Chaos and Weather

Guest Essay by Kip Hansen – 25 July 2020

 

“The pioneering study of Lorenz in 1963 and a follow-up presentation in 1972 changed our view on the predictability of weather by revealing the so-called butterfly effect, also known as chaos. Over 50 years since Lorenz’s 1963 study, the statement of “weather is chaotic’’ has been well accepted.”  Thus begins the abstract of a recent paper titled “Is Weather Chaotic? Coexisting Chaotic and Non-Chaotic Attractors within Lorenz Models”  [link to .pdf   link to PowerPoint presentation

The authors include B.-W. Shen, R. A. Pielke Sr., X. Zeng, J.-J. Baik, S. Faghih-Naini, J. Cui, R. Atlas, and T. A. L. Reyes.    Readers who follow the field of Chaos at the specialty group Chaotic Modeling and Simulation  will be familiar with Shen and Zeng.  Those who follow climate issues will recognize Roger Pielke Sr.

Here are the cites and links for studies by Edward N. Lorenz referenced in the above: 

Lorenz, E., 1963a: Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130-141.

Lorenz, E. N., 1972: Predictability: Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? Proc. 139th Meeting of AAAS Section on Environmental Sciences, New Approaches to Global Weather: GARP, Cambridge, MA, AAAS, 5 pp.

Edward Norton Lorenz: ”His discovery of deterministic chaos “profoundly influenced a wide range of basic sciences and brought about one of the most dramatic changes in mankind’s view of nature since Sir Isaac Newton,” according to the committee that awarded him the 1991 Kyoto Prize for basic sciences in the field of earth and planetary sciences.”    [source]

Shen et al. (2020) is a very interesting and deep study that attempts to answer what appears at first to be a simple question:

Is Weather Chaotic?

Now, as in all of my essays regarding Chaos and Climate:

Chaos & Climate – Part 4: An Attractive Idea

Lorenz validated (at Judith Curry’s  Climate Etc.)

[Note: Due to server changes over the years, some illustrations in these essays may appear as blank spaces.  Clicking on the blank space may bring up the missing image in a new tab/window.]

It is vitally important to realize that there are two distinct definitions of Chaos (and its adjective form – Chaotic).  Merriam-Webster has finally caught up with the science and offers this:

Chaotic —  adjective

cha·​ot·​ic | \ kā-ˈä-tik  \

Definition of chaotic

1: marked by chaos or being in a state of chaos : completely confused or disordered a chaotic political race After he became famous, his life became even more chaotic. They may look chaotic and barbaric, but scrums are a critical and strategic part of the game, and they unfold and escalate according to hockey’s venerated, unwritten rules of engagement.— David Fleming To the uninitiated visitor, the seemingly chaotic energy of a typical Thai market may give the impression of a free-for-all, …— Diane Ruengsom

2 mathematics : having outcomes that can vary widely due to extremely small changes in initial conditions In other words, what comes out of the program’s equations is extremely sensitive to what goes in. And that, as any mathematician would recognize, is one of the hallmarks of chaotic systems.— Ingrid Wickelgren A physical system—a weather system, say—is chaotic if a very slight change in initial conditions sends the system off on a very different course. — Physics Today

Shen et al. in this study   (and other earlier papers) are trying to get a handle on the question posed.  They want to know if the chaos that Lorenz (definition 2) found in his early toy weather model, which led to the accepted concept that “weather is chaotic” meant that weather (as we experience it in the real world day-to-day, week-to-week and month-to-month) is really chaotic (as in definition 1 – completely confused or disordered, random, stochastic and in longer time sense, unpredictable).

Some people have an understanding of “generalized, high-dimensional Lorenz Models (GLM)” – they can wade through the fascinating  published study (again, here).  The rest of us might have an easier time with the PowerPoint presentation (here), though it is no walk in the park either.

Here I will show a couple of their figures and comment to make them intelligible in light of my own five earlier Chaos and Climate essays and then wrap up with Shen et al.’s Bottom Line points.

This figure illustrates the three types of solutions found within their 3 Dimensional Lorenz Model. 

The first (panels a and d) is a Point Attractor – the Wiki gives examples here.  The important thing to understand is that no matter where the model is  started (Initial Conditions – or IC), the system (represented by the blue dots (so closely spaced they form a line) in (a) start at the end of what appears to be the tail, and converge on the solid blue spot on the left.  In (d) the same system starts mid-range, jumps up to a high range, then drops and begins to cycle up-and-down, converging on a single value.  (I covered this in my essay Chaos & Climate – Part 2: Chaos = Stability)

Panels (b) and (e) illustrate a system that enters into a chaotic state – a wholly  deterministic but essentially unpredictable two-lobed chaotic attractor.   Looking at panel (b) alone, one might fool oneself into thinking that this is a periodic system – it is not.  The sequential numeric results – each iteration – do not go around the two lobes like a record needle on an LP vinyl record.  Panel (e) shows that this system starts like panels (a) and (d) but instead of settling down to a single value, it increases steadily until it breaks into chaos around the x-axis value of 18 or so.  I used the following illustration using Robert May’s Population Dynamics formula to produce this:

 The red-circled portion is a bit of “nearly periodic”, nearly repeating pattern.

Lastly, Shen et al.’s (c) and (f) show a truly periodic attractor.  Periodic attractors can have any number of periods, or repeating values, as I showed here:

Shen’s panel (f), for example,  seems to have a period of six. 

Co-existing Solutions

This is Shen’s Figure 4 – showing the results of 256 differing solutions from 256 different Initial Conditions (ICs).  They find that some of the ICs produce chaotic orbits with a recurring “saddle point” and some of the ICs produce non-chaotic obits that eventually approach one or the other of two stable point attractors. 

The import of this is Shen et al.’s conclusion that:

In this study, we provide a report to: (1) Illustrate two kinds of attractor coexistence within Lorenz models (i.e., with the same model parameters but with different initial conditions). Each kind contains two of three attractors including point, chaotic, and periodic attractors corresponding to steady-state, chaotic, and limit cycle solutions, respectively. (2) Suggest that the entirety of weather possesses the dual nature of chaos and order associated with chaotic and non-chaotic processes [my bold – kh], respectively. Specific weather systems may appear chaotic or non-chaotic within their finite lifetime. While chaotic systems contain a finite predictability, non-chaotic systems (e.g., dissipative processes) could have better predictability (e.g., up to their lifetime).

The refined view on the dual nature of weather is neither too optimistic nor pessimistic as compared to the Laplacian view of deterministic unlimited predictability and the Lorenz view of deterministic chaos with finite predictability.”

And further report that:   

“The refined view may unify the theoretical understanding of different predictability within Lorenz models with recent numerical simulations of advanced global models that can simulate large-scale tropical waves beyond two weeks (e.g., Shen 2019b; Judt 2020).”

Cites:
Shen, B.-W., 2019b: On the Predictability of 30-Day Global Mesoscale Simulations of African Easterly Waves during Summer 2006: A View with the Generalized Lorenz Model.   Geosciences 2019, 9, 281.   https://doi.org/10.3390/geosciences9070281

Judt, F., 2020: Atmospheric Predictability of the Tropics, Middle Latitudes, and Polar Regions Explored through Global Storm-Resolving Simulations. Journal of The Atmospheric Sciences, 77, 257-276. https://doi.org/10.1175/JAS-D-19-0116.1

I encourage readers to at least make an attempt at reading and understanding this study and its implications for weather (and thus, maybe, climate) prediction.

# # # # #

Discussion:

In this section, I discuss my own observations on the issues raised by Shen et al. (2020).  These are not to be confused with the findings and opinions of the authors of Shen et al.

  1.  As in all of these studies of Chaos – the study of non-linear dynamical systems — which in many cases might more correctly be labelled “chaos in numerical models” – it is imperative not to confuse the resultant numerical chaos (chaotic results) with the real world results.  For instance, Robert May’s Population Models (see  “Simple Mathematical Models With Very Complicated Dynamics  June 1976 Nature 26(5560):457  DOI: 10.1038/261459a0”  )  However, natural non-linear dynamical systems do produce in the  real world the phenomena similar to those seen in numerical models of non-linear dynamical systems. 
  • Those who have read my series on Chaos and Climate (links at beginning of this essay) have already been exposed to the ideas that Chaos produces stability (single-point attractors), periodicities, and chaos (deterministic chaos, which is intrinsically unpredictable).  All three types of solutions are derived from the exact same formulas while changing inputs (see the bifurcation diagram and illustration below).  Inside the chaotic region of solutions to a single dynamical system, one again finds areas of periodicity.  These are marked by the vertical colored lines passing through the system plot at 2, 4 6, 8 points – the periodicities.

