River-enhanced non-linear overtide variations in river estuaries

Guo, Leicheng; Zhu, Chunyan; Cai, Huayang; Wang, Zheng Bing; Townend, Ian; He, Qing

doi:https://doi.org/10.5194/hess-2021-75

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https://doi.org/10.5194/hess-2021-75

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26 Feb 2021

| 26 Feb 2021

Status: this preprint was under review for the journal HESS but the revision was not accepted.

River-enhanced non-linear overtide variations in river estuaries

Leicheng Guo, Chunyan Zhu, Huayang Cai, Zheng Bing Wang, Ian Townend, and Qing He

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 well-documented, 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 M₄ 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 M₄ 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 M₄ variations through strong river-tide interaction. River flow enhances the effective bottom stress and dissipation of the principal tides, and reinforces energy transfer from principal tide to overtide. The two-fold effects explain the nonlinear M₄ 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.

Received: 08 Feb 2021 – Discussion started: 26 Feb 2021

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Leicheng Guo, Chunyan Zhu, Huayang Cai, Zheng Bing Wang, Ian Townend, and Qing He

Status: closed

RC1:
'Comment on hess-2021-75', Xiao Hua Wang, 08 Apr 2021

Review of ‘River-enhanced non-linear overtide variations in river estuaries’ by Guo et al (hess-2021-75)

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 non-linear 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 non-linear 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 493-503.

According to this paper, it seems to argue that it is the first time that this two-folds 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 Ganges-Brahmaputra-Meghna 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 non-linear 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 non-linear 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 Ganges-Brahmaputra-Meghna 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 493-503. 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 non-stationary harmonic analysis based on the Complex Demodulation method (Bloomfield, 2004) can be applied here to water level time series.

Line 458-459: Can you use your model results to explain why R2T ratio close to unity (not other values), benefits maximal M4 overtide generation?

Line 493-504: 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., Jalón-Rojas, I., Wang, X.H., Ritchie, E.A., 2020. Influence of Seasonal River Discharge on Tidal Propagation in the Ganges-Brahmaputra-Meghna Delta, Bangladesh. J. Geophys. Res. Ocean. 125, 1–19. https://doi.org/10.1029/2020JC016417

Citation: https://doi.org/10.5194/hess-2021-75-RC1
- AC1:
  'Reply on RC1', Leicheng Guo, 16 Apr 2021
  
  Response to online comments from Xiao Hua Wang, UNSW
  Review of ‘River-enhanced non-linear overtide variations in river estuaries’ by Guo et al (hess-2021-75)
  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 non-linear 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 non-linear 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 493-503.
  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 Ganges-Brahmaputra-Meghna 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 above-mentioned nonlinear overtide changes were predominantly reported in large, river- and tide-influenced estuaries and deltas, but less in many other tide-dominated estuaries with relatively smaller river discharge. We think that it maybe because the river discharge in tide-dominated 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 tide-dominated 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 two-folds 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 Ganges-Brahmaputra-Meghna 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 Ganges-Brahmaputra-Meghna 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 non-linear 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 non-linear 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 Ganges-Brahmaputra-Meghna 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 493-503. 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 (low-frequency) 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 high-frequency 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 Ganges-Brahmaputra-Meghna 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 river-tide 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 non-stationary 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 458-459: 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 two-fold 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 tide-averaged sediment transport, we had also detected that the tide-averaged sediment transport flux (induced by river flow, tidal asymmetry, and river-tide interactions) tends to be maximal when the river-enhanced residual current velocity equals to the magnitude of the tide-induced current velocity (e.g., the velocity amplitude of M₂ 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 493-504: This is not correct. See the above major comments.
  A: The texts in this paragraph has been thoroughly rephrased as 'Note that the above-mentioned nonlinear overtide changes were predominantly reported in large, river- and tide-influenced estuaries and deltas, but less in many other tide-dominated estuaries with relatively smaller river discharge. We think that it maybe because the river discharge in tide-dominated 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 tide-dominated estuaries are relatively shorter in physical length compared with tidal wavelength, hence the tidal river-tidal 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., Jalón-Rojas, I., Wang, X.H., Ritchie, E.A., 2020. Influence of Seasonal River Discharge on Tidal Propagation in the Ganges-Brahmaputra-Meghna 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.
  
  Citation: https://doi.org/10.5194/hess-2021-75-AC1
  - 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 555-558, 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.
    
    Citation: https://doi.org/10.5194/hess-2021-75-RC2
    
    AC2: 'Reply on RC2', Leicheng Guo, 28 May 2021
    
    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 555-558, 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.
    A: We have looked at the paper Elahi et al. (2020) carefully and was indeed inspired and have more thoughts on this issue. Firstly, Elahi et al. (2020) adopted the decomposition method proposed in Buschman et al. (2009) to calculate the friction under different river discharge (Eq. 1-4 in Elahi et al., 2020), and then used the decomposition to explain the overtide changes. However, it is noteworthy that the method was originally proposed to examine subtidal friction changes, in order to explain subtidal water level variations related to low-frequency tides like MSf etc. The method is not to explain the dynamics related to high-frequency tidal changes like M₄, because the energy transfer to the higher and lower-frequency tidal components by friction differs greatly.
    Secondly, the optimum condition proposed in Elahi et al. (2020) was a balance between the river-induced residual velocity and tidal velocity components. It explains the location of the maximum overtide amplitude within an estuary under specififed river discharge. The residual velocity increases in the landward direction, while the tidal velocity decreases. Thus the location of the balance varies for different river discharge scenarios. In other words, this balance may not be reached at the same time throughout the entire estuary. The optimum river discharge condition thus is not the same at different locations. As the overtide amplitude varies significantly along an estuary, a local optimum balance is not necessary to indicate the overall overtide generation and dissipation for the entire estuary as a whole. In this work, we looked at the integrated overtide energy by taking the entire estuary into consideration, and found a universal maximal threshold. In this case we use the river discharge to tidal discharge ratio at the mouth section of the estuary as an indicator of the optimum threshold for the estuary as a unit system. A R2T ratio=1 at the mouth of an estuary characterizes identical river flow velocity and tidal component velocity at the mouth (as that in Elahi et al., 2020). Overall, Elahi et al. (2020) examined the local optimum conditions while this work looked at the global maximum. As tidal waves are long waves, the integrated optimum and local peaks in overtide amplitude are simply two different ways of considering the variations in the relationship. In the revision, we will have a look at the spatial variations of the friction and the local overtide optimum conditions as suggested by the reviewer to see the differences.
    
    Citation: https://doi.org/10.5194/hess-2021-75-AC2
RC3:
'Comment on hess-2021-75', Anonymous Referee #2, 27 Apr 2021
This manuscript discusses the effect of river-tide 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 tide-river 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 u|u| 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 491-492 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 554-556: ‘In this work we see that the quadratic bottom stress term also leads to significant M4, through river-tide 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 friction-dominated 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 184-195 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 11-13 (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:

u|u| = 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 11-13 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. 12-13). 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 184-191: I don’t think this gives a proper reflection of the literature. Some of the analytical or semi-analytical 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 226-241: 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 263-264: 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. 11-13.

Section 3.1: I missed the calibration or setting of friction parameter. How was this done?

Ln 360-363: 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 424-435: 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 525-527: you should either prove this or don’t mention it.

