the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 06 Jun 2024
A rainfall-tracking travel time distribution model to quantify mixing and storage release preference in a large shallow lake by two-year stable isotopic data
Abstract. Lake Taihu is the largest eutrophic lake in China that is shallow and connected to a dense river network. Severe eutrophication is frequently observed in Lake Taihu due to excess pollutant loadings. Understanding water cycle dynamics is essential for investigating this problem. The travel time distribution (TTD), residence time distribution (RTD) and storage selection (SAS) function describing how water is stored, mixed and released in the lake provide fundamental information on water cycle dynamics. In this study, a rainfall mixing model is established and coupled with the age master equation model to estimate the time-variant TTDs and RTDs of rainwater, river water and all water in Lake Taihu based on the two-year high-resolution isotopic data. In the rainfall mixing model, a novel rainfall mixing factor is introduced to quantify the mixing of rainwater with older lake water. The estimated range of travel time varies between 2–4 months and 7–9 months, depending on lake volume. Lake Taihu shows an inverse storage effect, i.e., the release preference shifts toward younger water when lake volume is large. The results of rainfall mixing model reveal distinct patterns and control factors between the TTDs of rainwater and river water, with rainwater contributing to 15 %–25 % of outflow. Then, the SAS functions of rainwater and river water are analyzed, which are controlled by different source zones and flow patterns. Temporal variation of spatial distribution of deuterium isotope composition illustrates storage release preference is controlled by the variation of horizontal flow paths and velocities in the lake.
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RC1: 'Comment on egusphere-2024-1513', Anonymous Referee #1, 11 Jul 2024
Review of “A rainfall-tracking travel time distribution model to quantify mixing and storage release preference in a large shallow lake by two-year stable isotopic data” by Mao et al submitted to Hydrology and Earth System Sciences
In my opinion, the manuscript will be a great addition to the journal and be of interest to its readers. Mao et al targets the variations in time and in space of residence time and travel time at Lake Taihu, China, by processing and analyzing measurements of deuterium in the lake. Temporal scales in lakes have been correlated to water quality issues, therefore knowing how long a water parcel remains in the lake depending on the meteorological conditions will provide a clearer picture on how to better manage water quality in Taihu Lake. For example, their results show that depending on the inflow of water (through rivers and rain) the travel time can decrease by half.
However, I recommend the manuscript to be brought back to the authors for revisions for the following reasons:- There have been publications mentioning Lake Taihu’s residence time. I am thinking for example of Xu et al 2009 (https://doi.org/10.4319/lo.2010.55.1.0420), Xu et al 2015 (https://doi.org/10.1021/es503744q) and Paerl et al 2014 (https://doi.org/10.1371/journal.pone.0113123). Interestingly the value of residence time is not consistent among these publications, but they remain below the 1 year mark. I think the manuscript would benefit by including a comparison and a possible explanation of the discrepancy between the publications.
- There are portions of paragraph in Section 5 that appear contradictory to the results described in the abstract. For example, at the end of Section 5.2.1, lines 477-478, Mao et al states that “older water parcels may have greater chances leaving the system than younger water parcels”. However, in the abstract it is stated that “the release preference shifts toward younger water when lake volume is large”. A reorganization of the subsections would help to make things clearer.
- In the introduction, lines 51-52, the authors mention that only Smith et al. 2018 applied the travel time distribution theory to a lake. Unless the authors were implicitly focusing on the usage of isotopic compound, I disagree with the authors. Temporal scales have been addressed in previous work in lakes. For example, Rueda and Cohen, 2005 (https://doi.org/10.4319/lo.2005.50.5.1638) looked at variation in space and time of residence time in an embayment of Lake Ontario, Canada.
- The authors mention the spatio-temporal variations of the time scales are controlled by horizontal velocities on several occasions (in the abstract and in the conclusion), they did not discuss past work identifying the flow field at Lake Taihu. The manuscript would benefit from including comparisons with past work on average flow field in Lake Taihu Liu et al. 2018 (doi:10.3390/w10060792) shows for example a rather complex average flow field, which would definitely hinder the flow of water from inflow to outflow.
I noticed a couple of typos in the text:
- Figure 2: There is no sampling point 2. I assume it is supposed to be where sampling point 32 is.
- Line 304: If I understand properly, s_rain(t,tau) is the volume of rain water aged tau, and not s(t,tau).
- Line 499: please replace “board range” with “broad range”.
- Line 505: please replace “Lake” with “lake”.
Citation: https://doi.org/10.5194/egusphere-2024-1513-RC1 - AC1: 'Reply on RC1', Rong Mao, 15 Jul 2024
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RC2: 'Comment on egusphere-2024-1513', Paolo Benettin, 19 Jul 2024
This manuscript introduces a travel time model that deals with the problem of tracking water age in hydrologic systems characterised by multiple input fluxes. This is relevant for lake studies, which typically have 2 inputs (river inflow and rainfall). The authors apply the model to the case study of Lake Taihu, China, and calibrate the model parameters against 24 monthly values of mean isotope composition of the lake water. The authors use the calibrated model to discuss water age dynamics and rain/river water partitioning within the lake and the output fluxes.
