Negative Differential Resistance, Instability, and Critical Transition in Lightning Leader

Gou, Xueqiang; Xin, Chao; Xu, Liwen; Yuan, Ping; Zhang, Yijun; Cheng, Mingli

doi:https://doi.org/10.5194/npg-2024-15

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https://doi.org/10.5194/npg-2024-15

Preprints

09 Aug 2024

| 09 Aug 2024

Status: this preprint is currently under review for the journal NPG.

Negative Differential Resistance, Instability, and Critical Transition in Lightning Leader

Xueqiang Gou, Chao Xin, Liwen Xu, Ping Yuan, Yijun Zhang, and Mingli Cheng

Abstract. The phenomena of leader extinction and restrike during lightning events, such as multiple strokes in ground flashes or recoil leaders in cloud flashes, present significant challenges. A key aspect of this issue involves the discussion of the channel’s negative differential resistance and its instability. From the perspective of bifurcation theory in nonlinear dynamics, this paper posits an inherent consistency among the channel’s negative differential resistance, channel instability, and the critical transition from insulation to conduction. This study examines the differential resistance characteristics of the leader-streamer system in lightning development. We correlate the differential resistance characteristics of the leader-streamer channel with the channel’s state and instability transitions, investigating the critical current and potential difference conditions required for the stable transition of the leader-streamer channel.

Received: 05 Jul 2024 – Discussion started: 09 Aug 2024

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Xueqiang Gou, Chao Xin, Liwen Xu, Ping Yuan, Yijun Zhang, and Mingli Cheng

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RC1:
'Comment on npg-2024-15', Anonymous Referee #1, 10 Nov 2024 reply

I recommend the manuscript be accepted as is, pending the minor revisions below:
-I would recommend increasing the size of Figure 1. At its present size, it is difficult to read without zooming in very closely.
-In line 88, is the parameter a just a numerical parameter, or does it have a name or definition?
-In lines 91-93, the change in potential profile against current is discussed in Figure 1. Within these lines, there is mentioned a "certain state" and a "certain limit," as well as a "completely different state" before reaching steady state. Be more specific with what these thresholds and states really are, and if possible postulate on how they might come to be.

Reply

Citation: https://doi.org/10.5194/npg-2024-15-RC1
- AC1:
  'Reply on RC1', xueqiang gou, 07 Dec 2024 reply
  Dear Reviewer,
  We sincerely appreciate your valuable comments and suggestions for improving our manuscript. Below are our detailed responses to your specific concerns:
  Regarding the size of Figure 1
  
  Reviewer’s comment:
  I would recommend increasing the size of Figure 1. At its present size, it is difficult to read without zooming in very closely.
  Our response:
  
  We greatly appreciate your feedback regarding the size of Figure 1. We agree that the current size might affect readability, and we will adjust the figure size in the revised manuscript to ensure that the fonts and data points are more clearly visible. Additionally, we will optimize the resolution of the figure legend and annotations to better present the data and dynamic behavior.
  Definition of parameter a (Line 88)
  
  Reviewer’s comment:
  In line 88, is the parameter aa just a numerical parameter, or does it have a name or definition?
  Our response:
  
  Thank you for your comment on the definition of parameter a. We would like to clarify that a is a control parameter introduced in the normalized model referenced from Agop et al. (2012). It is used to describe the nonlinear characteristics of the system, and its specific roles include:
  Determining the range of the negative differential resistance (NDR) region;
  Controlling the possibility of bistability in the system;
  Influencing the formation of bifurcation points and hysteresis phenomena.
  In this study, we adopted this theoretical model and used a as a normalized control parameter to illustrate the relationship between negative differential resistance and critical transitions. However, our study does not involve detailed physical derivations or mechanisms of a, as the focus is on its role in describing the conditions for critical transitions.
  To avoid potential misunderstandings, we will supplement the following clarification about aa in the revised manuscript:
  a is a key parameter controlling the strength of nonlinear feedback and critical bifurcation behaviors in the system, determining the possibility of bistability and its triggering conditions;
  In our analysis, a is adopted from an existing theoretical model and is not the main focus of this study.
  Clarification of “certain state” and “certain limit” (Lines 91-93)
  
