After investigating the issue in some detail and actually doing the relevant surface integral for emission increase for a 1K increase in temperature ("odd" as some may find such an approach) , I found that the error associated with using the global mean temperature for the case in question was fairly small (probably a few percent at most). I described why this was so in my last post.
But apart from the variation of base temperature across the globe, shown in this "Temperature VS Latitude" graphic by Robert Rhode, there is a separate issue that also impacts the amount by which radiative emission changes at each location: the amount of temperature change at a given location.
For my previous analysis, in order to isolate the effect of base temperature on total radiative emission change by the earth (the issue originally suggested by climate scientist Roger Pielke, Sr in this post), I made the (admittedly artificial assumption) that temperature went up uniformly by 1K at each location across the earth's surface.
But, in reality a change in the global mean temperature is not the result of a uniform change in temperature across the earth’s surface. In fact, the actual change depends on location. For example, over the past half-century (1955 – 2005), the global mean surface temperature went up by 0.59 deg C, but the changes that were used to calculate that mean depended significantly on the location, as shown in the following NASA graphic:
Figure 3. Temperature change over the past 50 years based on local linear trends
(found on this NASA page)
Note: The black line on the right shows the change in the annual mean for each latitude over the period 1955- 2005 . Though the error bars (standard deviation) for each latitude are not shown, for the analysis below, they were calculated from the 2-D temperature map at the left.
So, the new question I would like to consider is the following:
Does using the change in the global mean temperature rather than the temperature changes as a function of location (shown by the NASA graphic) yield a significantly different answer for "total change in radiative emission by the earth'?
The answer may come as a surprise to some , but “No, it does not."
In fact, if one considers both the details of base temperature as a function of location and the details of temperature change as a function of location in order to calculate the total radiative emission change, one gets a result that is within about 3 % (with a similar associated uncertainty) of the result that one gets if one instead simply uses the global mean temperature and change in the global mean temperature to calculate the total emission change.
Effect of varying temperature change on the surface integral of radiative emission
We can estimate how much radiative emission changes across the entire earth's surface in response to temperature changes across the earth by doing a similar surface integral to the one I did for the original analysis.
But this time, when we do the surface integral, we simultaneously take into account both the base temperature as a function of location and the change in temperature (over some time period) as a function of location (as shown in the above NASA graphic)
For this new integral, we again assume that base temperature depends on surface location according to our previous piecewise linear approximation to the “Temperature VS Latitude” relationship, given on this annotated “Temperature VS Latitude" graphic (original graphic by Robert Rhode), but instead of assuming the 1K change in temperature at all locations across the globe, we now use the temperature change as a function of latitude over the past half-century, shown by the black line on the above NASA "Temperature Change" graph on the right.
As before, we use the “differential of radiative emission” -- proportional to “T^3(dT)” -- to approximate the emission change for a small change in temperature at each location (as indicated in the previous post, for a temperature change on the order of 1 degree K, this approximation introduces very little error into the result). Also as before, the temperature “T” at each location is taken from the central “Temperature vs Latitude” graph (or more precisely, the piecewise linear approximation shown superimposed on the graph).
But this time, instead of assuming that the temperature change “dT’ is 1K for each location, we use the temperature change for each latitude over the past half-century (1955- 2005) shown by the NASA graphic.
We then integrate“T^3(dT)” over the earth’s surface (for latitude-dependent "T" and " dT ") and compare this result to the result obtained by assuming that 1) every square meter of the earth has undergone an increase in temperature of 0.59C (the change in the global mean temperature over the past 50 years, based on linear trend) and 2) every square meter of the earth has emitted at the global mean temperature ( 287.4K -- ie, before the slight increase).
Note: a piecewise linear approximation to the NASA “Temperature Change VS Latitude” curve was used to do the integrals here (admittedly lots of pieces and lots of integrals!). As a check on this approximation, the average over the earth’s surface of temperature change from 1955-2005 was calculated using this piecewise linear approximation. This average was found to be within 0.3% of the 0.59C value that is given by NASA on their graphic (just above the 2-D temperature map on the left above), so the approximation used was indeed quite good.
This new result for "surface integral of T^3(dT) " -- taking into account both the details of base temperature as a function of position on the surface of the earth (represented by the “Temperature vs latitude” curve) and the variation in temperature changes (shown by the “Temperature change Vs Latitude” curve) -- differed by only 3.3% from (was about 3.3% less than) the surface integral obtained by assuming that 1) every square meter of the earth emitted at the global mean temperature and 2) every square meter of the earth underwent an increase in temperature of 0.59C.
Note: Details of the approximations for T and dT (as a function of co-latitude), radiative emission integrals (increase over past 50 years), and global averages of T and dT using the approximations are provided here.
