The Alongshore Tilt of Mean Dynamic Topography and its Implications for Model Validation and Ocean Monitoring

Renkl, Christoph; Oliver, Eric C. J.; Thompson, Keith R.

doi:https://doi.org/10.5194/egusphere-2024-1489

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https://doi.org/10.5194/egusphere-2024-1489

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28 May 2024

| 28 May 2024

The Alongshore Tilt of Mean Dynamic Topography and its Implications for Model Validation and Ocean Monitoring

Christoph Renkl, Eric C. J. Oliver, and Keith R. Thompson

Abstract. Mean dynamic topography (MDT) plays an important role in the dynamics of shelf circulation. Coastal tide gauge observations in combination with the latest generation of geoid models are providing estimates of the alongshore tilt of MDT with unprecedented accuracy. Additionally, high-resolution ocean models are providing better representations of nearshore circulation and the associated tilt of MDT along their coastal boundaries. It has been shown that the newly available geodetic estimates can be used to validate model predictions of coastal MDT variability on global and basin scales. On smaller scales, however, there are significant variations in alongshore MDT that are on the same order of magnitude as the accuracy of the geoid models.

In this study, we use a regional ocean model of the Gulf of Maine and Scotian Shelf (GoMSS) to demonstrate that the new observations of geodetically referenced coastal sea level can provide valuable information also for the validation of such high-resolution models. The predicted coastal MDT is in good agreement with coastal tide gauge observations referenced to the Canadian Gravimetric Geoid model (CGG2013a) including a significant tilt of alongshore MDT along the coast of Nova Scotia. Using the validated GoMSS model and several idealized models, we show that this alongshore tilt of MDT can be interpreted in two complementary, and dynamically consistent, ways: In the coastal view, the tilt of MDT along the coast can provide a direct estimate of the average alongshore current. In the regional view, the tilt can be used to approximate upwelling averaged over an offshore area. This highlights the value of using geodetic MDT estimates for model validation and ocean monitoring.

Received: 19 May 2024 – Discussion started: 28 May 2024

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Christoph Renkl, Eric C. J. Oliver, and Keith R. Thompson