Ln 531-535: I can’t follow this. Again this refers to the local discussion I commented on earlier.
Citation: https://doi.org/10.5194/hess-2021-75-RC3
- AC3:
  'Reply on RC3', Leicheng Guo, 28 May 2021
  RC3: 'Comment on hess-2021-75', Anonymous Referee #2, 27 Apr 2021
  This manuscript discusses the effect of river-tide 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.
  A: The reviewer's concerns are noted. However, they are too vague and open ended for us to be able to respond directly to their concerns. In this work, there are two main findings: one is the spatially non-uniform and non-linear changes in overtide amplitude in response to different river discharges, and one is the explanation of the integrated maximal overtide energy using a threshold of R2T=1. The first finding can be partially validated by similar results detected in Amazon (Gallo and Vinzon, 2005), in Columbia Estuary (Jay et al., 2014), Changjiang Estuary (Guo et al., 2015), St. Lawrence Estuary (Matte et al., 2013), and Ganges Delta (Elahi et al., 2020). These case studies only show tidal changes under one or two river discharge conditions, while the modeling work in our study extends this to a range of river discharges from 0 to 90000 m³/s, thus providing a more complete picture. We have also argued in the discussion section why similar phenomenon have not widely reported in many other tide-dominated (small) estuaries given tides are a very basic research theme that have been studied for centuries. "We think that it maybe because the river discharge in tide-dominated estuaries is comparatively 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 tide-dominated 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."
         As to the second main finding, it is mainly derived based on the model results in this work, which will inevitably depend on the model settings. However, we note that the findings Elahi et al. (2020), examining the Ganges, are consistent with our finding. Elahi et al. (2020) suggested an optimal balance featured by equal river-induced mean current velocity and tidal-induced current velocity magnitude, whereas we found that R2T=1 when looking at the mouth section, suggesting a level of consistency between the two approaches. To illustrate the validity of this finding, we are modeling more estuaries with varying shape and geometry and will include the results in the revised work as supporting information. Preliminary indications from the additional runs suggest that the results do not change the findings or conclusions of the main text.
  
  (2) The conclusions about the spatial characteristics of M4 are not supported by the results, which are integrated over the length of the channel.
  A: This maybe a misunderstanding. In the Conclusion section, we had stated that 'While the principal M₂ tide is increasingly dissipated as the R2T ratio increases from zero, the M₄ overtide amplitude decreases in the upper part of estuaries but increases in the lower part of estuaries'. This statement can be clearly seen from Figure 3b and Figure 4c.
         Following that, we concluded that 'With increasing R2T ratio, the total energy of M₄ overtide integrated throughout the estuary first increases and reaches a peak when the R2T ratio approaches unit.' This latter statement refers to the integrated results, which is clearly supported by Figure 6.
         The above two statements are not the same and should not be conflated.
  
  (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.
  A: The reviewer's concerns are noted. However, they are too vague and open ended for us to be able to respond directly to their concerns. It has been widely accepted and understood that 1) the significant overtide M₄ in shallow water environments is generated by the nonlinear processes like advection and friction etc.; 2) the energy of M₄ derives from its parent tides M₂ (Parker, 1991; Wang et al., 1999 etc); and 3) river discharge enhances tidal dissipation of M₂ predominantly through the friction effect (Horrevoets et al., 2004). Hence it logically makes sense that the overtide will depend on the generation and dissipation, both of which depends on the nonlinear mechanisms.
  
  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.
  A: It is true that the locally generated M₄ tide propagates in the estuary as a wave. However, M4 was not imposed at the seaward boundary in this modeling study. As M₄ is generated inside the estuary, it propagates, but also continues to be generated by the transfer of energy from higher harmonics further upstream. Hence the M₄ amplitude does not necessarily reduce in the landward direction in the same way as M₂. Both model results and observed results in many actual estuaries confirm the spatial variations of the M₄ tide, first increasing from the river mouth and then decreasing in the landward direction, e.g., in the Amazon and Changjiang estuaries. We do not expect anything else than this behavior, as at the mouth the M₄ was not imposed at the boundary. The amplitude of M₄ tide is not merely determined by the local generation and we do not agree with reviewer's statement "locally strong generation of M₄ results in a locally strong M₄ with some spatial smoothing due to the propgation". The local generation will for sure also influence the M₄ amplitude further landwards.
         We see that in some very special cases, e.g. estuaries that are not dominated by friction, are shorter, or where wave reflection is important, as mentioned by the Reviewer, different patterns may indeed emerge. However, we make no claim that our results apply to these cases. We simply note that friction has been shown to be one of the most important mechanisms in damping incoming astronomical tides in many real-word estuaries. The overtide generation still occurs in a frictionless environment, but its amplitude becomes smaller and the nonlinear behavior of along-channel change in response to increasing river discharge does not show up (see Figure S4 in the Supporting Information). Wave reflection is not expected to be large in estuaries with high river discharge which will damp the incoming tides to a large degree. This study and the associated findings and statements should be applicable to large river estuaries with considerable river discharge variations, but maybe not to some special systems, such as a barrier dam within tidal wave limit that causes wave reflection etc.
  
  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 tide-river 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 u|u| 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.
  A: There maybe another misunderstanding. When we talked about dissipation, we mainly referred to the dissipation of M₂, not dissipation of M₄. As mentioned above, the M₄ is generated by the transfer of energy from M₂, the parent tide of M₄. So M₄ generation inside estuaries is one the dominant processes. Additionally, M₂ was predominantly dissipated inside estuaries owing to friction and river discharge, hence dissipation of the M₂ tidal energy in the landward direction will affect the energy available to M₄, and then affect the generation of M₄. This balance of M₂ dissipation and M₄ generation then controls the net amplitude of M₄.
  
  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.
  A: The local M₄ amplitude is indeed controlled by both the local generation and the propagation of its component generated in the more seaward domain. However, we did not 'draw conclusion on how the local R2T affect the local M₄ generation'. As can be seen in response to previous questions, in this study we firstly detected nonlinear variations of M₄ amplitude along the estuary under different river discharge, which depends on the local M4 generation and large-scale propagation. Following that, we integrated the M₄ energy throughout the estuary and found a maximal value when scaled with the R2T at the mouth section. Hence, the threshold R2T=1 is not a local influence, but represents the overall impact of river on tides and an intermediate balance between the strength of river and tidal forcing. The findings in this study do indeed mainly apply to long estuaries with friction as an important controlling mechanism in tidal wave propagation. We have clarified that in the discussion to get rid of such confusion, e.g., at the end of section 4.1, we added that 'Hence, the findings in this work mainly apply to long estuaries with significant river discharge and friction control on tidal propagation in which wave reflection is limited. '
  
  Ln 491-492 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.
  A: The sentence in line 491-492 was 'These field data and model results confirm that the findings regarding the spatial dependence of overtide on river discharge are likely to be ubiquitous for river estuaries.' It said nothing about tidal phase. We do not address the phase of M₄. To get rid of any possible confusion, we rephrased the sentence as 'These field data and model results confirm that the findings regarding the spatial dependence of overtide amplitude on river discharge are likely to be ubiquitous for river estuaries.'
  
  ln 554-556: ‘In this work we see that the quadratic bottom stress term also leads to significant M4, through river-tide interaction’: this is stated as the main novelty, but is not new.
  A: The whole sentence mentioned here was 'In this work we see that the quadratic bottom stress term also leads to significant M₄, through river-tide interaction, i.e., between a river-enhanced mean current and M₂ current.' We did not emphasize that this is the main novelty. To get rid of any potential confusion, the sentence is rephrased as 'Additionally, the quadratic bottom stress term also leads to significant M₄, through river-tide interaction, i.e., between a river-enhanced mean current and M₂ current (Wang et al., 1999).'
  
