26 Feb 2021
26 Feb 2021
Riverenhanced nonlinear overtide variations in river estuaries
 ^{1}State Key Lab of Estuarine and Coastal Research, East China Normal University, Shanghai 200241, China
 ^{2}Department of Hydraulic Engineering, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft 2600GA, the Netherlands
 ^{3}School of Marine Engineering and Technology, Sun YatSen University, Guangzhou, China
 ^{4}Marine and Coastal Systems Department, Deltares, Delft 2629 HV, the Netherlands
 ^{5}School of Ocean and Earth Sciences, University of Southampton, Southampton, UK
 ^{1}State Key Lab of Estuarine and Coastal Research, East China Normal University, Shanghai 200241, China
 ^{2}Department of Hydraulic Engineering, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft 2600GA, the Netherlands
 ^{3}School of Marine Engineering and Technology, Sun YatSen University, Guangzhou, China
 ^{4}Marine and Coastal Systems Department, Deltares, Delft 2629 HV, the Netherlands
 ^{5}School of Ocean and Earth Sciences, University of Southampton, Southampton, UK
Abstract. Tidal waves traveling into estuaries are modulated in amplitude and shape due to bottom friction, funneling planform and river discharge. The role of river discharge on damping incident tides has been welldocumented, whereas our understanding of the impact on overtide is incomplete. Inspired by findings from tidal data analysis, in this study we use a schematized estuary model to explore the variability of overtide under varying river discharge. Model results reveal significant M_{4} overtide generated inside the estuary. Its absolute amplitude decreases and increases in the upper and lower parts of the estuary, respectively, with increasing river discharge. The total energy of the M_{4} tide integrated throughout the estuary reaches a transitional maximum when the river discharge to tidal mean discharge (R2T) ratio is close to unity. We further identify that the quadratic bottom stress plays a dominant role in governing the M_{4} variations through strong rivertide interaction. River flow enhances the effective bottom stress and dissipation of the principal tides, and reinforces energy transfer from principal tide to overtide. The twofold effects explain the nonlinear M_{4} variations and the intermediate maximum threshold. The model results are consistent with data analysis in the Changjiang and Amazon River estuaries and highlight distinctive tidal behaviors between upstream tidal rivers and downstream tidal estuaries. The new findings inform study of compound flooding risk, tidal asymmetry, and sediment transport in river estuaries.
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Leicheng Guo et al.
Status: final response (author comments only)

RC1: 'Comment on hess202175', Xiao Hua Wang, 08 Apr 2021
Review of ‘Riverenhanced nonlinear overtide variations in river estuaries’ by Guo et al (hess202175)
This paper uses a 1D estuary model to explore the variability of overtide under varying river discharge. Model results show that significant M4 overtide is generated inside the estuary. Its amplitude decreases and increases in the upper and lower parts of the estuary, respectively, with increasing river discharge. More importantly, the paper shows that the total energy of the M4 tide integrated throughout the estuary reaches maximum when the river discharge to tidal mean discharge (R2T) ratio is close to unity. The paper is a good contribution to improve the understanding of nonlinear overtides behaviors in river estuaries. A key result of this work is the two folds role of river discharge on tides. It appears that the paper does have something new to add to existing and recent literature of the topic (see below), namely, the authors have conducted analytical analysis of three nonlinear terms and their relative contributions to the M4 generation. Further they have found spatial variability of maximum M4 along the river under various river discharges. Based on these two, I consider the paper to be published subject to major revision.
Major comments
However, they have missed one of key references below that let to an incomplete literature review in Introduction and incorrect discussion from line 493503.
According to this paper, it seems to argue that it is the first time that this twofolds role is shown in literature. However, I would like to point out that we have recently demonstrated this phenomenon in our study on tidal propagation in the GangesBrahmaputraMeghna river system, published in Journal of Geophysical Research Oceans last year (Elahi et al., 2020). As in the present study, a threshold river discharge dictates the generation and dissipation of overtides beyond the middle of the GBMR estuary.