  Reviewer’s comment:
  In lines 91-93, the change in potential profile against current is discussed in Figure 1. Within these lines, there is mentioned a "certain state" and a "certain limit," as well as a "completely different state" before reaching steady state. Be more specific with what these thresholds and states really are, and if possible postulate on how they might come to be.
  Our response:
  Thank you for pointing this out. We realize that our description of “certain state” and “certain limit” in the current text may lack clarity. Below, we provide a more specific explanation of these terms and their corresponding thresholds and states:
  "Certain state":
  
  This refers to the system’s initial stable state (e.g., point A in Figure 1), where the channel current monotonically increases with voltage, and the system remains in a single stable state.
  "Certain limit":
  
  This refers to specific threshold values of voltage or current (e.g., points B and D in Figure 1) where the system undergoes bifurcation, transitioning from one stable state to another. These limits correspond to the critical points of the system’s dynamics.
  "Completely different state":
  
  This refers to the new stable state reached after bifurcation (e.g., point C in Figure 1), where the current-voltage relationship changes significantly, and the system transitions to a distinctly different stable state.
  Regarding the mechanism behind these states and limits, the analysis in this study is based on the theoretical model referenced. Below is a brief explanation of their physical implications:
  In the negative differential resistance region ($\frac{d\varphi }{dJ}<0$ ), the system exhibits a dynamically unstable region (represented by the dashed line in Figure 1), where unstable states cannot persist.
  Within this region, the system can exist in two possible stable states, depending on the historical path or initial conditions.
  This behavior is a hallmark of hysteresis: as voltage gradually increases, the system remains in the initial stable state (e.g., point A) until it reaches a critical point (e.g., point B), where it transitions to another stable state (e.g., point C). Conversely, as voltage decreases, the system stays in the higher stable state until it reaches another critical point (e.g., point D), where it returns to the initial stable state (e.g., point A).
  In the revised manuscript, we will provide a clearer description of these phenomena and explicitly annotate the stable states, critical points, and hysteresis behavior in Figure 1’s caption.
  
  Summary
  We greatly appreciate your valuable feedback, which has helped us improve the clarity and scientific rigor of our manuscript. In the revised version, we will:
  Adjust the size and resolution of Figure 1 to enhance its readability;
  Supplement the background and role of parameter aa in our analysis;
  Clarify the physical meaning of “certain state,” “certain limit,” and “completely different state” in both the text and Figure 1’s caption.
  We hope these revisions address your concerns. If you have any further suggestions, we would be grateful to consider them.
  Sincerely, On behalf of all co-authors
  For detailed responses with equations and proper formatting, please refer to the attached supplement file (PDF)
  
  Reply
  
  Citation: https://doi.org/10.5194/npg-2024-15-AC1
RC2:
'Comment on npg-2024-15', Anonymous Referee #2, 25 Nov 2024 reply
The manuscript "Negative Differential Resistance, Instability, and Critical Transition in Lightning Leader" aims to correlate the differential resistance characteristics of the leader-streamer channel with its dynamical states and transitions. I found the manuscript not fully convincing in terms of connecting theoretical concepts with practical ones, since a direct connection between the two proposed frameworks is not straightforward. Furthermore, the quality of the figure is not adequate for a scientific paper, as well as also the text is not fluent and with several typos and mismatching. I would suggest a careful revision of the manuscript before it can be accepted for publication in NPG.
Major comments
1. The main concern regards the connection between the formalism introduced in Section 2.2 and the formalisms and results presented in Sections 2.4 and 3. Indeed, what has been introduced in Section 2.2 is a classical theory of bifurcation for autonomous dynamical systems being written as a time-evolution mapping (continuous in this case) with a not implicit dependence on time in the forcing term. Conversely, what is introduced Eq. (10) is a non-autonomous dynamical system whose implicit variable is not time but one of the state variables (I). Thus, the connection among fixed points, instability, and other types of concepts cannot be simply ruled out. What the authors introduced in Eq. (10) is a mathematical description of the manifold or a dynamical bifurcation scenario for, at least, a 2-D dynamical system described by the state variables U and I. The authors need to carefully address these concepts and revise accordingly the manuscript by possibly considering a 2-D dynamical system of the form
dU/dt = f(U, I, u_parameters)

