The article has been substantially re-written, which makes the points a bit clearer (although I still don't think it goes far enough). The re-writing however introduces quite a few internal inconsistencies that I caught on a not entirely careful read, to be point where in my opinion it ends up being worse the initial submission... (so in that regard I disagree with the referee)

Some of these are noted below; the authors probably do want to go through it a few more times, as I make no promises I have caught everything (since I didn't do a very careful read). The linenumbers refer to the line numbers in the tracked manuscript version.

* general formatting: Copernicus now only offer reduced copy-editing. There is still type-setting available, but I am not sure various formattings, spellings, bibliography entries etc. are checked for, and there are a whole load of things I spotted below that the authors should fix themselves.

* general framing and the detection per level: As a dynamicist I still think the eddy detection per level is a misleading thing to do, but I note from this second reading that from the second paragraph in the introduction that there might not be an alternative choice as such with e.g. Ice-Tethered Profilers. Then I wonder if a better way of justifying the present choice taken in the article as something like

- ITPs detect and attribute eddy signals per depth
- since this work may want to compare with observational data, the present article takes the choice of comparing things per depth following established convention
- comment that it is known that geostrophic eddies can and do have a vertical structure/coherency, the present attribution probably over-estimates numbers, but that is just not an aspect considered in this work

I don't think it is a very good convention but it is a convention, which allows some deflection of the criticism. I just think rather than not saying anything and let other people pick at holes, it's probably better to highlight the issues, shut down that discussion and move on. This should/could be done at the introduction.

* sentence of line 145: Changing the sea-ice also changes the "generation" mechanism and instability characteristic at least from a linear point of view (because of changing the boundary condition), which is not discussed. This is also in one of Gianluca's papers (don't remember if it is the one cited here).

* sentence of line 147: Unnecessary leap-frogging sentence (and elsewhere). Why not just do something like "Eddies may persist beyond months at subsurface since they are shielded..." or similar?

* line 166: "ALONG the same line..."

* line 162 and 154: "ON THE one hand..." and "ON the other hand..."

* line 177: comma after "Canadian Basin" to separate the two clauses.

* line 179: Sentence beginning "In this paper..." defines what "eddy" is being referred to but should be much further up surely (although probably not right at the beginning of the introduction section). A ton of talk so far about "eddies" and only now are "eddies" being defined, which is out of order.

* line 185: Remove "Besides" (don't need it)

* line 190: There are words missing before "the Lagrangian framework". I don't know what is missing, but the sentence does work at the moment.

* line 190: "Lagrangian framework" is misleading, because Okubo-Weiss is an Eulerian measure. What is being done here is a detection by an Eulerian framework, but some quantities that are Lagrangian in nature such as implied eddy centre (of mass? area/volume?) are provided. It is also contradictory with what is written in line 298 later in the article.

* line 195 and elsewhere: The paper uses "Canada basin" and "Canadian Basin (CB)", which seems more confusing than necessary (and annoyed me to no end actually particularly later on in the article when it seems to be used inconsistently). See later...

* line 195: Dash line inconsistent length; probably fix the first one to be consistent with the rest of sentence, and be consistent elsewhere in the article.

* line 204: Want mathfont for z* (as "$z^*$")

* line 208: Floating comma after "Fig. S1"

* line 211: "..., WITH AN ASSOCIATED COMPUTATIONAL cost that allows for decadal integradtionS."

* line 213: "...viscosity AND diffusivity..."

* line 215: Reference should be "de Lavergne" (lower case "d"). In bibtex this would be done with

author = {{de Lavergne}, C. and ....}, or author = "{de Lavergne}, C. and ..."

* line 222: Inconsistent unit formatting even within the same sentence (change to W m-1 K-1 or N / m2, choose one and stick with it).

* line 227: "At THE Bering strait..."

* line 237: comma after "observations"

* line 238: "Across the Arctic... by AROUND -7% in September and -16% in March ON AVERAGE."

* line 240: French spelling floating around, "Mars" -> "March"

(Elsewhere in the article some of the sentences have a French syntax style to it. That is not a big issue as such, but it sticks out at the moment at least to me.)

* line 242: "sea ice" what? (cover? thickness? volume?)

* line 242: comma after "model"

* line 245: "...in THE supplementary MATERIAL"

* line 248: "...successfully REpresents..."

* line 253: remove ", that allows and sustains this vertical temperature structure,", don't think that's needed.

* line 254: comma after "BG"

* line 255: I thought the preferred way nowadays is to not use "psu" as a unit (since concentration is dimensionless). Consider "g kg-1" or "g / kg", just be consistent with choices made elsewhere.

* Eq. 1: Inconsistent forcing of "x" and "y" to e.g. line 289 and 290 (don't use \mathrm, \rm or whatever is being used here).

* line 292: "Eddies have to meet the following condition TO BE RETAINED IN THE CENSUS:..."

* Eq. 2: Inconsistent mathfont as above.

* below eq. 2: \alpha used but not defined here any more. Move some relevant text from below to be closer to eq. 2 (it's just some empirical user-specified parameter right, so just say it).

* line 298: "Eulerian over Lagrangian approach" is currently internally inconsistent with what is written in the introduction with "Lagrangian framework".

* line 308-309: If framing as "we take current approach to be consistent with what ITPs do" mentioned above, sentence here should probably be moved or repeated in the introduction also.

* line 314: "have an elliptic shape" -> "be elliptical in shape"

* line 316: "...an eddy of ABOUT 7.5 to 10km..."

* line 321: suggest "...metrics investigated (THE mean eddy radius...) are robust to..." and remove an internal bracket, because the bracketed content is the subject of the sentence ("metrics") being expanded, and sentence makes sense without the bracketed content in principle.

