Guest Essay by Kip Hansen
“The climate system is a coupled non-linear chaotic system, and therefore the long-term prediction of future climate states is not possible.”
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 third 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 two parts of this series are: Chaos and Climate – Part 1: Linearity and Chaos & Climate – Part 2: Chaos = Stability Today’s essay concerns Period Doubling leading to Chaos and what chaos means for climate modeling — please note that it is a [really] long essay.
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 in the Author’s Comment Policy section at the end of this essay.
The Period Doubling Problem
The classical maths formula used to demonstrate the principles of Chaos Theory is the Logistic Equation, which I have used throughout this series.
The biologist Robert May uses it as a “demographic model analogous to the logistic equation”…where “xn is a number between zero and one that represents the ratio of existing population to the maximum possible population”. The parameter r, is reproductive rate, expressed in whole numbers. In the logistic map, we focus on the interval [0, 4]. Doing so produces the ubiquitous bifurcation diagram for the logistic equation, or logistic map, which shows this characteristic as the parameter r is changed:
Following are time series of the results at various values of r – corresponding to the colored vertical lines:
For the time being, we will ignore the sections covered almost solid with grey, and concentrate on points intersected by the colored lines. We see that increasing r to 3.1 creates a saw-tooth graph with a period of 2. At 3.5, the period doubles to 4, then rapidly doubles again to 8 and then 16 (there is a point for 32 and 64 etc.). Additionally, the magnitude values (the x) goes from a narrow range of 20% of the unity at Period 2 to a whopping 45% of the unity (entire range) at Period 4.
Were we looking at the dynamics of wind over a new airplane wing design, we would first see a tiny inexplicable vibration (as the value of our r barely exceeded 3), followed by a definite shaking at r = 3.1, then watch as the thing shakes itself to pieces as r continues to increase.
Some might think that this is somehow a “feedback”, a “feedback loop” or a “runaway feedback loop”. They would be incorrect. The result — our poor hypothetical airplane wing literally shaking itself to bits — certainly looks similar, but the cause is quite different. This is a ubiquitous feature of chaotic non-linear dynamical systems, represented by the bifurcation diagram. In a sense, there is no cause other than the nature of the system itself.
Remember, there will be a cause for the increasing factor r – but an increase in r – let’s say a doubling from 1 to 2 – does not cause instability nor chaos – only an increase in the magnitude of x (see the small inserted image in the larger image above). Increasing r from 2 to 2.5 has the same innocuous effect, the magnitude of x is increased. The simple fact of increasing of r does not cause period doubling itself – as I showed in Part 2 of this series, it leads to stability at higher values of x until, that is, the value of r begins to be > 3, at that point we see the beginnings of the process of period doubling leading to chaos.
[There are many non-linear chaotic dynamical systems in the physical world, they all have their own set of parameters and formulas, and have their own circumstance at which the system enters the realm of period doubling leading to chaos – it is only the logistic equation in which the magic number is 3.]
Note as well that there are rather odd bits here: at 3.8+ there is a window with a period of 3, which cascades into a period of 6, then 12, then 24…..the small bifurcation seen near the bottom of the brown line near 3.8 – if magnified – looks precisely like an inverted version of the whole diagram – a feature called self-similarity, which we will not discuss here.
Do we see this in the real world? Yes we do – boom and bust animal populations, economics (see logistical map for a modified Phillips curve), in fluid flows, in the vibrations of motion systems, in irregular heart rate leading to life-threatening conditions. Period doubling cascades are common and can be quite destructive in physical systems.
This type of phenomena may have been responsible for the failure of the Tacoma Narrows Bridge (1940), about which the Wiki states: “In many physics textbooks, the event is presented as an example of elementary forced resonance, with the wind providing an external periodic frequency that matched the bridge’s natural structural frequency, though the actual cause of failure was aeroelastic flutter.” Note that the wind was only blowing 40 mph, in the nautical world known as a fresh gale (through which I have sailed too many times for comfort).
Cardiac specialists have been working on using chaos theory, and period doubling leading to chaos, in investigating heart beat irregularities, such as cardiac dysrhythmias, ventricular fibrillation and pulseless ventricular tachycardia. Here is a portion of my recent ECG, showing the electrical impulses as my heart beats four times:
I am assured that it is just as it ought to be. However, things can and do go wrong:
The Fast Heartbeat above (tachycardia) appears to be a doubling of heart rate. Shannon Lin, at UC Davis, reports in a paper titled ”Chaos in Human Systems” that “In the case of an arrhythmia, electrocardiograms (ECG’s) are implemented to measure the electrical currents produced by the heart. After reviewing the data, doctors were able to manipulate the heart’s beating through a chaos control program.” For more details, see Controlling Cardiac Chaos by Garfinkle et al.
