by Tomas Milanovic
There are scientists who equate chaos to randomness. I’d put that category at 90%.
There are scientists who equate chaos with Lorenz. They have seen the butterfly attractor picture one day or the other. They know that chaos is not randomness but not much more. I’d put that category at 9%.
There are then scientists who know what is chaos and really understand it. I’d put that category at 1% and much less for the climate scientists.
The chaos one could and should we be talking about as far as climate is concerned is spatio-temporal chaos.
What is known as chaos theory and often associated with Lorenz was actually discovered by Poincare 100 years ago and it is TEMPORALchaos. It is a paradox, but chaos was first discovered by Poincare in a Hamiltonian system which has been considered for centuries as the perfect deterministic clockwork – the celestail mechanics. Poincare has proven that a gravitational 3 body system was chaotic and unpredictable. Actually it is not even predictable statistically (e.g you can not put a probability on the event “Mars will be ejected from the solar system in N years”).
Scientists having been busy discovering relativity and QM (Poincare too); they have been ignoring these results for 60 years. Then Lorenz found chaos in fluid dynamics and the temporal chaos theory started slowly developing.
The most important point that everybody who wants to understand something about temporal chaos theory should understand that it is all about geometry in a finite dimensional phase space. In other words it deals mathematically with systems of non linear ODE where all unknowns are coordinates of the phase space and the state of the system is perfectly defined by a point P(t) in the phase space by giving its coordinates (degrees of freedom). If this rings a bell with hamiltonian mechanics, it is good as it should.
All the “advanced” concepts (bifurcations, shifts, attractors, fractals) are children of temporal chaos theory. The simple rule of thumb is that if there is only time dependence, then the chaos can be explained by chaos theory. Chaos theory doesn’t apply at all to the problems that bring us here, and here is why.
There is something much more complicated and qualitatively radically different from the temporal (Lorenzinan) chaos – the spatio-temporal chaos. There is no established spatio-temporal chaos theory. It is cutting edge and a few people have worked on this only for a few decades. Spatio-temporal chaos deals with the dynamics of SPATIAL PATTERNS. Mathematically we deal with fields described by non linear PDEs; Navier Stokes equation is an example.
Spatio-temporal chaos is as far from the temporal chaos theory as QM is from classical mechanics.
The biggest difficulty comes from the fact that we lost this convenient finite dimensional phase space. That’s why almost nothing transports from temporal chaos to spatio-temporal chaos. There are no attractors, bifurcations and such. The whole mathematical apparatus has to be invented from scratch and it will take decades. To know the state of the system, we must know all the fields at all points – this is an uncountable infinity of dimensions. As the fields are coupled, the system produces quasi standing waves all the time. A quasi standing wave is a spatial pattern that oscillates at the same place repeating the same spatial structures in time. However in spatio-temporal chaos these quasi standing waves are not invariants of the system on the contrary to the attractors which are the invariants of the temporal chaos. They live for a certain time and then change or disappear altogether.
You can see spatio-temporal chaos if you look at a fast mountain river. There will be vortexes of different sizes at different places at different times. But if you observe patiently, you will notice that there are places where there almost always are vortexes and they almost always have similar sizes – these are the quasi standing waves of the spatio-temporal chaos governing the river. If you perturb the flow, many quasi standing waves may disappear. Or very few. It depends.
Weather and climate are manifestations of spatio temporal chaos of staggering complexity because there is not only Navier Stokes equations, but there are many more coupled fields. ENSO is an example of a quasi standing wave of the system.
Of course I hope that the reader now knows that ENSO cannot be explained by something depending on time only (like indexes, time series and such) because if it could, we would have classical temporal chaos where space doesn’t matter. We would have solved the problem long times ago. But as ENSO is a pattern resulting of interaction of ALL fields in the system, it vitally depends on how these fields interact in space. That’s why all interpretations of ENSO (and other multidecadal quasi standing waves) are failing – people are using functions (series) that depend on time only which cannot clearly encode all the spatial interactions.
There are a few exceptions like Tsonis. I have written a long post in the Tsonis thread so won’t repeat. But Tsonis makes a step towards spatio-temporal chaos by considering that there are several interacting waves what is equivalent to introduce some dose of spatial interaction. Of course as Tsonis considers only 5 waves, it is a rather rough way to discretize space over the whole planet but it is a beginning.
The best way to imagin a full spatio-temporal chaos theory is to imagine that there is a different chaotic oscillator like the Lorenz butterfly) at every point of space (so there is an infinity of them) and that they are all coupled strongly with each other in a non linear and time dependent way. I am not saying that there can’t be some simplifications but nobody knows today. The only thing I am reasonably sure of is that there will be no progress in understanding be it via chaos or not as long as people will insist on the crutches of functions/series that are only time dependent.
That’s why it is completely incorrect to say that climate is a boundary value problem.
To illustrate what the REAL problem of climate dynamics is, I have posted in the Tsonis thread a link to this paper :http://amath.colorado.edu/faculty/juanga/Papers/PhysicaD.pdf
Despite the fact that this paper finds a MAJOR result and is the right paradigm for a study of spatio temporal chaotic systems at all time scales so also for climate, I suspect that nobody has read it.