We also investigated the nonlinear terms related to bottom friction by applying a numerical model setup (Delft3d) and following the methods of Goddin (1999) and Buschman et al. (2009). Our results show that the threshold river discharge produces maximum amplitude of frictional coefficient in the nonlinear term development, resulting in the maximum generation of overtides. The spatial variations of overtides with river discharge are also apparent in the short length of estuary in the GangesBrahmaputraMeghna delta (< 300 km).
For this reason, I believe that it would be pertinent for the authors to cite our work (Elahi et al., 2020) in both the introduction and discussion of the present study, particularly in line between 493503. That will enrich the discussion of results and increase the applicability of the study findings around the globe.
Other comments
Line 178: Not true. Although conventional harmonic analysis may not accurately resolve tidal constituents, a nonstationary harmonic analysis based on the Complex Demodulation method (Bloomfield, 2004) can be applied here to water level time series.
Line 458459: Can you use your model results to explain why R2T ratio close to unity (not other values), benefits maximal M4 overtide generation?
Line 493504: This is not correct. See the above major comments.
Line 538: maybe > may be
Line 556: SI?
Xiao Hua Wang
SARCCM, UNSW
Reference:
Elahi, M.W.E., JalónRojas, I., Wang, X.H., Ritchie, E.A., 2020. Influence of Seasonal River Discharge on Tidal Propagation in the GangesBrahmaputraMeghna Delta, Bangladesh. J. Geophys. Res. Ocean. 125, 1–19. https://doi.org/10.1029/2020JC016417

AC1: 'Reply on RC1', Leicheng Guo, 16 Apr 2021
Response to online comments from Xiao Hua Wang, UNSW
Review of ‘Riverenhanced nonlinear overtide variations in river estuaries’ by Guo et al (hess202175)
This paper uses a 1D estuary model to explore the variability of overtide under varying river discharge. Model results show that significant M4 overtide is generated inside the estuary. Its amplitude decreases and increases in the upper and lower parts of the estuary, respectively, with increasing river discharge. More importantly, the paper shows that the total energy of the M4 tide integrated throughout the estuary reaches maximum when the river discharge to tidal mean discharge (R2T) ratio is close to unity. The paper is a good contribution to improve the understanding of nonlinear overtides behaviors in river estuaries. A key result of this work is the two folds role of river discharge on tides. It appears that the paper does have something new to add to existing and recent literature of the topic (see below), namely, the authors have conducted analytical analysis of three nonlinear terms and their relative contributions to the M4 generation. Further they have found spatial variability of maximum M4 along the river under various river discharges. Based on these two, I consider the paper to be published subject to major revision.
A: Thank you for the summary and the encouragement.
Major comments
However, they have missed one of key references below that let to an incomplete literature review in Introduction and incorrect discussion from line 493503.
A: Thank you for suggesting the new article. We had included the new reference and related discussion in the Introduction and Discussion texts.
Specifically, in the second paragraph of the Introduction, we added that 'Elahi et al. (2020) documented the impact of river discharge in controlling the balance between the generation and dissipation of quarterdiurnal tides in the GangesBrahmaputraMeghna Delta. They suggested the presence of a critical river discharge threshold as an optimal condition for overtide generation'.
The discussion paragraph in section 4.1 is rephrased as 'Note that the abovementioned nonlinear overtide changes were predominantly reported in large, river and tideinfluenced estuaries and deltas, but less in many other tidedominated estuaries with relatively smaller river discharge. We think that it maybe because the river discharge in tidedominated estuaries is overall small and rarely reaches a magnitude that exceeds R2T=1. Therefore, the role of river discharge in stimulating tidal wave deformation and overtide generation has been widely observed and confirmed (when R2T<1), whereas further changing behaviors under R2T>1 are far less prevalent and hence less well documented. Another explanation is that most tidedominated estuaries are relatively shorter in physical length compared with tidal wavelength, hence the distinction between tidal river and tidal estuary and the spatially nonlinear overtide variations are less apparent compared with that in long estuaries with profound river influences.'
