the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 25 Sep 2025
Thermodynamic Concepts used in Physical Oceanography
Abstract. The thermodynamic concepts that are used in physical oceanography are reviewed, including the several different types of salinity, and how the First Law of Thermodynamics is derived. Different types of temperature are discussed, leading to potential enthalpy and Conservative Temperature, because of the need to accurately quantify the ocean's role in transporting heat. A key aspect of a thermodynamic variable is the extent of its non-conservation when mixing occurs at a given pressure. Methods are presented that quantify the amount of non-conservation of several thermodynamic variables, and these are illustrated in the global ocean. There has been confusion in the literature about the meaning of the salinity and temperature variables carried by ocean models, and here we explain why even in older ocean models that carry the EOS-80 equation of state (rather than TEOS-10), the model's salinity is Preformed Salinity and the model's temperature variable is Conservative Temperature. The thermodynamic reasoning that leads to the concept of neutral surfaces is reviewed, along with thermobaricity, cabbeling, the dianeutral motion caused by the ill-defined nature of neutral surfaces, and Neutral Surface Potential Vorticity.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Interactive discussion
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RC1: 'Comment on egusphere-2025-4556', Stephen Griffies, 22 Oct 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-4556/egusphere-2025-4556-RC1-supplement.pdf
- AC1: 'Reply on RC1', Trevor McDougall, 28 Feb 2026
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CC1: 'Comment on egusphere-2025-4556', Geoff Stanley, 05 Nov 2025
Review of "Thermodynamic Concepts used in Physical Oceanography" by Trevor McDougall (2025) for Ocean Science
Geoff Stanley
Canadian Centre for Climate Modelling and Analysis
November 5, 2025
General Comments
================
This review paper covers a big topic in Physical Oceanography that is often overlooked and/or misunderstood. The author has led progress in this field, working with a relatively small group of collaborators, for ~40 years, and so possesses a great store of knowledge that is vital to share with the wider community in an accessible format. This is a welcome review that goes a long way towards that end. I recommend it be published with "Minor Revisions".
Specific Comments - Major
=========================880 - 882 - "at such an extremum of CT on a neutral surface, a new branch of the γ(S_A, Θ) function arises". It's not the extrema where new branches arise, but at the (contours through the) saddle points. The only way that the extrema can bound a space where a branch of the function lives is if there are only two extrema (a max and a min) and then there is only one branch and we're back to being single-valued. It's saddles (and islands) that open up new branches. What I think you mean here is "an extremum of CT on a neutral trajectory" not a neutral surface. But this in general this is incorrect -- unless you have very luckily chosen the neutral trajectory that goes through the saddle of CT on the neutral surface containing that neutral trajectory. Otherwise and generically, the neutral trajectory will cross the contours of CT on this neutral surface that go through the saddle of CT, and it is these contours that are the boundaries between different branches of the multi-valued function, so this neutral trajectory will actually go through at least 3 branches, not 2. And more importantly, for these generic neutral trajectories (ie that do not go through the CT saddle on a neutral surface), the extremum of CT does not occur at a point where the function switches branches. The picture I have in mind here is of a typical saddle, going up (warmer) to the east and west, and going down (colder) to the north and south; then draw neutral trajectories as straight north-south lines. Any neutral trajectory that does not go through the saddle will have its maximum somewhere inside the lobe to the east or west of the saddle.
950: "This is guaranteed, ... has been proven untrue". I didn't notice this until now, but this is actually not a proof. I agree up to and including that ∮ (α/β) dΘ = 0. I agree that if α/β is a function of S and Θ only then this integral is zero. But α/β being a function also of pressure does not necessarily make the integral non-zero --- because the pressure field could be carefully tied to the S and Θ fields. Looking ahead to your next paragraph, it's actually the neutral helicity H being non-zero that means neutral surfaces don't exist. And H is basically T_b times the triple scalar product ∇P⋅∇S×∇Θ. So we could conceivably have T_b ≠ 0 (ie α/β is a function of pressure) and the triple scalar product = 0, so that H = 0, and neutral surfaces exist (locally). So the reason neutral surfaces don't exist is because both Tb ≠ 0 (ie α/β is a function of pressure) and the triple scalar product ≠ 0. The argument about ∮ (α/β) dΘ only considers that Tb ≠ 0, so it turns out to not be a proof. It feels like ∮ (α/β) dΘ ought to be non-zero, but that is just the same feeling that the triple scalar product is non-zero.
How to fix this? The easiest is to delete everything from "Let us begin" to "proven untrue" and move "Figure 1 ..." to the next paragraph. If you want to keep this ∮ (α/β) dΘ discussion (it is illuminating, after all), then I think you'd need to include some of the above, stating that a contradiction was not in fact reached and that we must go further to examine the neutral helicity.Section 14 -- I would love to see some more discussion here about avenues for future work (or perhaps avenues that are known to not prove fruitful) in seawater thermodynamics, particularly on the salinity and temperature variables and the equation of state. As young Trevor is getting younger every day, and there are not very many other people who have worked on these topics with him, this seems a good location to impart some direction for the field. For instance, do you think any heat-like variable will ever surpass Conservative Temperature? Oceanographers forty years hence would like to know.
