TITLE:
Dimensionality reduction of chaos by feedbacks and periodic forcing is a source of natural climate change
Phil Salmon
Abstract
The role of chaos in the climate system has been dismissed as high dimensional turbulence and noise, with minimal impact on long-term climate change. However theory and experiment show that chaotic systems can be reduced or “controlled” from high to low dimensionality by periodic forcings and internal feedbacks. High dimensional chaos is somewhat featureless. Conversely low dimensional borderline chaos generates pattern such as oscillation, and is more widespread in climate than is generally recognised. Thus, oceanic oscillations such as the Pacific Decadal and Atlantic Multidecadal Oscillations are generated by dimensionality reduction under the effect of known feedbacks. Annual periodic forcing entrains the El Niño Southern Oscillation. Numerous oceanic and Arctic centered climate oscillations have recently been observed to exhibit delayed synchrony in the manner of a “stadium wave”. Here, three studied systems with chaotic-nonlinear dynamics (physico-chemical and biological) are reviewed which show dimensionality reduction as a source of emergent pattern. This alongside mathematical studies of chaotic Lorenz models whose trajectories escape under controlling influences from chaotic to non-chaotic attractors – incurring a dimensional haircut in the process. Control of chaos by external and internal influences results in lowered phase-space dimensionality with emergent dissipative structure. Dimensionality reduction is proposed as a source of natural climate change. Thus the effect of astrophysical periodic forcing and oceanic and other feedbacks in climate is not so much to generate chaos as to lower dimensionality in a chaotic system, engendering the export of entropy and emergent regular spatiotemporal pattern.
Introduction
Climate: chaotic disorder or order out of chaos?
How important to climate is chaos? Climate is a global scale phenomenon of fluid flow, involving both liquid and gas, and in fluid flow systems chaos and turbulence are frequently present. In the attribution of causes to climate change, chaos is often dismissed as adding only “white” noise to climatic time-series (Penland 2003). The atmosphere is recognised as exhibiting high dimensional dynamics in general (Zeng et al. 1992), albeit with episodes of local low dimensionality limited in space and time (Patil et al. 2001, Oczkowski et al. 2005). Chaos is on the other hand invoked to amplify climate alarm by positing scenarios of system “tipping points” such that an anthropogenic climate forcing could bring unpredictably enlarged catastrophic change arising from the nonlinear-chaotic nature of the system (Schneider 1997, Peters et al. 2004).
Regarding the role of chaos in climate, a central question is that of dimensionality, which is the focus of the present study. This is a mathematical term meaning the number of dimensions of the phase space of the system, effectively the number of degrees of freedom or variables. (What is phase space? Imagine a graph plot with each axis being a variable of a system, or “dimension”. Two axes are easy enough to envisage – your regular x-y graph. More axes – then we’re into the multiverse of chaos and need our imagination; or alternatively, the right mathematical tools.) Low dimensional chaotic systems – referred to as “borderline chaos” – are rich in emergent pattern such as periodic oscillations and the presence of attractors, which are geometric structures in the phase space towards which the system’s trajectory is attracted. High dimensional chaos on the other hand ensues where the number of variables is greater, and complex unpredictable behaviour such as fluid turbulence occurs. It is relatively featureless with less emergent pattern, and resembles noise. Phenomena of nonlinear pattern formation that could play a role in changing climate on longer timescales are associated with low order (low dimensional) chaos near the boundary of chaotic transition (Hopf bifurcation, Wouapi et al. 2019). However the fluid flow phenomena collectively referred to as “climate” are sometimes considered to be associated with high order (high dimensional) chaos and turbulence which is not fertile of pattern on larger spatiotemporal scales. Hence the dismissal of chaos in climate as mere noise.
Despite chaos and nonlinearity being the unseen entity in the room in regard to the general climate narrative, which is justifiably focused on global change related to greenhouse gasses and other factors, climate science is a broad church and many researchers have indeed demonstrated that the paradigm of chaos and nonlinear dynamics can usefully be applied to certain climate and weather systems. And further, that the question of dimensionality of climatic chaos might not be as clear cut as the above simplistic characterisation might make it appear. Both Peters et al. (2004) and Rial et al. (2004) drew analogies between nonlinear-chaotic behaviour in various physical and biological systems and climate, showing that nonlinear processes spanning many spatial scales can cause unpredictable or even catastrophic changes in systems (note the mathematical term “catastrophe” is not necessarily as baleful as its general usage might sound – here it means abrupt discontinuous change). Shen (2019) investigated a specific atmospheric system – African Easterly Waves (AEWs) connected with the African Easterly Jet (AEJ), showing that the quasi-regular atmospheric waves could be well modelled as a Lorenz type chaotic-nonlinear model in which the waves emerge as a periodic limit cycle. This work led to the important insight that such atmospheric system behaviour tracked a mixture of chaotic and non-chaotic attractors. “The entirety of weather possesses a dual nature of chaos and order with distinct predictability” (Shen 2021). Without any structure or predictability, weather forecasters would be without a profession. A chimera of chaotic and orderly is something that emerges in several studied natural systems (Shen et al. 2021, 2022). This links to work already mentioned by Patil and others on local low dimensionality (Patil et al. 2001, Oczkowski et al. 2005) meaning localities in the atmosphere in space and time where chaos dimensionality is reduced, its behaviour regularised and hence the predictiveness of the system – a practical forecasting outcome of such studies – is extended to time periods of up to a month, much longer than possible for a purely chaotic (high dimensional) nonlinear system. Periodicity was also the subject studied by Franz and Zhang (1995) who recognised evidence of periodicity in both current and historic climate indices and developed a climate-like Lorenz-based model of geostrophic atmospheric circulation to investigate specifically the effect of such applied periodic forcing. They found that periodic forcing could both decrease or increase chaos dimensionality depending on factors such as frequency. Through these and other studies we will see that chaos can be controlled.
The Lorenz model of chaos
Much current work on modelling chaos and nonlinear dynamics in climate can be traced back to Ed Lorenz (1963) who in his paper “Deterministic Nonperiodic Flow” demonstrated with one of the first ever computer simulations of a climate system, a chaotically evolving climate from invariant inputs which was “nonperiodic”; that is, there was no repeating pattern. The emerging pattern was endlessly new. While generated on 1950’s computer technology (the Royal McBee LGP-30), what Lorenz showed with this simulation was of profound importance – that pattern can emerge apparently from nowhere in a climate-like chaotic nonequilibrium system (Gollub et al. 1999). Lorenz’ simple models have remarkable predictive value. For example, figure 1 shows a plot of the “El Niño” Southern Oscillation (ENSO) in the form of the evolution of sea surface temperatures (SSTs) within the Niño 3.4 region of the eastern equatorial Pacific in the last few decades. This is compared to two Lorenz model plots of a single phase space dimension with time; first the original nonlinear convection simulation of Lorenz (1963) which is effectively a numerical solution of Rayleigh-Benaud convection equations (e.g. Salzman 1962). And secondly, the Lorenz model system of Franz and Zhang (1995) under periodic forcing showing transient chaos alternating with resonant periodic orbits (please see their paper for the parameters employed). These three plots show remarkable similarity and together illustrate the shared characteristic of oscillatory pattern from low dimensional chaos. The simple Lorenz model simulates the ENSO quite well. We will return to these Lorenz nonlinear dynamic models.
