A recurrence analysis of chaotic and non-chaotic solutions within a generalized nine-dimensional Lorenz model

Reyes, Tiffany; Shen, Bo-Wen

doi:10.1016/j.chaos.2019.05.003

Cited by 16 publications

(7 citation statements)

References 29 publications

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“…The second kind of attractor coexistence with limit cycle and steady solutions was first documented by Shen [22]. Such coexistence was further discussed using the 9DLM with r = 1600 and 128 different ICs in Figure 9 of [52]. From a practical perspective, better predictability can be obtained for non-chaotic solutions with insensitivity to the initial conditions.…”

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confidence: 84%

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On the Predictability of 30-Day Global Mesoscale Simulations of African Easterly Waves during Summer 2006: A View with the Generalized Lorenz Model

Shen

2019

Geosciences

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Recent advances in computational and global modeling technology have provided the potential to improve weather predictions at extended-range scales. In earlier studies by the author and his coauthors, realistic 30-day simulations of multiple African easterly waves (AEWs) and an averaged African easterly jet (AEJ) were obtained. The formation of hurricane Helene (2006) was also realistically simulated from Day 22 to Day 30. In this study, such extended predictability was further analyzed based on recent understandings of chaos and instability within Lorenz models and the generalized Lorenz model. The analysis suggested that a statement of the theoretical predictability of two weeks is not universal. New insight into chaotic and non-chaotic processes revealed by the generalized Lorenz model (GLM) indicated the potential for extending prediction lead times. Two major features within the GLM included: (1) three types of attractors (that also appeared in the original Lorenz model) and (2) two kinds of attractor coexistence. The features suggest a refined view on the nature of weather, as follows: The entirety of weather is a superset that consists of chaotic and non-chaotic processes. Better predictability can be obtained for stable, steady-state solutions and nonlinear periodic solutions that occur at small and large Rayleigh parameters, respectively. By comparison, chaotic solutions appear only at moderate Rayleigh parameters. Errors associated with dissipative small-scale processes do not necessarily contaminate the simulations of large scale processes. Based on the nonlinear periodic solutions (also known as limit cycle solutions), here, we propose a hypothetical mechanism for the recurrence (or periodicity) of successive AEWs. The insensitivity of limit cycles to initial conditions implies that AEW simulations with strong heating and balanced nonlinearity could be more predictable. Based on the hypothetical mechanism, the possibility of extending prediction lead times at extended range scales is discussed. Future work will include refining the model to better examine the validity of the mechanism to explain the recurrence of multiple AEWs.

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confidence: 84%

“…In other words, both chaos and order may coexist. Such coexistence suggests better predictability for non-chaotic processes if we can identify them in advance (e.g., [52]).…”

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confidence: 99%

On the Predictability of 30-Day Global Mesoscale Simulations of African Easterly Waves during Summer 2006: A View with the Generalized Lorenz Model

Shen

2019

Geosciences

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“…The latter is defined when a trajectory returns to the neighborhood of a previously visited state. Recurrence may be viewed as a generalization of "periodicity" that braces quasi-periodicity and chaos [24,27,28].…”

Section: The First Kind Of Butterfly Effect (Be1)mentioning

confidence: 99%

Three Kinds of Butterfly Effects within Lorenz Models

et al. 2022

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Within Lorenz models, the three major kinds of butterfly effects (BEs) are the sensitive dependence on initial conditions (SDIC), the ability of a tiny perturbation to create an organized circulation at large distances, and the hypothetical role of small-scale processes in contributing to finite predictability, referred to as the first, second, and third kinds of butterfly effects (BE1, BE2, and BE3), respectively. A well-accepted definition of the butterfly effect is the BE1 with SDIC, which was rediscovered by Lorenz in 1963. In fact, the use of the term “butterfly” appeared in a conference presentation by Lorenz in 1972, when Lorenz introduced the BE2 as the metaphorical butterfly effect. In 2014, the so-called “real butterfly effect”, which is based on the features of Lorenz’s study in 1969, was introduced as the BE3.

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“…(2) the aforementioned three types of solutions; (3) hierarchical spatial scale dependence (e.g., [43,44]); and (4) two kinds of attractor coexistence [9,18,20,45,46]. Additionally, aggregated negative feedback appears within high-dimensional LMs when the negative feedback of various smaller scale modes is accumulated to provide stronger dissipations, requiring stronger heating for the onset of chaos in higher-dimensional LMs.…”

Section: Coexisting Attractors and Multistability Within The Glmmentioning

confidence: 99%

The Dual Nature of Chaos and Order in the Atmosphere

et al. 2022

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In the past, the Lorenz 1963 and 1969 models have been applied for revealing the chaotic nature of weather and climate and for estimating the atmospheric predictability limit. Recently, an in-depth analysis of classical Lorenz 1963 models and newly developed, generalized Lorenz models suggested a revised view that “the entirety of weather possesses a dual nature of chaos and order with distinct predictability”, in contrast to the conventional view of “weather is chaotic”. The distinct predictability associated with attractor coexistence suggests limited predictability for chaotic solutions and unlimited predictability (or up to their lifetime) for non-chaotic solutions. Such a view is also supported by a recent analysis of the Lorenz 1969 model that is capable of producing both unstable and stable solutions. While the alternative appearance of two kinds of attractor coexistence was previously illustrated, in this study, multistability (for attractor coexistence) and monostability (for single type solutions) are further discussed using kayaking and skiing as an analogy. Using a slowly varying, periodic heating parameter, we additionally emphasize the predictable nature of recurrence for slowly varying solutions and a less predictable (or unpredictable) nature for the onset for emerging solutions (defined as the exact timing for the transition from a chaotic solution to a non-chaotic limit cycle type solution). As a result, we refined the revised view outlined above to: “The atmosphere possesses chaos and order; it includes, as examples, emerging organized systems (such as tornadoes) and time varying forcing from recurrent seasons”. In addition to diurnal and annual cycles, examples of non-chaotic weather systems, as previously documented, are provided to support the revised view.

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A recurrence analysis of chaotic and non-chaotic solutions within a generalized nine-dimensional Lorenz model

Cited by 16 publications

References 29 publications

On the Predictability of 30-Day Global Mesoscale Simulations of African Easterly Waves during Summer 2006: A View with the Generalized Lorenz Model

On the Predictability of 30-Day Global Mesoscale Simulations of African Easterly Waves during Summer 2006: A View with the Generalized Lorenz Model

Three Kinds of Butterfly Effects within Lorenz Models

The Dual Nature of Chaos and Order in the Atmosphere

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