Specific Comments - Minor
=========================
171-174 - I would like to see further discussion here about how it is that S_A, T, P can be considered state variables. In particular, it jumps out that T is different from S_A and P, in that P and S_A appear as dP and dS_A, whereas T appears as T multiplying dη --- yet S_A, T, and P are all state variables on an equal footing in the end.229 - here F^Q is taken to be the molecular flux of heat, but on line 209 it also included the radiative flux of heat
252 - 261 - I quite liked this discussion with some surprising thermodynamic properties of seawater! However, if I understand correctly, the change of E caused by a hydrostatic pressure change of 1000dbar is not actually due to the pressure change, but due to moving the fluid parcel through the gravitational field until a 1000dbar change has occurred. The word "hydrostatic" does a lot of work on line 260. In contrast, the enthalpy change due to pressure is direct; putting seawater in a soda bottle and squeezing it, without moving it up or down, would change the enthalpy, but very little change to the total energy E (only due to changes of u, which as you say are much smaller). I'd recommend clarifying this point.
314 - For this discussion of why E is not conserved despite dE/dt equaling a divergence, it may be good to link to Figure 8 here.
Eq 18 - this is the first use of η̂ . Probably just need to explain what the hat notation means, presumably a function of (S_A, Θ, P).
364-5 - Do you really know this for ALL liquids?
431 - "The Gibbs function is unknown and unknowable to the extent ..." This always sounds to me like there is some true value of the Gibbs function out there, but we just cannot and will never be able to determine it. Wouldn't the commonly used phrase "arbitrary up to the addition of ..." be better? (Also, "the extent" doesn't indicate that the expression that follows is meant to be in addition to g.)
492 - this can't be right, because setting m₁ = m₂ yields 1/8 = 1/2.
Eq (26) and (27) - Would it be worth commenting on how mixing two parcels of equal θ (or Θ) but different S_A leads to the mixture having different θ (or Θ)?
559-563 - this "In conclusion" paragraph is a bit repetitive of what was just stated at the start of the previous paragraph (line 551 - 552), which already was a recap of what was said on lines 514-517 (albeit just for CT, not specific volume). So maybe there could be some streamlining.
577-9 - This also requires assuming ΔS_A = 0, right?
713 - "~2%" --- should this be "~2% or greater"? Otherwise, a lot is resting on how accurate that "~" means!
Eq (36), 829 - I really dislike ρ^Θ to refer to potential density. It looks like it has something to do with potential/Conservative Temperature. Just saying!
835 - 838 - To get that these three quantities vary quadratically in space requires assuming that that P and Θ vary linearly in space. This applies to both line 838 and 837. I think that should be stated. (Also line 925.)
892 - "Θ is the thickness-weighted Conservative Temperature of density coordinate averaging". This might be interpreted as doing TWA using in-situ density, which of course is not so different from geopotential-based averaging. This mistake has been made in the literature. So I'd recommend changing "density" to "neutral density".
931 - 932 - "epineutral front" does not appear to be used in the literature (a google scholar search for this term in quotes returns only this preprint), so it is worth defining it here or using another term.
932 - 936: I'm a bit skeptical of this idea that the area integrated effect of thermobaricity and cabelling is independent of K and ∇ₙΘ, from Klocker and McDougall (2010; "Influence...", p 1698). Their argument relies on assuming the epineutral Θ flux is constant with distance across the front. So at the very least that assumption should be stated here. But, how good is that assumption? The relevant integral is ∫ K (∂Θ/∂y|ₙ)² dy, and KM10 take K (∂Θ/∂y|ₙ) out of the integral, which feels like a major leap, and I think is not well justified by KM10 (though I may be missing something, not having reviewed that whole paper). So I am questioning whether it is a good idea to promote this idea in this review paper.
957 - 958 - "the scalar product ... with its curl must be zero everywhere in order for all the neutral tangent planes to join up" -- I fear many readers will interpret this as "H = 0 implies neutral surfaces exist". The correct interpretation, assuming we are on Earth which has islands and not on an aquaplanet, is "H = 0 is a necessary condition for neutral surfaces to exist".
972 - 973 - the "That is, ..." part is weaker than what comes before it. The second part, that H = 0 requires ∇Θ × ∇S_A be in the plane of constant P, is easy to see and should be stated first, I think. The first part, that ∇Θ × ∇S_A is parallel to ∇P × ∇ρ, requires some additional steps and so should come second, at least, if not with the extra steps. The steps I think are: show that ∇θ ⋅ ∇P × ∇ρ = ρ_S ∇θ ⋅ ∇P × ∇S = 0 and similarly show ∇S ⋅ ∇P × ∇ρ = 0, so ∇P × ∇ρ is orthogonal to both ∇Θ and ∇S, and so conclude.
981 - This is only about the geostrophic velocity, right? If so, suggest changing "Eulerian mean horizontal velocity" -> "Eulerian mean geostrophic velocity". Also, this paragraph could use a citation.
1014 - "but the data fills only a few percent of the area described by connecting every data point to every other point". It's not clear what this means: what is the area of "the data" that forms the numerator of this ratio? The denominator is has a well-defined area, but the area of a finite set of points is 0.
1039 - 1041 - This makes it sound like Stanley et al. (2021) reduced the number of iterations required to converge (which is true) by using the Poisson formulation (which was not the cause of the reduced iterations). The cause of the reduced iterations was using the better "vertical solver", ie doing carefully what you described a few lines above for the Klocker et al (2009) algorithm. I suggest replacing "finds the increment of height on the (𝑥, 𝑦) vertical cast corresponding to this increment Φ(𝑥, 𝑦) in ln𝜌ˡ" with "converts Φ(𝑥, 𝑦) to a height increment using the local stratification", and also inserting something like that text that was replaced to go after "Poisson equation solver" and before "converging to the final...".
