Alternative title: “Standing on the shoulders of Giant Bob”
Guest post by Phil Salmon
Introduction
One of the themes to emerge from the climate debate here on WUWT, concerns “chaos” and nonlinear system dynamics and pattern. Anyone acquainted at all with the nature of dynamical chaos and nonlinear / non-equilibrium pattern formation, and who also has an interest in the scientific questions about climate, cannot fail to sense that dynamical chaos has to be an important player in climate. Simply on account of the huge complexity of climate over the expanse of earth’s surface and deep time, and also the obvious impossibility of equilibrium in a rotating system with continuous substantial imbalances of heat and kinetic energy.
However, a “sense” is hardly adequate scientifically; it is necessary to go further than this and forge some kind of physical and mathematical model or hypothesis which can be tested. But here one runs into the problem of chaotic systems being .. well, chaotic and unpredictable; indeed for some the movement of a system into the chaotic region represents falling off the edge of the world of scientific testability and orthodox Popperian experimental investigation. Is it a contradiction in terms to imagine that you can study chaos scientifically and mathematically? The scientific community at large – not only climate science – while giving lip service to chaotic pattern formation as a real phenomenon, generally shrinks back from serious engagement with it, back into the comfortable regions of tidy linear and equilibrium equations.
However there does exist a well-established science of physical and mathematical study of chaotic, nonlinear systems, in which a wide range of nonlinear pattern forming systems are well understood and characterized. But owing to the human tendency to associate in closed communities – nowhere more in evidence than in the multi-faceted scientific world, there is in my view too little engagement between the chaos and nonlinear dynamics experts and scientists in a wide range of natural sciences whose studied systems are – unknown to both sides – accurately and usefully characterized by well-researched nonlinear pattern systems.
It is the purpose of this article to propose a well-known experimental “nonlinear oscillator”, namely the Belousov-Zhabotinsky chemical reaction, as an analogy – in terms of its dynamics and spatio-temporal pattern – for the El Nino Southern Oscillation (ENSO) system in the equatorial Pacific Ocean. This would characterize and alternation between El Nino and La Nina as a nonlinear oscillator. The definitive work of Bob Tisdale on the ENSO is used to liken the alternating multi-decadal periods of eE Nino and La Nina dominance (the PDO) as the two wings of the Lorenz butterfly attractor.
The term “chaos”, while a common shorthand for a class of phenomena and systems, is not a very accurate or helpful one. Chaos itself, strictly speaking, is truly chaotic and not a very fruitful area of mathmatic investigation. A system passes from the region of linear dynamics through “fringes” or borderlands of mathematical bifurcation before reaching full blown chaos, and it is in these marginal and transitional borderlands where the interesting phenomena of strange attractors and spontaneous pattern formation arise. But it is hard to find a convenient single word that takes its place – it is easier to say “chaos” than “nonlinear pattern formation in far-from-equilibrium dissipative systems”.
Even “nonlinear”, while better than “chaos”, is still inadequate: there are plenty of physical and mathematical systems which are clearly not “linear” but not related to non-equilibrium emergent pattern formation. A relative of mine – a TV weatherman in Monterrey, California for many years before his retirement – pointed this out to me, that it is not necessary to invoke nonlinear pattern formation to account for acute sensitivity to initial conditions – a simple high power relationship is sufficient for this. Acute sensitivity to initial conditions does indeed characterize many nonlinear systems – indeed, one popular metaphor for chaotic systems is the “butterfly wing” effect – namely that a butterfly wing’s disturbance of the air in one place can result in massive changes in weather systems a continent away. The butterfly wing analogy was coined by Edward Lorenz – a pioneer in mathematical study of non-equilibrium pattern system and also a meteorologist – we will return to Lorenz later. However this sensitivity does not uniquely define the type of system we are considering. (The “butterfly wing” metaphor is now inseparable from the actor Jeff Goldblum and his rather inane use of the phrase in the Jurassic Park films.)
If I had to propose an alternative to “chaotic” as a general short term for such systems with spontaneous nonlinear pattern dynamics, I would go for something like “non-equilibrium pattern” systems.
One of the most helpful references I have found on the subject of non-equilibrium pattern systems is the PhD thesis of a chemical engineer Matthias Bertram, entitled “controlling turbulence and pattern formation in chemical reactions” – previously posted on his web site but now reposted on Google docs:
Matthias uses the term “pattern formation in dissipative systems”. To quote from the introduction of this thesis:
“The concepts of self-organization and dissipative structures go back to Schrodinger and Prigogine [1–3]. The spontaneous formation of spatio-temporal patterns can occur when a stationary state far from thermodynamic equilibrium is maintained through the dissipation of energy that is continuously fed into the system. While for closed systems the second law of thermodynamics requires relaxation to a state of maximal entropy, open systems are able to interchange matter and energy with their environment. By taking up energy of higher value (low entropy) and delivering energy of lower value (high entropy) they are able to export entropy, and thus to spontaneously develop structures characterized by a higher degree of order than present in the environment.”
The author goes on to analyze several experimental non-equilibrium pattern systems, including the Belousov-Zhabotinsky reaction. He outlines the essential conditions for the operation of a nonlinear oscillator such as a far from equilibrium state, and an “excitable medium”, that is, a medium within which localized positive feedbacks can be initiated and run their course according to their associated refractory period. We will return to these parameters when we consider the ENSO.
The Belousov-Zhabotinsky reaction
There is a helpful short introduction to the Belousov-Zhabotinsky (“BZ”) reaction on Wikipedia:
http://en.wikipedia.org/wiki/Belousov%E2%80%93Zhabotinsky_reaction
Have a look at this youtube video shown in figure 1:
Figure 1A. A video of the BZ nonlinear oscillation in a stirred beaker.
Figure 1B. A graph of light transmissivity over time, illustrating the BZ reaction (from Wikipedia DE)
What you are looking at is the Belousov-Zhabotinsky (BZ) reaction in a stirred beaker. It is striking in that the beaker’s liquid contents oscillate between a dark blue colour and clear transparency, for multiple cycles. Most of us can recall school chemistry lessons from the (more or less) distant past, where we saw reactions such as the titration of potassium permanganate with hydrogen peroxide, causing a beaker or tube full of liquid to change from dark purple color to clear, or vice versa. But not many of us probably saw the oscillating BZ reaction with a tube of liquid oscillating between the two starkly contrasting states. The BZ reaction “involves several reagents and various intermediate species; the central reaction step is the oxidation of malonic acid by bromate, catalyzed by metal ions” (Matthias Bertram 2002). The system is jumping between two states looking for equilibrium but finding it in neither.
This is intriguing to watch but what is going on here, and what significance does it have to climate, to the behavior of atmospheres and oceans?
The BZ reaction is a gateway to a whole branch of science which is, to repeat, still very incompletely explored and whose significance is under-appreciated. The two individuals, Boris Belousov and Anatol Zhabotinsky, who established their eponymous reaction, have an interesting history which has some resonance with the politics of climate science. Boris Belousov accidentally came across the oscillating reaction in Soviet Russia during the early 1950’s (one of the important and long undiscovered Soviet scientific discoveries that also included the “Ilissarov frame” orthopedic method for making new bone by gradual movement apart of fractured bone ends). Belousov’s attempts to publish this finding were rejected repeatedly, on the grounds of the familiar “where’s the mechanism?” argumentum ad ignorantium. In 1961 a graduate student Anatol Zhabotinsky took up and ran with the discovery, but it was not until an international conference in Prague in 1969 that the reaction became widely known, two decades after its inception.
The BZ reaction is a “reaction-diffusion system”. It is a non-equilibrium pattern phenomenon known as a nonlinear oscillator; there are certain prerequisites for such a system to develop:
- The system is far from equilibrium
- It is an open system with a flow through of energy (dissipative)
- The system has an “excitable” medium
The BZ reaction meets these requirements sufficiently to set off nonlinear oscillation. Note that condition 2 is only partly and temporarily met – a tube of chemicals is not really open; however the availability of reagents makes the system for a limited time behave like an open system until the reagents become exhausted.
The BZ reaction in a thin film
There are many types and flavours of the BZ reaction. In the first example we saw the reaction in a beaker: however when the reaction is carried out in a thin film, a new element arises: instead of the solution changing colour en-masse, the colour changes are associated with intricate evolving patterns such as radiating ripples and spirals.
You can search for “BZ reaction” on youtube and find many examples of attractive moving patterns, some with musical accompaniment. One of these is given in the link below:
Figure 2. Three animations of the BZ reaction in a thin film, showing evolving spatiotemporal waves and patterns and alternations of dominant colour phase.
This link presents three BZ thin film animations. In the first, regions of orange and pale blue colour repeatedly expand and contract, encroaching on each-other reciprocally, such that looking at the dish as a whole, the predominant colour alternates between orange and pale blue. The second animation is one where typical BZ fringe and spiral patterns in dark and light purple radiate from various centers. If you look in the bottom left corner, a tongue of darker purple periodically grows and recedes. The third animation is a slower moving version of the first – if you have the patience to watch all of it, again there is an overall pattern of alternation between orange and pale blue as the predominant colour.