Figure 1. (a) The evolution of sea surface temperatures in the El Niño Southern Oscillation (ENSO) system within the Niño 3.4 region of the east Pacific from 2000-2020. Filled blue when below zero (La Nina conditions) and red when above (El Niño conditions). Reproduced with permission from the National Centers for Environmental Information (NCEI) on behalf of the National Oceanic and Atmospheric Administration (N.O.A.A.) in the USA. (b) The nonlinear model run of Lorenz (1963) convective model equations solved for y – the constant proportional to temperature difference between ascending and descending convective currents – based in turn on a numerical solution of the Saltzman convection equations (Salzman 1962). Numbers in the lower bar are number of model iterations. Published 1963 by the American Meteorological Society and reproduced with permission. (c) Plot of the equivalent constant y from Lorenz’ model adapted by Franz et al. (1995) with the added application of periodic forcing (see reference for parameter values). The plot resembles transient chaos alternating with periodic orbits. Reprinted figure with permission Franz M, Zhang M. Physical Review E., 52(4), 3558, 1995. Copyright (2023) by the American Physical Society. Note the resemblance in form of the wavetrains in a, b and c.
How many things can cause climate change? Can climate change itself?
There is no doubt that turbulent high dimensional chaos is widespread in the ocean and atmosphere. Based on this picture, while the role of chaos is acknowledged in climate, it is frequently regarded as merely a source of short term oscillations superimposed on longer term secular trends that have attribution to linear non-chaotic factors. Climate change on a larger spatiotemporal scale is predominantly attributed to atmospheric causes such as greenhouse gasses, notably anthropogenic CO2. Additional contributory attributed causes have been proposed, such as astrophysical cycles of solar activity (Vinos 2022) and other natural cyclicity arising from lunar, orbital-gravitational (Scafetta 2010), magnetic and cosmic ray phenomena (Svensmark et al. 1997). Volcanic activity and associated atmospheric sulphates have also been invoked (Ge et al. 2016). Cloud cover globally is as a very strong and possibly adaptive climatic force (Lindzen et al. 2001). Ocean circulation is observed to exhibit nonstationary oscillatory phenomena such as the Pacific Multidecadal Oscillation (PMO) and the Atlantic Decadal Oscillation (AMO) and others, and these are also posited as a major source of at least regional climate change (Wyatt et al. 2012). And finally, chaotic processes resulting in emergent spatiotemporal pattern in ocean and atmosphere are proposed as a source of internally driven climate change, including by Carl-Gustaf Rossby, discoverer of the eponymous inertial waves arising in a rotating fluid that occur in earth’s atmosphere and ocean (Palmer 1998). Climate has the unmistakable log-log fractal signature of chaotic-emergent processes, as shown for instance by the saw-tooth climate wavetrain over glacial-interglacial cycles of the ice core record from the Antarctic Vostok station (figure 2, Petit et al. 1999).
Figure 2. The saw-tooth climate wavetrain over the four most recent glacial-interglacial cycles of the ice core record from the Antarctic Vostok station (Petit et al. 1999). Temperature is reconstructed from oxygen isotope ratio (18/16). The temperature wavetrain exhibits the log-log fractal-like amplitude-frequency pattern of chaos-generated emergent pattern. A periodicity of about 100,000 years is evident due to the external periodic forcing exerted by the Milankovitch orbital cycles. (Kindly shared under Creative Commons.)
Agents causing climate change may do so via interaction with the chaotic-nonlinear nature of the climate system; this applies to atmospheric radiative forcing from CO2 and other greenhouse gasses, and to astrophysical periodic forcings. Researchers have noticed apparent correlations between certain time-series of climatic data and astrophysical phenomena, such as solar cycles of various period (Vinos 2022), lunar tidal cycles and others (Scafetta 2010). In the very long term (tens to hundreds of millennia) it is well established that one class of astrophysical cycles, the Milankovitch orbital oscillations of eccentricity, obliquity and precession are responsible for controlling the timing of the attractor-switching flicker between glacial and interglacial during the Pleistocene period. Prior to the mid-Pleistocene “revolution” or transition (MPT) 900,000 years ago the glacial cycling approximately followed obliquity periodicity of 41,000 years while since then it has followed intermittent obliquity cycles (Huybers and Wunsch 2005) entrained by the 100,000 year period of eccentricity oscillation (Lisiecki 2010).
Oscillations in the climate systems – how real are they?
However a model of strong external periodic forcing of climate by solar-astrophysical phenomena requires similarly strong correlation between the forcing agent and the responsive climate time series. And such correlation has proved elusive (Chiodo et al. 2019, Power et al. 2021). Even Milankovitch forcing of glacial-interglacial attractor-flicker is problematic from a mechanistic viewpoint as the recorded switches between states fail to follow exactly the Milankovitch cycles but show some intermittency and imprecision in matching (e.g. Maslin et al. 2005). Does absence of exact correlation refute the proposed Milankovitch forcing hypothesis?
“Correlation is not causation”: transient and intermittent correlations between astrophysical cycles and climate time series fall short of demonstration of causal link between the same. And weakness of the temporal correlations is not the only challenge faced by such hypotheses of astrophysical climate causation. Another criticism of such hypotheses is that the energy imparted to the climate system by astrophysical forcings falls well short of that which could be expected to enforce the climate shifts in question.
The paradigm of the climate system as a periodically forced chaotic-nonlinear oscillator could potentially address both the above criticisms of hypothesised astrophysical forcing of climate; namely, that (a) the timing isn’t right and (b) the forcing doesn’t have enough strength. An extensive literature outside of the climate field – of research into chemical and physical chaotic-nonlinear test systems – has provided some answers to these questions. For instance, exact timing correlation between forcer and forced response only exists in one class of periodically forced chaotic system – that of strong periodic forcing. In strong forcing, the frequency of the forcer is imposed on the forced system whose own natural frequency is overwhelmed. For instance, a pendulum or swing pushed regularly by an outside forcer. Then the outcome is simple resonant frequency-locking. The El Niño Southern Oscillation (ENSO) and its entrainment to the annual cycle is an example of this, as will be discussed later. In the case of weak forcing, however, correlation between forcer and forced system becomes transient and intermittent and sometimes the emergent system time series bears little if any resemblance to the periodic forcing that nonetheless caused it. So exact correlation between the two is not mandatory in order for there to be a role of periodic forcing. To quote Lin et al. (2004):
“The entrainment to the [strong] forcing can take place even when the oscillator is detuned from an exact resonance… In this case, a periodic force with a frequency ff shifts the oscillator from its natural frequency, f0, to a new frequency, fr, such that ff / fr is a rational number m:n. When the forcing amplitude is too weak this frequency adjustment or locking does not occur; the ratio ff / fr is irrational and the oscillations are quasi-periodic.”
