Revisiting Lorenz’s Error Growth Models: Insights and Applications

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Title: Revsiting Lorenz’s Error growth models

Dear Editor,

This paper is relevant to investigate Lorenz’s growth models with nonlinear terms. The investigation of quadratic and cubic non-linear terms is done using the non-dissipative Lorenz model. The first Lorens model is logistic ODEs, while the second model can be reformulated into the first. However, both models have a major difference, which is the critical point. Using in real world weather data, two models have been studied numerically to characterise errors. So, this suggests that the scale of initial errors and defined thresholds influences predictability horizons. Moreover, this study compares Logitic ODE and Logitic map to demonstrate chaotic behaviours with large integration time steps. Results are shown in Table 1 and Table 2. Table 1 is a comparative analysis of logitic ODE, logitic map and non-dissipative Lorenz model.

Further, this article is well-written, clear and well-organized. Therefore, I advise publishing this article after some refinement.

1-      In the text, the author mentions Figures 1, 2a, 2b, 3 and 4 in lines 52, 61, 76,129 and 163. However those are not included in the text.

2-      Please check references, especially in line 40 Lorenz 1984 and in line 243, Pedlosky 1971 doesn’t seen in the text.

Best regards

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Review of the paper “Revisiting Lorenz’s Error Growth Models”

byBo-Wen Shen

 

This study is dedicated to description of Lorenz’s error growth models (quadratic/cubic) and to describe mathematical relationship between them and non-dissipative Lorenz 1963 model. In particular, it is shown how the cubic model can be transformed in the form the Logistic ODE.

 

The article is well written, the introduction is too short, but it is compensated by brief description of the current state-of-art in Sect. 2 (incluring appropriate references) the conclusion is formulated logically. Overall, the topic is relevant and, as I understand, corresponds to the scope of the journal. 

 

The content of the work as a whole does not reflect new results, and is a certain combination of a review of classical information about Lorentz models and the author’s achievements in the field of their study. (E.g., “A variable transformation shows that the cubic model can be converted to the same form as the Logistic ODE” is in general not new result but is postulated in the Abstract, etc.) That is why the work is replete with references to one’s own works (self-citation). As I understand it, this formulation of the problem generally corresponds to the scope of the “Encyclopedia”.

 

Therefore, I propose to accept the article after minor revision.

 

I have two remarks:

 

1. The title of the article is too abstract and does not reflect the specific goals and objectives of the work. Please, be more concrete.

2. In conclusion, I would like to see at least a brief remark regarding the application of Lorentz models and specifically the new knowledge about transformations described by the author, in the real world. What is implication of the study?

 

A few minor notes (refer to lines):

 

96 “and supported by other references” - the phrase suggests a link to other works

 

Figs. 3 and 4 – there are no plots numbering (a,b,c…)

 

183 “Consequently, the time…” – this phrase refers to Fig. 5b? This should be added.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Review

for the article «Revisiting Lorenz’s Error Growth Models»

author: Bo-Wen Shen

General

The article is devoted to the current problem of proving the stability of a differential equation or system of differential equations.

The author refers to the Lorenz model, which describes the dynamics of the atmosphere. We will not discuss the adequacy of the Lorenz model itself now. Let us consider purely formally that we have a certain system of ordinary nonlinear differential equations. The question of the stability of the solution of this system is posed, i.e. its dependence on the initial conditions.

I have no questions about the formulation of the problem. I have a comment on the form of presentation of the material and its presentation. It is presented too briefly. The presentation requires the reader to refer to the sources. Therefore, I would like the article to be self-sufficient. The author should briefly introduce the reader to the essence of the problem. The main question: how is the Lorenz equation system related to the logistic equation? Why is the error in the system of equations determined by this equation?

Specific comments

1.    Line after 59. This study has nothing to do with the predictability of weather forecasts. Therefore, it is better not to talk about the weather, but to talk about the stability of the mathematical model of Lorenz, although it was proposed as a model of atmospheric dynamics. Or, as was said above, to present the material differently.

2.    The line before formula (5). The value on the right in formula (5) is more correctly called the relative growth rate.

3.    Equation (6) is simply postulated.

4.    Line 93. λ represents the relative growth rate.

5.    In Table 1, the discrete logistic ODE number should be (15), not (10).

6.    I couldn't derive equation (21).

Conclusion

The article requires revision and re-review.

