Gravity disturbance driven ocean circulation
 Department of Oceanography, Naval Postgraduate School, Monterey, CA 93943, USA
 Department of Oceanography, Naval Postgraduate School, Monterey, CA 93943, USA
Abstract. The Earth true gravity (g) has been simplified in oceanography and meteorology into the standard gravity g_{s} (= g_{0}k, g_{0} = 9.81 m s^{2}) with k the unit vector perpendicular to the spherical surface or the normal g_{n} [= g(φ)K] with K the unit vector perpendicular to the ellipsoidal surface. The gravity disturbance (δg = g – g_{n}) due to nonuniform Earth mass density is totally neglected. After including the gravity disturbance into the combined SverdrupStommelMunk equation for ocean circulation, the volume transport streamfunction (Ψ) is driven by both gravity disturbance forcing (GDF) and surface wind forcing (i.e., curl τ) with τ the surface wind stress. The nondimensional F number (i.e., ratio of global GDF versus global curl τ) is estimated as 0.6918 using three publicly available datasets in climatological, geodetic, and oceanographic communities. Such an Fvalue (0.6918) clearly shows the comparable GDF and surface wind stress curl in driving ocean circulation, and the urgency to include the gravity disturbance in ocean dynamics. Besides, this study also cleared up some misconceptions in gravity related valuables such as vertical, geopotential, marine geoid, and dynamic ocean topography.

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Preprint
(1054 KB)
Peter C. Chu
Interactive discussion
Status: closed

RC1: 'Comment on os202212', Anonymous Referee #1, 21 Mar 2022
This paper makes the claim that the neglect of spatial variations in the Earth's gravity field due to the inhomogeneous composition of the Earth leads to substantial revisions to the classical solution for the Sverdrup/Stommel/Munk solution for the depthintegrated circulation of the oceans. If true, this would certainly be a noteworthy result. However, having read the paper several times, and also the recent papers by the same author on the modifications to the equations of motion and oceanic/atmospheric Ekman layers due to the neglect of the same effect, I'm afraid that I am left scratching my head and wondering whether I'm missing something fundamental?
The key question is what one adopts as the vertical coordinate in the equations of motion? Perhaps I was fortunate as a graduate student to have sat through the lectures of Carl Wunsch, but I have always understood a constant "z surface" to represent a timemean equipotential that accounts for the gravitational and centrifugal forces, that surface being a "bumpy spheroid" due to inhomgeneities in the Earth's gravity field. Under this convention, the only departures of a zsurface from an equipotential are due to the temporal variations in the gravitational field, i.e., the tidal forces.
I accept that the above assumption may not be spelt out explicitly in many text books, and that these variations in the gravity field are neglected from the vast majority of (if not all) computational climate models. However, it strikes me as rather odd to define a z surface as either a spherical or spheriodal reference surface and then incorporate additional horizontal gravitational forces, as is done here and in the author's other recent papers. I am pretty sure the community does not have in mind that following a z surface from boundary, to the centre, of the Indian Ocean requires one to climb roughly 100m against gravity.
I have not worked through the details, but related to the final point above is that the assumption of a rigid lid at z=0 in the derivation of the Sverdrup/Stommel/Munk equation (19) is unjustified given the order 100m variations in sea surface elevation in the authors coordinate system.
I am also missing a good physical explanation in the paper of how the additional torques arise to drive the additional depthintegrated circulation? In equation (19), the additional source of vertical vorticity arises through the projection of the baroclinic production of vortictiy onto the vertical component of the vorticity equation. However, if a z surface is defined as an equipotential, then this term vanishes identically, calling into question the statements made in the abstract  the solution should not depend fundamentally on the choice of coordinate system.
So, in summary, I'm afraid that I cannot recommend this manuscript for publication as I feel that the results rely on a particular and, in my honest opinion, rather odd choice of coordinate system. If I have missed something fundamental, then I apologise in advance and am happy to stand corrected.
Finally, I note that there is quite a lot of overlapl between this and the author's three previous papers on a similar topic, especially in the preliminary material. If the manuscript is published, then I would suggest pruning the material down to focus on that which is novel to this contribution.

AC1: 'Reply on RC1', Peter Chu, 29 Mar 2022
The comment was uploaded in the form of a supplement: https://os.copernicus.org/preprints/os202212/os202212AC1supplement.pdf

AC1: 'Reply on RC1', Peter Chu, 29 Mar 2022

RC2: 'RC1: 'Comment on os202212', David Marshall', David P. Marshall, 23 Mar 2022
I had meant to post my review as a signed review, but unfortunately ticked the wrong box. I've been told that the system does not allow the editorial/technical support team to modify this status, so am reposting my review here.
This paper makes the claim that the neglect of spatial variations in the Earth's gravity field due to the inhomogeneous composition of the Earth leads to substantial revisions to the classical solution for the Sverdrup/Stommel/Munk solution for the depthintegrated circulation of the oceans. If true, this would certainly be a noteworthy result. However, having read the paper several times, and also the recent papers by the same author on the modifications to the equations of motion and oceanic/atmospheric Ekman layers due to the neglect of the same effect, I'm afraid that I am left scratching my head and wondering whether I'm missing something fundamental?
The key question is what one adopts as the vertical coordinate in the equations of motion? Perhaps I was fortunate as a graduate student to have sat through the lectures of Carl Wunsch, but I have always understood a constant "z surface" to represent a timemean equipotential that accounts for the gravitational and centrifugal forces, that surface being a "bumpy spheroid" due to inhomgeneities in the Earth's gravity field. Under this convention, the only departures of a zsurface from an equipotential are due to the temporal variations in the gravitational field, i.e., the tidal forces.
I accept that the above assumption may not be spelt out explicitly in many text books, and that these variations in the gravity field are neglected from the vast majority of (if not all) computational climate models. However, it strikes me as rather odd to define a z surface as either a spherical or spheriodal reference surface and then incorporate additional horizontal gravitational forces, as is done here and in the author's other recent papers. I am pretty sure the community does not have in mind that following a z surface from boundary, to the centre, of the Indian Ocean requires one to climb roughly 100m against gravity.
I have not worked through the details, but related to the final point above is that the assumption of a rigid lid at z=0 in the derivation of the Sverdrup/Stommel/Munk equation (19) is unjustified given the order 100m variations in sea surface elevation in the authors coordinate system.
I am also missing a good physical explanation in the paper of how the additional torques arise to drive the additional depthintegrated circulation? In equation (19), the additional source of vertical vorticity arises through the projection of the baroclinic production of vortictiy onto the vertical component of the vorticity equation. However, if a z surface is defined as an equipotential, then this term vanishes identically, calling into question the statements made in the abstract  the solution should not depend fundamentally on the choice of coordinate system.
So, in summary, I'm afraid that I cannot recommend this manuscript for publication as I feel that the results rely on a particular and, in my honest opinion, rather odd choice of coordinate system. If I have missed something fundamental, then I apologise in advance and am happy to stand corrected.
Finally, I note that there is quite a lot of overlapl between this and the author's three previous papers on a similar topic, especially in the preliminary material. If the manuscript is published, then I would suggest pruning the material down to focus on that which is novel to this contribution.

