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.”
– IPCC TAR WG1, Working Group I: The Scientific Basis
Introduction:
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.
I hope that before reading this essay, which is Part 3, that you have first read, in order, Parts 1 and 2.
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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If all the interactions are physically determined, the result is also determined. That different initial conditions yield different results is required. As much as you try to separate the colloquial and mathematical definitions of chaos, the word still poisons the thought process. Systems are chaotic only in that they are very difficult to understand (including the initial conditions), and we have a LONG way to go.
Really glad Nick pointed out model generation of ENSO, albeit always at the wrong time and strength. This is important. We will eventually be able to model the climate, the sooner if we quit wasting time on layer by layer radiative transfer and the Carbon fetish.
Yea , I was impressed when I first learned that extremely simple equations could generate strange chaotic behaviors back in the 70s , and without question weather , and therefore its running average , climate , is chaotic . Which makes weather an endlessly fascinating study , tho not one I’m particularly interested in .
But that has virtually nothing to do with the determinants of mean global temperature which I thought was the bone of contention . That is totally constrained by very simple gas law like relationships . The temperature of a gray body in our orbit is ~ 278.6 +- 2.3 from peri- to ap- helion . Our spectrum as seen from outside apparently yields a somewhat lower equilibrium . None the less our surface temperature is about 3% , 10K , warmer than the gray body value with fluctuations over our lifetimes a 10th of that .
I have never seen a quantitative explanation within the GHG spectral paradigm for even that 3% .
There is no doubt that this essay covers many interesting and pertinent points. However, a quick glance through it showed two examples that I found grating and which halted careful reading.
First, the use of the example of airflow buffering a wing to illuminate the period doubling of the logistic equation seems suspect. One should first demonstrate, or point to reasonable pieces of the literature on aerodynamics, showing that the equations of airflow around an elastic body actually simplify, under some assumptions, to the logistic equation. Otherwise this is just hand waving. It may have been better to use only examples from population growth.
Second, the Tacoma Narrows collapse is a topic with a long and rich history of analysis. Citing it as an example of resonance, as many physics texts do, is simply wrong. Neither is it a case of aerodynamic flutter. To be so would demand that the fundamental periods of two of its elastic modes would converge on one another and allow one mode to absorb energy from the airflow and then feed this energy into the destructive mode. I don’t know of any evidence showing this to have occurred at Tacoma Narrows. Instead, the scale model result, made during the original investigation of the disaster, showed the destructive torsional mode to exhibit negative damping beginning at low wind speeds. It didn’t appear suddenly. The interesting question is: how did the bridge enter the torsional mode, which had not been observed before, during its final hours?
These may seem like pedantic points, but a long and complex essay is only made more so, and its thesis made less convincing, by including examples that aren’t pertinent.
Reply to kevin kilty ==> Sorry if I was not clear with the airplane wing example — it is not meant to illustrate the logistic formula doubling — it is an example of real world periodic doubling that occurs ubiquitously in dynamical systems, the turbulence problem when dealing with fluid flow — of which airflow over a wing is an example.
As for the Tacoma Bridge, the behavior has the appearance of a chaotic system at work, and thus my use of the word “may”, and, as you say, it is oft mentioned in physics texts. (I am an essayist, not a bridge engineer).
It might be that had the bridge disaster been investigated in the 1980/1990s, when the engineering world was beginning to understand the implications of Chaos Theory, the results might have been different.
Just a bit of nit picking but Tacoma Bridge was more related to Catastrophe Theory than Chaos Theory. see
https://www.physicsforums.com/threads/chaos-theory-vs-catastrophe-theory.668707/
Reply to son of mulder ==> I’m uncertain if catastrophe theory is all that separate — it is a particular occurrence within the realm of dynamical system studies — and while a particular thing, may not be any different than a single identified behavior within the larger realm. Chaos Theory, which can be broadly defined as “the study of dynamical systems, particularity those highly sensitive to initial conditions”, is itself not very well bounded.
