We thank the referee for the review and comments!

- Line 142: produce -> produces

Indeed, fixed.

- Figure 5: Please use a different color scale for the current density because now shades of the surface are confusing the values.

We are now using an isoluminant CET colormap to decouple the colormap value from shading channel for this figure:

- Line 230, a single distance function: please include a reference here on how this is computed.

We added a reference to a general definition for SDFs and the specific one used in this study. 

- Line 235, Figure 10: do you mean Figure 8a?

Fig. 8a works to demonstrate this as well, changed!

- Line 249: Figure 8 -> Figure 8a

- Line 251: Figure 8 -> Figure 8c and d

Indeed, thank you for pointing both of these out!

- Line 256: delete comma.

Done, thank you!

Thank you for the detailed and constructive comments! We provide a point-by-point response below, interleaved with the original comments.

Other comments/questions:

1) Line 5-6 - The abstract says “This work presents a complementary method by noting that the magnetic field structures near reconnection lines exhibit two-dimensional features that can be identified in a suitably chosen local coordinate system.”  I think more care is needed here.  This sentence is certainly true for the work presented here, but I do not think the community would believe that it is a universal truth as the sentence implies.  There are numerous review papers on fully 3D reconnection (authors include Pontin, Priest, etc.) that describe spine-fan reconnection, etc.  If fully 3D reconnection happened in the Vlasiator simulations (which they could for a different set of solar wind parameters), I suspect the approach employed in this study would not be successful; at the very least it would take further research to determine this which is not being carried out in this study.  So, I suggest capturing the nuance by rewriting this sentence, and moreover going over the rest of the paper to fix any other sentence with similar issues.  I think it would be relevant to cite some 3D reconnection review articles and specifically point out that it is unknown whether (or unexpected that) the approach used here would work for such cases.

2) Line 8-9 - Same comment - the dimensionalities of the structures are 2D in this simulation, which “support[s] the use of such coordinate systems”, but this is only for the present simulation parameters.

Response to comments 1) & 2): We clarified this point in the abstract with modifying the latter half as:

This work presents a complementary method for quasi-2D structures in 3D settings by noting that the magnetic field structures near reconnection lines exhibit two-dimensional features that can be identified in a suitably chosen local coordinate system. We present applications of this method to a hybrid-Vlasov Vlasiator simulation of the Earth's magnetosphere, showing the complex magnetic topologies created by reconnection for simulation dominated by quasi-2D reconnection. We also quantify the dimensionalities of magnetic field structures in the simulation to justify the use of such coordinate systems.

3) Line 14 - The flux function has been used to find magnetic topologies in 2D much further back than 2017; it probably goes back to the 1960s.

This is true. We now cite Sonnerup, 1970 for an early example employing a flux function in this context and Servidio+2009 for the specifics of determining 2D nulls:

In a two-dimensional configuration, the problem is tractable via the use of a flux function (see Sonnerup, 1970 for an early example, with details given in, e.g., Servidio+2009). Recent examples of flux function use include [original references]. A generalized flux function can be defined…

4) Line 23-24 - It is probably best to say “separatrices (also called separator surfaces or separatrix surfaces)” since the alternate name for it is used later in the sentence and paper.

Thank you, a good suggestion - incorporated.

5) Line 32-44 - I think it would be relevant in the context of this paragraph to cite the Dorelli and Bhattacharjee, doi:10.1029/2008JA013410, 2009, paper.

Thank you, we added a point about Dorelli and Bhattacharjee.

6) Line 49 - The X-point topology being retained in the presence of an out-of-plane magnetic field goes all the way back to the original paper by Parker, JGR, 62, 509, 1957 - see Fig. 4.

Thank you, included a reference to Parker as well.

7) Line 52-56 - I understand the authors statement about defining an ““in-plane” null line”, but there are some subtleties that arise here.  For a 3D magnetic field, one can definitely take a plane and find where an in-plane null is.  However, if one takes a different plane that goes through that null (i.e., rotated from the initial plane), then typically the in-plane null in the different plane would not be in the same place in the different plane.  This means that the in-plane null is not an absolute quantity, it is a function of the plane itself.  I think this is actually a rather important issue, because the answer for where the in-plane null is located can change drastically with the plane in use, and which plane is “correct” has been the subject of scrutiny and is still not well understood.  (Such literature could also be cited here.)  Because of this, any statement about in-plane nulls must be taken with a grain of salt because it is not known which plane is the right plane to use.  I don’t think this has to be a major issue in this study, but I do think it should be conveyed to the reader so they are aware of this issue.