dI/dt = g(I, U, i_parameters)
where u_parameters and i_parameters refer to the bifurcation parameters leading eventually to critical transitions in the system.
2. The second main concern is related to Figure 2. Indeed, what the authors reported is valid for bi-stable dynamical systems which are described by a double-well potential function. It is not straightforward the connection with Eq. (1) and the system introduced in Line 103 which seems to be more similar to a hysteresis cycle. Which are the stable and unstable fixed points in your system? If φ is treated as a parameter the system admits 3 fixed points provided that J≠0 and J is real. However, limit cycles could emerge when crossing the complex plane (Hopf bifurcation). Thus, more careful analysis of the bifurcations should be carried out.
3. The third main concern is related to the presentation of the results and the overall structure of the manuscript. The authors need to carefully revise the manuscript to improve the quality of the figure as well as to check the consistency of the different typesettings of the text, typos, references, etc. Please find a list below.
Check the font size for subsections

Check the font type for references through the text (sometimes italics, sometimes not)

All figures need to be improved for quality

Line 84: mismatching between φ and that used in Eq. (1)

Figure 1: increase font and labels

Line 104: missing definition of φ

Line 105: formally, the condition for fixed points should be met not for all J but for a specific solution J* or something similar

Line 106: missing definition of what the subscript J means (I assume derivative with respect to J)

Line 106: missing space and capital letter "if we let"

Line 110: the assumption is not straightforward and the connection between Eq. (1) and Line 103 is missing

Line 139: please delete double point.

Line 140: please delete the period before introducing the equation.

Line 145: which type of fit is used?

Figure 3: missing space Fig3

Figure 4: missing space Fig4

Figure 5: missing space Fig5.

Reply
Citation: https://doi.org/10.5194/npg-2024-15-RC2
- AC2:
  'Reply on RC2', xueqiang gou, 07 Dec 2024 reply
  Dear Reviewer：
  Thank you very much for your detailed comments and constructive suggestions. Below are our responses to your specific points, particularly regarding Figure 2 and its relationship to Equation (1), as well as clarifications on other parts of the manuscript:
  Connection between the Formalism in Section 2.2 and the Results in Sections 2.4 and 3
  
  Reviewer’s Comment:
  The main concern regards the connection between the formalism introduced in Section 2.2 and the formalisms and results presented in Sections 2.4 and 3. Indeed, what has been introduced in Section 2.2 is a classical theory of bifurcation for autonomous dynamical systems... Conversely, what is introduced in Eq. (10) is a non-autonomous dynamical system whose implicit variable is not time but one of the state variables (I). Thus, the connection among fixed points, instability, and other types of concepts cannot be simply ruled out. What the authors introduced in Eq. (10) is a mathematical description of the manifold or a dynamical bifurcation scenario for, at least, a 2-D dynamical system described by the state variables U and I. The authors need to carefully address these concepts and revise accordingly the manuscript by possibly considering a 2-D dynamical system of the form
  dU/dt = f(U, I, u_parameters)
  
     dI/dt = g(I, U, i_parameters)
  Our Response:
  Thank you for your valuable feedback. We would like to clarify the following points:
  Nature of Equation (10):
  
     Equation (10) is not intended as a description of a dynamical system or non-autonomous system. Rather, it represents the calculation of the differential resistance dU/dI in the lightning channel. The purpose of this equation is to analyze the stability and critical transition points of the system by examining the sign of the differential resistance.
     The misunderstanding may have arisen due to the text not clearly explaining that Equation (10) is a static tool used for stability analysis, not a model for the system’s time evolution.
  Improving the Theoretical Connection:
  
     We realize that the connection between the theory in Section 2.2 and the practical analysis in Sections 2.4 and 3 could be better clarified. To address this:
     In the revised manuscript, we will provide a clearer explanation of Equation (10), emphasizing that it is used to study the stability of fixed points in the system, and how the sign of the differential resistance indicates stability or instability.
     We will also remove any references to "dynamical systems" or "time evolution" to avoid confusion and ensure that Equation (10) is interpreted as a stability analysis tool rather than a description of system dynamics.
  Figure 2 and Its Relationship to Equation (1)
  