  Section 3.2: I don’t see the hypothesis underlying this section. Your case without convergence still features a friction-dominated 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.
  A: Section 3.2 was dedicated to showing the possible impact of width convergence on the tidal changes. After comparing the results in Figures 3 (convergent estuary) and 4 (rectangular estuary), we see that the longitudinal changing behaviors of both M₂ and M₄ are similar, other than the firstly amplified M₂ and larger M₄ amplitude in the convergent estuary. The incoming M₂ tide was predominantly damped inside the rectangular estuary. The latter differences are understandable because width convergence enhances tidal amplification. Given the spatial changing behaviors are the same for both the convergent and non-convergent estuaries, the discussion in the sections following Section 3.2 are based on the results in the non-convergent estuary, in order to isolate the impact of width variations and consequent tidal wave amplification. Overall we think the message in Section 3.2 is helpful to indicate the sensitivity of the overtide changes to different shaped estuaries.
  
  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 184-195 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.
  A: The reviewer did not point out and we do not see why Eq. 3 and 4 are inconsistent. In lines 184-195, the point being made is that analytical solutions of governing tidal dynamic equations may not fully capture the nonlinear dynamics, which are important for overtide generation. The analytical solutions adopted scaling analysis in order to simplify the governing equations, e.g., linearize the friction term. The scaling analysis mentioned here is not to evaluate the relative importance of M₄ to M₂. In this work we used a numeric model which includes all the nonlinear processes represented in the governing equations and can thereby reproduce their non-uniform spatial changes under different river-tidal conditions. But we accept that both analytical and numerical approaches have their own advantages.
         ln line 283, the whole sentence was 'The bottom friction term is approximately expanded into a bottom shear stress term and a term considering depth variations, as the two terms on the right hand of Eq. (7), respectively, according to Godin and Martinez (1994), given the tidal amplitude to water depth ratio (|η|/h) is generally smaller than one', so the scaling was used to show that the tidal wave amplitude is relatively smaller with respect to the water depth, so the bottom friction can be expanded as that shown in Eq.7 according to Godin and Martinez (1994). Given M₂ is the main tidal constituent, adding another M₄ will not fundamentally change the tidal amplitude to water depth ratio (|η|/h).
         The Eq. 3 and 4 did not include the M₄ component because they were used to represent the condition at the river mouth. In this study only an astronomical M₂ tidal constituent was prescribed at the seaside boundary and M₄ tides are generated inside the estuary. Moreover, including the M₄ component in Eq. 3 and 4 will not fundamentally change the results in Eq. 5-7 because M₂ amplitude is a factor larger than that of M₄. To get rid of the confusion, we have added a sentence in the revision 'Note that the internally generated M₄ component is so far not included and shown in Eq. 3 and 4, but this does not fundamentally change the results, given that the amplitude of M₂ is much larger than M₄.'
  
  Eq 10 and therefore 11-13 (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:
  u|u| = 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 11-13 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. 12-13). 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.
  A: The above questions focus on the decomposition method and the associated results in Figure 5. We have given more thoughts to the methods to compare the contribution of different nonlinear terms (Eq. 11-13) and we noticed that the issue of different scaling may undermine the quality of the analysis. Given the results in Figure 5 are largely in line with previous studies- showing that the friction term dominates the overtide change behavior- this does not add much new understanding. In this revision we have therefore removed the discussion on the contribution of the nonlinear terms (section 3.3 and Figure 5) and associated texts on the methods to get rid of confusion. The main findings of this work are then 1) highlighting the spatially nonlinear changes of overtide amplitude under varying river discharges and 2) identification of the threshold condition controlling the maximal overtide generation.
  
  Other remarks
  Ln 184-191: I don’t think this gives a proper reflection of the literature. Some of the analytical or semi-analytical 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.
  A: Thank you for the suggestion. We have revised the sentences into 'Analytical models usually assume tidal propagation as a single wave component, based on simplified tidal dynamic equations after scaling analyses, e.g., adopting a linear assumption or a nonlinear expansion of the friction term (Green, 1837; Kreiss, 1957; Jay, 1991; Parker, 1991; Friedrichs and Aubrey, 1994; van Rijn, 2011). Further improved analysis method can take into account of more than one tidal component and robust approximation of the nonlinear friction term and their impact on morphodynamic changes (Lanzoni and Seminara, 1998; Ridderinkhof et al., 2014; Alebregtse and de Swart, 2016; Chernetsky et al., 2010; Dijkstra et al., 2017).' The suggested references are included in the reference.
  
  Ln 226-241: 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?
  A: Indeed, the tidal hydrodynamics within an estuary can adapt to any arbitrary bathymetry. However, this may only be stable if the bed is erodible. That is why, in this work we first run a morphodynamic simulation to obtain a close-to-equilibrium bed profile for the defined tidal forcing conditions and then use the equilibrium bed profile in further tidal simulations. We believe this treatment is preferred to minimize the impact of an arbitrary bathymetry on the tidal dynamics, thus highlighting the effect of river discharge. Even though, our previous study (Guo et al., 2016) did show that the overall overtide changes in response to river discharge variations exhibit similar spatial pattern under initial and equilibrium bed profiles (see Figure R1 below).
  
  Figure R1. Longitudinal changes of M₂ and M₄ amplitude modeled in a 560-km long rectangular estuary forced by different river discharge under primary initial bed profile (dotted lines) and equilibrium bed profiles (solid lines) (from Guo et al., 2016)
  
  Ln 263-264: how can the depth be constant after the morphodynamic computation?
  A: It is a mistake in using the word 'constant'. What we mean was that the water depth will not change over time anymore. So the sentence is corrected as 'As the bed level is prescribed as an equilibrium profile, the water depth h remains unchanged'.
  
  Eq 8: brackets missing in the cosine.
  A: A bracket is added.
  
  Eq 13: is a minus missing in the first term or did I get confused with the sign of u0?
  A: The minus sign depends on the reference direction of current. A minus sign for u₀ is needed if assuming flooding currents are defined positive. So a minus sign was added in Eq. 4. The Eqs. 11-13 were removed in this revision to get rid of confusion.
  
  Ln 332: why do you need the M4 amplitude and phase? It does not appear in Eq. 11-13.
  A: Both M₂ and M₄ amplitude and phases are output of the harmonic analysis, but only M₂ harmonics are used in Eq. 11-13.
  
  Section 3.1: I missed the calibration or setting of friction parameter. How was this done?
  A: As a schematized model was employed in this study, we did not do calibration of the friction parameter against field data. In the section 2.1, we added that 'A uniform friction coefficient of Chézy value of 65 m^1/2/s was used, which leads to predominantly landward tidal damping within the schematized estuary, comparable to what is observed in reality (see Figures 1 and 3).'
  
  Ln 360-363: why include S2 now? This seems inconsistent with the entire method section.
  A: Yes, the major part of this study focused on the M₂ and M₄ changes. The S₂ tide was mentioned here to show that other compound tide like MS₄, which was generated by M₂-S₂ interaction, exhibits similar changes as M₄ in response to river discharge (Figure 3c).
  
  Fig 5a: you find a contribution from bottom friction while there is no discharge. Is this the effect of tidal return flow?
  A: Yes, the u₀ velocity is nonzero in the no river discharge scenario, owing to a return flow under the predominantly progressive wave condition.
  