Another youtube video of a thin film BZ reaction is given in the link below; while it is tediously slow and would have benefited from acceleration, it shows nicely the radiating BZ patterns characterized by alternation between orange and pale blue as the predominant color.
http://www.youtube.com/watch?v=S20Jsfu9rkQ
Figure 3. Another animation of the BZ reaction in a thin film showing travelling patterns and alternating phases.
During some parts of these BZ sequences, especially of the first animation, you have the feeling that you could be watching one of Bob Tisdale’s animations of the temporal evolution of sea surface temperatures (SSTs), such as that occurring in the equatorial Pacific with alternating el Nino and La Nina cycles: such an animation is given (By Bob) in the link below:
Figure 4. An animation of sea surface temperature anomalies in the Pacific during the transition from el Nino to La Nina systems during 1997 – 1999 (from Bob Tisdale’s blogspot), from web page: http://bobtisdale.blogspot.com/2010/12/enso-related-variations-in-kuroshio.html
If one focuses on the south eastern Pacific off the Peruvian coast, where the alternating tongues of warm and cool surface water characterize respectively the alternating en Nino and La Nina, the analogy to the BZ reaction is particularly compelling.
The ENSO as a nonlinear oscillator?
However beyond an intriguing qualitative visual similarity, what basis is there for proposing that the ENSO could constitute the same type of nonlinear oscillator as the BZ reaction? Please note that I am not proposing that chemical reactions play a role in the ENSO – no, chemical potentials in the BZ reactor are matched by thermodynamic potentials in the atmosphere-ocean system. Specifically we can return to the question of the essential pre-requisites that the BZ system meets to operate as a nonlinear oscillator; how would the ENSO system also meet these pre-requisites?
1. A system far from equilibrium
At least this one is a no-brainer. Solar energy input is very unequally distributed on the earth’s surface, maximally at the equator and minimally at the poles. Add to this the rotation of the earth and associated day-night cycle, and oblique axis rotation causing reciprocal summer and winter in north and south hemispheres, and ocean circulation, and it soon becomes clear that equilibrium is never remotely approached. (In fact, a world with atmosphere, ocean and heat flux in equilibrium is a nightmare to contemplate, with stagnant anoxic seas and stale motionless air.)
2. An open, dissipative system
The global climate system is open, as it receives heat input from the sun which (Leif Svalgaard notwithstanding ) is subject to minor periodic fluctuation. Heat is also radiated out to space. Heat energy enters and leaves the system; thus it is dissipative.
3. A system with an excitable medium
This is perhaps the most critical requirement. “Excitable” implies that an induced change at one location sets in motion a positive feedback which results in local amplification and propagation of the induced change – for instance taking the form of a travelling wave in the BZ reaction. This is not a wave in the sense of an energy wave through water or air that merely transmits energy, but a wave in which a spreading reaction is stimulated generating new local energy with the propagating wave. A cascade of chemical reactions in the BZ reaction constitutes this excitability. This positive feedback is limited and runs its course – characterised by a refractory period – but its operation is sufficient to drive and sustain the nonlinear oscillation, and in some cases to generate complex spatiotemporal patterns.
How could such excitability exist in the equatorial Pacific where the ENSO takes place? To discuss this question I need to refer to an exchange I had a few months ago with Bob Tisdale on a thread here at WUWT. The topic was one of these chicken-and-egg discussions of what drives the ENSO, either top-down by trade winds for instance, or bottom up by variation in deep upwelling. I posed the question to (who better?) Bob Tisdale, suggesting that the spread of both the el Nino and the La Nina, could involve a time-limited positive feedback. The nature of these positive feedbacks is indicated in the two diagrams below.
Figure 5. The La Nina positive feedback: enhanced Peruvian cold upwelling sharpens the equatorial Pacific east-west pressure gradient, driving stronger trade winds which propel further upwelling.
Figure 6. The el Nino positive feedback: decreased upwelling weakens the trade winds which propel the upwelling.
Please note that in the schematic systems in figures 5 and 6 it is not really relevant which comes first – changes in the trade winds or in upwelling. They are linked in a feedback loop. The analogy that I had in mind was of the on-shore and off-shore breezes that occur in summer in temperate coastal locations such as the British Isles. Here, in the day, increasing land temperature warms the surface air, causing it to decrease in density and rise, drawing in on-shore winds from the sea. Conversely at night, the land temperature quickly cools, increasing surface air density such that the wind is reversed to an off-shore breeze. (By contrast the air temperature over the sea is relatively constant). It was this essential mechanism that I suggested for the equatorial Pacific ENSO system, that the upwelling off Peru associated with the start of a La Nina cycle, in cooling the east Pacific surface layer air, creates a higher air pressure or density to the east that acts to drive east-to-west (easterly) trade winds (of the type that propelled Thor Heyerdahl and his companions on their epic Peru to Indonesia crossing of the Pacific on their “Kon Tiki” balsa wood raft, recapitulating the voyages millennia earlier of Polynesian mariners and ocean island settlers). These energised trade winds will push Pacific surface equatorial water westwards, adding impetus to the Peruvian upwelling by drawing eastern Pacific deep water toward the surface in a conveyer-belt like fashion. Thus the full cycle of a positive feedback illustrated in figure 5.
Conversely, during an el Nino cycle, upwelling is slowed or interrupted, resulting proximally in increased solar heating of more static, less mixed surface water in the Pacific east. This will decrease the temperature and pressure east-west difference, sapping force from the trades and resulting in doldrum conditions of decreased winds. The weakened trades will then slow the upwelling conveyor, connecting a feedback cycle that moves toward interrupted upwelling and a rapid spread of warm surface water from the east Pacific (figure 6).
It was a big moment for me when Bob Tisdale replied to the affirmative, agreeing that a time-limited positive feedback did indeed drive the onset of el Nino and La Nina, until both ran their course, reaching, to quote the term Bob used, “saturation”. Of course the whole system involves more complexity than this idealised system – there are periods of neither el Nino nor La Nina, or of modified, “Modoki” el Nino systems. However for me Bob’s positive reply was very important because the final piece of the jigsaw for this BZ-reaction analogy fell into place. Now I had my excitable or reactive medium. So it began to become clearer that the ENSO can indeed be characterised as a nonlinear oscillator, analogous to the BZ reaction-diffusion system.
3. The attractors and longer term pattern of ENSO (the PDO)
A feature of non-equilibrium pattern systems and their spatio-temporal evolution is an attractor. An attractor is a subset of the (often multidimensional) phase space that characterises a system, towards which the evolving system state converges. When an attractor takes on a complex fractal form it becomes a “strange attractor”. The strangeness of attractors does not however mean that they are not well understood – on the contrary, many different classes of attractor have been identified and studied mathematically.
A somewhat dry and technical description of attractors is given in wikipedia:
http://en.wikipedia.org/wiki/Attractor
In the context of our analogy of the ENSO as a nonlinear oscillator, a particularly interesting type of nonlinear attractor is the Lorenz attractor. Figure 7 below shows the time plot of phase space displacement of a Roessler and a Lorenz attractor. In figure 8, the phase space trajectory plot is given for the two corresponding attractors. The Lorenz attractor displays phase space “tearing” into two separate domains, while the Roessler attractor is characterised by phase space folding. The bilaterally torn attractor is sometimes referred to as the Lorenz “butterfly”.
(The chaos butterfly is rehabilitated! Providing one understands that one is referring to the Lorenz butterfly attractor, not the spurious “butterfly wing” effect.)
Of course, the Lorenz and Roessler attractors are simple classic types of nonlinear attractor. The Lorenz attractor exhibits oscillation of a fractal nature on more than one scale: the fine scale oscillation itself oscillates over a longer time period between higher and lower values of the phase space parameter on the y axis. More complex versions of both attractors exist – and many further types also. Figure 9 shows two examples, a Roessler attractor which shows tearing like a Lorenz attractor, and a folded chaotic BZ reactor attractor which kind of looks like a cross between a Roessler and a Lorenz.
Figure 7. The time plot of phase space displacement of a Roessler and a Lorenz attractor.
Figure 8A. The phase space trajectory plot of the Roessler attractor (folding)
Figure 8B the Lorenz attractor (tearing).
Figure 9A. A half inverted torn chaos solution to a Roessler attractor
Figure 9B. a folded chaotic BZ attractor.
A note on reading the literature on chaos and non-equilibrium pattern dynamics. Only pay minimal attention to the text and even less to the maths. Just look at the pictures. It is the spatiotemporal multidimensional patterns that are the unifying and compelling feature, and it is pattern analogies between disparate systems which reveal the unifying pattern processes at work. In the above figures I have not defined the parameters in the x and y axis – they don’t really matter.