I think it is the case that some Chaotic systems may be Catastrophic but one could have a Catastrophic system that is not Chaotic. eg stretching elastic beyond its elastic limit would be catastrophic but not chaotic. The Tacoma bridge falls into this category I suspect. As I said nit picking.
The chaos model is constructed based on the understanding that physical processes and systems are incompletely and insufficiently characterized and unwieldy. This motivated the acknowledgment of a scientific [logical] domain characterized by accuracy that is inversely proportional to the product of time and space offsets from an observer’s frame of reference and enforced… encouraged by the scientific method. Science is corrupted through assumptions/assertions of uniformity, linearity, and independence, and disrupted by conflation through correlation.
Climate is the average weather statistics in a frame of reference over some indefinite (i.e. unpredictable) period. We are fortunate to live in a system with semi-stable processes over relatively — to human life — long, defined periods.
Human life is a chaotic process with a known source (i.e. conception) and an unpredictable sink (i.e. death or catastrophic cohesive change).
Let’s not dispute that climate modeling has about a 40 year history. That over this period, modeling teams have employed many very intelligent mathematicians, computer programmers, and sundry climate-type experts. I mean, what else have governments spent those billions on?
In performing and publishing hindcasts, the teams are explicitly stating their belief that models represent actual climate. They are being used by politicians and their green cheerleaders to push the CAGW meme. [Useful word, that meme.]
All the verbiage about “realizations” being nothing more than the computational result of known climate relationships just confuses the central issue: Should we fundamentally alter our various economies, forms of government, and social compacts to limit emissions of CO2 and other gasses.
Dance around all you want about chaos theory, nonlinear mathematical relationships, conformance of the climate system to equations and whatnot; the proof is in the pudding. And I eat pudding.
This has been a test. Nothing but a test. We now return you to your regular programming (of obfuscation).
Charlie Climate
Oops! For the umptheenth time. Ta-da! I’m actually Charlie Skeptic!
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.
if that were the case it would imply that climate is not bounded at all. T
he data shows it is, ergo some sort of prediction is in theory possible.
In practice we don’t know how to do it, sure..;)
Reply to Leo Smith ==> well, sure — however, I do say “meaningful projections or predictions of future climate states”. I can correctly point out, with certainty, that there will be weather tomorrow, and climate 100 years from now.
If I were a betting man, which I am not, I would say that the Climate 100 years from now will resemble very closely the climate we have now, with some local and regional variations,
Well there you go! You made a meaningful prediction!
First of all, don’t take this as criticism. Fantastic post, and if more people understood its message there wouldn’t be a single windmill left standing.
My point was broadly this: chaos means we don’t know and can’t know exactly. Well hello. Welcome to real life. It doesn’t mean we don’t know at all, however.
Take my favourite analogy of a car crash. No one knows where the pieces will end up, or in what shape they will be, but we can predict fairly accurately where they will NOT end up. Which is why there are various bits of fencing on race tracks in various places, but none round Taco Bell’s 15 miles away. Car crash bits won’t travel that far.
This is a problem I have very often with people who are not engineers. Even scientists are upset by the inference that not to know exactly, is not to not know at all.
Engineers never know exactly. They live in the real world.
The public do not. Even the educated liberal leaning middle classes who admire education and think they actually have some. Some tells them that truth is relative to culture, and they believe it, which is fair enough, but then they misunderstand what that means. They think that means that one truth is as good as any other. And there is no underlying Truth there at all.
They hear that all scientific theories are amenable to change, and cannot be proven to be true, and they misunderstand that, to think that you can make up any theory and its as good as any other theory. As long as it can’t be proved to be wrong. Like the existence of God or Man Made Climate Change.
So I just didn’t want another Great Misunderstanding to start here. For sure Chaos means we can’t exactly predict. But systems with overall negative feedback are bounded we may orbit strange attractors, but we do not fly off the graph paper altogether, or, if we do, we would have done it years ago.