Thus strong periodic forcing of a nonlinear oscillator causes frequency-locking to the forcer frequency while weak forcing can result in more complex and varying frequencies not necessarily easily traceable to the forcing periodicity. And regarding the strength of the input signal, the energy and work causing evolving changes in such systems can be internal, from the system’s internal dynamics (e.g. Lisiecki 2010). The external periodic forcing even of small energy inputs can pace or entrain the much greater internal excitable forces of the system.
“Strong or weak” are perhaps an over-simplification of the effects of periodic forcing on a chaotic system. Franz and Zhang (1995) made a comprehensive mathematical study of the effect of periodic forcings of various frequency on the Lorenz chaos parameter model: they showed that in most cases the outcome of periodic forcing was to entrain the chaotic system into oscillations and resonant periodic orbits – effects consistent with a dimensional reduction. Conversely however, at certain forcing frequencies chaos was induced, not suppressed, in the system, indicating local dimensional increase. We will return to this. However both Lin et al. (2004) and Franz and Zhang (1995) show that oscillations or more complex behaviours induced in a system by external periodic forcing cannot generally be expected to have the same time-period as the periodic forcing itself. A different paradigm is needed to investigate what role external astrophysical periodic forcing might have on the emergent dynamics of the climate system.
Dimensional reduction: reducing a system from high to low dimensional chaos by periodic forcing or feedback
The focus of this study is the dimensionality of chaotic systems such as the climate and the ocean. Specifically, that high-dimensional turbulence in a system can be made to transition to lower dimensional “borderline” chaos by the application of external periodic forcing and internal feedbacks. In the atmosphere, as mentioned above, high dimensional chaos is the rule – Zeng et al. (1992) for instance showed that chaotic attractors of studied atmospheric systems had “very high” fractal dimension. However it has become clear that general turbulent high dimensionality can be reduced locally, a phenomenon known as “local low dimensionality”. For instance in the North Pacific “storm track” upper tropospheric eddy energy can be dissipated in quasi-static waves (Okzkowski et al. 2005). Such local “extinctions” of atmospheric chaos-dimensionality typically last a few hours or days only. A metric of local low dimensionality called “Ensemble dimension” or E-dimension was developed (Patil et al. 2001, Okzkowski et al. 2005). This is in the province of weather forecasting. It is proposed herein that local decreases in dimensionality also take place in the ocean, where the slower tempo of oceanographic processes result in emergent oscillations of much longer periodicity, to multi-decadal at least, such that they can be a factor in naturally generated climate change.
To examine in general the control of chaos dimensionality, an overview of several studies will be given that employ Lorenz’ model of convection (1963), based on three differential convection equations. In addition to these mathematical studies, three experimental studies, two physico-chemical and one biological, will also be discussed to show the structure and pattern consequences of dimensional reduction in systems where the chaotic-nonlinear dynamics are more or less well understood. The process of dimensional reduction has been studied in experimental chemical systems extensively by authors such as Bertram (2002). Our first examined experimental system is an example of this – the oxidation of CO catalysed on a platinum surface (Pollman et al. 2001, Bertram et al. 2003). The second is a photonic system involving lasers and optical fibers, another useful and much-studied experimental model of nonlinear-chaotic dynamics in which such dimensional suppression can occur (Brunner et al. (2018). The third system is a biomedical study into the 3D architecture of trabecular (cancellous) bone in human patients affected by a congenital condition Juvenile Pagets Disease (JPD).
Thus both mathematical Lorenz models and experimental and biological systems are unified by the phenomenon of control factors influencing dimensionality of chaos, leading to consistent structural and pattern outcomes. Generalities can be observed both between such systems, and between these systems and the climate system, based on certain universalities in the behaviour of chaotic systems, regardless of their context and spatiotemporal scale. Such generalities illustrate an important general property of nonlinear-chaotic systems, that of hierarchical scale dependence (Shen 2016) where system behaviours in for example Lorenz models of varying dimensionality show self-similar structures at differing scales, in the manner of fractal patterns, with close correlations between system parameters across scales. Universalities and self-similarities between chaotic systems means that study of such systems in the real world necessarily encompasses many diverse scientific fields; Peters et al. (2004) commented that studies of nonlinear dynamics involving spatial connectivity and contagious processes had hitherto “rarely ventured beyond traditional disciplinary boundaries”. These authors thus sought to extend and connect analyses of such systems “through time and across space”, focussing on cases such as wildfires, desertification and infectious diseases. Chaotic system behaviours do show remarkable consistency across scales: for instance the chaotic phenomenon of flicker, where a system’s phase space has two strong attractor basins between which it periodically jumps (Dakos et al. 2013) can occur ten times a second in a lightbulb and ten times in a million years in earth’s Pleistocene glacial-interglacial flicker.
Is chaos in the climate high dimensional in nature (akin to turbulence), and thus unlikely to add more than high frequency noise to spatiotemporal trends? Or does it exhibit low dimensionality to an extent sufficient to generate longer term pattern that could contribute to changing climate? And is there a path from high to low dimensionality?
Hypothesis: Dimensionality reduction – a source of natural climate change
In a complex system characterized by high-dimensional chaos and turbulence, the introduction of external periodic forcing and/or feedback mechanisms can reduce the chaotic behaviour from a higher to a lower-dimensional kind. This in turn leads to the emergence of patterns and oscillations through nonlinear pattern formation and the export of entropy.
Chaos refers to a state of seemingly random, complex, non-repeating and highly sensitive behaviour in a deterministic system (Thomas 1999). In high-dimensional chaos, the system exhibits complex dynamics in a phase space possessing large number of dimensions, making it difficult to identify any underlying patterns or structures. Turbulence is a type of high dimensional chaotic behaviour commonly observed in fluid dynamics, characterized by irregular fluctuations and eddies.
Periodic forcing involves introducing external inputs or perturbations into the system at regular intervals. By imposing periodicity, the system is no longer left to evolve freely but experiences recurring influences. Feedback occurs when a portion of the system’s output is fed back as an input, influencing its subsequent behaviour. Feedback mechanisms can be positive (amplifying the effects) or negative (dampening the effects), depending on the nature of the interactions. When a fluid system possesses positive feedbacks enabling it to generate and propagate wave oscillations, then it is described as an excitable medium. In delayed feedback, where the system forces itself after a delay, the feedback becomes a form of periodic forcing. Thus we can see that feedbacks and periodic forcing are closely linked and sometimes are just different aspects of the same thing.
The introduction of periodic forcing and/or feedback can disrupt the chaotic behaviour of a system by constraining its dynamics. The periodic inputs or feedback act as control signals, guiding the system towards specific states or attracting it to certain regions of its phase space. This constraint effectively reduces the dimensionality of the system’s behaviour, bringing it into the region of borderline chaos that is most fertile in emergent pattern. The reduction of chaos and the emergence of patterns are accompanied by the export of entropy (Prigogine 1975, Bertram 2002). This means that the system expels or dissipates its disorder, leading to a more ordered and structured state.
