dioxide, some by Nobel Prize-winners, all published in well-known peer-reviewed journals, available for individual scrutiny on Beck's website, have been suppressed by the IPCC in order to persist in their claim that carbon dioxide in the atmosphere is "well-mixed", so that you can calculate its "radiative forcing" from the supposedly constant figure.
The relationship between the additional radiation at the top of the atmosphere (The radiative forcing) DF in W/sqm and the additional concentration of carbon dioxide C in parts per million by volume, over the reference level Co is given by the formula.
DF = 5.35 ln C/Co
where ln is the logarithm to the base e
Beck shows that the concentration of carbon dioxide in the atmosphere varies between 280 ppmv and 400pmmv, or even more, depending on the time, place, wind direction.
Again, no distribution function of the quantity lnC/Co is available, but it must certainly be skewed in the direction of the lower values of C, those below the arithmetical average because of the logarithmic relationship. This means that calculation of "radiative forcing" from the supposed "constant" or "well-mixed" average can be guaranteed to give an incorrectly high figure.