Status: closed

RC1:
'Comment on egusphere-2024-1489', Chris Hughes, 27 Jun 2024

This paper covers quite a lot of ground. It presents a model calculation of the mean dynamic topography along the Scotian Shelf - Gulf of Maine region, including comparison with tide gauge observations and interpretation in terms of terms in the equation of motion along the coastline (which in the model is at a mean depth of 23.4 m). These sections of the paper are a nice piece of work in themselves, similar in spirit to the work of Lin et al. (2015) as cited by the authors.
However, the scope is much broader, including a theoretical section relating the alongshore sea level slopes to offshore processes, sections on the Stommel model and on Csanady's Arrested Topographic Wave, and interpretation in terms of a regional average of upwelling, followed by a test of this interpretation using diagnostics of the time-varying component of the ocean model. I find this broader aspect of the paper unconvincing, and the presentation rather disorganised. The maths appears to be correct (except that the wind stress terms should all be divided by ρ₀ and the quickly-abandoned nonlinear terms are incorrect), but the interpretation and link with other models is not clear and, in particular, the description of the relevant diagnostic as an area-averaged upwelling does not seem appropriate. Accordingly, my recommendation for the paper is that it needs a major revision.
Interpretation and link to idealised models
The crucial diagnostic derived in section 3 is -(u - u_g+ u_gb)•∇H, which is described as an upwelling, presumably on the basis that the terms other than u_g•∇H represent upslope flows. However, (u_g- u_gb)•∇H represents the offshore geostrophic flow relative to the bottom (i.e. a thermal wind referenced to the bottom), which on an f-plane has no associated dw/dz, and u is the total, depth-averaged flow, which includes the wind-driven Ekman flow - another component which need not involve any vertical motion. More insight comes if we write u = u_g + u_E , separating the depth-averaged flow into geostrophic and frictional (Ekman) components (the latter includes the effect of wind stress, bottom stress, and lateral friction). The important quantity can now be rewritten as -(u_E + u_gb)•∇H, representing the combination of Ekman and bottom geostrophic onshore flows. In shallow water for example, where we would expect the onshore wind-driven Ekman term to be increasingly balanced by offshore Ekman flow due to bottom (and lateral) friction, this term tends to zero as that balance is established, although the exchange of water between upper and lower Ekman layers represents a downwelling. Equally, a deep water balance of onshore wind-driven Ekman flow and offshore barotropic flow would clearly be a downwelling flow, but again this term would be zero. In short, it cannot meaningfully be described as an upwelling.
The discussion of the Stommel model seems irrelevant. All models are consistent in that the sea level slope at the boundary is related to the difference between wind stress and frictional stress at the boundary (this is simply the boundary condition), but beyond that the Stommel model depends essentially on beta - the boundary current represents a balance between bottom stress curl and the beta term - so the f-plane derivation of section 3 is not relevant.
The Arrested Topographic Wave mode, while consistent with section 3, is barotropic and, in the light of the section 7 results which show the term related to stratification to be dominant, it seems to add little of relevance.
Maths
The derivation of (22) is correct, but very roundabout, with a number of approximations introduced gradually through the derivation. It is in fact a slight rewrite of quite a standard equation (the use of the boundary condition being the main innovation). If we remove the nonlinear terms from (3) (these are incorrect because the depth average of, for example, u squared is not the square of depth-averaged u), and note that the term in brackets on the left is p_b/gρ₀, replace h with H (an approximation used later in the paper), and introduce a streamfunction such that ρ₀Hu = k x ∇ψ, (3)xH becomes
(A) f∇ψ = ρ₀∇χ + H∇p_b - τ
(τ represents all the friction terms).
Dividing (A)S by H then taking the curl gives the barotropic potential vorticity equation, which integrates to a form of (22) when the boundary condition is used to replace the wind stress integral (it is helpful to work in terms of depth-integrated pressure P instead of sea level at the boundary, noting that P = ρ₀χ + Hp_b). This provides a much more straightforward derivation, without the "upwelling" interpretation.
Organisation
It seems odd to have the derivation of (22) at the beginning of a paper which then focuses on the time-mean flow and the boundary interpretation. It would be much more helpful to have a self-contained "steady state, boundary interpretation" part of the paper, then move to the "regional, time dependent" ideas and diagnostics. I'm not sure how useful (22) actually is (it seems to be a way of assessing which part of the dynamics that needs ultimately to be balanced by a bottom pressure term, has not been balanced by it until the sidewall is reached, thus needing a pressure gradient (sea level slope) along the sidewall), but it is certainly interesting, and particularly interesting that the χ term plays such a big role. In many ways this is almost 2 separate papers, but I do see the sense in keeping them together, as long as the logical progression is made clearer.
Minor issues
The description of how the permanent tide is accounted for is confusing (I sympathise! It is hard to explain this issue clearly). I suggest using (1) as the basis throughout, and explaining how h_e and N are calculated by correcting GPS heights and geopotential heights from tide-free to mean-tide system, and then applying (1), rather than saying (1) is applied then corrected.
Line 138 - note that MDT and model MSL differ by an unknown constant (dynamically irrelevant) offset.
Equation 3 and line 168 - a full definition of χ is needed (I think it is the Mertz and Wright one, which is actually depth-integrated PE anomaly divided by ρ₀), the nonlinear terms should be removed (and I would recommend going straight to H instead of h), the wind stress term should be divided by ρ₀, and a reference should be given for the source of the equation (as noted above, there are other simplifications of presentation that could be made too).
Line 179 - this seems to be a definition of bottom pressure torque rather than the JEBAR term, which is better defined in the quotation used later.
Line 202 - "wind setup" suggests the effect of a wind blowing towards, not along the coast.
Line 254 - "corrected for" seems wrong here - u* is the total flow minus the geostrophic flow relative to that at the bottom.
Line 607-8 (regarding the scale factor) - but what would be an appropriate value to use for "water depth at the coast", when it is not in a model with a fixed, finite sidewall?
In conclusion, I see that the authors have done a lot of work to interpret the coastal sea level signals they are investigating. The data analysis is good, the topic and results are interesting if not completely conclusive, and the cited literature is appropriate - I would like to see this paper published. But it does need some streamlining and reorganising to make it clearer what has actually been shown, and to improve the logical flow of the ideas.

Citation: https://doi.org/10.5194/egusphere-2024-1489-RC1
- AC1: 'Reply on RC1', Christoph Renkl, 01 Sep 2024
  
  Dear Chris Hughes,
  Thank you very much for your constructive comments and feedback on our paper! Following your suggestions, we reorganized the manuscript, simplified the maths, and clarified the
  
  interpretations and links between the considered models as described in the attached document. Given the considerable overlap between the comments by both reviewers, we also include the responses to Reviewer 2.
  Many thanks on behalf of the co-authors,
  Christoph Renkl
  
