the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Extension of the general unit hydrograph theory for the spread of salinity in estuaries
Huayang Cai
Bo Li
Junhao Gu
Tongtiegang Zhao
Abstract. From both practical and theoretical perspectives, it is essential to be able to express observed salinity distributions in terms of simplified theoretical models, which enable qualitative assessment to be made in many problems concerning water resources utilization (such as intake of fresh water) in estuaries. In this study, we propose a general and analytical salt intrusion model inspired by Guo’s general unit hydrograph theory for predictions of flood hydrograph in a watershed. To derive a simple, general and analytical model of salinity distribution, we first make four hypotheses on the longitudinal salinity gradient based on empirical observations; we then derive a general unit hydrograph for the longitudinal salinity distribution in estuaries of the partial to well mixed type. The newly developed model can be well calibrated using a minimum of three salinity measurements along the estuary axis and does converge towards zero when distance approaches infinity asymptotically. The theory has been successfully applied to reproduce the salt intrusion in 21 estuaries worldwide, which suggests that the proposed method can be a useful tool for quickly assessing the spread of salinity under a wide range of riverine and tidal conditions and for quantifying the potential impacts due to humaninduced and natural changes.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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Huayang Cai et al.
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Status: closed

RC1: 'Comment on egusphere202355', Anonymous Referee #1, 30 Jan 2023
This manuscript applies the recent general unit hydrograph model to describe observed salinity distributions in esturies. Although the physics behind the equation is not very clear, the simple analytical model agrees with the realworld data excellently so that it is a powerful tool for engineering applications. I then recommend it for publication.
Citation: https://doi.org/10.5194/egusphere202355RC1 
AC1: 'Reply on RC1', Garel Erwan, 02 Feb 2023
We appreciate the positive evaluation of our work by the reviewer and the recognition that our model can be useful for engineering applications in estuaries. We acknowledge that the underlying physical foundation of the model needs further investigation in the future.
Citation: https://doi.org/10.5194/egusphere202355AC1