AC2: 'Reply on RC2', Peter Chu, 29 Mar 2022
The comment was uploaded in the form of a supplement: https://os.copernicus.org/preprints/os202212/os202212AC2supplement.pdf

RC4: 'Reply on AC2', David P. Marshall, 31 Mar 2022
Thank you for your responses to my review.
I want to clarify to one critical point in the author's response, which is the statement "The z surface can never be defined as an equipotential surface".
With respect, can I ask why not?
I believe this is precisely what the ocean modelling community does, indeed the second reviewer (RC3) makes precisely the same point.

AC3: 'Reply on RC4', Peter Chu, 02 Apr 2022
Dear Professor Marshall,
Thank you very much for your quick response.
The statement should be clarified by "The z surface can never be defined as an equipotential surface of the true gravity (i.e., the true equipotential surface)."
The critical point is that the gravity used in oceanography and meteorology is not the TRUE GRAVITY.
The two attached figures illustrate the difference between the normal gravity which is called the effective gravity and used in oceanography and meteorology, and the true gravity which is the most important variable in geodesy.
Figure A shows the main features of the effective gravity [g(φ)K]: (1) it is obtained from the solid Earth with rotation and uniform mass density; (2) the unit vector K is perpendicular to the z surface (z = constant) and points the normal vertical; (3) the z surface is the normal horizontal and coincides with the normal geopotential surface; (4) any movement on the z surface (i.e., normal geopotential surface) is not against the normal gravity.
Figure B shows the main features of the true gravity [g(λ, φ, z) = g(φ)K + δg ]: (1) it is obtained from the solid Earth with rotation and nonuniform mass density; (2) the true gravity has never been used in oceanography and meteorology; (3) the true gravity vector g(λ, φ, z) is perpendicular to the true geopotential surface such as the geoid surface, which represents the true horizontal; (4) any movement on the true geopotential surface is not against the true gravity; (5) any movement on the zsurface is against the true gravity. An additional force, the gravity disturbance, shows up in the zsurface momentum equations.
I would like to know your opinion if the ocean dynamics needs to be advanced due to the fact that water particle is not against the true gravity while it climbs roughly 100 m following the geoid surface from boundary, to the center, of the Indian Ocean.
Best regards,
Peter Chu
.

AC3: 'Reply on RC4', Peter Chu, 02 Apr 2022

RC4: 'Reply on AC2', David P. Marshall, 31 Mar 2022

AC2: 'Reply on RC2', Peter Chu, 29 Mar 2022

RC3: 'Comment on os202212', Anonymous Referee #2, 31 Mar 2022
I have read the manuscript by Peter Chu, and while I found it quite thoughtprovoking, I am forced to conclude that it is actually quite misleading and do not recommend publication in its present form. It seems to me that the mistake the author is making is to formulate the equations of motion in spherical coordinates from the beginning. This is not my understanding of how the equations of motion used by models of the atmosphere and ocean are formulated. Rather, these models use a coordinate system in which the vertical direction is defined as being perpendicular to geopotential surfaces so that gravity always points along the vertical direction with no horizontal component. The resulting coordinate system is orthogonal and curvilinear with the horizontal surfaces varying in distance from the centre of earth in response to variations in the geopotential. The usual practice in the modelling community is to approximate this curvilinear coordinate system by spherical coordinates. It seems to me that the onus is on the author to show that the terms that are neglected when this approximation is made are important and should not be neglected. It should be noted that starting from an orthogonal, curvilinear coordinate system in which the horizontal surfaces correspond to geopotential surfaces, and then approximating the resulting system of equations using spherical coordinates, is not the same as formulating the governing equations in spherical coordinates from the beginning, as the author insists on doing.
It is worth noting that I have no argument with equation (12) in the manuscript which is written in vector form. One consequence of this equation is that the equilibrium state is the one in which isopycnal surfaces coincide with geopotential surfaces, as implied by equation (12) when the pressure gradient term is balanced by the term involving gravity, corresponding to hydrostatic balance. In the coordinate system used by ocean modellers, the equilibrium state corresponds to horizontally uniform stratification.
Another issue I have with the manuscript is the way in which the author evaluates the Jacobian term in his equation (19) using data directly from the World Ocean Atlas without comment. At the very least, one needs to ask what coordinate system is being used by the World Ocean Atlas and whether this is the same as the coordinate system being used in equation (19). Indeed, is it appropriate to simply insert data from the World Ocean Atlas directly into the Jacobian operator? I also feel that the author is being too relaxed in his treatment of the hydrostatic approximation since this strictly applies only in the coordinate system in which gravity acts in the vertical direction. However, these are minor points compared to what I have written in the first paragraph of my review.
In summary, I cannot recommend publication of this manuscript in its present form and I believe the argument being put forward by the author is flawed. At the very least, the author needs to formulate the governing equations in the orthogonal, curvilinear coordinate system in which gravity always acts in the vertical (z) direction. He then needs to consider the terms that are neglected when this coordinate system is approximated by spherical coordinates. Since these terms will, at best, involve accelerations terms arising from the curved surfaces that correspond to geopotential surfaces, I cannot see that these terms can be important or that models, as currently formulated, are fundamentally wrong. If the author believes otherwise, the onus is on him to show readers what these terms are, and why they are important.