  Citation: https://doi.org/10.5194/egusphere-2024-1489-AC1
RC2:
'Comment on egusphere-2024-1489', Anonymous Referee #2, 12 Jul 2024
This is an interesting extension of earlier studies comparing the geodetic and ocean model approaches to the estimation of the alongshore tilt of MDT. The paper is broken into two main sections – the first looking at the geodetic estimates to validate a regional model, and the second considering how the alongshore slope can provide information on the shelf circulation.
Overall the first section, using geodetic estimates of MDT to validate the GoMSS regional model, is a worthwhile addition to the literature. However I’m left with the sense that spatially-sparse geodetic measurements with relatively high uncertainties are of limited utility in validating a high-resolution model such as GoMSS. I think that the authors slightly overstate the agreement between the model and observations. Qualitatively there is broad agreement, but only when some measurements are excluded and other discrepancies explained.
The second section, deriving and describing the relationship between the alongshore tilt of MDT and the coastal and regional circulation, is less compelling. The coastal view, as it is described by the authors, is similar to the momentum balance presented by Lin et al., but improved through the use of a higher-resolution model. This is of value in terms of understanding the tilt of MDT and, perhaps, in terms of monitoring the alongshore flow. The value of the regional view is less clear.
The layout of the manuscript is challenging. Whilst I think I understand why the derivations are presented up front (sections 3 and 4), these sections are lengthy and heavy going. The results and interpretation sections consequently appear somewhat lost.
I suggest the following to improve the manuscript:
Consider reducing the derivations and/or moving them to an appendix. Whilst I won’t try to fault the math, the derivations are convoluted and rely on many assumptions along the way. Much of this is published elsewhere and I can’t help but think that it could be simplified.

I’m not convinced that the regional view is adding to the story here. It seems more of a theoretical study that doesn’t sit well with the rest of the paper. If it is to remain, the utility of an integrated measure of upwelling needs to be demonstrated.

The final paragraph of section 5 (lines 446 to 451) would benefit from some edits. The “good agreement” is qualitative and subject to a number of caveats (for example, the removal of some stations, and the uncertainty in the geodetic measurements). Also I think that the statement that “The agreement gives confidence that GoMSS captures the mean circulation, including the effect of tidal rectification…” is a little misleading. I believe that the set down at Yarmouth is the evidence of tidal rectification, but I struggle to see anything in the geodetic measurements to really support this.

In conclusion I think that the paper is interesting and worthy of publication. However I think that it needs some significant reformatting to improve the readability, and perhaps a reduction in scope to focus on the key messages.
Citation: https://doi.org/10.5194/egusphere-2024-1489-RC2
- AC2: 'Reply on RC2', Christoph Renkl, 01 Sep 2024
  
  Dear Reviewer 2,
  Many thanks for your valuable comments and suggestions on our paper! We addressed your concerns as described in the attached document. Given the considerable overlap between the comments by both reviewers, we also include the responses to Reviewer 1.
  Many thanks on behalf of the co-authors,
  Christoph Renkl
  