So, in other words, if we assume that both “temperature” and “temperature change” across the surface have the given latitudinal dependencies (given by our two graphics), we get an answer for the “total change in radiative emission across the surface of the earth” that is within 3.3% of the answer obtained using the global average temperature and change in the global average temperature (based on linear trend). The associated uncertainty (see below) is just a few percent.
Uncertainty associated with the above "3.3% difference" result
There is (what turns out to be a relatively small) uncertainty associated with this result. In my previous post, I considered the uncertainty introduced into the surface integral of T^3 by use of my piecewise linear approximation to the “Temperature vs Latitude” graph to represent base temperatures across the earth's surface. I determined that this uncertainty was not likely to be greater than a few percent. I also explained why using the mean temperature value for each latitude rather than the values scattered about the mean introduced a very small error into the integral calculation (of order 0.1%).
For the analysis above, there is an additional uncertainty associated with using the "Temperature change vs Latitude" relationship shown on the NASA graphic (black curve) rather than the individual temperature changes experienced by locations at each latitude. The issue is actually similar to what we encountered before: ie, the use of a mean value rather than the individual values that went into the mean.
In this case, the (black) curve shows the mean temperature change for each latitude. Though the error bands are not shown on the NASA graph at the right, they can be (and were) readily calculated from the temperature map at the left and when this is done and the "scatter" for each latitude was taken into account, it was found to make very little difference.
Why using the NASA "Temperature Change vs latitude" graph introduces very little error into the result
There is a very simple reason why using the mean temperature change for each latitude (shown by the black Temperature vs Latitude curve) yields very nearly the same answer as using the individual temperature change values that went into that mean:
As a first approximation, one can assume that all the locations at a given latitude emit at the same base temperature. Though this is not necessary to our final conclusion, it is actually a very good approximation, given the very small base temperature deviation relative to the mean base temperature at each latitude -- and the assumption makes it very clear why using the mean temperature change at each latitude is justified for the purpose at hand.
If one makes the above assumption that all the locations at a given latitude emit at the same base temperature (the mean value for that latitude T_o), one finds that the total amount by which radiative emission at that latitude changes is then proportional to the sum of the individual temperature changes experienced by all locations at that latitude. But the latter sum is just the mean temperature change for that latitude (dT_o) multiplied by the number of locations (N) -- ie, " N (dT_o) "
So, the emission change in that case is equal to
N (T_o)^3 (dT_o)where " T_o " is the mean base temperature for that latitude and " N " is the number of locations included in the sum (and we have ignored the Stefan Boltzmann constant, emissivity and a factor of 4)
In other words, under the assumption that all locations at a particular latitude emit at the same temperature T_o, we find that the total emission change is precisely what one gets if one assumes that all locations at that latitude have experienced a temperature change equal to the mean change for that latitude (ie, the value given by the black NASA curve on the right above).
In reality, the situation is more involved, since the locations at a given latitude do not all emit at precisely the same base temperature (ie, not all emit at precisely the mean T_o, though the vast majority are not far from it).
But (for a given latitude), if one simultaneously considers the (slight) deviation of the base temperature of each location from the mean " T_o " (for that latitude) and the deviation of the temperature change for each location from the mean temperature change " dT_o " (for that latitude), the result is not much different.
For the mathematically inclined, I provide the details of the latter analysis here (Note: after bringing up the first page, click on the small "tempchange (Set)" graphic to view the next page, of which there are 7).
Climate Scientist Raymond Pierrehumbert made the following comment on Rabett Run regarding the above "issue":
A lot of you guys are getting somewhat led up a garden path. There's no business about "scientists missing an error of 17%" going on no matter how you do the arithmetic. Climate models DO NOT CALCULATE THE ENERGY BUDGET USING A SINGLE GLOBAL MEAN TEMPERATURE,... Climate models do a radiative transfer calculation several times a day at each gridpoint, incorporating the full variation of temperature.
The sort of thing you guys are talking about only tell you how big the errors are in the most primitive blackboard-type zero-dimensional climate calculation, where indeed you do do the energy budget in terms of a global average temperature. The fact that the errors made by doing so are so small is in fact why you can get pretty far with such simple calculations, especially on a planet like Earth or Venus with a thick atmosphere and or ocean to redistribute heat and make temperature more uniform.
But on the issue of "scientists missing an error of 17%", I would remark that after looking into this issue in some detail, I don't even have to take Pierrehumbert's word for it. I know he is right: There is no business about "scientists missing an error of 17%" (or anything even close) going on no matter how you do the arithmetic. Claims to the contrary simply do not hold up under a careful analysis of the issue.
And you can indeed get pretty far with some relatively simple calculations in this case.