## Averaging

The layer model assumes that heat fluxes in and out of the Earth must be in balance exactly. What the layer model is after is an average temperature over the entire globe and over time. This is a reasonable assumption to make if we are willing to wait long enough for our answer to be right. On long enough timescales there is simply nowhere else for the heat energy to go; what comes in must go out. As we look in more detail, however, there are all kinds of wild imbalances. It takes heat every spring to warm lakes and ocean water and to melt the snow. There is also an imbalance between the heat fluxes on the long time-average, because of heat transport from equatorial regions and the higher latitudes to the north and south.

So is an eternal, unchanging, averaging model a reasonable one for a world that is bouncing around like Jell-O? Can you construct the average of the whole system by using averages of the pieces of the system? Or will averaging change the answer? The term for this possibility is biasing. Biasing issues come up a lot in the natural sciences. In principle, there could be a problem with averaging IR energy fluxes because they are a nonlinear function of temperature. One way to see the nonlinearity is to look at the equation and see that the light flux is proportional to temperature to the fourth power, and not to the first power, which would be linear

W m2K4

Another way would be to notice that a plot of the function is not a straight line (Fig. 6.1). Let's say we wanted to estimate the average energy flux of a planet that had two sides, one at 300 K (rather like Earth) and the other at 100 K (much colder than anywhere on Earth). The outgoing energy flux from the cold side would be about 6W/m2, and from the warm side 459 W/m2. The average energy flux would be 232 W/m2. Let's now m2

Temperature (K) (b) 600-,-

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Fig. 6.1 An example of how averaging can bias the results of a nonlinear system such as blackbody radiation energy flux a T4. In (a) we average across a huge temperature range, and the flux we get from combining areas of 100 and 300 K is very different from what we would expect if we averaged the temperature first to 200 K and then computed the answer. In (b) we see that over the temperature range of normal Earth conditions, the blackbody radiation energy flux is closer to linear, so averaging would be less of a problem, than in (a).

average the temperatures, run the average T through a T4, and try to get the same answer. The average temperature is 200 K, and the predicted energy flux is 91 W/m2. We'd be off by more than a factor of two. You can see the effect graphically as the difference between the straight line and the curve in Fig. 6.1a.

For the terrestrially normal range of temperatures (Fig. 6.1), we are zooming in on the nonlinear function enough that it looks much straighter than when we considered an absurdly wide temperature range. Still, it could be an important effect, and we certainly wouldn't want to neglect it in the global warming forecast. It doesn't appear that the biasing is so bad as to undermine the principles demonstrated by the layer model, however, because in this case the function turned out to be fairly linear.

Biasing could arise when calculating a time average from a time series of measurements, as if air temperatures were measured more often during daytime than at night, for example. The problem can be corrected for, as long as it is recognized. There are many other nonlinearities in the climate system, pairs of variables that are related to each other in nonlinear ways. For example, many effects of the wind, such as exchanging heat with the ocean, depend nonlinearly on wind speed.