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 — adjectivecha·ot·ic | kā-ˈä-tik

Definition ofchaotic

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/geosciences9070281Judt, 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.

- 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.**

**# # # # #**