1085-1086 - I find this surprising that g⁻¹ N² / (αₚ Θ_z - βₚ S_z) would be essentially unaffected by heave. If the p derivatives were not present on the α and β, then this ratio would just be a constant; but isn't the linear combination that uses αₚ and βₚ totally different from that using α and β ?
1092 - 1093 - Should "dashed" be "solid" here? To get from cast A to 2' on cast B, you need to go on the solid curve.
Also, "pint" -> "point". I could use a pint at this point, thank you!1245 - "This area of oceanographic research has attracted very little research and very few papers, and any conclusions drawn above about this topic must be regarded as preliminary; as unfinished business." I think this is overly pessimistic. True, there are relatively few papers on this topic, so maybe you mean that existing papers on the topic are "preliminary" to all the future papers that will come. But I think most people would read this as saying the existing papers on the topic are "preliminary" in that they are themselves incomplete and not very good. That is hardly the case!
Technical Corrections
=====================
166 - volumes -> volume282 - this is the first use of Θ. Should be explained what it is.
289, 306, ... - Leibnitz -> Leibniz
336 - "(R.02)" -> (19) ?
439 - 441: "Dear reader ... trash bin where it belongs." Excellent! No change needed here. ;)
482 - in -> is
491 - m₁ and m₁ -> m₁ and m₂
523 - evaluate -> evaluated
523 - "term in pressure integral of..." sounds like it's talking about the integrand, but you actually mean the integral itself. So delete "term in"
532 - "that of" -> "that, of"
556 - that -> the
558 - ";" -> ","
685 - "terms entropy" -> "terms of entropy"
707 - "(and to Absolute Salinity)" -> "(and Absolute Salinity)"
709 - named -> namely
726 - delete ")"
742 - "2 1/2 %" looks rather odd. Suggest "2.5%"
748 - effects -> effect
750 - ":-" -> ":"
835 - Eq (38) isn't about neutral variations, so change "three" to two" and delete "(38)".
838 - delete "an"
880 - "neutral surfaces" -> "neutral trajectories"
986 - The expression here only involves Pressure and CT, so delete "salinity".
1024 - Citation typo: "Jackett and McDougall (1985)" -> "Jackett and McDougall (1997)".
1029, 1043 - the ω looks like it has an overbar on it. Can that be removed?
1050 - "to a few by 10⁻⁹ m s⁻¹" --- delete "by"
1071 - It seems like "the vertical distance between adjacent neural surfaces" is collapsing the world over. Brainrot is taking hold. "neural" -> "neutral" ;-)
1238 - "closer" -> "is closer"
Citation: https://doi.org/10.5194/egusphere-2025-4556-CC1 - AC3: 'Reply on CC1', Trevor McDougall, 28 Feb 2026
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RC2: 'Comment on egusphere-2025-4556', Julian Mak, 02 Jan 2026
I read this from fresh and made some comments, then went back to cross-check with the other referee comments and it seems my comments largely echo theirs (which I hope means I am not spouting completely rubbish). Some of their comments I will repeat below. There are quite a few comments, but most of them are clarifications and writing.
- Comment about targeted audience: As discussed with the author already, there was a suggestion about this being an entry for the encyclopedia of geophysics or whatever it is called. The present content and presentation is targeting a sufficiently specific audience for that. I don't mean this as a criticism at all however: technical subjects require using technical language and content, and some things do need to be "worked on" to be understood, and this is probably one of those. While it is true some presentation and points here could be clarified (probably will some additional schematic diagrams maybe), I personally would think it is a shame if the technical content is removed/lessened in favour of a wider appeal.
- I agree with referee 1's point about "salinity" being placed where it is, it did seem slightly odd to me. The author's call on this: does going after the current section 4 "potential and conservative variables" work?
- line 75: "Pawlowicz eT al, 2012" (missing a "t" in "et")
- line 80 and elsewhere: Most usages of "which" I would personally have used "that" (or put a comma before the "which" and a comma to wrap the end of the secondary clause to denote the things between the commas as additional material supporting the principal clause). The author's call on whether to adjust accordingly.
- current section 3: Referee 1 raises the point about "quasi-static" vs. "reversible", would probably be good to made the point here (since we are going reasonably technical already).
- line 175: Comma after "FTR" to break up the sentence a bit.
- line 177: Comma after "change" to break up the sentence a bit.
- line 181: Comma after close bracket.
- line 183: I would remove the commas surrounding \mu but put comma after "represents" instead.
- line 185: "Fundamental Thermodynamic Relation" -> "FTR", since it's been defined already.
- line 193: Comma before "which".
- line 194: More of a comment here since the important part of the statement is true. Landau & Lifshitz (1959) seems a bit of an old reference (and I don't agree it is the clearest text on the subject; David Tong's lecture notes on thermodynamics and/or statistical physics now turned into books spring to mind).
- line 197: Echoing referee 1's point, I suggested "quality" -> "standard" or just remove that adjective, since there are perfectly good quality fluids textbooks that don't deal with thermodynamics.
- line 211: "obtaining" -> "resulting in"
- line 215: I would do "...models; such equations..." since the two parts are sufficiently connected.
- line 235: "BE positive" feels more natural to me (because of the presence of "both" I think)
- paragraph including line 235: Echoing referee 1's point, can probably expand this a bit more with some additional linking equations.
- line 269: Would add a comma after "flux".
- line 283: Would add a comma after "quantity".