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The paper provides a thorough mathematical analysis of Lorenz's error growth models, specifically focusing on the Logistic ordinary differential equation (ODE) and its connections to the non-dissipative Lorenz 1963 model. The analysis show that the importance of continuous dependence on initial conditions (CDIC) in predicting horizons and the chaotic behaviors exhibited by the Logistic map. The mathematical relationships between the Logistic ODE, Logistic map, and the Lorenz model are explained. Overall, the paper provides a valuable contribution to the understanding of Lorenz's error growth models and their mathematical relationships. I recommend accepting the manuscript after some revisions. Some comments as following:

1.         The introduction effectively sets the stage for the paper, offering a concise overview of Lorenz error growth models. However, the article references a limited body of literature and covers a relatively narrow scope. It is recommended that the content be expanded. It would benefit from a brief mention of the implications of these models in broader scientific or practical contexts to engage a wider audience.

2.         The analysis of growth rates is thorough, with a clear presentation of figures illustrating the concepts. But how does the paper define the growth rate? What is the difference between linear growth rate and nonlinear growth rate in physical? It would be beneficial to include a detailed definition of these growth rates in the context of current research and discussion about the physical differences between linear growth rate and nonlinear growth rate.

3.         Line 59, “Common methods for assessing predictability horizons include 1) analyzing growth rates”. Based on your full text expression, my understanding is that the growth rate here is not the error growth rate, but the rate of change in state amplitude. Thus, why does growth rate affect predictability? Is there any relevant literature to support it? The speed of growth does not indicate the speed of error growth, nor does it affect predictability.

4.         Line 64, the manuscript describes starting from small positive initial values, how much is it taken? Small relative to whom?

5.      Please indicate in the text what  in Equation 6 means.

6.      In describing the error growth rate of a dynamical system, the Lyapunov exponent is a linear result, while Ding and Li (2007) proposed a nonlinear localized Lyapunov exponent for describing the nonlinear error growth rate of a dynamical system, and a saturated relative error theorem for quantitatively estimating the predictable period. The authors may refer to these.

Chen, B., Li, J., & Ding, R. (2006). Nonlinear local Lyapunov exponent and atmospheric predictability research. Science in China Series D: Earth Sciences, 49, 1111-1120.

Ding, R., Li, J., & Li, B. (2017). Determining the spectrum of the nonlinear local Lyapunov exponents in a multidimensional chaotic system. Advances in Atmospheric Sciences, 34, 1027-1034.

Ding, R., & Li, J. (2007). Nonlinear finite-time Lyapunov exponent and predictability. Physics Letters A, 364(5), 396-400.

 

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

This paper can be accepted.

Author Response

Thanks very much!

Reviewer 2 Report

Comments and Suggestions for Authors

The author responded all my comments.

I recommend to publish.

Author Response

Thanks very much!

Reviewer 3 Report

Comments and Suggestions for Authors

The article may be accepted for publication in this form. 

Comments for author File: Comments.pdf

Author Response

Thanks for your comments.

The following two excerpts, taken from our conclusions and responses to the comments from the previous review round, illustrate that the Logistic equation has been hypothetically utilized to examine the errors associated with weather predictions (e.g., Lorenz 1969a, 1982).

 

(I)

The empirical nature of these error growth models presents challenges in establishing mathematical and physical consistency with real-world data. Lorenz himself acknowledged in 1982 that the inclusion of the nonlinear quadratic term in the error growth model is reasonable, but not readily verifiable (Lorenz 1982). Over the past decades, significant efforts have continued to justify this nonlinear quadratic term (e.g., Nicolis 1992; Shen 2020). In this study, we first reiterated the connections among the sigmoid, hyperbolic tangent, and hyperbolic secant squared functions, and progressively provided the links between the Logistic ODE and the non-dissipative Lorenz 1963 model. While both are idealized, the Lorenz 1963 model is fundamentally derived from the Rayleigh Benard convection partial differential equations. To the best of our knowledge, this is the first attempt to integrate Lorenz’s error growth model with his 1963 model.

 

(II)

As discussed, while the Logistic model has been used to estimate error growth, Lorenz acknowledged that the quadratic hypothesis is not verifiable (as also shown in Figure R1 below). Although Nicolis (1992) attempted to examine the relationship between the Logistic model and the Lorenz 1984 model (that is not derived from physics-based PDEs), this study is the first to illustrate the mathematical relationship between the Logistic model and the non-dissipative Lorenz model (that is derived from the  Rayleigh Benard convection equations without dissipative terms).

Reviewer 4 Report

Comments and Suggestions for Authors

The author has answered my questions, and I suggest accepting the manuscript.

Author Response

Thanks very much!