Tuesday, February 15, 2011

Comments on Spatio-Temporal Chaos

Some comments from a guest post on Dr Curry's site. I think she has a couple dueling chat bots who've taken up residence in her comments (see if you can guess who they are). This provides a bit more motivation for getting to the forced system results we started talking about earlier. The paper and discussion that Arthur Smith links is well worth a read (even though it isn't actually responsive ; - ).

Tomas – you claimed to focus on my comment, but *completely ignored* the central element, which you even quoted:
“small random variations in solar input (not to mention butterflies)” [as what makes weather random over the long term]
Chaos as you have discussed it requires fixed control parameters (absolutely constant solar input) and no external sources of variation not accounted for in the equations (no butterflies). You gave zero attention in your supposed response to my comment to this central issue. Others here have been accused of being non-responsive, but I have to say that is pretty non-responsive on your part.
The fact is as soon as there is any external perturbation of a chaotic system not accounted for in the dynamical equations, you have bumped the system from one path in phase space to another. Earth’s climate is continually getting bumped by external perturbations small and large. The effect of these is to move the actual observed trajectory of the system randomly – yes randomly – among the different possible states available for given energy/control parameters etc.
The randomness comes not from the chaos, but from external perturbation. Chaos amplifies the randomness so that at a time sufficiently far in the future after even the smallest perturbation, the actual state of the system is randomly sampled from those available. That random sampling means it has real statistics. The “states available” are constrained by boundaries – solar input, surface topography, etc. which makes the climate problem – the problem of the statistics of weather – a boundary value problem (BVP). There are many techniques for studying BVP’s – one of which is simply to randomly sample the states using as physical a model as possible to get the right statistics. That’s what most climate models do. That doesn’t mean it’s not a BVP.

This isn’t anything new – almost every physical dynamical system, if it’s not trivially simple, displays chaos under most conditions. Statistical mechanics, one of the most successful of all physical theories, relies fundamentally on the reliability of a statistical description of what is actually deterministic (and chaotic – way-more-than-3-body) dynamics of immense numbers of atoms and molecules. This goes back to Gibbs over a century ago, and Poincare’s work was directly related.
Tomas’ comments about the 3-body system being 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”” is true in the strict sense of the exact mathematics assuming no external perturbations. That’s simply because for a deterministic system something will either happen or it won’t, there’s no issue of probability about it at all. 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.
and see some of the discussion that followed in that journal (comments linked on that Project Euclid page).

Arthur Smith, while that is a very good paper that you linked (thank you for finding one that everyone can access), it only had a very short section on ergodic theory, and you’re back to the same hand-waving analogy about statistical mechanics and turbulent flows. The [lack of] success for simple models (based on analogy to kinetic theory btw) for turbulent flows of any significant complexity indicates to me that I can’t take your analogy very seriously.
Where’s the meat? Where’s the results for the problems we care about? I can calculate results for logistic maps and Lorenz ’63 on my laptop (and the attractor for that particular toy exists).
A more well-phrased attempt to explain why hand-waving about statistical mechanics is a diversion from the questions of significance for this problem (with apologies to Ruelle): what are the measures describing climate?
If one is optimistic, one may hope that the asymptotic measures will play for dissipative systems the sort of role which the Gibbs ensembles have played for statistical mechanics. Even if that is the case, the difficulties encountered in statistical mechanics in going from Gibbs ensembles to a theory of phase transitions may serve as a warning that we are, for dissipative systems, not yet close to a real theory of turbulence.
What Are the Measures Describing Turbulence?


  1. I should probably add a note about my personal bias on this topic. To me stochastics are ugly, and dynamics are beautiful. It seems like a sin to destroy insight into the dynamics by assuming them away.

  2. Also, "chaoplexic cathedrals lacking predictive credibility." That popped into my head yesterday while I was shoveling $h1t (no joke!). Need to stop reading climate blogs...

    This is actually something Tobis is worried about too:
    I have a great deal of doubt that building ESMs is a good use of scientific time and effort.

    I asked Chip whether these efforts weren't vulnerable to an accusation that they have too many degrees of freedom and not enough constraints, which he freely admitted. Nevertheless he insisted that ESMs were bound to produce interesting results. I am not at all convinced that this is possible.

    The insiders are worried that's where we're headed, the outsiders and old men think we're already there.