This is a fair point. We have included a disclaimer on this as follows:

It should be noted, however, that the notion of an in-plane null depends on the definition of the plane, and it is not clear what plane should be chosen for this decomposition. This work proposes one possible selection of a useful coordinate system via variance analysis for this purpose. Other selections of a coordinate system are possible, as shown by e.g., Genestreti et al 2018.

8) Line 57-58 - It is not clear to me that it is crucial to define a local coordinate system to ensure an accurate description of magnetic field structures.  Most of the examples provided here are coordinate system independent - a null is a null in any coordinate system, a current sheet is a current sheet in any coordinate system, etc.  Perhaps the authors could expand upon or solidify the statement being made here?

Admittedly this is an unintentionally broad claim -a suitably-selected local coordinate system would rather be much more useful for analysis than an awkwardly-chosen one; this was not meant as a statement that the physical structures would depend on the coordinate systems. Rewritten as:

For magnetic field structures such as \eg current sheets (CSs), null lines, and flux ropes, it is useful to select a suitable coordinate system that can exploit the properties of the structure to simplify the problem at hand.

9) Line 58-59 - The sentence starting “Methods such as” would typically warrant the inclusion of some references; perhaps a review article would be sufficient.

Introduced references to Sonnerup 1998, Paschmann 1998.

10) Line 60 - There are other recent in-depth studies comparing hybrid MVA coordinate systems, such as Genestreti et al., doi:10.1029/2018JA025711, 2018.  There are others that may be relevant too.

Thank you, included a reference to Genestreti here.

11) Line 73 - “to a local” (“a” is missing)

Thank you, fixed.

12) Line 76 - “in this simulation.” (“simulation” is missing)

Thank you!

13) Line 99 - Could the authors clarify what they mean by saying the method only gives information about the direction but not the sign?  

The directions are obtained from the GGT/GTG matrices as eigenvectors of the respective matrices. Eigenvectors can be multiplied by any scalar and remain eigenvectors, and after normalization, the directions are defined up to a constant of \pm 1. For example, in the ideal 1D current sheet, the MGA L direction could as well be chosen to point along either lobe field. We reformulated the sentence as:

However, as these methods obtain the initial directions as eigenvectors that are defined up to a real scalar, the directions (after normalization) are defined up to a factor of $\pm1$.

14) Line 134-141 - I know the information is in other papers, but it would be nice to include the boundary conditions. 

Expanded the description of boundary conditions in the following paragraph that describes the simulation run employed.

15) Line 188-189 - Could the authors clarify what is meant by the “centres of current sheets”?  In an idealized current sheet (like a Harris sheet), the symmetry of the set up makes the centre a robust physical quantity.  In an actively reconnecting magnetotail, undoubtedly the perfect symmetry would be broken.  Is the centre signifying the half distance between the current sheet edges, or the maximum in the current, or something else?

We find the B_L = 0 surface approximates well the peak current in the Vlasiator simulation, given the limited resolution, even during active reconnection and flapping of the magnetotail current sheet. We now acknowledge in text that this is not strictly correct in general, but sufficient for our purposes.

16) Line 192 - Remove “The” before “Figure 5…”

Thank you, fixed!

17) Figure 5 - Are the local boundary normal vectors plotted at random locations or is there a method to their locations?  It might be nice to share with the reader how those positions were chosen.

The placement of the vectors can be described as arbitrary (and now described as such in the caption). A fixed number of vectors is plotted automatically at original mesh locations on the sheet by the VisIt software.

18) Figure 6 caption - It says the null lines are colored by partial B_N / partial L in both plots. Is this correct for panel (a) given that it is not in LMN coordinates?

The figure has been amended to give partial B_z/partial r in panel (a).

19) Line 210-211 - Is fixing the orientation of neighbouring coordinate systems done by hand or using an algorithm?  It might be nice to let the reader know.

This is done by hand, examples are given further down in the text. We expand on this point here as follows:

…which can be fixed by re-orienting neighbouring coordinate systems. In the magnetotail, for example, the choice we make is to choose L to point sunward, so that if the constructed LMN system has initially L tailwards, we multiply the L and N vectors by -1 at these cells. Automatic determination of consistently-oriented neighbourhood charts could be considered in a future update.

20) Line 235 - Most journals require figures to be referenced in the order they are presented.  Here, Figure 10 is referred to before Figure 8 and 9.

Changed to refer to Figure 8, as it also demonstrates the FOTE method in Vlasiator data (albeit a test case).

