Referee report on 'Potential mapping method for the steady-state magnetosheath model' by Y. Narita et al.

This manuscript describes a methodology for mapping magnetosheath locations relative to specific empirically-based models for the bow shock and magnetopause into an equivalent magnetosheath location with boundaries described by confocal paraboloids. Analytic solutions for plasma streamlines (and potential) and magnetic fields (and magnetic potential) can then be conformally mapped to a space bounded by more realistic boundaries.

In general, this article does not represent a significant advancement. It reads more like an Appendix of a larger study, with the Appendix detailing a technique to map locations between confocal, parabolic boundaries and empirically-based boundaries. While this technique is similar to previous efforts as described by Soucek and Escoubet [2012], Trattner et al., JGR [2015], and others, there is no effort here to demonstrate that this particular mapping technique better matches observations than previous techniques.

Some of the references to empirical models of boundary shapes/sizes are inconsistent with the description provided here, or are examined under extremely specific solar wind conditions, or do not properly represent the knowledge of the physical boundaries far down the flank. Specifically,

1. The Farris et al., JGR [1991] empirical bow shock model is not a paraboloid model. It is an ellipsoid model (eccentricity of 0.81), describing the bow shock on the dayside. It is not a proper representation of the far flank bow shock.
2. The Cairns et al., JGR [1995] paraboloid bow shock model also does not properly represent the far flank bow shock. The distant bow shock shape approaches that of a hyperboloid.
3. The authors have selected a very specific exponent for the Shue et al., JGR [1997] model (alpha = 0.5) in an effort to show that the analytic model is 'simple'. The solar wind conditions for which this exponent is applicable (from the Shue et al. mode) is not often encountered (IMF Bz > +8 nT, with specific values of solar wind dynamic pressure in order that alpha = 0.5).

Additional references to analytic models of the magnetosheath magnetic field (using expansions in Legendre polynomials) that make use of flexible magnetopause and bow shock boundary models (e.g., Romashets and Vandas, JGR, [2019]) should be provided and discussed.

Although the claim is made in the manuscript that this technique can be applied to arbitrary boundary model shapes, it is not demonstrated that under general circumstances, the equations can be written in a closed form.

The technique described relies on determining the (straight line, or minimum) distance from a given point within the magnetosheath to the magnetopause. This is along the normal direction from the magnetopause. However, Lines 109-110 state that the task is to find the shell variable 'v' and the connector variable 'u' in the empirical magnetosheath. However, while the connector variable 'u' of the empirical magnetosheath is normal to the magnetopause surface, it is not a straight line through the magnetosheath – and doesn’t represent the minimum distance from the given point to the magnetopause. In other words, the distance from the magnetopause extends over a (narrow) range of connector variable 'u' values.

The rationale for the methodology described is confusing. For most implementations, the solar wind parameters are known, and the corresponding parameters at a given point within the magnetosheath are desired. However, the methodology here is to start with known parameters at a given place within the magnetosheath (relative to empirical models), conformal map to a location relative to the KF paraboloid boundaries, calculate the 'u' and 'v' values and determine the B-field, streamline, and potentials. The solar wind drivers appear to be missing. It appears that part of this strategy is based on the Toepfer et al. [2022] motivation; but a clear description for the order of steps for this technique is missing.

Several of the equations presented are incorrect. For example, Eq.5 is infinite everywhere, due to the denominator. How are Eqs.20 and 23 are used to derive Eq.24? Why do the units not match for the terms within Eq.39? How do Eqs.31-32 lead to Eq.33 when ymp=0?

General comments

This manuscript proposes a method for generalizing the mapping of flow lines and magnetic field in the magnetosheath. The proposed method aims to be computationally inexpensive and generalizable, which is desirable for many applications including statistical studies. However, a number of critical issues need to be addressed if the manuscript is to be considered for publication. The results presented are not general enough to be of use for actual applications; thus, the manuscript represents no (or very minor) scientific advancement. The novelty of the study needs to be explained, in particular by a more focused comparison with previous works. Furthermore, the method should be presented more clearly; it is currently difficult to tell if the proposed method is incorrect or if the presentation is unclear and contains too many mistakes (typos and inconsistencies). Please find below my detailed comments and suggestions for improvement.

Specific comments

Concerning the whole manuscript

Coordinate system: The authors introduce a new, non-parabolic coordinate system (u,v,ϕ) in which the potentials for the velocity and magnetic field are to be expressed. However, it is not mentioned whether the new coordinate system is orthogonal; in fact, from Fig. 3 it appears that the grid in the right panel is not. The parabolic coordinate system (used in KF94) is orthogonal by construction, and the gradient and Laplace equation in this system are defined and given explicitly (Eq. (8)-(9) in the KF94 paper). In this manuscript, however, the potentials are obtained from the shell and connector variables v and u in the new coordinate system (according to line 274-285 and 289). The authors do not define the gradient and curl in this system and thus the evaluation of Eq. (1) is not defined in the manuscript.