According to this paper, it seems to argue that it is the first time that this twofolds role is shown in literature. However, I would like to point out that we have recently demonstrated this phenomenon in our study on tidal propagation in the GangesBrahmaputraMeghna river system, published in Journal of Geophysical Research Oceans last year (Elahi et al., 2020). As in the present study, a threshold river discharge dictates the generation and dissipation of overtides beyond the middle of the GBMR estuary.
A: Thank you again for pointing out this. We noticed the study and the findings in Elahi et al. (2020), and had changed the tone in this manuscript to get rid of the confusion. In the first paragraph of the Discussion section 4.1, we had added 'In the GangesBrahmaputraMeghna Delta, numerical model results under varying constant river discharge also suggested enhanced quarterdiurnal tides in the lower delta and a transition from increase to decrease in the upper regions of the delta with increasing river discharge (Elahi et al., 2020).'
We also investigated the nonlinear terms related to bottom friction by applying a numerical model setup (Delft3d) and following the methods of Goddin (1999) and Buschman et al. (2009). Our results show that the threshold river discharge produces maximum amplitude of frictional coefficient in the nonlinear term development, resulting in the maximum generation of overtides. The spatial variations of overtides with river discharge are also apparent in the short length of estuary in the GangesBrahmaputraMeghna delta (< 300 km).
For this reason, I believe that it would be pertinent for the authors to cite our work (Elahi et al., 2020) in both the introduction and discussion of the present study, particularly in line between 493503. That will enrich the discussion of results and increase the applicability of the study findings around the globe.
A: We have read very carefully the suggested article and included the citation and discussion of it in the Introduction and Discussion sections. Modeling results in terms of the longitudinal amplitude variations of the quarterdiurnal tides (D4, in Figure 7d in Elahi et al. 2020) are similar with Figure 3b in this work, other than the large D4 amplitude at the mouth of the GBMD. The results and findings in this work with a case of the Changjiang River estuary are thus overall in line with that in Elahi et al. (2020) in GBMD, which also justify the findings from the schematized modeling. We are happy to see the two works output consistent results thus provide more confidence on the findings. Beyond that, this work moves a little bit forward in terms of identifying the maximal threshold (R2T=1) and the contribution of the three nonlinear terms in the tidal dynamic equations. The decomposition method proposed in Buschman et al. (2009) and used in Elahi et al (2020) refers to the subtidal friction term which is used to explain the (lowfrequency) subtidal water level variations. It is different from methods in this work that is used to quantify the contribution of the friction, advection, and discharge gradient terms on the highfrequency overtide behaviors. Discussion and citation to Elahi et al. (2020) are included in the revision. For example, we added that 'Elahi et al. (2020) documented the impact of river discharge in controlling the balance between the generation and dissipation of quarterdiurnal tides in the GangesBrahmaputraMeghna Delta. They suggested the presence of a critical river discharge threshold as an optimal condition for overtide generation' and 'Bushman et al. (2009) and Elahi et al. (2020) employed similar decomposition method of the subtidal friction term to quantify the relative importance of river, tide, and rivertide interaction on subtidal water level variations.' in the Introduction section.
Other comments
Line 178: Not true. Although conventional harmonic analysis may not accurately resolve tidal constituents, a nonstationary harmonic analysis based on the Complex Demodulation method (Bloomfield, 2004) can be applied here to water level time series.
A: We have modified the sentence as 'Conventional harmonic analysis may not accurately resolve tidal properties for a given river discharge magnitude owing to the nonstationary variations (Jay and Flinchem, 1997), although there were attempts to use continuous wavelet transform (Jay et al., 2014; Guo et al., 2015) and complex demodulation method (Bloomfield, 2013) as complementary approaches to resolve tidal species instead of individual constituents. In addition, modeling is another method to examine tidal dynamics (Elahi et al., 2020)' to better clarify the meaning.
