In the previous posting about the RSS AMSU temperatures from May 2010, I have made one obvious observation explicit. If you look at the

RSS monthly anomalies,you will notice that the month-on-month variations of the temperatures in the first column with temperatures - namely the global mean temperature anomaly - are much smaller than the variations in the other columns, especially the fifth (Northern) and sixth (Southern) polar temperature columns.

It is often said that the temperature changes in the polar regions are more pronounced than they are in other regions. But is it true? I made a comment - a guess - that the main reason why the first column is changing much less violently is simply that it describes a larger area of the Earth's surface, namely almost the whole world, and the local fluctuations are therefore averaged out more accurately.

If you assume that there are N quantities that have the same statistical properties such as the typical variation in time or the width of their distribution, the average of these N numbers will have a variation that is sqrt(N) times bigger than one such quantity - which means sqrt(N) times smaller than this quantity multiplied by N.

I have explained this basic point about statistics in a previous article.

Because the 60°N-82.5°N strip of the Earth covers about 1/16 of the Earth's surface, we may say that the whole surface is 16 times bigger so the global temperature variations may be expected to be approximately sqrt(16) = 4 times smaller than the Northern polar ones.

JollyJoker has stated an alternative hypothesis in the fast comments: the variations of the polar monthly anomalies are bigger because the Arctic, with its feedbacks, amplifies the global temperature swings. He supported this statement by a "visual inspection". Who is right? Which effect is more important? Can we test these hypotheses?

You bet. There's a lot of data to test such hypotheses.

So I made several calculations. First, I decided to find the best linear fit that allows you to predict the Northern polar monthly temperature anomalies out of the global ones. You assume that there is a linear relationship of the form

TOf course, this relationship surely doesn't hold exactly. But you may try to find the best values of "a,b", by the method of least squares: we want to find the best linear fit. We're mainly interested in the value of the coefficient "b" - an amplification factor._{North polar}(t) = a + b T_{global}(t).

JollyJoker's hypothesis - and the generic lore you may often hear about the Arctic - would lead you to believe that "b" is very high, something like 3 or 5. How much is "b" actually? Well, Mathematica allows you to find the best fit by a single command (the Mathematica code may be found in the fast comments under the RSS AMSU article).

The answer is that if you substitute the RSS monthly data since 1979, the coefficient is only "b=1.54" for the North polar region. It means that you get the best prediction of the North polar monthly temperatures, assuming that you know the global ones, if you multiply the changes of the global temperature by the factor of 1.54.

That's slightly greater than one but not by much. The opinion that this "slightly above one" number is just noise is supported by the Southern polar result which should a priori be expected to be similar: however, the corresponding Antarctic value of "b" is just 0.22 - it is actually much smaller than one! It means that the Antarctica apparently suppresses the global temperature swings. Its temperatures try "not to care" about the global changes and they're 78% successful in this goal.

I suspected that a significant portion of these numbers is due to the overall trend: the Arctic has been warming much faster than the Antarctica (three times or so). So I also tried to "detrend" the data, i.e. to subtract the linear curves. That's equivalent to finding the best linear fits for the month-on-month jumps. Such an alternative task is much more sensitive to higher-frequency, shorter-term changes of the temperature.

As you might expect, both results went closer to one. It was 1.26 for the Northern polar strip and 0.35 for the Southern polar strip. It's still true that the North is somewhat above one while the South is significantly below one.

If we focus on the North for the rest of this text, are the factors of 1.54 or 1.26 enough to explain the more significant fluctuations in the Northern polar column relatively to the global column? The answer is a resounding No. If you sum the 376 squared month-on-month jumps (between 377 monthly data), you will obtain

5.65 (°C)A huge difference between the global and the local data, indeed. Divide the numbers by 376, to find the average squared jump, and take the square root. You will find out that the typical month-on-month jumps - the root mean squares (or quadratic means; I like the term "Pythagorean averages") - are^{2}for the global

191.71 (°C)^{2}for the North polar strip

105.7 (°C)^{2}for the South polar strip.

0.123°C globalThe Northern and Southern polar strips have 5.8 and 4.4 times bigger variations than the whole surface, respectively! These factors are much greater than 1.54 or 1.26. And of course, the situation is even worse for the Antarctic region because the global temperature didn't help us much to predict the local temperature: recall that "b" was smaller than one.

0.714°C North

0.530°C South

It means that the large month-on-month variations of the Northern or Southern polar strips are completely dominated by noise. In fact, note that at the beginning, I predicted that just by counting the areas, the Arctic variations should be 4 times stronger than the global ones. This seems to agree with the data (5.8 or 4.4) extremely well.

So JollyJoker has been misled by his "visual inspection". It's not true that the polar regions amplify the changes of the global mean temperature by a significant factor. Most of the "largeness" of the local polar variations are due to the noise - also known as the "weather" - that has nothing to do with the changes of the global mean temperature.

In fact, if you calculate the typical temperature variations at different places than the Arctic, but you consider comparably large regions, your conclusions won't be too much different. You may just notice that the continental climate suffers from larger variations while the coasts experience smaller variations (largely because of the water's heat capacity). But the polar regions are not "strikingly" different from comparably large regions in the moderate zone or elsewhere.