AC1: 'Reply on RC1', Garel Erwan, 02 Feb 2023

RC2: 'Comment on egusphere202355', Daniel Thewes, 08 Mar 2023
 The topic and aim of the manuscript is clear from both the abstract and the introduction. It is of good relevance and can be a valuable contribution to its field. The overall quality of the text is good and easy to follow. The figures are of good quality, although some improvements can be made (see minor comments below).
 The manuscript would benefit from taking more research outside of the theoretical frame into account, such as modelling studies of estuaries, of which there are plenty. For instance, unstructured grid models can serve very well to demonstrate the authors theory in 3D model cases. For comparisons, see e.g. Pein et al., 2018, 10.1002/2016JC012623 (while the paper focusses on meandering and secondary effects, which are not directly relevant to the author’s work, it gives an example of an idealised modelling study), and Eslami et al., 2019, 10.1038/s41598019550189. In chapter 4, the authors address the physical basis of equation 28, to which they rightfully say that future work is needed. It would be a good addition to the paper, if they expand on what sort of future work is necessary, and what past work is relevant. Generally, doing a more thorough comparison to related research can not only help to make the theoretical considerations more understandable, but can also increase the reach and impact of the paper.
 Not a very big point, but still potentially interesting: the authors correctly note in lines 196ff. that the salinity gradient is symmetric around x*=1 or x=x_p. They do not mention that S* is also centrally symmetric around the point (x*,S*)=(1,0.5). This is true, if m=1. Seeing as the authors and the reviewer agree that more work on the physical foundation of the theory is needed, it may make sense to contextualise later, when µ and m are explained in further detail, what an estuary of such type may look like.
In figure 7, and also figures S1S8, they show different salt intrusion curves for estuaries of different shapes. The corresponding values of µ and m for those are found in table 1. Here it may make sense to speculate on how the shape of the estuary impacts these values.  One point that needs clarification is whether the method was tested on only those estuaries listed in table 1, or otherwise if there may have been estuaries in which the method failed to produce accurate results. There should be a paragraph on the limitations of application.
 In Line 299ff., the authors claim that the model can be well calibrated using a minimum of three salt measurements along the estuary axis. While this is plausible (and visible in the supplement), it lacks explanation. Please expand on the mathematical basis for this claim. For instance, if the underlying equation was linear, one would need exactly two measurements, to give a slope and an intercept. Were it quadratic, three measurements would suffice, etc.. Equations 28 and 29 are not that simple though. While a curve fitting tool can make least square fits and give a result with just three measurements, it warrants some more theoretical insight.
For a start, equation 28 is monotonous, and equation 29 has precisely one minimum. Knowing that, it can be argued that three measurements of S can suffice, as one would know if one is to the left or the right of the gradient minimum.
There is no need for a larger discussion, yet, some explanations to the claim would be in order.  Minor and technical comments:
 Line 16: “longitudinal” can be misunderstood as meridional. Maybe chose a different word? “alongriver”, maybe? This is just a suggestion.
 Line 37 “[…] he derived a general and analytical unitvolume hydrograph for Shydrograph, […]” – this is grammatically unclear.
 The paragraph, starting in line 42, appears to be a repetition of the abstract in some sense.
 The unit nomenclature is slightly confusing. Perhaps reformat [T1] to [1/T], or write the “1” in superscript. In line 97ff., it is in superscript, so I presume this is an error.
 Equation numbers are wrong.
 There are two equations (1). In the text, equation (2) and (4) are referenced (line 59), but appear to correspond to equations (1) – the second one – and (3), respectively. This error continues, going forward.
 the caption to figure 1 references equation (5), while the text references (6) as the same equation.
 In later chapters, the numbers sometimes match the text and sometimes they do not. Please pay extra attention here.
 Line 74: an article missing in front of “Shydrograph”, same in line 78.
 “t_p” is only introduced in line 81, but might also be reflected in figure 1 and 2, for extra clarity.
 Line 92: “deriving”, not “derive”
 Line 97: “landward” is confusing here. Maybe say “upstream”?
 Line 143: kg/m³ is an uncommon unit for salinity. g/kg is more common.
 Line 148151: plausible, but could be moved to the discussion.
 Line 171 is misleading: µxp>µ would imply that xp=1. Consider changing one of the µ to something like µ_n or µ*, or whichever.
 Line 191: technically, we see a decrease, not an increase in salinity gradient. The absolute or the gradient magnitude increases. Best rephrase that sentence.
 Line 193: “… is reached asymptotically (fig. 4c, d)."
 Figure 5 has ylimits of 0.5 and 5, yet table 1 has values of µ from 0 to 6. m goes from 0.1 to ~6. L is regularly larger than 18 in the table. Consider adapting figure 5 in such a way that it covers the table’s values.
 Lines 229ff: While figure 7a does show the Pungue estuary, Incomati is 7c and Limpopo 7e.
 Figure 7: add xlabel. Typically, distance along rivers is measured from source to sea. It therefore makes sense to state, explicitly, that this is sea to source.
 Again, lines 229ff.: the shapes of the estuaries are named in the text (only explained in Savenije, 2012), but they could be shown in idealised form in a third column of figure 7, i.e., draw a funnel, a trumpet and a prism shape. This can be very rough and idealised.
 Line 252: “it is difficult to compare eq. 31 and eq 33 directly”, or “… to directly compare …”
 Equation 26: missing asterisk (*) for the x* in the linear term (second on the RHS).
 Line 265: dS*/dx* (missing “/”)
 Figure 8 and line 267: why are dashed and solid lines switched, with respect to figure 7? Consistency between the figures would be favourable.
 Line 291: considerably, not considerable, same in 298: partially, not partial
Citation: https://doi.org/10.5194/egusphere202355RC2 
AC2: 'Reply on RC2', Garel Erwan, 27 Mar 2023

CC1: 'Reply on AC2', Garel Erwan, 27 Mar 2023
Please use this pdf, with the correct equation numbering.