The Lorenz attractor and the ENSO
Does the time plot of the Lorenz attractor in figure 7 (b), with its higher and lower frequency components, remind you of anything? The wavetrain appears to spend alternating periods oscillating in a higher and a lower region of the y axis. Here again our discussion turns to the definitive work by Bob Tisdale on the ENSO. Bob’s recent posting on WUWT (reposted from his own blogspot) entitled “Integrating ENSO: multidecadal changes in sea surface temperature” had the subtitle “Do multidecadal changes in the strength and frequency of el Nino and La Nina events cause global sea surface temperature anomalies to rise and fall over multidecadal periods?”. A link to this article (pdf) is:
This tour-de-force of the ENSO and its controlling influence on global SSTs demonstrated how, over the past century, there have been alternating periods of about three decades duration during which the el Nino and La Nina systems are reciprocally dominant. Two plots from Bob’s article are shown below in figure 10.
Figure 10a shows the ENSO oscillations exhibiting alternating periods of higher and lower elevation on the y axis (Nino SST 3.4 anomalies), although with far more noise than the tidier level-switching oscillation of the Lorenz attractor. The Nino 3.4 plot thus resembles a very untidy or chaotic Lorenz attractor time plot of the type shown in figure 7b. The alternating periods dominated by the el Nino (1910-1944, 1976-2009) and by La Nina (1945-1975) represent the two wings of the Lorenz butterfly. Thus this period-alternation between a generally warming el Nino dominated phase and a cooling La Nina dominated phase, fits in with the description of the ENSO system as a nonlinear oscillator, of the BZ reaction type, and characterised by a torn attractor of the Lorenz – or possibly modified torn Roessler – variety. It is also known as the Pacific decadal oscillation, or PDO.
Figure 10A. The Nino 3.4 SST anomalies from 1910 to the present, averaged into roughly 30 year periods by Bob Tisdale.
Figure 10B. Global SST compared to period-averaged Nino 3.4 anomaly. Both from “Multidecadal changes in sea surface temperature” by Bob Tisdale.
Is the PDO the Lorenz butterfly attractor of the ENSO?
Closely linked to the ENSO is the PDO – indeed Bob Tisdale asserts that the PDO is an epiphenomenon of the ENSO. His recent posting on multidecadal variation in SSTs elucidates this relationship, showing the PDO to essentially comprise alternating periods of el Nino and La Nina dominance. On the basis of the proposal presented here that the ENSO is a nonlinear oscillator, we can suggest further that the alternating “PDO” phases are the paired “butterfly wings” of a Lorenz attractor characterising the ENSO.
Figure 11. Could the Pacific Decadal Oscillation (PDO) represent the operation of a Lorenz “butterfly” torn attractor on the ENSO?
Periodic forcing of the ENSO nonlinear oscillator
At this point, some of you may be saying “hold on a moment – I’m not convinced by this BZ reaction analogy. Most of the BZ reactions (e.g. shown on youtube) show spiral and fringe patterns that are not at all persuasive analogies to the shifting regional patterns of ocean surface temperatures”. You would have a point. However it is necessary at this stage to introduce another class of nonlinear oscillators – the periodically forced nonlinear oscillator. The BZ reactions that were referred to above, and shown in the attached movies, are all unforced examples. These unforced BZ reactions oscillate and their own natural frequency, and are indeed often characterised by such radiating spiral and fringe patterns. But the spatiotemporal patterns can change profoundly when the BZ reaction is subject to periodic forcing. Figure 12, provided by Matthias Bertram’s PhD thesis, shows a series of spatial patterns from a BZ reaction which is catalysed by a light sensitive metal catalyst, then subject to various regimes of periodic forcing by light pulses. The first case (a) is unforced and looks like many of the youtube BZ reaction animations. However a wide range of different patterns is observed (b-g) when different periodic forcings are applied.
Figure 12. A BZ reaction with a light-sensitive metal catalyst, showing spatially extended nonlinear oscillator patterns. Case (a) is unforced; all the remaining are subject to different amplitudes and frequencies of light pulse periodic forcing. Taken from the PhD thesis of Matthias Bertram.
Anna Lin et al. (2004) looked further at the role of periodic forcing in the light-sensitive BZ reaction. The BZ system in the absence of forcing oscillates at its natural frequency. When forcing was applied by periodic light flashes, they found a difference in the kind of response depending on whether the forcing was strong or weak. To quote the authors:
“The entrainment to the forcing can take place even when the oscillator is detuned from an exact resonance [refs]. In this case, a periodic force with a frequency f(f) shifts the oscillator from its natural frequency, f(0), to a new frequency, f(r), such that f(f) / f(r) is a rational number m:n. When the forcing amplitude is too weak this frequency adjustment or locking does not occur; the ratio f(f) / f(r) is irrational and the oscillations are quasi-periodic. In dissipative systems frequency locking is the major signature of resonant response.”
So with strong forcing, “frequency locking” occurs and there is a clear relationship between the frequencies of the periodic forcing and of the BZ systems responsive forced oscillation. However when the forcing is weak, the reaction’s responsive frequency shows a much more complex relation to the forcing frequency, and its resultant oscillations can be described as “quasi-periodic”.
Returning to the ENSO, how could the equatorial Pacific nonlinear oscillator be periodically forced? Periodic forcing of the oceans and of climate in general is a frequent topic of posts at WUWT. There are many such known and potential sources of periodic forcing over a wide range of time-scales. The Milankovich orbital related cycles operate over periods of 105 years to decades and centuries (in the case of resonant harmonics of orbital oscillations). Then there is oscillation in solar output from the 11 year sunspot cycles to the longer periodicities such as the Gleissburg cycles. One persuasive source of PDO forcing is solar-barycentric, as outlined by Sidorenko et al. (2010), the movement of the solar system barycenter around the sub-Jupiter point (center of gravity of a solar system containing only the sun and Jupiter):
This periodic asymmetry in the solar orbit has shown a wavelength and inflection points similar to the PDO cycle in the last two centuries.
Turning to the oceans and the thermo-haline circulation of deep ocean currents, it is well known that the strength of cold water downwelling at the key sites such as the Norwegian sea is subject to significant variation – indeed after a period of a few decades of relative weakness, Norwegian sea downwelling has recently strengthened (Nature, 29 November 2008, doi:10.1038/news.2008.1262 – link in references). Once could go on. There is no shortage of potential sources of periodic forcing of the atmosphere-ocean system, either of the equatorial Pacific or indeed globally.
If the PDO represents the operation of the ENSO Lorenz attractor, then the periodicity of the PDO should tell us if the system is unforced or forced and frequency locked – in which cases it would have regular periodicity, or if it is weakly periodically forced, in which case an irregular wavelength might be expected. Jacoby et al. 2004 traced the PDO oscillations over the last 400 years, using oak tree rings on the Russian Kurille Islands:
http://www.wsl.ch/info/mitarbeitende//frank/publications_EN/Jacoby_etal_PPP_2004.pdf
A PDO wavetrain is clearly discernible but the wavelength varies from 30-60 years. The PDO thus appears to be a real multidecadal oscillation but it is not frequency locked, showing frequency variation. This points to the PDO arising from a weakly periodically forced ENSO. Mantua et al. (2002) also review data on palaeo-records of the PDO, concluding that its wavelength varies from 50-70 years. They concluded that the causes of the PDO are unknown.
http://www.atmos.washington.edu/~mantua/REPORTS/PDO/JO%20Pacific%20Decadal%20Oscillation%20rev.pdf
Thus the PDO seems to be almost but not quite regular – apparently aiming for a 60 year cycle but fluctuating from it. This could be evidence of periodic forcing of the ENSO system that close to the boundary between “weak” and “strong” forcing. Of course, these suggestions about sources of periodic forcing of the ENSO and PDO are speculative. If, as set out by Lin et al. (2004), in the case of a weak periodic forcing of a nonlinear oscillator such as the BZ reactor, the relation between a putative forcing frequency f(f) and the responsive frequency f(r) is irrational, this complicates the search for conclusive proof of such a link. However the PDO’s apparently limited departure from 60 year periodicity might suggest a forcing near the boundary of strong and weak, and therefore an intermittent frequency locking.
Conclusions
- Owing to the far-from-equilibrium state of the earth’s atmosphere and ocean climate system, the a priori case for the operation of non-equilibrium/nonlinear pattern dynamics is strong.
- The Belousov-Zhabotinsky reaction-diffusion system in a thin film is a compelling model of a nonlinear oscillation arising spontaneously in a far-from-equilibrium spatially-extended system, with apparent similarities to the ENSO sea surface temperature spatio-temporal oscillation in the equatorial Pacific.
- The apparent positive feedbacks (spatio-temporally limited) associated with the initiation of both el Nino and La Nina systems, linking Peruvian coast deep upwelling with equatorial trade winds, qualify the equatorial Pacific as an excitable medium, a key pre-requisite of an oscillating reaction-diffusion system such as the BZ reaction. The open and dissipative nature of the climate and ocean meet another such requirement.
- Of the class of known attractors of nonlinear oscillatory systems, the Lorenz and possibly Roessler attractors bear similarities to the attractor likely responsible for the alternating phases of La Nina and el Nino dominance that characterise the ENSO and constitute the PDO.
