Knot Probability of Random Magnetic Field Lines

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the manuscript “Knot Probability of Random Magnetic Field Lines”, the authors consider the knotting probability of magnetic field lines based on the knowledge of the knotting probability of random walks and propose applying it to knot formation in solar magnetic field lines. They derive an equation for knot probability as a function of the distance between the footpoints of magnetic field lines in sunspots, showing that knot complexity increases with distance. They also propose a relationship between knot complexity and magnetic energy, suggesting that magnetic field topology influences the dynamics of active regions and solar flares. 

The work is presented clearly; however, I have doubts about the underlying idea of magnetic field lines being randomly distributed on the Sun. While this hypothesis may be partially valid within a specific range of scales, it is not the case when considering active regions where the magnetic fluxes are organized into coherent structures at various scales. Indeed, the solar magnetic field is governed at all fluid scales by magnetohydrodynamic (MHD) equations, which impose physical constraints not found in the purely random walks considered in the manuscript.

The authors have not yet validated their modeling equations, leaving some underlying assumptions unverified. This work has the potential to be highly valuable, particularly if the authors test their equations to establish the range of applicability for randomly distributed magnetic field lines. Before considering publication, I strongly encourage the authors to support their findings with observational data of the Sun’s magnetic fields or, if that is not feasible, with numerical simulations or an appropriate solar magnetic field model.

Author Response

Reviewer 1:

In the manuscript “Knot Probability of Random Magnetic Field Lines”, the authors consider the knotting probability of magnetic field lines based on the knowledge of the knotting probability of random walks and propose applying it to knot formation in solar magnetic field lines. They derive an equation for knot probability as a function of the distance between the footpoints of magnetic field lines in sunspots, showing that knot complexity increases with distance. They also propose a relationship between knot complexity and magnetic energy, suggesting that magnetic field topology influences the dynamics of active regions and solar flares. 

The work is presented clearly; however, I have doubts about the underlying idea of magnetic field lines being randomly distributed on the Sun. While this hypothesis may be partially valid within a specific range of scales, it is not the case when considering active regions where the magnetic fluxes are organized into coherent structures at various scales. Indeed, the solar magnetic field is governed at all fluid scales by magnetohydrodynamic (MHD) equations, which impose physical constraints not found in the purely random walks considered in the manuscript.

 

Authors:

Thank you for this question. What we refer to as “random loops” are not elementary random curves, but rather structured as you expected. Random loops have coherent loop topology just like you said, while the geometry of their curves are locally random. There have already been a wide range of studies based on the random loop model in similar systems, including observation support to random loop model for active regions published on ApJ (Fractality of Magnetic Helicity Distribution in the Solar Corona, https://iopscience.iop.org/article/10.3847/1538-4357/adaaed). We have explained in detail in line 71-89 about this.

There is not conflict between random loop model and dominant MHD flow, because according to the dynamo theory, there is always local fluctuation in addition to global flow, which can be expressed as a superposition of random waves. According to the ApJ paper we cited in line 84, it is appropriate to model the magnetic field lines between sunspots as random loops.

 

Reviewer 1:

The authors have not yet validated their modeling equations, leaving some underlying assumptions unverified. This work has the potential to be highly valuable, particularly if the authors test their equations to establish the range of applicability for randomly distributed magnetic field lines. Before considering publication, I strongly encourage the authors to support their findings with observational data of the Sun’s magnetic fields or, if that is not feasible, with numerical simulations or an appropriate solar magnetic field model.

 

Authors:

Thank you for the advice. In fact, there have already been published observations verifying the random loop model for active regions, and we have cited it in line 84. In the cited paper, researchers derived new scaling law based on the random loop model, and it is verified by observation to the newly-emerged active region. This is a direct support to the random loop model for magnetic field lines between sunspots. We hope with these explanations, the model now appears acceptable to the reviewer.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper posits that solar magnetic field lines may locally behave like random curves. Introducing statistical physics methods to the study of the solar magnetic field could be beneficial. However, several critical aspects require substantial improvement before the manuscript can be considered for publication.