Once upon a time, two friends and I, in a less than sober evening at one of the aforesaid friends house, decided to solve the N body problem on his newly acquired computer thing. They were into astronomy. I wrote code. On that machine, BASIC.
Three hours later ‘orbits’ was running. Stepwise integration of IIRC about a 9 body problem. 9 celestial bodies of varying masses were given random positions and velocities in a 3D space, and the iterative output was plotted in 2D as tracks on the screen.
Three hours later I knew why the solar system was the way it was, with all the planets more or less in a plane, and all almost phase locked in harmonically related orbits.,
Because no other configuration was stable. Planets and asteroids off the ecliptic got thrown into deep space. Deeply elliptical orbits resulted in either loss of planets altogether, or if one was asteroid mass, cometary orbits developing.
My point is this: That exercise showed me that a very real law of nature is simply this: unstable shít doesn’t persist (in time). Nature abhors instability. Car crashes soon stop. The law of entropy tends towards dull wet rainy Sunday afternoons, not Saturday Night Fever. Darwin’s ‘law’ is misunderstood, It never was ‘survival of the fittest’ it was merely ‘elimination of that simply too dysfunctional to live long enough to bear offspring’ .
And that is why politics is what it is. It panders to the illusions of generations of people who have merely managed to steal enough food, avoid gross disease and accident, long enough to shack up in some sweaty bed, and do natures little belly dance, to produce yet another generation of pathetic inadequate ignorant excuses for humanity.
Nature demands no more.
However, it does seem that a few percent who are slightly more than that in every generation seem to promote group survival, so here we are. Genetically condemned to be hated smartasses. Until needed.
And getting stuff right, is our game. So: A minor correction to your excellent piece. We can predict chaotic systems, especially natural ones, that we know have been around a long time. They must be bounded by overall strong negative feedback. Nature does require that they be unchanging, merely that they have ‘temporal persistence’. That they not be so unstable as to cease to be, like a Norwegian Blue. The roulette wheel always stops on some number.
We ought to be able to come up with e.g. two surface temperatures (high and low) beyond which the climate can never ever go, because negative feedback simply won’t permit it. We won’t be able to predict where, at any given moment, it will be, but we should be able to define limits.
IN the end we know that (apart from residual geothermal heat content) the earth derives all its energy by daytime radiation from the sun, and loses it all by (night-time) radiation into space. What happens in between is locally interesting, but globally irrelevant. The strength of BOTH radiations is essentially a function of the albedo. The case for CO2 induced warming is that somehow CO2 becomes an asymmetrical albedo, letting high intensity short daytime wavelengths through, but absorbing longer wavelengths which comprise the night time radiation.
But that doesn’t make a lot of sense either. If the atmosphere is warmer due to CO2, it will in the end radiate to space more. its a fourth power law, which is massively non-linear even at 293°K.
Of course all this wonderfully chaotic turbulence and convection that moves bits of atmosphere and water in and out of high and low radiation places, and distributes energy round the planet is going to seethe and boil so to speak and have hot spots and cold spots and cyclones and so on. But we aren’t interested in short term local variations, but in long term systemic ones. Overall, how hot (or cold) COULD it get before T⁴ simply overwhelmed any possible albedo variations?
You and I have a gut instinct that the answer is ‘not very much upwards’ simply because its never been very much warmer than it is today, although it has been somewhat. And ‘rather a lot downwards’ because extensive historical glaciation is as near a fact as we ever have to deal with.
In fact we as a species probably don’t need to know anything beyond that. There’s plenty of land mass up towards the North Pole where people could exist quite comfortably if people got a lot warmer. WE are more in short supply of places to live in the Northern hemisphere was glaciated down to 50°N or whatever.
So the problem ceases to be the one the IPCC concentrates on, and becomes a perceptual and political one. How can we convince the Great Unwashed that they have been essentially lied to, by the very people they allow to form their opinions and the mores of the society they found themselves in? Could they, or a significant fraction of them, be educated in the basic principles of Chaos theory and system analysis to understand these things for themselves?