In a chaotic system high dimensionality is not a fait accompli; it can be changed. Lowering of dimensionality in a chaotic system is often referred to as “control” of chaos, in engineering for example. In the following sections we will examine dimensionality reduction in chaos, first in mathematically modelled Lorenz systems, secondly in real world chemical, physical and biological systems, and finally assess whether this “dimensional haircut” plays a role in the climate system.
Dimensionality reduction: mathematical examples from the Lorenz model
A simple model system that exhibits high-dimensional chaos is the Lorenz system (Lorenz 1963) based on three nonlinear partial differential equations governing Rayleigh-Benard convection. These equations are as follows:
dx/dt = – (sigma x) + (sigma y)
dy/dt = – (xz) + (rho x) – y
dz/dt = (xy) – (beta z)
Here x, y, and z are variables, and sigma, rho, and beta are parameters. This classic Lorenz model – and derivatives thereof – has proved an extremely rich resource for researchers to investigate chaotic dynamics. A laboratory of chaos in three equations. The Rayleigh number r in the context of Lorenz models represents the intensity of convection for instance arising from heating. These models match real world convective systems (such as a saucepan of water heated on a cooker hob) in that there exists a “critical Rayleigh number” rc at which the system transitions from laminar to chaotic turbulent flow. The critical Rayleigh number can give important information about a system’s stability and its tendency to chaotic or non-chaotic behaviour.
1. The effect of feedbacks on Lorenz models of chaos.
Studies of the Lorenz models clarify the roles of feedback, positive and negative, on chaotic dynamics. Negative feedback generally stabilises or controls chaotic systems, a kind of phase space “sheep dog” corralling the trajectory to restrict it to a certain region of the phase space. Work on 5-dimensional (Shen 2014) and 7-dimensional (Shen 2016) Lorenz models showed that predictability and stability are increased by adding terms for a nonlinear negative feedback loop, or “damping”. Shen’s 5D Lorenz model (Shen 2014) is described by the author as a coupled dissipative system, implying that a degree of negative feedback is “baked in” or an intrinsic part of the model. Insights gained by Shen in these Lorenz models were applied to real weather systems (Shen 2019) such as African Easterly Waves (AEWs) and the Madden-Julian Oscillation (MDO) showing that realistic processes of dissipation or “friction” which represent nonlinear negative feedback, stabilise the systems into steady state and more predictable behaviour. This can explain why “practical” weather predictability sometimes exceeds the “intrinsic” predictability based theoretically on chaos models.
Positive feedbacks on the other hand cause “escapes” – that is, the trajectory will break out of its original phase space region into a neighbouring region or attractor (Shen 2014, 2016). Thus positive feedbacks destabilize a chaotic system moving it to a different phase space location while negative feedbacks stabilise the trajectory, corralling it within a restricted region. For example positive feedback will move a system from a chaotic phase space region to an attractor or a periodic orbit conferring on it oscillatory or otherwise patterned (non chaotic) behaviour accompanied by an export of entropy. These escapes also represent a dimensionality reduction.
It is interesting to note that the negative feedback within the Lorenz models (Shen 2014, 2016) increases their stability in another way also, by increasing the critical Rayleigh number rc of transition to chaos. While positive feedback lowers dimensionality from turbulent-like towards borderline chaos, negative feedback effectively increases the dimensionality needed for transition to full chaos (positive Lyapunov exponent), increasing the size of the borderline chaos regime in the system phase space. Nonlinear negative feedback in such a thermal convective system pushes higher the critical Rayleigh number, meaning effectively that it is harder to keep such systems chaotic/turbulent and more likely that they will show laminar quasi-stable low-dimensional behaviour. In this way feedbacks extend the reach of patterned “borderline-chaotic” low dimensional behaviour in a system like climate. Therefore the effects of both positive and negative feedbacks is often to reduce chaos dimensionality, although conversely a system in a high dimensional region of its phase space can be maintained there by the stabilising effect of negative feedback.
2. The effect of periodic forcing on Lorenz models of chaos
Franz and Zhang (1995) recognised the presence of periodicity both in the climate record and in potential astrophysical forcing of earth’s climate. They sought to simulate this by adding periodic forcing to the Lorenz model, making an extensive survey of the parameter space including a wide range of forcing frequencies. An externally forced Lorenz system can be described as “non autonomous”. These authors showed that, depending on frequency, periodic forcing can act in a way similar to both negative and positive feedback. That is, it can either stabilise a system within a region of phase space, chaotic or non-chaotic, like negative feedback, or it can destabilise a system so that it escapes from a chaotic phase space region to a low dimensional attractor or a non-chaotic region. (Which of these happens depends on relative frequency of the forcing and forced system and resonant effects.) Thus at certain frequencies periodic forcing, like feedback, can reduce the dimensionality of a chaotic system. But at others – it could keep it high. “Periodic forcing is able to both create and suppress chaos by resonant interaction” (Franz and Zhang 1995). These authors showed that their Lorenz models with periodic forcing were always dissipative – the same as Shen showed for Lorenz models with nonlinear feedback loops. This dissipative character shows a similarity between the effects on Lorenz chaotic models between feedback and periodic forcing – on chaos dimensionality at least. It also shows the relevance of these models to climate which is also a dissipative system.
An important parameter in the modelling of chaotic systems is the Lyapunov dimension, named after the founder of the field of mathematics based “on the stability of movement” (Lyapunov 1950). It is related to the Lyapunov exponent, a kind of movement differential, the degree by which two closely neighbouring trajectories in a system phase space diverge from each-other. Chaotic systems have a positive Lyapunov exponent. Lyapunov dimension is related to the sum of positive Lyapunov exponents and corresponds directly with the focus of the present study – the dimensionality of a chaotic system. Franz and Zhang’s non-autonomous Lorenz model with varying forcing periodicity showed borderline low dimensional chaos and simple periodic attractors with Lyapunov dimension around 1 while higher dimensional chaos with more complex strange attractors at Lyapunov dimension around 2. The authors concluded that it was likely that periodic forcing – such as from the annual cycle – did generally reduce chaos dimensionality in the climate (otherwise there would be too little predictiveness); which “suggests that periodic processes such as annual or diurnal cycles should not be omitted even in simple climate models”.
3. The opposite of dimensionality reduction: control of chaos to keep it high
The study of a Lorenz model by Capeans et al. (2017) had a goal that was the opposite of dimensionality reduction; instead it was to maintain high dimensionality. This can be the goal in engineering where sustaining high-order chaos can improve combustion in an engine or prevent undesirable mechanical resonances (Schwartz et al. 1996). Capeans et al. (2017) developed a “partial control” method to keep their Lorenz model in the phase space regime of chaos and to prevent departures or escapes to lower dimensional attractor regions. The partial control method by Capeans et al. (2017) actually reduced chaos dimensionality somewhat, while keeping the system still chaotic, away from non-chaotic attractors. A plot of their Lorenz system in 3D – in figure 3 – shows graphically in blue the expanded trajectory of the system in high dimensional turbulent chaos, and in red the trajectory in reduced phase space of the classic Lorenz “butterfly” attractor, with lowered dimensionality – while still remaining chaotic.
