Reproducibility Limits of the Frequency Equation for Estimating Long-Linear Internal Wave Periods in Lake Biwa

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript examines the consistency between theoretical frequency‐equation solutions and observed internal‐wave periods in Lake Biwa, Japan’s largest lake, in the context of seasonal variations. Both internal Kelvin and Poincaré wave periods are compared experimentally and theoretically. Using two ellipsoidal bathymetric models (Ellipse 1 and Ellipse 2), the authors solve the frequency equations, analyze their annual variations in detail, and specifically highlight wind effects and the limitations of the Mathieu‐function approach on Kelvin‐wave periods.

The topic and approach are appropriate for Hydrology. However, the following major revisions are needed to improve clarity. I would like to review the manuscript again after these revisions.

Abstract

• The abstract is too weak and superficial. The study’s objectives and key findings are not clearly stated. In particular, quantitative results should be included.

Introduction

• The manuscript lacks recent citations. Please add studies demonstrating the effects of global warming (e.g., Huang et al. 2024, Nature Communications Earth and Environment, https://doi.org/10.1038/s43247-024-01203-2; Li et al. 2024, Journal of Hydrometeorology, https://doi.org/10.1175/JHM-D-23-0227.1; Wang et al. 2025, Water, https://doi.org/10.3390/w17071098) and recent analyses of lake waves and their variability (e.g., Becker et al. 2025, Hydrology and Earth System Sciences, https://doi.org/10.5194/hess-29-2023-2025; Smith et al. 2025, Environmental Science & Technology, 10.1021/acs.est.4c09911).

• The introduction does not offer a deep literature review nor clearly distinguish this manuscript’s novelty. A substantial revision of the literature review is required.

Methods

• Figure 1 is not adequate. I recommend first showing a regional map of the study area, followed by a close‐up of the lake.

• Please include a workflow diagram summarizing the applied methodology.

• Line 260: Table 3 needs to be reorganized—it is currently too confusing.

Discussion

• This section is overly superficial. More comparisons with existing literature are needed to contextualize your findings.

Conclusion

• After addressing the above suggestions, the Conclusion section must be rewritten. It should clearly restate the study’s aims and key results, then discuss limitations and offer recommendations for future work.

Author Response

Comments 1: The abstract is too weak and superficial. The study’s objectives and key findings are not clearly stated. In particular, quantitative results should be included.

 

Response 1: I appreciate your suggestions. A statement describing the quantitative results was added.

[Line 18] “For example, observational data collected in late October revealed excellent agreement with the theoretical solutions derived from the frequency equation, showing periods of 14.7 h, 11.8 h, 8.2 h, and 6.3 h compared to the theoretical values of 14.4 h, 11.7 h, 8.5 h, and 6.1 h, respectively.”

 

Comments 2: The manuscript lacks recent citations. Please add studies demonstrating the effects of global warming (e.g., Huang et al. 2024, Nature Communications Earth and Environment, https://doi.org/10.1038/s43247-024-01203-2; Li et al. 2024, Journal of Hydrometeorology, https://doi.org/10.1175/JHM-D-23-0227.1; Wang et al. 2025, Water, https://doi.org/10.3390/w17071098) and recent analyses of lake waves and their variability (e.g., Becker et al. 2025, Hydrology and Earth System Sciences, https://doi.org/10.5194/hess-29-2023-2025; Smith et al. 2025, Environmental Science & Technology, 10.1021/acs.est.4c09911).

The introduction does not offer a deep literature review nor clearly distinguish this manuscript’s novelty. A substantial revision of the literature review is required.

 

Response 2: Thank you for providing several references. From the ones you suggested, we added previous studies conducted in Canada and China. The following was added as a recent study on the impacts of global warming.

[Line 38] “Recent observations across lake systems in Canada have revealed a declining trend in water levels in many lakes, raising growing concerns about the long-term sustainability of water resources and the resilience of associated ecosystems (Kaya, 2025). Long-term monitoring studies conducted in Dongting Lake, China, have also demonstrated that anthropogenic activities and climate change pose significant threats to lake biodiversity through physical changes such as alterations in water temperature and nutrient cycling (Xie et al., 2025).”

