Guest Essay by Kip Hansen
Introduction: (if you’ve read the previous installments, you may skip this intro)
The IPCC has long recognized that the Earth’s climate system is a coupled non-linear chaotic system. Unfortunately, few of those dealing in climate science – professional and citizen scientists alike – seem to grasp the full implications of this. It is not an easy topic – not a topic on which one can read a quick primer and then dive into real world applications. This essay is the fourth in a short series of essays to clarify the possible relationships between Climate and Chaos. This is not a highly technical discussion, but a basic introduction to the subject to shed some light on just what the IPCC might mean when it says “we are dealing with a coupled non-linear chaotic system” and how that could change our understanding of the climate and climate science. The first three parts of this series are: Chaos and Climate – Part 1: Linearity ; Chaos & Climate – Part 2: Chaos = Stability ; Chaos & Climate – Part 3: Chaos & Models. Today’s essay concerns the idea of chaotic attractors, their relationship to climate concepts, and a short series wrap up.
Definitions: (if already understand the first sentence below, you may skip the rest of this section)
It is important to keep in mind that all uses of the word chaos (and its derivative chaotic) in this essay are intended to have meanings in the sense of Chaos Theory, “the field of study in mathematics that studies the behavior of dynamical systems that are highly sensitive to initial conditions”. In this essay the word chaos does not mean “complete confusion and disorder: a state in which behavior and events are not controlled by anything” Rather it refers to dynamical systems in which “Small differences in initial conditions …yield widely diverging outcomes …, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable.” Edward Lorenz referred to this as “seemingly random and unpredictable behavior that nevertheless proceeds according to precise and often easily expressed rules.” If you do not understand this important distinction, you will completely misunderstand the entire topic. If the above is not clear (which would be no surprise, this is not an easy concept), please read at least the wiki article on Chaos Theory. I give a basic reading list at the end of this essay.
Climate Attractors: An Attractive Idea
In the field known as Chaos Theory, the study of dynamical systems sensitive to initial conditions, there is a phenomenon known as an attractor. Here I give the definition of this concept from the venerable Wiki:
“…an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed.
In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space. In physical systems, the n dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; … If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions, An attractor can be a point, a finite set of points, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor. …. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory.”
In previous parts of this series, I have shared examples and images of various attractors. The household funnel is the simplest physical example. When held with the spout pointing down, any object entering the mouth of the funnel tends down and out the spout. Exact placement in the funnel mouth doesn’t matter, all points lead to the spout.
The funnel represents a type of attractor called a point attractor. Once the system enters the attractor, the value evolves towards this single point.
A cyclical attractor might have two or three values (ranges in some cases), cycling between them. We see this in certain values of the Bifurcation Diagram, expressed as the Period Doubling that leads to chaos.
From Part 2:
When we graph this equation — x → r x (1 – x) — with a beginning “r” of 2.8, and an initial state value of 0.2, this is what we find:
Even though the starting value for x is 0.2, iterating the system causes the value to x to settle down to a value between 0.6 and 0.7 – more precisely 0.64285 — after 50 or so iterations. Jumping in at the 50th iteration, and forcing the value out of line, down to 0.077 (below) causes a brief disturbance, but the value of x returns to precisely 0.64285 in a short time:
Kicking the value out of line upward at year 100 has a similar result. Adjusting the “r”, the forcing value, down a bit at year 150 brings the stable attractor lower, yet the behavior remains stable, as always.
In the above example, the attractor of the system is a single value, to which the numerical value tends to evolve even when perturbed. In other systems, the graphed values might appear to spiral in to a single point or travel in complicated paths that eventually and inevitably lead to a single point.
Following from the Bifurcation Diagram, one sees easily that at some values of “r” the system becomes cyclic, with periods of 2, 4, 8, 16 as “r” increases until chaos ensues, yet past that point one still finds points, values of “r”, where the period is 3 then 6, 12, 24. Each vertical slice through the diagram presents one with the attractor for that value of “r”, which could be represented by their own geometric graphic visualization.