  Reviewer’s Comment:
  The second main concern is related to Figure 2. Indeed, what the authors reported is valid for bi-stable dynamical systems which are described by a double-well potential function. It is not straightforward the connection with Eq. (1) and the system introduced in Line 103 which seems to be more similar to a hysteresis cycle. Which are the stable and unstable fixed points in your system? If φ is treated as a parameter the system admits 3 fixed points provided that J≠0 and J is real. However, limit cycles could emerge when crossing the complex plane (Hopf bifurcation). Thus, more careful analysis of the bifurcations should be carried out.
  Our Response:
  Thank you for your comments on Figure 2. We would like to clarify the following points regarding the figure and its relationship to Equation (1):
  Core Concept of Equation (1):
  
     Equation (1) defines the relationship between normalized voltage and normalized current , where parameter determines the strength of the system's non-linearity and the potential for bistable behavior. By varying , Equation (1) can produce the following behaviors:
       Monostable behavior: When is small, the system exhibits a single stable state, with a monotonically increasing relationship between voltage and current.
  Bistable behavior: As increases, the system can exhibit bistability, with two stable points and one unstable region (negative differential resistance).
  Clarification of Figure 2:
  
     Figure 2 is a landscape diagram that visually represents the transition between stable and unstable states under different conditions, which we associate with the bistable behavior described by Equation (1). The figure shows how the system transitions between two stable states, with the critical points   and indicating where the system undergoes a shift from one stable state to another. These critical points are not the stable states themselves, but rather the thresholds at which the transition occurs.
     We will revise the manuscript to clarify that   and   are critical points, not stable states, and that they mark the boundary between different stable states in the system.
     We will also explain how the hysteresis cycle observed in Figure 2 corresponds to the negative differential resistance region in Equation (1).
  Universality and Literature Support:
  
     Figure 2 highlights the universality of bistable behavior, which is commonly observed in various natural systems such as ecological systems (e.g., Scheffer's studies) and lightning channel dynamics. This figure visually demonstrates the critical transition phenomenon that can also be found in other complex systems.
     We will include references to relevant literature (such as Scheffer et al.) to support the broader applicability of this model and its connection to critical transitions in complex systems.
  Presentation and Structure of the Manuscript
  
  Reviewer’s Comment:
  The third main concern is related to the presentation of the results and the overall structure of the manuscript. The authors need to carefully revise the manuscript to improve the quality of the figure as well as to check the consistency of the different type settings of the text, typos, references, etc. Please find a list below.
  Our Response:
  Thank you for pointing out areas for improvement in the manuscript presentation. We will address the following points in the revised version:
  Figure Quality:
  
     We will improve the resolution of all figures (Figure 1, Figure 2, Figure 3, Figure 4, and Figure 5) and increase the font size and clarity of the labels, particularly in Figure 2.
  Text Consistency:
  
     We will fix any mismatched symbols, such as the inconsistency between in Line 84 and its usage in Equation (1).
     We will clarify the definitions of all symbols (e.g., in Line 104, in Line 106).
     We will also check the overall text formatting, correcting any typos, reference formatting issues, and other minor inconsistencies.
  Clarification of Figure 2:
  
     We will add more detailed descriptions to the figure caption to clarify how Figure 2 relates to Equation (1), as well as its role in illustrating bistable behavior and its physical significance.
  Conclusion
  We appreciate your constructive feedback, which has been instrumental in improving the manuscript. In the revised version, we will:
  Provide clearer explanations of Equation (10) and its role in analyzing stability transitions.
  
  Clarify the relationship between Figure 2 and Equation (1).
  
  Improve the manuscript’s presentation, including figure quality and text formatting, to ensure clarity and consistency.
  
  We hope these revisions will address your concerns. Thank you again for your valuable input, and we look forward to your further suggestions.
  Best regards,
  On behalf of all authors
  
  For detailed responses with equations and proper formatting, please refer to the attached supplement file (PDF)
  
  Reply
  
  Citation: https://doi.org/10.5194/npg-2024-15-AC2
RC3:
'Comment on npg-2024-15', Anonymous Referee #3, 10 Dec 2024 reply

The manuscript explores stability and critical transitions in lightning discharge channels using concepts from nonlinear dynamics, particularly bi-stable systems. I recommend acceptance contingent upon revision. While Figures 1 and 2 effectively demonstrate the theoretical principles, they remain too abstract and do not correspond directly with the specific dynamics of lightning channels discussed. I suggest adding figures that depict the stability profiles derived from equations 9 and 10, as these directly describe the behavior of lightning systems. This enhancement will clarify the instability mechanisms within real lightning channels, making the application of theoretical models more comprehensible and scientifically rigorous.

