The authors have made substantial changes to the manuscript. The mathematical derivations are now more succinct, and the rearrangement makes the logic of the arguments much clearer. I would like to thank them for these clarifications and responses to my (and the other reviewer's) earlier comments. The paper is greatly improved.
However, I do still have some significant problems with the interpretation of the "regional view" analysis, which I feel really does still need some further changes before it can be considered ready for publication. There is also one issue with the coastal interpretation, and some other minor issues which I list below.
The maths of the regional view is correct, and the analysis is interesting. In particular, it is intriguing that the JEBAR term appears so dominant in this analysis, in both mean state and time dependence, when the coastal balance is so dominated by the winds. I am persuaded that the presentation of the Csanady model is worthwhile for the insight it gives into both wind forced and "downstream" responses for the barotropic case, as a contrast to what is actually seen in the model.
The problem is with the interpretation as "upwelling", and the descriptions in various places referring to vortex stretching. I have tried for quite some time to find a good physical interpretation of the balance that has been presented, and have not come up with anything clearer, but what is clear is that w* is not related to upwelling in any meaningful sense.
To make this clear, consider a few examples.
1) Consider a wind stress which does not intersect the coast, and has no curl. This will drive a recirculating Ekman flux with no divergence, and so the only flow will be this Ekman flux. From (19), u* is then the Ekman flux divided by H, and w* from (20) can be positive or negative, or zero, depending on how the topography is orientated relative to the Ekman flux. In this scenario there is no upwelling or downwelling. There will also be no coastal sea level signal. So w* can be nonzero when w is zero.
2) Close to the coast in the diagnostics presented here, the primary balance is between wind stress (an Ekman flux away from the coast) and pressure gradient (a geostrophic flow towards the coast), with bottom stress a minor contributor in most places, and no depth-average flow toward or away from the coast. It is also close to barotropic, so geostrophic u and bottom geostrophic u are equal in (19) making u* a flow along the coast (along isobaths), and therefore w* is zero from (20). However, this represents an upwelling flow as the geostrophic flow toward the coast is at greater depth than the Ekman flux away. So w can be zero in an upwelling flow.
3) In a region where the flow does not reach the seafloor (so the geostrophic component can be computed by thermal wind relative to the bottom), and a wind-driven Ekman flow away from the coast is balanced by a geostrophic flow towards the coast, the depth-averaged u and bottom geostrophic u in (19) would both be zero, so u* would be minus the thermal wind flow, making w* -ve from (20). But in this region there is no vertical velocity (no convergence of the Ekman flux), and the oceanward surface Ekman flux balanced by a coastward geostrophic flow at greater average depth actually implies an upwelling somewhere coastward of this region. So w* can be nonzero when w is zero, and the flow is not confined to the surface Ekman layer.
4) The diagnostics shown in Fig. 8 illustrate a case in which the JEBAR term (coming from geostrophic u minus bottom geostrophic u in (19), or thermal wind flow relative to the bottom) dominates, and is larger than the depth-averaged flow effect. In that case, from (19) and (20), w* is negative as observed if the thermal wind flow is towards the coast. With the wind-driven Ekman flow being away from the coast, this is clearly an upwelling scenario similar to 3) above.
5) In the Csanady model the JEBAR term is absent. In this case, it is only the depth-averaged flow that matters in (19), and w* can be diagnosed from the streamlines in Fig. 5, being positive in the upper part of the figure where the depth-integrated flow is toward the coast and negative in the lower part of the picture. But there is no wind stress curl in this configuration, so w is zero near the surface, and a downslope geostrophic flow requires vortex stretching, and hence w negative through the interior of the flow where the geostrophic flow is downslope, and vice versa. The right hand panel shows that this geostrophic flow is (in the sense of towards or away from the coast) in the opposite direction to the depth-integrated flow in the northern region and in much of the southern region, so the actual w in a large part of the flow has the opposite sign to w*. On the other hand, with southward geostrophic currents everywhere, the bottom Ekman flux will be away from the coast everywhere and thus w will be negative in the bottom Ekman layer. If this layer has sufficient weight in the chosen form of vertical average, it will make w* have the same sign as vertically averaged w in the southern region, but will exacerbate the mismatch in the north. So a full solution can have different relationships between w* and w in different regions.
From the above examples it should be clear that there is no meaningful sense in which positive w* can be held to represent upwelling. It seems more likely to be a downwelling, but is not in general related to the vertical velocities in any consistent way. In the same way, w* has no consistent relationship to Ekman pumping or vortex stretching (see example 1 above), and interpreting the curl(tau/H) term as Ekman pumping is incorrect, since it includes both wind stress curl and a term tau x grad(H) which represents the Ekman flux across depth contours, which need not be associated with any vertical velocity.
It is unfortunate that there seems to be no simple interpretation of u*.grad(H), but the results associated with it do seem very interesting. In particular, Fig. 8 shows that the term which is absent in the Csanady model is actually dominant in this simulation, which is interesting for such a shallow region (perhaps a measure of the strong freshwater influence from river input?). It is a particularly striking contrast to Lentz (2024). I suggest removing any mention of upwelling, and removing the w* notation which misleads through its implied relationship to w. Other, similar phrases such as "area integrated vortex tube stretching" (line 489) and "associated net Ekman pumping velocity" (line 333, see also 397) should also be removed as they are inaccurate. Similarly, the introduction of xi (line 495) is unhelpful, and the diagnostics in Fig. 8 should be presented in terms of sea level change, to remain consistent with other diagnostics, and to ensure that the scale is meaningful.
I would like to see these results kept in the paper because they are very interesting, even if a clear interpretation cannot be reached. But it is important that the misleading interpretations be removed.
Another small but important issue is in the interpretation of the coastal balance, starting around line 425. Here, and elsewhere, it is stated that bottom friction partly balances wind stress. In fact, Fig. 7 shows that it acts in concert with wind stress in most regions. This is consistent with the fact that the Nova Scotia Current flows in the opposite direction to the wind stress.
Minor points
Line 35: In fact Huang only says the accuracy can be at the centimetre level for low-lying coastal regions, and errors can be decimetric on mountainous coasts. It would be worth qualifying this statement.
Lines 265 and 353: As above, the curl(tau/H) term is a combination of a "torque" (in the same sense as used in the term bottom pressure torque) and an Ekman flux across depth contours.
Line 379: "Convergence of this onshore flow implies downwelling that is balanced by return flow in the
frictional bottom boundary layer" is confusing. The wind-driven Ekman flux at the coast implies downwelling but this is only partially balanced by a frictional return flow. Some of the return flow is geostrophic, and some does not return but flows along the coast. The total onshore flow (which is what matters for u* in this model) is not balanced by any offshore flow by definition, but by a divergence of the alongshore flow. Please clarify.
Line 437-8: It is only the boundary condition of the Csanady model that is justified by these diagnostics, not its application in the interior.
Line 502: typo "with and"
Line 518-9: I don't follow this statement. Is it in relation to the curl(tau/H) term again? If so, it is a misinterpretation as the vorticity input is only due to curl(tau).
Line 554: please change "wind setup" to "coastal sea level slope due to alongshore wind stress" |
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 - ug + ugb)•∇H, which is described as an upwelling, presumably on the basis that the terms other than ug•∇H represent upslope flows. However, (ug - ugb )•∇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 = ug + uE , 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 -(uE + ugb )•∇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 pb/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∇pb - 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 + Hpb). 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 he 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.