Is There a Global Average Temperature? Part 2

In a previous post I gave my rebuttal to a common objection to climate science that there is no global average temperature (or that it's incalculable or meaningless). At the time I was unaware of a paper published by Essex and McKitrick[1] on the subject, published in 2007 (hereafter EMA07). I've since read it, and while I don't think my previous post needs to change in response to it, I do think it may be worthwhile to update that post with responses specifically tailored to this paper. Others have already responded this paper (it's 18 years old), most notably at RealClimate and Rabett Run, and they do a thorough job of responding to the more technical aspects of this paper. I don't think I can add anything here that you wouldn't be better served reading there, but I do have a couple thoughts about this that I think would be helpful.

Averaging Intensive Variables

The main argument of this paper appears to be centered on the distinction between two types of variables: extensive and intensive. In their words, extensive variables are additive - volume, mass, energy, etc. Intensive variables are not additive; their values are independent of size, and so adding them makes no sense - temperature, pressure, etc. And of course this is true. Let's take as a hypothetical example, two  1-liter containers of water, container A at 20 C and container B at 80 C. You can add their masses (2 kg) and their volumes (2 liters) but adding their temperatures makes no sense. Combining the two containers would not produce a liquid at 646 K. From here things get a little weird. I'll quote:

A sum over intensive variables carries no physical meaning. Dividing meaningless totals by the number of components cannot reverse this outcome.

In the abstract I can see that point. It's meaningless to add temperatures in the way I described above. Combining two liquids doesn't add temperature volume or mass. But is it true that "dividing by the number of components can't revers this outcome?" Seems to me that's something we can test experimentally. First, let's note that adding the temperatures of containers of the same size and dividing by the number of containers produces a calculation with the units of temperature; so far, so good. But as EMA07 show, that's not sufficient to know that the resulting calculation is actually meaningful. However, let's take our container A and container B and mix them with no heat loss. What would the resulting temperature be? Well the experimental answer is that it will show up in "middle" between the two. With no heat loss, the mixture will have a temperature of 50 C. And it turns out that we can reliably calculate this temperature, and here's how. Let's call the temperature of first container Ta and the second container Tb. The temperature of the mixture is Tm. So:

Tm = (Ta + Tb)/2
Tm =  (20 + 80)/2 = 50 C

In other words, this is an average of the temperatures of the two containers. And it turns out that this works for any number of containers of the same size and the same contents (or same heat capacity). So we can verify experimentally that averaging temperature can produce meaningful results. And we can take this a step further. Let's make container A two liters instead of one. Now we have two containers of differing volumes. This doesn't prevent us from calculating the resulting temperature of the mixture with no heat loss. We can simply do a weighted mean. Using Ma and Mb for the masses of these containers:

Tm = (Ta*Ma + Tb*Mb)/(Ma + Mb)
Tm = (20*2 + 80*1)/(2 + 1) = 40 C

So we can validate experimentally that calculating mean temperatures and even weighted mean temperatures produces meaningful results. Adding the temperatures has no physical meaning, but dividing the totals by the number of components does (or at least can) reverse the outcome.

Types of Averages

Now I want to be fair here. EMA07 do acknowledge this, though to me they explicitly contradict their claim that the math of averaging temperature produces meaningless results. Here's their words:

After putting them in thermal contact they will equilibrate at the common temperature (Ta + Tb)/2. If on the other hand they equilibrate reversibly, i.e. while producing work, their common final temperature will be √ TaTb.

Fair enough, but I don't think this gets EMA07 out of what I think is a real mess. They point out there are many types of averages, or many types of "middles" that can be calculated. It's a fair point that the type of average should be stated, though I think it's fair that if you just say "average" you're talking about the arithmetic mean and not any of the other types of averages or "middles" that could be calculated. But the point is that these types of calculations exist because they each have uses in varying contexts, and many of these can be verified experimentally. Scientists know when to use (Ta + Tb)/2 and when to use √ TaTb. And when you use the right calculation you get meaningful results.

EMA07 goes into a fair amount of detail to show that different kinds of averages produce differing results and can produce differing trends. I don't see any need to weigh in on the details here, because I think this would miss the larger point. The point is that these types of averaging methods exist for different uses, and the fact that you could use a different averaging method to get differing results or contradictory trends doesn't mean that the choice to use the arithmetic mean when calculating GMST anomalies vs some other method is arbitrary (or that resulting calculations aren't meaningful). The choice between doing incorrect math and doing correct math isn't arbitrary. To calculate average daily temperature, most datasets use Tavg = (Tmax + Tmin)/2, and that can be verified experimentally as being meaningful and useful[2]. It's not like there's good reason to use √ Tmax*Tmin, and the decision to use the former instead of the latter is just an arbitrary decision.

It's also fair to say that there can be multiple calculated "middles" that are both valid and different from each other but still meaningful. There are good reasons to calculate both mean and median, for instance, and the differences between the calculated values can provide helpful information.

The Meaning of Global Average Temperature

Buried under the layers of weirdness in this paper, I think there may be a decent point. Global average temperature is just one metric we can use to evaluate the impact humans are having on the climate, and it's probably by far the most common way of evaluating global climate, but I don't think it's fair to say that there's a precisely proportional relationship between GMST and the excess energy being trapped within the climate system (nor do I think scientists are suggesting this). Surface water and 2m air have different heat capacities for one, but not all the excess heat is at the surface. Some of it transported deeper in the oceans, some goes to evaporate water, some goes to melt ice, etc. So we should be careful not to make GMST mean more than it does. But Rasmus Benestad has a really good point that I think justifies putting this metric as front and center in discussions of climate change.

The whole paper is irrelevant in the context of a climate change because it missed a very central point. CO2 affects all surface temperatures on Earth, and in order to improve the signal-to-noise ratio, an ordinary arithmetic mean will enhance the common signal in all the measurements and suppress the internal variations which are spatially incoherent (e.g. not caused by CO2 or other external forcings).

This coupled with the fact that we have good results back to 1850 I think is a compelling case for continuing to use this metric. However, there are other metrics that can be and are used, like ocean heat content, the volume of the oceans (ocean water experiences thermal expansion with warming), sea level rise, global ice balance, and the height of the tropopause. But the problem for EMA07 is that all these metrics are telling virtually the same story. The climate is warming in response to the increase in greenhouse gases (partially masked by aerosol pollution). So if you don't like GMST, pick another one. Nothing changes.

References:

[1] C. Essex, R. McKitrick, B. Andresen: Does a Global Temperature Exist?; J. Non-Equil. Thermod. vol. 32, p. 1-27 (2007).
https://www.rossmckitrick.com/uploads/4/8/0/8/4808045/globtemp.jnet.pdf

[2] Wang, K. Sampling Biases in Datasets of Historical Mean Air Temperature over Land. Sci Rep 4, 4637 (2014). https://doi.org/10.1038/srep04637