  Ln 495 ‘majority of estuaries’: I don’t think this is obviously true. This would at least need a reference.
  A: The sentence was rephrased and references are included to get rid of confusion. In the revised work, the new sentence is 'Note that the above-mentioned nonlinear overtide changes were predominantly reported in large, mixed river-tide energy estuaries and deltas, e.g., Amazon (Gallo and Vinzon, 2005), Changjiang (Guo et al., 2015), and Ganges (Elahi et al., 2020), but less in many other tide-dominated estuaries with relatively smaller river discharge, although the importance of overtide in controlling tidal asymmetry and residual transport were widely reported, e.g., in Humber in the UK (Winterwerp, 2004) and western Scheldt Estuary in the Netherlands (Wang et al., 2002).'
  
  Ln 424-435: 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.
  A: We aimed to stress the role of nonlinearity of the friction in controlling river-tide interaction, thus a comparison of the model results between the nonlinear (default) and linear friction was conducted. As the discussion of the different contribution of the nonlinear terms (original section 3.3) was removed in this section, the texts on the discussion of the sensitivity to linear friction have been moved into section 4.2 as part of the discussion instead of result.
  
  Ln 525-527: you should either prove this or don’t mention it.
  A: The changing behavior of MS₄ was shown in Figure 3c. The sentence was removed to be more focused on overtide M₄.
  
  Ln 531-535: I can’t follow this. Again this refers to the local discussion I commented on earlier.
  A: The sentence 'In the lower part of estuaries where the incident tides are less dissipated, river flow plays a more important role in reinforcing the effective bottom friction. As a result, dissipation of the principal tide is more prominent in the upper part of estuaries, while tidal energy transfer and overtide generation are more substantial in the lower part of estuaries (Figures 8)' was to stress the different impact of river flow on the tidal changes in seaward and landward part of an estuary. As we had argued in the response to previous questions, the spatial variations of overtide M₄ along the estuary are a combined result of dissipation of M₂ and generation of M₄. To further clarify this point, the sentence is rephrased in this revision as 'In the seaward part of estuary where the incident tidal waves are less dissipated by increasing river discharge, the role of the river discharge in reinforcing the effective bottom friction and enhancing overtide generation is more pronounced. Whereas, in the landward part of the estuary, river discharge plays a prominent role in causing more dissipation of the principal tide. As a result, the tidal energy available to be transferred to overtide is constrained, thus overtide amplitude reduces with increasing river discharge in the landward regions (Figures 8).
  
  Citation: https://doi.org/10.5194/hess-2021-75-AC3
RC4:
'Comment on hess-2021-75', 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?

Citation: https://doi.org/10.5194/hess-2021-75-RC4
- AC4: 'Reply on RC4', Leicheng Guo, 28 May 2021
  
  RC4: 'Comment on hess-2021-75', 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?
  A: Indeed the schematized model can be used to explore the sensitivity to other convergence rates. In this work, we mainly present results in a rectangular channel and a convergent channel, with dimension comparable to the Changjiang Estuary. Gallo and Vinzon (2005) have modeled the tides in the Amazon Estuary and their modeling results are cited and highlighted in this work (see Figure 6). Moreover, Elahi et al. (2020) also undertook a similar modeling study of the Ganges Delta and showed consistent results with this work. Instead of focusing on specific sites, we used schematic models to investigate the broad implications for similar estuaries. The main objective was to reveal the general pattern of the relative changes in the overtide amplitude in response to different magnitude of river discharge. We have not sought to reproduce tidal propagation changes in specific system, in which the absolute amplitude and longitudinal changes will very much depend on the regional changes in channel width, depth and bed slope etc. The channel pattern in the schematized model, e.g., reduced and increased M4 amplitude in the landward and seaward parts of estuaries, is overall consistent with the actual data analysis in Amazon, Changjiang, Columbia etc., justifying the model results in this work. As a response to this question and the following one, we have constructed more models with varying width convergence rates and preliminary simulations show similar patterns as found for the rectangular channel, although the absolute amplitude varies slightly. These extra model results will be included in the revised supporting information of the revised manuscript. We have added a sentence to note the similar pattern as convergence varies in section 2.2 'Sensitivity tests examining the influence of the rate of convergence indicate that the patterns are similar to those for the rectangular channel case, with some small variations in the amplitudes of the various components'.
  
  (2) To what extent the R2T value (= 1) is appliable since morphology plays a role?
  A: Morphology is indeed an important factor affecting tidal propagation and changes in estuaries. However, the difference in morphology will not alter the main findings in this work, because: 1) the model results in both rectangular and convergent channels show similar changing patterns of M₄ (see Figures 3 and 4), and 2) we use an equilibrium bed profile for tidal simulations, and have shown that the bed profile will not change the patterns of longitudinal change (see Figure R1 above). We will include more model results in Figure 6 to confirm the R2T threshold in the revised manuscript.
  
  (3) Give more details about the benefits of maximal overtide amplitude.
  A: In the first paragraph of Section 4.3 we discuss the relevance and implications of the maximal threshold conditions. The finding of a maximal overtide under intermediate river flow condition (or equal river and tidal forcing strength) has broad implications for estuary management. For example, one implication is for the residual sediment transport which tends to be the largest when the river flow velocity is equal to a tidal velocity. Such a residual sediment transport tends to lead to the largest seaward sediment flushing and consequently the deepest equilibrium depth (Guo et al., 2016).This threshold condition occurs because river discharge dissipates tides and the impact of river-tide interaction on creating residual sediment transport changes nonlinearly with rising river discharge. These arguments have been incorporated into the discussion section of the revised manuscript.
  
  (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?
  A: River discharge affects the friction mainly via the velocity magnitude. River discharge enlarges the current velocity, and the friction the moving flow feels accordingly. The impact of river flow in inducing more dissipation of astronomical constituents was exerted via the effective friction term (Horrevoets et al., 2004).
  Morphology affects the longitudinal variations of river flow velocity and landward decrease in tidal prism and tidal velocity. The morphology can therefore influence the location of maximum overtide, and this impact was reflected in the changing the strength balance between river and tidal forcing.
  The location of maximal frication may have implications for the occurrence of maximal overtide amplitude. Elahi et al. (2020) explained that the place with equal river flow velocity and tidal velocity is an optimum condition for overtide amplitude in the case of Ganges Delta. Inspired by that, more general analysis will be added in the revised manuscript to further quantify the controlling impact of friction on the peak overtide amplitude.
  
  (5) How is the M4 identified from the model? Or which term represents the M4 in the model?
  A: M₄ tide was not imposed at the seaside boundary in the model simulations. However, M₄ was generated and detected inside the estuary owing to the river-enhanced nonlinear dynamics. The M₄ amplitude is analyzed by a harmonic method based on the modeled time series of water levels and currents. The M₄ component was not included in the Eqs. 3 and 4, because the equations were used to explain how overtide could be generated via the nonlinear terms (Eqs. 8 and 9).
  
  (6) What is the role of the tide in this study?
  A: In this work we focused on the impact of river forcing on the tides. Tides at the seaside boundary of the estuary are expected to influence the absolute overtide amplitude inside the estuary, but not on the non-dimensional results (M₄/M₂ ratio) presented in Figure 6. Including more tidal component, e.g., S₂, induces generation of other compound tide such as MS₄ that exhibit similar spatial variations as M₄(see Figure 3c). To test the sensitivity to tides, we ran additional simulations by increasing the M₂ amplitude at the boundary. Model results show increased overtide amplitude inside the estuary, and larger river discharge is needed to meet the R2T=1 threshold, which is understandable. These extra simulations and discussions will be included in the revised manuscript.
  