Reply

Citation: https://doi.org/10.5194/npg-2024-15-RC3
- AC3:
  'Reply on RC3', xueqiang gou, 13 Dec 2024 reply
  Response to Reviewer’s Comments
  
  Dear Reviewer
  Thank you for your insightful comments and suggestions regarding our manuscript. We greatly appreciate your feedback, as it has helped us refine our presentation and ensure clarity. Below, we address your specific concerns regarding the interpretation of Equations (9) and (10) and the associated visual representation:
  Response to Concern on Equations (9) and (10)
  Reviewer’s Comment:
  You suggested that Equations (9) and (10) describe a non-autonomous dynamical system and could benefit from a stability analysis involving a two-dimensional dynamical system representation.
  Our Response:
  We appreciate your perspective and would like to clarify the purpose of Equations (9) and (10). These equations are not intended to represent a dynamical system, either autonomous or non-autonomous. Instead, they are static expressions for calculating the differential resistance $\frac{\text{d}U}{dI}$, a key parameter used to analyze stability in lightning channels. Specifically:
  $\frac{dU}{dI}$ captures the relationship between voltage and current in the channel. Its sign indicates stability: regions where$\frac{\text{d}U}{dI}>0$ correspond to stable states, while regions where $\frac{\text{d}U}{dI}<0$ correspond to unstable states.
  These equations are used to identify critical transitions in stability, providing a direct measure of the conditions under which negative differential resistance arises.
  To avoid potential misunderstanding, we will revise the manuscript to better emphasize the role of Equations (9) and (10) as static analytical tools rather than dynamic models.
  Response to Concern on the Representation of Stability in Figures**
  Reviewer’s Comment:
  You suggested adding figures to illustrate the stability profiles derived from Equations (9) and (10), as this would enhance clarity regarding the instability mechanisms in lightning channels.
  Our Response:
  We agree that visual representation is crucial for understanding the implications of the analysis. In fact, the variation of differential resistance $\frac{\text{d}U}{dI}$ with current is already presented in Figure 4, where we illustrate how $\frac{\text{d}U}{dI}$ changes under different conditions (e.g., channel lengths). This figure effectively visualizes the transitions between stable $\frac{\text{d}U}{dI}>0$ and unstable $\frac{\text{d}U}{dI}<0$ regions, offering a clear connection between theoretical predictions and practical dynamics of lightning channels.
  However, based on your suggestion, we will enhance the manuscript as follows:
  Provide additional context in the caption of Figure 4 to explicitly highlight how it relates to the stability analysis and the underlying mechanisms of negative differential resistance.
  
  Ensure the discussion in the text explicitly connects Figure 4 with the findings from Equations (9) and (10), reinforcing their practical relevance to lightning channel dynamics.
  
  Conclusion
  We are grateful for your thoughtful suggestions, which have helped us identify areas where additional clarity can be provided. By refining the explanations of Equations (9) and (10) and enhancing the context around Figure 4, we aim to address your concerns effectively. We hope these revisions meet your expectations, and we remain open to further suggestions for improvement.
  Thank you again for your valuable feedback.
  Best regards
  On behalf of all authors
  For detailed responses with equations and proper formatting, please refer to the attached supplement file (PDF)
  
  Reply
  
  Citation: https://doi.org/10.5194/npg-2024-15-AC3

Xueqiang Gou, Chao Xin, Liwen Xu, Ping Yuan, Yijun Zhang, and Mingli Cheng

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Short summary

Our research examines lightning's complex behavior by studying changes in its electrical pathways under different conditions. We found that lightning channels switch between stable and unstable states based on their length and surrounding electric fields. This helps explain why lightning often reactivates after a brief pause and offers new insights into these processes. Our findings could improve lightning prediction and protection, benefiting scientific understanding and public safety.


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