  3. Interesting points.
    A.Smith only shows his ignorance of the issue and still keeps talking about "perturbations".
    Despite a good albeit much too general link heavily relying on ergodicity in its arguments, he didn't understand yet that there are no "perturbations" in a chaotic system.
    While there may and must be an external force or energy supply to the system, the dynamical trajectory is just a solution of the ODEs where everything is already accounted for .
    What is a "perturbation" of a 3 body system ?
    A 4th body ?
    That's ridiculous because then we'd have a 4 body problem and not a 3 body problem.
    And a 4 body problem would again be as chaotic as a 3 body system just more complex with 1 equation more and still no "perturbations" .

    I am interested by your "forced" variation of Lorenz .

  4. So Joshua what is the progress with the "forced" Lorenz ?

    Btw when you say that you want to examine a periodic "forcing" of the Lorenz system , do you mean that you want to solve the 3 Lorenz equations with rho explicitely given as sin(W.t+f) ?
    I ask because the Lorenz system is already (per definition) forced and the "forcing" is the rho term .
    I vaguely remember that in another post that I can't find you said something to the effect that this was NOT what you were going to do .

    But you have not really the choice . If you want to have a variable "forcing" and still stay with the Lorenz model , you have only one solution - make rho a time function instead of a constant .
    Anything else would only look like a Lorenz system from the mathematical point of view but the underlying physical system would be completely different and possibly even non existent .
    In the latter case , the (mathematical) results would have no physical interpretation and the (mathematical) exercice would have little use in physics .

  5. Hi Tom,

    The forced system is still on my queue, but it keeps getting pushed down the stack by "real" life.

    I agree with you about several things. Adding a non-homogeneous term the way I was planning will break the connection of our maximal truncation to the original physical system of interest. I still think it's useful to play with mathematical toys this way. Though, you're right about being circumspect about what insights we draw from such "playing" towards the behavior of real systems.

    Since Dewitt linked that old discussion, I had the idea after browsing it to demonstrate some of the points G. Browning was trying to make using the Lorenz '63 system with non-homogeneous forcing. Possibly even a pseudo-proof-of-concept about our competing conjectures, but that one's a bit more speculative. I'm not sure I can do it with such a simple toy ; - )

    Just to keep track, some possibly relevant discussion at Lucia's: [1] [2] [3]

  6. Hello Joshua

    For the variable "foprcing" I linked you http://arxiv.org/abs/chao-dyn/9405012 precisely because this is the only way to get a true Lorenz with a true variable "forcing" .
    Anything else would not represent a real flow that the Lorenz equations approximate .

    As I didn't visit the thread you mention after my intervention , I didn't even know that there were "competing conjectures" :)

    Just to be clear - my "conjecture" is a very qualitative way to examine the origin of chaos (not turbulence which is just a particular case of chaos !) in fluid flows .

    For that I pondered 3 things we know :

    - The Kolmogorov turbulence theory which I call privately "statistics of whorls" makes a homogeneity and isotropy assumption .
    As the real flows are neither homogeneous nor isotropic (unless in some particular cases) , I didn't expect any deep answers anyway . Correspondingly I am not surprised that the theory breaks at small scales and that the Kolmogorov scale is not some magical scale where any numerical attempt yields a perfectly correct and robust solution .

    - When one starts with a Rayleigh Taylor system in equilibrium , one can compute N-S on Kolmogorov scales untill one gets blue in the face and always gets the same answer - nothing happens .
    As this is not what is observed , people studied the system at vastly sub Kolmogorov scales . Actually they didn't even use N-S , they used the real microscopical molecular forces and potentials at angstroem scales .
    The WHOLE system was about 0,01 ┬Ám :)
    Well not only did they get the Rayleigh-Taylor instability but also produced the following chaotic flow which was better described than what N-S did (after an artificial perturbation by hand which was necessary to get something else than f=0 solution) .

    - At the molecular scale it is known that a hard billiard ball model is chaotic (and ergodic) .

    OK so now I displayed a , for me uncharacteristic , utter lack of mathematical rigor and qualitatively concluded that the origin of chaos in fluids is the upscaling of the molecular chaos at scales where N-S is not valid because the continuum approximation is trivially wrong .
    To use an image - I see it like the molecular chaos which is "chanelled" towards bigger scales where it is "shaped" by the specific form of the N-S .
    So N-S is in this picture a kind of meta boundary condition which forces the molecules to only specific collective behaviours at large scales . On the extremes those collective behaviours are very wise (laminary , Reynolds 0) or unwise and stochastic (full blown turbulence , Raynolds infinite) .
    And then there are all the cases with Raynolds between 0 and infinity which is the "normal" chaos that we see around us .