Some dynamical systems do the opposite – no matter where you start them, one or more values races off to infinity.
And some attractors, when viewed as plotted graphics are fantastically varied and beautiful to look at:
© Creative Commons
Lorenz’s famous Butterfly Attractor (named for the two reasons 1. It looks a bit like a butterfly’s wings and 2. In honor of the Butterfly effect), is often used as a proxy for “the attractor” of Climate (with the initial cap).
See this animated here.
This error appears many times in the literature and in “popular science” explanations of both Climate and Chaos. The latest version making the rounds, recently posted to blog comments repeatedly, are two related videos (parts of a 9 chapter film) from Jos Leys, Étienne Ghys and Aurélien Alvarez at chaos-mbath.org. The films are lovely and very well made, well worth watching. However, though they specifically explain that the Lorenz attractor is not in any way a representation of the climate, “In 1963, Edward Lorenz (1917-2008), studied convection in the Earth’s atmosphere. As the Navier-Stokes equations that describe fluid dynamics are very difficult to solve, he simplified them drastically. The model he obtained probably has little to do with what really happens in the atmosphere.”, they go on to use it and the Lorenz Mill to make the suggestion that climate is predictable based on the finding that some of the features of the Lorenz Attractor and the Lorenz Mill are statistically probabilistic, hence predictable. In the second (Chapter 8) film, they specifically claim:
“Take three regions on the Lorenz attractor (they could represent conditions of hurricane, drought or snow). If we measure the proportions of the time that trajectories with different initial conditions spend within these regions, then we find that for all trajectories, these proportions converge to the same numbers, even if the order in which the trajectories encounter the three regions is incomprehensible. …. By refocusing on statistical issues, science can still make predictions!”
Readers who want the full blood-and-guts version of why this is nonsensical (other than in a trivial way) can read Tomas Milanovic’s Determinism and predictability over at Judith Curry’s excellent blog, Climate Etc. (Be sure to go through and read all the comments from Tomas Milanovic, David Young and Michael Kelly). Those with more pragmatic tastes (and a more common, lower, level of understanding of higher maths) can read my post (also at Dr. Curry’s) Lorenz validated.
Let me just make a couple of obvious points for those who don’t have the time to watch the two 13 minute films or read the two Climate Etc. posts.
- The Lorenz Attractor has [almost] nothing to do with climate or weather in the form used by Lorenz. “The Lorenz attractor arises in a simplified system of equations describing the two-dimensional flow of fluid with uniform depth and imposed temperature difference between the upper and lower surfaces.” — Richard McGehee
- One must use very specific parameters to get the Lorenz equations to produce the Lorenz Attractor – other parameters produce single point attractors,.
- Looking at arbitrarily selected “regions” of the Lorenz Attractor – and saying “they could represent conditions of hurricane, drought or snow” is disingenuous. The attractor has no snow, no rainfall or drought (as the equations are about fluid flow in two-dimensions under temperatures differences, it might describe some thing resembling a hurricane, if applied to a real physical system, such as the famous washtub experiments of atmospheric circulation). Regions of the Lorenz Attractor do not represent weather of any kind whatever.
- Probabilistic analysis of the Lorenz attractor is interesting to mathematicians – but not to weathermen or climate scientists
- The real world climate is chaotic, complex, bounded, multi-dimensional, and, if it has attractors, they will be themselves exist in multiple phase spaces – as Tomas Milanovic points out “the ability to compute phase space averages for particular attractor topologies changes nothing on the fact that the system is still chaotic and will react on perturbations in an unpredictable way over larger time scales.”
- We have absolutely (literally absolutely) no idea what the precise, or even an, attractor for the weather or climate system might look like, separate from the long-term historic climate record. We have no reason to believe it would be statistically smooth or even if it would be amenable to statistical analysis.