I am getting old and cynical. Somehow this excerpt from Raymond Chandler’s ‘the Long Goodbye’ seems apt enough to finish with:
He wasn’t listening. He was frowning at his own thoughts. “There’s a peculiar thing about money,” he went on. “In large quantities it tends to have a life of its own, even a conscience of its own. The power of money
becomes very difficult to control. Man has always been a venal animal. The growth of populations, the huge costs of wars, the incessant pressure of confiscatory taxation – all these things make him more and more venal. The average man is tired and scared, and a tired, scared man can’t afford ideals. He has to buy food for his family. In our time we have seen a shocking decline in both public and private morals. You can’t expect quality from people whose lives are a subjection to a lack of quality. You can’t have quality with mass production. You don’t want it because it lasts too long. So you substitute styling, which is a commercial swindle intended to produce artificial obsolescence. Mass production couldn’t sell its goods next year unless it made what it sold this year look unfashionable a year from now. We have the whitest kitchens and the
most shining bathrooms in the world. But in the lovely white kitchen the average American housewife can’t produce a meal fit to eat, and the lovely shining bathroom is mostly a receptacle for deodorants, laxatives, sleeping pills, and the products of that confidence racket called the cosmetic industry. We make the finest packages in the world, Mr. Marlowe. The stuff inside is mostly junk.”
People are susceptible to Marketing. That’s why Marketing has temporal persistence. Shiny thing make it all better.
http://www.thedailymash.co.uk/news/business/shiny-thing-make-it-all-better-201001282420
Its not enough to know the truth. Getting people to believe it is the truth is the harder problem, when all they really want to do is believe in comfortable lies.
Anyway. the truth is that chaos or not, we should be able to come up with a ‘climate won’t get much worse (or better) than X’ . That problem I will leave to you.
The problem that interests me, because no one else is really working on it, is ‘How will we convince them its true, when we do find it?’
Or even whether in fact its worth telling them at all, rather than letting the sheer instability of Western society cease to have temporal persistence, because its simply too dysfunctional?
Societies and civilizations are themselves chaotic systems. Perhaps in the end the cultural mores that allowed our Western society to flourish – the ability to trust strangers, to trust banks, to trust people with education – will in fact be it’s downfall. Perhaps we trust entirely too much.
And yet if we disassemble trust, civilisation will collapse.
Well that’s my thought for the day. Don’t give up the analysis just because its hard. Lies are easy, te truth is often hard.
And I’ll go back to pondering the psychology and philosophy of the human mind, to see if there is a better way than telling well crafted lies, to influence it.
Nature does NOT require that they be unchanging, …sorry., I hare the inability to edit posts.
Leo,
Your response in 25 words or less: The probability of a climate “Tipping Point” is vanishingly small because it would have already happened if it was probable that CO2 could cause it.
I agree with your cynical view of the general situation. Perhaps it is because as we get older and come to terms with our mortality that we are less patient and forgiving of the BS that is a part of the games played by the young.
Leo “I am getting old and cynical”
I would rather say that you are getting old and wise. Thanks for the post
Reply to Leo ==> See the upcoming Part 4 of this series for bounded — this is a very strong inescapable observational fact.
Thank you for the Chandler….you might have guessed from my writing style that I am a fan … re-read The Long Goodbye earlier this year.
Chaotic dynamics of the dissipative nonlinear climate system, combined with the hugeocean heat capacity and strong temperature gradients, means that climate changes by itself. It does not need external forcing to change.
External climate forcing can happen, but climate can also change with external forcing from internal dynamics only. It os quite possible that all 20th century warming was from internal dynamics only.
Chaos as illucidated by Lorenz means the climate system can be expected to continually fluctuate in surface temperature without the need of external forcing.
When you hear mention of forcing of climate then this means “we don’t understand chaos”.
The application of Prigogine’s nonlinear thermodynamics to atmospheric radiative and water vapour feedbacks, would probably lead to the conclusion that Miskolczi came to. It’s not about getting sums right, it’s about finding the right metaphor or analogy.