  Citation: https://doi.org/10.5194/hess-2021-75-AC4

Status: closed

RC1:
'Comment on hess-2021-75', Xiao Hua Wang, 08 Apr 2021

Review of ‘River-enhanced non-linear overtide variations in river estuaries’ by Guo et al (hess-2021-75)

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 non-linear 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 non-linear 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 493-503.

According to this paper, it seems to argue that it is the first time that this two-folds 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 Ganges-Brahmaputra-Meghna 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 non-linear 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 non-linear 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 Ganges-Brahmaputra-Meghna 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 493-503. 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 non-stationary harmonic analysis based on the Complex Demodulation method (Bloomfield, 2004) can be applied here to water level time series.

Line 458-459: Can you use your model results to explain why R2T ratio close to unity (not other values), benefits maximal M4 overtide generation?

Line 493-504: 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., Jalón-Rojas, I., Wang, X.H., Ritchie, E.A., 2020. Influence of Seasonal River Discharge on Tidal Propagation in the Ganges-Brahmaputra-Meghna Delta, Bangladesh. J. Geophys. Res. Ocean. 125, 1–19. https://doi.org/10.1029/2020JC016417

Citation: https://doi.org/10.5194/hess-2021-75-RC1
- AC1:
  'Reply on RC1', Leicheng Guo, 16 Apr 2021
  
  Response to online comments from Xiao Hua Wang, UNSW
  Review of ‘River-enhanced non-linear overtide variations in river estuaries’ by Guo et al (hess-2021-75)
  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 non-linear 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 non-linear 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 493-503.
  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 Ganges-Brahmaputra-Meghna 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 above-mentioned nonlinear overtide changes were predominantly reported in large, river- and tide-influenced estuaries and deltas, but less in many other tide-dominated estuaries with relatively smaller river discharge. We think that it maybe because the river discharge in tide-dominated 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 tide-dominated 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 two-folds 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 Ganges-Brahmaputra-Meghna 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 Ganges-Brahmaputra-Meghna 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 non-linear 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 non-linear 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 Ganges-Brahmaputra-Meghna 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 493-503. 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 (low-frequency) 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 high-frequency 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 Ganges-Brahmaputra-Meghna 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 river-tide 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 non-stationary 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 458-459: 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 two-fold 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 tide-averaged sediment transport, we had also detected that the tide-averaged sediment transport flux (induced by river flow, tidal asymmetry, and river-tide interactions) tends to be maximal when the river-enhanced residual current velocity equals to the magnitude of the tide-induced current velocity (e.g., the velocity amplitude of M₂ 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 493-504: This is not correct. See the above major comments.
  A: The texts in this paragraph has been thoroughly rephrased as 'Note that the above-mentioned nonlinear overtide changes were predominantly reported in large, river- and tide-influenced estuaries and deltas, but less in many other tide-dominated estuaries with relatively smaller river discharge. We think that it maybe because the river discharge in tide-dominated 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 tide-dominated estuaries are relatively shorter in physical length compared with tidal wavelength, hence the tidal river-tidal 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., Jalón-Rojas, I., Wang, X.H., Ritchie, E.A., 2020. Influence of Seasonal River Discharge on Tidal Propagation in the Ganges-Brahmaputra-Meghna 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.
  
  Citation: https://doi.org/10.5194/hess-2021-75-AC1
  - 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 555-558, 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.
    
    Citation: https://doi.org/10.5194/hess-2021-75-RC2
    
    AC2: 'Reply on RC2', Leicheng Guo, 28 May 2021
    
    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 555-558, 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.
    A: We have looked at the paper Elahi et al. (2020) carefully and was indeed inspired and have more thoughts on this issue. Firstly, Elahi et al. (2020) adopted the decomposition method proposed in Buschman et al. (2009) to calculate the friction under different river discharge (Eq. 1-4 in Elahi et al., 2020), and then used the decomposition to explain the overtide changes. However, it is noteworthy that the method was originally proposed to examine subtidal friction changes, in order to explain subtidal water level variations related to low-frequency tides like MSf etc. The method is not to explain the dynamics related to high-frequency tidal changes like M₄, because the energy transfer to the higher and lower-frequency tidal components by friction differs greatly.
    Secondly, the optimum condition proposed in Elahi et al. (2020) was a balance between the river-induced residual velocity and tidal velocity components. It explains the location of the maximum overtide amplitude within an estuary under specififed river discharge. The residual velocity increases in the landward direction, while the tidal velocity decreases. Thus the location of the balance varies for different river discharge scenarios. In other words, this balance may not be reached at the same time throughout the entire estuary. The optimum river discharge condition thus is not the same at different locations. As the overtide amplitude varies significantly along an estuary, a local optimum balance is not necessary to indicate the overall overtide generation and dissipation for the entire estuary as a whole. In this work, we looked at the integrated overtide energy by taking the entire estuary into consideration, and found a universal maximal threshold. In this case we use the river discharge to tidal discharge ratio at the mouth section of the estuary as an indicator of the optimum threshold for the estuary as a unit system. A R2T ratio=1 at the mouth of an estuary characterizes identical river flow velocity and tidal component velocity at the mouth (as that in Elahi et al., 2020). Overall, Elahi et al. (2020) examined the local optimum conditions while this work looked at the global maximum. As tidal waves are long waves, the integrated optimum and local peaks in overtide amplitude are simply two different ways of considering the variations in the relationship. In the revision, we will have a look at the spatial variations of the friction and the local overtide optimum conditions as suggested by the reviewer to see the differences.
    
    Citation: https://doi.org/10.5194/hess-2021-75-AC2
RC3:
'Comment on hess-2021-75', Anonymous Referee #2, 27 Apr 2021
This manuscript discusses the effect of river-tide 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 tide-river 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 u|u| 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 491-492 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 554-556: ‘In this work we see that the quadratic bottom stress term also leads to significant M4, through river-tide 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 friction-dominated 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 184-195 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 11-13 (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:

u|u| = 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 11-13 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. 12-13). 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 184-191: I don’t think this gives a proper reflection of the literature. Some of the analytical or semi-analytical 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 226-241: 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 263-264: 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. 11-13.

Section 3.1: I missed the calibration or setting of friction parameter. How was this done?

Ln 360-363: 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 424-435: 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 525-527: you should either prove this or don’t mention it.

Ln 531-535: I can’t follow this. Again this refers to the local discussion I commented on earlier.
Citation: https://doi.org/10.5194/hess-2021-75-RC3
- AC3:
  'Reply on RC3', Leicheng Guo, 28 May 2021
  RC3: 'Comment on hess-2021-75', Anonymous Referee #2, 27 Apr 2021
  This manuscript discusses the effect of river-tide 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.
  A: The reviewer's concerns are noted. However, they are too vague and open ended for us to be able to respond directly to their concerns. In this work, there are two main findings: one is the spatially non-uniform and non-linear changes in overtide amplitude in response to different river discharges, and one is the explanation of the integrated maximal overtide energy using a threshold of R2T=1. The first finding can be partially validated by similar results detected in Amazon (Gallo and Vinzon, 2005), in Columbia Estuary (Jay et al., 2014), Changjiang Estuary (Guo et al., 2015), St. Lawrence Estuary (Matte et al., 2013), and Ganges Delta (Elahi et al., 2020). These case studies only show tidal changes under one or two river discharge conditions, while the modeling work in our study extends this to a range of river discharges from 0 to 90000 m³/s, thus providing a more complete picture. We have also argued in the discussion section why similar phenomenon have not widely reported in many other tide-dominated (small) estuaries given tides are a very basic research theme that have been studied for centuries. "We think that it maybe because the river discharge in tide-dominated estuaries is comparatively 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 tide-dominated 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."
         As to the second main finding, it is mainly derived based on the model results in this work, which will inevitably depend on the model settings. However, we note that the findings Elahi et al. (2020), examining the Ganges, are consistent with our finding. Elahi et al. (2020) suggested an optimal balance featured by equal river-induced mean current velocity and tidal-induced current velocity magnitude, whereas we found that R2T=1 when looking at the mouth section, suggesting a level of consistency between the two approaches. To illustrate the validity of this finding, we are modeling more estuaries with varying shape and geometry and will include the results in the revised work as supporting information. Preliminary indications from the additional runs suggest that the results do not change the findings or conclusions of the main text.
  