Given all that, the idea that the climate system might have the physical equivalent of a chaotic attractor, even if it is a strange attractor, is still quite appealing to many. If it did, and we could discover it, mathematically or physically, we might then attempt some kind of statistical analysis of it to have some idea of the probabilities of what climate might do in the future. But only probabilities, and “probabilities of what” is highly uncertain. Remember, the climate covers the whole planet, and while we are mostly interested in what takes place close to the surface, it happens at all levels – a huge complicated area in both space and time. The possibility of analysis that would reveal useful statistical probabilities for even general climate issues such as hard or mild northern hemisphere winters in anything but the near-present, certainly less than a decade, is unlikely.
Probabilities might be interesting mathematically. Every gambler knows the probabilities of his game – the chances – and knows that probabilities are not predictions or projections – a bet on lucky number 17 still has a one in 38 chance of winning a payout of 36x on every spin of the roulette wheel in Las Vegas – knowing the probabilities doesn’t give him any insight into what the spin will bring. The action of the ball in a roulette wheel is chaotic in the sense of sensitivity to initial conditions – the exact speed of the spin of the wheel, the force the croupier gives to the ball, the exact point of release of the ball and its exact relationship to the spinning wheel (which spins in the opposite direction to that of the ball) at that precise moment. The balls subsequent motion depends then on the exact conditions, speed and angle, when it leaves the track and strikes the first deflector – and while that motion will be entirely deterministic, it simply sets the initial conditions for the next contact of the ball with another deflector or separator. The path of the ball during this spinning and bouncing is chaotic. Rather quickly the ball runs out of energy and the ball is captured by one of the 38 numbered pockets in the wheel. In a fair wheel, with a large enough number of spins, the results are normally spread between all of the 38 possibilities, each coming up 1/38th of the time. The probabilities can be perfectly known, yet the outcome in any one spin cannot be predicted – we can however, predict the outcome of ten thousand spins – more-or-less 1 in 38 for each number. Such a probability prediction only allows the gambler not to make stupid mistakes – like thinking three reds in a row means the next spin must be black. Such a set of generalized probabilities would be useless for climate or weather. (I would, however, like to read an essay on the potential usefulness of climatic probabilities – what kind of probabilities some think might be discovered and how we might use them to our benefit.)
(You would be surprised by how many instructional videos there are on systems for beating the casinos at roulette – all of them showing remarkable results. Yet the makers of the films are not retired millionaire gamblers and one wonders why they don’t just tour casinos and make a mint with their own systems?)
Even if, by some quirk of fate, we were able to stumble upon the structure of the multi-phasic attractor of Earth’s climate in the present day, which could then somehow magically be analyzed for statistical probabilities in a useful spatial and temporal way, such as seasonally for a specific region over the next decade, they would still just be probabilities, with only one actuality allowed. After that, the minute alterations of the ever-changing initial conditions and determining parameters of the system would lead to unpredictable differences in the attractor or even a shift to a new attractors altogether. These issues make the possibility of long-term useful predictions of the climate impossible.
But can we make any useful predictions about the climate? Of course we can!
If there is a shift in the northern jet stream, we can predict things about near-term European seasonal weather. If an El Niño develops, we can predict certain general weather and climate conditions. If there is a persistent blocking high in one area, weather is predictably affected downstream. Where does our ability to make these predictions come from? From models? Only models of the past – looking at the historical climate, recognizing patterns and associations, checking them against the records, and using them to make reasonable guesses about what might be coming up in the near future.
An aside about Hurricane Forecasting using Models
We can make weather/climate predictions about the near-future in some cases – hurricane-path prediction models are “pretty good” out several days, certainly good enough to issue warnings and for localities to make preparations, with a current average track accuracy of a bit better than +/- 50 nautical miles at 24 hours out. The error increases with time – at 48 hours 75 miles, at 72 hours 100 miles, at 5 days it is 200 miles. These results are about one half of the track errors in 1989. This accuracy was enough to warn the barrier islands of Brevard County, Florida (Cape Canaveral, Cocoa Beach, Patrick Air Force Base) for the recent major Hurricane Matthew – the islands were evacuated based on 24 hour predictions of a direct hit.