- line 289 and 306: "Leibniz" (no "t")
- line 292 and eq 16: I would have "Eq. (16):", then show equation, and then start the text again.
- eq 16: Echoing referee 1's point about definition of control volume and probably having a schematic.
- line 307 and eq 17: As above, show the equation and then justify the steps, having "Eq. (17):", show equation, and then start the text again.
- text after eq. 17: (Geoff's comment) Useful to link to Fig 8.
- eq 18: Echoing all comments, define the meaning of hats here (bring/repeat some of the text from line 829 forward to here). Also needs a full stop to end the equation.
- line 334: No "expression (R.02)" here.
- line 345: Suggest comma before "while".
- line 355: Suggest comma after "correctly".
- line 365: Comma before "while".
- line 367: Common to replace the semi-colon.
- Section 6 title: Doesn't really talk or define temperature? If this section included section 7 then it's probably fine.
- line 417: Suggest comma before "and".
- line 436: Sentence "Because the four coefficients..." is redundant since it is covered by the sentence before already.
- line 439 to 441: Authors call on this, I just think this sentence is unnecessary.
- line 461: Instead of "t" (which is time), suggest \Delta T or \delta T but define it explicitly.
- line 475: Remove first two "nor"s, and "nor potential temperature" -> "or potential temperature", ending the sentence there (the bit after the semi-colon is unnecessary).
- line 491: Second "m_1" -> "m_2"
- line 492: I assume since m_1 + m_2 = m, so either
m_1 = m_2 = 1/2 m => 1/2 m_1 m_2 / m^2 = 1/2 * (1/2 m)^2 / m^2 = (1/2)^3 = 1/8, or
1/2 m_1^2 / (2 m_1)^2 = 1/2 * 1/4 = 1/8,
so I think one of Geoff's comments can be ignored? - line 533: "on" -> "upon"?
- section 9.1: I thought the point about BGC affecting salt content and therefore density is interesting, may want to echo this at the end (see a later comment).
- line 726: Remove floating closed bracket.
- line 742: "2 1/2%" -> "2.5%".
- line 746: missing degree sign in "2C".
- line 756: Comma before "and", otherwise too many "and"s floating around.
- line 760: Full stop replaced by colon since opening the list.
- line 763-764: Round brackets instead of square brackets probably.
- line 786: Comma before "a".
- line 798: Remove "of".
- line 815: add "= 0" to in-line equation, and probably a comma as well.
- line 829: "...pressure \tilde{P}, T_b is the..., AND the over-hats AGAIN indicate that..."
- line 845: Just because it looks a little odd having a sentence start with a lower case letter, would suggest passive voice as "A variable called orthobaric density \rho_v, which is a function..., was introduced by de Szoeke et al (2000)."
- line 892: "neutral" density?
- line 924: Semi-colon to colon.
- line 932: Clarify and/or expand what is meant by "epineutral" (echoing Geoff's comment).
- 1st paragraph of sec 12: See Geoff's comment about this not being a "proof", and clarify accordingly, and some content in the paragraph of line 971.
- eq 47: Suggest comma instead, and move the bracketed text before to after the equation and remove the brackets (state first, then define).
- line 1080: NSPV is already defined so can just use it, remove expanded text.
- line 1093: Remove brackets (to be consistent with two points before).
- eq 52: Needs a comma after the zero vector.
- eq 53: Suggest full stop and starting next sentence with "Taking the epineutral gradient of this equation, WE find that..."
- eq 54: Need a comma and a full stop here.
- line 1106: "its" -> "the"
- line 1108: Can just use NSPV again without the expansion (or just don't use the acronym at all actually).
- eq 55: Need a comma and a full stop here.
- line 1113: No "Eq. (113)" here.
- line 1122: Can just use NSPV again without the expansion.
- eq 57: Needs a comma after the zero vector.
- eq 59: Needs a comma after the zero vector.
- line 1135: Can just use NSPV again without the expansion.
- eq 61: Is "\kappa" defined? Might be good to redefine it as a reminder if so.
- eq 65: As above, state equation, replace full stop with comma, and move the bracketed text to after the equation removing the square brackets.
- sentence of line 1177: Would suggest moving or repeating sentence further up as some sign-posting for the reader.
- line 1200: Would add comma before brackets and remove the brackets.
- line 1225: Commas surrounding "which depends on the electrical conductivity of seawater" since this is the secondary clause.
- line 1244-1245: "...very little research, and... as preliminary AND unfinished business"
- section 14: Echoing Geoff's point, I would have liked to see some key summaries and some call-to-action by providing/highlighting some research outlook directions (e.g. I thought the BGC model coupling to salinity was interesting) the author would like to see in due course. It makes the end a bit less abrupt, and I think that while a review a paper certainly looks to the past, some vision towards the future would be desirable.
- Figures: Doesn't need to go in the back even for the draft.
- Figure 2 and 3: If possible please make the black dots bigger to be consistent with Figure 5.
- Figure 4: We general prefer authors to not use jet/rainbow colour schemes now. Would suggest a Red/Blue one (for positive and negative values) or similar.
- Figure 12: Would prefer not jet/rainbow, but this might be more difficult to achieve.
Citation: https://doi.org/10.5194/egusphere-2025-4556-RC2 - AC2: 'Reply on RC2', Trevor McDougall, 28 Feb 2026
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EC1: 'Comment on egusphere-2025-4556', Julian Mak, 02 Jan 2026
Mostly presentation comments, so please do go ahead with the revisions accordingly. There is quite a few points in the three reports combined, but some of these are repeated, and the review paper is also fairly long.
