the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 09 Aug 2024
Negative Differential Resistance, Instability, and Critical Transition in Lightning Leader
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.
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Status: open (until 20 Dec 2024)
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RC1: 'Comment on npg-2024-15', Anonymous Referee #1, 10 Nov 2024
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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.
Citation: https://doi.org/10.5194/npg-2024-15-RC1 -
RC2: 'Comment on npg-2024-15', Anonymous Referee #2, 25 Nov 2024
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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.
Citation: https://doi.org/10.5194/npg-2024-15-RC2
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