21) Line 244-245 - Are the details of the initial condition omitted because they are thought not to matter in the results?  Do they matter?  Might it be useful to include specifically what was simulated?

The initial conditions were designed to produce a small perturbation to the vacuum superposition, to break the initial current-free condition. The system is an unscaled terrestrial dipole, with plasma parameters within reasonable parameters for outer magnetosphere, and the simulation was run for milliseconds at a coarse resolution. Considering, for example, the upper bound of initial velocity of 750km/s, the maximum distance for magnetic field advection over 7 milliseconds would be ~5 km. This is small compared both to the domain size of 2*105 km3 and the maximum grid resolution of 3125 km, so we do not expect significant propagation in the system. The description of the initial condition has been added to the manuscript.

22) Line 250 and Figure 8(c) caption- I was hoping for more of an explanation of why the authors think they are obtaining a few extra null line segments not accounted for by the analytical model, but nothing was provided.  Maybe it’s not that important?  Or maybe it is important because if the algorithm is giving spurious results then the same could happen in the Vlasiator simulations.  Could the issue be related to the initialization?  Was this tested?

As shown with the projection, these are valid X-line detections in the sense of in-plane null, albeit in a very strong guide field. As far as we can tell, these are produced by the small nudge of time propagation, likely from the initial gradients in velocity, as these features are not visible when inspecting the initial state using an LMN coordinate system purely from MGA analysis (as the hybrid LMN system is not strictly valid at t = 0). 

23) Line 252 - I don’t think “a” is necessary before “FOTE-detected”

Indeed: followed by a plural “nulls”.

24) Line 259 - I wasn’t sure what “prominent cones” referred to.  Could the authors point to it using XYZ coordinates?

“Prominent cone-shaped artefacts” would be more descriptive. Since these are removed via filtering, they are not shown in the present figures. The artefacts relate to defining the LMN system from the vacuum superposition field and orienting it with the L.Z > 0 condition, as described. The isocontour surfaces of B_L = 0 due to the L direction flipping are roughly conical: Below, for reference, B_L = 0 surfaces in purple, clipped to X > 0, for this choice of L vector orientation, with the oriented L vectors in green. The artefacts from the orientation of the L vector show up as conical surfaces.

Manuscript amended as to read:

For this particular case, the LMN-contouring of the superposition field produces prominent conical artefacts (not shown) where the coordinate system is not oriented consistently with this choice, and the data to be contoured is filtered to lie within 1 cell of a FOTE null line as a way to choose compact neighborhoods for regularized orientation.

25) Line 259-260 - What does it mean that the data was “pre-filtered”?

“Filtered” would be sufficient, indeed. Expanded a bit on this process as follows:

and the data to be contoured is filtered to lie within 1 cell of a FOTE null line as a way to choose compact neighborhoods of regularized orientation.

26) Figure 10 caption, line 3 - What technique is used to draw in-plane magnetic field lines?  It looks like a quiver plot, which would be fine.  It’s not done using the flux function, right?

The fieldlines are produced by Matplotlib streamline plot on the in-plane magnetic field components, indeed not via a flux function. This is now noted in the text.

27) Line 268 - Should (a) and (b) instead be (c) and (d)?

Indeed! Corrected.

28) Line 268 - What about the flap is “pathological”?  Is the implication that it is numerical rather than physical?

Pathological in the context of the previous method, since that uses Bz as the current sheet normal component, even when the flap is vertical. This has been clarified in the text.

29) Line 273 - Do the authors have nothing else to say about the stray detections?  Is this a major problem or a minor problem?  Does it have implications about the robustness of the technique?

In general, the FOTE method is limited by the first-order approximation, and as such it is definitely not an exact method. We added to the manuscript:

These strays are found to be a minor issue, showing up on the current sheet as isolated cells flagged as containing null lines, while the true detections form coherent structures.

30) Line 277 - It would be good to give XYZ coordinates to point the reader to where to look in Fig. 10.

Added example coordinates of X = -17 RE, Y=3 RE, also annotated on the figure.

31) Line 282 - “on Figure 11” should be “in Figure 11”

Thank you, corrected.

32) Line 284 - There should be a period at the end of the equation.

Thank you, corrected.

33) Line 285 - It is not clear to me why sinuous X and O lines would be indicative of an FTE.  I would think an FTE would have no problem being perfectly straight.  If it is bent, I could picture one region of curvature, but it’s not clear to me why they would have a sinuous shape with multiple “wavelengths”.  