AC5: 'Reply on RC3', Peter Chu, 04 Apr 2022
Thank you very much for reviewing my manuscript.
Response to the Geopotential Surface and Vertical Direction
“I have read the manuscript by Peter Chu, and while I found it quite thoughtprovoking, I am forced to conclude that it is actually quite misleading and do not recommend publication in its present form. It seems to me that the mistake the author is making is to formulate the equations of motion in spherical coordinates from the beginning. This is not my understanding of how the equations of motion used by models of the atmosphere and ocean are formulated. Rather, these models use a coordinate system in which the vertical direction is defined as being perpendicular to geopotential surfaces so that gravity always points along the vertical direction with no horizontal component.”
I disagree.
Oceanographers have used the geoid for several decades, but almost no one recognizes that the geoid surface represents the true horizontal. Perpendicular to the geoid surface is the true vertical. The geoid surface is a TRUE GEOPOTENTIAL SURFACE at the top ocean. The global geoid data are publicly available from gravity models of geodetic community such as the EIGEN6C4, which shows that the geoid surface varies from 106.2 m to 85.83 m.
However, the geopotential and geopotential surface used in oceanography and meteorology are the normal geopotential and normal geopotential surface, but not the TRUE GEOPOTENTIAL and TRUE GEOPOTENTIAL SURFACE.
The two attached figures illustrate the difference between the normal gravity which is called the effective gravity and used in oceanography and meteorology, and the true gravity which is the most important variable in geodesy.
Figure A shows the main features of the effective gravity [g(φ)K]: (1) it is determined from the solid Earth with rotation and uniform mass density; (2) the unit vector K is perpendicular to the z surface (z = constant) and points the normal vertical; (3) the z surface is the normal horizontal and coincides with the normal geopotential surface; (4) any movement on the z surface (i.e., normal geopotential surface) is not against the normal gravity.
Figure B shows the main features of the true gravity [g(λ, φ, z) = g(φ)K + δg ]: (1) it is determined from the solid Earth with rotation and nonuniform mass density; (2) the true gravity has never been used in oceanography and meteorology; (3) the true gravity vector g(λ, φ, z) is perpendicular to the true geopotential surface such as the geoid surface, which represents the true horizontal; (4) any movement on the true geopotential surface is not against the true gravity; (5) any movement on the zsurface is against the true gravity. An additional force, the gravity disturbance, shows up in the zsurface momentum equations.
Response to the Coordinate System
“The resulting coordinate system is orthogonal and curvilinear with the horizontal surfaces varying in distance from the centre of earth in response to variations in the geopotential. The usual practice in the modelling community is to approximate this curvilinear coordinate system by spherical coordinates. It seems to me that the onus is on the author to show that the terms that are neglected when this approximation is made are important and should not be neglected. It should be noted that starting from an orthogonal, curvilinear coordinate system in which the horizontal surfaces correspond to geopotential surfaces, and then approximating the resulting system of equations using spherical coordinates, is not the same as formulating the governing equations in spherical coordinates from the beginning, as the author insists on doing.”
I disagree.
The ocean dynamics to include the effect of the gravity disturbance should be coordinate independent. I use the vector form to redrive the combined SverdrupStommelMunk equation (see Supplement).
Response to the Hydrostatic Balance
“It is worth noting that I have no argument with equation (12) in the manuscript which is written in vector form. One consequence of this equation is that the equilibrium state is the one in which isopycnal surfaces coincide with geopotential surfaces, as implied by equation (12) when the pressure gradient term is balanced by the term involving gravity, corresponding to hydrostatic balance. In the coordinate system used by ocean modellers, the equilibrium state corresponds to horizontally uniform stratification.”
“I also feel that the author is being too relaxed in his treatment of the hydrostatic approximation since this strictly applies only in the coordinate system in which gravity acts in the vertical direction. However, these are minor points compared to what I have written in the first paragraph of my review.”
The vertical direction depends on the gravity. The vertical is in the zdirection for the normal (effective) gravity (shown in Fig. A of the Supplement), and in the direction normal to the true geopotential surface such as the geoid for the true gravity (shown in Fig. B of the Supplement). Since normal gravity is just an approximation of the true gravity, replacement of the normal gravity by the true gravity in ocean dynamics becomes necessary.
Equation (15) is the hydrostatic equilibrium with the true gravity. I explain it clearer in the Supplement.
Response to the Data Format
“Another issue I have with the manuscript is the way in which the author evaluates the Jacobian term in his equation (19) using data directly from the World Ocean Atlas without comment. At the very least, one needs to ask what coordinate system is being used by the World Ocean Atlas and whether this is the same as the coordinate system being used in equation (19). Indeed, is it appropriate to simply insert data from the World Ocean Atlas directly into the Jacobian operator?”
I agree. Both geoid (N) (from the EIGEN6C4) and insitu density data (ρ) (from WOA18) are represented in spherical coronates. The spherical coordinates are only used to estimate the gravity disturbance forcing (GDF).
Response to Your Recommendation
“In summary, I cannot recommend publication of this manuscript in its present form and I believe the argument being put forward by the author is flawed. At the very least, the author needs to formulate the governing equations in the orthogonal, curvilinear coordinate system in which gravity always acts in the vertical (z) direction. He then needs to consider the terms that are neglected when this coordinate system is approximated by spherical coordinates. Since these terms will, at best, involve accelerations terms arising from the curved surfaces that correspond to geopotential surfaces, I cannot see that these terms can be important or that models, as currently formulated, are fundamentally wrong. If the author believes otherwise, the onus is on him to show readers what these terms are, and why they are important.”
The ocean dynamics to include the effect of the gravity disturbance should be coordinate independent (see Supplement).
The spherical coordinate system is used because the geoid and density data are represented in the spherical coordinates.
After reading my responses, you may change your recommendation.

RC6: 'Reply on AC5', Anonymous Referee #2, 05 Apr 2022
In his reply to my review, the author states that “Oceanographers have used the geoid for several decades, but almost no one recognizes that the geoid surface represents the true horizontal“. On this important point I disagree. For independent evidence, I refer the author to the book “Ocean Dynamics” by Olbers, Willebrand and Eden and published by Springer. Their Figure 2.9 corresponds exactly to the author’s Figure 1a, showing that they are aware of the geoid. More to the point, they state on page 45 that “It is hence very convenient and useful to use a coordinate system which has phi = constant as one coordinate surface (phi is the geopotential). For orthogonal coordinates, gravity must thus coincide exactly with one coordinate direction, i.e. g = (0,0, g). The geopotential is then dependent on the vertical coordinate z. Referring the potential to the mean surface, i.e. phi(z = 0) = 0, we have phi (z) = gz. The geopotential is thus the work which must be applied to lift a unit mass from z = 0 to height z.“
Regarding the Sverdrup/Stommel/Munk problem, the issue is the direction that is used for the vertical. In the coordinate system used by the author, this is not the same as in the coordinate system I describe above, or as used in the standard Sverdrup/Stommel/Munk problem. In the latter, there is no horizontal component of gravity. The different vertical directions lead to different torque balances in the vertical direction.
Regarding the coordinate system used by the World Ocean Atlas, having talked to observationalists, I am assured that they regard horizontal surfaces as coinciding with geopotential surfaces and hence use the coordinate system I describe above.
I am afraid I stand by my original review. The mistake being made by the author is to work in spherical coordinates from the beginning, whereas the coordinate system used by modellers and observationalists is an orthogonal, curvilinear coordinate system in which the vertical direction is perpendicular to geopotential surfaces. As such, I cannot recommend publication of the manuscript. The author could, nevertheless, make a very useful contribution by writing an authoritative manuscript dealing with these issues. But the author needs to be clear about what coordinate system is being used by modelers and observationalists. It is not the coordinate system he uses in his submitted manuscript.