I anticipate that, while I appreciate this work, I am afraid there are major technical issues in the solution proposed by the authors that prevents publication of the current version of the paper, but I think that similar results produced under alternative – and possibly simpler solutions may be worth of publication.
The authors touch on a very interesting and challenging problem that is multiple-input tracking in lumped models. Most of the water transit time literature (see the review work that myself and many colleagues have written in 2022, https://doi.org/10.1029/2022WR033096) dealt with systems with one single input, i.e. rainfall. Therefore, the effort put in place by the authors fills an important gap and is to be credited. The text, while needing some English proofread, is understandable and the approach undertaken by the authors appears rigorous and is described in full detail.
A key point of the paper is a new solution to the tracking of multiple sources within a lumped model. The authors call their solution “Rainfall mixing” or “rainfall tracking”. I think this solution is problematic on some fronts:
- The manuscript seems to often confound space with age. Lumped age models do not explicitly account for space. The effects of physical processes (like advection and dispersion) can be effectively reproduced by using a lagged transit time distribution or SAS function, but those processes are not directly modelled. Similarly, water age models do not account for any physical mixing within the storage and they only prescribe how the output fluxes remove waters of different ages from the storage (which is why the community tends to speak about “random sampling” rather than “well mixing”). The idea that rainfall water is well mixed with pre-existing lake water while river water is not is unsuitable to this lumped framework and should rather be translated into SAS language and equations. Translating the different transport processes of two very different inputs into a single SAS function may be challenging and highlights the limitations of lumped water age models.
- The rainfall mixing approach is difficult to understand and after reading it multiple times I still am not sure I could follow all the steps. My understanding is that ultimately the “candidate” beta-shaped SAS functions are checked at any time step and if some constrain is not respected, they are modified. The effect of this transformation is difficult to follow, but I think it generally increases the amount of young water released to the outflow, to effectively simulate the “preferential” release of rain water. The manuscript should clarify the effect of this modification more explicitly, for example by showing some examples of candidate and modified SAS function or by showing a simulation with and without those constraints. I think it is also important that this more complex approach is justified in terms of model performance, i.e. that the model with the modification performs significantly better than a “traditional” model.
- Monthly time steps for the solution of the water age balance using the Euler Forward scheme is potentially coarse. I invite the authors to verify that the numerical accuracy is not compromised by the use of large time steps.
I believe these major issues need to be addressed before discussing additional minor comments. However, I reiterate that an improved or simplified multiple-input tracking system would likely make the paper worthy of publication.
Citation: https://doi.org/10.5194/egusphere-2024-1513-RC2 -
AC2: 'Reply on RC2', Rong Mao, 24 Jul 2024
We sincerely appreciate your valuable comments. After several days of careful analysis, we are confident in informing you that we have addressed all of your major comments, and will make thorough revisions based on them. Your valuable suggestions have greatly helped improve the quality of our manuscript.
Below is a brief summary of the modifications we will make in the revised manuscript. For detailed replies to your comments, please refer to the attached file.
(1) We derived a new analytical solution for the SAS function that successfully integrates the transport of rainwater and river water into a single SAS function, as you suggested. Therefore, steps for checking the 'candidate' beta-shaped SAS functions can be removed. We defined a new variable, the "event rain water threshold age", which is crucial for determining the shape of the new SAS function.
With this new SAS function, the TTD, RTD, and SAS function can be calculated using methods from traditional TTD literature. After that, the volumes of rain water and river water in the lake and outflows can be partitioned using the rainfall mixing factor. Therefore, the new SAS function will make our model simpler, easier to understand, and more straightforward to implement.
(2) Regarding the issue of large time steps, we have conducted an error analysis for different time steps: dt=1,0.5,0.25, 0.125 months. The deuterium concentration increases slightly with smaller time steps. Based on the results of error analysis, we will revise our manuscript using the model with a smaller time step of dt=0.25 month and add a paragraph to discuss the error caused by different time steps.
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EC1: 'Reply on AC2', Damien Bouffard, 30 Jul 2024
Dear Authors,
I would appreciate it if you could elaborate further on your specific responses to the first and second points raised by the reviewers. As it stands, it is difficult to understand how the manuscript will be modified based on these two highly important points.
Thank you.