Line 458459: Can you use your model results to explain why R2T ratio close to unity (not other values), benefits maximal M4 overtide generation?
A: In the first paragraph of the discussion section 4.2 Role of river discharge, we have argued that the river discharge has twofold impact on the incident tides, i.e., enhancing tidal energy dissipation (damping) and transferring to higher frequencies (deformation). Hence quantitatively we conclude that an intermediate river discharge would benefit maximal overtide generation because it will not dissipate the astronomical tides too much and enhance the nonlinear friction effect in stimulating tidal energy transfer to overtide frequencies. Qualitatively, the model results suggest that the maximum threshold is around R2T=1 (Figure 6). However, it is technically challenging to explain why the threshold R2T ration is 1, but not other values like 0.5 or 2. In studying tideaveraged sediment transport, we had also detected that the tideaveraged sediment transport flux (induced by river flow, tidal asymmetry, and rivertide interactions) tends to be maximal when the riverenhanced residual current velocity equals to the magnitude of the tideinduced current velocity (e.g., the velocity amplitude of M_{2} tide) (Guo et al., 2014, 2016, JGR:Earth Surface). We think a R2T ratio equal to unit (or similarly identical mean current velocity and tidal velocity) reflects a delicate balance between river and tidal forcing, while larger river discharge (R2T>1) or stronger tidal discharge (R2T<1) leads to deviation from the threshold.
Line 493504: This is not correct. See the above major comments.
A: The texts in this paragraph has been thoroughly rephrased as 'Note that the abovementioned nonlinear overtide changes were predominantly reported in large, river and tideinfluenced estuaries and deltas, but less in many other tidedominated estuaries with relatively smaller river discharge. We think that it maybe because the river discharge in tidedominated estuaries is overall small and rarely reaches a magnitude that exceeds R2T=1. Therefore, the role of river discharge in stimulating tidal wave deformation and overtide generation has been widely observed and confirmed (when R2T<1), whereas further changing behaviors under R2T>1 are far less prevalent and hence less well documented. Another explanation is that most tidedominated estuaries are relatively shorter in physical length compared with tidal wavelength, hence the tidal rivertidal estuary distinction and the spatially nonlinear overtide variations are less apparent compared with that in long estuaries with profound river influences.'
Line 538: maybe > may be
Line 556: SI?
A: Changes are made as suggested. SI is the abbreviation of Supporting Information.
Reference:
Elahi, M.W.E., JalónRojas, I., Wang, X.H., Ritchie, E.A., 2020. Influence of Seasonal River Discharge on Tidal Propagation in the GangesBrahmaputraMeghna Delta, Bangladesh. J. Geophys. Res. Ocean. 125, 1–19. https://doi.org/10.1029/2020JC016417
A: The suggested reference is included and cited in the revised work.

RC2: 'Reply on AC1', Xiao Hua Wang, 17 Apr 2021
The authors have not addressed my comment on R2T=1 for maximum M4 generation satisfactorily. The R2T ratio close to unity for maximal M4 generation can be explained by explaining the optimum generation of quadratic frictional coefficient terms in the algebraic development discussed in Godin (1999) as we did in Table 5 in our paper. In lines 555558, the authors mentioned that the quadratic bottom stress term leads to significant M4 generation. The authors can use their model results to analyze the relationship between the R2T ratio and quadratic frictional terms for the generation of M4 tide to explain the unity of the R2T ratio.
It also should be noted that R2T =1 is not applicable to different estuaries (or indeed at different locations in an estuary) for maximum D4 generation. Our paper has demonstrated that an optimum balance between residual velocity and tidal velocity components is also found at R3 and R5 for the Q20 and Q40 scenarios, respectively.