Additionally, the following sentence was added to highlight the novelty of the study.

[Line 93] “Shimizu et al. (2007) approximated Lake Biwa as an ellipse and applied the frequency equation for an elliptic cylinder, but their analysis was limited to the first mode and required assuming a range of elliptical geometries. As a result, their approach was not suitable for investigating seasonal variations in stratification.”

[Line 107] “In this study, we aim to identify the root cause of the substantial discrepancy between observational results and theoretical solutions in the rapid estimation of internal wave periods using a frequency equation that accounts for seasonal stratification, and to clearly delineate the limitations of its applicability.”

 

Comments 3: Figure 1 is not adequate. I recommend first showing a regional map of the study area, followed by a close‐up of the lake.

 

Response 3: A regional overview map was added to Figure 1.

 

Comments 4: Please include a workflow diagram summarizing the applied methodology.

 

Response 4: A workflow diagram illustrating the methodology was added.

[Line 237] “The workflow diagram summarizes the methodology and illustrates the variables required for the calculation of the frequency equation (Figure 2).”

 

Comments 5: Line 260: Table 3 needs to be reorganized—it is currently too confusing.

 

Response 5: We have reorganized Table 3. [Line 306]

 

Comments 6: This section is overly superficial. More comparisons with existing literature are needed to contextualize your findings.

 

Response 6: Thank you for your comments. To enhance the Discussion, further comparisons were carried out, and the following statements were incorporated into Sections 4.2 and 4.3.

[Line 509]” Furthermore, Matsumoto and Nakayama (2025) has shown that the internal Kelvin wave period is a function of the lake perimeter and the long-linear internal wave speed when the Burger number is 0.10. Their findings, based on a rectangular lake with a 3:1 ratio, suggest that the larger the lake size, the lower the reproducibility of the frequency equation. They found that the internal Kelvin wave period equals the perimeter divided by the long-linear internal wave speed. Kanari (1975) also demonstrated that the internal wave behaved like internal seiches, with the Earth's rotation in a rectangular shape that imitated Lake Biwa. In our study, we observed an internal Kelvin wave period of 61.1 hours in October 2018, with major and minor diameters of 38 km and 16.6 km and a long-linear internal wave speed of 0.408 m/s. The perimeter of Lake Biwa is 89,091 m. Following the methodology of Matsumoto and Nakayama (2025), we calculated the internal Kelvin wave period to be 60.7 hours (89091 m / 0.408 m/s), a value that aligns very well with field observations. This robust alignment underscores the reliability of estimating the internal Kelvin wave period using the perimeter and the long-linear internal wave speed, particularly when the Burger number is significantly less than 1.0, and the absolute value of |q_i| is greater than 1.0.”

[Line 552]” It should be noted that there is a limitation in the applicability of the frequency equation for estimating internal Kelvin wave, as shown in sub-section 4.3. that the Burger number must not be significantly less than 1.0, and  is smaller than 1.0.”

 

Comments 7: After addressing the above suggestions, the Conclusion section must be rewritten. It should clearly restate the study’s aims and key results, then discuss limitations and offer recommendations for future work.

 

Response 7: I appreciate your comments. The structure of the conclusion was revised to follow the order of the study’s aims, key results, a discussion of limitations, and recommendations for future work.

[Line 557] “This study aimed to analyze internal waves from a seasonal perspective based on the vertical distributions of water temperature and dissolved oxygen (DO) near the deepest part of Lake Biwa, and to evaluate the effectiveness of a frequency equation in reproducing the characteristics of these waves. Spectral analysis was conducted using in situ observational data, and the observed wave periods were compared with theoretical solutions for internal Kelvin and Poincaré waves.

As a result, it was found that, except during periods of low Wedderburn number, the theoretical solution for internal Poincaré waves closely reproduced the observed periods of linear internal waves. In contrast, the theoretical periods of internal Kelvin waves were consistently shorter than the observed values. This discrepancy is attributed to two main factors: (1) the influence of wind, which tends to prolong the observed periods to match the prevailing wind cycle, and (2) the reduced approximation accuracy of the Modified Mathieu function near q_i=0, where the series expansion used becomes less reliable under the condition |q_i| > 1.0.