  (2) The conclusions about the spatial characteristics of M4 are not supported by the results, which are integrated over the length of the channel.
  A: This maybe a misunderstanding. In the Conclusion section, we had stated that 'While the principal M₂ tide is increasingly dissipated as the R2T ratio increases from zero, the M₄ overtide amplitude decreases in the upper part of estuaries but increases in the lower part of estuaries'. This statement can be clearly seen from Figure 3b and Figure 4c.
         Following that, we concluded that 'With increasing R2T ratio, the total energy of M₄ overtide integrated throughout the estuary first increases and reaches a peak when the R2T ratio approaches unit.' This latter statement refers to the integrated results, which is clearly supported by Figure 6.
         The above two statements are not the same and should not be conflated.
  
  (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.
  A: The reviewer's concerns are noted. However, they are too vague and open ended for us to be able to respond directly to their concerns. It has been widely accepted and understood that 1) the significant overtide M₄ in shallow water environments is generated by the nonlinear processes like advection and friction etc.; 2) the energy of M₄ derives from its parent tides M₂ (Parker, 1991; Wang et al., 1999 etc); and 3) river discharge enhances tidal dissipation of M₂ predominantly through the friction effect (Horrevoets et al., 2004). Hence it logically makes sense that the overtide will depend on the generation and dissipation, both of which depends on the nonlinear mechanisms.
  
  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.
  A: It is true that the locally generated M₄ tide propagates in the estuary as a wave. However, M4 was not imposed at the seaward boundary in this modeling study. As M₄ is generated inside the estuary, it propagates, but also continues to be generated by the transfer of energy from higher harmonics further upstream. Hence the M₄ amplitude does not necessarily reduce in the landward direction in the same way as M₂. Both model results and observed results in many actual estuaries confirm the spatial variations of the M₄ tide, first increasing from the river mouth and then decreasing in the landward direction, e.g., in the Amazon and Changjiang estuaries. We do not expect anything else than this behavior, as at the mouth the M₄ was not imposed at the boundary. The amplitude of M₄ tide is not merely determined by the local generation and we do not agree with reviewer's statement "locally strong generation of M₄ results in a locally strong M₄ with some spatial smoothing due to the propgation". The local generation will for sure also influence the M₄ amplitude further landwards.
         We see that in some very special cases, e.g. estuaries that are not dominated by friction, are shorter, or where wave reflection is important, as mentioned by the Reviewer, different patterns may indeed emerge. However, we make no claim that our results apply to these cases. We simply note that friction has been shown to be one of the most important mechanisms in damping incoming astronomical tides in many real-word estuaries. The overtide generation still occurs in a frictionless environment, but its amplitude becomes smaller and the nonlinear behavior of along-channel change in response to increasing river discharge does not show up (see Figure S4 in the Supporting Information). Wave reflection is not expected to be large in estuaries with high river discharge which will damp the incoming tides to a large degree. This study and the associated findings and statements should be applicable to large river estuaries with considerable river discharge variations, but maybe not to some special systems, such as a barrier dam within tidal wave limit that causes wave reflection etc.
  
  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 tide-river 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 u|u| 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.
  A: There maybe another misunderstanding. When we talked about dissipation, we mainly referred to the dissipation of M₂, not dissipation of M₄. As mentioned above, the M₄ is generated by the transfer of energy from M₂, the parent tide of M₄. So M₄ generation inside estuaries is one the dominant processes. Additionally, M₂ was predominantly dissipated inside estuaries owing to friction and river discharge, hence dissipation of the M₂ tidal energy in the landward direction will affect the energy available to M₄, and then affect the generation of M₄. This balance of M₂ dissipation and M₄ generation then controls the net amplitude of M₄.
  
  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.
  A: The local M₄ amplitude is indeed controlled by both the local generation and the propagation of its component generated in the more seaward domain. However, we did not 'draw conclusion on how the local R2T affect the local M₄ generation'. As can be seen in response to previous questions, in this study we firstly detected nonlinear variations of M₄ amplitude along the estuary under different river discharge, which depends on the local M4 generation and large-scale propagation. Following that, we integrated the M₄ energy throughout the estuary and found a maximal value when scaled with the R2T at the mouth section. Hence, the threshold R2T=1 is not a local influence, but represents the overall impact of river on tides and an intermediate balance between the strength of river and tidal forcing. The findings in this study do indeed mainly apply to long estuaries with friction as an important controlling mechanism in tidal wave propagation. We have clarified that in the discussion to get rid of such confusion, e.g., at the end of section 4.1, we added that 'Hence, the findings in this work mainly apply to long estuaries with significant river discharge and friction control on tidal propagation in which wave reflection is limited. '
  
  Ln 491-492 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.
  A: The sentence in line 491-492 was 'These field data and model results confirm that the findings regarding the spatial dependence of overtide on river discharge are likely to be ubiquitous for river estuaries.' It said nothing about tidal phase. We do not address the phase of M₄. To get rid of any possible confusion, we rephrased the sentence as 'These field data and model results confirm that the findings regarding the spatial dependence of overtide amplitude on river discharge are likely to be ubiquitous for river estuaries.'
  
  ln 554-556: ‘In this work we see that the quadratic bottom stress term also leads to significant M4, through river-tide interaction’: this is stated as the main novelty, but is not new.
  A: The whole sentence mentioned here was 'In this work we see that the quadratic bottom stress term also leads to significant M₄, through river-tide interaction, i.e., between a river-enhanced mean current and M₂ current.' We did not emphasize that this is the main novelty. To get rid of any potential confusion, the sentence is rephrased as 'Additionally, the quadratic bottom stress term also leads to significant M₄, through river-tide interaction, i.e., between a river-enhanced mean current and M₂ current (Wang et al., 1999).'
  
  Section 3.2: I don’t see the hypothesis underlying this section. Your case without convergence still features a friction-dominated 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.
  A: Section 3.2 was dedicated to showing the possible impact of width convergence on the tidal changes. After comparing the results in Figures 3 (convergent estuary) and 4 (rectangular estuary), we see that the longitudinal changing behaviors of both M₂ and M₄ are similar, other than the firstly amplified M₂ and larger M₄ amplitude in the convergent estuary. The incoming M₂ tide was predominantly damped inside the rectangular estuary. The latter differences are understandable because width convergence enhances tidal amplification. Given the spatial changing behaviors are the same for both the convergent and non-convergent estuaries, the discussion in the sections following Section 3.2 are based on the results in the non-convergent estuary, in order to isolate the impact of width variations and consequent tidal wave amplification. Overall we think the message in Section 3.2 is helpful to indicate the sensitivity of the overtide changes to different shaped estuaries.
  