- It is possible that the ENSO / PDO system might be periodically forced; the significant but limited variation of the time-period of the PDO evidenced in the palaeo-record of the last few centuries suggests a forcing strength close to the threshold required for frequency locking.
- If the ENSO and PDO can be characterised as a nonlinear oscillator with a Lorenz type attractor, one might speculatively extend the analogy more widely to the earth’s climate as a whole, and such features as the alternation between glacial and interglacial states (during a glacial epoch such as the present one).
- It is hoped that scientists and mathematicians with expertise in non-equilibrium pattern systems, such as reaction-diffusion oscillatory systems, might bring their analytical techniques to bear on the study of the earth’s atmosphere, oceans and climate. In this way the hypotheses presented here could be confirmed or refuted, and perhaps the nature and identity of the significant drivers of climate could be found.
Postscript
What implications does this paper have for anthropogenic global warming (AGW), if any? It was not written primarily to address the AGW issue. CO2 is not mentioned. However there are some indirect implications. The finding that Bob Tisdale’s observation of alternating periods of el Nino and La Nina dominance – in other words the PDO – is well described by a nonlinear oscillator driven by a torn Lorenz (or Lorenz-Roessler) attractor, give Bob’s conclusions greater “real-world” plausibility. (Nonlinear attractors are a common feature of the real world.) It is also a riposte to those who argue against the reality of the PDO or AMO (Pacific decadal oscillation, Atlantic multidecadal oscillation) on the grounds that a credible mechanism does not exist. It does!
One important mathematical aspect of a nonlinear oscillator with an attractor is its “Lyapunov stability”. Alexander Lyapunov, from Yaroslavl, Russia, established a century ago the maths of stability of both linear and nonlinear systems, such that a nonlinear system such as an oscillator is characterised by a “Lyapunov exponent”. The full works on this are given here:
http://cobweb.ecn.purdue.edu/~zak/ECE_675/Lyapunov_tutorial.pdf
The maths here is all way over my head – I’m a “mere” biologist! Essentially the Lyapunov exponent assesses how strong or “attractive” the attractor is – i.e. how strong a perturbation of the system is needed to move it – unwillingly – away from its attractor. More expert mathematic analysis of the ENSO as nonlinear oscillator would include derivation of the Lyapunov exponents. This would tell us the stability of the system and its resistance to change due to any outside influences.
The global circulation models (GCMs) are essentially linear. That presumably is why they generally fail to reproduce the ENSO and PDO. (If they show any nonlinear behaviour it is probably more by accident than design.) It remains to be seen whether climate and ocean modelling – of the ENSO or of larger parts of the global climate, which used a nonlinear oscillator as a starting point, would be more effective.
Post-postscript
Mathematical / computer modelling of a nonlinear oscillator such as the BZ reaction is not too difficult (for people into that kind of thing) and well established. The “Brusselator” – so named for being invented at the Free University of Brussels (VUB) is a good example:
http://en.wikipedia.org/wiki/Brusselator
References
Controlling turbulence and pattern formation in chemical reactions. Matthias Bertram, PhD thesis, Berlin, 2002. https://docs.google.com/leaf?id=0B9p_cojT-pflY2Y2MmZmMWQtOWQ0Mi00MzJkLTkyYmQtMWQ5Y2ExOTQ3ZDdm&hl=en_GB
G. Nicolis and I. Prigogine, Self-organization in Nonequilibrium Systems (Wiley, New York, 1977).
E. Schroedinger, What is Life ? (Cambridge Univ. Press, 1944).
P. Glandsdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations (Wiley, New York, 1971).
The ENSO-Related Variations In Kuroshio-Oyashio Extension (KOE) SST Anomalies And Their Impact On Northern Hemisphere Temperatures. Bo Tisdale, from the web page: http://bobtisdale.blogspot.com/2010/12/enso-related-variations-in-kuroshio.html
Integrating ENSO: Mutidecadal variation in sea surface temperature. Bob Tisdale.
Pdf of this article: https://docs.google.com/leaf?id=0B9p_cojT-pflYjYyMTdkYzItMDMwOS00MjFjLWJmYTAtMzdjYjM1YjhhMmFj&hl=en_GB
Resonance tongues and patterns in periodically forced reaction-diffusion systems. Anna Lin et al., DOI: 10.1103/PhysRevE.69.066217, Cite as: arXiv:nlin/0401031v1 [nlin.PS].
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Indeed, indeed! One tires of reading qvetches about how small are the TSI changes, how noisy the GCR/temperature link, etc. But there’s something happening here …
One of Joe’s more interesting segments evah! Note also that he sez this La Nina’s second year is likely to be howlin’ cold — and then the subsequent El Ninos get even colder (centered on SE USA).
Ted Turner will be wimpering in Atlanta. Judith won’t be too happy, either!
This is all quite superb thinking and pattern-spotting.
As for a label, how about the NEPS acronym: “Non-Equilibrium/nonlinear Pattern Systems”?
Puts me in mind of some of the “Constructal Law” thinking. Even the IEEE is getting in on it:
http://www.constructal.org/img/ieee_comp_and_pack_cover_fig8.png
Of course writing this :
A note on reading the literature on chaos and non-equilibrium pattern dynamics. Only pay minimal attention to the text and even less to the maths. Just look at the pictures.
in a post dedicated in non linear dynamics is not a good beginning and is bound to lead to misconceptions .
The pictures of a tenis ball and of the sun look strikingly similar yet there is no nuclear fusion going on in the tenis ball 🙂
In summary I would say :
1) Yes , ENSO and all pseudo periodical climatic structures are chaotic
2) No , this has nothing to do at all with the Lorenz or Rossler attractor
To develop the arguments it is necessary to use and understand some mathematical concepts . Sorry but it is not more possible to understand non linear dynamics without maths than to understand quantum mechanics without maths .
1)
To the first argument , there is this paper (chosen among dozens of papers dealing with the same matter) : http://www.environment.harvard.edu/docs/faculty_pubs/tziperman_chaos.pdf
Basically the authors conclude : An analysis of a time series from an ENSO prediction model is consistent with the low-order chaos mechanism
The term to be stressed is the low-order and deserves some development .
The temporal chaos theory , non linear dynamics , low-order chaos (all these terms are synonymous) studies the time evolution of a system in a low-dimensional phase space .
The phase space is a space where the system states are represented by a point whose coordinates are the dynamical variables . So it is an abstract structure which has nothing to do with the usual space we all know .
For instance the Lorenz system has 3 independent variables so its phase space has the dimension of 3 (it is just a coincidence that it is the same dimension like the usual space) .
The phase space of ENSO (see linked paper) has a dimension smaller than 8 what means that the number of independent variables driving its dynamics is smaller than 8 hence the term “low dimensional chaos” even if its dimension is much greater than the Lorenz system .
Caveat : the chaos theory allows to estimate how long a time series must be to get a robust estimate of the dimension of the phase space as well as of the attractors (if they exist) .
The 100 years of data for ENSO is shorter than the necessary length so the estimation is not very robust . It is a strong suggestion but not a certainty . We will need a couple of centuries more of data to get a robust answer .
2)
Why ENSO , even if it is chaotic what it is with a high probability , has nothing to do with Lorenz or any other popular attractor ?
Well an attractor is a subspace of the phase space . So it is something that you will never see because it doesn’t happen in the ordinary space .
The dimension of the Lorenz attractor is a number between 1 and 2 .
So it is more than a “curve” but less than a “surface” .
That’s why it is called fractal or strange .
Moreover all these attractors strange or not , happen in a finite dimensional phase space as already explained in 1) above .
But what is the phase space of any real oceanic oscillation ?
Well as this oscillation is an evolution of a spatial structure in time , or in other words an evolution of the field f(x,y,z,t) , the phase space is infinite dimensional .
It is easy to understand intuitively why – there is an infinity of points (x,y,z) and each of these points oscillates .
The spatial structure known as ENSO is the spatial result of an interaction of an infinity of oscillators (one in each point) .
This is called spatio-temporal chaos and is completely different from the temporal (Lorenzian) chaos .
More specifically there are no attractors in spatio-temporal chaos (how could one define or make a “picture” of an infinite dimensional structure ?) . Also almost none of the results of the temporal chaos theory transport to the spatio-temporal domain .
That’s why the Lorenz attractor , a child of finite dimensional dynamics dependent on time only has nothing to do with ENSO , a child of infinite dimensional spatio-temporal chaos .
Those who have followed the arguments as far as here will certainly be able now to point out an apparent contradiction between what I wrote in 1 and what I wrote in 2 .
Indeed in 1 I have written that ENSO was a low dimensional temporal chaos while in 2 I have written that ENSO was an infinite dimensional spatio-temporal chaos which had nothing to do with temporal chaos .
As the explanation of why this is not really a contradiction allows a rather deep insight in the spatio-temporal chaos , I will do it right away .