When I say that Period Doubling Bifurcations are ubiquitous in dynamical systems, I am not exaggerating – try this simple internet search for ‘images bifurcation diagram’. Clicking through to the origins of the resulting images will give you some idea – they are found everywhere there are non-linear dynamical systems – biology, evolution, chemistry, physics, mathematics, heat flow, fluid flow, fluid mixing, heart rate manipulation, the study and function of brain neurons, anti-control of DC motors, various physical oscillators, the mapping and control of epidemics of diseases such as measles, mechanical engineer concerns of vibrations in structural beams and such esoteric topics as “Chaos Appearance during Domain Wall Motion under Electronic Transfer in Nanomagnets” (really…).
As laid out in Part 2 of this series, engineers know about chaos and go to great lengths to keep their systems within the parameters of stable regimes. As one engineering paper puts it “Chaos is undesirable in most engineering applications. Many researchers have devoted themselves to find new ways to suppress and control chaos more efficiently.”
Period Doubling Leads to Chaos
Earlier in this essay series, I quoted Edward Lorenz writing: “a phenomena that later came to be called ‘chaos’ – seemingly random and unpredictable behavior that nevertheless proceeds according to precise and often easily expressed rules”.
He uses the term chaos to refer to processes that “appear to proceed according to chance even though their behavior is determined by precise laws” and stretches the definition to include “phenomena that are slightly random, provided their much greater apparent randomness is not a by-product of their slight true randomness. That is, real-world processes that appear to be behaving randomly – perhaps the falling leaf or the flapping flag – should be allowed to qualify as chaos, as long as they would continue to appear random even if any true randomness could somehow be eliminated.”
[ This definitional problem is exacerbated by the use of several other terms – nonlinearity, nonlinear dynamics, complexity, and fractality – which are often used today synonymously with chaos in one sense or another. This on top of the fact that Chaos Theory is a misnomer – it is not a single theory, but a broad field of study, and concerns systems that are entirely deterministic. ]
Our bifurcation diagram shows what happens when a nonlinear dynamic system is pushed past a certain point – whether it be in population dynamics, aerodynamic flow or bridge building. Cascading period doubling leading to chaos – seemingly random and unpredictable behavior – the nearly solid grey portion of the Logistic Map. Some chaotic systems exhibit period halving cascades followed by stability followed by period doubling cascades. The truth is that this behavior is NOT random at all, rather it is strictly deterministic, but, at any and all given points in the chaotic realm, all future individual values are unpredictable, they cannot be determined without actually calculating them.
Hidden in the chaos regime are areas of periodic behavior, perfectly orderly. Also note that the values of x are constrained — at an r value of 3.7, x will not be below 0.2 or above 0.9 (on a unity scale). Allowing the r to exceed 4 however, allows any value of x across the entire scale, all or nothing, and everything in-between.
Even more weirdly, when the data points in the chaotic realm are looked at in different ways, say through a time series of the value difference of each succeeding point, or in more dimensions, very intricate and mathematically beautiful relationships are seen – called Strange Attractors – “a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.”
The Roessler Attractor is an example.
The Roessler Attractor was designed by Otto Rössler in 1976, “but the originally theoretical equations were later found to be useful in modeling equilibrium in chemical reactions.”
Dave Fultz (1921-2002), worked in the University of Chicago’s famous Hydrodynamics Lab, where “Before the advent of sophisticated numerical modeling, Dave cleverly devised and systematically exploited a number of laboratory analogs to gain insight into many complex atmospheric processes, most significantly the atmospheric general circulation. His ‘dishpan’ experiments provided tangible examples of otherwise poorly understood physical processes.” In his dishpan, he found not only the order seen in atmospheric processes, such as the jet stream but also things that disturbed him “For an organized person, chaos is both an object of fascination because it’s so different, and also of apprehension.” Raymond Hide, at Cambridge, did similar work, which included these images of his basic dishpan apparatus and some of the results, including a chaotic state on the right.
These physical experiments were nearly simultaneously replicated in numerical models, including those famously done by Edward Lorenz.
The Chaos Problem in Climate Models
“The climate system is a coupled non-linear chaotic system” and when one models it, the model is made up, mathematically, of various formulas for the non-linear dynamics of fluid motion, heat transfer and the like.