Figure 3. A Lorenz system in a 3D phase space showing in blue a high dimensional chaotic-turbulent trajectory, and in red the regularised portion of that 3D phase space described by a reduced dimensional system under external controlling influences (Capeans et al. 2017) displaying a classic Lorenz butterfly-shaped attractor. This figure nicely illustrates the consequences of dimensionality change in a Lorenz chaotic system. Reproduced with permission of the Royal Society under open access.
In summary, the introduction of feedback and/or periodic forcing in a complex system with high-dimensional chaos and turbulence can bring about a dimensional “haircut”; that is, the dimensional reduction of chaotic behaviour toward borderline chaos accompanied by the emergence of patterns and the export of entropy. Through this process, the system transitions from a state of unpredictability to one characterized by coherence, order, and oscillatory behaviour.
Dimensionality reduction: real world examples from physics, chemistry and biology
1. 2D patterns of oxidation of carbon monoxide on a platinum catalyst.
Bertram and others have developed a model system which views, using photoemission electron microscopy (PEEM), 2D spatiotemporal patterns generated by the oxidation of carbon monoxide CO catalysed on a platinum (Pt) crystal surface (Pollmann et al. 2001, Bertram et al. 2003). Areas of the crystal surface occupied by oxygen and CO appear as different light intensities and these respective areas change with time, generating moving 2D patterns. This system clearly shows the suppression of chemical turbulence by the application of global delayed feedback. Feedback was generated using the PEEM image, electronically time-averaging the imaged intensity of the O-CO reaction on the Pt surface and converting this intensity into a control signal to adjust the input flow rate of CO, one of the system’s gaseous reagents. So the changing CO-O2 spatial patterns on Pt changed the CO input rate. Parameters of this feedback were adjusted, namely intensity and delay. Changing spatiotemporal 2D patterns were observed as turbulence was suppressed by adding feedback. First intermittent turbulence was seen, followed by emergent pattern such as oscillatory standing waves, cellular structures, and phase clusters. An example of patterns induced in this system by applied feedback of differing delay time is shown in figure 4. The system with no feedback showed high dimensional chaos similar to noise, while those with variable feedback applied exhibited distinct pattern. To quote Bertram et al. 2003:
“In this paper, feedback-induced pattern formation in CO oxidation on Pt~110! has been experimentally studied in a parameter regime where the unperturbed reaction exhibited chemical turbulence. By applying strong global delayed feedbacks, turbulence could be completely suppressed, leading to uniform oscillations. A large variety of complex spatiotemporal patterns was found when global delayed feedback was used to bring the system to the boundary between regular and chaotic dynamics.”
Figure 4. Photoemission electron microscopy (PEEM) images of 2D patterns, with false colour added, exhibited by CO oxidation catalysed on a platinum surface with (a) no applied feedback where turbulence is established; and (b-f) feedback applied with differing intensity and time delay resulting in lower dimensional emergent oscillatory pattern. Reprinted (part of figure 9) with permission from Bertram M, Beta C, Pollmann M, Mikhailov AS, Rotermund HH, Ertl G. Physical Review E. 67(3): 036208, 2003. Copyright (2003) by the American Physical Society.
The authors were able to replicate their observations in a simulated model based on the Ginzburg-Landau equation. The authors commented that “the observed effects of pattern formation near the edge of chaos may be typical for a broad class of reaction-diffusion systems”. This has been confirmed by other authors such as Vanag et al. (2000) in chemical oscillatory systems such as the thin film Belousov-Zhabotinsky (BZ) reaction.
2. Light pulse evolution in a photonic delay system within optical fibers
Our second experimental example is photonic delay systems. These involve the sending of light pulses, such as from semiconductor lasers, along optical fibers. A controlled delay is sometimes introduced into a photonic signal by means of photonic delay lines or delay circuits with different path lengths – such as with an added series of loops as in a roller-coaster. Photonic delay systems are used for instance in telecommunications, signal processing and radar systems. These systems have proved to be a rich source of experimental data on chaotic behaviour and its modification by applied feedbacks – where again dimensionality reduction causes emergent pattern (Brunner et al. 2018).
Photonic delay systems generate complex spatiotemporal phenomena such as “chimeras” and “solitons”. In a network of oscillating lasers, some lasers will synchronize their oscillations, while others will remain unsynchronized or show chaotic behaviour; this coexistence of synchronized and incoherent dynamics within the same system is called a chimera state. And solitons are localized wave packets that can maintain their shape and velocity while propagating through a medium – over further distances than would normally be expected. Dissipative solitons rely on the gain (amplification) in the medium they travel through to compensate for the energy lost due to dispersion and other causes, allowing them to persist and propagate over long distances through the medium without spreading out or losing their shape. So it’s a way of getting optical signals to travel further without degrading.
Figure 5. In a photonic delay system, application of delayed feedback can transform the spatiotemporal development of the system – shown here as the evolution upward in the vertical direction – from one of high order chaotic noise (c) with short scale stochastic variation to one exhibiting longer term wavelike oscillation (d) – described by the authors as “breathing”. This partial figure is reproduced from Brunner D, Penkovsky B, Levchenko R, Schöll E, Larger L, Maistrenko Y. Chaos: An Interdisciplinary Journal of Nonlinear Science. 28(10): 103106, 2018, with the permission of AIP Publishing.
Brunner et al. show the evolution with time of the 2D cross-section of an optical chimera – figure 5 – where again the application of delayed feedback changes the trajectory from small scale high dimensional stochastic noise to larger scale wavelike oscillation emergent from low dimensional chaos. So this is another experimentally demonstrated example of dimensionality reduction caused by feedback.