Emergent nonlinearity is also behind the truth of Lovelock’s Gaia hypothesis in which the global flora reacts to changes e.g. relaxed CO2 starvation to modify the environment to its own advantage.
The prediction of climate depends upon a realization that we are dealing with an open system.
There is no greenhouse effect. The temperature of the warmest month has not increased in the southern hemisphere in the last seventy years as seen here: http://www.esrl.noaa.gov/psd/cgi-bin/data/timeseries/timeseries.pl?ntype=1&var=SST&level=2000&lat1=0&lat2=-90&lon1=0&lon2=360&iseas=0&mon1=0&mon2=11&iarea=1&typeout=2&Submit=Create+Timeseries
Meteorologists have long known that surface temperature varies with geopotential height at 500 hPa. Geopotential height reflects the temperature of the air below the point of measurement.
It’s a characteristic of the atmosphere that most of the variation in its water content occurs close to the surface. As the air cools overnight it voids moisture. As it warms during the day clouds evaporate.
The temperature of the air above the near surface layers tends to be driven by its ozone content rather than surface temperature and the more so in high pressure cells of descending air.
The day to day variation in the temperature of the air aloft is much greater than at the surface. Because the water vapour content of the upper air is relatively invariable, as its temperature changes so does the volume of moisture that exists in the condensed, frozen form that we see as clouds. Clouds can reflect up to 90% of incident sunlight. This is why there is this relationship between surface temperature and geopotential height.
Gordon Dobson was in the forefront of the investigation of modes of natural climate change in the first half of the nineteenth century. When Dobson used a spectrophotometer to measure the ozone content of the air he noticed that as total column ozone increases, surface pressure falls away and the tropopause is lower by as much as 2-3km. This situation results in Jet Streams.
Secondly Dobson noticed that the ozone content of the air increases in high latitudes.
In 1956, Dobson was amazed to observe the ozone hole at Halley Bay, Antarctica, reversing the patterns of ozone accumulation that he had seen in the northern hemisphere.
In point of fact, the entire southern hemisphere is something of an ozone hole due to the very active descent of ozone deficient air from the mesosphere that occurs over the Antarctic continent. The rate of descent varies over time and with it the ozone content of the air globally. This is where the system is most open to external influences.
If we leave the subject of climate variation to myopic specialists with inflated egos but little common sense, we are doomed to thrash about in ignorance and superstition forever. More at https://reality348.wordpress.com/
Just found this delightfully frank little gem in a paper I am touring [Design and Simulation of a H.264 AVC Video Streaming Model, Doggen, JeroenVan der Schueren, Filip January 31, 2008].
Page 2, Section 2:
“Simulation is an approach which can be used to predict the behavior and performance of large, complex stochastic systems [2]. The development of an accurate simulation model requires extensive resources. When a model is not very accurate, one can make the wrong conclusions from the simulation results. The basic problem is that every simulation model is inherently wrong, ranging from lightly flawed up to totally wrong. As a result the simulation outcome is only as good as the model and it is still only an estimate of a possible projected outcome.”
I had read this post and Willis’ slightly more recent one on cloud feedback, then read this paper’s intro paragraph and almost hurt myself with a laughter.
The good news according to the IPCC and GISS is that climate models are so much better than models in other more readily verifiable arenas of science… the problems that researchers in other disciplines have with modeling these types of systems don’t occur in climate modeling. \
Great post thanks Kip
The same principles apply to economics? I sense yes. Economists have real incentive to predict accurately, but wow, do they get it wrong, most of the time.
Reply to Michael Carter ==> There are economists that consider the stock markets and national economies to be nonlinear dynamical systems — thus chaotic.