And probably only few would understand the importance of both the result and of the paradigm. Of course the climate is more difficult than even a network of chaotic oscillators because, among others, the coupling constants vary with time and the uncoupled dynamics of the individual oscillators are not known.
Also the quasi ergodic assumption taken in the paper is not granted for the climate.
Yet even in the general case it appears completely clearly that the system doesn’t follow any dynamics of the kind “trend + noise” but on the contrary presents sharp breaks , pseudoperiodic oscillations and shifts at all time scales. Of course the behaviours in the case when the coupling constants vary will be much more complicated and are not studied in the paper.
Unfortunately people working on these problems are not interested by the climate science and those working in climate science are not even aware that such questions exist , let alone have adequate training and tools to deal with them.
Concerning these paradigm issues, this belongs obviously to the unresolved questions and as far as I am aware, it is only on blogs and among others on your blog that they are discussed.
TM’s Summary 2/16
Main points for the summary:
1)
I commend Jstults for excellent and relevant contributions. He has a good knowledge of the litterature but most importantly he is able to manipulate chaotic ODEs. SpenceUK also added good contributions. Dan Hughes blog on numerical solutions of Lorenz equations is a good read.
2)
This is NOT about numerical models. This cannot be about numerical models.
I hope that by now most have understood that from the mathematical point of view temporal chaos theory is about solutions of non linear ODEs and spatio-temporal chaos about solutions of non linear PDEs.
The former which is much older than the latter (Poincaré 100 years ago on hamiltonian conservative systems and Lorenz 50 years ago on 2D fluid dynamics) is a good introduction to important concepts and mathematical tools but of little to no help in climate matters.
As numerical models cannot find solutions of any system of non linear ODEs or PDEs because the system is simply spatially too huge and all the equations are not known anyway, they have no relevance to what I discuss here.
If I attempt to characterise what they are in my eyes, I would say that they are simulators of the evolution of the system under approximate constraint of conservation laws.
But as R.Hilborn has rightly written “The dynamically allowed space is much smaller than the space that is allowed by the conservation laws”.
Btw I recommend R.Hilborn’s excellent textbook (http://www.amazon.com/Chaos-Nonlinear-Dynamics-Introduction-Scientists/dp/0198507232/ref=cm_cr_pr_product_top) for anybody who would like to go a bit farther than the basics of the non linear dynamics.
From that follows that whatever states the numerical simulation computes, it cannot be sure that they are dynamically allowed. Many of them may very well be just plausible states of the fields but the system will never visit them because they are dynamically forbidden.
This poses the question of the metrics of the states (how do we define a state of the system so that this definintion leads to a meaningful metrics) which is another debate.
3)
There is a fundamental difference both mathematically and physically between temporal chaos and spatio temporal chaos. Judith rightly notes that few of the climate scientists have knowledge about temporal chaos let alone spatio temporal chaos. Even Tsonis and Swanson are not really experts of chaos theory but their paradigm (coupled oscillators) is identical to the spatio temporal chaos paradigm. That is why their work is qualitatively different from the “orthodox” school.
My personal opinion is that I do not believe that numerical models (GCM) can give meaningful support or development to their work but I do not know if they believe it themselves. For that we’d need their opinion.
4)
There is still the old school that continues to equate chaos with randomness. I am not sure that they are willing to learn modern physics so it is certainly not blog discussions that would convince them.
Characteristic of this school is the following quote :
But as soon as you add any sort of noise, your perfect chaotic system becomes a mere stochastic one over long time periods, and probabilities really do apply.
A nice review of the relationships between chaos, probability and statistics is this article from 1992:
“Statistics, Probability and Chaos” by L. Mark Berliner, Statist. Sci. Volume 7, Number 1 (1992), 69-90.
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ss/1177011444
I suspect that these people didn’t really read the link.
The part relating to stochasticity admits that it is merely a qualitative overview and references the fundamental papers among which Ruelle and Eckmann.
It has apparently escaped to the author of the quote that I have linked the R&E paper in the very first post and he certainly didn’t read it.
What the Berliner’s summary says is that :
IF we have a temporal chaotic system and IF this system is ergodic THEN a stochastical interpretation is possible
Unfortunately neither of the ifs is valid for weather/climate.
Despite this rather obvious point, these people still talk about “perturbations”.
Actually the chaos doesn’t exist for them because there are “perturbations”.
This is a complete misunderstanding of chaos theory.
There are no “perturbations” inside a chaotic system – a solution of the dynamical equations is what it is and all the “perturbations” are already accounted for.
The system cannot be decomposed in a linear way in a sum : nice smooth if possible deterministic solution + noise or “perturbation”.
Of course the external energy supply which is necessary to produce chaos is not necessarily constant. It may even be considered random. This doesn’t imply in any way that the system suddenly becomes random too and none of the quoted papers says anything approaching.
A kind of randomness or more precisely the existence of an invariant (of the initial conditions and of time!) probability distribution of the states exists only for ergodic systems.
But the ergodic property is NOT a given.
Even in temporal chaos some systems are ergodic and some are not.
In spatio-temporal chaos the question is fully open especially as a complete ergodic theory of spatio-temporal systems doesn’t exist yet.
In any case the ergodicity has nothing to do with “perturbations” or variations of the external energy fluxes.