(Quite long, but quite amusing:)
As part of our on-going obsession with Essex and McKitrick's Taken by Storm we return to their briefing. Not that we have made no progress friends, we now know that by temperature, they do not mean something read by your average garden thermometer holding concrete figure, but something much more amusing and a lot less useful. Having confused us, their mother-in-laws and definitely McKitrick, they then choose to cast aspersions on global temperature measurements, calling it the king of temperatures, "T-Rex". As we will see old friend T is not adverse to taking a bite out of their butts. This is fair because E&M have sown a lot of confusion in their jeremiad
First, two definitions: An extensive parameter depends on the size of the system, an intensive one does not. Volume and energy are extensive, temperature and pressure are intensive. We can boil Essex and McKitrick down to two statements. By the end of this post, mom will be able to tell you why both are wrong. The first is:
...the physical equations involving temperature assign no meaning to a sum of temperatures on their own, only in conjugate pairs with extensive variables. So here is a question: if the mean is the “total T”, divided by the number of observations n, and “total T” (i.e. the sum of some temperature numbers) does not have a physical significance, what does the mean mean?
and the second is
If the physical context does not imply one type of average then mathematics steps in with an infinity of ad hoc possibilities. It includes an infinite variety of weights, exponents, algebraic formulas, functions and other mathematical cookware, all chosen to reduce a list of numbers to a single value. When any one of these cookware rules is used, it is normally presumed wishfully that the number produced is “representative.” But in reality an averaging rule can be found to rationalize any value in the list from the largest to the smallest as the “representative. Only in the case of a particular physical context can we select one from this range that has a specific physical meaning. Everything else is just statistics. Everything else is ad hoc. But no such physical rules are prescribed for intensive variables on their own. In the absence of physical guidance, any rule for averaging temperature is as good as any other.
But there is a physical context that implies the arithmetic average is the proper one for temperatures in a homogeneous medium like the atmosphere, and there are several types of averages which are idiotic for averaging temperatures if you use the wrong scales as Essex and McKitrick do. Tim Lambert has pretty well taken apart Figure 2 and 3 in the Taken by Storm briefing, showing the fine touch of Essex and McKitrick with averages.
Unfortunately, either Essex or McKitrick or both do not understand zero and negative numbers. You know where my money is. Let us look at the various types of averages.
For a set of n measurements (a1, a2, a3, a4.....) the arithmetic mean is the sum divided by n. For an arithmetic average, it does not matter where the zero of the scale is, e.g. whether you use Celcius or Kelvin (no ethical scientist uses Fahrenheit or Rankine. Engineers OTOH.......).
The geometric mean is the nth root of the product of all the measurements. IF one of the measurements is zero, then the geometric mean is zero no matter what the other values are. If n is even should you pick the even or odd root? If there is an odd number of negative measurements the root is imaginary, and so on. Clearly there is a real danger in using the geometric mean if you are using a temperature scale where the possibility of zero or negative measurements exists.
To calculate the harmonic mean you find the average of the inverses (1/ai) and then take the average of that. Clearly, if one of the ai is zero, the harmonic mean blows up, and you would be a lot better off.
The root-mean-square average is the square root of the average of the squared measurements (ai^2). Let us see how that does on a couple of simple cases: The RMS average of (+2,+2) is +2, so is the RMS average of (+2,-2) is +2....Hmmm.
But let us not bother with such trivialities. Let us look at the first example that Essex and McKitrick propose.
Consider a system consisting of a cup of coffee, at 33 degrees C, and a cup of ice water at 2C. What is the “one” temperature that describes both these liquids, at this moment, as they stand? Clearly there isn’t one temperature, there are two. The physics does not say in any way that there is a single temperature for the whole. But if you are interested in climate you might say that there is one temperature anyway, if your thinking has been clouded by the T-Rex obsession. To such a person combining the temperatures of the ice water and coffee into one number would be no different than combining the temperatures of the poles and equator into one number. Of course you can do it. You have an infinity of choices, but the physics doesn’t say which one to use. Fine then: pick an averaging rule. Better yet, pick four out of the infinity of choices. Then ask whether this system is “warming” or “cooling” as the liquids relax to room temperature. As is conventional, cooling will mean our average is declining, while warming will mean the average is rising.They then calculate the various averages for the coffee and the ice water warming separately up to a room temperature of 20 C. They claim to plot the predictions of the arithmetic average, the harmonic, the RMS and what they call radiation (proportional to T^4 following the Stefan-Boltzmann law). The only one they get right is the arithmetic average (see below). The figure to the left shows what they calculated. I have added the lines showing the temperatures of the coffee and the ice water
Essex and McKitrick then riff about how the radiation curve is higher than the RMS, and the arithmetic average, and the harmonic is lower (yr. hmble. hare added the geometric average for the heck of it)
So, let us ask is there anything that sets the arithmetic average apart? Why yes, happy that you asked. How about if you had first equilibrated the ice water and the coffee, and then let them warm up to room temperature. You would get the blue dots which overlay the arithmetic average. That seems to me a fine physical reason for preferring the arithmetic average.
But wait, what hath McKitrick wrought? He calculated the "Radiation" curve using Celcius. Any General Chemistry student knows you have to use absolute temperatures when dealing with thermo, maybe even Essex knows that when he is not hunting T Rex, but McKitrick? Hmmm. When you use Kelvin, you get the dashed brown line, which is pretty close to the arithmetic average. Tell you what, let us use Kelvin for all of the averages.
That looks a bit different does it. If you excuse the bad graphics (still learning, advice sought), and blow the image up a bit you will see that the ONLY average that did not change is the arithmetic average.
Seems to me another pretty good reason for using the arithmetic average to characterize temperature. What is more, a lot of the blather about how different the averages are, well, has gone poof.
But wait, we have more! E&M spent a long time pointing out that to "properly" average an intensive quantity, you have to multiply it by an extensive property and you then average the extensive product. Energy appears to fit that bill, remember that for something homogeneous, E = Cv T, where Cv is the specific heat per mole, or m^3, or whatever, and T is in Kelvin. Another good reason to use Kelvin and not that effete Celcius. Now there can be problems if Cv is a function of temperature, but for the atmosphere, it is not the case, or more accurately put, not significantly so.