3. Trabecular bone architectural change from chaotic to linear-parallel under influence of cell-to-cell feedback
Our third example is a biological / medical one – that of a peculiar form of osteoporosis and its effect on the architecture of trabecular (“spongy”) bone at certain locations. The bones of mammals and many other vertebrates undergo continual remodelling in which osteoblast cells form new bone and osteoclast cells destroy (“resorb”) existing bone. 20% of the human skeleton by mass comprises the honeycomb-like trabecular bone which fills the spaces of the large bones such as the bulbous ends of the femur, tibia, and other long bones as well as scapulae and iliac crests (shoulder and pelvic flat bones). These bones would be too expensive metabolically to fill with solid bone – and they would be too heavy. Trabecular bone has a chaotic architecture like a foam which is a chaotic attractor resulting from the nonlinearity of the remodelling process (Salmon 2015) – indeed a laminar-turbulent transition occurs at the growth plates where growing bone is first formed (Salmon et al. 2023). This chaotic foam structure allows mechanical forces to be resisted from all directions, although sometimes trabeculae are aligned preferentially along dominant axes of loading since there is a mechanically adaptive component to the chaotic remodelling (Huiskes et al. 2000). The two processes of bone formation and resorption are linked in a feedback where the osteoblast cells which form new bone signal to osteoclast cells to initiate bone resorption following the formation cycle. One of the most significant recent discoveries of bone cell biology was that of the cytokine signalling protein “RANKL” (e.g. Yasuda et al. 1998) – easier to say than “receptor activator of nuclear factor kappa beta (NFkB) ligand”. RANKL from osteoblasts signals the corresponding RANK receptor on the surface of osteoclast cells activating them to begin a cycle of resorption. A signal to destroy what they have just built. And what goes around comes around – osteoclasts also return the compliment by signalling to osteoblasts to form bone after they have resorbed it (Walker et al. 2008). A signal to rebuild what they have just destroyed. This is a positive feedback – osteoblasts signalling osteoclasts and vice versa. And as we have seen – positive feedback and a chaotic system would be unlikely to coexist. Therefore to maintain the chaotic architecture, the osteoblast cells secrete another signalling molecule called Osteoprotegerin or OPG (Lacey et al. 1998). OPG competitively binds the RANK receptor on osteoclasts, thus blocking the RANKL cytokine which the same osteoblasts also secrete. This doing of two mutually opposing things is quite normal in biology, it’s the Yin and Yang from which emergent complexity arises as recognised by Alan Turing in his reaction diffusion complexity model (Turing 1952). So OPG restrains the formation-resorption cycle from runaway positive feedback. The blocking of positive feedback by OPG is in turn a negative feedback – that could be called damping or friction; or dissipation.
In the genetic disease Juvenile Pagets Disease (JPD), otherwise known as Idiopathic Hyperphosphatasia, OPG production is reduced or stopped by a mutation (Cundy et al. 2002). So the inhibition of the coupled formation-resorption loop is removed, resulting in runaway positive feedback between formation and resorption mediated by RANKL. Blood markers of both activities are shown to be greatly elevated in JPD patients (Cundy et al. 2002). In terms of the phase space of the emergent architecture of the trabecular bone, the negative feedback exerted by OPG was controlling the system trajectory within a chaotic orbit (Salmon 2004) – as seen in the Lorenz models. Removing OPG activity releases positive feedback resulting in an escape from the chaotic to a regular periodic orbit, and the architectural consequences of this change are shown in figure 6. Dimensionality reduction has again taken place. Normal trabecular architecture at the iliac crest and elsewhere is a chaotic labyrinthine form (high dimensional) while the same site in an OPG patient shows highly abnormal parallel plate arrangement of the trabeculae (low dimensional). The resulting bone architecture in OPG patients is severely osteoporotic, anisotropic and mechanically compromised, resulting in higher risk of fracture. This chaotic to regular pattern change is not too dissimilar to the transition seen at the surface of the platinum crystal when viewing by electron microscope the distributions of O and CO in Bertram and Pollmann’s (Bertram et al. 2003) feedback-driven suppression of chaos in that experimental system as discussed above (figure 4).
Figure 6. (A) A clinical iliac crest biopsy from a teenager suffering from JPD (where the gene for OPG is inactivated) shows unusual parallel trabecular structure indicative of a transition from a chaotic to a regular nonlinear pattern. (B) In contrast, the iliac crest biopsy from a normal adult imaged by microCT (SkyScan 1072) shows typical chaotic labyrinthine trabecular structure at this site. Magnification for A and B, ~x10. Reproduced with permission from Salmon P (2004) Journal of Bone and Mineral Research. 19(5): 695-702. © 2004 American Society for Bone and Mineral Research.
What is interesting about this trabecular bone and JPD example is that the consequences to chaos dimensionality are seen for both positive and negative feedbacks. As shown by Shen (2014, 2016) negative feedbacks control the system trajectory restricting it to stay within its phase space neighbourhood – the phase space sheep dog. Thus chaotic systems are kept chaotic and likewise non chaotic systems made to stay that way – by negative feedback. Positive feedback by contrast causes a system to escape from a chaotic attractor to a regular periodic orbit, accompanied by dimensionality reduction.
Dimensionality reduction in the climate system
As we have seen, two things can “de-dimensionalise” chaos: external periodic forcing (Eiswirth et al. 1988, Franz and Zhang 1995, Lin et al. 2000, 2004) and internal feedback (Vanag et al. 2000, Pollmann et al. 2001, Bertram 2002, Bertram et al. 2003, Shen 2014, 2016, Brunner et al. 2018). In the climate system, both of these are available to provide the dimensional haircut that moves the system from high order turbulence to low-dimensional chaos and emergent pattern.
Oceanic oscillations such as the Pacific Decadal Oscillation (PDO) and the Atlantic Multidecadal Oscillation (AMO) are both driven by feedbacks associated with the oceanic process of high latitude downwelling and formation of cold and dense bottom water. The AMO is the better understood system and is linked to the Atlantic Meridional Overturning Circulation (AMOC) that is driven by a feedback that operates in the following way. The region of the Caribbean and Gulf of Mexico possesses excess warmth and salinity due to interhemispheric heat piracy importing heat trans-equatorially via the afferent Caribbean current. Water from this Caribbean region is transported by the part of AMOC referred to as the North Atlantic Drift or “Gulf Stream” north-eastward across the Atlantic and toward the Arctic. When this water reaches the Norwegian Sea it cools, upon which its still elevated salinity confers on it exceptionally high density, driving its downwelling to the ocean floor (“deep water formation”) and southward. The southward transport of this formed bottom water impels further the North Atlantic Drift through conservation of water volume and mass. This is referred to as the “salt-advection” feedback (Weijer et al. 2019) and is the propeller of the AMOC. The positive salt-advection feedback makes the Atlantic ocean an excitable medium on the large spatiotemporal scale of the AMO and AMOC.
It is likely that other ocean basins are also made excitable by analogous salt-advection feedbacks. However the exceptional strength and significance of the AMOC is due to the Atlantic being meridionally bounded by land. The AMOC results in the cyclical phenomenon of the Atlantic Multidecadal Oscillation (AMO) which is defined as an index related to the averaged sea surface temperatures (SSTs) of the North Atlantic, detrended to remove long term secular changes (Van Oldenborgh et al. 2009). Evidence was presented by Levitus et al. (2009) that the oscillation of the AMO index is linked to variation in the strength of transport of warm Caribbean water across the North Atlantic to the Arctic; they showed close correlation between seawater temperatures at 100m depth in the Barents Sea with the AMO over the prior half century. This data is shown in figure 7.
Figure 7. Black line: monthly seawater temperature (degrees C) in the Barents Sea at 100 – 150 m depth, from 1900 to 2006. Years without all 12 months of data are excluded. Red line: the Atlantic Multidecadal Oscillation Index. From: Levitus S, Matishov G, Seidov D, Smolyar I. Geophysical Research Letters 36(19), L19604, 2009; reproduced with permission of the American Geophysical Union.
