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Weather critical exponents

How quickly is the weather changing? What is the right probabilistic distribution for the apparently random history of temperatures at a given location? Consider the deviation "Delta T(d)" of the daily temperature (on day "d") from the long-term average. Compute the correlation coefficient "C(s)" between "Delta T(d)" and "Delta T(d+s)", two temperature variations separated by "s" days.

Govindan et al. (some of the authors being famous people who promoted fractals) showed in Phys. Rev. Let. (and in a book edited by Murray "SantaFe" Gell-Mann) that there is apparently universal scaling

  • C(s) = #.s^{-gamma}

where the exponent "gamma" seems to be universal, between 0.6 and 0.7, independently of the location (at least as long as it is a continental station). The universal exponent could very well be 0.65. (This counting is somewhat analogous to the CMB scale-invariant spectrum but the exponent differs.) The law seems to hold when "s" is a couple of days or ten years - in fact, no violation of the law is known from the data, not even at very long time scales. This critical exponent is what I call an interesting insight about temperature dynamics.

The authors demonstrate that most climate models give a very different exponent which is usually closer to an experimentally wrong value 0.5, and moreover lead to results that depend on the location: coasts are supposed to differ. Because the results of Govindan et al. imply that the climate models don't work - and moreover, more concretely, overestimate the trends, the consensual scientists such as William immediately know what to think about the paper:

  • But... is [the paper] any good? Weeeeeelllll... probably not. This is yet more of the fitting power laws to things stuff. They use "detrended fluctuation analysis" (DFA) which I don't understand, but that doesn't matter, we'll just read the results.

Of course, the result that William sees at the end of the paper is that the models give wrong exponents and their prediction of global warming is thus unjustified. This could mean that the predicted global warming will be smaller than one predicted by IPCC 2001, and therefore William knows what to think about the paper even though he does not understand a word.

I added the boldface because William's innocently honest description of the "mainstream" climate edition of the "scientific method" is refreshing. William continues with some amount of nonsensical criticism - such as that it is strange that they included Prague as a representative city :-) - and then he promoted a paper by Fraedrich and Blender (FB), also in Phys. Rev. Let.

These Gentlemen offer a surprising conclusion that the scaling exponent should be around 0.5 for inner continents and 1 for the oceans which William, of course, immediately accepts. Why? Because it would help his global warming beliefs.

I have not analyzed the data in detail, but the FB statement seems to contradict something I would call a physics intuition. Oceans or continents can change the (dimensionful) timescales of exponentially decaying processes or the overall size of the temperature fluctuations, but they should not change the (dimensionless) critical exponents of the power laws.

It should not be surprising that the original team, Bunde et al., published a one-page comment also in Phys. Rev. Let. about FB which shows that the FB results contradict both their analysis as well as the initial data. Of course, William won't inform you about such a thing and he will erase every comment on his blog that would try to link to the new corrected paper by Bonde et al. - which is what he did to my comment. He's just damned scared that all these flawed scientific assertions will be revealed.

Instead, he is going to convince you that the critical exponents (and probably also the rest of physics) are uninteresting because they have already validated their friends' models and no amount of heresy such as the critical exponents - or publications in Phys. Rev. Let. that no true AGW believer would ever read - can change the holy word. ;-)

Meanwhile, the people who are still able to use their brains may compute the critical exponents in the statistical climate data and falsify most of the climate models that are being used today. Noise is not always the same thing as another noise, and there are scientific methods to determine whether two "noises" match. Cosmologists have been using these methods for more than a decade to analyze the CMB. The modern alchemists of course don't want to hear about the methods that have the power to show that some models are simply wrong and the "wrongness" can't be hidden behind the apparent "randomness" because when investigated scientifically, "randomness" is not universal. The very purpose of science is to uncover the layer of randomness and see the patterns that can be expressed by quantitatively measurable and predictable numbers.

Finally, there have been quite many different papers that show that the climate models fail to reproduce the observed temperatures, for example paper from Boston University here; or a paper by Douglass et al.

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snail feedback (8) :

reader William M. Connolley said...

Hi Lubos. Good to see you posting on this yourself rather than spamming my blog. Yes, you're banned there: you've been naughty.

As to the substance: you claim I don't mention the Bunde rejoinder: this is odd because I wrote At this point Govindan drops out, but some of the original authors reply, saying that (i) the scaling isn't 0.5 at K; and (ii) it isn't 0.5 at other interior points too (they pick yet another scatter of random stations). F+B reply, that (i) Oh yes it is (ii) maybe its the fitting interval: they use 1-15 years; the others are using 150-2500 days. On (i), looking at the pics, I'm with F+B and I can't see what the others are up to.

Oddly enough though, *you* don't seem to have found space to have mentioned F+B's reply to Bunde.

But onto substance: you think that F+B contradicts intuition: to the contrary, as I point out F+Bs results are exactly what you would expect: over the oceans, high A (~ 0.9) and over the continental interiors, low A (~ 0.5) and in between, mixed A (~ 0.65). Why is this exactly what you expect? Because the ocean has a long memory but the land doesn't.