RC2: 'Reply on AC1', Xiao Hua Wang, 17 Apr 2021

AC1: 'Reply on RC1', Leicheng Guo, 16 Apr 2021

RC3: 'Comment on hess202175', Anonymous Referee #2, 27 Apr 2021
This manuscript discusses the effect of rivertide interactions on the generation of overtides, specifically the M4 tide. First, several mechanisms causing the M4 tide in the 1D shallow water equations are computed and analyzed following the method of Gallo and Vinzon (2005). Second, and their main finding, is that the total energy in the generated M4 tide varies with the river discharge and displays a maximum for an, in their range, intermediate discharge. This is further explained conceptually.
I like the idea of the main finding that the energy in the generated M4 tide varies with river discharge and displays a maximum and I think such a thing would be an insightful finding. However, I do not think the conclusions are actually valid and certainly not sufficiently demonstrated. To summarize my main comments (full details are given below): (1) I have good reasons to think that the conclusions are actually only valid for a few cases that look very much like the chosen case study and carry little generality for other estuaries. (2) The conclusions about the spatial characteristics of M4 are not supported by the results, which are integrated over the length of the channel. (3) I think the explanation of the maximum of M4 energy for intermediate discharge as balance between dissipation and generation is incorrect and actually caused by a different mechanism.
Furthermore, the method employed by the authors is shaky: the key equations that much of the results rely on contain multiple quite essential errors and the case study is very specific (also see details below).
This leads me to the recommendation to reject this paper.
Main comments about the conclusions
 In fact you study the transfer of energy from two harmonic components (subtidal and M2) to another (M4). The generated M4 tide in itself is a wave that may propagate according to its own dynamics. This highlights two big problems with the present analysis:
Firstly, your case is friction dominated and for a long estuary without reflection at the head of the estuary. This means that travelling waves will decay. Hence, a small increment in M4 tide generated in some location will not propagate very far. Thus, you dominantly see that locally strong generation of M4 results in a locally strong M4 with some spatial smoothing due to the propagation. I expect that this is totally invalidated in estuaries that are not dominated strongly by friction everywhere or which are shorter and reflect the incoming wave. Hence, your results only represent a small portion of all estuaries. Some of the strongly converging, less frictional branches of the Yangtze estuary itself (which are not considered in this study) could already be a counterexample. Hence, I am of the opinion that a ‘general theory’ as presented here is not so useful and one could just as well study the handful of actual estuaries satisfying it.
Secondly, you explain the maximum in integrated M4 energy for river discharge as a balance between dissipation of the river flow on the M2 tide vs generation of M4 by tideriver interaction (Fig 8). This is not necessarily true. What you actually find is redistribution from the subtidal and M2 water motion to other frequencies. This happens primarily through the term uu in the bottom friction, which you may easily show has a maximum for (approximately) R2T=1. So the actual generation of M4 has a maximum. Dissipation is an additional effect but I would guess it is not essential.
 In section 4.1 and figure 7 you then draw some of the main conclusions on how the local R2T affects the local M4 generation. This is not addressed by your theory, which considers the total integrated M4 energy. Therefore this conclusion is not supported by your results. I expect that this conclusion indeed works in the friction dominated – long estuary setting here but not in general, where the M4 may propagate.
 Ln 491492 actually address the phase of the M4 (relative to the M2). You don’t show any results related to the phase, so this conclusion cannot be drawn.
 ln 554556: ‘In this work we see that the quadratic bottom stress term also leads to significant M4, through rivertide interaction’: this is stated as the main novelty, but is not new.
 Section 3.2: I don’t see the hypothesis underlying this section. Your case without convergence still features a frictiondominated M2 tide. Since the M2 is similar to the case with convergence, I don’t see why the M4 generation should be so different. In any case, just one example of a case without convergence does not prove much. This section does not add anything for me.
Main comments about the method
 Equations (3) and (4) are inconsistent. You assume that only a subtidal and M2 water motion are present, but the numerical computation also allows for all overtides. Implicitly, you assume here that all overtides are much smaller than subtidal and M2, i.e you employ scaling (you do this explicitly on ln 283). This is weird, because in ln 184195 you argued why models based on scaling analyses are not good enough for your study and you need to use a fully numerical model. If you want to do this, I’d recommend using scaling analysis formally in the analysis. This becomes problematic when the M4 tide is not small compared to M2.