Citation: https://doi.org/10.5194/egusphere-2025-4556-EC1 -
AC4: 'Reply on EC1', Trevor McDougall, 28 Feb 2026
Thank you. I have now changed the manuscript in accordance with the suggestions of all three, Griffies, Stanley and Mak.
Citation: https://doi.org/10.5194/egusphere-2025-4556-AC4
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AC4: 'Reply on EC1', Trevor McDougall, 28 Feb 2026
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2025-4556', Stephen Griffies, 22 Oct 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-4556/egusphere-2025-4556-RC1-supplement.pdf
- AC1: 'Reply on RC1', Trevor McDougall, 28 Feb 2026
-
CC1: 'Comment on egusphere-2025-4556', Geoff Stanley, 05 Nov 2025
Review of "Thermodynamic Concepts used in Physical Oceanography" by Trevor McDougall (2025) for Ocean Science
Geoff Stanley
Canadian Centre for Climate Modelling and Analysis
November 5, 2025
General Comments
================
This review paper covers a big topic in Physical Oceanography that is often overlooked and/or misunderstood. The author has led progress in this field, working with a relatively small group of collaborators, for ~40 years, and so possesses a great store of knowledge that is vital to share with the wider community in an accessible format. This is a welcome review that goes a long way towards that end. I recommend it be published with "Minor Revisions".
Specific Comments - Major
=========================880 - 882 - "at such an extremum of CT on a neutral surface, a new branch of the γ(S_A, Θ) function arises". It's not the extrema where new branches arise, but at the (contours through the) saddle points. The only way that the extrema can bound a space where a branch of the function lives is if there are only two extrema (a max and a min) and then there is only one branch and we're back to being single-valued. It's saddles (and islands) that open up new branches. What I think you mean here is "an extremum of CT on a neutral trajectory" not a neutral surface. But this in general this is incorrect -- unless you have very luckily chosen the neutral trajectory that goes through the saddle of CT on the neutral surface containing that neutral trajectory. Otherwise and generically, the neutral trajectory will cross the contours of CT on this neutral surface that go through the saddle of CT, and it is these contours that are the boundaries between different branches of the multi-valued function, so this neutral trajectory will actually go through at least 3 branches, not 2. And more importantly, for these generic neutral trajectories (ie that do not go through the CT saddle on a neutral surface), the extremum of CT does not occur at a point where the function switches branches. The picture I have in mind here is of a typical saddle, going up (warmer) to the east and west, and going down (colder) to the north and south; then draw neutral trajectories as straight north-south lines. Any neutral trajectory that does not go through the saddle will have its maximum somewhere inside the lobe to the east or west of the saddle.
950: "This is guaranteed, ... has been proven untrue". I didn't notice this until now, but this is actually not a proof. I agree up to and including that ∮ (α/β) dΘ = 0. I agree that if α/β is a function of S and Θ only then this integral is zero. But α/β being a function also of pressure does not necessarily make the integral non-zero --- because the pressure field could be carefully tied to the S and Θ fields. Looking ahead to your next paragraph, it's actually the neutral helicity H being non-zero that means neutral surfaces don't exist. And H is basically T_b times the triple scalar product ∇P⋅∇S×∇Θ. So we could conceivably have T_b ≠ 0 (ie α/β is a function of pressure) and the triple scalar product = 0, so that H = 0, and neutral surfaces exist (locally). So the reason neutral surfaces don't exist is because both Tb ≠ 0 (ie α/β is a function of pressure) and the triple scalar product ≠ 0. The argument about ∮ (α/β) dΘ only considers that Tb ≠ 0, so it turns out to not be a proof. It feels like ∮ (α/β) dΘ ought to be non-zero, but that is just the same feeling that the triple scalar product is non-zero.
How to fix this? The easiest is to delete everything from "Let us begin" to "proven untrue" and move "Figure 1 ..." to the next paragraph. If you want to keep this ∮ (α/β) dΘ discussion (it is illuminating, after all), then I think you'd need to include some of the above, stating that a contradiction was not in fact reached and that we must go further to examine the neutral helicity.Section 14 -- I would love to see some more discussion here about avenues for future work (or perhaps avenues that are known to not prove fruitful) in seawater thermodynamics, particularly on the salinity and temperature variables and the equation of state. As young Trevor is getting younger every day, and there are not very many other people who have worked on these topics with him, this seems a good location to impart some direction for the field. For instance, do you think any heat-like variable will ever surpass Conservative Temperature? Oceanographers forty years hence would like to know.
Specific Comments - Minor
=========================
171-174 - I would like to see further discussion here about how it is that S_A, T, P can be considered state variables. In particular, it jumps out that T is different from S_A and P, in that P and S_A appear as dP and dS_A, whereas T appears as T multiplying dη --- yet S_A, T, and P are all state variables on an equal footing in the end.229 - here F^Q is taken to be the molecular flux of heat, but on line 209 it also included the radiative flux of heat
252 - 261 - I quite liked this discussion with some surprising thermodynamic properties of seawater! However, if I understand correctly, the change of E caused by a hydrostatic pressure change of 1000dbar is not actually due to the pressure change, but due to moving the fluid parcel through the gravitational field until a 1000dbar change has occurred. The word "hydrostatic" does a lot of work on line 260. In contrast, the enthalpy change due to pressure is direct; putting seawater in a soda bottle and squeezing it, without moving it up or down, would change the enthalpy, but very little change to the total energy E (only due to changes of u, which as you say are much smaller). I'd recommend clarifying this point.