(I should draw the map of the typical month-on-month variations of the local temperature anomalies at some point. Sorry that I don't have it now. It's something I want to see.)

**Big picture**

So the right qualitative picture of the regional temperatures is the following:

You may imagine that the Earth's surface, with its 510 million squared kilometers, is divided to 300 "squares". Each of them is a 1300 km by 1300 km square whose area is 1.69 million squared kilometers. Of course, you shouldn't imagine that there are canonical, sharp boundaries between these regions: a sphere can't be divided to squares, after all. Instead, the boundaries are fuzzy - much like the "cells in the phase space", if you're a physicist - and the purpose of the picture is to calculate the number of "independent regions" that have their own weather rather than to discuss their detailed geography.

If the distance between two places is much shorter than 1300 km, you should appreciate that their monthly temperatures will be significantly correlated. If the distance between two places is much greater than 1300 km, you should appreciate that their temperatures will behave almost independently. I chose the number 1300 km as a good representative to engineer a consistent and simple picture that doesn't need a terribly sophisticated mathematics; in a different quantitative treatment, the typical distance at which the correlations will go away will probably be similar to 1000-2000 km.

So as long as you consider regions much larger than 1.69 million squared kilometers, the typical temperature variations will be a decreasing function of the area. The characteristic quadratic means of the month-on-month temperature jumps will behave as

Delta T = 2 °C / sqrt(area/1.7 mil kmThe minimum characteristic month-on-month temperature jump (calculated from the anomalies) is 2 °C. A jump of this magnitude is relevant for the local weather at one particular place and for areas that are smaller than 1.7 million squared kilometers. When you consider larger areas, the noise tends to average out, because of the basic laws of statistics.^{2}).

Once you get to the whole Earth's surface which is 300 times bigger than our elementary square, the temperature variations will drop from 2 °C to 2 °C divided by sqrt(300). That's equal to 0.12 °C, as calculated at the beginning. Agreed. Of course, I chose the figure 1300 km so that these figures agree almost exactly but they're roughly consistent with other checks, too.

I claim that this picture is compatible with all basic statistical characteristics of the regional temperature series.

**White vs red noise: autocorrelations in space vs time**

There is one disclaimer I should add: when I say that the local weather - or the weather in the 300 elementary squares of the Earth - is just "noise", it's important to realize that I am not saying anything about the color of this noise i.e. about its autocorrelations in time. It's clear that the local weather is mostly "white noise": what happens in Moscow on May 9th this year is pretty much uncorrelated to the same day in the last year.

As you're getting to larger areas, the color of the noise is getting more red. That's because of the "inertia" or long-term persistence of the atmosphere. If you classify the colors of the noise to "mostly white" and "mostly pink/red", according to an arbitrary but fixed threshold for the critical exponent, you will find out that the threshold for this exponent depends on the area of the surface you study.

At short enough periods of time, the variations are so dramatic that you may imagine that the noise is white. It is pink for long enough periods of time and there is a transition somewhere in between: the transition is determined by a time scale where "mostly white" switches to "mostly pink/red". This time scale is getting shorter if you consider larger areas.

In fact, there should exist another power law that determines how the crossover timescale between the white and pink/red noise depends on the area. It's plausible that this time scale is simply inversely proportional to the area - so that the 2+1-dimensional volume of spacetime is fixed. ;-) I haven't checked this thing.

It's also important to note that if we didn't consider month-on-month changes but changes over different basic intervals of time that is K times longer (or shorter) than one month, the analysis would have to be carefully redone. Various quantities would scale as different powers of K: some of them could remain constant, others would increase sqrt(K) times, and so on.

When I talk about different "pixels" in time, it's also important to say that the independence of the regions only holds at short enough time scales. At long enough time scales, the exchange of heat may become sufficient to keep the globe more aligned. In other words, the effective number of "300 cells per surface of the Earth" would become smaller if you considered e.g. decadal averages rather than monthly averages.

At any rate, the noise i.e. the weather and its partial averaging out over large enough space, large enough time, or large enough spacetime, and the statistical properties of such averaged numbers are completely paramount for the understanding of a vast majority of the statistical features of the RSS temperature tables and all the correlations you may try to find in these tables.

**Summary**

There is no "climate" in the RSS temperature tables from 1979 to 2010: everything is "weather". No simple relationship that assumes that the temperature on several sufficiently distance places are hugely correlated - or even amplifications of one another - works well enough for it to be accurate enough or useful. Instead, the assumption that the individual places behave independently is much more compatible with all the basic statistical features of the data.

There's only "weather" which is pretty much random while the "climate" is just whatever you get by averaging the "weather" - and its variations are dominated by the statistically inevitable residuals from the changes of the "weather" rather than by independent simple linear relationships that would be characteristic for the "climate".

Because a basic point of the "science of climate change" is to completely neglect the weather, such a science can't be compatible with the observations of the temperatures at time scales that are comparable to 30 years. In this sense, the whole "science of climate change" - whose very goal is to pretend that the weather and meteorology don't exist or don't impact the global averages - is a pseudoscience.

And that's the memo.

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