We have seen that the main difficulty of spatio-temporal chaos is its infinite dimensionality due to the dependence on spatial coordinates .
The ordinary temporal chaos theory can only deal with finite dimensional phase spaces .
So clearly one can try to get a finite phase space dimension by discretizing the continuous space .
By cutting up the space in a finite number of grids and nodes , the number of oscillators becomes now finite (instead of an infinite number of points we get a finite number of nodes) and hence the phase space is now finite too .
The good old temporal chaos theory can now be applied on this new system of N coupled oscillators .
And what one hopes is that this approximation will give results that will not be too far from the results that would be given by a true spatio-temporal chaos theory which doesn’t exist yet and won’t probably exist for decades .
This paper : http://amath.colorado.edu/faculty/juanga/Papers/PhysRevLett_96_254103.pdf shows how it works in reality .
So now you also understand what the authors of the paper in 1 really did – they discretized the space .
Admittedly they discretized it rather dramatically because they reduced this huge thousands km by thousands km area to a single number , namely the average sea surface temperature which then depends only on time and no more on space .
So if one admits that this single number still contains traces of the real infinite dimensional spatio-temporal chaos , one may look for them with classical temporal chaos tools .
It is similar to an even more dramatical discretization that all climate “scientists” do when they reduce the area of the whole Earth to a single number – average global temperature .
However if somebody said that this extreme discretization is a rather brutal way to rape the subtle spatially correlated data and that it could potentially destroy the important signals , I would have to agree .
Thanks for all the positive and helpful feedback and criticism. Makes the enterprise seem worthwhile. No time now to address all the issues, just a random sample:
Mike Haseler says:
January 25, 2011 at 3:24 am
Your point about 1/f noise and its fractal nature is useful. One of the characteristics of a nonequilibrium pattern system is what is called “log-log” power law distribution – which amounts to what you explained – it looks similar in the large scale as the small scale. The book “Deep Simplicity” by John Gribben, Random House NY, explains this well.
Murray Duffin says:
January 25, 2011 at 10:29 am
Thanks for your suggestions about possible forcing periodicities.
Bob Tisdale says:
January 25, 2011 at 3:54 am
The “sloshing” of warm west Pacific water eastwards in an el Nino is presumably linked to the weakened trade winds – so my overall approximate scheme survives? Thanks for the clarification about PDO and PDV, I had seen the terms but was not clear what PDV was.
PaulM says:
January 25, 2011 at 7:28 am
Yes negative feedback is an important aspect of nonlinear systems. It goes alternatively by the names of dissipation or friction or damping. This “friction” indeed generated emergent complex pattern, while a dominance of positive only feedback tends to establish monotonous regular oscillation. So negative and positive feedback are in a sort of balance. Someone correctly mentioned the heartbeat as an example of a nonlinear oscillator. This is one dominated by positive feedbacks, thus – fortunately for most of us – it is regular (I wouldnt like to experience an ECG like the ENSO time plot!) Matthias Bertram in his thesis goes on to study the Pt-catalysed oxidation of CO as an example of the role of feedback – if you download his thesis from the link provided (8MB) and read this section it should be helpful. Thanks.
wayne says:
January 25, 2011 at 2:34 pm
Thanks for the positive comments. I tried to explain in the intro that it was not chaos per se but the border region before chaos where the nonlinear complex pattern formation emerges.
TomVonk says:
January 26, 2011 at 4:30 am
This is exactly the expert input I was hoping to attract – even if critical. OK the ENSO attractor is multidimensional, fractal and thus strange. However presumably one can choose how many dimensions to look at and study. If one chooses just 2-3 then maybe a Lorenz attractor can be an approximation or starting point.
Thanks again for all responses, I’ll return soon when I have time,
Phil Salmon
(“phlogiston”)
This makes iceages and other extreme events more understandable. So much change with so little drivers. I’ve always had trouble understanding how such large changes can occure. We need more discussion down this path.
Thank you for this fascinating and important article. It really does seem clear that the current Climate is determined by which attractor the weather is orbiting right now, and the future Climate by which other attractor the weather will shift to next. It also seems really clear there are two groups of attractors – Glacial and Inter-glacial. It also seems clear that Climate models developed without this insight will not be useful predictors of Climate, and further it is entirely possible that no model will be useful in that way.
TomVonk says:
January 26, 2011 at 4:30 am
“It is similar to an even more dramatical discretization that all climate “scientists” do when they reduce the area of the whole Earth to a single number – average global temperature .
However if somebody said that this extreme discretization is a rather brutal way to rape the subtle spatially correlated data and that it could potentially destroy the important signals , I would have to agree .”
It is not only brutal, but senseless. In studying the cross-regional coherence of the multidecadal signal components of temperature series, I find not only diminution with distance, but occasional phase-reversals as well as rather abrupt curtailments. This is consistent with the widely-noted inconsistent global effects of ENSO. GST is indeed an unscientific way of characterizing the chaotic thermalization and flux of solar energy through the climate system.
Fascinating post. Thank you for your time and effort in putting it together. Well done!
Anthony –
I was going to post this as a comment on your excellent ENSO piece, but the darned thing grew too big and I thought I’d send it here instead. Apologies if this is outside what you intended your email for.
To begin:
Anthony –
To be honest, I would have thought all this is pretty obvious. I had not heard of B-Z reactions, but it is clear that this oscillation from a one state to another state occurs. Thanks for giving me a name for it.
I’ve been asking here for quite a while if anyone could point me to ANY explanation of what causes the ENSO oscillations. Every online source just talks about ENSO as if it is a primary forcing/causative agent, and none of them talk about what causes IT.
Your and Bob’s description here is essentially what I was trying to put into words, without me knowing of the existence of the B-Z reaction. I could see in Bob’s graphics of the ENSO exactly what you described. But you two are able put it all in scientific terms and with a higher level of understanding.
To me talking about “averages” in a dynamic system is massively oversimplifying the system, dumbing it all down. Converging all that is going on at one moment into one temperature data point is ludicrous for understanding because “the devil is in the details” – i.e., the more local events/effects/values. I have always seen Standard Deviation as the two averages that together can be combined to make up an overall average, but the real goodies are in the high and low and what oscillations are going on around those two values. That seems to be essentially what the B-Z reaction deals with – the oscillations around two states, the wings of the Lorenz “attractor” – the higher SD and the lower SD. For some reason this seems obvious.
And in my questioning of the ENSO’s cause, my mind – like yours and Bob’s – went to upwellings of deep cold waters and some mechanism in the system that allowed the upwellings to occur in an oscillatory manner. And, conversely, to “put a lid on it.” The heat – or lack of it – for El Niño and La Niña – had to come from somewhere, and the only source of heat/cold was down below. Thus it had to either be a separate heat/cold source that itself was oscillating, or it had to be a submerging of heat/cold that was oscillating. (I toyed with the idea of heat input from sea vents, but then I would have to explain why the output of those oscillated, and there just isn’t anything known about them at the present. But they are adding heat to the system, and the amount is not known at present. I presently doubt this amount is significant, but will keep an eye on it.)
I looked at ENSO as a resonance – that the system was going in and out of resonance. Audible resonance and mechanical/vibratory resonance – if too much energy is being put into the system – will overshoot and drop out of resonance, dropping the state to chaos and back the starting point again (more or less). I do not know the names for any of it, but it happens. When the resonance occurs each time, it is as if the system organizes momentarily, but the overloading of input energy means it can’t maintain that state for long.
I am very pleased that you and Bob have gone in this direction (as if you cared specifically for my vote!…LOL). I believe there is a paper in this for you two.
I am also very happy to see you mention the PDO in this, too, because IMHO it is a longer-period version of the same kind of system. I would suggest that rather than being the wings of the Lorenz “butterfly” attractor, that it might stand alone. I suggest this because it is possible that at the latitudes of the PDO the energy input is also present – but of a lower magnitude. This would suggest longer periods of oscillation, which is what we see. Also, the Mid-Latitude Cells might be the direct output mechanism for the PDO as opposed to the Hadley Cells and the Intertropical Convergence Zone with ENSO. I believe the two B-Z systems can be somewhat independent.
…As a last mention, I would suggest a look at the very similar effects of “cymatics” which is the study of sound vibrations and the patterns they form on Chladni plates – thin vibrating surfaces. They look VERY MUCH like Bertram’s patterns. And I believe the two are related. Slight variations in sound frequencies or energy input make quite a change in the patterns, similar to the forced samples of Bertram. Light and sound are all EM frequencies, after all. Thus far cymatics has not looked at the effects in a dynamic system, however. That would open up all kinds of new avenues of exploration. (There are also implications for atomic theory and possibly even cosmological theory, IMHO.) My understanding is that it all has to do with local energy states and nodes. It might be worth noting that the patterns in cymatics tend to “stay put,” with only brief transition states. The “system” wants, it seems, to be at some more organized state. This might well be an important feature – that chaos isn’t necessarily the norm. Weather – the small-scale version of climate – seems to always actually be more or less organized, with highs and lows and trade winds, etc. A starting point: http://tiny.cc/2f0nw.
