Natural Variability, Attribution and Climate Models #8

More on Natural Variability over Long Timescales

In #7 we looked at Huybers & Curry 2006 and Pelletier 1998 and saw “power law relationships” when we look at past climate variation over longer timescales.

Pelletier also wrote a very similar paper in 1997 that I went through, and in searching for who cited it I came across “The Structure of Climate Variability Across Scales”, a review paper from Christian Franzke and co-authors from 2020.

To summarize, many climatological time series exhibit a power law behavior in their amplitudes or their autocorrelations or both. This behavior is an imprint of scaling, which is a fundamental property of many physical and biological systems and has also been discovered in financial and socioeconomic data as well as in information networks. While the power law has no preferred scale, the exponential function, also ubiquitous in physical and biological systems, does have a preferred scale, namely, the e-folding scale, that is, the amount by which its magnitude has decayed by a factor of e. For example, the average height of humans is a good predictor for the height of the next person you meet as there are no humans that are 10 times larger or smaller than you.

However, the average wealth of people is not a good predictor for the wealth of the next person you meet as there are people who can be more than a 1,000 times richer or poorer than you are. Hence, the height of people is well described by a Gaussian distribution, while the wealth of people follows a power law.

Brief Digression to explain “e-scaling”

For readers unfamiliar with calculus, “e-folding” or 1/e seems like a strange language. In brief, when we look at continuous change we use calculus and one number that jumps out of lots of natural relationships is e. This number is approximately 2.78.

Most people have come across the idea of a “half life” for radioactive decay. For example, plutonium 238 has a half-life of 88 years. After 88 years you have 1/2 of the original quantity (the rest and decayed into uranium 234). After another 88 years, 176 years in total, you have 1/4. And so on.

It seems natural to use the number 2, but it just turns out that in nature e=2.78 is more common as the factor. So we talk about “the time to decay by 1/e” or “e-folding”.

Just think “half life”, it’s close enough.

End of Digression

It’s a challenge to summarize a 40-page review paper. I’ll just pick out a few points that help add a few jigsaw pieces to the map.

Remember that the biggest problem we encounter in understanding natural climate variability is a lack of quality data prior to the last 100 years where we have been adding CO2 to the atmosphere, and therefore adding a “driving force” in temperature.

Power law scaling variation over the globe

Here’s their calculation of a parameter in power law scaling over the globe for the last 100 years of temperature history:

Now this parameter is a little confusing. It’s the “Hurst parameter”.

When it’s close to 0.5 the power law scaling is about zero (the value of b in the last article is zero), so this is effectively random uncorrelated noise.

When it’s 0.9, the power law scaling is strong (the value of b in the last article is 0.8).

The simple explanation of this is over land there is much less persistence of temperature data than over the ocean (I also mentioned this in #7 without digging into it in any detail). This is intuitively obvious. The ocean stores energy because the heat capacity of the ocean is much higher than land. So temperature fluctuations over land aren’t stored (aren’t correlated to the last time period) whereas temperature fluctuations over the ocean are stored.

Power Laws in Rainfall and Temperature

The first graphic below (a) is the time series of rainfall in one location in China over 50 years. The second graphic (b) is comparing that data to what we call a “normal distribution” (also known as Gaussian) - black curve, and to a power law (red).

The idea is that rainfall often follows a power law. This is intuitive. Most people have experienced light rain, heavy rain (10x), super heavy rain (100x).. and depending on where you live the multiple can be substantial. But we don’t experience this in temperature - if normally it’s 20’C in a given month, and occasionally it’s 30’C in that month, we don’t expect 100’C.

So in looking at their data and explanation I’m holding onto the thought that they are discussing two completely different processes - temperature and rainfall will follow different scaling laws.

Our choice of precipitation data (Figure 1a) exhibits the typical intermittent behavior with no or only very little precipitation on most days interspersed with an occasional extreme event. Hence, precipitation is a climatological variable that is highly episodic. Consequently, the distribution of precipitation is much more heavy tailed than a Gaussian distribution (Figure 1b). Thus, large values are much more likely than in the case of variables that are Gaussian distributed; the Gaussian distribution decays much faster than a power law. The tails of many precipitation distributions, as well as of other climatological quantities, decay according to a power law. This power law relation between intensity and probability of occurrence constitutes a scaling relationship.

