Scary scaling









Lumo said...
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Lumo said...
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CapitalistImperialistPig said...

Bellete - I don't know what Lubos had to say in his erased comments, but I would like someone to address a comment he made on his blog: 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.

You, and the original authors, both say gamma = .5 corresponds to white noise, and that larger values correspond to memory, which seems to contradict Lumo, but could you make the argument a bit more directly? What exactly is Lumo missing here?

Arun said...

Lubos provide a link to http://arxiv.org/abs/physics/0305080

Arun said...

I also find this, emphasis mine:


Power-Law Persistence in the Atmosphere: Analysis and Applications

Authors: Armin Bunde, Jan Eichner, Rathinaswamy Govindan, Shlomo Havlin, Eva Koscielny-Bunde, Diego Rybski, Dmitry Vjushin


We review recent results on the appearance of long-term persistence in climatic records and their relevance for the evaluation of global climate models and rare events.The persistence can be characterized, for example, by the correlation C(s) of temperature variations separated by s days.We show that, contrary to previous expectations, C(s) decays for large s as a power law, C(s) ~ s^(-gamma). For continental stations, the exponent gamma is always close to 0.7, while for stations on islands gamma is around 0.4. In contrast to the temperature fluctuations, the fluctuations of the rainfall usually cannot be characterized by long-term power-law correlations but rather by pronounced short-term correlations. The universal persistence law for the temperature fluctuations on continental stations represents an ideal (and uncomfortable) test-bed for the state of-the-art global climate models and allows us to evaluate their performance. In addition, the presence of long-term correlations leads to a novel approach for evaluating the statistics of rare events.

Arun said...

There is a description of the DFA algorithm in section II of this:


How does one apply this to the weather, on the scale of days to years? How does one take out the annual periodicity, for instance?

Arun said...

I think Belette's A is ( 1 - gamma / 2).

Thus physics/0208019 which finds gamma = 0.4 for small-island weather stations, has A = 0.8, which tends to agree with F+B about the oceans.

Belette said...

Arun, CIP - the idea that the ocean has a "memory" is a commonplace, and the name you want is Hasselmann (Tellus, 1976, p473 is probably the first). The idea being that the ocean integrates short term noise to give you a red spectrum.

As to what Lubos is missing - its more what he is assuming. The very idea that there is this universal power law is deeply dodgy, and probably simply wrong - I suggest reading the F+B paper for that.

Wolfgang said...


it seems that you and Lubos are talking about different things.

You write that exponent = 1/2 is white noise, but this is of course not true for Lubos' exponent.

(A previous comment mentions the same issue.)

In addition, it seems to me that
your post is inconsistent:

"A ~ 0.5 is no memory (white noise); A ~ 1 is long memory (red noise). [..]
They discover that the GCMs [..] get A ~ 0.5. Which [..] means that the GCMs overestimate the trends. [..] But... this is not news."

Either A = 0.5 is white noise or trends but not both.

EliRabett said...

Arun they killed the annual cycle by finding an average temperature Tave for each day, and then working with DELTA Ti - Ti-Tave.

Power laws are numerology without a theoretical basis. What worries me about these exercises is that the data clearly has curvature (on a log-log plot!) for most of the stations, and that means that the fit to an exponent depends on where you start your fit.

Belette said...

Wolfgang - well, I started this! If Lubos wants to use a different nomenclature thats his (and perhaps his readers!) problem.

Govindan (and F+B) are both using S(f) ~ s^-B, with B = 2A - 1 (they use A as the primary measure). So A = 0.5 is B = 0. And G as Arun says. Its on p2 of F+B with refs.

As for overestimating the trends... I'm just reporting what G et al. say. I'm not quite sure why they think A = 0.5 indicates overestimation... you're welcome to read the paper and explain their reasoning if you like! My point was that GHGs only is well known to overestimate the trends, so there is nothing new there.

Eli - I don't think this is numerology. The F+B paper shows that the A's follow a physically plausible pattern: what you would expect, in fact. And (F+B fig 1) they clearly are straight (ish) lines over a fair portion of the time range - 1-15y.

Arun said...

From F+B "Long time memory on centennial time scales is found only with a comprehensive ocean model".

Are the oceans the reason we can meaningfully talk of climate?

Belette said...

Arun - the point about the ocean as opposed to the land (though I'm sure RP Sr would dispute this...) is that being a large fluid reservoir it has a "memory" and can thus support variability on long timescales. Whereas the land, being fairly boring (at least as represented in GCMs) can't.

So we would have a climate if the ocean temperatures were fixed to be annually-repeating (as they can be in atmos-only GCM runs); but we can have a climate with rather more interesting long-term fluctuations with an ocean too.

Potentially the same could be true at even longer timescales with large amounts of land ice (LGM; D-O cycles) though this is less clear.