Yes. Economies are very sensitive to nonrandom, unpredictable events. Take the Bali bar bombing. One event directly effecting a few people effects the psychology of many more that sends the economy into nosedive, effecting millions. We can go back even further: what events lead to the decision by the bombers to do what they did? We may even have to go back to events that occurred in their childhood
I am using economics as it displays the principle in a clearer manner. I see it in ecology. We cannot predict populations of organisms in the natural world very far out into the future. There are too many threads in the web and too much dependency
Replacing the inappropriately chosen term ‘Chaos’ with
Nexus meaning a connection or series of connections linking two or more things or events
could avoid persistence of confusion at least when referring to the weather events.
Reply to vukcevic ==> Well, yes — unfortunate choice to describe a system that is entirely deterministic.
“… inappropriately chosen term …“
“Well, yes — unfortunate choice”
Great example; “Chaos is undesirable in most engineering applications.”
No sheet, Sherlock ; )
I say change it, and suggest ‘chaosh’ because it’s available and everyone familiar with the matter will realize right away what is being referred to, and why the change is being adopted. In fact, rather than going through all that explaining about it in your next last installment ; ) I suggest simply saying ‘chaos’ is misleading, so you’re going to use ‘chaosh’ for the rest of the article instead to avoid confusion. .
And, if you can trigger a shift in general usage thereby . . it’s Chaosh!
The most important practical conclusion from chaos / nonlinear system theory is that the null hypothesis of climate is that it will always be changing. And that this change is internal, not needing external forcing. The energy needed for this comes from the vast heat capacity of the oceans, whose own dissipative nonlinear circulation dynamics serve up continual climate change.
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There is also a fractal character to natural chaotic climate change. Fractal pattern looks the same at different spatial scales e.g. the fern leaf. So chaotic oscillation can give (with periodic forcing)) Milankvich glacial cycles but also “micro-interglacials” or DO events within glacial periods and also the millenial, century decade and interannual climate variations.
The Lorenz attractor, often associated with a chaotic system oscillating between two states, when plotted with time has a pattern very similar to many climate time plots, such as PDO and alternation between el Nino and La Nina dominated periods in the Pacific.
The implication is to make it harder to claim that warming over the last 100 years has been in any way anomalous.
We are used to thinking of anomalies relative to a linear, fixed temperature.
With chaos, variation is the baseline.
What is the anomaly? That is the interesting question.
http://www.physics.emory.edu/faculty/weeks//research/time/tseries1/vlorenz.gif
http://ccreweb.org/documents/physics/chaos/LorenzCircuit3_files/image028.gif
http://blogs.mathworks.com/images/cleve/odes_intro_01.png
+1
I have been challenged to produce ‘chapter and verse’ from the IPCC TAR WG1, Working Group I: The Scientific Basis that states “The climate system is a coupled non-linear chaotic system, and therefore the long-term prediction of future climate states is not possible.” Please help!
Reply to Krov Menuhin ==> The link is given next to the image at the very top of the essay: to make this easier, here it is:
https://www.ipcc.ch/ipccreports/tar/wg1/501.htm
Sixth bullet point down in that document — second sentence.
This is not the only place that this is stated, but that is a simple one to find.
you just need to click on the provided link just below the quote
https://www.ipcc.ch/ipccreports/tar/wg1/501.htm (which appears to be the executive summary)
and search for the sentence.
IPCC pays lip service to the chaotic nature of the system, but nonetheless makes linear models with climate-like noise added.
I would just like to correct ‘Philohippous’ when he said that stalls in a P51 according to the pilots manual states ‘never let the plane stall as recovery is unlikely’. The manual in fact states that stalls are “comparatively mild” and “recovery when you release stick and rudder is almost instant’. I have stalled the P51 a number of times, there is plenty of warning and recovery is simple.
Kip
Another good text getting to the heart of chaos and its implications is “Deep Simplicity” by John Gribben:
https://www.amazon.co.uk/Deep-Simplicity-Complexity-Emergence-Penguin/dp/0141007222
Reply to ptolemy2 ==> Thank for the link to John Gribbin’s book — haven’t read it, but it was on the Good Reads reading list for Chaos.