Shen et al. have found the same in simple Lorenz models and in generalized multi-dimensional Lorenz weather models and have found that a single system can simultaneously contain both chaotic and non-chaotic regions, “Each kind contains two of three attractors including point, chaotic, and periodic attractors corresponding to steady-state, chaotic, and limit cycle solutions, respectively.”  Some of these solutions are/should be/could be  predictable to some extent. Shen et al. believe “that [their model] can simulate large-scale tropical waves beyond two weeks”.   Maybe they can. It is a start, at least. 

  • At the conclusion of my earlier essays on Chaos and Climate, my Bottom Line was:

“It is the patterns of the past, repeating themselves over and over, that inform us in the present about what might be happening next.  Remember, chaotic systems have rigid structures, they are deterministic, and Chaos Theory tells us we can search for repeating patterns in the chaotic regimes as well.”

This, to me, appears validated somewhat by what Shen and his co-authors have found in their  generalized, multidimensional Lorenz models and, maybe, in the large scale weather phenomenon known as  “African Easterly Waves (AEWs)”.

Shen at al. find what I would have expected.  It is reassuring though that they do find two different kinds of chaotic attractors in their nonlinear dynamical system models – generalized   multidimensional Lorenz models.  This finding validates that weather models, at least, are truly Chaos- Theory-chaotic.

# # # # #

Author’s Comment:

It is encouraging to see that serious climate scientists are pursuing the very underlying nature of weather and climate,  acknowledging that they are nonlinear dynamical systems that have all the classic features of Chaos. 

I am not surprised that Shen, Pielke, and the other authors are encouraged by finding that they might be able to predict at least large scale weather features, such as African Easterly  Waves more than two weeks into the future.  That feat, if true, exceeds the expected limit for weather prediction.  They are doing it through pattern-recognition, of course, but it is still a real feat.

Until Climate Science, as a whole, fully recognizes climate as a non-linear dynamical system, and understands the implications of its deep chaotic nature, there will be little progress made in long-term prediction.  Currently, CliSci is stuck on the idea that “averaging” multiple chaotic outputs to find “ensemble means” actually tells us something other than the trivial “mean” of those particular runs of that particular model with its particular parameter inputs.  That idea is nonsensical.

Lastly, a couple more reference links:

Gleick, J., 1987: Chaos: Making a New Science, Penguin, New York, 360 pp. 

Lorenz, E., 1963b: The predictability of hydrodynamic flow. Trans. N.Y. Acad. Sci., Ser. II, 25, No. 4, 409-432.

Read widely, think for yourself and think critically.

# # # # #

135 thoughts on “Chaos and Weather

  1. IPCC’s TAR Report Chapter 14
    https://www.ipcc.ch/site/assets/uploads/2018/03/TAR-14.pdf
    Page 774 section 14.2.2.2 which among other things, says:

    “In climate research and modelling, we should recognise that we are dealing with a coupled non-linear chaotic system, and therefore that the long-term prediction of future climate states is not possible.”

    Besides that, I really liked your pendulum example from one of your previous offerings:

    • Manic Climate modeler says you just have to statistically average the motions of these 6 individual pendulums to get a prediction (projection) of the overall trajectory of the pendulum ensemble system .

      • tom0mason…you just have to statistically average the motions of these 6 individual pendulums to get a prediction (projection)…

        OK, you do that with your stock market model.

      • Am *average* trajectory tells you nothing about the trajectory of each individual pendulum other than there might be some congruent points in the trajectory of each (? points of attraction).

        In addition, the pendulums are forced into a damped movement because of friction and the damped movement is not synchronous in time, i.e. each pendulum traces a different damped path as time moves forward.

        Neither is there any external energy added to the pendulums as there is with weather and climate.

        Main point – an average trajectory for these six pendulums is as useless as the “average global temperature”.

        • Steve Case and Tim Gorman,
          Glad to see you got my (sarcastic) message.
          Averaging chaotic (or near chaotic) behavior is as useful as chocolate rocking-horse shi excrement.
          With chaotic progression absolute initial conditions and the minutia of all the variables matter.

          • It’s even worse when the initial conditions change from minute-to-minute, hour-to-hour, day-to-day, week-to-week, and year-to-year. A model primed with Initial Condition Set A will give one answer, the same model when primed with Initial Condition Set B (say 24 hours later) will give a different answer. If the model does *not* give a different answer then something is wrong.

  2. “There are doing it through pattern-recognition, of course, but it is still a real feat.“

    Did you mean They are…?

    • Derg ==> Thanks, appreciate readers who really read closely. Auto-spelling correctors coupled with the difficulty of editing one’s own work (one always already knows what one meant to say and so reads it that way) fresult in these little errors. Thanks for pointing it out.

      • Kip
        I presume that “fresult” is a newly invented word that means a no-cost result. 🙂

      • Kip,
        I find that copying the contents into a different kind of document, resulting is different line lengths, a different font point size, maybe a different font, often lets me see errors that were invisible (to me) where I created the text.

          • oops — published too soon.

            I use several text processing programs — each with different spell checkers.

            In the end — it is down to my inattention to the smallest of small details.

  3. I’m surprised by the authors saying this:
    “Steady state or nonlinear periodic solutions have no (long-term) memory
    regarding their initial tiny perturbations
    Ø initial tiny perturbations completely dissipate”

    It seems to me that weather would be hysteresic.

    • Robert ==> The key to that fact is “Steady state or nonlinear periodic solutions” — in these cases (and they are pat and parcdel of all non-linear dynamic systems) the Initial Condition, the starting point does not effect long-term behavior.

      I demonstrated this in Chaos & Climate – Part 2: Chaos = Stability. A system that (under a particular set of parameters) produces a single-point attractor goes to that single point no matter where it is started. even attempting to kick it out of its pattern mid-stream only temporarily effects it, it reverts to the single point solution affect a few iterations.

      See the paragraph starting with “Even though the starting value for x is 0.2…” and its illustration.

      Changing the parameters — the other values — will result in a different single-point, a periodocity, or chaos.

      We see this in the real weather — weather is as predictable as it is because these little differences day to day don’t greatly change the seasonal patterns. This is why Part 2 talks about chaotic stability — hugely important to understand how the chaotic nature of weather and climate often demands and enforces stability as opposed to instability.

      • Kip,

        “A system that (under a particular set of parameters) produces a single-point attractor goes to that single point no matter where it is started”

        The problem is that this statement assumes there are no new inputs to the system or no changes to the system inputs after the starting point. Yet this simply doesn’t apply either to weather systems or the global climate system. Small changes in the sun’s output, in cloud cover, in vegetation cover, in other weather systems (i.e. moving low and high pressure system), etc, will all change the the single point attractor location. And it is that attractor point that actually spells out the weather.

        It’s why even 24hr or 48hr weather predictions can be wrong. Heck, even 8hr predictions can be wrong.

        “even attempting to kick it out of its pattern mid-stream only temporarily effects it, it reverts to the single point solution affect a few iterations.”

        This just isn’t the case with weather and climate models. The chaotic model can be predicting 100% rain in the next 24 hr period and can be totally wrong for any number of “mid-stream” reasons. And the rain may never actually happen at all.

        I agree that a static model behaves as you describe. Weather and climate are never static, however. They are dynamic. And just a little dynamism can change your single point attractor. It’s worse for climate models than weather models. Climate models don’t handle dynamism very well, see clouds for instance.

        • Tim ==> You are quite right — but you may be “zooming in” too far.

          The point of Chaotic Stability is the point you are making — but you may be “too close” to see it. Despite “Small changes in the sun’s output, in cloud cover, in vegetation cover, in other weather systems (i.e. moving low and high pressure system), etc, will all change the the single point attractor location.”

          There is no “attractor location” — that is some weather modellers pipe dream. No one thinks that the weather system is that fine-grained as a system. .

          The stability in weather and in climate is that we seldom (but not never) see snow in the Hudson Valley of New York in July. The Global Average Surface Air Temperature has been stable within a few degrees C for millennia (as far as we can tell from here) and what excursions there have been are to Ice Ages and back again (which MAY be a two-pole climate “attractor”) . Long-term pseudo-stabilities appear as deserts and rain forests (which, over time, change their climatic character).

          And, absolutely, climate models/the climate system is dynamic — it is a non-linear dynamical system and all that that fact implies.

          • Kip,

            “The Global Average Surface Air Temperature has been stable within a few degrees C for millennia (as far as we can tell from here) and what excursions there have been are to Ice Ages and back again (which MAY be a two-pole climate “attractor”) .”

