On the Determination of the 3D Velocity Field in Terms of Conserved Variables in a Compressible Ocean

Round 1

Reviewer 1 Report

Let me start by saying that I find the paper impressive, and I support publication after minor revision. It shows the beauty of the concepts of Lorentz APE and how the velocity field should lie in the intersections between two iso-surfaces of conserved quantities.  But it also shows in detail and honesty, the ambiguity of these approaches when they are supposed to be applied to more realistic flows as of compressible oceans. I followed the math and I have not found any flaws. However, the feeling from the reading is that until one tests the different suggestions to extract the 3D velocity field, we cannot really tell which of them are indeed practical. Especially, what seems most concerning is the sensitivity of the structure of the Bernoulli potential to the different approximations/assumptions. Nonetheless, the paper goes in depth through the different plausible evaluations of the 3D flows and analyses their advantage and disadvantages, as well as their relations to previous results. By that the author offers the community a set of evaluation tools for the 3D compressible flow velocity field of which he himself about to test in his future work. I would like however to ask the author to be more detailed when explaining the equations and to assume less prior knowledge. Please write explicitly what each of the terms and subscripts mean, and if possible, provide these as close as possible to the equations. I found myself confused about the meaning of some of the subscripts and the meaning of some of the terms in the equations (or understood only later when I reading further through the text).

Author Response

Response to Reviewer

I am thankful for the overall support and positive comments, and pleased to see that the referee appreciates the beauty and usefulness of APE theory. As per the referee’s request, I have tried to improve clarity wherever I believe that things could be made clearer. I have also added a few illustrations as per Reviewer 2’s request. I have also expanded the introduction to provide a better overview of the underlying issues. I hope that the outcome is more satisfactory.

Reviewer 2 Report

 

Review of “On the determination of the 3D velocity field in terms of conserved variables in a compressible ocean” by Rémi Tailleux.

 

This manuscript is a thorough theoretical analysis describing how the flow can be described without the need for a reference level (the so-called level of no-motion).

 

This work tackles an interesting and timely topic of ocean physics. Despite the clear effort of the author, I still feel that the manuscript is hard to read. In particular the use of a large number of variables for the demonstration and the lack of schematic to build an intuitive understanding of their relations, and of their links with more classical ocean variables, make the manuscript hard to read. Finally the lack of applications makes it also hard to follow.

 

Hence, I recommend this work for major revision. I am convinced that the author will be able to quickly improve the manuscript and to make it publishable.

 

 

Major Comments:

1) I find that there is a missed opportunity to show the velocity field obtained by the two methods (ideal fluid and compressible seawater). In general an application would be useful (even if local, e.g., North Atlantic, or idealized, e.g., based on prescribed fields), showing the circulation and the associated heat and freshwater transport. I suspect that an application in the context of the Southern Ocean would also be interesting. I understand that the application will require a significant work. However I feel that some, even partial, applications are needed.

 

2) The manuscript suffers from lack of visual supports to guide the reader. There is a lot of variable that are defined. Their relation to each other are sometimes hard to follow (or to remember). I would suggest the author to add visual supports to the thorough analysis, such as figures or schematics.

 

 

Specific Comments:

1) L.38-39: Note that from a theoretical perspective, rather than a level of no-motion, a level of known-motion is enough. And this level exists thanks to the ARGO deep displacement (e.g., ANDRO database). This allows for the  reconstruction of the 3D geostrophic velocity field (e.g., Colin de Verdière and Ollitrault, 2016 and Colin de Verdière et al, 2019), for instance.

    

2) l.142: I might be misinterpreting something, but (17) still derives from geostrophic equilibrium so that it is not superior in this respect to (13). I understand the usefulness of the functional relationship, however the relation between (15) and (16) still apply, isn't it? It might be the way the results are presented rather than anything else, but the section 2.5 starting by "while", suggest that the issue on the relation between (15) and (16) will be solved by the end of the section. Whereas it is not, to the best of my understanding.

 

3) Tab.1: Would this be possible to have a schematic or an example to visualize their relation and contexts when they match/mismatch?

 

4) L.286-291: This is quite intriguing (and highly interesting)... If geostrophic, and assuming the non-divergence of the flow, the (vertical shear of) vertical velocity is quite well constraint by the horizontal flow : f dz w = beta v. Would that suggest that these two solutions differ by a constant on the vertical velocity?

 

 

Minor Comments:

l.23: Replace “the simulated of the 3D oceanic velocity field” by “the simulated 3D oceanic velocity field”.