AC6: 'Reply on RC6', Peter Chu, 08 Apr 2022
Thank you very much for your quick response to my reply.
Response to the ‘Geoid Surface Represents the True Horizontal’
“In his reply to my review, the author states that ‘Oceanographers have used the geoid for several decades, but almost no one recognizes that the geoid surface represents the true horizontal’. On this important point I disagree. For independent evidence, I refer the author to the book “Ocean Dynamics” by Olbers, Willebrand and Eden and published by Springer. Their Figure 2.9 corresponds exactly to the author’s Figure 1a, showing that they are aware of the geoid.”
My statement ‘Oceanographers have used the geoid for several decades, but almost no one recognizes that the geoid surface represents the true horizontal’ has two components: (1) oceanographers have used the geoid for several decades, (2) the geoid surface represents the true horizontal.
It is not surprised that Fig. 2.9 in the book “Ocean Dynamics” by Olbers, Willebrand and Eden (2012) corresponds exactly to Fig.1a in my manuscript. Because similar figures about geoid were published by oceanographers earlier such as ‘Fig. 5 EGM96 geoid height N …’ in the paper:
Wunsch C., and D. Stammer (1998), Satellite altimetry, the marine geoid, and the oceanic general circulation. Annu. Rev., Earth. Planet. Sci., 26, 219253.
The geoid surface is treated as a reference surface, but not a true horizontal surface.
Response to the Geopotential
“It is hence very convenient and useful to use a coordinate system which has phi = constant as one coordinate surface (phi is the geopotential). For orthogonal coordinates, gravity must thus coincide exactly with one coordinate direction, i.e. g = (0,0, g). The geopotential is then dependent on the vertical coordinate z. Referring the potential to the mean surface, i.e. phi(z = 0) = 0, we have phi (z) = gz. The geopotential is thus the work which must be applied to lift a unit mass from z = 0 to height z.”
I disagree. Because this statement is valid only for the effective gravity, not for the true gravity.
In the book “Ocean Dynamics” by Olbers, Willebrand and Eden (2012), and any other similar books such as “Principles of Large Scale Numerical Weather Prediction, by Phillips in “Dynamic Meteorology” (1973 edited by Morel) pages 27, “Geophysical Fluid Dynamics” (1986) by Pedlosky pages 1719, “AtmosphereOcean Dynamics” by Gill (1982) pages 7374, and “Atmospheric and Oceanic Fluid Dynamics” by Vallis (2006) pages 5457, the effective gravity g_{eff} (or called the normal gravity in geodesy) is used with the corresponding effectivegeopotential Φ_{ef}_{f}, and effectivegeopotential coordinate (coincidence with the Earth ellipsoidal surface).
However, the true gravity g is the summation of the effective gravity g_{eff} and the gravity disturbance δg, g = g_{eff} + δg, with the corresponding true geopotential, Φ = Φ_{eff} – T . Here, T is the gravity disturbance potential.
Response to the Sverdrup/Stommel/Munk Problem
“Regarding the Sverdrup/Stommel/Munk problem, the issue is the direction that is used for the vertical. In the coordinate system used by the author, this is not the same as in the coordinate system I describe above, or as used in the standard Sverdrup/Stommel/Munk problem. In the latter, there is no horizontal component of gravity. The different vertical directions lead to different torque balances in the vertical direction.”
I agree.
Response to the Recommendation
“I am afraid I stand by my original review. The mistake being made by the author is to work in spherical coordinates from the beginning, whereas the coordinate system used by modellers and observationalists is an orthogonal, curvilinear coordinate system in which the vertical direction is perpendicular to geopotential surfaces. As such, I cannot recommend publication of the manuscript. The author could, nevertheless, make a very useful contribution by writing an authoritative manuscript dealing with these issues. But the author needs to be clear about what coordinate system is being used by modelers and observationalists. It is not the coordinate system he uses in his submitted manuscript.”
Thank you very much for your critics and recommendation. I will revise the manuscript thoroughly according to the critics of yours and the other two reviewers.

AC6: 'Reply on RC6', Peter Chu, 08 Apr 2022

RC6: 'Reply on AC5', Anonymous Referee #2, 05 Apr 2022

AC5: 'Reply on RC3', Peter Chu, 04 Apr 2022

RC5: 'Comment on os202212', Anonymous Referee #3, 02 Apr 2022
I have read the first several pages of this manuscript and I think it is incorrect and cannot be published. The author uses an oblate ellipsoidal coordinate system and takes into account the variations of the gravitational potential along a surface of constant height in this coodinate system. This then allows the calculation of the GDF which depends also on the gradients of in situ density along this same coodinate ellipsoidal surface.
I ask the author to consider the following situation where the planet is an aqua planet, and the ocean is not in motion. This requires that in situ density is constant at each point on the real geoid surface (not the ellipsoidal approximation to it). The author's GFD is however nonzero and large in this situation; that is, his equation (22). But this turns out only to be that he has not chosen his vertical distance to be measured from the real geoptential. Rather he has chosen the zero of his height to be in an ellipsoidal surface. So his equations show substantial motion, but we know that there should be no motion.
This siimple thought experiment shows that the manuscript is flawed.
The development of the equations with respect to the geoid is done in textbooks, for example in the early pages of the text "Fundamentals of Ocean Climate Models" by S. M. Griffies, published in 2004. These ocean models do not put the ocean in motion if the in situ density is constant on geoptential surfaces.

AC4: 'Reply on RC5', Peter Chu, 03 Apr 2022
General Response
When I am a reviewer, I read paper many times, identify the merit (especially the creativity)/flaw, and make positive or negative recommendation after careful thought.
I am amazed that you have made the negative recommendation after only reading first several pages of the manuscript. Your negative recommendation is on the base of a simple thought experiment.
Response to the Simple Thought Experiment
“ I ask the author to consider the following situation where the planet is an aqua planet, and the ocean is not in motion. This requires that in situ density is constant at each point on the real geoid surface (not the ellipsoidal approximation to it). The author's GFD is however nonzero and large in this situation; that is, his equation (22). But this turns out only to be that he has not chosen his vertical distance to be measured from the real geopotential. Rather he has chosen the zero of his height to be in an ellipsoidal surface. So his equations show substantial motion, but we know that there should be no motion.”
I disagree.
In Equation (22), the Jacobian of the insitu density (ρ) and the gravity disturbance (T) is the projection of the vector product of (del ρ) and (del T) on the zdirection. Consider that the insitu density is constant at each point on the true geopotential surface, i.e., the isopycnal surface coincides with the true geopotential surface. This requires that the two vectors (del ρ) and (del T) are parallel. Their vector product is zero,
(del ρ) ×(del T) = 0
which leads to
GDF = 0
which shows that the GDF does not drive any motion in this simple thought experiment. It is the opposite outcome as you thought. It demonstrates the merit of the manuscript (also see Supplement).
Response on the Geopotential and Geopotential Surface
“The development of the equations with respect to the geoid is done in textbooks, for example in the early pages of the text "Fundamentals of Ocean Climate Models" by S. M. Griffies, published in 2004. These ocean models do not put the ocean in motion if the in situ density is constant on geopotential surfaces.”
The geopotential and geopotential surface used in oceanography and meteorology including in the text "Fundamentals of Ocean Climate Models" by S. M. Griffies are the normal geopotential and normal geopotential surface, but not the TRUE GEOPOTENTIAL and TRUE GEOPOTENTIAL SURFACE.
The two attached figures illustrate the difference between the normal gravity which is called the effective gravity and used in oceanography and meteorology, and the true gravity which is the most important variable in geodesy.
Figure A shows the main features of the effective gravity [g(φ)K]: (1) it is determined from the solid Earth with rotation and uniform mass density; (2) the unit vector K is perpendicular to the z surface (z = constant) and points the normal vertical; (3) the z surface is the normal horizontal and coincides with the normal geopotential surface; (4) any movement on the z surface (i.e., normal geopotential surface) is not against the normal gravity.
Figure B shows the main features of the true gravity [g(λ, φ, z) = g(φ)K + δg ]: (1) it is determined from the solid Earth with rotation and nonuniform mass density; (2) the true gravity has never been used in oceanography and meteorology; (3) the true gravity vector g(λ, φ, z) is perpendicular to the true geopotential surface such as the geoid surface, which represents the true horizontal; (4) any movement on the true geopotential surface is not against the true gravity; (5) any movement on the zsurface is against the true gravity. An additional force, the gravity disturbance, shows up in the zsurface momentum equations, such as in Equation (18) of the manuscript.
Finally, I hope you may change your recommendation after reading my responses.