  Citation: https://doi.org/10.5194/egusphere-2024-1489-AC2

Status: closed

RC1:
'Comment on egusphere-2024-1489', Chris Hughes, 27 Jun 2024

This paper covers quite a lot of ground. It presents a model calculation of the mean dynamic topography along the Scotian Shelf - Gulf of Maine region, including comparison with tide gauge observations and interpretation in terms of terms in the equation of motion along the coastline (which in the model is at a mean depth of 23.4 m). These sections of the paper are a nice piece of work in themselves, similar in spirit to the work of Lin et al. (2015) as cited by the authors.
However, the scope is much broader, including a theoretical section relating the alongshore sea level slopes to offshore processes, sections on the Stommel model and on Csanady's Arrested Topographic Wave, and interpretation in terms of a regional average of upwelling, followed by a test of this interpretation using diagnostics of the time-varying component of the ocean model. I find this broader aspect of the paper unconvincing, and the presentation rather disorganised. The maths appears to be correct (except that the wind stress terms should all be divided by ρ₀ and the quickly-abandoned nonlinear terms are incorrect), but the interpretation and link with other models is not clear and, in particular, the description of the relevant diagnostic as an area-averaged upwelling does not seem appropriate. Accordingly, my recommendation for the paper is that it needs a major revision.
Interpretation and link to idealised models
The crucial diagnostic derived in section 3 is -(u - u_g+ u_gb)•∇H, which is described as an upwelling, presumably on the basis that the terms other than u_g•∇H represent upslope flows. However, (u_g- u_gb)•∇H represents the offshore geostrophic flow relative to the bottom (i.e. a thermal wind referenced to the bottom), which on an f-plane has no associated dw/dz, and u is the total, depth-averaged flow, which includes the wind-driven Ekman flow - another component which need not involve any vertical motion. More insight comes if we write u = u_g + u_E , separating the depth-averaged flow into geostrophic and frictional (Ekman) components (the latter includes the effect of wind stress, bottom stress, and lateral friction). The important quantity can now be rewritten as -(u_E + u_gb)•∇H, representing the combination of Ekman and bottom geostrophic onshore flows. In shallow water for example, where we would expect the onshore wind-driven Ekman term to be increasingly balanced by offshore Ekman flow due to bottom (and lateral) friction, this term tends to zero as that balance is established, although the exchange of water between upper and lower Ekman layers represents a downwelling. Equally, a deep water balance of onshore wind-driven Ekman flow and offshore barotropic flow would clearly be a downwelling flow, but again this term would be zero. In short, it cannot meaningfully be described as an upwelling.
The discussion of the Stommel model seems irrelevant. All models are consistent in that the sea level slope at the boundary is related to the difference between wind stress and frictional stress at the boundary (this is simply the boundary condition), but beyond that the Stommel model depends essentially on beta - the boundary current represents a balance between bottom stress curl and the beta term - so the f-plane derivation of section 3 is not relevant.
The Arrested Topographic Wave mode, while consistent with section 3, is barotropic and, in the light of the section 7 results which show the term related to stratification to be dominant, it seems to add little of relevance.
Maths
The derivation of (22) is correct, but very roundabout, with a number of approximations introduced gradually through the derivation. It is in fact a slight rewrite of quite a standard equation (the use of the boundary condition being the main innovation). If we remove the nonlinear terms from (3) (these are incorrect because the depth average of, for example, u squared is not the square of depth-averaged u), and note that the term in brackets on the left is p_b/gρ₀, replace h with H (an approximation used later in the paper), and introduce a streamfunction such that ρ₀Hu = k x ∇ψ, (3)xH becomes
(A) f∇ψ = ρ₀∇χ + H∇p_b - τ
(τ represents all the friction terms).
Dividing (A)S by H then taking the curl gives the barotropic potential vorticity equation, which integrates to a form of (22) when the boundary condition is used to replace the wind stress integral (it is helpful to work in terms of depth-integrated pressure P instead of sea level at the boundary, noting that P = ρ₀χ + Hp_b). This provides a much more straightforward derivation, without the "upwelling" interpretation.
Organisation
It seems odd to have the derivation of (22) at the beginning of a paper which then focuses on the time-mean flow and the boundary interpretation. It would be much more helpful to have a self-contained "steady state, boundary interpretation" part of the paper, then move to the "regional, time dependent" ideas and diagnostics. I'm not sure how useful (22) actually is (it seems to be a way of assessing which part of the dynamics that needs ultimately to be balanced by a bottom pressure term, has not been balanced by it until the sidewall is reached, thus needing a pressure gradient (sea level slope) along the sidewall), but it is certainly interesting, and particularly interesting that the χ term plays such a big role. In many ways this is almost 2 separate papers, but I do see the sense in keeping them together, as long as the logical progression is made clearer.
Minor issues
The description of how the permanent tide is accounted for is confusing (I sympathise! It is hard to explain this issue clearly). I suggest using (1) as the basis throughout, and explaining how h_e and N are calculated by correcting GPS heights and geopotential heights from tide-free to mean-tide system, and then applying (1), rather than saying (1) is applied then corrected.
Line 138 - note that MDT and model MSL differ by an unknown constant (dynamically irrelevant) offset.
Equation 3 and line 168 - a full definition of χ is needed (I think it is the Mertz and Wright one, which is actually depth-integrated PE anomaly divided by ρ₀), the nonlinear terms should be removed (and I would recommend going straight to H instead of h), the wind stress term should be divided by ρ₀, and a reference should be given for the source of the equation (as noted above, there are other simplifications of presentation that could be made too).
Line 179 - this seems to be a definition of bottom pressure torque rather than the JEBAR term, which is better defined in the quotation used later.
Line 202 - "wind setup" suggests the effect of a wind blowing towards, not along the coast.
Line 254 - "corrected for" seems wrong here - u* is the total flow minus the geostrophic flow relative to that at the bottom.
Line 607-8 (regarding the scale factor) - but what would be an appropriate value to use for "water depth at the coast", when it is not in a model with a fixed, finite sidewall?
In conclusion, I see that the authors have done a lot of work to interpret the coastal sea level signals they are investigating. The data analysis is good, the topic and results are interesting if not completely conclusive, and the cited literature is appropriate - I would like to see this paper published. But it does need some streamlining and reorganising to make it clearer what has actually been shown, and to improve the logical flow of the ideas.