 

Eqs (3-4): It should be mentioned, even if it is obvious, that we are at equilibrium for both momentum and density. Giving an idea on the regime over which this set of equations are valid is needed.

 

Eq (2): The absence of [ (u . grad) . u ] should, at least, be commented. It is not obvious that they would not contribute to some extent to the equilibrium.

 

l.118: \rho_z should be defined. I understand that it is (\partial \rho / \partial z), but this notation was not previously introduced.

 

l.159: Replace “the” by “The”.

 

l.164,184,187,206,Tab.1,230: Replace throughout the text “P vector” by “P-vector”. (I note that in the discussion you prefer the use of “vector P”, which is fine for me, but somehow inconsistent with the rest of the manuscript. You might want to make it consistent throughout the manuscript.)

 

l.232: v_n and v_s have not be defined.

 

l.339: Replace “is it” by “it is”.

 

l.355: Replace “lest” by “least”.

 

l.356: Remove “they”.

 

References:

Colin de Verdière, A., and M. Ollitrault, 2016: Direct determination of the world ocean barotropic circulation. J. Phys. Oceanogr., 46, 255–273, https://doi.org/10.1175/JPO-D-15-0046.1.

Colin de Verdière, A., T. Meunier, and M. Ollitrault, 2019: Meridional overturning and heat transport from Argo floats displacements and the planetary geostrophic method: Applications to the subpolar North Atlantic. J. Geophys. Res. Oceans, 124, 6270–6285, https://doi.org/10.1029/2018JC014565

Author Response

Response to Reviewer 2

Review of “On the determination of the 3D velocity field in terms of conserved variables in a compressible ocean” by Rémi Tailleux.

This manuscript is a thorough theoretical analysis describing how the flow can be described without the need for a reference level (the so-called level of no-motion).

This work tackles an interesting and timely topic of ocean physics. Despite the clear effort of the author, I still feel that the manuscript is hard to read. In particular the use of a large number of variables for the demonstration and the lack of schematic to build an intuitive understanding of their relations, and of their links with more classical ocean variables, make the manuscript hard to read. Finally the lack of applications makes it also hard to follow.

Hence, I recommend this work for major revision. I am convinced that the author will be able to quickly improve the manuscript and to make it publishable.

I thank the referee for the generally supportive comments and suggestions for improvements, which I did my best to implement.

Major Comments:

1) I find that there is a missed opportunity to show the velocity field obtained by the two methods (ideal fluid and compressible seawater). In general an application would be useful (even if local, e.g., North Atlantic, or idealized, e.g., based on prescribed fields), showing the circulation and the associated heat and freshwater transport. I suspect that an application in the context of the Southern Ocean would also be interesting. I understand that the application will require a significant work. However I feel that some, even partial, applications are needed.

It is important to note that the primary aim of the paper is to explain how to define the concepts of Bernoulli function, potential vorticity, and density, which are central to the theoretical determination of the absolute velocity field, to the case of compressible seawater, which has never been satisfactorily been done before. In other words, the main result is about explaining how to improve the tools. How to use the new tools to compute hopefully improved estimates of the absolute velocity field is of course the end goal of the exercise, but I am still a few months away from being able to provide a thorough and well tested application of the method. On the other hand, I agree that some illustration of the quantities discussed is useful, and I have constructed a figure illustrating the differences between the different definitions of the Bernoulli functions. I hope that the referee can accept that this is ok for the time being even if quite short of what is being asked.  

2) The manuscript suffers from lack of visual supports to guide the reader. There is a lot of variable that are defined. Their relation to each other are sometimes hard to follow (or to remember). I would suggest the author to add visual supports to the thorough analysis, such as figures or schematics.

Thank you for the suggestion. I have added a figure that I hope is helpful.

 

Specific Comments:

1) L.38-39: Note that from a theoretical perspective, rather than a level of no-motion, a level of known-motion is enough. And this level exists thanks to the ARGO deep displacement (e.g., ANDRO database). This allows for the  reconstruction of the 3D geostrophic velocity field (e.g., Colin de Verdière and Ollitrault, 2016 and Colin de Verdière et al, 2019), for instance.

Thank you for drawing my attention to these papers. Although I knew about the first one, I must admit that I wrote the paper overlooking the possibility that the use of ARGO deep displacement might be sufficient to constrain the absolute geostrophic velocity field.

 

2) l.142: I might be misinterpreting something, but (17) still derives from geostrophic equilibrium so that it is not superior in this respect to (13). I understand the usefulness of the functional relationship, however the relation between (15) and (16) still apply, isn't it? It might be the way the results are presented rather than anything else, but the section 2.5 starting by "while", suggest that the issue on the relation between (15) and (16) will be solved by the end of the section. Whereas it is not, to the best of my understanding.

