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Chaotic Modeling and Simulation International Conference

CHAOS 2020: 13th Chaotic Modeling and Simulation International Conference pp 805–825Cite as

Is Weather Chaotic? Coexisting Chaotic and Non-chaotic Attractors Within Lorenz Models

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Part of the book series: Springer Proceedings in Complexity ((SPCOM))

Abstract

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 in J. Atmos. Sci. 20:130–141, [1]) study, the statement of “weather is chaotic” has been well accepted. Such a view turns our attention from regularity associated with Laplace’s view of determinism to irregularity associated with chaos. Here, a refined statement is suggested based on recent advances in high-dimensional Lorenz models and real-world global models. 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, 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.

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Notes

  1. 1.

    The term “dimension” is conventionally used for a system of ODEs [31, 32]. In this study, the 5DLM, 7DLM, and 9DLM are referred to as high-dimensional or high-order Lorenz models [12].

  2. 2.

    Similar findings for the dependence of various solutions (i.e., chaotic and limit cycle solutions) on the strength of heating were also reported using a two-layer, quasi-geostrophic model that describes the finite-amplitude evolution of a single baroclinic wave by Pedlosky and Frenzen [34].

  3. 3.

    In our 5D-, 7D-, and 9DLMs, we can obtain closed form solutions of trivial and non-trivial equilibrium points and use them to verify the numerical solutions of equilibrium points.

  4. 4.

    By comparison, chaotic models in Lorenz [52,53,54] were not derived from physics-based partial differential equations.

  5. 5.

    Negative feedback can be found within the so-called Lorenz-Stenflo system that extends the 3DLM with one additional ODE containing one additional mode that takes rotation into consideration [55,56,57].

  6. 6.

    A torus is defined as a composite motion with two (or more) oscillatory frequencies whose ratio is irrational [8].

  7. 7.

    The coexistence of chaotic and quasi-periodic orbits has been recently documented in a modified Lorenz system by Saiki et al. [58].

  8. 8.

    Such a dependence on initial conditions, close to (or away from) the non-trivial equilibrium point, can be shown by the following YouTube video for a double pendulum (between 1:00 and 1:20).

    https://www.youtube.com/watch?v=LfgA2Auyo1A.

  9. 9.

    As a result, we agree with Prof. Arakawa that the predictability limit is not necessarily a fixed value [46].

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Acknowledgements

We thank reviewers of the manuscript, the editor, and Drs. M. Alexander, R. Anthes, B. Bailey, J. Buchmann, D. Durran, M. Ghil, F. Judt, B. Mapes, Z. Musielak, T. Krishnamurti (Deceased), C.-D. Lin, T. Palmer, J. Pedlosky, J. Rosenfeld, R. Rotunno, I. A. Santos, C.-L. Shie, S. Vannitsem, and F. Zhang (Deceased) for valuable comments and discussions. We appreciate the eigenvalue analysis provided by Mr. N. Ferrante. We are grateful for support from the College of Science at San Diego State.

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Authors and Affiliations

  1. San Diego State University, San Diego, USA

    Bo-Wen Shen, S. Faghih-Naini, J. Cui & T. A. L. Reyes

  2. CIRES and ATOC, University of Colorado At Boulder, Boulder, USA

    R. A. Pielke Sr.

  3. The University of Arizona, Tucson, USA

    X. Zeng

  4. Seoul National University, Seoul, South Korea

    J.-J. Baik

  5. University of Bayreuth, Bayreuth, Germany

    S. Faghih-Naini

  6. Friedrich-Alexander University Erlangen-Nuremberg, Erlangen, Germany

    S. Faghih-Naini

  7. North Carolina State University, Raleigh, USA

    J. Cui

  8. University of Maryland, Baltimore County, Baltimore, USA

    R. Atlas

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  1. ManLab, Technical University of Crete, Chania, Greece

    Christos H. Skiadas

  2. Department of Management Science and Technology, Hellenic Mediterranean University, Agios Nikolaos, Crete, Lasithi, Greece

    Yiannis Dimotikalis

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Shen, BW. et al. (2021). Is Weather Chaotic? Coexisting Chaotic and Non-chaotic Attractors Within Lorenz Models. In: Skiadas, C.H., Dimotikalis, Y. (eds) 13th Chaotic Modeling and Simulation International Conference. CHAOS 2020. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-030-70795-8_57

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