Citation: https://doi.org/10.5194/egusphere-2024-1513-EC1 -
AC3: 'Reply on EC1', Rong Mao, 30 Jul 2024
Dear Editor,
We apologize for not clearly describing in AC2 how we plan to revise our manuscript based on RC2's first and second comments. Our previous responses to these two comments were included on the fourth and fifth pages of the attachment in AC2. To make our responses and revisions clearer, we have rewritten a detailed reply as follows:
Overall, we have proposed a new SAS function based on the original model. This new SAS function will effectively address the technical issues raised in RC2's first and second comments and will meet RC2's requirement for a simpler model, as stated in the last sentence of RC2's comments. The following are our point-by-point responses and revisions regarding the first and second comments (RC2’s comments are in italics):
1. Point-by-point reply to the first comment
We fully accept RC2's first comment and will reply and make revisions from the following three aspects:
(1) “The manuscript seems to often confound space with age. Lumped age models do not explicitly account for space. The effects of physical processes (like advection and dispersion) can be effectively reproduced by using a lagged transit time distribution or SAS function, but those processes are not directly modelled. Similarly, water age models do not account for any physical mixing within the storage and they only prescribe how the output fluxes remove waters of different ages from the storage (which is why the community tends to speak about “random sampling” rather than “well mixing”).”
Reply and revisions: In the methodology section (section 3) of our manuscript, the discussion on spatial mixing is mostly found in section 3.2.2. In section 3.2.2, we admit that we often conflated the spatial mixing processes of rain water and river water with the lumped model. To clarify our model's description, we will divide section 3.2.2 into two subsections: 3.2.2.1 and 3.2.2.2. Section 3.2.2.1 will focus on the spatial mixing process of rain water and river water in Lake Taihu, without discussing the lumped model. Section 3.2.2.2 will discuss how we developed the lumped rainfall mixing model, based on this spatial mixing process. This will prevent confusion between the spatial mixing process and the lumped model in our manuscript. It's important to emphasize that our lumped model is based on the real mixing processes in Lake Taihu, making it physically meaningful. Moreover, our rainfall mixing model is essentially a lumped model, using the rainfall mixing factor to encapsulate the spatial rainfall mixing process. The rainfall mixing factor quantifies the fraction of rainwater aged \tau in the mixed water that is older than or equal to \tau.
(2) "The idea that rain water is well mixed with preexisting lake water while river water is not is unsuitable to this lumped framework and should rather be translated into SAS language and equations."
Reply and revisions: In the new Section 3.2.2.2, we will translate the rainfall mixing process into SAS language: this means that rainwater aged \tau in outflow rivers is randomly sampled from the rain water aged \tau in the mixed lake water that is older than or equal to \tau. This random sampling process is based on the fact that the rainfall mixing factors for rain water aged \tau in the mixed lake water older than or equal to \tau and in the outflow water older than or equal to \tau are the same
Additionally, in our original manuscript, we have already expressed the rainfall mixing model in the form of lumped equations, that is, in equations (11)-(13). These equations are lumped equations because they do not include flow details. The quantities in equations (11)-(13) are lumped quantities that can be linked to the lumped water age model presented in equation (7). In the revised manuscript, we will present these three equations in Subsection 3.2.2.2 to emphasize the lumped nature of the equations in the rainfall mixing model.
(3) "Translating the different transport processes of two very different inputs into a single SAS function may be challenging and highlights the limitations of lumped water age models."
Reply and revisions: Our new SAS function effectively addresses this comment. As noted in AC2, this new function successfully integrates the transport of rain water and river water into a single SAS function, making our model simpler, easier to understand, and more straightforward to implement. This new SAS function will be introduced in Section 3.2.3, which will cover the incorporation of the rainfall mixing model with the age master equation model. The original text in Section 3.2.3 will be removed.
2. Point-by-point reply to the second comment
"The rainfall mixing approach is difficult to understand and after reading it multiple times I still am not sure I could follow all the steps. My understanding is that ultimately the “candidate” beta-shaped SAS functions are checked at any time step and if some constrain is not respected, they are modified. The effect of this transformation is difficult to follow, but I think it generally increases the amount of young water released to the outflow, to effectively simulate the “preferential” release of rain water. The manuscript should clarify the effect of this modification more explicitly, for example by showing some examples of candidate and modified SAS function or by showing a simulation with and without those constraints. I think it is also important that this more complex approach is justified in terms of model performance, i.e. that the model with the modification performs significantly better than a “traditional” model."
Reply and revisions: This issue will no longer exist in our revised manuscript, as this comment was based on our old method in old section 3.2.3. In the revised manuscript, we will use our newly proposed SAS function, which will be introduced in section 3.2.3. Since our new SAS function provides direct calculations, RC2' confusion about the old SAS function's calculation methods will no longer be an issue. Specifically, the new SAS function eliminates the need to check the “candidate” beta-shaped SAS functions at each time step.
In our old method, we needed to check the beta-shaped SAS function because it might not accurately capture the fraction of event young rain water in outflow rivers (see Figure 1 in the attachment file of AC2). Instead, by introducing the event rain water threshold age in our new SAS function, we avoid this step. Furthermore, using this new SAS function, as introduced in AC2, ensures that the model implementation remains consistent with the “traditional” lumped water age model, making our approach simpler and easier to understand.
Finally, I hope this response clarifies our replies in AC2.
Citation: https://doi.org/10.5194/egusphere-2024-1513-AC3
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AC3: 'Reply on EC1', Rong Mao, 30 Jul 2024
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EC1: 'Reply on AC2', Damien Bouffard, 30 Jul 2024
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