314 - For this discussion of why E is not conserved despite dE/dt equaling a divergence, it may be good to link to Figure 8 here.
Eq 18 - this is the first use of η̂ . Probably just need to explain what the hat notation means, presumably a function of (S_A, Θ, P).
364-5 - Do you really know this for ALL liquids?
431 - "The Gibbs function is unknown and unknowable to the extent ..." This always sounds to me like there is some true value of the Gibbs function out there, but we just cannot and will never be able to determine it. Wouldn't the commonly used phrase "arbitrary up to the addition of ..." be better? (Also, "the extent" doesn't indicate that the expression that follows is meant to be in addition to g.)
492 - this can't be right, because setting m₁ = m₂ yields 1/8 = 1/2.
Eq (26) and (27) - Would it be worth commenting on how mixing two parcels of equal θ (or Θ) but different S_A leads to the mixture having different θ (or Θ)?
559-563 - this "In conclusion" paragraph is a bit repetitive of what was just stated at the start of the previous paragraph (line 551 - 552), which already was a recap of what was said on lines 514-517 (albeit just for CT, not specific volume). So maybe there could be some streamlining.
577-9 - This also requires assuming ΔS_A = 0, right?
713 - "~2%" --- should this be "~2% or greater"? Otherwise, a lot is resting on how accurate that "~" means!
Eq (36), 829 - I really dislike ρ^Θ to refer to potential density. It looks like it has something to do with potential/Conservative Temperature. Just saying!
835 - 838 - To get that these three quantities vary quadratically in space requires assuming that that P and Θ vary linearly in space. This applies to both line 838 and 837. I think that should be stated. (Also line 925.)
892 - "Θ is the thickness-weighted Conservative Temperature of density coordinate averaging". This might be interpreted as doing TWA using in-situ density, which of course is not so different from geopotential-based averaging. This mistake has been made in the literature. So I'd recommend changing "density" to "neutral density".
931 - 932 - "epineutral front" does not appear to be used in the literature (a google scholar search for this term in quotes returns only this preprint), so it is worth defining it here or using another term.
932 - 936: I'm a bit skeptical of this idea that the area integrated effect of thermobaricity and cabelling is independent of K and ∇ₙΘ, from Klocker and McDougall (2010; "Influence...", p 1698). Their argument relies on assuming the epineutral Θ flux is constant with distance across the front. So at the very least that assumption should be stated here. But, how good is that assumption? The relevant integral is ∫ K (∂Θ/∂y|ₙ)² dy, and KM10 take K (∂Θ/∂y|ₙ) out of the integral, which feels like a major leap, and I think is not well justified by KM10 (though I may be missing something, not having reviewed that whole paper). So I am questioning whether it is a good idea to promote this idea in this review paper.
957 - 958 - "the scalar product ... with its curl must be zero everywhere in order for all the neutral tangent planes to join up" -- I fear many readers will interpret this as "H = 0 implies neutral surfaces exist". The correct interpretation, assuming we are on Earth which has islands and not on an aquaplanet, is "H = 0 is a necessary condition for neutral surfaces to exist".
972 - 973 - the "That is, ..." part is weaker than what comes before it. The second part, that H = 0 requires ∇Θ × ∇S_A be in the plane of constant P, is easy to see and should be stated first, I think. The first part, that ∇Θ × ∇S_A is parallel to ∇P × ∇ρ, requires some additional steps and so should come second, at least, if not with the extra steps. The steps I think are: show that ∇θ ⋅ ∇P × ∇ρ = ρ_S ∇θ ⋅ ∇P × ∇S = 0 and similarly show ∇S ⋅ ∇P × ∇ρ = 0, so ∇P × ∇ρ is orthogonal to both ∇Θ and ∇S, and so conclude.
981 - This is only about the geostrophic velocity, right? If so, suggest changing "Eulerian mean horizontal velocity" -> "Eulerian mean geostrophic velocity". Also, this paragraph could use a citation.
1014 - "but the data fills only a few percent of the area described by connecting every data point to every other point". It's not clear what this means: what is the area of "the data" that forms the numerator of this ratio? The denominator is has a well-defined area, but the area of a finite set of points is 0.
1039 - 1041 - This makes it sound like Stanley et al. (2021) reduced the number of iterations required to converge (which is true) by using the Poisson formulation (which was not the cause of the reduced iterations). The cause of the reduced iterations was using the better "vertical solver", ie doing carefully what you described a few lines above for the Klocker et al (2009) algorithm. I suggest replacing "finds the increment of height on the (𝑥, 𝑦) vertical cast corresponding to this increment Φ(𝑥, 𝑦) in ln𝜌ˡ" with "converts Φ(𝑥, 𝑦) to a height increment using the local stratification", and also inserting something like that text that was replaced to go after "Poisson equation solver" and before "converging to the final...".
1085-1086 - I find this surprising that g⁻¹ N² / (αₚ Θ_z - βₚ S_z) would be essentially unaffected by heave. If the p derivatives were not present on the α and β, then this ratio would just be a constant; but isn't the linear combination that uses αₚ and βₚ totally different from that using α and β ?
1092 - 1093 - Should "dashed" be "solid" here? To get from cast A to 2' on cast B, you need to go on the solid curve.