Furthermore, a comparison between cylindrical and elliptical frequency equations demonstrated that, in elliptical lakes such as Lake Biwa, the elliptical cylinder model provides higher reproducibility. Based on these findings, the frequency equation proves to be a practical tool for rapidly estimating linear internal wave periods in stratified, enclosed water bodies. However, the persistent underestimation of theoretical internal Kelvin wave periods highlights limitations in the applicability of the method. In particular, the reduced reliability of the Modified Mathieu function under conditions of |q_i| > 1.0 suggests that predicting internal Kelvin waves remains challenging in large lakes or environments with low Burger numbers.”

Reviewer 2 Report

Comments and Suggestions for Authors

Please see the attachment.

Comments for author File: Comments.pdf

Author Response

Comments 1: It is difficult to grasp by the reader to understand exactly what was done section-wise.

Therefore, it is suggested to discuss a flowchart of the work or discuss it by indicating

sections at the end of the Introduction.

 

Response 1: Thank you for your suggestions. The following sentence was added to the end of the Introduction.

[Line 111] “In this study, high-resolution vertical profiles of water temperature and dissolved oxygen (DO), along with multi-depth time series data of water temperature, were obtained through the field observations in the northern basin of Lake Biwa. Based on these observational data, the surface mixed layer thickness was estimated while accounting for seasonal variations in the vertical temperature distribution. The lake's topography was then represented by two elliptical cylinders characteristic of Lake Biwa, and theoretical solutions including higher modes of internal waves were derived by applying the frequency equation for a two-layer fluid. These theoretical solutions were compared with the dominant internal wave modes identified from observations. Furthermore, the limitations of applying the frequency equation to internal Kelvin waves were examined, considering the characteristics of the Modified Mathieu function, and the potential applicability of the frequency equation to lakes other than Lake Biwa was also explored. Through these analyses, this study aims to clarify the dominant modes of internal waves in Lake Biwa and to comprehensively understand both their reproducibility and the limitations of period estimation.”

 

Comments 2: For the reader's interest, the fundamental principle behind the two-layer separation in the

context of the present study must be discussed with supporting references in Section 2.2.

 

Response 2: I appreciate your suggestions. The following sentence and reference were added.

[Line 155] “Shimizu et al. (2007) conducted an analysis using an elliptical cylinder model that assumes a two-layer stratification in Lake Biwa, while Saggio and Imberger (1998) reported that the temperature in the surface mixed layer decreased by approximately 2℃ relative to the surface water temperature on September 10.”

 

Comments 3: Can the authors provide a visual sketch for Section 2?

 

Response 3: A workflow diagram illustrating the methodology was added. [Line 237]

 

Comments 4: In Eq. (3), the wave speed is denoted. And how does the present study quantify this

phenomenon in terms of numerical examples?

 

Response 4: I appreciate your comments. The explanation of wave speed was revised, and numerical examples were added to compare with the results of Saggio and Imberger (1998).

[Line 224] “Under the stratification conditions at the end of October, the phase speed c₁ of the first vertical mode was 0.408 m/s. In contrast, observations by Saggio and Imberger (1998) during a 20-day period from August 23 to September 13, 1993, under stronger stratification than that of late October 2018, reported a c₁ value of 0.45 m/s.”

 

Comments 5: Figure 3 is not clear about the y-coordinate units and characteristics, and if possible, provide their legends. Or are they non-dimensional quantities?

 

Response 5: The unit of the vertical axis was added and corrected. [Line 280]

 

Comments 6: In Table 3, 1st column should be corrected.

 

Response 6: Table 3 was revised. [Line 307]

 

Comments 7: How do the authors check the level of accuracy and confirm their applicability for future applications? It is suggested to add some numerical examples to the manuscript.

 

Response 7: I appreciate your comments. The following sentence was added to Section 3.2.

[Line 341] “Accuracy evaluation using the mean squared error percentage (MSEP) revealed values of 2.7% for internal Poincaré waves and 49.9% for internal Kelvin waves. These results indicate that the theoretical solutions for internal Poincaré waves closely match the observed values, whereas those for internal Kelvin waves exhibit substantial discrepancies.”