Detailed comments

L.2
The authors state that "the solar magnetic field on small scales can be seen as nearly random." This assertion is crucial but lacks sufficient justification. What is the physical length scale below which the magnetic field can be reasonably approximated as nearly random? Without this clarification, the foundational premise of the model is weak. Furthermore, the authors did not state that the results are based on this assumption, except in the abstract.

L.109
The authors claim that "if we consider the field line as a discretized random walk, its step number N is N∝ L²." While the proportionality is plausible, the paper fails to connect the step length of the discretized random walk to any meaningful physical scale in the solar corona. What physical quantity does this step length represent? The absence of this connection significantly weakens the model's physical relevance.

L.57
The authors assume that the field lines behave similarly to random curves, but they do not explain how or at what scales the influence of MHD equations can be ignored.

L.69
What is meant by “local scale”?

L.182
What are the previous observations, and how do they support this point?

L201

The results do not seem to support the claim that “there are frequent reconnections to change the knot type.” Could you please clarify this?

L.202-204
The authors state that “the specific field line knotting cannot be directly observed in great detail with current tools.” What is the scale of the structures that cannot be observed? Do the authors imply that these structures can only be detected in the NLFFF based on higher resolution magnetograms? If so, the scale of these fields is much smaller than that of sunspots.

Author Response

Reviewer:

This paper posits that solar magnetic field lines may locally behave like random curves. Introducing statistical physics methods to the study of the solar magnetic field could be beneficial. However, several critical aspects require substantial improvement before the manuscript can be considered for publication.

 

Detailed comments

 

L.2
The authors state that "the solar magnetic field on small scales can be seen as nearly random." This assertion is crucial but lacks sufficient justification. What is the physical length scale below which the magnetic field can be reasonably approximated as nearly random? Without this clarification, the foundational premise of the model is weak. Furthermore, the authors did not state that the results are based on this assumption, except in the abstract.

 

Authors:

The formulation that the magnetic field can be seen as a combination of global flow and local random fluctuation is the fundamental idea in the dynamo theory and mean-field theory, which is widely accepted in solar MHD dynamics, please see equation 6 and 7. 

There have already been published observation result on ApJ (Fractality of Magnetic Helicity Distribution in the Solar Corona, https://iopscience.iop.org/article/10.3847/1538-4357/adaaed) supporting the random loop model on the scale of newly-emerged active regions, please see line 83-89.

 

Reviewer:

L.109
The authors claim that "if we consider the field line as a discretized random walk, its step number N is N∝ L²." While the proportionality is plausible, the paper fails to connect the step length of the discretized random walk to any meaningful physical scale in the solar corona. What physical quantity does this step length represent? The absence of this connection significantly weakens the model's physical relevance.

Authors:

Thank you for the question, we have explicitly stated that L is proportional to the distance between the two foot-points of sunspot.

 

Reviewer 2:

L.57
The authors assume that the field lines behave similarly to random curves, but they do not explain how or at what scales the influence of MHD equations can be ignored.

Authors:

The influence of MHD equation does not conflict to the random waves. As explained above, the MHD flow is decomposed into the combination of global flow and local fluctuation.

 

Reviewer:

L.69
What is meant by “local scale”?

Authors:

Thank you for the question. We have explained in detail about this in the text. By “local” we refer to a segment of the curve, in comparison to the global structure, which is a loop.

 

Reviewer:

L.182
What are the previous observations, and how do they support this point?

Authors:

Thanks, we have added the explicit reference (C E Parnell, ApJ, 2000) on the fact that the energy distribution crosses several magnitudes, see line 201-202.

 

Reviewer:

L201

The results do not seem to support the claim that “there are frequent reconnections to change the knot type.” Could you please clarify this?

Authors:

Thank you for pointing this out. The main result in this paper, does not have a direct connection to the frequency of the reconnection. We have deleted the word “frequent”. According to observations, we know that there are reconnections, but the frequency is unknown. Thank you.

 

Reviewer:

L.202-204
The authors state that “the specific field line knotting cannot be directly observed in great detail with current tools.” What is the scale of the structures that cannot be observed? Do the authors imply that these structures can only be detected in the NLFFF based on higher resolution magnetograms? If so, the scale of these fields is much smaller than that of sunspots.

