Coexistence of Chaotic and Non-chaotic Orbits in a New Nine-Dimensional Lorenz Model

Shen, Bo-Wen; Reyes, Tiffany; Faghih-Naini, Sara

doi:10.1007/978-3-030-15297-0_22

Cited by 12 publications

(18 citation statements)

References 11 publications

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“…Recently, the 5DLM has been re-derived and analyzed in detail by Moon et al [15] and Felicio and Rech [16], showing the model's robustness. Furthermore, the 5DLM has been extended to higherdimensional LMs (e.g., [17][18][19][20]) and a generalized LM (e.g., [21,22]).…”

Section: The Five-dimensional Lorenz Model (5dlm)mentioning

confidence: 99%

A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006

Reyes¹,

Shen²

2020

Current Topics in Tropical Cyclone Research

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Accurate detection of large-scale atmospheric tropical waves, such as African easterly waves (AEWs), may help extend lead times for predicting tropical cyclone (TC) genesis. Since observed AEWs have comparable but slightly different periods showing spatial and temporal variations, local analysis of frequencies and amplitudes of AEWs is crucial for revealing the role of AEWs in the modulation of TC genesis. To achieve this goal, we investigate the recurrence plot (RP) method. A recurrence is defined when the trajectory of a state returns to the neighborhood of a previously visited state. To verify implementation of the RP method in Python and its capability for revealing a transition between different types of solutions, we apply the RP to analyze several idealized solutions, including periodic, quasiperiodic, chaotic and limit cycle solutions, and various types of solutions within the three- and five-dimensional Lorenz models. We then extend the RP analysis to two datasets from the European Centre for Medium-Range Weather Forecasts global reanalysis and global mesoscale model data in order to reveal the recurrence of multiple AEWs during summer 2006. Our results indicate that the RP analysis effectively displays the major features of time-varying oscillations and the growing or decaying amplitudes of multiple AEWs.

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Section: The Five-dimensional Lorenz Model (5dlm)mentioning

confidence: 99%

A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006

Reyes¹,

Shen²

2020

Current Topics in Tropical Cyclone Research

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“…Upscale transfer is a major feature in turbulence models. However, it is not clear whether the 1969 model is a turbulence model because the a conservative partial differential equation with a As a result of the above four features within the GLM, a refined view of the nature of weather is proposed [22,23] as follows: The entirety of weather is a superset that consists of both chaotic and non-chaotic processes. In other words, both chaos and order may coexist.…”

Section: A Brief Review Of Lorenz (1969)mentioning

confidence: 99%

“…As shown in Figure 5 with r = 680, the 256 orbits could be classified as chaotic orbits and steady-state solutions, indicating the coexistence of two types of solutions. A detailed analysis of the dependence of chaotic and non-chaotic solutions on ICs using the ensemble modeling approach with various Ns and Rs can be found in Figure 5 of [23]. The second kind of attractor coexistence with limit cycle and steady solutions was first documented by Shen [22].…”

mentioning

confidence: 94%

“…To address this question, the predictability problem that has been outstanding for decades and only partially resolved must be revisited [8,9]. Since this is not an easy task, we attempt to address this problem based on recent global mesoscale modeling studies [6,[10][11][12][13][14][15], a 10-year analysis of global reanalysis data [16], and generalized Lorenz modeling studies [17][18][19][20][21][22][23]. While short-term hurricane predictions have been extended to produce promising results at 15-30 days scales, long term climate TC simulations have been shown to provide improved temporal and spatial locality for simulated TCs.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Such a view turns our attention from regularity associated with Laplace's view of determinism to irregularity associated with chaos. In contrast to single type chaotic solutions, recent studies using a generalized Lorenz model (Shen 2019a, b;Shen et al 2019) have focused on the coexistence of chaotic and regular solutions that appear within the same model, using the same modeling configurations but different initial conditions. The results suggest that the entirety of weather possesses a dual nature of chaos and order with distinct predictability.…”

mentioning

confidence: 99%

Is Weather Chaotic? Coexisting Attractors, Multistability, and Predictability

Shen

Pielke

Zeng

et al. 2021

Preprint

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Since Lorenz’s 1963 study and 1972 presentation, the statement “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. In contrast to single type chaotic solutions, recent studies using a generalized Lorenz model (Shen 2019a, b; Shen et al. 2019) have focused on the coexistence of chaotic and regular solutions that appear within the same model, using the same modeling configurations but different initial conditions. The results suggest that the entirety of weather possesses a dual nature of chaos and order with distinct predictability. Furthermore, Shen et al. (2021a, b) illustrated the following two mechanisms that may enable or modulate attractor coexistence: (1) the aggregated negative feedback of small-scale convective processes that enable the appearance of stable, steady-state solutions and their coexistence with chaotic or nonlinear limit cycle solutions; and (2) the modulation of large-scale time varying forcing (heating). Recently, the physical relevance of findings within Lorenz models for real world problems has been reiterated by providing mathematical universality between the Lorenz simple weather and Pedlosky simple ocean models, as well as amongst the non-dissipative Lorenz model, and the Duffing, the Nonlinear Schrodinger, and the Korteweg–de Vries equations (Shen 2020, 2021). We additionally compared the Lorenz 1963 and 1969 models. The former is a limited-scale, nonlinear, chaotic model; while the latter is a closure-based, physically multiscale, mathematically linear model with ill-conditioning. To support and illustrate the revised view, this short article elaborates on additional details of monostability and multistability by applying skiing and kayaking as an analogy, and provides a list of non-chaotic weather systems. We additionally address the influence of the revised view on real-world model predictions and analyses using hurricane track predictions as an illustration, and provide a brief summary on the recent deployment of methods for multiscale analyses and classifications of chaotic and non-chaotic solutions.

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Coexistence of Chaotic and Non-chaotic Orbits in a New Nine-Dimensional Lorenz Model

Cited by 12 publications

References 11 publications

A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006

A Recurrence Analysis of Multiple African Easterly Waves during Summer 2006

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

Is Weather Chaotic? Coexisting Attractors, Multistability, and Predictability

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