Thus the salt-advection feedback in the AMOC is reducing the chaos-dimensionality of Atlantic ocean circulation and coupled atmospheric system allowing the emergent oscillatory pattern that we call the AMO (and other related oscillations – see Wyatt et al. 2012). This dimensionality reduction is also evident in the oscillations in the photonic delay system (Brunner et al. 2018) with application of feedback, where in figure 5(d) one can see a resemblance to the time series of oceanic oscillations such as the PDO or AMO. There is a universality to chaotic pattern-formation processes – what Shen (2016) called hierarchical scale dependence. Further work in support of the AMO being an emergent nonlinear oscillation was provided by Vidal-Henriquez et al. (2017) in a special edition of the Chaos journal commemorating the 100th birthday of Ilia Prigogine. Modelling a Martiel-Goldbeter reaction-diffusion-advection system, these authors showed that a convective instability in the vicinity of a boundary can become a continuous source of waves. Thus deep water downwelling associated with salt-advection feedback in the Norwegian Sea near to coastlines becomes a source of oceanic multidecadal oscillation. The authors applied their conclusions to social signalling between amoebae – but why not apply it to the Atlantic ocean as well? Taking this conclusion further – we can predict that any significant occurrence of oceanic vertical mixing over the whole deep ocean water column, such as major upwelling and downwelling sites, especially near a bounding coastline, will be associated with a multi-annual or multidecadal oceanic oscillation. This oscillation arises from dimensionality reduction caused by the positive salt-advection feedback, and also from the findings of Vidal-Henriquez et al. (2017) about convective instability near a boundary. A possible mechanism for such oscillation is periodic or quasi-periodic fluctuation in the strength and flow rate of the upwelling or downwelling, with climatic consequences attendant on the resulting effects on ocean circulation.
Likewise the El Niño Southern Oscillation (ENSO) is based on the excitability of the Bjerknes feedback (Bjerknes 1969) coupling the equatorial trade winds with Peruvian coast deep upwelling. El Niño as named centuries ago by Peruvian fishermen means the little boy in Spanish, implying the Christ Child, due to the usual peak of a warming El Niño cycle around Christmas. Thus even the name of this climatic oscillation tells us that it is phase-locked to the annual cycle. The manner in which the Bjerknes-linked excitability of the eastern equatorial Pacific is entrained to annual phase-locking was set out analytically by Tzipermann et al. (1995, 1997 1998). They summarise it as follows (Tzipermann et al. 1998):
“El Niño events owe their name to their tendency to be locked to the seasonal cycle. A simple explanation is proposed here for the locking of the peak of ENSO’s basin-scale warming to the end of the calendar year. The explanation is based on incorporating a seasonally varying coupled ocean–atmosphere instability strength into the delayed oscillator mechanism for the ENSO cycle. It is shown that the seasonally varying amplification of the Rossby and Kelvin ocean waves by the coupled instability forces the events to peak when this amplification is at its minimum strength, at the end of the calendar year. The mechanism is demonstrated using a simple delayed oscillator model…”
We have already seen from Franz and Zhang (1995) the similarity between these authors’ periodically forced Lorenz model and the recorded SSTs in the ENSO region – in figure 1c. Franz and Zhang’s (1995) results thus give some support the conclusion of Tzipermann et al. (1995, 1997, 1998) that ENSO is a nonlinear oscillator forced by the annual cycle.
The annual cycle and associated Rossby and Kelvin waves are clearly of sufficient strength for this periodic forcing of a nonlinear oscillator to be of the strong type, as discussed above. The oceans are thus evidently subject to periodic astrophysical forcing from the annual cycle. Further longer term lunar and solar oscillations and their harmonics could also potentially play a forcing role in atmospheric and ocean circulation systems (Scafetta 2010, Vinos 2022). For example White and Liu (2008) found evidence of a signal from the 11-year solar sunspot cycle in the wavetrain of ENSO in the equatorial Pacific. This would be a case of weak periodic forcing where identification of the sunspot forcing influence required analysis of several harmonics (White and Liu 2008). Feedback driven excitability of atmosphere and ocean combined with external periodic forcing, converts chaos into continuous change and oscillation in climate on many timescales – as predicted by Lorenz (1963).
Wyatt and co-workers have demonstrated a large number of climatic indices centered on the oceans and Arctic, particularly (but not only) in the northern hemisphere, which include the AMO and PDO and which oscillate with a similar 50-80 year (“multidecadal”) frequency (Wyatt et al. 2012). The synchronised and staggered waveform of these oscillations led to authors to name it the “stadium wave” – a propagating network of linked oscillations such that a higher order wave appears to be riding on all the waves, in a form of modulation, as shown in figure 8, reproduced with permission from Wyatt et al. 2012. Some of these oscillations are linked to the cyclicity of the AMOC as described above, others to different oscillatory feedback driven systems of the Pacific and Indian oceans (e.g. Bjerknes 1969). That an oscillation in one ocean basin can transmit oscillatory impulse to a neighbouring ocean has been shown by Van Oldenborgh et al. (2009). This paints a picture in which the AMO and PDO, rather than being isolated examples of oceanic oscillation, instead are members of a global network of coupled atmosphere-ocean oscillations which are synchronised.
Figure 8. Normalised indices of oscillatory climate metrics from Wyatt et al. 2012. The synchronised and staggered form of these waves led to authors to name it the “stadium wave” – a propagating network of linked oscillations. The abbreviations refer to atmosphere-ocean indices: -NHT – (inverted) Northern Hemisphere Temperature, -AMO – (inverted) Atlantic Multidecadal Oscillation, AT – Atmospheric-Mass Transfer Anomalies, NAO – North Atlantic Oscillation, NINO 3.4 – El Niño Southern Oscillation region 3.4 surface temperature, NPO – North Pacific Oscillation Index, PDO – Pacific Decadal Oscillation, ALPI – Aleutian Low Pressure Index. Reproduced from Wyatt MG, Kravtsov S, Tsonis AA. Climate Dynamics. 38:929-49, 2012, with the permission of Springer Nature.
Discussion
Mathematically modelled chaos based on the Lorenz model aligns well with experimental studies and clinical observations, in showing structure and pattern consequences of changed chaos dimensionality, brought about by feedbacks and periodic forcing. Chaotic systems are far from uniform but are very diverse in their structure and dynamics. In the present study some examples of this are given and some of the terminology and system players in chaos and nonlinear dynamics are explained, with a view to making the analysis of chaos somewhat more transparent and accessible. Chaos is a large part of what happens in the natural world, and dimensionality of one of the keys to making sense of this.
The literature on well-characterised theoretical and experimental chaotic nonlinear systems makes it clear that chaotic dynamics can be controlled, specifically by control of the dimensional level. Shen (2014, 2016) took Lorenz’ original 3D chaos model (Lorenz 1963) and extended it to 5D and 7D allowing the incorporation of nonlinear feedback loops that made the systems dissipative in character mimicking the thermal dissipation and damping of natural systems such as the climate. These studies showed that negative and positive feedback, while having distinct effects on chaotic systems, both act in general to diminish high dimensional chaos and increase low dimensional, more patterned and more predictable system behaviours. Likewise Franz and Zhang’s (1995) comprehensive study of non-autonomous Lorenz models showed that periodic external forcings have a strong effect on system dimensionality, sometimes stabilising a state of high dimensional chaos but more often causing destabilising escapes or stabilising instead low dimensionality. So in aggregate the effects of periodic forcing were more likely to lower than to raise chaotic-nonlinear dimensionality – as evidenced by low dimensionality in the climate system itself (Franz and Zhang 1995).