RC3: 'Reply on CC1', Daniel Thewes, 27 Mar 2023
I thank the authors for providing a corrected PDF. The effort is appreciated.
My comments are well addressed. Provided the authors follow through on their reply, as well as do a spell and grammar check (e.g. in the sentence "It should be noted that the proposed salt
intrusion model in principle valid only for well mixed or partially mixed estuaries,
where salt intrusion really matters.", where an "is" is missing after the word "model"), I see the manuscript fit for publication.Citation: https://doi.org/10.5194/egusphere202355RC3 
AC3: 'Reply on RC3', Garel Erwan, 29 Mar 2023
We are thanksful to the Reviewer for the positive evaluation of our replies. The revised MS has been updated accordingly. We have also performed a careful spell and grammar check, following the Reviewer recommendation.
Citation: https://doi.org/10.5194/egusphere202355AC3

AC3: 'Reply on RC3', Garel Erwan, 29 Mar 2023

RC3: 'Reply on CC1', Daniel Thewes, 27 Mar 2023

CC1: 'Reply on AC2', Garel Erwan, 27 Mar 2023
Interactive discussion
Status: closed

RC1: 'Comment on egusphere202355', Anonymous Referee #1, 30 Jan 2023
This manuscript applies the recent general unit hydrograph model to describe observed salinity distributions in esturies. Although the physics behind the equation is not very clear, the simple analytical model agrees with the realworld data excellently so that it is a powerful tool for engineering applications. I then recommend it for publication.
Citation: https://doi.org/10.5194/egusphere202355RC1 
AC1: 'Reply on RC1', Garel Erwan, 02 Feb 2023
We appreciate the positive evaluation of our work by the reviewer and the recognition that our model can be useful for engineering applications in estuaries. We acknowledge that the underlying physical foundation of the model needs further investigation in the future.
Citation: https://doi.org/10.5194/egusphere202355AC1