            There is no such thing as a “Global Average Surface Air Temperature”. Climate is determined by the temperature envelope and not the average temperature, be it a local averge, a regional average, or a global average. Two widely separated points on the globe can have the exact same average temperature while having vastly different temperature envelopes. Taking their average tells you exactly nothing about what is going on locally, regionally, or globally.

            If you are getting the same average temp for an ice age and a warming period within a few degC then something isn’t kosher. For the temperature envelope at any specific point on the globe simply cannot be the same during an ice age and a warming period.

            If the average global temp for an ice age and for a warming period is the same within a few degC then exactly what is all the alarmism about? That we will never have another ice age? I’d like to see the climate models try to prove that one!

          • Tim ==> When we are looking for Earth-like planets, we look for planets with a broad swath of land that has a generalized temperature around 15 degrees C. Earth has been so for a long time (and just barely got back up there according to NASA GISS.

            I am no fan of Global Average Temperature by any measure…which you will know if you’ve read any of my stuff here….

            None of this has anything to do with the main thrust of the study under consideration or this essay.

        • In Northern Virginia, we’ve been watching the day-to-day failure of weather predictions for the past two or three months. For weeks, the daily and ten-day forecasts showed 80% or higher probability of rain, and the sky was cloudless. No rain anywhere near us. Then a week or so ago, it rained on two consecutive nights when the probability given in each day’s forecast was zero. A couple of days ago, we had a 10% chance, and had one of the more torrential downpours I’ve ever seen. Easily 0.5 inches in ten minutes. It was so heavy that the water broke through a leak repair in our kitchen wall, and we had to deploy towels and bowls to keep up with it. Weather Underground hastily upped the chance of rain to 90%, and under predicted the total by about an inch. There’s nothing new about this inability to predict our weather.

          • Michael Kelly ==> You certainly have that right — not even with minute-to-minute weather radar is weather prediction accurate on a very local scale. A huge thundershower forced my wife and I to pull to the side on the NY State thruway, a mile from home. At home, my son was disappointed that his garden got zero rain.

      • But we know in fact that in the long run climate does change greatly and climate is just an average of the weather. The best model which describes long term fluctuations considers both the minor perturbations and forcings changing the attractor.
        https://www.tandfonline.com/doi/pdf/10.3402/tellusa.v34i1.10782
        https://journals.aps.org/pre/abstract/10.1103/PhysRevE.101.062145
        https://www.youtube.com/watch?v=HbJ_I3xbIMg
        https://www.nap.edu/read/10136/chapter/5

        Shouldn’t dampening and resonance effects of be important in weather prediction?

        • Robert W. Turner ==> If we knew the exact right model for weather prediction — we’d be able to predict it!

          The two-well resonance concept may just “look similar” — no idea if it is a valid approach for weather prediction. Somehow I doubt that applying knowledge of atomic level Brownian motion to huge systems, such as the Atlantic Ocean, will produce actionable knowledge.

          The similarity seems to be that these systems have two semi-stable phases and they shift one to another.

  4. Nicely done Kip:

    Minor technical issue: Your charts are being rendered on my ancient version of Linux Firefox as an empty image frame followed by the actual image. No information lost (I think), but a bit disconcerting. I had to look at the page html source to figure out that much. If you know what the problem is and it’s fixable at your end, you might want to fix it. But I imagine it’s fine with (some? all?) other browsers. Personally, I’ve long since given up expecting web pages to render sanely if done in anything more “sophisticated” than HTML 4.01 Transitional.

    Serious question: Given than chaotic phenomena seem to be a bit difficult (i.e. more or less impossible) to predict, is there any way to actually test whether climate is a chaotic phenomenon?

    • Don K ==> WUWT is back on WordPress and the current posting interface is different than the last version. I am uncertain exactly what it is doing but my versions of Firefox seem to render the page properly. The older linked essays, however, do show some problems, as I noted.

      Microsoft Edge shows the phenomenon you mention.

      I’ll dig into the code and see what’s up.

      * * * * *
      “. . .is there any way to actually test whether climate is a chaotic phenomenon?”

      Essentially, this is what Shen et al. is about, though they start with checking to see if weather is really chaotic in nature. They have answered “Yes” – and hope that the repeating patterns (single point attractors and periodocities) will allow predicting weather phenomena out past the apparent, accepted two-week limit.

    • Don K ==> Formatting Issue — as they say in deep tech circles “We Are Working On It!”

      • Kip: “We Are Working On It!” — whatever you did seems to have worked. … For me at least … For others? Who knows? It’s not like html is fully determinant — Tim Berners-Lee’s probable initial intentions notwithstanding.

        “repeating patterns (single point attractors and periodocities) will allow predicting weather phenomena out past the apparent, accepted two-week limit.”

        Isn’t predicting weather for relatively long terms based on patterns what Joe Bastardi is about? At least sort of. Personally, I find Bastardi kind of offputting so I don’t follow his stuff closely. But that doesn’t mean he’s wrong, or his approach is without merit.

        • Don K ==> “Working on It” — the never-tiring Charles-the-Moderator has taken care of it for me…something to do with a new system for inputting new essays on the server.

          Predicting the weather from long-term patterns and short-term actualities is what ALL weather forecasting is about, You know that moist warm air heading east towards your town will rise as it crosses the local hills, cool off, and bring potential rain showers….

          Bastardi is a great speaker, heard him at the Heartland confernec ein Washington last year. He doesn’t do my area’s weather.

  5. I’ve never understood people’s fascination with the butterfly effect.

    The concepts of ill-conditioned and ill-posed systems (systems whose outcomes are extremely sensitive to slight changes in starting conditions, and thus in practise are unpredictable) have known about for more than a century. They are not quite the same as chaotic systems, but their conclusions are the same. Some systems are near impossible to predict in the long term.

    • Killer ==> Read Lorenz’ 1972 paper (link near the beginning of the essay) he clears up a lot of misconceptions about the Butterfly Effect.

      • It just occurred to me that we have a human “wing flap” event to look at that should have produced something noticeable. The date of the event was October 30, 1961.

        Was there any unusual weather after that date? I haven’t looked, but I know that many here have weather data bases that should yield some information – if any is to be found.

        The event was the detonation of the RDS-220, popularly known as the “Tsar Bomba” – a Soviet nuclear device yielding the energy equivalent of 50 million tons of TNT. That’s far more than the entire combined energy yield of all of the explosives used in all wars in history. It’s also more than twice the energy yield of the 1980 Mount St Helens eruption. Further, all of that energy was liberated in roughly one tenth of a microsecond, making it the third most powerful event ever to occur on Earth after the collision that formed the Moon and the asteroid impact that killed the dinosaurs.

        That’s a big flap. What were the results?

  6. I remember attending a talk about thirty years ago on randomness (it seem to me that the term chaos has simply replaced the word randomness. Is there a difference?). The gist if the talk was that in some forms of randomness, the random points were bound to a center point, but over time, that center point changed location, randomly. Fractal theory was still in vogue, so I considered you could scale up the randomness into ever-larger fractals.

    WRT reality, the end result was, if you could determine the center point, you could put boundaries on your predictions, and improve upon them. And that would work until it failed, which meant you had to find a new central point.

    How any of this could help anyone, I haven’t a clue., but it was interesting.

    • jtom ==> You sound like you’d benefit from reading either my Chaos Series (links in essay) of even one of the major popular science books on Chaos, such as

      Gleick, J., 1987: Chaos: Making a New Science, Penguin, New York, 360 pp. (link at end of essay).

      Randomness — stochastic processes — are very very different than Chaos Theory “chaotic processes”which are entirely deterministic but unpredictable.

      Repeated coin tosses are truly random, there is no pattern and no predicting what the next toss will be (other than 50-50).

      In Chaos, the next “flip” (next data point) is not predictable either — but is strictly determined by the system.

      • Kip: While I get your point about a classic coin flip (with a fair coin) being random, one could argue that it is actually deterministic and predictable. If you know the specific physical parameters – coin weight, diameter, flipping impact force, area, point of contact, height above landing point, etc.) you could create a mathematical model to accurately predict the outcome. It follows that you could construct a machine that would be capable of reproducing these specific parameters to flip a coin that would always result in “heads”.

        In fact, it is well known in the gambling industry that it is very difficult to create devises that have truly random outcomes. Consider that the balls used in lottery picking machines are carefully guarded and frequently measured for weight, diameter, roundness, etc. to ensure against tampering. There is a lot of literature regarding tests for randomness and discussion about how difficult it is to create random number sets that pass them.