    Hence the conjecture that the ORIGIN of chaos in fluids is at scales where N-S breaks down .
    However to be complete , I immediately add that even if N-S is not the origin , it is still necessary to shape this molecular chaos in specific forms like f.ex a sea surface under a storm .

  7. Tom, sorry, I'm not sure why, but the spam filter keeps quarantining your comments.

    Your longer explanation (and this emphasis: the conjecture that the ORIGIN of chaos in fluids is at scales where N-S breaks down) makes more sense to me than the short comment at Lucia's. Especially the part about needing the initial perturbation to set off the RT instability.

    The group that did that paper I linked about the Richtmeyer-Meshkov instability also did some converged solutions of Rayleigh-Taylor instabilities. I think the results high-light that continuum breakdown is important (but not always). Since we *know* NS underpredicts shock thickness and entropy generation (because of continuum breakdown) the shock-driven RM instability results where they weren't able to converge a numerical solution (even in distribution after ensemble averaging) provide a meaningful contrast to the RT results where they could get converged solutions (and good match to the experimental results, which are all over the place because of sensitivity to particular equipment set-ups / data collection procedures, and they were able to diagnose this convincingly). This is why I said to SteveF that I think your conjecture is right for certain flows, but not others.

    It could come down to definitions too. If "chaos" is defined as sensitivity to infinitesimal perturbations, then there could be turbulent flows (like the RT example) where you get turbulence but not chaos because below a certain length-scale the perturbation doesn't matter. For a continuum equation set, you can't get more infinitesimal than molecular motions.

    So, with that definition of chaos and distinction from turbulence, I think I'd agree with your conjecture. Not sure though, I have to think about it some more.

    As always, thank you for taking the time with your comment to make me think!

  8. You are right Joshua , there is this matter of definition .

    While we are in a perfectly well defined and studied domain with the temporal chaos , e.g it is mathematically a study of a system of N non linear ODE and physically a study of orbits and volumes in the phase space (Lyapounov coefficients , mixing etc) , it is much more fuzzy and muddy for spatio-temporal "chaos" .

    I have always found it extremely dangerous and misguiding to use the same word (chaos) for 2 very different processes . I have also written several posts on this subject .
    I am pretty sure that 99,9 % of people don't realize that the spatio-temporal "chaos" happens in an infinite dimensional Hilbert space where elements are not points whose coordinates are the degrees of freedom (easy to understand) but FIELDS in an uncountably dimensional space of functions (a bit harder unless one was trained in QM like I was) .

    That's why I find it difficult (and often ill defined) to talk about "chaos" in fluids . It is impossible to visualize . When one talks about "attractors" then it means a set of FUNCTIONS - dense in the sense of the L² norm and one can't make a picture of that .
    Etc .

    So yes , when one discusses N-S , turbulence and "chaos in fluids" , then I guess that everybody has just a vague and intuitive notion of what he is talking about and that's why misunderstandings are common .

    For my part there is still the important notion of sensibility to initial conditions but the mathematical treatments are much more hardcore than in the temporal chaos .
    In temporal chaos you just measure ||f(x0,t) - f(x1,t)|| with the same f (the system's dynamics) and an euclidian norm in a finite dimensional space . Looks easy enough .
    In spatio-temporal chaos you measure ||Fx0(t) - Fx1(t)|| with different initial fields (Fx1 and Fx0) and an L² norm in an infinite dimensional Hilbert space .
    That's harder .

    What it gives when one is doing it seriously is f.ex : https://www.ima.umn.edu/preprints/Jan83Dec83/21.pdf .
    I am linking it just as an example of a seminal paper dealing with this important matter of "chaos" in fluids .
    Of course you will note that in this case there is no vagueness and everything is well and sharply defined :)

  9. This comment in the Quarterly Journal of the Royal Meteorological Society from 2003 is relevant to the silliness of statistical mechanics / thermodynamics hand-waving about turbulent flows:

    In a recent paper, Shimokawa and Ozawa (2002) formulate the hypothesis that a nonlinear system is likely to move to a state of maximum entropy production by perturbation. They base this hypothesis on results pertaining to transitions among multiple steady states of thermohaline circulation, obtained from an oceanic general-circulation model. In this comment it is shown that their result linking the increase of entropy production to a state of dominant stability does not re ect a universal trend but is, rather, system-specific.
    In conclusion, a general thermodynamic principle underlying the evolution of nonlinear systems such as atmospheric circulation and related processes seems not to be available at the present time.

    Borrowing unwarranted credibility seems to be a standard rhetorical tactic for alarmists.