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Average temperature vs average irradiance

Vincent Gray and Alan Siddons have been emphasizing an important point that the (arithmetic) average temperature is not the relevant quantity that should be substituted into various calculations of the heat and energy budget of Earth.

Why? Because what matters for the energy budget is the average radiated (or absorbed) energy. According to the Stefan-Boltzmann law, the energy radiated by a black body per unit surface area (also called "radiant emittance") is proportional to the fourth power of the absolute temperature.

When we divide the solar constant by four, to obtain 342 Watts per squared meter, we really average the irradiance over latitudes, seasons, and parts of a day. If the Earth were a black body, that would be equivalent to averaging the fourth power of the absolute temperature.

But the average value of the fourth power of temperature is something different than the fourth power of the average temperature!
The IPCC often seems to neglect this point. Similar non-linearities were also emphasized by Essex, McKitrick, and Andresen in their article "Does a global temperature exist?"

You might think that the difference between the two types of averages - average temperature and the temperature calculated from the average irradiance corresponding to individual temperatures - is negligible. Well, it is not. I ask you for some patience to look at the situation quantitatively.

The Stefan-Boltzmann equation is the following:
j = σ T⁴.
The irradiance j is proportional to the fourth power of the absolute temperature T. The proportionality constant σ (sigma) is called the Stefan-Boltzmann constant and it equals 5.67 x 10-8 W/(m²K⁴), i.e. 56.7 nanowatt per squared meter and per quartic Kelvin.

Now, let us calculate a bit to see that we are on the same page. Imagine that the average temperature is 15 °C i.e. 273+15 = 288 Kelvin. Here are a few calculators if you need one. Take the fourth power of 288 and multiply by the constant above. You should get 390.08 Watt/m². Because of the (mostly natural) greenhouse effect, albedo, and related effects, the relevant average solar irradiance is about 342 Watt/m² but the qualitative essence of the argument below is unaffected.

Non-uniform Earth: the numbers change

However, let us now appreciate that the temperature is not uniform all over the Earth and throughout the years and days. The non-uniformities come from
  • climate zones - latitude etc.
  • albedo variations
  • weather - random local, temporary variations
  • seasons
  • day-and-night cycles
Each of these effects creates non-uniformities. For the sake of simplicity, let us only consider climate zones, realizing that the discrepancy revealed by such a gedanken experiment will be underestimated. Also, we approximate the world by 4 climate zones of the same area whose temperatures are
  • 313 K (40 °C)
  • 293 K (20 °C)
  • 283 K (10 °C)
  • 263 K (-10 °C)
The arithmetic average of these four temperatures is 288 K (15 °C).

The figures in Celsius degrees are included for your better understanding. Let us consider the Kelvin figures to be exact. Now, take the fourth power of these four temperatures, multiply them by the Stefan-Boltzmann constant, and calculate their average. You should get
(544.2 + 417.9 + 363.7 + 271.3)/4 =
= 1597.04 / 4 = 399.26 Watt/m²
That differs by 9 W/m² from the figure that we calculated from the arithmetic average of the temperatures! Now, 9 W/m² is a lot. It is more than five times the change of the irradiance (1.6 W/m²) blamed on greenhouses gases added since 1750!

One thing to notice is that the hot regions contribute more - by a factor of two! (540/270) - to the energy budget which is much stronger difference than what you would expect if you treated the temperature linearly. The fourth power becomes rapidly increasing at higher temperatures. And on the contrary, the same absolute temperature variations of cool regions are much less important for the energy budget than those of the hot regions.

Evolving non-uniformities

The discrepancy above is indisputable but you might think that it doesn't matter because when we differentiate it with respect to time, it goes away because it is time-independent. But it doesn't go away because the amount of non-uniformities has been changing since 1750, too. To see how much it matters, let us consider other four temperatures whose average is 288 K (15 °C), namely
  • 310 K (37 °C)
  • 292 K (19 °C)
  • 284 K (11 °C)
  • 266 K (-7 °C)
Relatively to the previous list of four temperatures, I have reduced the variation. The hottest temperature was reduced by 3 °C, the coolest one was raised by 3 °C, and the middle temperatures were shifted by 1 °C in the direction that reduces the differences. You can check that the average is still 288 K (15 °C).

Repeat the exercise with the fourth powers. You should obtain 397.14 W/m². That's nearly by 2 W/m² less than the previous non-uniform calculation. This figure still exceeds the whole contribution to the energy budget attributed to man-made greenhouse gases since 1750.

Now, if you assume that the zones that led to the 399.26 W/m² result were relevant in 1750 and you want to keep the same incoming energy 399.26 W/m² in 2008 while having the same temperature differences as in the list (310,292,284,266), you will have to increase the latter four figures by something like 0.4 °C. Check the math. The conclusion is the following:
Even if the (arithmetic) average temperature increases by 0.4 °C or so, the total energy budget of the Earth can be completely unaffected as long as the combined non-uniformities of the temperature (from climate zones; regional albedo variations; day-and-night differences; seasons; and weather) decrease by those 3 °C described above - from ±25 °C to ±22 °C, so to say.
So in reality, the arithmetic average temperature may be increasing, variations (and storminess, weather extremes, and other things that have been described as "bad") may be decreasing, and the combination of these two is no evidence of a man-made forcing simply because the overall energy budget is completely unchanged!

And indeed, there are hints, to say the least, that the day-night differences, winter-summer differences, polar-tropical differences, and albedo non-uniformities have been decreasing in the last century or so. Someone should refine the calculations above.

There are two basic lessons to be learned from this exercise:
  1. The impact of nonlinearities shouldn't be neglected and climatology should carefully observe the evolution of the differences between climate zones; seasons; weather variations; regional changes of albedo; day-and-night differences.
  2. In the calculations of forcings, it is not the arithmetic average of temperatures that should be substituted but rather the fourth root of the arithmetic average of the fourth powers of the (absolute) temperatures. In this way, the bulk of the problems discussed in the previous point - and in this whole article - can be circumvented.
And that's the memo.

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