        • Rick ==> They have yet to accomplish such a feat — no accurate coin flip machine — no models that predict the outcomes (other than 50-50).

          Chaotic systems are not RANDOM — just unpredictable . . . . they are entirely deterministic.

          • But, there seems to also be randomness in the climate system. An unusually large solar flare, a large volcanic eruption, a large meteor strike, are examples of new inputs that, at least according to current views, can result in rather long term changes. Current astrophysics says the sun is not only dynamic but evolving. If true, there are likely to be very long term, no return accepted, changes in this planet’s weather systems – without any changes in the physics that determines just what occurs from what input.

          • Kip: I don’t disagree, but are you saying a coin flip is deterministic but chaotic or random? Is it a difference without a distinction?

          • Rick ==> A coin flip is RANDOM but NOT CHAOTIC.

            To understand the difference is the problem that CliSci has….it keeps confusing the two.

            Read or re-read my series on Chaos and Climate — read the Wiki Chaos Theory entry — Read Glieck.

            It is NOT easy to understand until one immerses oneself a bit into the topic – – – -once a non-linear dynamical system enters into the Chaotic Real, (region), the subsequent results are “random” iand unpredictable n that they are unpredictable without calculating them individually in order. But they are entirely deterministic.

            Using the exact same formula and exact inputs (precisely exact) and the exact same calculator (even down to the exact operating system and sometimes processor) and doing exactly 1972 iterations, all the data points will be exactly the same and the next answer, the next data point, will be exactly the same.

            This is not true with any real coin toss — you will never get the exact same sequence over a long period — it is truly random in practicality.

        • Rick ==> The gambling industry only needs approximately random outcomes — the odds are in their favor sufficiently to overcome any small deviations — say favoring the roulette wheel number 33 — which no gambler will ever accidentally discover anyway.

          Now, random number sets that are absolutely truly random is a problem if you are a NSA cryptologist. (I was once-upon-a-time a cryptologist — my official desk sign said “Mr. Crypts” — it was a rather odd outfit).

  7. A great piece of writing Kip Hansen. Well said!

    Without a basic understanding of the chaos in the system, predictions, or error filled analysis of inaccurate global factors and many ‘parameterized’ elements, fill the modeled projections, can not show the probable trajectory of the climate — EVER! Statistical averaging within the models removes the very chaotic (and noise) signals that needs to be analysed!

    Until Climate Science, as a whole, fully recognizes climate as a non-linear dynamical system, and understands the implications of its deep chaotic nature, there will be little progress made in long-term prediction. Currently, CliSci is stuck on the idea that “averaging” multiple chaotic outputs to find “ensemble means” actually tells us something other than the trivial “mean” of those particular runs of that particular model with its particular parameter inputs. This idea is nonsensical.

    Is the very nub of it!

  8. Chaos has an order obfuscated through incomplete or insufficient characterization and unwieldy processes.

    • And I would humbly offer, look at https://en.wikipedia.org/wiki/Fractal where they say “Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractals.[34]”
      Basically from relatively simple equations chaotic behavior is observed and can often be plotted as fractals. Fractals from chaos gives regular patterns of self similarity.

      • tom ==> Hmmmm …. sorta true. Like graphs of strange attractors, it requires very very specific parameters to get a “cool strange attractor” — trust me, in the 80’s I spent night after night trying to get some particular attractor that was shown in some textbook, with the proper formula, but no indication of the exact values need to achieve the graph…..

        It is because the fractal images are interesting that most graphs are of fractals — not because fractals just pop out every time you approach a non-linear dynamical system.

        That said, once one it attuned to SEE fractals in the real world, they are everywhere.

        While living at sea, I noticed that the wave pattern on the sea was the same no matter what scale it was look at — satellite images have the same pattern as looking down from the top of my mast. Wave patterns are fractal.

  9. Are glaciations and interglacials the product of chaos? I just don’t think so.

    And it is absolutely obvious to anybody with a modicum of brain that the flap of a butterfly’s wings in Brazil DOES NOT set off a tornado in Texas. What a ridiculous notion.

    Chaos is overrated and used as an excuse for things we don’t have a good knowledge of, starting from how tornadoes originate.

    • Javier ==> You should read Lorenz on that — the link is in the essay, but here it is again: Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?

      Lorenz’ rhetorical question is not meant to be taken quite so literally and he is happy to discuss it.

      The real topic is that of Sensitivity to Initial Conditions in Non-linear Dynamical Systems.

      PS: No one really thinks that a butterfly wing flag causes tornadoes……

      • “PS: No one really thinks that a butterfly wing flag causes tornadoes……”

        Some people do, believe me. Probably some of the same people who think breaking and burning things is protected speech.

        • Jeff ==> or that wearing a paper mask will protect you (or others) from infection by viruses.

          • The masks are not intended to protect you from GETTING covid – they are intended to limit the flow of YOUR outgoing breath. No mask, 3 feet. With a mask 2 1/2 inches. Walk into a walk-in freezer and breath. You will see where your breath goes. Put on a mask and look again.
            A huge number of people are infected but don’t know it. The masks are to limit THOSE people from EXHALING the virus. Since no one knows who that is, we are told to all wear masks.
            Doctors us the N95 masks to keep from getting infected. But their exhaust ports are open, so if a doctor is sick the mask is not preventing him from getting it – it is permitting him to SPREAD it.

          • John Shotsky ==> This is a controversial issue — if you wish to read the medical science on the issue, you can see this peer-reviewed recent meta-study May 2020 :

            A Xiao J, Shiu E, Gao H, et al. Nonpharmaceutical Measures for Pandemic Influenza in Nonhealthcare Settings—Personal Protective and Environmental Measures. Emerging Infectious Diseases. 2020;26(5):967-975. doi:10.3201/eid2605.190994.

            Link to the paper on the CDC web site is: https://wwwnc.cdc.gov/eid/article/26/5/19-0994_article

            Note: The journal is Emerging Infectious Diseases and the paper is published on the web site of the US C.D.C.

            The money quote from their abstract:

            “Although mechanistic studies support the potential effect of hand hygiene or face masks, evidence from 14 randomized controlled trials of these measures did not support a substantial effect on transmission of laboratory-confirmed influenza. We similarly found limited evidence on the effectiveness of improved hygiene and environmental cleaning. ”

            Please note that Covid-19 is a corona-virus caused illness, as are all the influenzas.

          • This thing about a person with zero symptoms spreading virus does not consider the false positive rate of tests. Take a population ‘real’ level of virus as 2% and a test that is 98% specific. If you test 1000 people, you will get 20 ‘real’ cases and 20 false positives. If the population rate goes down to 0.5%, you will get still get 20 false positives,and 5 real cases. Similarly, if the sensitivity of the test is 98%, you will miss 2 real cases for a 2% prevalence. Given the plethora of companies making tests on the fly, the sensitivity and specificities are all over the map. See the Spectator article for a very good discussion – hope its not paywalled.

            https://www.spectator.co.uk/article/how-many-covid-diagnoses-are-false-positives-

          • Fran ==> Its worse then we thought. Not only that, but they are often not testing for active cases, but anyone who has or has had Covid…..

            Many estimate that nearly everyone has been exposed – if not already, then soon, they will have been. That’s the nature of influenza pandemics…..and there is no getting around that.

            That fact is why we’ve never had any masking or social distancing for past pandemics – – – doesn’t really help in the long run.

    • Tornadoes are caused by being in an Ice Age.

      Global climate is not chaotic in terms human lifetimes.
      Weather is chaotic in terms human lifetimes.

      An Ice Age is dry, and drier world has more tornadoes, hurricanes, severe
      weather, more extremes in terms of hot days and cold days.
      In terms of dryness, in terms of other interglacial periods, we currently fairly dry.
      In terms of analogy and double pendulums and global climate, it seems we have cooling of deep oceans from cold water falling mostly in Arctic and geothermal heating of the ocean floor. And in timescales only being starting in centuries into thousands of years.
      In terms of global weather and analogy of double pendulums you tropical ocean heat engine and the El Niño & La Niña which changes within human lifetimes. And other weather and ocean patterns.

      • gbaikie ==> Using terms with acknowledged definitions is important. Shen et al. ask a valid question, and answer it in the affirmative — weather is Chaos-Theory-chaotic — which also means we see stability and repeated patterns as well as unpredictable chaos,

        That the climate is relative stable is a feature of the chaotic system that makes up climate.