The difference of ~50 miles is illustrated here – note the times of the two images – the path projected in the left image is three hours earlier than the right image — the difference in the path can best be seen in the blow-ups in the upper-left of each of the two images:
Live television broadcasters called this “the little 11th hour shift” that saved Cape Canaveral – the center of Matthew shifted east 20-30 miles, making the difference between the direct landfall of the eye of a major hurricane on the highly developed barrier islands and the effects of a near-miss pass 30 miles off-shore. You can watch this evolve in an animation in the National Hurricane Center’s archive of Hurricane Matthew.
How can we best predict future climates?
I maintain that the best chance of determining the probabilities of climate long-term future outcomes lies in the past, not in mathematical, numerical modeling attempts to predict or project the future. We know to varying degrees of accuracy, temporally and spatially, what the climate was in the past, it has had tens of thousands of years to go through its iterations, season to season, year to year, and has left evidence of its passing. The past shows us the actual boundaries, the physical constraints of the system as it really operates.
Some maintain that because we are changing the composition of the atmosphere by adding various GHGs, mostly CO2, that the present and future, on a centennial scale, are unique and therefore the past will not inform us. This is trivially true, the present is always unique (there is only one, after all). But similar atmospheric conditions have existed in the past. Has this exact set of circumstances existed in the past? No. If nearly these circumstances had existed, would this tell us what to expect? No again, climate is chaotic, and profoundly dependent on initial conditions.
This has nothing to do with the question of whether or not, or how much, increasing CO2 concentrations will add energy (by retention) to the climate system. That question is simply a matter of physics – if GHGs block outgoing radiation of energy, then the blocked energy will remain in the system until such time as a new equilibrium is reached. What the effects of that energy retention will be are what the various branches of science are investigating. Making early decisions and assumptions — no matter how reasonable they appear — would be an error – along the lines of those made in physics regarding the expansion of the universe.
So, why study the past to know the future? It is my view, shared by others, that the climate system is bounded – limited in its possibilities – and that these boundaries are “built-in” to the dynamical climate system. From the historical record, the climate system has an apparent or seeming overall attractor, one could say, outside of which it cannot go (barring something like a catastrophic meteor strike). Included in that attractor are the two long-term states known as Ice Ages and Interglacials, between which the climate switches, much like a two-lobed chaotic attractor. We have little understanding of what causes the shift, but we know it takes place and how long interglacials of the past have lasted. We also know that during the past interglacials, the average surface temperature of the earth has been remarkably stable – staying within a range of 2 or 3 degrees, producing a period during which Mankind has thrived (for better or for worse), with apparent Warm Periods and Little Ice Ages (cooler periods). There is no evidence other than the historic record for labeling this the or an attractor of the system — but it has the appearance of one.
This sounds a bit like I am saying that we can’t predict the far-future climate because of chaos therefore we must look to the [chaotic] past to predict the climate. Almost, but no prize. It is the patterns of the past, repeating themselves over and over, that inform us in the present about what might be happening next. Remember, chaotic systems have rigid structures, they are deterministic, and Chaos Theory tells us we can search for repeating patterns in the chaotic regimes as well.
Of course, this is exactly how weather forecasting was done prior to the advent of computers. The experience of the weatherman, well educated in the past patterns for his/her region, would look to the available data on regional temperatures, air pressures, cloud type and cover and wind directions, and give a pretty good guess at the coming day’s and week’s weather. The weatherman knew of bounds of weather for his locality for the calendar date, and with his knowledge of the weather patterns for his area, could feel confident of his general forecast.