The Heat Transfer formulas are given as:
Note on Stefan-Boltzman: “Thermal radiation at equilibrium was studied by Planck by using equilibrium thermodynamic concepts. The thermal properties of the gas of photons are well-known. One of them, the Stefan-Boltzmann law gives the value of the energy flux in terms of the temperature of the emitter through a power law: σT4.” “…the classical scheme is no longer applicable [as when the radiation is not in equilibrium due to the presence of thermal sources or temperature gradients] and it then becomes necessary to employ a nonequilibrium theory. A first attempt to describe non-equilibrium radiation could be performed via nonequilibrium thermodynamics. Nevertheless, some of the laws governing the behavior of thermal radiation are non-linear laws whose derivation is beyond the scope of this theory which provides only linear relationships between fluxes and forces.” — Nonequilibrium Stefan-Boltzmann law, Pérez-Madrid_ and Rubí (2010)
And there is the Boltzman Transport Equation (BTE), which describes the statistical behaviour of a thermodynamic system not in equilibrium, the classic example of which is a fluid with temperature gradients, such as an ocean or an atmosphere, causing heat to flow from hotter regions to colder ones. “The equation [Boltzman Transport Equation] is a nonlinear integro-differential equation, and the unknown function in the equation is a probability density function in six-dimensional space of a particle velocity and position. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.” Writing out the formula will not enlighten us here, however, I point out that the equation is a nonlinear stochastic partial differential equation, since the unknown function in the equation is a continuous random variable. […in a stochastic … process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve..]”
And, as applies also in climate: Newton’s law of cooling:
“Convective cooling is sometimes described as Newton’s law of cooling: The rate of heat loss of a body is proportional to the temperature difference between the body and its surroundings.”
“However, by definition, the validity of Newton’s law of cooling requires that the rate of heat loss from convection be a linear function of (“proportional to”) the temperature difference that drives heat transfer, and in convective cooling this is sometimes not the case. In general, convection is not linearly dependent on temperature gradients, and in some cases is strongly nonlinear. In these cases, Newton’s law does not apply.” (additional link).
The Navier–Stokes equations [which describe the motion of viscous fluid substances and are used to model things such as the weather and ocean currents] “are nonlinear partial differential equations in the general case and so remain in almost every real situation. … The nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the turbulence that the equations model.”
I’m sure those of you who read here often recognize the significance of these formulas/laws. It can be disturbing to realize that Stefan-Boltzman, Newton’s Law of Cooling and Navier–Stokes equations of fluid dynamics are not, in fact, linear but are, out there where the oceans meet the atmosphere, all non-linear in nature and behavior. These are among the many nonlinear dynamical systems involved in climate modelling.
The linear versions used in everyday science, climate science included, are often somewhat simplified versions of the true phenomena they are meant to describe – simplified to remove or constrain the non-linearities. In the real world, the non-equilibrium world, climate phenomena behave non-linearly, in the sense of non-linear dynamical systems. Why then do we use these simplified formulas if they do not accurately reflect the real world? It is because the formulas that do accurately describe the real world are non-linear and far too difficult—or impossible — to solve – and even when solvable, produce results that are under common circumstances, in a word, unpredictable and highly sensitive to initial conditions. Not all the formulas can be simplified adequately to remove the non-linearity.
These examples are given to illustrate, to repeat again and reinforce, that many of the physical principles and mathematical formulas used to represent them in General Circulation Models to predict weather, climate and climate change, in their original and proper forms, are nonlinear, as they represent physical nonlinear dynamical systems.
The result of this situation is model simulations that look like this multi-model ensemble of winter surface air temperatures in the Arctic, overlaid with a CRUT2.0 version of the observed temperature record:
The accompanying text indicates that “All three runs of [one of the models] ( magenta thin lines with open triangles…)… started from relative warm states, contrary to simulations from other models and observations. The sea ice simulation by this model apparently shows inappropriate initialization for simulating the climate of the twentieth century (Zhang and Walsh 2006). Another explanation is that the model is still in a nonequilibrium state (Y. Yu 2005, IPCC workshop, personal communication). Because of this, the results from FGOALS-g1.0 are excluded from the statistics and discussions in the next sections.” In other words, even though the overall picture is that of 20 models each run multiple times, each returning classic nonlinear, chaotic results, the FGOALS runs were so far out of sync with the others (for reasons not fully understood), that they simply had to be thrown away – not because that is a bad model – but because of the basic non-linear nature of the physics of the climate modeled – the physics are extremely sensitive to initial conditions and when we are doing “just the maths” the result – the chaos in their nature — is only barely constrained.
We can see that taken all together, there appears to be a sort of greenish/bluish concentrated band that runs from 1880 to 2000, starting at -0.3 and running up to +0.3 which appears about 0.4 degrees wide. I suspect that it is based on–is an artifact of– some agreement in parameters between the models. This should not be mistaken for an agreed upon prediction/projection. The standard practice in Climate Science is to “average” [sort of] all those squiggly lines (model run outputs) and call that a “projection”. For why this is absurd, you’ll have to read Real Science Debates Are Not Rare a guest post here at WUWT by Dr. Robert G. Brown from October 2014. [I recommend reading Brown’s essay, without qualification, to anyone interested in any field of science – not just climate.]
This discussion of models is not in any way intended to be an attack on models in general but only to point out that the results returned by such models are the output of coupled nonlinear chaotic systems and thus return wildly different results for the same problem, with the same physical formulas, using slightly differing models of the same climate system, from essentially the same starting conditions.