Citation: https://doi.org/10.5194/egusphere-2024-1489-RC1
- AC1: 'Reply on RC1', Christoph Renkl, 01 Sep 2024
  
  Dear Chris Hughes,
  Thank you very much for your constructive comments and feedback on our paper! Following your suggestions, we reorganized the manuscript, simplified the maths, and clarified the
  
  interpretations and links between the considered models as described in the attached document. Given the considerable overlap between the comments by both reviewers, we also include the responses to Reviewer 2.
  Many thanks on behalf of the co-authors,
  Christoph Renkl
  
  Citation: https://doi.org/10.5194/egusphere-2024-1489-AC1
RC2:
'Comment on egusphere-2024-1489', Anonymous Referee #2, 12 Jul 2024
This is an interesting extension of earlier studies comparing the geodetic and ocean model approaches to the estimation of the alongshore tilt of MDT. The paper is broken into two main sections – the first looking at the geodetic estimates to validate a regional model, and the second considering how the alongshore slope can provide information on the shelf circulation.
Overall the first section, using geodetic estimates of MDT to validate the GoMSS regional model, is a worthwhile addition to the literature. However I’m left with the sense that spatially-sparse geodetic measurements with relatively high uncertainties are of limited utility in validating a high-resolution model such as GoMSS. I think that the authors slightly overstate the agreement between the model and observations. Qualitatively there is broad agreement, but only when some measurements are excluded and other discrepancies explained.
The second section, deriving and describing the relationship between the alongshore tilt of MDT and the coastal and regional circulation, is less compelling. The coastal view, as it is described by the authors, is similar to the momentum balance presented by Lin et al., but improved through the use of a higher-resolution model. This is of value in terms of understanding the tilt of MDT and, perhaps, in terms of monitoring the alongshore flow. The value of the regional view is less clear.
The layout of the manuscript is challenging. Whilst I think I understand why the derivations are presented up front (sections 3 and 4), these sections are lengthy and heavy going. The results and interpretation sections consequently appear somewhat lost.
I suggest the following to improve the manuscript:
Consider reducing the derivations and/or moving them to an appendix. Whilst I won’t try to fault the math, the derivations are convoluted and rely on many assumptions along the way. Much of this is published elsewhere and I can’t help but think that it could be simplified.

I’m not convinced that the regional view is adding to the story here. It seems more of a theoretical study that doesn’t sit well with the rest of the paper. If it is to remain, the utility of an integrated measure of upwelling needs to be demonstrated.

The final paragraph of section 5 (lines 446 to 451) would benefit from some edits. The “good agreement” is qualitative and subject to a number of caveats (for example, the removal of some stations, and the uncertainty in the geodetic measurements). Also I think that the statement that “The agreement gives confidence that GoMSS captures the mean circulation, including the effect of tidal rectification…” is a little misleading. I believe that the set down at Yarmouth is the evidence of tidal rectification, but I struggle to see anything in the geodetic measurements to really support this.

In conclusion I think that the paper is interesting and worthy of publication. However I think that it needs some significant reformatting to improve the readability, and perhaps a reduction in scope to focus on the key messages.
Citation: https://doi.org/10.5194/egusphere-2024-1489-RC2
- AC2: 'Reply on RC2', Christoph Renkl, 01 Sep 2024
  
  Dear Reviewer 2,
  Many thanks for your valuable comments and suggestions on our paper! We addressed your concerns as described in the attached document. Given the considerable overlap between the comments by both reviewers, we also include the responses to Reviewer 1.
  Many thanks on behalf of the co-authors,
  Christoph Renkl
  
  Citation: https://doi.org/10.5194/egusphere-2024-1489-AC2

Christoph Renkl, Eric C. J. Oliver, and Keith R. Thompson

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It is shown that newly available geodesy combined with suitably located coastal tide gauges is accurate enough for validating model predictions of coastal mean dynamic topography (MDT). Moreover, alongshore differences of MDT can provide estimates of (1) average alongshore current and (2) area-integrated nearshore circulation. Thereby estimates of MDT from geodesy and tide-gauges have value for ocean monitoring.

Short summary

Mean dynamic topography (MDT), describes variations in the mean sea surface height above a reference surface called geoid. We show that MDT predicted by a regional ocean model, including a significant tilt along the coast of Nova Scotia, is in good agreement with estimates based on sea level observations. We demonstrate that this alongshore tilt of MDT can provide a direct estimate of the average alongshore current, and can also be used to approximate upwelling averaged over an offshore area.


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