Looking at averages, like climatologists do, is missing the dynamism in the system, trying to simplify it – apparently because that is all the level they can deal with at our present stage of knowledge and present capacity to understand complexity. I can’t blame them for wanting to, to see what it tells them – but then they should acknowledge that it is probably oversimplifying things and that we shouldn’t read too much into the “averages.” Averaging water things down. That should be obvious. But it is the exact details of what is going on that allows us to move forward in our understanding. Averages can’t do that.
Sorry for the length of this comment…
TomVonk says:
January 26, 2011 at 4:30 am
Of course writing this :
“A note on reading the literature on chaos and non-equilibrium pattern dynamics. Only pay minimal attention to the text and even less to the maths. Just look at the pictures.”
in a post dedicated in non linear dynamics is not a good beginning and is bound to lead to misconceptions .
Yes I was asking for trouble to add that paragraph, probably a mistake. In my case I don’t have much choice – as a biologist most of the maths in papers on nonlinear dynamics is beyond me. But I dont let this stop me reading the papers. I try to qualitatively understand the main features of the systems being described. One of the points of my article was to promote some more engagement between the field of nonlinear “chaotic” system dynamics and mathematics, and climate / oceanographic research. The consensus here and your view also is that the ENSO is indeed some kind of nonlinear oscillator. What is needed is for those knowledgable about the analysis of attractors and nonlinear oscillators to do some work with (e.g.) the ENSO and characterise it accurately, for instance as I suggested in the paper, finding the Lyapunov exponents.
I do feel there is in scientific research too much of “you in your small corner and I in mine”. Getting interdisciplinary engagement means people trying to get past technical jargon and maths and, even if they understand less than 50% of a paper in an alien discipline, still try to get something from it.
In summary I would say :
1) Yes , ENSO and all pseudo periodical climatic structures are chaotic
2) No , this has nothing to do at all with the Lorenz or Rossler attractor
To develop the arguments it is necessary to use and understand some mathematical concepts . Sorry but it is not more possible to understand non linear dynamics without maths than to understand quantum mechanics without maths .
As shown by Bob Tisdale (referenced in the article) the ENSO is characterised by alternating periods predominantly favouring the warming el Nino evens and cooling La Nina events. Such segregation into two phase space domains looks like a torn Lorenz type attractor. Note that there are many variations of each attractor type such as the Lorenz and Roessler – I included in the article a Roessler attractor that was torn and looked very much like a Lorenz. As for dimensionality, you could look at as many variables as you like, but the normal practice in the case of ENSO (see Bob Tisdale’s postings) is just two dimensions, time, and the sea surface temperature for a defined region of the equatorial Pacific (e.g. Nino 3.4). Looking just at this would allow someting like a Lorenz or Roessler to operate. But I defer to your greater knowledge about the mathematical nature of attractors.
To the first argument , there is this paper (chosen among dozens of papers dealing with the same matter) :
http://www.environment.harvard.edu/docs/faculty_pubs/tziperman_chaos.pdf
Basically the authors conclude : An analysis of a time series from an ENSO prediction model is consistent with the low-order chaos mechanism. The term to be stressed is the low-order and deserves some development. The temporal chaos theory , non linear dynamics , low-order chaos (all these terms are synonymous) studies the time evolution of a system in a low-dimensional phase space .
If so many authors have studied the chaos structure of ENSO and found it to be a “low order chaos mechanism”, then why do Mantua et al. (2002) who review data on palaeo-records of the PDO, (finding that its wavelength varies from 50-70 years) conclude that the causes of the PDO are unknown?
http://www.atmos.washington.edu/~mantua/REPORTS/PDO/JO%20Pacific%20Decadal%20Oscillation%20rev.pdf
This is an example of the disconnect between disciplines that I was trying to address. The field of nonlinear chaos knows about nonlinear oscillators, and has studied ENSO. Some match-making is needed here.
Why by the way does your whole posting not include the word oscillator? The hypothesis I advanced concerned a specific well characterised system, the nonlinear oscillator operating in a reaction-diffusion system, of which the BZ reaction is the reference example.
Caveat : the chaos theory allows to estimate how long a time series must be to get a robust estimate of the dimension of the phase space as well as of the attractors (if they exist) .
The 100 years of data for ENSO is shorter than the necessary length so the estimation is not very robust . It is a strong suggestion but not a certainty . We will need a couple of centuries more of data to get a robust answer .
Yes indeed, more time is needed with good data to characterise the ENSO, and climate variation in general. There is a paper on “Topological characterisation of spatio-temporal chaos:
http://math.fau.edu/kalies/papers/tcstc.pdf
which shows a relation between a nonlinear dynamic time plot and a corresponding chaos topology map. What would be perhaps appropriate here would be to apply such an approach to take for instance a time plot of ENSO (e.g. Nino 3.4) and convert it into a chaos topology map. This might illucidate the nature of the attractors. (A similar approach could also be taken to something like the Vostok and other ice cores of the last few glacial-interglacial cycles.)
2)
Why ENSO , even if it is chaotic what it is with a high probability , has nothing to do with Lorenz or any other popular attractor ?
Well an attractor is a subspace of the phase space . So it is something that you will never see because it doesn’t happen in the ordinary space .
The dimension of the Lorenz attractor is a number between 1 and 2 .
So it is more than a “curve” but less than a “surface” .
That’s why it is called fractal or strange .
Moreover all these attractors strange or not , happen in a finite dimensional phase space as already explained in 1) above .
But what is the phase space of any real oceanic oscillation ?
Well as this oscillation is an evolution of a spatial structure in time , or in other words an evolution of the field f(x,y,z,t) , the phase space is infinite dimensional .
It is easy to understand intuitively why – there is an infinity of points (x,y,z) and each of these points oscillates .
The spatial structure known as ENSO is the spatial result of an interaction of an infinity of oscillators (one in each point) .
This is called spatio-temporal chaos and is completely different from the temporal (Lorenzian) chaos .
More specifically there are no attractors in spatio-temporal chaos (how could one define or make a “picture” of an infinite dimensional structure ?) . Also almost none of the results of the temporal chaos theory transport to the spatio-temporal domain .
That’s why the Lorenz attractor , a child of finite dimensional dynamics dependent on time only has nothing to do with ENSO , a child of infinite dimensional spatio-temporal chaos .
I made the point in the introduction to the paper that the term “chaos” is an incorrect generalisation of the systems we are looking at, i.e. the BZ reaction and ENSO. Pure chaos is turbulent and not rich in emergent pattern. However it is the transient region where Hopf bifurcation begins where emergent pattern formation occurs. We are talking here about spontaneous pattern formation in a non-equilibrium dissipative sysem. This is not chaos.
It makes no sense to consider every molecule of water in the equatorial Pacific as an independent player and with a dimension to itself in the phase space. Sea water is liquid, not gaseous. Even in the atmosphere with the trade winds that are integral to the ENSO system, air atoms and molecules are influenced by their neighbors. We talk about ocean currents and atmospheric winds, not just about independent water and air molecules.
ENSO is not turbulence. Again turbulence is chaos and we are not interested in chaos here. ENSO has a repeating spatiotemporal structure (call it discretised if you like), the large tongue of alternatively cold or warm water at the eastern equatorial Pacific. It resembles the alternating and repeating 2-phase patterns of the BZ reactor, but not chaotic turbulence. It is a function of other discretised entities, Peruvian coast upwelling (as part of the oceanic thermo-haline circulation – THC – system of deep currents) and the equatorial trade winds – both entities comprising more than one molecule. Of course we deal with discretised entities because that is how they behave in the real world.
Once we recognise discretised elements operating in the ocean and atmosphere, such as periodic upwelling in a certain restricted region and an alternating wind patter in a restricted region, then this reduces the number of dimensions of our phase space to a much more manageable number. I disagree with your conclusion that the ENSO is “a child of infinite dimensional spatio-temporal chaos”.
I do agree that discretising the sea surface temperature – or troposphere temperature – on a much larger scale and over the whole world is a “brutal” generalisation that misses most or all of the important detail and the operation of independent systems. But we can recognise particular oceanic or atmospheric entities operating approximately as discreet entities.
@phlogiston:
Modeling/analyzing this doesn’t need to include the entire globe, IMHO.
I don’t know enough about the maths to contribute anything, but in terms of concept, I can see quite a bit of all of this.
Most of the time I disagree that ocean currents are created by winds, which is what I keep reading is what the oceanographers say. But if there is one region where I might agree, it would be there where ENSO happens, along the Equator in the Pacific Ocean, where the N & S Hadley cells might push the water toward the Equator before the winds go vertical in the Intertropical Convergence Zone. That would tend to recycle heat back to the Equator, in both the atmosphere and hydrosphere.