AC4: 'Reply on RC5', Peter Chu, 03 Apr 2022

EC1: 'Conclusion on os202212', Karen J. Heywood, 08 Apr 2022
I am grateful to the author and three referees for participating in this debate.
It is always good to return to first principles and critically examine the fundamental equations and assumptions upon which our theories and science build. I therefore thank the author for stimulating this discussion and challenging our established theories.
However the three reviewers all question the approach and the arguments made, in particular the coordinate system and the logical deductions the author makes. I did not find the author’s responses to these concerns convincing. Restating what is said in the paper does not address the issue of whether the basic arguments are correct.
Therefore I will not be requesting a revised version of the paper to be submitted for consideration in Ocean Science. I note that the reviewers made constructive suggestions to reframe this work and I encourage the author to consider their comments carefully in taking this work forward.
 AC7: 'Reply on EC1', Peter Chu, 09 Apr 2022
Interactive discussion
Status: closed

RC1: 'Comment on os202212', Anonymous Referee #1, 21 Mar 2022
This paper makes the claim that the neglect of spatial variations in the Earth's gravity field due to the inhomogeneous composition of the Earth leads to substantial revisions to the classical solution for the Sverdrup/Stommel/Munk solution for the depthintegrated circulation of the oceans. If true, this would certainly be a noteworthy result. However, having read the paper several times, and also the recent papers by the same author on the modifications to the equations of motion and oceanic/atmospheric Ekman layers due to the neglect of the same effect, I'm afraid that I am left scratching my head and wondering whether I'm missing something fundamental?
The key question is what one adopts as the vertical coordinate in the equations of motion? Perhaps I was fortunate as a graduate student to have sat through the lectures of Carl Wunsch, but I have always understood a constant "z surface" to represent a timemean equipotential that accounts for the gravitational and centrifugal forces, that surface being a "bumpy spheroid" due to inhomgeneities in the Earth's gravity field. Under this convention, the only departures of a zsurface from an equipotential are due to the temporal variations in the gravitational field, i.e., the tidal forces.
I accept that the above assumption may not be spelt out explicitly in many text books, and that these variations in the gravity field are neglected from the vast majority of (if not all) computational climate models. However, it strikes me as rather odd to define a z surface as either a spherical or spheriodal reference surface and then incorporate additional horizontal gravitational forces, as is done here and in the author's other recent papers. I am pretty sure the community does not have in mind that following a z surface from boundary, to the centre, of the Indian Ocean requires one to climb roughly 100m against gravity.
I have not worked through the details, but related to the final point above is that the assumption of a rigid lid at z=0 in the derivation of the Sverdrup/Stommel/Munk equation (19) is unjustified given the order 100m variations in sea surface elevation in the authors coordinate system.
I am also missing a good physical explanation in the paper of how the additional torques arise to drive the additional depthintegrated circulation? In equation (19), the additional source of vertical vorticity arises through the projection of the baroclinic production of vortictiy onto the vertical component of the vorticity equation. However, if a z surface is defined as an equipotential, then this term vanishes identically, calling into question the statements made in the abstract  the solution should not depend fundamentally on the choice of coordinate system.
So, in summary, I'm afraid that I cannot recommend this manuscript for publication as I feel that the results rely on a particular and, in my honest opinion, rather odd choice of coordinate system. If I have missed something fundamental, then I apologise in advance and am happy to stand corrected.
Finally, I note that there is quite a lot of overlapl between this and the author's three previous papers on a similar topic, especially in the preliminary material. If the manuscript is published, then I would suggest pruning the material down to focus on that which is novel to this contribution.

AC1: 'Reply on RC1', Peter Chu, 29 Mar 2022
The comment was uploaded in the form of a supplement: https://os.copernicus.org/preprints/os202212/os202212AC1supplement.pdf

AC1: 'Reply on RC1', Peter Chu, 29 Mar 2022

RC2: 'RC1: 'Comment on os202212', David Marshall', David P. Marshall, 23 Mar 2022
I had meant to post my review as a signed review, but unfortunately ticked the wrong box. I've been told that the system does not allow the editorial/technical support team to modify this status, so am reposting my review here.
This paper makes the claim that the neglect of spatial variations in the Earth's gravity field due to the inhomogeneous composition of the Earth leads to substantial revisions to the classical solution for the Sverdrup/Stommel/Munk solution for the depthintegrated circulation of the oceans. If true, this would certainly be a noteworthy result. However, having read the paper several times, and also the recent papers by the same author on the modifications to the equations of motion and oceanic/atmospheric Ekman layers due to the neglect of the same effect, I'm afraid that I am left scratching my head and wondering whether I'm missing something fundamental?
The key question is what one adopts as the vertical coordinate in the equations of motion? Perhaps I was fortunate as a graduate student to have sat through the lectures of Carl Wunsch, but I have always understood a constant "z surface" to represent a timemean equipotential that accounts for the gravitational and centrifugal forces, that surface being a "bumpy spheroid" due to inhomgeneities in the Earth's gravity field. Under this convention, the only departures of a zsurface from an equipotential are due to the temporal variations in the gravitational field, i.e., the tidal forces.
I accept that the above assumption may not be spelt out explicitly in many text books, and that these variations in the gravity field are neglected from the vast majority of (if not all) computational climate models. However, it strikes me as rather odd to define a z surface as either a spherical or spheriodal reference surface and then incorporate additional horizontal gravitational forces, as is done here and in the author's other recent papers. I am pretty sure the community does not have in mind that following a z surface from boundary, to the centre, of the Indian Ocean requires one to climb roughly 100m against gravity.
I have not worked through the details, but related to the final point above is that the assumption of a rigid lid at z=0 in the derivation of the Sverdrup/Stommel/Munk equation (19) is unjustified given the order 100m variations in sea surface elevation in the authors coordinate system.
I am also missing a good physical explanation in the paper of how the additional torques arise to drive the additional depthintegrated circulation? In equation (19), the additional source of vertical vorticity arises through the projection of the baroclinic production of vortictiy onto the vertical component of the vorticity equation. However, if a z surface is defined as an equipotential, then this term vanishes identically, calling into question the statements made in the abstract  the solution should not depend fundamentally on the choice of coordinate system.
So, in summary, I'm afraid that I cannot recommend this manuscript for publication as I feel that the results rely on a particular and, in my honest opinion, rather odd choice of coordinate system. If I have missed something fundamental, then I apologise in advance and am happy to stand corrected.
Finally, I note that there is quite a lot of overlapl between this and the author's three previous papers on a similar topic, especially in the preliminary material. If the manuscript is published, then I would suggest pruning the material down to focus on that which is novel to this contribution.

AC2: 'Reply on RC2', Peter Chu, 29 Mar 2022
The comment was uploaded in the form of a supplement: https://os.copernicus.org/preprints/os202212/os202212AC2supplement.pdf

RC4: 'Reply on AC2', David P. Marshall, 31 Mar 2022
Thank you for your responses to my review.
I want to clarify to one critical point in the author's response, which is the statement "The z surface can never be defined as an equipotential surface".
With respect, can I ask why not?
I believe this is precisely what the ocean modelling community does, indeed the second reviewer (RC3) makes precisely the same point.