        • Global climate is stable because it depends the massive ocean temperature which at moment has average temperature of about 3.5 C. And this ocean temperature in Ice Age has been within the range of about 1 to 5 C.
          And ocean at 1 to 5 C is cold ocean, and global climate with cold ocean are in an Ice Age.
          Earth history has had much warmer oceans and therefore was not in this global Icebox Climate.
          Or a warmer ocean globally makes the Earth less dry, and makes a more uniform global temperature {increases the average global air temperature}.

  10. The abstract concept of a mathematical attractor obfuscates the reality of a physical attractor. For the climate system, this physical attractor is the long term convergence to the laws of physics. Not only is an energy balance required, the change in entropy resulting from any change in balance must also be minimized.

    The fact that the long term ratio between the average surface emissions and the average total solar forcing is within 1% of the golden mean and independent of temperature, forcing or topography can indicate either a bizarre unexplained coincidence or is the signature of chaotic convergnce, as the golden mean often arises in the steady state solutions of chaotic systems.

    • co2isnotevil ==> In all my study of Chaos Theory, I’ve never run across the idea that “… the golden mean often arises in the steady state solutions of chaotic systems.”

      Can you give examples of references?

      • Kip,

        Fibonacci butterflies and fractal solutions arising from chaos (i.e. clouds) are examples where the golden mean can have relevance to chaos. Biological evolution also advances chaotically and this ratio similarly arises in converged biological solutions. Keep on mind that the chaos of climate is not about a chaotic steady state, but a chaotic path to a new steady state as conditions change where this path is modulated by the chaotic behavior of clouds.

        What makes this so interesting for the climate is that the converged ratio of surface emissions to incident solar energy is within 1% of the golden mean and is remarkably independent of temperature, forcing, latitude, topography or GHG concentrations. Something must be driving this ratio to its constant value, moreover; this ratio is converged to almost instantly relative to the time scales many believe relate to the climate’s response to change.

        I’ll even go out on a limb and make a prediction that when we discover an exoplanet whose steady state surface climate is chaotically converged to by clouds, this same ratio will emerge.

        • co2isnotevil ==> Interesting. Can you give specific examples of what you are saying? Just saying X converges to the Golden Mean doesn’t quite answer my question…..

          Have you got some links or other pointers to detailed examples?

          • Kip,

            If the golden mean is arising, it results from the chaotic self organized nature of the atmosphere by clouds. It’s more related to optimized self reorganization, then to the chaotic behavior of clouds that facilitates this reorganization. Although, in practice, you can’t have optimized self reorganization without some chaos to push you out of local minima. I’ve observed this developing place and route software, where adding the right amount of randomness in all the right places converges to better solutions. Examples of the golden mean arising from self organized structures is also wide spread in biology.

            I suspect that the goal of minimizing changes in entropy as the system changes state is what drives the atmosphere’s self reorganization towards some sort of optimum energy configuration, whose bulk properties can be characterized by the golden mean.

            Keep in mind that this is just a hypothesis, although I’ve yet to find a test that can falsify it and that’s not for the lack of trying.

          • co2 ==> Just one little link to an example — a paper, an essay, even a wiki article?

          • co2isnotevil ==> At last, a link to approximate Golden Means in Nature — it is a fascinating hobby to look for things that come close to the Golden Mean ratio . . . but it is not ubiquitous in Nature — for example, some flowers, not all flowers; some galaxies, but not all spiral galaxies,; some European faces, but not all faces; some body measurements, but not all bodies; this goes on for all the examples.

            Gleick discusses this in his book linked at the end of the essay.

          • Kip,

            I’ve had Gleick’s book for decades and have read all or parts of it several times since.

            Keep in mind that I’m not saying that I know for sure that the average ratio between the BB surface emissions and emissions at TOA MUST be the reciprocal of the golden mean. What I do know is that this ratio is the most tightly controlled and quickly converged relationship among any pair of climate related variables, its value, while highly dependent on the amount of clouds, is uncorrelated to any climate variable, its yearly average varies by only a few percent and that Earth’s long term average is equal to 1/g, where g is the golden mean, is well within the margin of error.

            I’m not saying that this demonstrably constant ratio must be 1/g, only that if it is, the reason is the chaotically self organized atmosphere between the BB emissions of the surface and space, where clouds manifest the chaos allowing an optimum state to emerge which I suspect is an organization of energy that minimizes the change in entropy as the system changes state.

            If you consider the state to be the temperature, and consider the hemispheres to respond independently, their response to seasonal change is to be continually changing state around some mean, so there’s reason for this natural system to converge towards an optimum energy configuration. Owing to asymmetries between hemispheres, even the global state is always varying around some mean. If the golden ratio has any significance, it’s only relevant is to the global mean and not the variability around it, other than to center the combined averages of all variability around some mean. None the less, the per hemisphere variables that have the closest average value to each other are the ratios between the BB missions of the surface and the emissions at TOA despite a significant difference in the average amount of clouds per hemisphere and the strong dependence of this ratio on the amount of clouds.

            Can you think of another reason for why this ratio would be so constant?

          • co2 ==> I have yet to see that clouds have any relationship to the Golden Mean — do you have some link or study that shows this relationship.

          • Kip,

            Clouds are a free variable relative to the energy balance which means that an energy balance can be established for any amount of clouds. The amount of clouds has no direct relationship to the golden mean or any other constant. Plotting the amount of clouds vs. the surface temperature is similar to what I would expect of a control signal adapting to external change.

            http://www.palisad.com/co2/sens/se/ca.png

            What it’s adapting to is water freezing at 273K and an ocean surface temperature saturation effect that occurs at about 300K. But why it’s adapting is unexplained. The surface emissions along the X axis are calculated with the SB Law from the reported temperatures, so these plots are effectively functions of state.

            The ratio of emissions at TOA to the SB emissions of the surface is a strong function of cloud coverage, since clouds emit at a much colder temperature than the surface below. In fact, the emissions at TOA I’m plotting are not reported values and are calculated as the cloud amount weighted contributions from cloud and the surface emissions, based on the BB emissions at their reported temperatures passing through the appropriate radiant transfer models and averaged over the relevant 30 km grid cells sampled at a 4 hour rate.

            If we plot the planets emissions at TOA vs. the same surface emissions for all the same points represented in the earlier scatter plot, we get this.

            http://www.palisad.com/co2/sens/se/po.png

            See how the result is almost perfectly linear, except for at the high end, where the ocean temperature saturation effect is kicking in. Notice how any non linear effects from freezing water (i.e. reflection from ice and snow) is exactly offset and linearized by the non linear response of the clouds? Is this just a coincidence too? The only consequence of linearizing this response is to maintain a constant ratio of surface emissions to emissions at TOA. That this ratio wants to be a constant should be clear, but if this constant must be golden, which we still don’t know for sure, only then can a connection between clouds and the golden mean be established.

            The little dots are monthly measurments, while the larger dots are averages over 3 decades of data. Measurements are across 2.5 degree slices of latitude from pole to pole.

          • co2 ==> Maybe you could whip all of your ideas and suspicions about clouds and the Golden Mean into an essay and throw it out to be crowd reviewed….?

          • Kip,

            What I talked about in the previous comments was but a tiny slice of the big picture which I discuss in more detail here:

            http://www.palisad.com/co2/gbm.pdf

            If you’re concerned that the gray body model is too simple, get over it since it’s the ONLY law of physics capable of quantifying the bulk behavior of the planet. Don’t be misled by the many layers of obfuscation and indirection used by the IPCC to quantify the climate, much of which even many skeptics have ignorantly embraced. It’s all a smoke screen designed to confound and confuse by making the bulk climate seem far more complicated than it really is.

            The gray body model predicts the observed cloud behavior I discussed in the prior comments and looking for that behavior was a test of that model. The prediction arises since given the strong effect clouds have on the effective emissivity, the observed cloud behavior is necessary for the relatively constant emissivity required for the gray body model to be valid everywhere from pole to pole, as it must be since the laws of physics are also the same from pole to pole.

            The path taken to arrive at a new steady state is indeed very complex, however; the requirements driving what that new steady state must be are quite simple. Conferring the complexity of the path to a new steady state on the new steady state itself blinds people to the simple elegance underneath the apparent complexity. It also leads to models attempting to predict the path under the assumption that a proper steady state will emerge and of course, it usually won’t.

            My purpose of introducing the gray body model is to replace the faulty feedback model, whose flaws I call out in the essay, with something that can accurately quantify the bulk behavior of the planet and that doesn’t require magic, arm waving or any undiscovered laws of physics.

