All functions belong to the firm of
Mclaren Maclaurin and Taylor in the short term
See UPDATE BELOW and FURTHER UPDATE
Further UPDATE: His original mistake has cost Eli no end of aggro, coupled with more than a bit of chocolate flung his way by Msr. Pielke, however, being a thinking bunny, he has gone back and thought. What we all have missed, and what explains the results in the chart below is that the Maclaurin series is a power series expansion around a point
where the function and its derivatives are evaluated at a not constantly changing. This continues at the return of Maclaurin and Taylor. In the end, using a linear approximation within the range of temperature found on Earth is perfectly reasonable no matter what the Old Guy says
Ethon flew back from Boulder where he had been keeping company with the math department while Roger Pielke Sr. was retaking Cal I. To tell the truth the JGR D editors could use a refresher too. Atmoz first pointed Eli to this, so when Eth came flying in Rabett Labs was up and running. Were Eli a cynical Brit he would think the old Guy might be the only person trying to enter the IPCC with honourable intentions:
which he elaborates in the comments at Atmoz (At does a pretty good job of demolishing this himself)
The definition of the global average surface temperature used by the IPCC and others can be expressed as
dH/dt = f -T’/λ
where H is the heat content of the land-ocean-atmosphere system, f is the radiative forcing (i.e. the radiative imbalance), T’ is the change global average surface temperature in response to the change in H, and λ is called the “climate feedback” parameter which defines the rate at which the climate system returns forcing to space as infrared radiation and/or as changes in reflected solar radiation (such as from changes in clouds, sea ice, snow, vegetation, etc).
There is a fundamental problems, however, with the use of this equation for the description for global warming.
T is defined in the above equation as a global proxy for the thermodynamic state of the climate system. As such, it must be tightly coupled to that thermodynamic state of the climate system. Specifically, in this context, T is the global average radiative temperature of the Earth’s surface since the outgoing radiative flux at the top of the atmosphere is determined to a large extent by the surface radiative temperature. However, this outgoing longwave radiation is proportional to the fourth power of T. T’ = +1 C in the polar latitudes in the winter, for example, would have much less of an effect on the change of longwave emission than T’=+1°C increase in the tropics. The spatial distribution of T’ matters, whereas the equation given above ignores the consequences of spatially varying values of T’.
Eli pointed out that since Maclauren and Taylor tell us something more
To assess the use of the global average of [T**4 - (T+T’)**4], the spatial map of this field should be presented. This would show where the change to the value of sigma T**4 is the largest. Please show this. The plot of “normal” and ‘average” radiative temperature is not what we are proposing.
Indeed, it is easy to show that weighting by (T+T’)**4 significantly emphasizes the lower latitudes, since the relationship is to the 4th power of temperature. I look forward to your analysis as we have recommended.
(T+T’)^4 = T^4*(1+T’/T)^4
(1+T’/T)^4 ~ 1 + 4 T’/T+ higher order smaller terms, and then
[T**4 - (T+T’)**4] = 4 T’UPDATE 2/6/08: As pointed out in the comments Eli screwed this up
[T^4 - (T+T’)^4] = T^4-T^4*(1 + 4 T’/T) = T^4* 4 T’/T= 4T'*T^3
good thing no one reads this blog. The change in emission depends on T^3 for a constant change T'. OTOH for the 250-350 K interval this is also pretty well approximated by a linear function in T.
so as long as T’ is small compared to T (1K/300K is pretty small) the change in radiation is linear in T’ EVERYWHERE so Roger’s ”
“Indeed, it is easy to show that weighting by (T+T’)**4 significantly emphasizes the lower latitudes, since the relationship is to the 4th power of temperature. I look forward to your analysis as we have recommended.”
is just wrong, as anyone who learned about series expansions of functions in Cal I would know. Given the back and forth with Roger Pielke Jr. about Hansen's 1988 scenarios being exponential or linear, this failing appears to be genetic.
Above, Eli has shown that the CHANGE, in blackbody emission for a body at any temperature T by a small amount T' is linear in the change in temperature. This means that if you are looking at changes in emission due to changes in temperature, it doesn't much matter if you take it as functions of T'^4 power or T'.
Yet, young bunnies, lets see how well a linear form fits the Stefan-Boltzmann curve itself over a reasonable temperature range of 250 to 320 K (-23 to +47 C, ethical science bunnies don't do no F).
The red line (the red S-B curve) fit's pretty well to a linear function of temperature (the straight line). Tant pis Roger.