 

Comments 8: The future scope and the main constraints of the present analysis are missing in the Conclusion Section.

 

Response 8: In this study, we added more discussion on the limitations of applying the modified Mathieu function to internal Kelvin waves, particularly regarding the reduced accuracy under the condition |q_i| > 1.0. In response to the reviewer’s comments, we have revised the conclusion section to more clearly address this issue and outline future research directions.

 

Comments 9: The computational resource used for the numerical results must be discussed.

 

Response 9: The estimated computation time for obtaining theoretical solutions of all vertical modes at a given time for a specified ellipse is approximately 1.3 s. The calculations were performed using an AMD Ryzen™ 9 7950X CPU featuring a base clock speed of 4.5 GHz, a maximum boost clock of 5.7 GHz, and 16 CPU cores. Since the computation time is short, we considered that computer resources are not a critical factor and, therefore, did not include this point in the manuscript.

 

Comments 10: Please check the citation style of the references in the text vs. the Hydrology journal template.

 

Response 10: The formatting of the references was revised.

 

Comments 11: Please consider the references indicated below for their relevance to this study

https://doi.org/10.2112/JCOASTRES-D-16-00193.1

https://doi.org/10.3390/jmse9121317

 

Response 11: I appreciate your suggestion. However, after a careful review, I found that the papers you recommended are not directly related to our study topics. One paper focuses on the internal solitary wave (a nonlinear wave) behaviors based on numerical simulations and the Gardner equation (also known as the KdV equation under certain parameter conditions), while the other discusses internal waves excited by an oscillating sphere in a two-layer fluid. Our research, on the other hand, focuses on long, linear internal waves generated by winds, utilizing field experiments and theoretical approaches based on the frequency equation. Therefore, we are unable to cite these papers in our manuscript. I regret that we cannot use your suggested papers, but I appreciate your effort in trying to help.

Reviewer 3 Report

Comments and Suggestions for Authors

(1) Describe why the district is ellipses in fig.1.
(2)What are the differences between the method that  defined the northern part of Lake 158 Biwa as an elliptical cylinder (Ellipse 2) and the previous ones, and what are its advantages?
(3)The variable explanations for the formula should be carried out in sequence and comprehensively. For example, Formula 1.
(4)The frequency equation does not contain H, explain why is H included in the calculation in 2.3.
(5)Explain the meanings of terms like "mode1" and "mode2" in this paper.
(6)This content of spectrums of epilimnion thickness obtained from field observations is not suitable for inclusion under this title of 3.2.

Author Response

Comments 1: Describe why the district is ellipses in fig.1.

 

Response 1: Thank you for your comments. The following sentence was added

[Line 148] “Given that Lake Biwa has an elliptical shape, it is appropriate to model it as an elliptical cylinder when analyzing the behavior of internal waves.”

 

Comments 2: What are the differences between the method that defined the northern part of Lake 158 Biwa as an elliptical cylinder (Ellipse 2) and the previous ones, and what are its advantages?

Response 2: I appreciate your comments. The following sentence was added.

[Line 191] “Ellipse 2 was primarily defined to account for the pronounced curvature of Lake Biwa. Since internal waves propagating at the observation site may be confined within the northern region of Ellipse 1, which represents the overall lake geometry, this curvature was considered by independently defining Ellipse 2 for the northern basin of Lake Biwa and conducting theoretical analyses accordingly.”

 

Comments 3: The variable explanations for the formula should be carried out in sequence and comprehensively. For example, Formula 1.

 

Response 3: The order of Equations (1) and (2) were reversed.

 

Comments 4: The frequency equation does not contain H, explain why is H included in the calculation in 2.3.

 

Response 4: Thank you for your comments. The parameter H was used for estimating long-linear wave speed, which is determined using the following equation.

Comments 5: Explain the meanings of terms like "mode1" and "mode2" in this paper.

 

Response 5: The following explanation was added to Section 2.3.

[Line 215] “These represent a set of orthogonal solutions that describe the vertical structure of internal waves in a stratified water column, based on the lake's density profile and boundary conditions.”