      • n.n

        It’s really one of several stable solutions depending on the possible range of cloud behavior. The golden mean solution is only potentially valid when the chaotically variable range of cloud absorption spans the absorption of surface emissions required to establish the golden mean as a solution for 1/e, where e is the effective emissivity of the surface, relative to space. No other planet or moon in our solar system meets this constraint, although Mars comes the closest and probably did in the distant past. In all the other cases, clouds either absorb nearly everything emitted by the surface or nothing at all.

        Note that the average fraction of the surface covered by clouds is also relatively constant, but not as constant as the converged ratio of surface emissions to solar forcing which is unambiguously highly dependent on the average fraction of the surface covered by clouds.

        Relative to balance, the fraction of the surface covered by clouds is a free variable, that is, balance can be achieved for any amount of clouds. That the average amount of clouds is also relatively constant tells me that the amount of clouds is not dictated by any requirements for balance and something else must be driving their steady state average behavior. Nothing else from either side of the debate can explain why the average cloud coverage is so constant, although Lindzen’s iris effect could be a manifestation of the same converged, golden mean solution that I’m noticing.

  11. Great, Kip! Here’s another example of chaos: here in Argentina the local newspaper has an alarming piece about “calentamiento global” (global warming) and then on the next page a story about how the snow storms in southern Argentina are the fiercest in more than 20 years.

    • Hah! Not really a chaotic phenomenon. It is utterly predictable. More like the cognitive dissonance endemic to the media.

  12. ‘Until Climate Science, as a whole, fully recognizes climate as a non-linear dynamical system, and understands the implications of its deep chaotic nature, there will be little progress made in long-term prediction. Currently, CliSci is stuck on the idea that “averaging” multiple chaotic outputs to find “ensemble means” actually tells us something other than the trivial “mean” of those particular runs of that particular model with its particular parameter inputs. That idea is nonsensical.’

    I think you nailed it, Kip. Although for “them”, alarmist, ect., it is not a recognition but an admission.

    • al ==> At least some like Shen, Zeng, Pielke and others are taking the question seriously. It is tremendously important and not just in the silly “oh, we’ll just average the chaotic outputs” sort of way.

    • al in kansas,

      The steady state climate is only non linear with regard to the T^4 relationship between temperature and W/m^2. When quantified in terms of average W/m^2 of solar forcing and average W/m^2 of emissions by the surface or the planet, the system is very linear. This is to be expected as the consequence of COE, since 1 Watt is 1 Joule per second and relative to steady state averages, time factors out, In addition, the average ratio of surface emissions to planet emissions is relatively constant at about 1 W/m^2 of planet emissions per 1.62 W/m^2 of surface emissions.

      Most of what we perceive as non linearities in the climate system are not in the steady state solution space, but are chaotic non linearities along the path to a new steady state and is the consequence of an atmosphere chaotically self organized by clouds.

      • to co2isnotevil 1) the first large paragraph of my comment was a quote of Kip’s final point, which I was agreeing with, so I think your reply is more pointed in his direction. 2)I must strongly disagree. There are no steady states in climate, just different time constants. Given the size of the error bars in all climate measurements and that the inputs are always in a state of flux, a true steady state is not possible. In other words, the system never gets to the end of the path. I also do not think that for a given set of fixed inputs that the end of the path (the “steady state”) would necessarily be the same every time for a given input. Namely, the end of the path is indeterminate. Yes, we can calculate some rough constraints, but too many people are way to confident in their understanding of “climate”, however they may happen to define it.

  13. NAO/AO anomalies can be predicted at any range once you know how the Sun drives them. For example I can safely predict widespread northern hemisphere heatwaves for late July and through August of 2045, which has the same Jovian t-square type as in the heatwaves of 1934, 1949, 1976 (pictured in the link below), 2003, and 2018. The positive NAO/AO anomalies occur from when the inner planets group on the Saturn line.

    https://www.linkedin.com/pulse/major-heat-cold-waves-driven-key-heliocentric-alignments-ulric-lyons/

    • Ulric ==> I’m afraid I won’t be around to confirm your prediction — don’t have the 25 years to spare. Interesting idea…..

      • That’s one very long range example, I regularly predict weekly NAO/AO anomalies a year or so ahead.

    • Ulrich
      Again by asserting that every smallest wiggle of the climate system is forced by solar variation, you are implying the climate system is passive, so external forcing can change it.

      This is wrong. Climate is active and excitable according to the well established theory of chaotic and nonlinear dynamics that Kip has described in his article.

      The CO2 crowd make the same mistake – by assuming climate is passive and only “forcing” from CO2 or methane or CFCs or particles or any other politically convenient entity, makes it move.

      Solar and other astrophysical forcing can interact with internal climate dynamics to cause oscillatory behaviour. A simple example of this is the tides.

      • Phil,

        The climate system is not passive, ocean phases respond inversely to changes in the solar wind strength, that’s why climate tipping points are impossible. I contend that much of the variability of the annular modes and the inverse response of the ocean phases would not even exist without daily-weekly variability of the solar wind. The major heatwaves and cold-waves are all discretely solar driven in the weeks which they occur, there’s no way that they would otherwise exist.

        The mistake which the ‘CO2 crowd’ make is in assuming that all the variability of weather and of the ocean phases, is internal, chaotic, and unforced. Which makes them attribute weather extremes to climate change, when in reality the solar driven weather extremes are the main agent of climate change. And it allows them to attribute recent global warming exclusively to rising CO2 forcing, but which in reality is largely the ocean phases shifting warmer in response to weaker solar wind states.

  14. A while back I found a good YT video on the bifurcation diagram, and its relation to the Mandelbrot set and the Feigenbaum constant, at the link below. By Veritasium.

  15. Everyone here is an example of the Butterfly Effect… Every thing you do matters! Every move you make, every action you take matters not just for you but your family, your home town and your country.

    Discover The Butterfly Effect by Andy Andrews

    • codetrader ==> Can’t say I ascribe to the “self-help guru” version of the Butterfly Effect any more than I do the the various misunderstandings of quantum mechanics theories.

      I may make a nice metsaphor from which some lessons may be learned, or taught, but the author has obviously not read Lorenz’ 1972 paper linked in the essay.

  16. Weather may appear to be chaotic, but the does not mean it actually is, in terms of either having random attributes or in which a small action can have potentially huge impact. I am more inclined to believe that our models are missing significant factors of heat transport. Do we have a good model for cloud formation? or how much heat is transported by evaporation? Is the electrical energy of lightning a factor? Chaos is just an excuse for an incomplete model

    • Joe Chang ==> You express the hope of all number junkies —- “Chaos must not actually be true — not in the real world!”

      That view has been found, over and over and over again, not to be true. Read Gleick or any of the recommended reading in my Chaos series.

      Watch the video above, this one https://youtu.be/ovJcsL7vyrk

      Shen et al. (which includes the very conservative Roger Pielke Sr.) address the question directly: Is Weather Chaotic? They answer in the affirmative. Note that this paper is not some post-doc’s one off — it is part of an ongoing investigation into non-linear dynamical systems.

      That said, you are certainly right that “models are missing significant factors of heat transport. Do we have a good model for cloud formation? or how much heat is transported by evaporation? Is the electrical energy of lightning a factor?”

      However, the concept that “Chaos is just an excuse for an incomplete model” is just as certainly wrong.

      In fact, CHAOS IS THE MODEL for weather and climate. The failure to truly comprehend and include the fact that weather and climate are, by their very nature, non-linear dynamical systems, and thus inherently, natively, Chaos-Theory-chaotic makes climate modelling specious.

    • Chaos and randomness aren’t the same thing. A chaotic system is deterministic which means that if you have absolutely perfect knowledge of the entire system, that is you know the location and velocity of every single atom, you could know the future of the system exactly. The whole premise of chaos theory is that tiny difference do make a huge difference and since we don’t have perfect knowledge the future state of the system is guaranteed to wildly diverge from predictions.

      So, yes, the issue is that we have an incomplete model of the earth’s atmosphere but that isn’t an excuse because a truly complete model is impossible and anything less than perfect knowledge isn’t good enough to keep the predictions from going completely off the tracks eventually. A model that accurately incorporates 90% of the relevant factors isn’t good enough. 99.9% wouldn’t be good enough. Only a 100% perfect model would work beyond the short term.

      • Aaron D. ==> Tiny differences in Initial Conditions produce vastly differing long term effects in the chaotic regions of non-linear dynamical systems — but have little effect in the long-term effects of the single-point attractor regions of the same system. This is the finding of Shen et al. and is a feature of all Chaotic systems. ICs have little effect in the initial stages of periodic regions that occur before the breakdown into chaos….where they start doesn’t change the eventual values of the periods — kicking the iteration out of sync with a weird value in the middle just acts like a new IC and after a bit the system stabilizes back to its period values. Once a non-linear dynamical system enters its chaotic regime — all bets are off — except for what we know to expect for Chaos.