Also, "pint" -> "point". I could use a pint at this point, thank you!1245 - "This area of oceanographic research has attracted very little research and very few papers, and any conclusions drawn above about this topic must be regarded as preliminary; as unfinished business." I think this is overly pessimistic. True, there are relatively few papers on this topic, so maybe you mean that existing papers on the topic are "preliminary" to all the future papers that will come. But I think most people would read this as saying the existing papers on the topic are "preliminary" in that they are themselves incomplete and not very good. That is hardly the case!
Technical Corrections
=====================
166 - volumes -> volume282 - this is the first use of Θ. Should be explained what it is.
289, 306, ... - Leibnitz -> Leibniz
336 - "(R.02)" -> (19) ?
439 - 441: "Dear reader ... trash bin where it belongs." Excellent! No change needed here. ;)
482 - in -> is
491 - m₁ and m₁ -> m₁ and m₂
523 - evaluate -> evaluated
523 - "term in pressure integral of..." sounds like it's talking about the integrand, but you actually mean the integral itself. So delete "term in"
532 - "that of" -> "that, of"
556 - that -> the
558 - ";" -> ","
685 - "terms entropy" -> "terms of entropy"
707 - "(and to Absolute Salinity)" -> "(and Absolute Salinity)"
709 - named -> namely
726 - delete ")"
742 - "2 1/2 %" looks rather odd. Suggest "2.5%"
748 - effects -> effect
750 - ":-" -> ":"
835 - Eq (38) isn't about neutral variations, so change "three" to two" and delete "(38)".
838 - delete "an"
880 - "neutral surfaces" -> "neutral trajectories"
986 - The expression here only involves Pressure and CT, so delete "salinity".
1024 - Citation typo: "Jackett and McDougall (1985)" -> "Jackett and McDougall (1997)".
1029, 1043 - the ω looks like it has an overbar on it. Can that be removed?
1050 - "to a few by 10⁻⁹ m s⁻¹" --- delete "by"
1071 - It seems like "the vertical distance between adjacent neural surfaces" is collapsing the world over. Brainrot is taking hold. "neural" -> "neutral" ;-)
1238 - "closer" -> "is closer"
Citation: https://doi.org/10.5194/egusphere-2025-4556-CC1 - AC3: 'Reply on CC1', Trevor McDougall, 28 Feb 2026
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RC2: 'Comment on egusphere-2025-4556', Julian Mak, 02 Jan 2026
I read this from fresh and made some comments, then went back to cross-check with the other referee comments and it seems my comments largely echo theirs (which I hope means I am not spouting completely rubbish). Some of their comments I will repeat below. There are quite a few comments, but most of them are clarifications and writing.
- Comment about targeted audience: As discussed with the author already, there was a suggestion about this being an entry for the encyclopedia of geophysics or whatever it is called. The present content and presentation is targeting a sufficiently specific audience for that. I don't mean this as a criticism at all however: technical subjects require using technical language and content, and some things do need to be "worked on" to be understood, and this is probably one of those. While it is true some presentation and points here could be clarified (probably will some additional schematic diagrams maybe), I personally would think it is a shame if the technical content is removed/lessened in favour of a wider appeal.
- I agree with referee 1's point about "salinity" being placed where it is, it did seem slightly odd to me. The author's call on this: does going after the current section 4 "potential and conservative variables" work?
- line 75: "Pawlowicz eT al, 2012" (missing a "t" in "et")
- line 80 and elsewhere: Most usages of "which" I would personally have used "that" (or put a comma before the "which" and a comma to wrap the end of the secondary clause to denote the things between the commas as additional material supporting the principal clause). The author's call on whether to adjust accordingly.
- current section 3: Referee 1 raises the point about "quasi-static" vs. "reversible", would probably be good to made the point here (since we are going reasonably technical already).
- line 175: Comma after "FTR" to break up the sentence a bit.
- line 177: Comma after "change" to break up the sentence a bit.
- line 181: Comma after close bracket.
- line 183: I would remove the commas surrounding \mu but put comma after "represents" instead.
- line 185: "Fundamental Thermodynamic Relation" -> "FTR", since it's been defined already.
- line 193: Comma before "which".
- line 194: More of a comment here since the important part of the statement is true. Landau & Lifshitz (1959) seems a bit of an old reference (and I don't agree it is the clearest text on the subject; David Tong's lecture notes on thermodynamics and/or statistical physics now turned into books spring to mind).
- line 197: Echoing referee 1's point, I suggested "quality" -> "standard" or just remove that adjective, since there are perfectly good quality fluids textbooks that don't deal with thermodynamics.
- line 211: "obtaining" -> "resulting in"
- line 215: I would do "...models; such equations..." since the two parts are sufficiently connected.
- line 235: "BE positive" feels more natural to me (because of the presence of "both" I think)
- paragraph including line 235: Echoing referee 1's point, can probably expand this a bit more with some additional linking equations.
- line 269: Would add a comma after "flux".
- line 283: Would add a comma after "quantity".
- line 289 and 306: "Leibniz" (no "t")
- line 292 and eq 16: I would have "Eq. (16):", then show equation, and then start the text again.
- eq 16: Echoing referee 1's point about definition of control volume and probably having a schematic.
- line 307 and eq 17: As above, show the equation and then justify the steps, having "Eq. (17):", show equation, and then start the text again.
- text after eq. 17: (Geoff's comment) Useful to link to Fig 8.
- eq 18: Echoing all comments, define the meaning of hats here (bring/repeat some of the text from line 829 forward to here). Also needs a full stop to end the equation.