The results of the above model run ensemble do not “predict” the known past with any degree of accuracy, missing even the obvious highs and lows. The major reason for this is not that the models are incorrect and incomplete – it is that that are correct enough to include at least some of the actual non-linearity of the real climate and thus produce results that are 1) wildly all over the place, dependent on initial conditions and 2) different every time they are run when using anything other than exactly identical initial conditions with no variations whatever – both of these are facets of the same gem – Chaos.
Thus, when viewed through the lens of Chaos Theory – the lens of the study of non-linear dynamical systems – to say “the [climate] models are in agreement….” is nearly nonsensical.
But, wait, some may say, look at that bluish-green band and the uplifting at the right side….surely that tells us something, that the models agree that temperature will be generally rising and rise faster closer to present time. The answer to this, from Chaos Theory, is to point out that parameters have been added to the basic equations to ensure this result – that the model has been “tuned” to “work” – tuned to at least generally produce this result because if the model doesn’t at least approximately reproduce the past, the known observational data, the model is deemed wrong – for the model to be judged correct, to be judged useful at all, it must produce this general picture to “agree with” a century of known observational data.
It is the tuning to produce a “match” to the recent past that constrains the models to produce increases with rising CO2 – it is simply part of the formula used to produce the model in all cases. In our example image below, if the models didn’t produce projections that looked enough like the 1980 to 1999 (past) section, they would have been re-tuned until they did so. Without this tuning, Chaos Theory tells us the models would give us output that looks more like the right side of this image, where the temperature is still chaotic, but just as likely to be down as up, as it has not been tuned/parameterized to rise automatically with CO2. (The blue line represents the mid-line of projections at year 2000 (the start of the “future’ for these runs).
I demonstrated how easily this is type of tuning is accomplished in Part 2 of this series, producing these two images:
The little top graph I created in ten minutes, I used the simplest of non-linear formulas (the logistic equation), writing code to slightly alter the forcing (the “r” in the formula) so that it increases and decreases minutely — by a mere 1/1000th a year, up or down in various time periods (roughly tuning my model to the Global Average Temperature Over Land 1856-2014 observations by guesstimate) and then adding to that result an addition varying randomly from 2 to 6 percent. Superimposing my tuned chaotic graph over the real observations shows the fit. Nothing proved here about the climate, only about how easy, how trivial, it is to parameterize even a known non-linear formula to simulate a known data set.
It is possible, we see, that the parameters, the tuning, of GCMs may represent the major control factor of the overall shape and direction of the model results.
I am quite sure that tuning/parameterizing a GCM is far more complex and difficult and, as we see in the example of Arctic Winter temperatures, not always that successful.
Some [on one side of the Climate Divide] characterize this tuning, this parameterization, of the climate models as a sort of cheating somehow. It is not. It is simply a necessary step if models are going to be useful for anything at all. It is because it is necessary that the true effects of such tuning-parameterization must be fully acknowledged when interpreting the results of model runs and ensembles – something that many believe is lacking in modern climate science discussions. That acknowledgement must accompany the acknowledgement of the true significance of the underlying non-linearities and thus the overall limitations of the models themselves.
Some climate scientists, mathematicians, and statisticians are of the opinion that it is just not possible to take models based on multiple coupled [interdependent] non-linear dynamical systems, each individually hugely dependent on their own initial conditions, give them a shake, and pour out meaningful projections or predictions of future climate states – or even the past or present. They feel it is even less likely that blending or averaging multiple model projections can produce results that will match any kind of objective reality – particularly of the future.
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Author’s Comment Policy:
First, 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 this 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.
Intro to Chaos Theory Reading List:
The Essence of Chaos — Edward Lorenz
Does God Play Dice ? — Ian Stewart
CHAOS: Making a New Science — James Gleick
Chaos and Fractals: New Frontiers of Science — Peitgen, Jurgens and Saupe
Additional reading suggestions at Good Reads (skip the Connie Willis novella)
Second, 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 paragraph from the top). That will save us all a lot of back and forth.
For those readers who feel unfulfilled, I promise that there will be a Part 4 of this series in which will talk about Chaotic Attractors a little more, then try to wrap all these concepts together and present my view of how Chaos Theory must inform our understandings of climate science.
I will try to answer your questions, supply pointers to more information, and chat with you about Chaos and Climate.
Thanks for reading.
[Disclosure: The trick with the ease of tuning the logistic formula to match crutem4 is just that [a trick] – but interestingly depends on aspects of Chaos Theory that make it possible. Notice that my little graph was not wildly all over the place, nor did I have to run it a thousand times to get one output that matched crutem4, as I would have had to do with a GCM. Ten CliSci Brownie Points and a Gold Star to the first reader to expose the trick in comments].
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