When they say “Gaussian” they are talking about the “normal distribution” or Bell curve.

Scaling over Different Timescales

Lovejoy and Schertzer (2013) and Lovejoy (2015b) postulated the existence of five distinct power law scaling regimes.

1. the weather regime with time scales from 6 hr up to 20 days with an exponent of 𝛽 ≈ 1.8

2. the macroweather regime with time scales between 20 days and 50 years and 𝛽 = 0.2

3. the climate regime with time scales between 50 and 80,000 years (includes glacial-interglacial cycles) and 𝛽 = 1.8

4. the macroclimate regime between 80,000 and 500,000 years and 𝛽 = 0.6

5. the megaclimate regime for time scales larger than 500,000 years which takes us to the limit of reliable proxies (Lovejoy & Schertzer, 2013) and 𝛽 = 1.8.

If you review this against figure 1 in the last article you can see there are differences and similarities.

Long Range Dependence Challenged

The paper notes some issues with this long range climate persistence:

..but since they were not derived from basic physical laws their use in climate research was originally, and continous to be, met with criticism (e.g., Klemes, 1974; Maraun et al., 2004; Mann, 2011). Long-range dependence also implies that even the most distant past still influences the current and future climate, which appears at odds with common intuition. Many geophysical equations of motion such as the Navier-Stokes or the primitive equations are usually Markovian, that is, their current state only depends on the immediately preceding state and not on states in the more distant past. Furthermore, they do not have memory terms..

The author of Mann 2011 is Michael E Mann, an ever-popular figure in climate discussions.

Mann:

Indeed, the existence of “long-range dependence” or LRD has sometimes been invoked (e.g. Koutsoyiannis et al. 2008) in support of the proposition that long-term temperature trends are a manifestation of intrinsic long-term natural variability of the climate system. Were that the case, the observed warming of the past century, rather than reflecting the response to human- caused increases in greenhouse gas concentrations, might instead just be the result of chance, long-term natural fluctuations. It is thus rather important to address the issue of whether or not there is support for interpreting global temperature fluctuations as the manifestation of LRD.

And later:

One limitation of previous analyses using coupled climate model simulations is that the physics of the models is sufficiently complex that it is difficult to interpret the results in terms of basic physical properties. There is some utility in turning instead to a far more basic class of climate models known as “Energy Balance Models”..

And so he looks at results from a simple model instead of a Global Climate Model (GCM).

..More generally, any statistical analysis of climate data must be informed by an understanding of the physics underlying the observational phenomena at hand. A statistical analysis devoid of physical understanding is prone to false, and in some cases, extremely misleading inferences—such as the conclusion that global warming can be dismissed as the random excursions of a process displaying LRD.

I agree with this last point.

At the same time, using a simple model to demonstrate that climate persistence has been overstated also seems to have a basic flaw. That is, coupling lots of processes that occur on different timescales (days, years, decades, millenia) requires a complex model and you might find that long term persistance occurs in certain cases. However, if you use a simple model that doesn’t include these coupled processes then you will likely reach a conclusion that might also be far from reality.

Conclusion

Many readers might be getting a little lost.

We are discussing an important but challenging point - how much does the climate vary in the absence of us burning fossil fuels (also in the absence of volcances and other “natural” forcing).

Simple persistent models (AR1) seem flawed. Power law scaling seems to exist, implying longer term persistence.

Different papers reach quite different conclusions about the power law values that exist over different timescales.

There are clearly geographical relationships - for example, the value over land is different from the value over the ocean.

Rainfall and temperature probably have very different scaling laws.

Some people have criticized longer term persistence, but it’s hard to weigh up the competing ideas.

We don’t have lots of data over more than a century, which makes this challenging.

Note for commenters - for those who don’t understand radiative physics but think they do and want to yet again derail comment threads with how adding CO2 to the atmosphere doesn’t change the surface temperature.. head over to Digression #3 - The "Greenhouse effect" and add your comments there. Comments here on that point will be deleted.

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References

The Structure of Climate Variability Across Scales, Christian Franzke et al, Reviews of Geophysics (2020)