 Eq 10 and therefore 1113 (i.e. the main decomposition that you rely on in the results) is wrong. This is not what Godin (1999) uses. You need to use a scaling factor U here:
uu = U(a*u_scaled + b*u_scaled^3),
such that u_scaled ranges between 1 and 1. On a more detailed level, the coefficient a, b you choose in Eq. 10 are Heron’s approximation while Godin (1999) argues that one should better use Chebyshev’s approximation.
 Eq 1113 contains another mistake: ‘theta’ is forgotten everywhere. Hence the phase information is lost. This is essential.
 I am not entirely convinced of the comparison (fig 5) between the ‘discharge gradient’ term (Eq 11) to the advection and friction terms (Eq. 1213). The first appears in the continuity equation and the latter terms in the momentum equation. To create the same unit for all terms, you scale with two different quantities, but why can I compare these? I know Gallo & Vinzon (2005) did the same, but to me this is a very inexact analysis. I think you may at most compare the results on order of magnitude and conclude that all terms are of a similar order of magnitude.
Other remarks
 Ln 184191: I don’t think this gives a proper reflection of the literature. Some of the analytical or semianalytical literature actually resolves (part of the) overtide and various nonlinear terms, e.g. Friedrichs & Aubrey (1988), Lanzoni & Seminara (1998), Ridderinkhof et al (2014), Alebregtse & de Swart (2016), Chernetsky et al (2010), Dijkstra et al (2017). Indeed full treatment of the nonlinearities is not done this way, but since the M4 tide is still generally small compared to the M2 tide, these methods could still work.
 Ln 226241: I don’t think a morphodynamic computation is necessary at all. One could just compute hydrodynamics for a given bathymetry (this would be different when computing sedimentation rates or such). If you do this: what is the final bathymetry?
 Ln 263264: how can the depth be constant after the morphodynamic computation?
 Eq 8: brackets missing in the cosine.
 Eq 13: is a minus missing in the first term or did I get confused with the sign of u0?
 Ln 332: why do you need the M4 amplitude and phase? It does not appear in Eq. 1113.
 Section 3.1: I missed the calibration or setting of friction parameter. How was this done?
 Ln 360363: why include S2 now? This seems inconsistent with the entire method section.
 Fig 5a: you find a contribution from bottom friction while there is no discharge. Is this the effect of tidal return flow?
 Ln 495 ‘majority of estuaries’: I don’t think this is obviously true. This would at least need a reference.
 Ln 424435: why discuss this here. You don’t seem to do this explicitly, so this is more a discussion to me. I don’t find this very insightful, because naturally the linearized friction does not contain any transfer of energy from one frequency to another.
 Ln 525527: you should either prove this or don’t mention it.
 Ln 531535: I can’t follow this. Again this refers to the local discussion I commented on earlier.

RC4: 'Comment on hess202175', Anonymous Referee #3, 29 Apr 2021
Finding M4 tide in the Changjiang and Amazon River estuaries, this manuscript discussed how the M4 is generated by different river discharges. This is an interesting work but the study of the mechanism seems not too stronge. I have few commons:
(1) If the morphology is schematized, the authors can also try other convergence ratios besides the prismatic model. Maybe Amazon model. Or how can the results relate to Amazon since the morphology comes from Changjiang?
(2) To what extent the R2T value (= 1) is appliable since morphology plays a role?
(3) Give more details about the benefits of maximal overtide amplitude.
(4) How does the river discharge affect the effective friction? Is the location of the maximum overtide amplitude related to the morphology? where the friction is maximum?
(5) How is the M4 identified from the model? Or which term represents the M4 in the model?
(6) What is the role of the tide in this study?
Leicheng Guo et al.
Leicheng Guo et al.
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