Turning to experimental studies, the oxidation of carbon monoxide catalysed on a platinum surface, and the photonic delay systems with lasers transmitting along optical fibers, demonstrate a short-circuiting of high-dimensional turbulent chaos by periodic forcing and feedbacks, reducing the studied systems to the low dimensional regime more fecund of emergent pattern. In engineering the goal is control of chaos dimensionality to suit design and process requirements. Interestingly, in some systems the goal is to maintain high dimensionality to sustain chaotic turbulent-like behaviour, and suppress low dimensional emergent pattern as an undesirable outcome (Capeans et al. 2017). The biological example presented – the chaotic architecture of trabecular bone transformed by a positive feedback-induced phase space escape to local low dimensionality and pathological parallel plate structure in JPD patients (Salmon 2004), is also a striking case of profound pattern change from dimensionality reduction, from the RANKL feedback loop. In normal bone without JPD, negative feedback in the form of OPG holds in check the positive feedback that uninhibited RANKL signalling by osteoblasts would produce. Here negative feedback is restricting the system trajectory to within a high dimensional chaotic region – the “phase space sheep dog”.
Dimensionality reduction is then shown to be a factor in the earth’s climate system in which many “barbers” – periodic forcings and internal feedbacks – bring about a dimensional haircut. Periodic forcings can be the annual cycle or solar and lunar cycles (Tzipermann et al. 1995, Franz and Zhang 1995, Vinos 2022). Feedbacks can be between ocean surface winds (e.g. Trade Winds) and coastal upwelling as in the Bjerknes feedback leading to the El Niño Southern Oscillation (ENSO), or between equator to pole transport of warm saline water and polar region cold water downwelling such as the “salt-advection” feedback driving the AMOC (Weijer et al. 2018). Rial et al. (2004) asserted that “feedbacks are the most likely processes behind most of the nonlinearities in the climate”, such as emergent oscillations. And the work by Wyatt and co-workers (2012) on the “stadium wave” of networked climate oscillations show that multidecadal oscillatory behaviour is very pervasive in the ocean-driven climate system.
One feature in common with many studied dynamic systems is transience and locality of dimensional reduction. A spatiotemporal alternation between order and chaos. Capeans et al. (2017) pointed to the widespread phenomenon of transient chaos, where a system’s trajectories jump between a chaotic region of its phase space and an external attractor, alternating between high and low dimensionality. Brunner (2018) demonstrated in photonic systems the existence of “chimeras” of coexisting chaotic and nonchaotic oscillating lasers. And in atmospheric dynamics Shen et al. (2021) in addressing the question “is weather chaotic?” reviewed research demonstrating chaotic-nonchaotic attractor coexistence within conservative Hamiltonian systems (Hilborn 2000) and the coexistence of chaotic and nonchaotic solutions in dissipative systems (Sprott et al. 2005). Shen et al. (2022) described dimensionality reduction in their Lorenz-based climate models as “a transition from a chaotic solution to a non-chaotic limit cycle solution”; that is, emergent periodicity. They concluded, “The atmosphere possesses chaos and order; it includes, as examples, emerging organized systems (such as tornadoes) and time varying forcing from recurrent seasons”. Franz and Zhang (1995) also showed transient chaos alternating with periodic orbits in their Lorenz model, shown here in figure 1 (c). Other authors describe the phenomenon of atmospheric local low dimensionality coinciding with local “extinctions” of chaos dimensionality from high to low accompanied by regular wave-like pattern and increased predictability (Patil et al. 2001, Ott et al. 2002, Oczkowski et al. 2005). This corrects an over-simplification or generalisation that dimensionality is either high or low generally in a system; it can vary locally from low to high, in space and time. The spatial mapping of the atmospheric “ensemble-dimension” by these authors had some equivalence to the mapping (in parameter space) of the Lyapunov dimension in the periodically forced Lorenz models of Franz and Zhang (1995).
There are very different timescales of dimensionality reduction in the atmosphere (a few days) and the oceans (multi-annual, multidecadal or longer). It is proposed here that the local dimensionality reduction hitherto studied in the atmosphere and weather, also takes place in the ocean giving emergent periodicity – with some attendant predictability – over those longer timescales. And specifically it is predicted that any significant oceanic vertical mixing over the whole deep ocean water column, such as upwelling and downwelling sites especially near a bounding coastline, represents convective instability near a boundary (Vidal-Henriquez et al. 2017) and thus will be associated with a multi-annual or multidecadal oceanic oscillation. Just as deep water formation in the Norwegian Sea is linked to the Atlantic Multidecadal Oscillation (AMO), and Peruvian coastal upwelling with the El Niño Southern Oscillation (ENSO).
The onset of chaos, from linearity via transitional bifurcations to eventually full turbulence, is well understood (e.g. Malomed et al. 1990). Turn on a tap slowly – first it drips, then flows in a smooth stream, then the stream wobbles, before disordered turbulence breaks out. Climate is about fluid flow and mixing in atmosphere and ocean, so obviously chaotic turbulence will be the rule. Therefore, the onset of chaos per se in climate is not really so surprising or important. What’s more interesting is the opposite, where turbulent chaos is reduced toward the border between chaotic and linear, where the phenomena of emergent pattern arise. Mathematically, turbulence is chaos with a large number of dimensions to its phase space, so moving from turbulence back toward borderline chaos means dimensionality reduction. In the atmosphere and ocean this brings about emergent spatiotemporal pattern – such as the multidecadal climatic oscillations of the Pacific, Atlantic and other oceans.
Emergent climatic oscillations and spatiotemporal patterns are dissipative structures, a concept originated by Prigogine (Prigogine et al. 1968, 1975), involving export of entropy. The climate comprises a set of dissipative structures involving air, water, water vapor and ice, a heat engine redistributing heat from equator to pole. These dissipative structures can be maintained – at least intermittently – in a low-dimensional chaotic regime by the controlling influence of external periodic forcing and internal feedbacks (e.g. Franz and Zhang 1995, Tzipermann et al. 1997). Therefore the appearance of regular spatiotemporal structure in climate – such as periodic or quasi-periodic oscillation in temperature or precipitation at any location – does not necessitate a search for a specific local cause of each individual feature of that structure, of each hill or valley in that landscape. Instead, such climate landscapes can be seen as the inevitable result of chaotic nonlinear dynamics, partially or intermittently controlled or “trimmed” to remain within the low dimensional regime where regular pattern emerges.
Acknowledgements
I am grateful to Evelyn Salmon for her helpful corrections and improvements on the manuscript text.
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