  17. “Currently, CliSci is stuck on the idea that “averaging” multiple chaotic outputs to find “ensemble means” actually tells us something other than the trivial “mean” of those particular runs of that particular model with its particular parameter inputs. That idea is nonsensical.”

    Robert Brown (rgb@duke) said much the same a few years ago on WUWT. I’ve not seen anything from him in recent years. Perhaps his masters at Duke shut him down on climate matters?

    • DHR ==> Yes, Brown was responding to some of my chaos essays and responses to them by others at the time, and I edited some of his comments into an essay ( here ).

      My last contact with him indicated that he was busy trying to use chaos and randomness to predict the future. Not a trival endeavor.

  18. None of this appears to have much to do with real weather or the climate. Except that it
    makes the existence of tipping points more likely. After all if the system has multiple stable
    points (whether periodic cycles, chaotic attractors, stationary points) then it becomes a lot
    easier to say that a small perturbation can move the system from one stable point to another.

    Remember that in a phase space the weather is represented by a single point and so there is no
    way that different bits of weather can exist in different basins of attraction. If you want to state
    that different portions of the globe have independent weather systems then you might as well do that using a low dimensional system and just change the parameters of the model from place to place.

    • Izaak Walton ==> No weather exists in phase space — weather is a real world non-linear dynamical system composed of more than one (at least two…) coupled major nonlinear dynamical systems (atmosphere and oceans).

      Tipping Points are an alarmist’s dream-state and exist only to scare little children and the feeble minded.

      The proof is thousands of years of (relatively) stab;e Earth Climate — shifting over millennia from Ice Ages to Warmer Periods and back.

      So far, there has been no HOT Periods.

      And no other apparent “tipping” — changes, ever so slowly, but no tipping.

      • What appears to be lost in all of these discussions is this:
        The sun is essentially a point source of energy to earth. The part of the earth nearest the sun rotates at 1000 mph. The two ends of the earth that are the spin axes rotates at 0 mph. That sets up a powerful force that essentially causes the jet streams – where the hot is separated from the cold by a fast stream of turbulent air.
        Now, if the earth was a perfect sphere, things would be a bit different, but land masses have gigantic mountain ranges that perturb the jet streams. THAT is your butterfly. And because earth’s tilt changes from day to day, there are changes in how and where these land masses affect the jet streams. Now, set a thermometer somewhere on earth, no matter how perfect, no matter how protected, and all you get is a set of numbers – not one of them related to climate, no matter how many years. Add millions of thermometers – does not matter. You can’t ‘measure’ climate in the face of this setup. Earth is basically a rotisserie – the sun never stops delivering energy – ever. That energy gets dispersed based on all the above factors. Average temperature? Not possible. Claims of CO2 causing any of this? No words come to mind, it is not even the flea on the elephant…not even the mite on the flea on the elephant. Not even the bacteria on the mite on the flea on the elephant…think about it.

        • “…the sun never stops delivering energy – ever.”

          Well … stars come in all shapes sizes, longevity, etc.

      • Kip,
        If weather doesn’t exist in phase space then the article you are discussing has zero
        relevance to reality since it is exclusively concerned with trajectories in phase space.
        The point is that for any dynamical system the entire system can be represented as
        a single point in phase space. If the weather is a dynamical system then it can be represented as a single trajectory in phase space and so it is either in the basin of attraction of a chaotic attractor or somewhere else. It can’t be in two places in phase space at once.

        Furthermore even in chaotic systems averages over trajectories (or along them) can
        provide useful information. Which is why climate is a lot more stable than the weather.

          • Kip,
            I did read the study. It doesn’t actually seem to actually present anything new. It
            was already well known that dynamical systems can have regions of multistability.
            It was similarly known that the higher dimensional Lorentz model had multistability
            and so again there is nothing new in this presentation. As far as I can tell the novel
            claim here is that different “weather systems” can coexist in different regions of phase space at the same time. Which might be true since nowhere to they define what a
            “weather system” is.

            Again for this paper to be useful you would have to show that the weather in different portions of the globe is uncoupled. Then and only then could you have different weather systems in different bits of phase space. But if the weather is uncoupled then there is the additional problem of proving that the same approximate model with the same parameters applies to each uncoupled system. But they don’t do that.

          • Izaak ==> Can’t really help you if that’s all you gained from the paper. It is not a paper of proofs, it is a report about their investigation and their sense what their findings might mean.

            If it means nothing to you, then it is to your loss.

          • Kip,
            As you say there are no proofs in the paper. And so any suggestions about what it
            might mean are basically just guesses based on your personal preferences. But the
            claim that the weather can be both chaotic and predictable can not be justified from
            their paper or their methods.

            To derive their generalised Lorentz method they start with a set of partial differential
            equations for the atmosphere of the entire globle. They then assume that the solution can
            be written as an infinite sum of Fourier like modes and then truncate this at some finite order. This results in a set of coupled ordinary differential equations for the time evolution of the amplitudes of each of the modes. Thus the terms in their equation do not correspond to the weather at some particular location in space but rather some global parameter such as the average wind speed in the atmosphere. Therefore in their model it is impossible to have different weather systems in different areas of the globe and so the weather is either chaotic or periodic but not both.

            If you wanted to claim that some aspects of local weather are more predictable than others that is fine and I wouldn’t object. But to show that you would need to start with a set of local equations for the weather and demonstrate that they are decoupled. And then analyse each local set individually. None of which they do. Nor is it possible to do so using their approach.

  19. A little OT, here, but long ago, someone told me that a proper definition of randomness depended upon statistics; and a proper definition of statistics depended upon randomness. Hence there is some circularity going on here which should bother us.

  20. “They want to know if the chaos that Lorenz (definition 2) found in his early toy weather model, which led to the accepted concept that “weather is chaotic” meant that weather (as we experience it in the real world day-to-day, week-to-week and month-to-month) is really chaotic (as in definition 1 – completely confused or disordered, random, stochastic and in longer time sense, unpredictable).”

    Well, I can predict with pretty good confidence that it is going to be very hot in my neck of the woods this time of year. Year after year. Like clockwork. There seems to be a pattern here.

    • Tom ==> You could be a weatherman [-person?] with your new-found skill — if we needed only predictions of seasons…..

      • Kip,

        There is a peer reviewed manuscript that will appear
        In the September issue of the journal Dynamics of Atmospheres and Oceans that mathematically proves that all global climate models are based on the wrong atmospheric system of equations so that results from those models can no longer be defended as to have any meaning.

        The Lorenz equations, though mathematically amusing, are nowhere close to the 3d atmospheric equations of motion.
        The best understanding of those equations is obtained thru the Bounded Derivative Theory introduced by Kreiss. That theory
        Is the basis of the new manuscript that introduces the correct dynamical system.

        Jerry

        • Jerry ==> Please stop back and leave a link when it is available. Or send me the link @ kip at the domain i4.net

  21. Kip
    Thanks for this excellent and much needed article.
    A couple of quick comments.
    Figure 1 e (time plot of the Lorenz or Roessler attractor) looks similar to a time plot of ENSO with periods of El Niño and La Niña. The resemblance is not a coincidence.

    In the second figure, the regions you circle in red are an example of “amplitude death”. This is a phenomenon that pops up in chaotic oscillator systems and results from delayed coupling or delayed feedback, something that the ocean always adds to the climate system. Here’s a couple of papers on amplitude death:

    https://www.researchgate.net/publication/7387084_Amplitude_death_in_coupled_chaotic_oscillators

    https://www.nature.com/articles/s41598-018-30026-3

    • Phil ==> Thanks for the links — I’ll check those papers out. The system used in the creation of Shen’s Figure 1 is a 3-dimensional Lorenz Model (for which I do not have the details — but they may be in the paper) and in my figure of May Island Squirrel Population (an imaginary island) I use Robert May’s logistic population model.

      I’m not sure either represents a “System of coupled oscillators” despite appearances of a similar phenomenon.

      Coupled Oscillator systems are pretty wild . . . . .

  22. The Bumper Sticker Science phrase that, “The weather is chaotic, but climate is not”, or, “Each trajectory (of the temperature, for example) might be chaotic, but the ensemble average is not”, seems to be related to the equally Bumper Sticker Science phrase, “The climate is the average of the weather.” That is, if you insist that Climate is the average of Weather and that Weather is chaotic and that Climate is not chaotic, then you must insist that the average of chaotic trajectories is not chaotic.