Thank you for pointing out the potential for confusion here. Whether (17) should be regarded as superior to (13) cannot be established theoretically as it depends on whether the assumed relationship between B(\rho,Q) is sufficiently well verified empirically. The advantage is that rho and Q are quantities that can be estimated from the stratification without the need to know the pressure field. If the use of ARGO mean displacements could be established to be a robust and accurate way to fix the reference level of the dynamic method, then (13) would still remain superior to (17) especially if it could be established empirically that the assumed relationship B(rho,Q) is not accurately verified in observations. I tried to rewrite this to make the issue clear.

3) Tab.1: Would this be possible to have a schematic or an example to visualize their relation and contexts when they match/mismatch?

I have created a figure for the Bernoulli function.  

4) L.286-291: This is quite intriguing (and highly interesting)... If geostrophic, and assuming the non-divergence of the flow, the (vertical shear of) vertical velocity is quite well constraint by the horizontal flow : f dz w = beta v. Would that suggest that these two solutions differ by a constant on the vertical velocity?

I am not sure I understand the point made by the referee as the linear Sverdrup balance is f dw/dz = beta v, not f dz w = beta v, which integrates as f (w1-w2) = beta int(z1,z2) v dz. Physically, I have updated the text by pointing out that if the horizontal flow is geostrophic in both cases, then the vertical velocities are simply given by w = u.Sh and w = u.Sa, where Sh and Sa are the slope vectors associated with Ph and Pa respectively. The difference in vertical velocity is therefore dw = u.(Sh-Sa) and can be traced back to the difference in the slope vectors Sh and Sa.

 

 

Minor Comments:

l.23: Replace “the simulated of the 3D oceanic velocity field” by “the simulated 3D oceanic velocity field”.

Done.

Eqs (3-4): It should be mentioned, even if it is obvious, that we are at equilibrium for both momentum and density. Giving an idea on the regime over which this set of equations are valid is needed.

I don’t think that the steady form of the density equation in the ideal fluid thermocline equations is really valid as written, and I don’t think that it can really be justified. I hope, however, that now that the meaning of density has been clarified, it will be possible to justify the steady-form of the density equation by invoking some form of time averaging based on the thickness-weighted averaging technique so as to avoid the introduction of eddy-correlation terms. As I haven’t fully understood how to do it properly yet, I have deferred a rigorous justification to a future study once I have properly understood it.  

Eq (2): The absence of [ (u . grad) . u ] should, at least, be commented. It is not obvious that they would not contribute to some extent to the equilibrium.

The referee may be right but the main aim of my paper is to discuss how to extend the ideal fluid thermocline equations to compressible seawater. The ideal fluid thermocline equations are based on strict geostrophic balance, and therefore are only aimed to describe balance flows with small Rossby number, which I have now pointed out in the revision.

l.118: \rho_z should be defined. I understand that it is (\partial \rho / \partial z), but this notation was not previously introduced.

Done.

l.159: Replace “the” by “The”.

Done. Thanks for pointing this out.

l.164,184,187,206,Tab.1,230: Replace throughout the text “P vector” by “P-vector”. (I note that in the discussion you prefer the use of “vector P”, which is fine for me, but somehow inconsistent with the rest of the manuscript. You might want to make it consistent throughout the manuscript.)

Done.

l.232: v_n and v_s have not be defined.

Now defined, thanks for pointing this out.

l.339: Replace “is it” by “it is”.

Done. Thanks for pointing this out.

l.355: Replace “lest” by “least”. 

Done. Thanks for pointing this out.

l.356: Remove “they”.

 Done. Thanks for pointing this out.

References:

Colin de Verdière, A., and M. Ollitrault, 2016: Direct determination of the world ocean barotropic circulation. J. Phys. Oceanogr., 46, 255–273, https://doi.org/10.1175/JPO-D-15-0046.1.

Colin de Verdière, A., T. Meunier, and M. Ollitrault, 2019: Meridional overturning and heat transport from Argo floats displacements and the planetary geostrophic method: Applications to the subpolar North Atlantic. J. Geophys. Res. Oceans, 124, 6270–6285, https://doi.org/10.1029/2018JC014565

Reviewer 3 Report

See attached

Comments for author File: Comments.pdf

Author Response

Response to Reviewer 3

Review on:  On the determination of the 3D velocity field in terms of conserved variables in a compressible ocean, by R. Tailleux

The paper presents a theoretical framework to reconstruct a three-dimensional velocity field from materially

conserved quantities in a two-component compressible ocean. The work generalises the work by Needler 1985.