At this point I would have written about the problematic essence of numerical climate models – Chaos and Sensitivity to Initial Conditions. I would have run some chaotic formulas, made tiny, tiny changes to a single initial condition and shown how those changes would make huge differences in outcome, then liken this to modern GCMs, general circulation models, the type of climate model which employs a mathematical model of the general circulation of a planetary atmosphere or ocean.
Serendipitously, a group at NCAR/UCAR did it for me and produced this image and caption (from a press release):
With the caption: “Winter temperature trends (in degrees Celsius) for North America between 1963 and 2012 for each of 30 members of the CESM Large Ensemble. The variations in warming and cooling in the 30 members illustrate the far-reaching effects of natural variability superimposed on human-induced climate change. The ensemble mean (EM; bottom, second image from right) averages out the natural variability, leaving only the warming trend attributed to human-caused climate change. The image at bottom right (OBS) shows actual observations from the same time period. By comparing the ensemble mean to the observations, the science team was able to parse how much of the warming over North America was due to natural variability and how much was due to human-caused climate change. Read the full study in the American Meteorological Society’s Journal of Climate. (© 2016 AMS.)”
The 30 North American winter projections were produced as part of the CESM-Large Ensemble project, running the same model 30 times with exactly the same parameters with the exception of a tiny difference in a single initial condition – “adjusting the global atmospheric temperature by less than one-trillionth of one degree”.
I will not repeat the essay here – but it contains what I would have written here. If you haven’t read it, you may do so now: Lorenz Validated.
Chaos Theory, and the underlying principles of the non-linearity of dynamical systems and ‘dependence on initial conditions’, inform us of the folly of attempting to depend on numerical climate models to project or predict future climate states in the long-term. The IPCC correctly states that “…the long-term prediction of future climate states is not possible.”
The hope that statistical analysis of climate model ensembles will produce pragmatically useful probabilities of long-term future climate features is, I’m afraid, doomed to disappointment.
Weather models today produce useful near-present, daily forecasts (and even weekly for large weather features) on local and regional levels and may produce useful short-term-future weather predictions. When coupled with informed experience from the past, weather/climate patterns, they may eventually provide regional next-season forecasts. The UK’s MET claimed this result recently, bragging of 62% accuracy in back-casting general winter conditions for the UK based on pattern matching with the NAO. Judith Curry’s Climate Forecast Applications Network (CFAN) is working on a project to make regional-scale climate projections. Success of these longer range projections depends in large part on the definition used for “useful forecasts”.
Hurricane path and intensity models have halved their error margins since 1990, achieving a useful average predicted-path accuracy of +/- 50 miles at 24 hours with an accuracy of +/- 200 miles at 5 days. Hurricane Matthew’s 11th hour shift may be an illustration of these models having nearly reached the limit of accuracy.
At the end of the day, a deep and thorough understanding of Chaos Theory, down at its blood-and-guts roots, is critical for climate science and should be included as part of the curriculum for all climate science students – and not just at the “Popular Science” level but at a foundational, fundamental level.
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Intro to Chaos Theory Reading List:
Additional reading suggestions at Good Reads (skip the Connie Willis novella)
Recent blog links:
At Climate Etc.:
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Author’s Comment Policy:
Since I will still be declining to argue, in any way, about whether or not the Earth’s climate is a “coupled non-linear chaotic system”, I offer the above basic reading list for those who disagree and to anyone who wishes to learn more about, or delve deeper into, Chaos Theory and its implications.
Also, before commenting about how the climate “isn’t chaotic”, or such and such data set “isn’t chaotic”, please re-read the Definitions section at the beginning of this essay (second section from the top). That will save us all a lot of back and forth.
I hope that before reading this essay, which is Part 4, that you have first read, in order, Parts 1 , 2, and 3 . As the essay Lorenz Validated was originally intended as part of this essay, it is suggested reading.
I will try to answer your questions, supply pointers to more information, and chat with you about Chaos and Climate.
Thanks for reading.
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