In terms of ENSO, this is a starting point for the weather systems in both the northern and southern hemispheres, iwhich to my mind simplifies the problem to be analyzed. It seems enlarging the analysis to the entire globe wouldn’t add anything to the study of it. I am a mechanical designer. Looked at as a driveR-vs-driveN system, I would only look at the driveN part of the system only to see what its needs are on the output end. I wouldn’t look even beyond the Pacific Mid-Latitude cells if I was analyzing this. It is not necessary to include anything beyond that. The Hadley cells-cum-IVZ is the entire system except for bleed-through to the Mid-Latitudes cell. The mid-latitudes are not demanding the heat energy output, although they accept it gladly, and take that heat energy off the hands of the Hadley cells in heat transfer.
So it seems that the system to be looked at is a lot less than the entire globe. ENSO is about as independent as you can get in looking at global climate systems. It is one end of the chain of heat energy events, and that is far simpler than if it were in the middle of the chain of events.
Now, the PDO is a different story in that respect. But in studying ENSO’s special case a lot should be able to be learned that would then apply to the PDO in its middle-of-the-chain position. And that position should be part of the reason why its oscillating frequency is so much lower, because of greater dissipation/blending of the heat being carried poleward, acting like a shock absorber in the system. The incoming heat from the Sun being less dense should be the main factor: lower solar W/m^2 would seem to mean a slower cycle time. Neither of the two factors would seem to contribute to quicker cycling, IMHO.
phlogiston says:
January 27, 2011 at 2:01 am
It is understandable that, as a biologist, you would seek holistic, non-mathematical explanations of physical phenomena such as ENSO and be satisfied with qualitative similarities. But physical science is much more demanding than that. It requires a detailed specification of the sources of energy and a rigorous determination of the effects produced thereby. That is what is meant by the term “dynamical mechanism.” Neither Tom Vonk, who looks at physics from the theoretical quantum-level upward, or a geophysicist like myself taking a meso- to global scale view of ENSO, finds your speculation credible. It is not a matter of compartamentalization of science into narrow disciplines that do not communicate with one another. Some odeas are just plainly inapplicable on both theoretical and empirical grounds. Purely phenomenolgical treatments of data, such as Bob Tisdale does superbly, may provide a good qualitative description, but they do not constitute bona fide physics.
It is at the analytic level that many of the connections you make simply fail. Despite fledgling attempts at modelling the variation of equatorial thermocline depth as low-order chaos, what drives the temperature variations seen in Nino3.4 has never been adequately explained. Nor has any physical connection between ENSO and the PDO ever been established. And THC is by definition the density-driven, as opposed to the enrmously mre energetic wind-driven, component of ocean currents; it does NOT drive upwelling or the mixing in the surface layer. I won’t touch on other miscomprehensions here. Suffice to say that the lapses are as glaring as would be my speculations on unsolved problems of biology.
I have 2 questions for Phil Salmon:
1) Do you believe that strange nonchaotic attractors play no role in terrestrial climate?
2) Do you believe that mathematical “proofs” that terrestrial climate is chaotic are based on tenable assumptions?
sky wrote, “In studying the cross-regional coherence of the multidecadal signal components of temperature series, I find not only diminution with distance, but occasional phase-reversals as well as rather abrupt curtailments.”
And what of interannual timescales?
TomVonk wrote, “More specifically there are no attractors in spatio-temporal chaos (how could one define or make a “picture” of an infinite dimensional structure ?) . “
Was this a slip?
sky says:
January 27, 2011 at 2:08 pm
phlogiston says:
January 27, 2011 at 2:01 am
It is understandable that, as a biologist, you would seek holistic, non-mathematical explanations of physical phenomena such as ENSO and be satisfied with qualitative similarities. But physical science is much more demanding than that. It requires a detailed specification of the sources of energy and a rigorous determination of the effects produced thereby. That is what is meant by the term “dynamical mechanism.” Neither Tom Vonk, who looks at physics from the theoretical quantum-level upward, or a geophysicist like myself taking a meso- to global scale view of ENSO, finds your speculation credible. It is not a matter of compartamentalization of science into narrow disciplines that do not communicate with one another. Some ideas are just plainly inapplicable on both theoretical and empirical grounds. Purely phenomenolgical treatments of data, such as Bob Tisdale does superbly, may provide a good qualitative description, but they do not constitute bona fide physics.
There is more in common between climate (the integrated ocean and atmosphere dynamics) and biology than you think. Its all very well for the “exact” sciences to look down their noses at biology – perhaps a consolation for the fact that there is much more money in the biosciences 🙂 . Both biological and climate processes are characterised by nonequilibrium pattern phenomena, and “bottom up” physical and mathematical analysis of climate that ignores such phenomena will only be exact in the sense of being exactly wrong.
I do not seek “non-mathematical” explanations, the maths of nonlinear dynamics is well established – all I seek to do is identify the processes at work – for instance proposing a common pattern mechanism (the nonlinear oscillator) between the BZ reactor and the ENSO – and leave the detailed maths to others. I have some experience in this field of nonlinear / nonequilibrium dynamics and have published in the peer reviewed literature on the subject.
Nor am I “satisfied with similarities”, Tom’s comparison of the sun to a tennis ball was funny but unfair. I stated in the article that similarity alone was insufficient, and went on to set out the conditions that the ENSO would have to satisfy to allow the operation of a BZ reaction-like nonlinear oscillator: a far-from-equilibrium condition, an open dissipative energy throughput and an excitable or reactive medium. I gave reasons and evidence that those conditions were satisfied. At the outset I made the clear distinction between non-linear pattern formation at the border of chaos, and full-blown chaos itself. Tom overlooked this distinction and tried to categorise ENSO as actual chaos (phase space tending to infinite dimensions) with which I disagreed.
Your advocacy of reductionism and disparaging use of the term “phenomenological” raise the important underlying issue of reductionism versus a heirarchical approach. (You use the word “holistic” tries to liken this to mystical new age like thinking). Reductionism gets you no-where in biology. It may not be stated out loud but practicing biologists realise that they are largely confined to speculative guesses based on phenomenological observations. Some systems such as molecular genetics allow the illusion of exactness but integrated up to the organism level, the predictiveness decays fast. Why else has an exponential increase in resources for molecular genetic research by pharmaceutical companies been accompanied by a drying up of drug candidate pipelines? Because the message has still to get through that reductionistic bottom-up biology does not work.
The problem (one of the problems) with reductionism is that it leads to denial of the real world. Calling an observation “phenomenological” is a form of denial. Something is observed, but “the models dont predict it” so the observation is denied. It is linked to argumentam ad ignorantium. We cant comprehend an observed phenomenon on the basis of our current understanding so we deny the existence of the phenomenon. Like the long resistance to the hypothesis of continental drift. Or the geologist Lyell, author of uniformitarian geology, instinctively rejecting Agassi’s claims to have evidence for former ice ages – on the basis that no mechanism could be envisioned for the climate fluctuating from colder to warmer and back.
In complex systems such as biology and climate it is not difficult to cast doubt on a complex phenomenon, by using obscurantist tactics and demanding complete and tidy, linear mathematical explanations prematurely. CAGW is a deeply reductionist body of theory, its bottom-up mechanism is the Arrhenius CO2 story, and the real world and climate are forced to fit the theory. Observations that contradict it are subject to obscurantist attacks and derided as “phenomenological” in denial of observed fact.
It is at the analytic level that many of the connections you make simply fail. Despite fledgling attempts at modelling the variation of equatorial thermocline depth as low-order chaos, what drives the temperature variations seen in Nino3.4 has never been adequately explained. Nor has any physical connection between ENSO and the PDO ever been established. And THC is by definition the density-driven, as opposed to the enrmously mre energetic wind-driven, component of ocean currents; it does NOT drive upwelling or the mixing in the surface layer. I won’t touch on other miscomprehensions here. Suffice to say that the lapses are as glaring as would be my speculations on unsolved problems of biology.
In the context of the BZ reactor, what do you mean by the analytic level? The dynamics and attractors of this system are well understood mathematically, and in the “Brusselator” they are accurately simulated. There is no need for bottom up simulation of atomic interactions. Just understanding of the nature of the non-equilibrium nonlinear pattern phenomena.
It is a reductionist oversimplification to say “the THC is driven by density only” or that “upwelling has nothing to do with THC”. I demonstrated in the article how upwelling and surface winds can be linked in a positive feedback. If we understand the system as an excitable medium with potential for positive feedbacks then it becomes redundant to talk about what drives what, what comes first the chicken or the egg.
“Nor has any physical connection between ENSO and the PDO ever been established.” A perfect example of reductionist arrogance. Bob Tisdale – the arch phenomenologician – demonstrates clearly the PDO and PDV being linked to the ENSO, but you deny this observation on the basis of an a priori demand for a complete finished physical explanation. This is flawed epistemology. To return to the biomedical / pharmaceutical analogy: some medicines like bisphosphonate drugs for osteoporosis for example are effective for a range of bone diseases and have been the standard treatment for decades. But the biology of how they work is incompletely understood – new aspects of the mechanisms are still being found. A drug industry demanding reductionist explanations for drug efficacy in advance, would have no medicines.