AC3: 'Reply on RC4', Peter Chu, 02 Apr 2022
Dear Professor Marshall,
Thank you very much for your quick response.
The statement should be clarified by "The z surface can never be defined as an equipotential surface of the true gravity (i.e., the true equipotential surface)."
The critical point is that the gravity used in oceanography and meteorology is not the TRUE GRAVITY.
The two attached figures illustrate the difference between the normal gravity which is called the effective gravity and used in oceanography and meteorology, and the true gravity which is the most important variable in geodesy.
Figure A shows the main features of the effective gravity [g(φ)K]: (1) it is obtained from the solid Earth with rotation and uniform mass density; (2) the unit vector K is perpendicular to the z surface (z = constant) and points the normal vertical; (3) the z surface is the normal horizontal and coincides with the normal geopotential surface; (4) any movement on the z surface (i.e., normal geopotential surface) is not against the normal gravity.
Figure B shows the main features of the true gravity [g(λ, φ, z) = g(φ)K + δg ]: (1) it is obtained from the solid Earth with rotation and nonuniform mass density; (2) the true gravity has never been used in oceanography and meteorology; (3) the true gravity vector g(λ, φ, z) is perpendicular to the true geopotential surface such as the geoid surface, which represents the true horizontal; (4) any movement on the true geopotential surface is not against the true gravity; (5) any movement on the zsurface is against the true gravity. An additional force, the gravity disturbance, shows up in the zsurface momentum equations.
I would like to know your opinion if the ocean dynamics needs to be advanced due to the fact that water particle is not against the true gravity while it climbs roughly 100 m following the geoid surface from boundary, to the center, of the Indian Ocean.
Best regards,
Peter Chu
.

AC3: 'Reply on RC4', Peter Chu, 02 Apr 2022

RC4: 'Reply on AC2', David P. Marshall, 31 Mar 2022

AC2: 'Reply on RC2', Peter Chu, 29 Mar 2022

RC3: 'Comment on os202212', Anonymous Referee #2, 31 Mar 2022
I have read the manuscript by Peter Chu, and while I found it quite thoughtprovoking, I am forced to conclude that it is actually quite misleading and do not recommend publication in its present form. It seems to me that the mistake the author is making is to formulate the equations of motion in spherical coordinates from the beginning. This is not my understanding of how the equations of motion used by models of the atmosphere and ocean are formulated. Rather, these models use a coordinate system in which the vertical direction is defined as being perpendicular to geopotential surfaces so that gravity always points along the vertical direction with no horizontal component. The resulting coordinate system is orthogonal and curvilinear with the horizontal surfaces varying in distance from the centre of earth in response to variations in the geopotential. The usual practice in the modelling community is to approximate this curvilinear coordinate system by spherical coordinates. It seems to me that the onus is on the author to show that the terms that are neglected when this approximation is made are important and should not be neglected. It should be noted that starting from an orthogonal, curvilinear coordinate system in which the horizontal surfaces correspond to geopotential surfaces, and then approximating the resulting system of equations using spherical coordinates, is not the same as formulating the governing equations in spherical coordinates from the beginning, as the author insists on doing.
It is worth noting that I have no argument with equation (12) in the manuscript which is written in vector form. One consequence of this equation is that the equilibrium state is the one in which isopycnal surfaces coincide with geopotential surfaces, as implied by equation (12) when the pressure gradient term is balanced by the term involving gravity, corresponding to hydrostatic balance. In the coordinate system used by ocean modellers, the equilibrium state corresponds to horizontally uniform stratification.
Another issue I have with the manuscript is the way in which the author evaluates the Jacobian term in his equation (19) using data directly from the World Ocean Atlas without comment. At the very least, one needs to ask what coordinate system is being used by the World Ocean Atlas and whether this is the same as the coordinate system being used in equation (19). Indeed, is it appropriate to simply insert data from the World Ocean Atlas directly into the Jacobian operator? I also feel that the author is being too relaxed in his treatment of the hydrostatic approximation since this strictly applies only in the coordinate system in which gravity acts in the vertical direction. However, these are minor points compared to what I have written in the first paragraph of my review.
In summary, I cannot recommend publication of this manuscript in its present form and I believe the argument being put forward by the author is flawed. At the very least, the author needs to formulate the governing equations in the orthogonal, curvilinear coordinate system in which gravity always acts in the vertical (z) direction. He then needs to consider the terms that are neglected when this coordinate system is approximated by spherical coordinates. Since these terms will, at best, involve accelerations terms arising from the curved surfaces that correspond to geopotential surfaces, I cannot see that these terms can be important or that models, as currently formulated, are fundamentally wrong. If the author believes otherwise, the onus is on him to show readers what these terms are, and why they are important.