* line 328: "Similarly" -> "Similar"

* line 331: Comma after "eddies"

* line 373: Either "...is said TO BE significant" or "...is significant" (remove "said")

* line 380 and later: Since the authors took the trouble to define "CB" why isn't it used here? Easy fix would be to have "Across the Canadian Basin (CB)" as a reminder, and then use CB throughout.

* Table 1: Above point about "psu" as a unit (also Fig. 4 and elsewhere)

* Table 1: Do use "\mathrm{th}" here probably, and elsewhere (e.g. line 396, Fig. 4)

* line 408: Probably "The vertical structure is CLOSE to the vertical structure..."

* line 417: "...ARE FOUND, forming together the pycnocline layer."

* line 428: "follows" -> "follow" (because "properties" is the relevant subject)

* line 437: "barely changes" makes "comparatively" redundant

* Fig. 6 caption: Missing a comma between "5,000" and "10,000"

* line 570: Pretty sure the acronym "PV" is no longer defined, so acronym used before definition.

* line 594: Comma in "9000" (e.g. inconsistent with "5,500" in the same sentence)

* line 596: Confusing use of sign, is it a "decrease of 20%" (so no need for "reduced"), "decreased to 20%", "reduced to -20%", or some theme and variations thereof? Offending article is probably that minus sign.

* line 599: inconsistent formatting on "s" (probably do want it roman)

* line 633: "...associated WITH the generation..."

* line 638 and later: Similarly for CB here.

* line 643, 645 and elsewhere: "decreases by -55%" means an increase of 55%, which I assume is not what is meant. Similarly for later and in the next paragraph.

* line 666: "fastened" means to "lock in" (e.g. "please fasten your seat belt", like "fast ice" is "locked in/anchored ice"), presumably "faster" is meant here?

* line 674: the "in particular" at the end is floating, either something like "...and IN PARTICULAR to changes in atmospheric forcing" and elsewhere, or just remove it (don't need it).

* line 678: "...the EDDY POPULATION density..." to mirror that in line 680

* line 685: "...and in the MIZ DUE to..."?

* line 691: Another confusing bit where "Canadian Basin" was defined to be "CB" (but not used in the first case)...

* line 696: ...but also we have "the CB" and "THE whole Canadian Basin" in the same sentence! What is going on? (I am going to guess the latter one is "THE whole Canada basin", noting the missing "the" and lower case "basin" consistent with somewhere around section 2 when it was defined).

* line 704: Previously was "Northwind" but now it's "NorthWind", why?

* line 708: "...a WEAKER increase in the number..."

* line 714: "...interpret THE OBSERVED changes."

* Fig. 10: Again, we have "Canadian Basin", "CB" and "Canada Basin" (with the capitalisation when it was introduced without). Please do something about this, because it's either confusing or annoying at present, and neither is a good option when it can be avoided with some internal consistency check...

* section 4: So now I no longer have faith in whether "Canadian Basin" here really means "Canada basin" or "CB". Please go through this again and make things internally consistent. I am going to guess it's actually "Canada basin" that is being referred to? Because "Canada basin" as used here encompasses the "BG" and "CB", but honestly who knows anymore...

* line 737: The lack of buoyancy signature seems suspicious/contradictory, since baroclinic eddies (by the definition of "baroclinic") requires them to have a buoyancy signature right (baroclinic torque would be \nabla p \times \nabla \rho, but no buoyancy anomaly means \nabla rho = 0 so there is no baroclinic torque...)? This either implies they are not in fact baroclinic (e.g. horizontal shear instabilities), the detection is not really detecting geostrophic eddies that this paper is supposed to be centred around, or some other thing. This one needs a serious attempt at commenting now that it has been mentioned...

* line 678: "BG" is used above, so internal inconsistency.

* line 778: Same point above regarding sea-ice and changing generation mechanism.

* line 783: "timeS" (because there are multiples of these)

* line 831: "PV" acronym used but not defined. A quick search indicates it is only used twice anyway, so could also just be expanded (as "potential vorticity" with no capitalisation for consistent with elsewhere in the article.)

* line 891: Please consider acknowledging the referees/editor if the authors think the comments have actually helped (even if the authors may not have enjoyed reading the comments...)

* bibliography: The entry formatting is not consistent at the moment, consider checking (e.g. Kinnard et al 2011 and Kubryakov et al 2021 have different capitalisations).

Just a bit more (line numbers refer to the ones in the tracked file):

* line 128: lowercase "g" for "gyre" (to be consistent with usage in line 130)
* line 184: for consistent should have a space between "W" and "m-1" to be consistent with usage everywhere else (e.g. line 192)
* line 185 and many times elsewhere (e.g. table 1, line 360, line 458, Fig 10 caption): the usage of $\cdot$ was fine before, not sure why it was removed, please restore or use $\times$ to have "a.b \times 10^c"
* (optional) line 260: would normally have the u and v italic rather than roman (so $u$ instead of $\mathrm{u}$, $\textnormal{u}$ or ${\rm u}$), but this is used consistently in the manuscript so could just ignore this comment.
* fig 3 caption: the $y$ in $\partial_y u$ should be italic (i.e. don't add anything to it or typeset it like I did)
* fig 4 caption: remove the dash before the "th" and straighten it like in the text (e.g. line 368). I normally do something like probably "$10^{\tiny \mathrm{th}}$
* fig 5 top row colobar units: inconsistent with usage elsewhere, want "$m^2 s^{-2}$"
* table 2: are the semicolons under $\Delta S$ and $\Delta T$ meant to be there?

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