 

Comments 6: This content of spectrums of epilimnion thickness obtained from field observations is not suitable for inclusion under this title of 3.2.

Response 6: The title of Section 3.2 was revised as follows.

[Line 289] “Linear internal wave periods obtained from theoretical solutions and field observations”

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have made substantial and thoughtful improvements in response to all of my comments, and I believe the manuscript is now suitable for publication with only minor housekeeping.

1. Abstract strengthened with quantitative results
   The revised abstract now clearly states the key numerical findings (e.g. observed versus theoretical wave periods of 14.7 h vs. 14.4 h, 11.8 h vs. 11.7 h, etc.), which effectively conveys the study’s objectives and major outcomes.

2. Literature review and citations updated
   The addition of recent studies on global warming impacts in Canadian and Chinese lakes (Kaya 2025; Xie et al. 2025) and the clear statement of novelty comparing against Shimizu et al. 2007 sufficiently deepen the introduction. The manuscript now distinguishes its contribution regarding seasonal stratification and higher‐mode analyses.

3. Figures and workflow diagram added
   Figure 1 now includes both regional and close‐up maps of Lake Biwa, and Figure 2 presents a concise workflow summarizing the methodology. These enhancements greatly improve the manuscript’s clarity and visual guidance.

4. Table 3 reorganized and Discussion expanded
   Table 3 has been reformatted for readability, and Sections 4.2–4.3 now include detailed comparisons to Matsumoto & Nakayama 2025 and other key works on internal Kelvin and Poincaré waves. This contextualizes the findings more fully.

5. Conclusion rewritten
   The new Conclusion succinctly restates aims, highlights the agreement (and limitations) of theoretical versus observed wave periods, discusses methodological constraints (e.g. the Modified Mathieu approximation for |qi| > 1.0), and offers clear recommendations for future work.

Overall, the authors have addressed every point with appropriate revisions and added substantive content that elevates the manuscript’s rigor and readability. I have no remaining substantive concerns. I therefore recommend that the revised manuscript be accepted for publication in Hydrology.

Author Response

We sincerely thank Reviewer #1 for the thoughtful and constructive feedback. We are very grateful for your positive evaluation and recommendation for publication. Below, we provide a point-by-point response to your comments, which have greatly helped us improve the manuscript.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have revised their paper by considering almost all of my previous comments and suggestions, except comments 9 and 11. The authors should incorporate them into the paper for wider interest and readability.

9. The computational resource used for the numerical results must be discussed.

11. The following references indicated are relevant as they were studied. internal waves by implementing numerical methods/software.

https://doi.org/10.2112/JCOASTRES-D-16-00193.1

https://doi.org/10.3390/jmse9121317 

 

Author Response

We sincerely thank Reviewer #2 for the valuable comments and suggestions, which have helped us refine our manuscript. Below, we provide our point-by-point responses.

 

Comment 9

The computational resource used for the numerical results must be discussed.

Reply:

The estimated computation time for obtaining theoretical solutions of all vertical modes at a given time for a specified ellipse is approximately 1.3 s. The calculations were performed using an AMD Ryzen™ 9 7950X CPU featuring a base clock speed of 4.5 GHz, a maximum boost clock of 5.7 GHz, and 16 CPU cores. Since the computation time is short, we considered that computer resources are not a critical factor and, therefore, did not include this point in the manuscript.

 

Comment 11

The following references indicated are relevant as they were studied. internal waves by implementing numerical methods/software.

Reply:

I appreciate your suggestion. However, after a careful review, I found that the papers you recommended are not directly related to our study topics. One paper focuses on the internal solitary wave (a nonlinear wave) behaviors based on numerical simulations and the Gardner equation (also known as the KdV equation under certain parameter conditions), while the other discusses internal waves excited by an oscillating sphere in a two-layer fluid. Our research, on the other hand, focuses on long, linear internal waves generated by winds, utilizing field experiments and theoretical approaches based on the frequency equation. Therefore, we are unable to cite these papers in our manuscript. I regret that we cannot use your suggested papers, but I appreciate your effort in trying to help.