- line 334: No "expression (R.02)" here.
- line 345: Suggest comma before "while".
- line 355: Suggest comma after "correctly".
- line 365: Comma before "while".
- line 367: Common to replace the semi-colon.
- Section 6 title: Doesn't really talk or define temperature? If this section included section 7 then it's probably fine.
- line 417: Suggest comma before "and".
- line 436: Sentence "Because the four coefficients..." is redundant since it is covered by the sentence before already.
- line 439 to 441: Authors call on this, I just think this sentence is unnecessary.
- line 461: Instead of "t" (which is time), suggest \Delta T or \delta T but define it explicitly.
- line 475: Remove first two "nor"s, and "nor potential temperature" -> "or potential temperature", ending the sentence there (the bit after the semi-colon is unnecessary).
- line 491: Second "m_1" -> "m_2"
- line 492: I assume since m_1 + m_2 = m, so either
m_1 = m_2 = 1/2 m => 1/2 m_1 m_2 / m^2 = 1/2 * (1/2 m)^2 / m^2 = (1/2)^3 = 1/8, or
1/2 m_1^2 / (2 m_1)^2 = 1/2 * 1/4 = 1/8,
so I think one of Geoff's comments can be ignored? - line 533: "on" -> "upon"?
- section 9.1: I thought the point about BGC affecting salt content and therefore density is interesting, may want to echo this at the end (see a later comment).
- line 726: Remove floating closed bracket.
- line 742: "2 1/2%" -> "2.5%".
- line 746: missing degree sign in "2C".
- line 756: Comma before "and", otherwise too many "and"s floating around.
- line 760: Full stop replaced by colon since opening the list.
- line 763-764: Round brackets instead of square brackets probably.
- line 786: Comma before "a".
- line 798: Remove "of".
- line 815: add "= 0" to in-line equation, and probably a comma as well.
- line 829: "...pressure \tilde{P}, T_b is the..., AND the over-hats AGAIN indicate that..."
- line 845: Just because it looks a little odd having a sentence start with a lower case letter, would suggest passive voice as "A variable called orthobaric density \rho_v, which is a function..., was introduced by de Szoeke et al (2000)."
- line 892: "neutral" density?
- line 924: Semi-colon to colon.
- line 932: Clarify and/or expand what is meant by "epineutral" (echoing Geoff's comment).
- 1st paragraph of sec 12: See Geoff's comment about this not being a "proof", and clarify accordingly, and some content in the paragraph of line 971.
- eq 47: Suggest comma instead, and move the bracketed text before to after the equation and remove the brackets (state first, then define).
- line 1080: NSPV is already defined so can just use it, remove expanded text.
- line 1093: Remove brackets (to be consistent with two points before).
- eq 52: Needs a comma after the zero vector.
- eq 53: Suggest full stop and starting next sentence with "Taking the epineutral gradient of this equation, WE find that..."
- eq 54: Need a comma and a full stop here.
- line 1106: "its" -> "the"
- line 1108: Can just use NSPV again without the expansion (or just don't use the acronym at all actually).
- eq 55: Need a comma and a full stop here.
- line 1113: No "Eq. (113)" here.
- line 1122: Can just use NSPV again without the expansion.
- eq 57: Needs a comma after the zero vector.
- eq 59: Needs a comma after the zero vector.
- line 1135: Can just use NSPV again without the expansion.
- eq 61: Is "\kappa" defined? Might be good to redefine it as a reminder if so.
- eq 65: As above, state equation, replace full stop with comma, and move the bracketed text to after the equation removing the square brackets.
- sentence of line 1177: Would suggest moving or repeating sentence further up as some sign-posting for the reader.
- line 1200: Would add comma before brackets and remove the brackets.
- line 1225: Commas surrounding "which depends on the electrical conductivity of seawater" since this is the secondary clause.
- line 1244-1245: "...very little research, and... as preliminary AND unfinished business"
- section 14: Echoing Geoff's point, I would have liked to see some key summaries and some call-to-action by providing/highlighting some research outlook directions (e.g. I thought the BGC model coupling to salinity was interesting) the author would like to see in due course. It makes the end a bit less abrupt, and I think that while a review a paper certainly looks to the past, some vision towards the future would be desirable.
- Figures: Doesn't need to go in the back even for the draft.
- Figure 2 and 3: If possible please make the black dots bigger to be consistent with Figure 5.
- Figure 4: We general prefer authors to not use jet/rainbow colour schemes now. Would suggest a Red/Blue one (for positive and negative values) or similar.
- Figure 12: Would prefer not jet/rainbow, but this might be more difficult to achieve.
Citation: https://doi.org/10.5194/egusphere-2025-4556-RC2 - AC2: 'Reply on RC2', Trevor McDougall, 28 Feb 2026
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EC1: 'Comment on egusphere-2025-4556', Julian Mak, 02 Jan 2026
Mostly presentation comments, so please do go ahead with the revisions accordingly. There is quite a few points in the three reports combined, but some of these are repeated, and the review paper is also fairly long.
Citation: https://doi.org/10.5194/egusphere-2025-4556-EC1 -
AC4: 'Reply on EC1', Trevor McDougall, 28 Feb 2026
Thank you. I have now changed the manuscript in accordance with the suggestions of all three, Griffies, Stanley and Mak.
Citation: https://doi.org/10.5194/egusphere-2025-4556-AC4
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AC4: 'Reply on EC1', Trevor McDougall, 28 Feb 2026
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