Generalization of the method: In section 3, the mapping algorithm is reduced to an axisymmetric geometry and the calculations are made specifically for the Farris (1991) bow shock and Shue (1997) models. Yet, it is stated in sections 1 and 4 that the method is easy to generalize to an arbitrary magnetopause shape. Would it be possible to express the derivations in more general terms? I suggest giving expressions on a form which allows for non-axisymmetric geometries and aberrated GSE coordinate systems (see for example the asymmetric magnetosheath thickness and aberrated x axis in https://doi.org/10.1002/jgra.50465), or otherwise indicate to the reader which modifications and transformations are necessary to generalize the method. Are there restrictions on the choice of magnetopause model? Does the method require an analytic expression for the magnetopause?

Figures 1 and 2 show examples of results of the mapping, but not in which way the proposed method is better than (or even different from) the method by Soucek (2012). They are also introduced very early in the manuscript, before the potential functions or the mapping procedure have been described, and do not contain much relevant information. I suggest replacing them with figures showing the steps of the mapping procedure and/or a comparison with the Soucek (2012) method.

Title

The phrase “Potential mapping method” can be interpreted as “Possible mapping method”. Perhaps the title could be rephrased to avoid misunderstanding.

Abstract

The abstract lacks a motivation for the study (a short background and a science question to be answered). It is very technical, especially the second sentence, and difficult to follow before reading the manuscript.

Section 1

This section needs to be more concise and explain the novelty of the proposed method. In particular, the advantages compared to Soucek an Escoubet (2012) need to be clearly explained. Furthermore, since this study is very similar to the present work, it would be informative if the manuscript contained a comparison between the two methods in cases where the difference between them is the most important, together with an explanation as to why the proposed method is advantageous.

The introduction should be restructured. For example, lines 36-38 (“While the radial mapping [...]”) say the same thing as lines 52-53 (“While the radial mapping [...]”). The “gap” that this study fills is also stated twice (line 25-26 and line 48-49).

Line 52-53: “While the radial mapping is nearly boundary-fitted on the dayside, the orthogonality of mapping degrades in the flank to tail region.”: this sentence is essential as it mentions the difference between the proposed method and the previous one. However, the term “orthogonality of mapping” should be better explained and perhaps illustrated in a figure. Currently, it is not clear what “orthogonality” refers to.

Section 2

The title should be more informative, for example indicating that the section reviews theory from previous works.

This section reviews previous results from KF94. It should begin with an introductory sentence explaining what the section contains. It was not entirely clear when the review of previous work ends and the new work starts. Also, the level of detail in the section seems a bit unnecessary. Would the authors perhaps consider referring to the works by Kobel and Flückiger (1994) and Guicking et al. (2012), instead of writing out all the equations in the main text? Alternatively, detailed equations could be placed in an appendix to improve the flow in the text.

Section 3

Section 3.1: This section contains confusing terminology and probably some typos. These include:

• Line 130: “magnetosheath-to-magnetopause distance”: the magnetosheath is a region bounded by the magnetopause, so this phrase does not make sense. Instead of magnetosheath, consider writing “position vector r” (since αemp is the distance from r to the magnetopause).
• “magnetosheath-normal direction”: → “magnetopause-normal direction” (line 136 and Figure 4 caption).
• “magnetopause thickness” → “magnetosheath thickness” (line 137, Figure 4 caption, line 250, possibly more places in the text).

Line 138-139: are these the only input parameters, or should the magnetopause and bow shock shapes also be regarded as input?

Also, the reason for defining a unit vector orthogonal to the magnetopause was not clear when reading the manuscript for the first (or second) time. The motivation for this choice should be emphasized in this section and better explained in the introduction.

Section 3.2-3.5: see above comments about generalization of the method. Equations (18)-(20) and (26)-(28) are specific to the Shue and Farris models; it should be clarified that they do not describe a generalized method.

Section 3.6: The derivations are very similar to what has already been done in section 3.2-3.5. For the sake of getting a better flow in the text, perhaps it would be possible to reduce the number of equations, or make an appendix with the details?

Section 3.8: Line 270-272: What is meant by the sentence “The mesh pattern [...]”? Why is this important?

Section 4

This section needs to be reworked when the above comments have been taken into account.

Line 292: “wider range” – compared to what? (This also appears in the abstract.)

Line 294: “The method is applicable to an arbitrary shape of magnetosheath domain”: in its current state, the method is specific to the Shue and Farris models, and thus this sentence is too strong (see above comments about generalization of the method).

Line 302-304: Here the authors mention non-orthogonality – is the coordinate system (u,v,ϕ) orthogonal in the magnetosheath?

Line 309-311: “In reality, non-axisymmetric [...]” – this should be expanded on or incorporated in other parts of the manuscript.

Technical corrections

Please check the language and grammar throughout the manuscript. For example:

• Line 4: “solution of Laplace equation” → “solution of the Laplace equation”
• Line 51: “magnetopause. But” → “magnetopause, but”
• Line 53: “orthogonality of mapping” → “orthogonality of the mapping”
• Line 63: “flow velocity is U is” →  “flow velocity U is”
• Line 164: “[...], the cylindrical distance” → “[...], and the cylindrical distance”
• Line 207: “such that relative distance” → “such that the relative distance”
• Line 277: “streamline” → “streamlines”
• Line 278: “nose of magnetopause” → “nose of the magnetopause”
• Line 293: “solar wind condition” → “solar wind conditions”