  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 184-195 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.
  A: The reviewer did not point out and we do not see why Eq. 3 and 4 are inconsistent. In lines 184-195, the point being made is that analytical solutions of governing tidal dynamic equations may not fully capture the nonlinear dynamics, which are important for overtide generation. The analytical solutions adopted scaling analysis in order to simplify the governing equations, e.g., linearize the friction term. The scaling analysis mentioned here is not to evaluate the relative importance of M₄ to M₂. In this work we used a numeric model which includes all the nonlinear processes represented in the governing equations and can thereby reproduce their non-uniform spatial changes under different river-tidal conditions. But we accept that both analytical and numerical approaches have their own advantages.
         ln line 283, the whole sentence was 'The bottom friction term is approximately expanded into a bottom shear stress term and a term considering depth variations, as the two terms on the right hand of Eq. (7), respectively, according to Godin and Martinez (1994), given the tidal amplitude to water depth ratio (|η|/h) is generally smaller than one', so the scaling was used to show that the tidal wave amplitude is relatively smaller with respect to the water depth, so the bottom friction can be expanded as that shown in Eq.7 according to Godin and Martinez (1994). Given M₂ is the main tidal constituent, adding another M₄ will not fundamentally change the tidal amplitude to water depth ratio (|η|/h).
         The Eq. 3 and 4 did not include the M₄ component because they were used to represent the condition at the river mouth. In this study only an astronomical M₂ tidal constituent was prescribed at the seaside boundary and M₄ tides are generated inside the estuary. Moreover, including the M₄ component in Eq. 3 and 4 will not fundamentally change the results in Eq. 5-7 because M₂ amplitude is a factor larger than that of M₄. To get rid of the confusion, we have added a sentence in the revision 'Note that the internally generated M₄ component is so far not included and shown in Eq. 3 and 4, but this does not fundamentally change the results, given that the amplitude of M₂ is much larger than M₄.'
  
  Eq 10 and therefore 11-13 (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:
  u|u| = 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 11-13 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. 12-13). 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.
  A: The above questions focus on the decomposition method and the associated results in Figure 5. We have given more thoughts to the methods to compare the contribution of different nonlinear terms (Eq. 11-13) and we noticed that the issue of different scaling may undermine the quality of the analysis. Given the results in Figure 5 are largely in line with previous studies- showing that the friction term dominates the overtide change behavior- this does not add much new understanding. In this revision we have therefore removed the discussion on the contribution of the nonlinear terms (section 3.3 and Figure 5) and associated texts on the methods to get rid of confusion. The main findings of this work are then 1) highlighting the spatially nonlinear changes of overtide amplitude under varying river discharges and 2) identification of the threshold condition controlling the maximal overtide generation.
  
  Other remarks
  Ln 184-191: I don’t think this gives a proper reflection of the literature. Some of the analytical or semi-analytical 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.
  A: Thank you for the suggestion. We have revised the sentences into 'Analytical models usually assume tidal propagation as a single wave component, based on simplified tidal dynamic equations after scaling analyses, e.g., adopting a linear assumption or a nonlinear expansion of the friction term (Green, 1837; Kreiss, 1957; Jay, 1991; Parker, 1991; Friedrichs and Aubrey, 1994; van Rijn, 2011). Further improved analysis method can take into account of more than one tidal component and robust approximation of the nonlinear friction term and their impact on morphodynamic changes (Lanzoni and Seminara, 1998; Ridderinkhof et al., 2014; Alebregtse and de Swart, 2016; Chernetsky et al., 2010; Dijkstra et al., 2017).' The suggested references are included in the reference.
  
  Ln 226-241: 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?
  A: Indeed, the tidal hydrodynamics within an estuary can adapt to any arbitrary bathymetry. However, this may only be stable if the bed is erodible. That is why, in this work we first run a morphodynamic simulation to obtain a close-to-equilibrium bed profile for the defined tidal forcing conditions and then use the equilibrium bed profile in further tidal simulations. We believe this treatment is preferred to minimize the impact of an arbitrary bathymetry on the tidal dynamics, thus highlighting the effect of river discharge. Even though, our previous study (Guo et al., 2016) did show that the overall overtide changes in response to river discharge variations exhibit similar spatial pattern under initial and equilibrium bed profiles (see Figure R1 below).
  
  Figure R1. Longitudinal changes of M₂ and M₄ amplitude modeled in a 560-km long rectangular estuary forced by different river discharge under primary initial bed profile (dotted lines) and equilibrium bed profiles (solid lines) (from Guo et al., 2016)
  
  Ln 263-264: how can the depth be constant after the morphodynamic computation?
  A: It is a mistake in using the word 'constant'. What we mean was that the water depth will not change over time anymore. So the sentence is corrected as 'As the bed level is prescribed as an equilibrium profile, the water depth h remains unchanged'.
  
  Eq 8: brackets missing in the cosine.
  A: A bracket is added.
  
  Eq 13: is a minus missing in the first term or did I get confused with the sign of u0?
  A: The minus sign depends on the reference direction of current. A minus sign for u₀ is needed if assuming flooding currents are defined positive. So a minus sign was added in Eq. 4. The Eqs. 11-13 were removed in this revision to get rid of confusion.
  
  Ln 332: why do you need the M4 amplitude and phase? It does not appear in Eq. 11-13.
  A: Both M₂ and M₄ amplitude and phases are output of the harmonic analysis, but only M₂ harmonics are used in Eq. 11-13.
  
  Section 3.1: I missed the calibration or setting of friction parameter. How was this done?
  A: As a schematized model was employed in this study, we did not do calibration of the friction parameter against field data. In the section 2.1, we added that 'A uniform friction coefficient of Chézy value of 65 m^1/2/s was used, which leads to predominantly landward tidal damping within the schematized estuary, comparable to what is observed in reality (see Figures 1 and 3).'
  
  Ln 360-363: why include S2 now? This seems inconsistent with the entire method section.
  A: Yes, the major part of this study focused on the M₂ and M₄ changes. The S₂ tide was mentioned here to show that other compound tide like MS₄, which was generated by M₂-S₂ interaction, exhibits similar changes as M₄ in response to river discharge (Figure 3c).
  
  Fig 5a: you find a contribution from bottom friction while there is no discharge. Is this the effect of tidal return flow?
  A: Yes, the u₀ velocity is nonzero in the no river discharge scenario, owing to a return flow under the predominantly progressive wave condition.
  
  Ln 495 ‘majority of estuaries’: I don’t think this is obviously true. This would at least need a reference.
  A: The sentence was rephrased and references are included to get rid of confusion. In the revised work, the new sentence is 'Note that the above-mentioned nonlinear overtide changes were predominantly reported in large, mixed river-tide energy estuaries and deltas, e.g., Amazon (Gallo and Vinzon, 2005), Changjiang (Guo et al., 2015), and Ganges (Elahi et al., 2020), but less in many other tide-dominated estuaries with relatively smaller river discharge, although the importance of overtide in controlling tidal asymmetry and residual transport were widely reported, e.g., in Humber in the UK (Winterwerp, 2004) and western Scheldt Estuary in the Netherlands (Wang et al., 2002).'
  