You also say in fact, no violation of the law is known from the data - this is odd, because if you read F+B, you'll find that their *observational* stuff finds variance from the supposed universal law. You describe F+Bs surprising conclusion that the scaling exponent should be - no, its not *should be*, they discover, observationally, what it *is*.

reader Luboš Motl said...

Dear William,

I am sure that you won't understand the reason, but the role of critical exponents is not to depend on whatever parameter such as position you want to invent, and the reply to FB whose existence you've tried to hide from your readers roughly 18 times, indeed, shows that the measured exponent is universal for continental stations and that the paper by FB you promote is not right.

Or do you really misunderstand that there is a reply to the comment by FB, a reply to reply? Then I guarantee to you that you are the last one, and even Quantoken has gotten it hours ago. Afraid to click, right? The truth is scary for some people.

I always had a link to FB on this page.

Your attempts to censor the actual research are completely childish because your pathetic blog has roughly 4% of the impact of and 14% of mine. You just can't win your game based on lies and a denial of facts, got it? What you're doing has nothing whatsoever to do with science.

All the best

reader Quantoken said...


The relationship that
C(s) ~ S^(-gamma)
Is completely wrong!

Too bad that you don't seem to have a rigorous enough math training to be able to see instantaneously and instinctively why it was wrong.

I do not need experimental data to see that it could not possibly be right. Plug in s = 1, and you get the correlation is exactly one, regardless of the value of gamma. That means the temperature deviation of B, exactly one day later from date A is completely correlated to that the previous day. Since temperature C, 2 days later will also be completely correlated to that of B, 1 day later, and B is completely correlated to A. Therefore C is completely correlated to A as well. Therefore temperature deviation of any one day will always be completely correlated to any other day.

That is obviously a wrong conclusion. Try to plug in s=0 and you can also see it's wrong. It gives 0 while you expect that C(s)=1 for s=0.

The statistical model of temperature deviation, because the temperature depends on it's immediate history, is clearly one of Markov Chain. I do not know why you have no idea?

The confinement condition must be:
C(0) = 1
C(infinity) = 0
C(s1+s2) = C(s1)*C(s2)

You can easily see that the correct relationship is:

C(s) = exp(-s*gamma)

That is the correct power decay law.


reader Quantoken said...

Lubos wrote:
"I have not analyzed the data in detail, but the FB statement seems to contradict something I would call a physics intuition. Oceans or continents can change the (dimensionful) timescales of exponentially decaying processes or the overall size of the temperature fluctuations, but they should not change the (dimensionless) critical exponents of the power laws."

Your intuition here is correct. But the correlation relationship, as I discussed, should be an exponentially decaying one, not the power law one as in the paper. The paper was wrong. And at the end of day, continent or ocean do change the time scale of the exponential decay, because of the difference of latent heat capacity.

I will not waste time to discuss how that time scale difference approves or disapproves the Global Warming theory. All I want to say is if one doesn't even know Markov Chain, then there is no credibility to any conclusion he draws out of any statistical calculation.


reader Luboš Motl said...

Dear Quantoken,

the point you are missing is that for the special value of the exponent, the correlation is not 1 but rather goes to an arbitrary constant # as clearly indicated. Of course, the power law breaks for s=0. I hope that you understand that it makes most of your comments irrelevant. Hope that everything is well.

All the best

reader Quantoken said...

The arbitrary multiplier constant # really does not matter once the function is NORMALIZED. So it is really not arbitrary at all, it's fixed by normalization requirement. And it can be absorbed into S itself by scaling S. At given C(s) = #*s^(-gamma) I can always find a s which leads to C(s)=1. It may not be exactly one day, but 13.35 hour. But doesn't matter. You can still derive the wrong conclusion and hence show the power law relationship is wrong.


reader Luboš Motl said...

Dear Quantoken,

you say that "you can still derive wrong conclusions and therefore the power law is wrong".

This sentence of yours is incorrect because *I* can't derive wrong conclusions. Only you can derive wrong conclusions, by using wrong reasoning, because you don't want to realize that the statement about the power law keeps the coefficient # arbitrary, so even if you imagine that the exponent "s" takes a particular value, it never implies that the correlation coefficient itself is one - it does not imply simply because "#" can be different.

But you would be right if you wrote that the exponent can never exceed certain interval. Your improved argument would give one such bound.

All the best

reader Quantoken said...

You can derive a wrong conclusion, starting from the wrong relationship which is the power law. It does not mean that you did anything wrong or that your logic is wrong. But merely means that your starting point, the power law, is wrong.
C(s) = #*s^(-gamma)
Regardless of what # is, you can always find a s0 satisfying:
C(s0) = #*s0^(-gamma) = 1
C(s) = #*s0^(-gamma)*(s/s0)^-gamma)
= 1*(s/s0)^(-gamma)

This is what I mean you can absorb # into s by scaling s, and hence make it disappear.

Once you get C(s0) = 1 for a none zero s0, you know it gives the wrong correlation relationship. 1 means an exactly correlation, for example proportionality.

Please take time to think this over. You are much slower than I expected in understanding this.


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