Authors:

We are only suggesting NLFFF method as one option, we did not make any claim that this is the only method. For example, there is also reconstruction method that take the Lorentz force into account, which may also work. By “specific field line knotting”, we refer to the self-linking and twist of the field line. It is not necessarily much smaller than the scale of sunspot, but will be better observed if the resolution to the magnetogram is enhanced. We have further explained this in the text, see Line 203-204.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I appreciate the authors’ effort to clarify the concept of "random loops" and the reference to Xiong et al. (2025) to justify using this model in the context of solar magnetic field lines. However, I would like to raise a few considerations.

1. In the revised version of the manuscript, the range of scales over which the random loop approximation remains valid does not clearly emerge. Providing a more explicit discussion on this aspect would strengthen the justification of the model.
2. I agree that the presence of local fluctuations is compatible with dynamo theory and with the decomposition of the magnetic field into a global flow plus a stochastic component. However, in active regions, the magnetic field is highly structured, and its configurations are governed by MHD equations, which may impose constraints on the assumption of local randomness. Clarifying how the model aligns with MHD solutions in active regions would be beneficial.

I appreciate the authors' efforts, but some aspects still require further clarification and more direct validation.

Author Response

Dear Reviewer,

Please see the attached pdf file for our reply, because we have drawn a diagram for illustration for you. We hope these clarifications can eliminate misunderstandings. Thanks.

Best Wishes

Authors

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have performed a solid revision of the text. They have significantly improved their paper, and I recommend the paper for publication. 

Comments on the Quality of English Language

 The current level of English is sufficient for understanding the content, but professional editing would elevate it significantly for publication.

Author Response

Reviewer 2:

The authors have performed a solid revision of the text. They have significantly improved their paper, and I recommend the paper for publication. 

 

Authors:

Thank you very much for the approval, we are grateful to it. 

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

I appreciate the effort to address the issue concerning the range of scales for the applicability of the random loop approximation. However, this point could still benefit from a more precise and quantitative discussion. In particular, it would be helpful to explicitly define the typical spatial scales - e.g., in megameters or in relation to sunspot separation - over which the approximation holds.

Moreover, it may be beneficial to highlight that the presence of field fluctuations in the dynamics and structure of coronal loops is consistent with MHD models developed for coronal loop systems (Nigro et al. 2008, ApJ 685(1) 606–621), which have also shown agreement with solar coronal observations (Cadavid et al. 2014, ApJ 795(1) 48). Including these supporting references would provide greater robustness to the claims regarding fluctuations that remain coherent within the framework of an MHD description of magnetic structures in the solar corona.

Author Response

Reviewer 1:

I appreciate the effort to address the issue concerning the range of scales for the applicability of the random loop approximation. However, this point could still benefit from a more precise and quantitative discussion. In particular, it would be helpful to explicitly define the typical spatial scales - e.g., in megameters or in relation to sunspot separation - over which the approximation holds.

 

Authors:

Thank you for your comment. We have added explicit description to the spatial scale in line 151-155: “The spatial scale for this equation's application is the same to the distance between foot-pints of the same sunspot, which may vary from $10^3$ to $10^5$ kilometers, and is proportional to $L$. So that when the span of the foot-point is large, is it more likely to observe complexly knotted field lines.”

 

Reviewer 1:

Moreover, it may be beneficial to highlight that the presence of field fluctuations in the dynamics and structure of coronal loops is consistent with MHD models developed for coronal loop systems (Nigro et al. 2008, ApJ 685(1) 606–621), which have also shown agreement with solar coronal observations (Cadavid et al. 2014, ApJ 795(1) 48). Including these supporting references would provide greater robustness to the claims regarding fluctuations that remain coherent within the framework of an MHD description of magnetic structures in the solar corona.

 

Authors:

Thank you for your comment. We have added this content by citing these two papers in line 105-107.

Round 4

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript has improved and the authors have adequately addressed all my comments and concerns. The clarifications provided, have strengthened the manuscript. I have no further comments and I recommend the manuscript for publication in its current form.