You seem to be exagerrating. The vast majority of the earth's surface
is typically between -15 and +35 at any time. Try it again with a
smaller range.
Then consider that most of the points you are considering will be in
the middle of the range:
===>
#!/usr/bin/env python
def av(temps):
return sum(temps)/len(temps)
def radav(temps):
temps_k = [temp + 273.16 for temp in temps]
fourth = av([temp_k ** 4 for temp_k in temps_k])
return fourth**0.25 - 273.16
def compare(temps):
r = radav(temps)
a = av(temps)
return (str(temps) + " => " + str(r) +" vs " +
str(a) + " : difference " + str(r-a))
alltemps = [
[-50,50],
[-35, 15],
[-32, -16, -8, -4, -2, 0, 2, 4, 8, 16, 32],
[-32, 32]
]
for tempset in alltemps:
print compare(tempset)
<===
yielding
====>
[-50, 50] => 12.8668749519 vs 0 : difference 12.8668749519
[-35, 15] => -6.50251402771 vs -10 : difference 3.49748597229
[-32, -16, -8, -4, -2, 0, 2, 4, 8, 16, 32] => 1.35422954286 vs 0 :
difference 1.35422954286
[-32, 32] => 5.46946364458 vs 0 : difference 5.46946364458
<====
Both of these effects make the effect you are worried about much smaller.
So for an inhomoegenous global average change of order 5 C, on a
planet slightly colder and more varied than ours designed for
computational convenience, you get a ten per cent difference.
Admittedly this is bigger than I expected. It is likely that the
models are a little bit more accurate on the one measure than the
other, but this is still, unfortunately, in the noise on predictive
time scales.
I guess the point of doing a simple average is pedagogical. People
might be even more skeptical if you have to redefine "average
temperature". I agree that T^4 is arguably a better quantity to be
averaging on physical grounds. In practice, it could be argued that it
doesn't tell a sufficiently different story to be worth bothering
about.
If the global mean sensitivity is really the number of concern, a
simple average makes the number look "worse" by probably less than
10%. You could argue, though, that these are simply different
currencies, and it is better to be consistent.
Exercise for the reader: construct a set of changes that has different
sign in the two measures, and a magnitude in excess of a degree. What
relevance do you expect such changes to have to realistic issues?
Second exercise: what happens if the changes are smaller? Does the
approximation get better? Why or why not? (I got this wrong at first,
which shows how rusty my math is, or as the brits would say, how rusty
my "maths" are.)
If you are using a Windows PC, Python is not preinstalled, but
installing is free and easy. Go to http://python.org .
mt
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