AC5: 'Reply on RC3', Peter Chu, 04 Apr 2022
Thank you very much for reviewing my manuscript.
Response to the Geopotential Surface and Vertical Direction
“I have read the manuscript by Peter Chu, and while I found it quite thoughtprovoking, I am forced to conclude that it is actually quite misleading and do not recommend publication in its present form. It seems to me that the mistake the author is making is to formulate the equations of motion in spherical coordinates from the beginning. This is not my understanding of how the equations of motion used by models of the atmosphere and ocean are formulated. Rather, these models use a coordinate system in which the vertical direction is defined as being perpendicular to geopotential surfaces so that gravity always points along the vertical direction with no horizontal component.”
I disagree.
Oceanographers have used the geoid for several decades, but almost no one recognizes that the geoid surface represents the true horizontal. Perpendicular to the geoid surface is the true vertical. The geoid surface is a TRUE GEOPOTENTIAL SURFACE at the top ocean. The global geoid data are publicly available from gravity models of geodetic community such as the EIGEN6C4, which shows that the geoid surface varies from 106.2 m to 85.83 m.
However, the geopotential and geopotential surface used in oceanography and meteorology are the normal geopotential and normal geopotential surface, but not the TRUE GEOPOTENTIAL and TRUE GEOPOTENTIAL SURFACE.
The two attached figures illustrate the difference between the normal gravity which is called the effective gravity and used in oceanography and meteorology, and the true gravity which is the most important variable in geodesy.
Figure A shows the main features of the effective gravity [g(φ)K]: (1) it is determined from the solid Earth with rotation and uniform mass density; (2) the unit vector K is perpendicular to the z surface (z = constant) and points the normal vertical; (3) the z surface is the normal horizontal and coincides with the normal geopotential surface; (4) any movement on the z surface (i.e., normal geopotential surface) is not against the normal gravity.
Figure B shows the main features of the true gravity [g(λ, φ, z) = g(φ)K + δg ]: (1) it is determined from the solid Earth with rotation and nonuniform mass density; (2) the true gravity has never been used in oceanography and meteorology; (3) the true gravity vector g(λ, φ, z) is perpendicular to the true geopotential surface such as the geoid surface, which represents the true horizontal; (4) any movement on the true geopotential surface is not against the true gravity; (5) any movement on the zsurface is against the true gravity. An additional force, the gravity disturbance, shows up in the zsurface momentum equations.
Response to the Coordinate System
“The resulting coordinate system is orthogonal and curvilinear with the horizontal surfaces varying in distance from the centre of earth in response to variations in the geopotential. The usual practice in the modelling community is to approximate this curvilinear coordinate system by spherical coordinates. It seems to me that the onus is on the author to show that the terms that are neglected when this approximation is made are important and should not be neglected. It should be noted that starting from an orthogonal, curvilinear coordinate system in which the horizontal surfaces correspond to geopotential surfaces, and then approximating the resulting system of equations using spherical coordinates, is not the same as formulating the governing equations in spherical coordinates from the beginning, as the author insists on doing.”
I disagree.
The ocean dynamics to include the effect of the gravity disturbance should be coordinate independent. I use the vector form to redrive the combined SverdrupStommelMunk equation (see Supplement).
Response to the Hydrostatic Balance
“It is worth noting that I have no argument with equation (12) in the manuscript which is written in vector form. One consequence of this equation is that the equilibrium state is the one in which isopycnal surfaces coincide with geopotential surfaces, as implied by equation (12) when the pressure gradient term is balanced by the term involving gravity, corresponding to hydrostatic balance. In the coordinate system used by ocean modellers, the equilibrium state corresponds to horizontally uniform stratification.”
“I also feel that the author is being too relaxed in his treatment of the hydrostatic approximation since this strictly applies only in the coordinate system in which gravity acts in the vertical direction. However, these are minor points compared to what I have written in the first paragraph of my review.”
The vertical direction depends on the gravity. The vertical is in the zdirection for the normal (effective) gravity (shown in Fig. A of the Supplement), and in the direction normal to the true geopotential surface such as the geoid for the true gravity (shown in Fig. B of the Supplement). Since normal gravity is just an approximation of the true gravity, replacement of the normal gravity by the true gravity in ocean dynamics becomes necessary.
Equation (15) is the hydrostatic equilibrium with the true gravity. I explain it clearer in the Supplement.
Response to the Data Format
“Another issue I have with the manuscript is the way in which the author evaluates the Jacobian term in his equation (19) using data directly from the World Ocean Atlas without comment. At the very least, one needs to ask what coordinate system is being used by the World Ocean Atlas and whether this is the same as the coordinate system being used in equation (19). Indeed, is it appropriate to simply insert data from the World Ocean Atlas directly into the Jacobian operator?”
I agree. Both geoid (N) (from the EIGEN6C4) and insitu density data (ρ) (from WOA18) are represented in spherical coronates. The spherical coordinates are only used to estimate the gravity disturbance forcing (GDF).
Response to Your Recommendation
“In summary, I cannot recommend publication of this manuscript in its present form and I believe the argument being put forward by the author is flawed. At the very least, the author needs to formulate the governing equations in the orthogonal, curvilinear coordinate system in which gravity always acts in the vertical (z) direction. He then needs to consider the terms that are neglected when this coordinate system is approximated by spherical coordinates. Since these terms will, at best, involve accelerations terms arising from the curved surfaces that correspond to geopotential surfaces, I cannot see that these terms can be important or that models, as currently formulated, are fundamentally wrong. If the author believes otherwise, the onus is on him to show readers what these terms are, and why they are important.”
The ocean dynamics to include the effect of the gravity disturbance should be coordinate independent (see Supplement).
The spherical coordinate system is used because the geoid and density data are represented in the spherical coordinates.
After reading my responses, you may change your recommendation.

RC6: 'Reply on AC5', Anonymous Referee #2, 05 Apr 2022
In his reply to my review, the author states that “Oceanographers have used the geoid for several decades, but almost no one recognizes that the geoid surface represents the true horizontal“. On this important point I disagree. For independent evidence, I refer the author to the book “Ocean Dynamics” by Olbers, Willebrand and Eden and published by Springer. Their Figure 2.9 corresponds exactly to the author’s Figure 1a, showing that they are aware of the geoid. More to the point, they state on page 45 that “It is hence very convenient and useful to use a coordinate system which has phi = constant as one coordinate surface (phi is the geopotential). For orthogonal coordinates, gravity must thus coincide exactly with one coordinate direction, i.e. g = (0,0, g). The geopotential is then dependent on the vertical coordinate z. Referring the potential to the mean surface, i.e. phi(z = 0) = 0, we have phi (z) = gz. The geopotential is thus the work which must be applied to lift a unit mass from z = 0 to height z.“
Regarding the Sverdrup/Stommel/Munk problem, the issue is the direction that is used for the vertical. In the coordinate system used by the author, this is not the same as in the coordinate system I describe above, or as used in the standard Sverdrup/Stommel/Munk problem. In the latter, there is no horizontal component of gravity. The different vertical directions lead to different torque balances in the vertical direction.
Regarding the coordinate system used by the World Ocean Atlas, having talked to observationalists, I am assured that they regard horizontal surfaces as coinciding with geopotential surfaces and hence use the coordinate system I describe above.
I am afraid I stand by my original review. The mistake being made by the author is to work in spherical coordinates from the beginning, whereas the coordinate system used by modellers and observationalists is an orthogonal, curvilinear coordinate system in which the vertical direction is perpendicular to geopotential surfaces. As such, I cannot recommend publication of the manuscript. The author could, nevertheless, make a very useful contribution by writing an authoritative manuscript dealing with these issues. But the author needs to be clear about what coordinate system is being used by modelers and observationalists. It is not the coordinate system he uses in his submitted manuscript.

AC6: 'Reply on RC6', Peter Chu, 08 Apr 2022
Thank you very much for your quick response to my reply.
Response to the ‘Geoid Surface Represents the True Horizontal’
“In his reply to my review, the author states that ‘Oceanographers have used the geoid for several decades, but almost no one recognizes that the geoid surface represents the true horizontal’. On this important point I disagree. For independent evidence, I refer the author to the book “Ocean Dynamics” by Olbers, Willebrand and Eden and published by Springer. Their Figure 2.9 corresponds exactly to the author’s Figure 1a, showing that they are aware of the geoid.”
My statement ‘Oceanographers have used the geoid for several decades, but almost no one recognizes that the geoid surface represents the true horizontal’ has two components: (1) oceanographers have used the geoid for several decades, (2) the geoid surface represents the true horizontal.
It is not surprised that Fig. 2.9 in the book “Ocean Dynamics” by Olbers, Willebrand and Eden (2012) corresponds exactly to Fig.1a in my manuscript. Because similar figures about geoid were published by oceanographers earlier such as ‘Fig. 5 EGM96 geoid height N …’ in the paper:
Wunsch C., and D. Stammer (1998), Satellite altimetry, the marine geoid, and the oceanic general circulation. Annu. Rev., Earth. Planet. Sci., 26, 219253.
The geoid surface is treated as a reference surface, but not a true horizontal surface.
Response to the Geopotential
“It is hence very convenient and useful to use a coordinate system which has phi = constant as one coordinate surface (phi is the geopotential). For orthogonal coordinates, gravity must thus coincide exactly with one coordinate direction, i.e. g = (0,0, g). The geopotential is then dependent on the vertical coordinate z. Referring the potential to the mean surface, i.e. phi(z = 0) = 0, we have phi (z) = gz. The geopotential is thus the work which must be applied to lift a unit mass from z = 0 to height z.”
I disagree. Because this statement is valid only for the effective gravity, not for the true gravity.
In the book “Ocean Dynamics” by Olbers, Willebrand and Eden (2012), and any other similar books such as “Principles of Large Scale Numerical Weather Prediction, by Phillips in “Dynamic Meteorology” (1973 edited by Morel) pages 27, “Geophysical Fluid Dynamics” (1986) by Pedlosky pages 1719, “AtmosphereOcean Dynamics” by Gill (1982) pages 7374, and “Atmospheric and Oceanic Fluid Dynamics” by Vallis (2006) pages 5457, the effective gravity g_{eff} (or called the normal gravity in geodesy) is used with the corresponding effectivegeopotential Φ_{ef}_{f}, and effectivegeopotential coordinate (coincidence with the Earth ellipsoidal surface).
However, the true gravity g is the summation of the effective gravity g_{eff} and the gravity disturbance δg, g = g_{eff} + δg, with the corresponding true geopotential, Φ = Φ_{eff} – T . Here, T is the gravity disturbance potential.
Response to the Sverdrup/Stommel/Munk Problem
“Regarding the Sverdrup/Stommel/Munk problem, the issue is the direction that is used for the vertical. In the coordinate system used by the author, this is not the same as in the coordinate system I describe above, or as used in the standard Sverdrup/Stommel/Munk problem. In the latter, there is no horizontal component of gravity. The different vertical directions lead to different torque balances in the vertical direction.”
I agree.
Response to the Recommendation
“I am afraid I stand by my original review. The mistake being made by the author is to work in spherical coordinates from the beginning, whereas the coordinate system used by modellers and observationalists is an orthogonal, curvilinear coordinate system in which the vertical direction is perpendicular to geopotential surfaces. As such, I cannot recommend publication of the manuscript. The author could, nevertheless, make a very useful contribution by writing an authoritative manuscript dealing with these issues. But the author needs to be clear about what coordinate system is being used by modelers and observationalists. It is not the coordinate system he uses in his submitted manuscript.”
Thank you very much for your critics and recommendation. I will revise the manuscript thoroughly according to the critics of yours and the other two reviewers.