  Ln 424-435: 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.
  A: We aimed to stress the role of nonlinearity of the friction in controlling river-tide interaction, thus a comparison of the model results between the nonlinear (default) and linear friction was conducted. As the discussion of the different contribution of the nonlinear terms (original section 3.3) was removed in this section, the texts on the discussion of the sensitivity to linear friction have been moved into section 4.2 as part of the discussion instead of result.
  
  Ln 525-527: you should either prove this or don’t mention it.
  A: The changing behavior of MS₄ was shown in Figure 3c. The sentence was removed to be more focused on overtide M₄.
  
  Ln 531-535: I can’t follow this. Again this refers to the local discussion I commented on earlier.
  A: The sentence 'In the lower part of estuaries where the incident tides are less dissipated, river flow plays a more important role in reinforcing the effective bottom friction. As a result, dissipation of the principal tide is more prominent in the upper part of estuaries, while tidal energy transfer and overtide generation are more substantial in the lower part of estuaries (Figures 8)' was to stress the different impact of river flow on the tidal changes in seaward and landward part of an estuary. As we had argued in the response to previous questions, the spatial variations of overtide M₄ along the estuary are a combined result of dissipation of M₂ and generation of M₄. To further clarify this point, the sentence is rephrased in this revision as 'In the seaward part of estuary where the incident tidal waves are less dissipated by increasing river discharge, the role of the river discharge in reinforcing the effective bottom friction and enhancing overtide generation is more pronounced. Whereas, in the landward part of the estuary, river discharge plays a prominent role in causing more dissipation of the principal tide. As a result, the tidal energy available to be transferred to overtide is constrained, thus overtide amplitude reduces with increasing river discharge in the landward regions (Figures 8).
  
  Citation: https://doi.org/10.5194/hess-2021-75-AC3
RC4:
'Comment on hess-2021-75', 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?

Citation: https://doi.org/10.5194/hess-2021-75-RC4
- AC4: 'Reply on RC4', Leicheng Guo, 28 May 2021
  
  RC4: 'Comment on hess-2021-75', 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?
  A: Indeed the schematized model can be used to explore the sensitivity to other convergence rates. In this work, we mainly present results in a rectangular channel and a convergent channel, with dimension comparable to the Changjiang Estuary. Gallo and Vinzon (2005) have modeled the tides in the Amazon Estuary and their modeling results are cited and highlighted in this work (see Figure 6). Moreover, Elahi et al. (2020) also undertook a similar modeling study of the Ganges Delta and showed consistent results with this work. Instead of focusing on specific sites, we used schematic models to investigate the broad implications for similar estuaries. The main objective was to reveal the general pattern of the relative changes in the overtide amplitude in response to different magnitude of river discharge. We have not sought to reproduce tidal propagation changes in specific system, in which the absolute amplitude and longitudinal changes will very much depend on the regional changes in channel width, depth and bed slope etc. The channel pattern in the schematized model, e.g., reduced and increased M4 amplitude in the landward and seaward parts of estuaries, is overall consistent with the actual data analysis in Amazon, Changjiang, Columbia etc., justifying the model results in this work. As a response to this question and the following one, we have constructed more models with varying width convergence rates and preliminary simulations show similar patterns as found for the rectangular channel, although the absolute amplitude varies slightly. These extra model results will be included in the revised supporting information of the revised manuscript. We have added a sentence to note the similar pattern as convergence varies in section 2.2 'Sensitivity tests examining the influence of the rate of convergence indicate that the patterns are similar to those for the rectangular channel case, with some small variations in the amplitudes of the various components'.
  
  (2) To what extent the R2T value (= 1) is appliable since morphology plays a role?
  A: Morphology is indeed an important factor affecting tidal propagation and changes in estuaries. However, the difference in morphology will not alter the main findings in this work, because: 1) the model results in both rectangular and convergent channels show similar changing patterns of M₄ (see Figures 3 and 4), and 2) we use an equilibrium bed profile for tidal simulations, and have shown that the bed profile will not change the patterns of longitudinal change (see Figure R1 above). We will include more model results in Figure 6 to confirm the R2T threshold in the revised manuscript.
  
  (3) Give more details about the benefits of maximal overtide amplitude.
  A: In the first paragraph of Section 4.3 we discuss the relevance and implications of the maximal threshold conditions. The finding of a maximal overtide under intermediate river flow condition (or equal river and tidal forcing strength) has broad implications for estuary management. For example, one implication is for the residual sediment transport which tends to be the largest when the river flow velocity is equal to a tidal velocity. Such a residual sediment transport tends to lead to the largest seaward sediment flushing and consequently the deepest equilibrium depth (Guo et al., 2016).This threshold condition occurs because river discharge dissipates tides and the impact of river-tide interaction on creating residual sediment transport changes nonlinearly with rising river discharge. These arguments have been incorporated into the discussion section of the revised manuscript.
  
  (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?
  A: River discharge affects the friction mainly via the velocity magnitude. River discharge enlarges the current velocity, and the friction the moving flow feels accordingly. The impact of river flow in inducing more dissipation of astronomical constituents was exerted via the effective friction term (Horrevoets et al., 2004).
  Morphology affects the longitudinal variations of river flow velocity and landward decrease in tidal prism and tidal velocity. The morphology can therefore influence the location of maximum overtide, and this impact was reflected in the changing the strength balance between river and tidal forcing.
  The location of maximal frication may have implications for the occurrence of maximal overtide amplitude. Elahi et al. (2020) explained that the place with equal river flow velocity and tidal velocity is an optimum condition for overtide amplitude in the case of Ganges Delta. Inspired by that, more general analysis will be added in the revised manuscript to further quantify the controlling impact of friction on the peak overtide amplitude.
  
  (5) How is the M4 identified from the model? Or which term represents the M4 in the model?
  A: M₄ tide was not imposed at the seaside boundary in the model simulations. However, M₄ was generated and detected inside the estuary owing to the river-enhanced nonlinear dynamics. The M₄ amplitude is analyzed by a harmonic method based on the modeled time series of water levels and currents. The M₄ component was not included in the Eqs. 3 and 4, because the equations were used to explain how overtide could be generated via the nonlinear terms (Eqs. 8 and 9).
  
  (6) What is the role of the tide in this study?
  A: In this work we focused on the impact of river forcing on the tides. Tides at the seaside boundary of the estuary are expected to influence the absolute overtide amplitude inside the estuary, but not on the non-dimensional results (M₄/M₂ ratio) presented in Figure 6. Including more tidal component, e.g., S₂, induces generation of other compound tide such as MS₄ that exhibit similar spatial variations as M₄(see Figure 3c). To test the sensitivity to tides, we ran additional simulations by increasing the M₂ amplitude at the boundary. Model results show increased overtide amplitude inside the estuary, and larger river discharge is needed to meet the R2T=1 threshold, which is understandable. These extra simulations and discussions will be included in the revised manuscript.
  
  Citation: https://doi.org/10.5194/hess-2021-75-AC4

Leicheng Guo, Chunyan Zhu, Huayang Cai, Zheng Bing Wang, Ian Townend, and Qing He

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https://doi.org/10.5194/hess-2021-75-supplement

Leicheng Guo, Chunyan Zhu, Huayang Cai, Zheng Bing Wang, Ian Townend, and Qing He

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Short summary

Overtide is a shallow water tidal component and its interaction with astronomical tides induces tidal wave deformation, which is an important process that controls sediment transport. We use a numerical tidal model to examine overtide changes in estuaries under varying river discharges and find spatially nonlinear changes and the threshold of an intermediate river that benefits maximal overtide generation. The findings inform management of sediment transport and flooding risk in estuaries.


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