The method allows to define different reconstructed velocity fields, for example depending on the form of the

Bernoulli’s function B used. The results look sound and the paper is overall well-written. The results are definitely of interest for the scientific community, and are likely to inspire further studies (non-uniqueness of the reconstruction). Consequently, I would recommend publication of the work. The author may want to consider the following minor points prior publication:

• The author should careful check that all symbols used are properly defined at the time they are used first.

For example, ρ⋆ is first used in equation (5) but only defined nearly a page later, line 96.

• The ‘steady-state’ hypothesis could be discussed in more details and emphased. It could be interesting

to discuss how the order of magnitude of the difference between various choices of B compares with the

errors made by all the other assumptions made (including steadiness). Obviously this will depend on the

‘flow regime’ (e.g. Ro, etc...), but some discussion could be initiated.

• The Data Availability Statement looks incomplete (stop at ‘at’)
• Some sentences are very long and could benefit from being split into shorter sentences to ease readability.

 

I thank the referee for supportive comments and constructive criticism.

• I have carefully checked the manuscript to ensure that notations are defined when first encountered.
• I have added some discussion of the steady-state hypothesis, which is indeed central to the whole framework.
• I have completed the ‘data availability statement’, thanks for pointing this out.
• I have tried to break down the longest sentences into shorter ones.

Round 2

Reviewer 2 Report

 

Review of “On the determination of the 3D velocity field in terms of conserved variables in a compressible ocean” by Rémi Tailleux.

 

This manuscript is a thorough theoretical analysis describing how the flow can be described without the need for a reference level (the so-called level of no-motion). This work tackles an interesting and timely topic of ocean physics.

 

I am fully satisfied with the new version of the manuscript. (I have two minor comments, though – please see below). I feel that the author has done an excellent job in clarifying a complex problem.

 

Hence, I recommend this work for publication.

 

 

Minor Comments:

 

l.51-53: I feel that this sentence might be both controversial (to some degree) and might be misinterpreted. First meso-scale eddies are partly (if not for a largely) geostrophic. I guess/understand that the author are referring about their transient behavior. (Although, does a real mean state exist?). However, Deep Argo Displacements are averaged over the Argo period in Colin de Verdière et al. Also, the treatment of Colin de Verdière et al elegantly restricts the reference velocities to their geostrophic component. However, I agree with the inherent limitations of their analysis. This sentence should be rewritten to be less controversial and more aligned to what was actually done when using these deep Argo displacements as the reference level.

 

l.20, 49, 52, and 465: I think that “Argo” should not be written in full caps (i.e., “ARGO”) but with a capital on the first letter (i.e., “Argo”).

Author Response

Comments and Suggestions for Authors

Review of “On the determination of the 3D velocity field in terms of conserved variables in a compressible ocean” by Rémi Tailleux

This manuscript is a thorough theoretical analysis describing how the flow can be described without the need for a reference level (the so-called level of no-motion). This work tackles an interesting and timely topic of ocean physics.

I am fully satisfied with the new version of the manuscript. (I have two minor comments, though – please see below). I feel that the author has done an excellent job in clarifying a complex problem.

Hence, I recommend this work for publication.

I am grateful to the referee for their last minor suggestion, which I full agree with, and for their supportive and constructive review overall.

 

Minor Comments:

 

l.51-53: I feel that this sentence might be both controversial (to some degree) and might be misinterpreted. First meso-scale eddies are partly (if not for a largely) geostrophic. I guess/understand that the author are referring about their transient behavior. (Although, does a real mean state exist?). However, Deep Argo Displacements are averaged over the Argo period in Colin de Verdière et al. Also, the treatment of Colin de Verdière et al elegantly restricts the reference velocities to their geostrophic component. However, I agree with the inherent limitations of their analysis. This sentence should be rewritten to be less controversial and more aligned to what was actually done when using these deep Argo displacements as the reference level.

I have modified said sentence as follows: "Nevertheless, some limitations remain as Argo floats displacements are also a priori impacted by energetic small scale and transient ageostrophic motions in addition to the geostrophic flow, which the authors sought to mitigate by averaging over the Argo period. Outside the Argo period, or when focusing on time snapshots, alternative/complementary approaches for constraining the unknown reference level are still needed."

l.20, 49, 52, and 465: I think that “Argo” should not be written in full caps (i.e., “ARGO”) but with a capital on the first letter (i.e., “Argo”).

Thanks, done.