    The only correct characterization off averaging is that the operation reduces the frequency and amplitude of the trajectories: nothing more, nothing less, and certainly does not introduce any Science in any way, shape, or form. The average is itself chaotic, as can be directly illustrated by use of an algebraic example.

    The length of the time interval of the “continuous dependence” region, more or less about 20 Lorenz Time Units for your basic methods of numerical integration, is in fact a function of the numerical methods used for the integration. The length can be extended to an arbitrary interval length by use of high-order discrete approximation to the continuous equations plus high-precision representations of numbers. Very, very high orders: like 1500th order by application of the Taylor-series approach, and thousands of digits for finite representations of numbers and doing arithmetic. See the results of this Google Scholar search.

    Chaos is the antithesis of randomness. Random covers all of phase space, every little bit of it, chaos is stuck to the trajectory.

    • Math was never my strongest suit. However, it is my recollection from the advance courses I did have that the solutions to the differential or partial differential that are used to mathematically model these types of systems may converge, diverge, or do neither. In some systems of equations there are mathematical proofs that the system has no solution. The simplest best known of these is the “three body” problem. Yes you can throw a super computer or even “Deep Thought” at it and get an approximation of an answer, but you may still get 7×8=42.

  23. Kip
    Your last figure is a nice diagram of Hopf bifurcations leading up to the transition to chaos. Generally it is at the border of transition to chaos, where the system is still low-dimensional, where the interesting emergent pattern phenomena occur; rather than in high dimensional turbulent “full blown chaos”.

    Reading some research in chemical engineering by Matthias Bertram and others, I came to realise that one scenario highly relevant to the climate system is the opposite of the Hopf diagram; that is, rather than the progress of an initially linear system over the threshold into chaos, the transition of an already chaotic turbulent system from high down to lower dimensionality, nearer to the Hopf boundary regime where pattern formation can occur. In other words, the reduction rather than the increase in chaotic behaviour and dimensionality.

    How can this occur? Bertram and others give some interesting examples from chemical engineering model systems on thin films, including the platinum-catalysed oxidation of CO and the Belousov-Zhabotinsky reaction. Bertram’s goal as an engineer was to control chaotic processes, and he shows two ways to do this: adding feedback, and adding periodic external forcing. To quote:

    Spontaneous pattern formation and spatiotemporal chaos (turbulence) are common features of spatially extended nonlinear systems maintained far from equilibrium. The aim of this work is to control and engineer such phenomena. As an example, the catalytic oxidation of carbon monoxide on a platinum (110) single crystal surface is considered. In order to control turbulence and to manipulate pattern formation in this reaction, two different control methods, global delayed feedback and periodic forcing, are employed.

    https://pure.mpg.de/rest/items/item_740669/component/file_740668/content

    http://www.fhi-berlin.mpg.de/complsys/Pdf/2001/pollmann-bertram-rotermund-01.pdf

    In a nutshell what they do is reduce the dimensionality of the chaotic system by either of these two factors, delayed feedback or periodic forcing. In this way they reduce the “chaoticness” of the system bringing it to the borderline chaos region where interesting and – for them – useful pattern and oscillation emerge.

    This made me think of oceanic systems where feedbacks are linked to oscillation. For instance ENSO. You have the Bjerknes feedback whereby Peruvian oceanic upwelling (linked to the Humboldt current) interacts with the trade winds to create intermittent positive feedback which reinforces both the upwelling and the trade winds. (The cold upwelling sets up a sea surface temperature gradient which impels the trade winds).

    Another longer term oscillation is the Atlantic Meridional Overturning Circulation (AMOC) which oscillates in strength, giving rise to the AMO – Atlantic Multidecadal Oscillation. Here again there is an intermittent positive feedback. The Gulf Stream increases the salinity of sea surface water in the far North Atlantic and the Norwegian Sea. As this water cools it becomes very dense, causing substantial downwelling all the way to the ocean floor resulting in the “deep water formation” that drives the ocean circulation system. This down welled water flows back south along the ocean floor, completing the 3D loop of the AMOC and reinforcing the Gulf Stream.

    In both these cases, ENSO and AMOC, it’s fair to say that the ocean circulation system in 3D is chaotic and turbulent. A glance at the NullSchool ocean circulation animations will confirm this:

    https://earth.nullschool.net/#current/wind/surface/level/orthographic=-40.76,39.11,587

    Applying the paradigm of Matthias Bertram to this, we can suggest that the presence of feedback in these oceanic systems – the Bjerknes feedback with ENSO and the salinity feedback with AMOC – is “reducing the dimensionality” of the turbulent chaotic circulation systems and causing quasi-regular oscillations to arise.

    And as Bertram also found with the BZ reaction, periodic forcing can also bring about emergent oscillation in a chaotic ocean system. ENSO is known to be phase-locked to the annual cycle such that El Niños typically happen at Christmas (this their name).

    Tziperman, Cane and Zebiak have shown how ENSO can be modelled as a delayed oscillator periodically forced by the annual cycle:

    https://journals.ametsoc.org/jas/article/54/1/61/24285

    Warmists try to write off chaos as just noise in the evolving climate system. They may be right if they are talking about only high dimensional turbulence. However they miss the fact that both feedbacks and external periodic forcing (from annual, solar and other astrophysical sources) can reduce the dimensionally of climate subsystems with the result of emerging pattern and oscillation. This can be on many timescales up to century and millennial.

    This process, the reduction of dimensionality of chaotic climate systems by feedback or periodic forcing, provides a paradigm to understand how observed fluctuations and oscillations occur in the climate where a direct proximal cause seems elusive. ENSO, AMO and PDO are some examples. Here a real case can be made for chaotic-nonlinear dynamics being the cause of much more substantive climate change than just short term noise. It makes such a model the null hypothesis for much natural ocean driven climate change that is observed over many time scales.

    • The notion that AMOC gives rise to the AMO and reinforces the Gulf Stream is one of many fairy tales of “climate science,” ensuing from simplistic “conveyor belt” conceptualization of oceanic circulation. The major surface currents are wind-driven and thermohaline circulation is merely a dynamical adjunct that does not form any systematically closed loop of mass transport.

    • When established oceanography for decades describes the thermohaline circulation and the AMOC, and then post-modern climate science-oceanography wants to cancel them, I’m sorry but the place for post-modern oceanography where the sun does not shine. I don’t think that carbonised science is good for anything.

      Of course it’s chaotic, not a simple loop – if you look carefully you’ll see several uses of the word “chaos” in my comment above. That’s why entrainment and reduction of dimensionality by feedbacks and periodic forcing are so important – or, I’m guessing, you didn’t get that part?

  24. Phil ==> Quite right — limiting one’s view of Chaos to focus on the odd behaviors in the chaotic realms really misses most of the features of Chaos including enforced stabilities and all the features of periodocities/ocillations and pattern repetition including fractals and fractal-like features in natural world non-linear dynamical systems.

      • Phil ==> Gee — I am uncertain about the “fractal nature” of pattern matching two disparate graphs whose patterns only match after “Ice core temperatures have been linearly adjusted to give the best fit to the modern data, by multiplying them by about 1.6 and subtracting about 0.2°C.”

  25. Question for the weather experts:
    Why is there a reporting anomaly every now and again? Here’s weather highs and lows at my zip for 3 days running (source – weather channel monthly tab):

    July 16: 92H, 75L
    July 17: 92H, 92L (hmmm)
    July 18: 94H, 72L

    This is the second anomaly I’ve seen in recent months – each favoring a higher average temp. Any ideas??

  26. Thanks for simplifying the argument (no pun intended). Even for those who do not follow the whole paper the conclusion:

    “Until Climate Science, as a whole, fully recognizes climate as a non-linear dynamical system, and understands the implications of its deep chaotic nature, there will be little progress made in long-term prediction. Currently, CliSci is stuck on the idea that ‘averaging’ multiple chaotic outputs to find ‘ensemble means’ actually tells us something other than the trivial ‘mean’ of those particular runs of that particular model with its particular parameter inputs. That idea is nonsensical.”

    is worth repeating ad nauseam.

    • Jerry ==> I’m not sure that there is any such thing as “2D turbulence”. What would that be in the Real World?

  27. Kip,

    Do you believe 2d turbulence is chaotic ?
    If so please read the Henshaw, Kreiss and Reyna manuscripts on mathematical estimates for solutions of the turbulence equations
    Jerry

Comments are closed.