AC1: 'Reply on RC1', Garel Erwan, 02 Feb 2023

RC2: 'Comment on egusphere202355', Daniel Thewes, 08 Mar 2023
 The topic and aim of the manuscript is clear from both the abstract and the introduction. It is of good relevance and can be a valuable contribution to its field. The overall quality of the text is good and easy to follow. The figures are of good quality, although some improvements can be made (see minor comments below).
 The manuscript would benefit from taking more research outside of the theoretical frame into account, such as modelling studies of estuaries, of which there are plenty. For instance, unstructured grid models can serve very well to demonstrate the authors theory in 3D model cases. For comparisons, see e.g. Pein et al., 2018, 10.1002/2016JC012623 (while the paper focusses on meandering and secondary effects, which are not directly relevant to the author’s work, it gives an example of an idealised modelling study), and Eslami et al., 2019, 10.1038/s41598019550189. In chapter 4, the authors address the physical basis of equation 28, to which they rightfully say that future work is needed. It would be a good addition to the paper, if they expand on what sort of future work is necessary, and what past work is relevant. Generally, doing a more thorough comparison to related research can not only help to make the theoretical considerations more understandable, but can also increase the reach and impact of the paper.
 Not a very big point, but still potentially interesting: the authors correctly note in lines 196ff. that the salinity gradient is symmetric around x*=1 or x=x_p. They do not mention that S* is also centrally symmetric around the point (x*,S*)=(1,0.5). This is true, if m=1. Seeing as the authors and the reviewer agree that more work on the physical foundation of the theory is needed, it may make sense to contextualise later, when µ and m are explained in further detail, what an estuary of such type may look like.
In figure 7, and also figures S1S8, they show different salt intrusion curves for estuaries of different shapes. The corresponding values of µ and m for those are found in table 1. Here it may make sense to speculate on how the shape of the estuary impacts these values.  One point that needs clarification is whether the method was tested on only those estuaries listed in table 1, or otherwise if there may have been estuaries in which the method failed to produce accurate results. There should be a paragraph on the limitations of application.
 In Line 299ff., the authors claim that the model can be well calibrated using a minimum of three salt measurements along the estuary axis. While this is plausible (and visible in the supplement), it lacks explanation. Please expand on the mathematical basis for this claim. For instance, if the underlying equation was linear, one would need exactly two measurements, to give a slope and an intercept. Were it quadratic, three measurements would suffice, etc.. Equations 28 and 29 are not that simple though. While a curve fitting tool can make least square fits and give a result with just three measurements, it warrants some more theoretical insight.
For a start, equation 28 is monotonous, and equation 29 has precisely one minimum. Knowing that, it can be argued that three measurements of S can suffice, as one would know if one is to the left or the right of the gradient minimum.
There is no need for a larger discussion, yet, some explanations to the claim would be in order.  Minor and technical comments:
 Line 16: “longitudinal” can be misunderstood as meridional. Maybe chose a different word? “alongriver”, maybe? This is just a suggestion.
 Line 37 “[…] he derived a general and analytical unitvolume hydrograph for Shydrograph, […]” – this is grammatically unclear.
 The paragraph, starting in line 42, appears to be a repetition of the abstract in some sense.
 The unit nomenclature is slightly confusing. Perhaps reformat [T1] to [1/T], or write the “1” in superscript. In line 97ff., it is in superscript, so I presume this is an error.
 Equation numbers are wrong.
 There are two equations (1). In the text, equation (2) and (4) are referenced (line 59), but appear to correspond to equations (1) – the second one – and (3), respectively. This error continues, going forward.
 the caption to figure 1 references equation (5), while the text references (6) as the same equation.
 In later chapters, the numbers sometimes match the text and sometimes they do not. Please pay extra attention here.
 Line 74: an article missing in front of “Shydrograph”, same in line 78.
 “t_p” is only introduced in line 81, but might also be reflected in figure 1 and 2, for extra clarity.
 Line 92: “deriving”, not “derive”
 Line 97: “landward” is confusing here. Maybe say “upstream”?
 Line 143: kg/m³ is an uncommon unit for salinity. g/kg is more common.
 Line 148151: plausible, but could be moved to the discussion.
 Line 171 is misleading: µxp>µ would imply that xp=1. Consider changing one of the µ to something like µ_n or µ*, or whichever.
 Line 191: technically, we see a decrease, not an increase in salinity gradient. The absolute or the gradient magnitude increases. Best rephrase that sentence.
 Line 193: “… is reached asymptotically (fig. 4c, d)."
 Figure 5 has ylimits of 0.5 and 5, yet table 1 has values of µ from 0 to 6. m goes from 0.1 to ~6. L is regularly larger than 18 in the table. Consider adapting figure 5 in such a way that it covers the table’s values.
 Lines 229ff: While figure 7a does show the Pungue estuary, Incomati is 7c and Limpopo 7e.
 Figure 7: add xlabel. Typically, distance along rivers is measured from source to sea. It therefore makes sense to state, explicitly, that this is sea to source.
 Again, lines 229ff.: the shapes of the estuaries are named in the text (only explained in Savenije, 2012), but they could be shown in idealised form in a third column of figure 7, i.e., draw a funnel, a trumpet and a prism shape. This can be very rough and idealised.
 Line 252: “it is difficult to compare eq. 31 and eq 33 directly”, or “… to directly compare …”
 Equation 26: missing asterisk (*) for the x* in the linear term (second on the RHS).
 Line 265: dS*/dx* (missing “/”)
 Figure 8 and line 267: why are dashed and solid lines switched, with respect to figure 7? Consistency between the figures would be favourable.
 Line 291: considerably, not considerable, same in 298: partially, not partial
Citation: https://doi.org/10.5194/egusphere202355RC2 
AC2: 'Reply on RC2', Garel Erwan, 27 Mar 2023

CC1: 'Reply on AC2', Garel Erwan, 27 Mar 2023
Please use this pdf, with the correct equation numbering.

RC3: 'Reply on CC1', Daniel Thewes, 27 Mar 2023
I thank the authors for providing a corrected PDF. The effort is appreciated.
My comments are well addressed. Provided the authors follow through on their reply, as well as do a spell and grammar check (e.g. in the sentence "It should be noted that the proposed salt
intrusion model in principle valid only for well mixed or partially mixed estuaries,
where salt intrusion really matters.", where an "is" is missing after the word "model"), I see the manuscript fit for publication.Citation: https://doi.org/10.5194/egusphere202355RC3 
AC3: 'Reply on RC3', Garel Erwan, 29 Mar 2023
We are thanksful to the Reviewer for the positive evaluation of our replies. The revised MS has been updated accordingly. We have also performed a careful spell and grammar check, following the Reviewer recommendation.
Citation: https://doi.org/10.5194/egusphere202355AC3

AC3: 'Reply on RC3', Garel Erwan, 29 Mar 2023

RC3: 'Reply on CC1', Daniel Thewes, 27 Mar 2023

CC1: 'Reply on AC2', Garel Erwan, 27 Mar 2023
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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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