The recognition of non-equilibrium pattern phenomena requires a step back from the complacency of reductionist bottom-up thinking. An integrated understanding of phenomena in complex systems allowing for emergent patterns should not be derided as “holistic” or phenomenological. A non-linear attractor is just as real as an electron or a photon.
Feet2theFire says:
January 27, 2011 at 12:25 pm
@phlogiston:
So it seems that the system to be looked at is a lot less than the entire globe. ENSO is about as independent as you can get in looking at global climate systems. It is one end of the chain of heat energy events, and that is far simpler than if it were in the middle of the chain of events.
Thanks for your engineer’s perspective, this apparent independence of the ENSO makes it a good system to study. For instance as a nonlinear oscillator.
because of greater dissipation/blending of the heat being carried poleward, acting like a shock absorber in the system.
Indeed dissipation (or “friction” or “damping”) are recognised as necessary elements of a nonlinear dynamic system and tend to favour complex patterns and attractors – by contrast a nonlinear oscillator dominated by positive feedbacks without damping becomes regular and monotonic, like the cardiac cycle.
Paul Vaughan says:
January 27, 2011 at 10:21 pm
I have 2 questions for Phil Salmon:
1) Do you believe that strange nonchaotic attractors play no role in terrestrial climate?
This sounds like a trick question. I deliberately focused on the ENSO system as a convenient model. The term chaotic is ambiguous for reasons already discussed. The appearence of ENSO oscillation alternating between el Nino dominated and La Nina dominated multidecadal regimes – documented by Bob Tisdale – led to the proposal of a torn Lorenz-like attractor. (Providing we can convincingly limit the system variables and phase-space dimensions to 2-3 only.)
2) Do you believe that mathematical “proofs” that terrestrial climate is chaotic are based on tenable assumptions?
I’m not really qualified to answer this – sorry for the cop-out! I was hoping that someone like you might take up a challenge like this.
Thinking out loud….
The mention in this post of “butterfly” brought to mind the second thing that Maunder is famous for (after the Maunder Minimum), and that is his butterfly pattern of sunspots. Sunspots do not occur at the solar equator nor close to the poles. And they also seem to be unequally distributed north or south in each sunspot cycle. Their distribution is not a whole lot different than the phase space displacement of a Lorentz attractor as shown in Figure 7 (b).
This suggests a limiting mechanism for sunspots. Among other things, I wonder if anyone has looked at the sequence of sunspots, if they alternate N-S or if they come in N groupings and then S groupings. But they do start out at higher latitudes and then the pattern works its way equator-ward over the life of each cycle.
All this seems to parallel the Lorentz attractor pattern, which may mean sunspots are a non-linear oscillation – perhaps even an oscillation within an oscillating pattern.
All this is based on similarities and speculation as to what is going on. There does not seem to be any explanation for the Maunder butterfly pattern. It is just accepted as the way things are, without understanding of the whys of it.
It would be interesting – if these are both BZR patterns – if ENSO’s manifestation was ON the equator and the Sun’s manifestation is in the mid-latitudes. On the other hand, (still speculating) perhaps the real thing going on on the Sun is actually on the equator but it is causing visual “symptoms” away from the equator. In that regard, perhaps internally the Sun has some equivalents to Hadley cells that transport heat (EM?) energy away from the equator. (Which would imply there is an internal ICZ as well…) I don’t know jack about solar mechanisms but have been reading up on sunspots, so my knowledge is piecemeal.
And BTW, I love the phrase in this post, “…the fine scale oscillation itself oscillates over a longer time period between higher and lower values of the phase space parameter…” This is how I see SDs in general, and in this Lorentz attractor it displays this idea magnificently.
Phil, have you reviewed the literature (which goes back decades) on modeling ENSO as a nonlinear oscillator? (recent example: Warren White at Scripps)
Clarification: I’m not suggesting this will be either a fruitful exercise or a constructive use of time, but there are bases to be covered (& perhaps courtesies to be paid) before suggesting a “new” approach.
I believe you have misunderstood sky’s comments, but you are absolutely correct in asserting that ignorance of scale in the study of spatiotemporal pattern & process is (on ethics alone) a condemnable failure. Disciplines like landscape ecology & advanced physical geography, out of sheer necessity in grappling with pervasively rich & thick complexity, attained enlightenment on extent decades ago, but my impression is that such an awakening has not yet occurred (or at least has not yet matured) in disciplines that have traditionally focused narrowly & rigidly on only grain (which is enough to get the job done when investigating relatively simple phenomena). Fortunately there are some bright mainstream lights like Franklin Schwing, who appear to be slowly but surely developing a handle on the nature of multiscale spatiotemporal variation, so perhaps (at least we can hope) there will be more widespread awareness of the spatiotemporal version of Simpson’s Paradox in the not so distant future.
@phlogistin said Januay 28, 3:01pm:
I got a good chuckle in reading the first of this, because my mind went exactly to uniformitarianism and Lyell and Agassiz – and then that is exactly where you took it. Then I laughed out loud. Uniformitarianism is based on denying phenomena. Its very premise is that “only what we see today could ever have happened in the past.” The operative word there is “see.” In you saying “we cant [sic] comprehend” you are in essence talking about seeing. Blinders exist in all of us – what we chose not to see, we simply can’t comprehend, or even apprehend.
I do like your points to sky about reductionism and phenomenology. Building up from iota is theoretically a good way to approach science, but grasping the context, the framework, the phenomena is of primary importance. Stem cells – what is it that allows them to become different cells? There is some pattern, some resonance, or (at the risk of sounding New Age) some gestalt/holistic construct that must inform the infinitesimals. There is something in flocks of birds and schools of fish that makes them move as if one body. Yes, the parts are important, too, and must be studied – but not to the exclusion of the bigger picture. Yet models must be built from the reductionist POV – the code insists on it. But how those subroutines fit together is the crux of making the models parallel the empirical world.
Sorry for my generalist/interdisciplanary-leaning dogma there, but I had to voice my agreement. Quantification and qualification in science both have their places. It doesn’t seem like any of the non-math/non-hard sciences have 100% understanding that. Modelers for sure don’t. Bottom-up science has nearly infinite paths upward from that bottom, and the chances of them choosing the correct path to an understanding of the whole is not good, to say the least. There has to be an idea of the target and the ability to recognize when the results are balderdash, and the humility to say, “Oops!”
Paul Vaughan says:
January 28, 2011 at 8:52 am
Phil, have you reviewed the literature (which goes back decades) on modeling ENSO as a nonlinear oscillator? (recent example: Warren White at Scripps)
I am certainly not the first to draw the obvious conclusion that climatic systems such as ENSO behave like nonlinear far-from-equilibrium pattern systems. You are right it would have been better to have mentioned some other such studies, e.g. Lockwood 2001:
http://onlinelibrary.wiley.com/doi/10.1002/joc.630/pdf
J. Lockwood, “Abrupt and sudden climatic transitions and fluctuations: a review” (
Int. J. Climatol. 21: 1153–1179, 2001) in which he actually refers to the BZ reaction and draws an analogy to climate systems:
“For example, the Belousov–Zhabotinsky reaction in inorganic chemistry …
Because the movement from one stable state to another, as the distance from equilibrium increases, depends on universal numerical features rather than the actual
mechanisms involved, it is not surprising that some of the curves look similar to climatological time series.”
Another is by A. Mary Selvam 2003, “The Dynamics of Deterministic Chaos in Numerical Weather Prediction Models “:
http://arxiv.org/ftp/physics/papers/0310/0310034.pdf
in which the author suggests the ENSO might be linked to “semi permanent dominant eddies (limit cycles) in the atmosphere boundary layer”, with the 5 year ENSO periodicity representing 40-50 days (the period of atmospheric general circulation oscillation) squared.
I believe you have misunderstood sky’s comments
If so then I apologise.
I agree with your comments about heirarchical levels of organisation, I read some years ago “The Cosmic Blueprint” by Paul Davies which also articulated these arguments nicely.
Its just possible that eventually climate science and the aftermath of AGW could be the trigger for more general acceptance over many scientific disciplines of, as you put it, “multiscale spatiotemporal variation” and a challenge to the dominance of reductionism.
Feet2theFire says:
January 28, 2011 at 8:24 am
Thinking out loud….
What you are suggesting about the space and time pattern of sun spots sounds like a “limit cycle”, a repeating pattern that “shakes down” out of infinite possibilites in a non-equilibrium pattern system.
http://en.wikipedia.org/wiki/Limit-cycle
Your biological analogies are persuasive. It reminds me of a passage in “Deep Simplicity” by John Gribben, where he talks about limit cycles. Apparently a sort of “chaotic network” with chaotic transmission between multiple nodes will converge to cycle between only a small subset of the nodes – the “limit cycle”. In one experimental example, 30,000 light bulbs (virtual or real) are so linked, and the resultant limit cycle was about 250 nodes only. The significance? There are about 30,000 genes in the human body, and the number of cell (tissue) types is 256.
Oh – that brings up post number 100 – you narcissist!!