AC6: 'Reply on RC6', Peter Chu, 08 Apr 2022

RC6: 'Reply on AC5', Anonymous Referee #2, 05 Apr 2022

AC5: 'Reply on RC3', Peter Chu, 04 Apr 2022

RC5: 'Comment on os202212', Anonymous Referee #3, 02 Apr 2022
I have read the first several pages of this manuscript and I think it is incorrect and cannot be published. The author uses an oblate ellipsoidal coordinate system and takes into account the variations of the gravitational potential along a surface of constant height in this coodinate system. This then allows the calculation of the GDF which depends also on the gradients of in situ density along this same coodinate ellipsoidal surface.
I ask the author to consider the following situation where the planet is an aqua planet, and the ocean is not in motion. This requires that in situ density is constant at each point on the real geoid surface (not the ellipsoidal approximation to it). The author's GFD is however nonzero and large in this situation; that is, his equation (22). But this turns out only to be that he has not chosen his vertical distance to be measured from the real geoptential. Rather he has chosen the zero of his height to be in an ellipsoidal surface. So his equations show substantial motion, but we know that there should be no motion.
This siimple thought experiment shows that the manuscript is flawed.
The development of the equations with respect to the geoid is done in textbooks, for example in the early pages of the text "Fundamentals of Ocean Climate Models" by S. M. Griffies, published in 2004. These ocean models do not put the ocean in motion if the in situ density is constant on geoptential surfaces.

AC4: 'Reply on RC5', Peter Chu, 03 Apr 2022
General Response
When I am a reviewer, I read paper many times, identify the merit (especially the creativity)/flaw, and make positive or negative recommendation after careful thought.
I am amazed that you have made the negative recommendation after only reading first several pages of the manuscript. Your negative recommendation is on the base of a simple thought experiment.
Response to the Simple Thought Experiment
“ I ask the author to consider the following situation where the planet is an aqua planet, and the ocean is not in motion. This requires that in situ density is constant at each point on the real geoid surface (not the ellipsoidal approximation to it). The author's GFD is however nonzero and large in this situation; that is, his equation (22). But this turns out only to be that he has not chosen his vertical distance to be measured from the real geopotential. Rather he has chosen the zero of his height to be in an ellipsoidal surface. So his equations show substantial motion, but we know that there should be no motion.”
I disagree.
In Equation (22), the Jacobian of the insitu density (ρ) and the gravity disturbance (T) is the projection of the vector product of (del ρ) and (del T) on the zdirection. Consider that the insitu density is constant at each point on the true geopotential surface, i.e., the isopycnal surface coincides with the true geopotential surface. This requires that the two vectors (del ρ) and (del T) are parallel. Their vector product is zero,
(del ρ) ×(del T) = 0
which leads to
GDF = 0
which shows that the GDF does not drive any motion in this simple thought experiment. It is the opposite outcome as you thought. It demonstrates the merit of the manuscript (also see Supplement).
Response on the Geopotential and Geopotential Surface
“The development of the equations with respect to the geoid is done in textbooks, for example in the early pages of the text "Fundamentals of Ocean Climate Models" by S. M. Griffies, published in 2004. These ocean models do not put the ocean in motion if the in situ density is constant on geopotential surfaces.”
The geopotential and geopotential surface used in oceanography and meteorology including in the text "Fundamentals of Ocean Climate Models" by S. M. Griffies are the normal geopotential and normal geopotential surface, but not the TRUE GEOPOTENTIAL and TRUE GEOPOTENTIAL SURFACE.
The two attached figures illustrate the difference between the normal gravity which is called the effective gravity and used in oceanography and meteorology, and the true gravity which is the most important variable in geodesy.
Figure A shows the main features of the effective gravity [g(φ)K]: (1) it is determined from the solid Earth with rotation and uniform mass density; (2) the unit vector K is perpendicular to the z surface (z = constant) and points the normal vertical; (3) the z surface is the normal horizontal and coincides with the normal geopotential surface; (4) any movement on the z surface (i.e., normal geopotential surface) is not against the normal gravity.
Figure B shows the main features of the true gravity [g(λ, φ, z) = g(φ)K + δg ]: (1) it is determined from the solid Earth with rotation and nonuniform mass density; (2) the true gravity has never been used in oceanography and meteorology; (3) the true gravity vector g(λ, φ, z) is perpendicular to the true geopotential surface such as the geoid surface, which represents the true horizontal; (4) any movement on the true geopotential surface is not against the true gravity; (5) any movement on the zsurface is against the true gravity. An additional force, the gravity disturbance, shows up in the zsurface momentum equations, such as in Equation (18) of the manuscript.
Finally, I hope you may change your recommendation after reading my responses.

AC4: 'Reply on RC5', Peter Chu, 03 Apr 2022

EC1: 'Conclusion on os202212', Karen J. Heywood, 08 Apr 2022
I am grateful to the author and three referees for participating in this debate.
It is always good to return to first principles and critically examine the fundamental equations and assumptions upon which our theories and science build. I therefore thank the author for stimulating this discussion and challenging our established theories.
However the three reviewers all question the approach and the arguments made, in particular the coordinate system and the logical deductions the author makes. I did not find the author’s responses to these concerns convincing. Restating what is said in the paper does not address the issue of whether the basic arguments are correct.
Therefore I will not be requesting a revised version of the paper to be submitted for consideration in Ocean Science. I note that the reviewers made constructive suggestions to reframe this work and I encourage the author to consider their comments carefully in taking this work forward.
 AC7: 'Reply on EC1', Peter Chu, 09 Apr 2022
Peter C. Chu
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