He wins points for using Katsushika Hokusai’s painting of The Great Wave on the cover. While in Japan in 1998, my wife and I accidentally visited a museum of his work. We were on our way to something else, Snow Monkeys, I think, and the bus or train let us off across the street from the museum. Stunning art.
Hokusai’s work is often used as an example of fractals, which some see in the fingers coming off of the top of the wave.
Thanks Kip, You’re right, Hokusai’s “The Great Wave” a truly iconic picture:
The little curly wavelets do indeed seem fractal-like. That makes it even more cool.
It made me wonder where I had seen it before. The band Keane must have used it as the basis for their album art for “Under the iron sea”:
I always have problem when chaos is presented under the perspective of temporal chaos only .
This is the case for the logistic equation, Lorenz system, N body system and more generally systems described by a finite number of ODE (Ordinary Differential Equations) .
This kind of Chaos is relatively well understood because every variable corresponds to a degree of freedom of the system so that we deal with systems described by a finite (often low) number of degrees of freedom .
One of the consequences is that temporal chaos attractors may be represented as curves/surfaces in phase space . The Lorenz system has 3 degrees of freedom so the attractor lives in an ordinary 3 D space .
Spatio temporal chaos (e.g a system’s dynamics depends not only on time but on time and space) whose typical representant is fluid dynamics and Navier Stokes équations is a completely different beast and very few results transport from temporal chaos to spatio temporal chaos .
The reason for that is that spatio temporal chaos is described by PDE (partial differential équations) and a PDE is equivalent to an infinity of ODEs .
Follows that spatio temporal chaotic systems have an infinity of degrees of freedom .
The degrees of freedom of spatio temporal chaotic system live in an infinite dimensionnal Hilbert space – they are functions and not simple variables x(t) like it is the case in temporal chaos .
This has among other for consequence that attractors are infinite sums of functions f(x,t) and cannot be represented by drawings in an ordinary finite dimensional R^n space .
Attractors in spatio temporal systems must be understood as superpositions of functions that oscillate both in time and space .
ENSO is an excellent example of a linear combination of an unknown number of such “elementary” functions .
By extension if an attractor existed for the climate (which is an example of spatio temporal chaos but much more complex than “simple” Navier Stokes) then this attractor would be a linear combination of a possibly infinite number spatio temporal “oscillation modes” .
Nobody knows if such an attractor exists and even if the proof of existence was brought in the next decades (I am convinced that it won’t happen) , we would still be unable to find the “elementary” spatio temporal oscillators especially if their number is infinite .
That’s why extreme caution is adviced when somebody tries to transport temporal chaos insights into the spatio temporal domain because such transport is most likely to be wrong .
The spatio temporal chaos is an extremely badly understood domain and we will need much time before some useful results are obtained .
A specific note for N.Stokes
What I said above applies to statistics of spatio temporal chaos too .
It is known that e.g ergodicity is a property that allows to have existence and to find statistical invariants for temporal chaos what gives foundation to your belief that we can compute probabilities of future states even if we are unable to make a deterministic prediction .
Unfortunately for you, this result doesn’t transport to spatio temporal systems . So to climate .
You cannot be sure that models (whatever models you use be it CFD or whatever) are able to find some combination of functions f(x,t) which would be an invariant probability of future states .
Especially then not if such an invariant doesn’t exist what can’t be excluded 🙂
Reply to Tomas ==> “By extension if an attractor existed for the climate (which is an example of spatio temporal chaos but much more complex than “simple” Navier Stokes) then this attractor would be a linear combination of a possibly infinite number spatio temporal “oscillation modes.” “ENSO is an excellent example of a linear combination of an unknown number of such “elementary” functions .” and “Nobody knows if such an attractor exists.”
Yes, I agree with these points, and have tried to lead the discussion in this direction…. I discuss Chaos Theory with climate because it is an important issue — just as system turbulence is with airplane design. You can’t really talk climate without understanding, in some sense, what it means that it is a coupled non-linear chaotic system.
I’ll try to wrap it up in the next, and last, part of this series.
Thanks for checking in and contributing.