Here, sinuous is meant as a descriptor of the observed behaviour of X and O lines (and FTEs), not as the indicator - we consider prominent O lines as the indicators. We rewrote the paragraph slightly to reduce repetition and to note this with more clarity:

“The FOTE method reveals a sinuous and alternating neutral line structure on the magnetopause flanks, shown in Figure~\ref{fig:flank-nulls}. The O-lines are marked with blue cells signifying dayside FTEs extending onto the flanks as a part of the sinuous pattern.”

34) Line 286 - Is there a reason that it says that the small values of |partial B_N / partial L| should be inspected instead of just inspecting them here?

This is meant as a general statement on the method, elaborated in the manuscript:

Regions with small values of $|\partial{}_LB_N|$ (grey) should be inspected for their validity, in general, as these signify low variability in the magnetic field. These regions may be susceptible to local, insignificant perturbations fulfilling the FOTE criterion.

35) Line 293 - What does “are a result in and of themselves” mean?  A result of what?  If the point is that it is useful to know what the dimensionality looks like in the magnetosphere, I would take it a step further and say this is a crucial part of the present study because I don’t think the analysis technique employed here would work unless the current sheets are 2D.  To :me, this is something that really must be shown in order to justify the analysis approach being used.

“Results in and of themselves” tries to refer to the relations of the dimensionality quantities and the features they exhibit in the simulation. However, these are scarcely explored in our context beyond the discussion of the lobe. We emphasised the quasi-2D prerequisite for the LMN systems here, and left the “results in and of themselves” statement out, as this is discussed later in any case:

The applicability of the LMN coordinate system should be inspected in terms of dimensionality of the magnetic structures, to ensure that the structures inspected are locally quasi-2D.

36) Line 300 - “that 1” should be “than 1”

Thank you, corrected!

37) Line 300-301 - I’m confused by the statement there.  The authors correctly point out that D1 + D2 + D3 = 1.  Thus, if D3 approaches 1, then D1 and D2 should go to zero.  Then, D3/D1 should go to infinity.  How could it be that D3/D1 > 1 is rare in the D3 goes to 1 limit then?

The ‘rarity’ concerns the occurrence of cells with D3/D1 > 1 as D3 approaches 1. The statement is perhaps more clear (and definitely stronger) when stated just in terms of the D3->1 corner of the histogram. Clarified in the manuscript as:

Magnetic structures exhibiting $D_3$ dominance appear to be limited: the occurrence of cells is increasingly rare as $D_3 \rightarrow 1$.

38) Line 301 - What does it mean to say D3 = D1 is explained by the lobe region.  What is the explanation?

This stripe is dominated by the contribution of the lobe regions, which indeed is not an explanation by itself. The text has been clarified in this respect.

39) Line 303-305 and 314-315 - I don’t think the statement here and the reference to Zeiler’s paper are generally valid.  It is certainly true that in the Zeiler paper, if reconnection of a Harris-type current sheet takes place in 3D then it doesn’t look too different than 2D.  However, that does not mean that reconnection without a guide field in general is essentially 3D; indeed there is spine-fan reconnection, etc., that is not 2D.

We removed the reference, this was indeed a bit of a dangling statement.

40) Line 308 - The is the first time it is stated that the algorithm being studied is fast and efficient.  How is it being determined that it is fast or efficient?  Compared to what - previous algorithms to find X-lines and O-lines?  

The algorithmic complexity is essentially O(N), with some number of operations required per cell, and the operation is spatially localised - no tracing required. Expanded the text as:

Here, we have presented two local methods to acquire X and O topologies of magnetic field from plasma simulations of Earth's magnetosphere, enabling fast and efficient identification of possible reconnection sites and flux transfer events. The algorithms work on $O(N)$ complexity, i.e., operating once per grid cell (and possibly using a contouring algorithm of similar complexity), without requiring field line or separator tracing such as used by e.g. Haynes+2010.

41) Line 325 - What does the phrase “topological connectivity” mean?

This means the topology of the output mesh after e.g. contouring operations. We elaborated this in the revision as:

… the FOTE method does not produce topological connectivity for the in-plane null lines, that is, it can be used to mark cells containing null lines, but not to which neighbouring cells those lines connect to. In contrast, the contouring method outputs line topologies from the isocontours of $B_L = 0 = B_N$.

42) Line 334 - I think the first sentence should say “in quasi-2D current sheets” or something like that.

We have now included this specifier, thank you.

43) Line 335 - It is up to the authors, but it seems like it would be nice for the authors to say more about the kind of insights they expect might be possible.

We have now noted the D1 \approx D3 characteristic of the tail lobes as an example.