Numerical Modeling of the Three-Dimensional Wave-Induced Current Field

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The proposed study focuses on the three-dimensional numerical modeling of coastal zone hydrodynamics, employing a recently developed analytical model that incorporates a three-dimensional formulation of radiation stress. By examining cross-shore and alongshore bathymetric variability, the work evaluates how variations in bottom friction, internal volumetric current transport, free surface elevation, and velocity distributions influence model predictions. Comparing 2D and 3D model solutions under increasingly complex bathymetric conditions, the authors find that while both approaches align under idealized, homogeneous settings, significant differences emerge when bathymetry becomes more complex—especially in the velocity fields and transport dynamics. Furthermore, alongshore variability introduces additional lateral gradients that are typically overlooked in simpler models, underscoring the risk of underestimating key hydrodynamic variables. The findings highlight that accounting for three-dimensional bathymetric variability is pivotal for enhancing model accuracy in coastal engineering and environmental management contexts, with the improved 3D framework offering clearer insights into circulation patterns and potential rip current formations in real-world coastal environments.

Overall, I appreciated how the introduction was clearly written, and the methodology, while generally well explained, occasionally came across as convoluted in its presentation. The results provided do seem to corroborate the authors main findings. However, some of the figures could benefit from improved editing to better convey the data. Despite these strengths, I am not entirely convinced that all relevant processes, particularly those related to turbulence and wave-current interactions in the boundary layer, are adequately accounted for in the model—an omission that may have important implications for the predicted current velocity profile. Moreover, the article should be better structured and could benefit from some rearranging of the subsections.

Main comments:

General comment: While the authors have introduced a novel turbulence modeling approach, the current formulation appears to omit certain critical processes—particularly those involving the interaction of waves and currents in the nearshore boundary layer. In high-energy surf zones, wave-induced turbulence, vortex shedding, and complex shear layers can significantly modify velocity distributions, sediment transport, and overall mixing. Relying solely on linear wave theory and simplified eddy viscosity coefficients in the horizontal and vertical directions may therefore underestimate key turbulence mechanisms and fail to capture the full range of wave–current interactions. As discussed in Marino et al.; 2024 (https://doi.org/10.5194/os-20-1479-2024) and Peruzzi et al.; 2021 (https://doi.org/10.1017/jfm.2021.605), more advanced closures or coupling strategies can substantially enhance realism and predictive capabilities. Incorporating these insights would likely improve the accuracy of the model by ensuring that turbulence generation and dissipation pathways. If they are not included, please discuss why.

Line 67: The numerical modeling approach, while featuring a novel approach to turbulent stress modeling, excludes higher-order wave-current and current-current interaction terms, and relies on linear wave theory—an approach that could limit fidelity in energetic, nonlinear surf zones. Although consistency in numerical formulations is important, real-world conditions frequently demand capturing the complexities of higher-order interactions. Future iterations integrating these elements could significantly enhance the model’s accuracy and applicability. Please discuss.

Line 96: Modeling turbulence through Reynolds-averaged equations with separate horizontal and vertical eddy viscosity coefficients is a practical approach. However, in highly stratified or near-surf zones, turbulence can exhibit strong variability not always captured by fixed or simple parameterizations—potentially affecting the accuracy of velocity and stress predictions. Please discuss.

Line 158: While the zero-equation turbulence model and the distinction between horizontal and vertical viscosity coefficients are practical for many coastal flows, they may oversimplify turbulence dynamics—especially under wave-breaking or strongly sheared conditions where Boussinesq’s hypothesis might be less accurate. Adopting more advanced turbulence closures could potentially refine the representation of Reynolds stresses and improve the fidelity of wave-induced current predictions. Please discuss.

Section 3.1.1 and 3.2.1 are repeated! The same chapters are presented twice.

Line 301: The adoption of the Arakawa C-type staggered grid is a solid choice for coastal hydrodynamic simulations, offering improved numerical stability and accuracy. However, in domains with complex or rapidly varying boundaries, specialized treatment for boundary conditions may be required to fully leverage the grid’s advantages. Can yoy please discuss about it?

Figure 3: Labels are too small

Section 4.2: This section should not stay in the Results and discussion section.

Author Response

Thank you very much for taking the time to review this manuscript.  I appreciate your kind words on the introduction and methodology. Please find the detailed responses below and the corresponding revisions highlighted in the re-submitted files.

Comment 1: Line 67: The numerical modeling approach, while featuring a novel approach to turbulent stress modeling, excludes higher-order wave-current and current-current interaction terms, and relies on linear wave theory—an approach that could limit fidelity in energetic, nonlinear surf zones. Although consistency in numerical formulations is important, real-world conditions frequently demand capturing the complexities of higher-order interactions. Future iterations integrating these elements could significantly enhance the model’s accuracy and applicability. Please discuss.

Response 1: I thank the Reviewer for noting the omission of higher‑order wave–current and current–current interaction terms and the reliance on linear wave theory. In the revised manuscript, the Analytical Model section has been expanded to (i) discuss the consequences of excluding higher interaction terms for energetic, nonlinear surf zones; (ii) outline how future work will incorporate third‑ and fourth‑order Stokes expansions to recover these effects under breaking‑wave conditions; and (iii) clarify the limitations of Airy (linear) wave theory in capturing nonlinear velocity products and roller dynamics, as well as potential remedies via weakly nonlinear or fully nonlinear wave formulations.

Comments 2: Line 96: Modeling turbulence through Reynolds-averaged equations with separate horizontal and vertical eddy viscosity coefficients is a practical approach. However, in highly stratified or near-surf zones, turbulence can exhibit strong variability not always captured by fixed or simple parameterizations—potentially affecting the accuracy of velocity and stress predictions. Please discuss.

Comments 3: Line 158: While the zero-equation turbulence model and the distinction between horizontal and vertical viscosity coefficients are practical for many coastal flows, they may oversimplify turbulence dynamics—especially under wave-breaking or strongly sheared conditions where Boussinesq’s hypothesis might be less accurate. Adopting more advanced turbulence closures could potentially refine the representation of Reynolds stresses and improve the fidelity of wave-induced current predictions. Please discuss.

Response 2 and 3:  I appreciate the comment on the use of Reynolds‑averaged equations with fixed horizontal and vertical eddy viscosities. The Analytical Model section now includes a dedicated discussion of how this zero‑equation closure may underrepresent turbulence variability in highly stratified or near‑surf regions, and it outlines prospects for adopting two‑equation or Reynolds‑stress transport closures to dynamically adjust eddy viscosities to local shear and buoyancy conditions.

Comments 4: Section 3.1.1 and 3.2.1 are repeated! The same chapters are presented twice.

Response 4: I thank the Reviewer for catching this oversight. In the revised manuscript, the duplicated subsection has been removed, and the remaining sections have been renumbered accordingly.

Comments 5: Line 301: The adoption of the Arakawa C-type staggered grid is a solid choice for coastal hydrodynamic simulations, offering improved numerical stability and accuracy. However, in domains with complex or rapidly varying boundaries, specialized treatment for boundary conditions may be required to fully leverage the grid’s advantages. Can yoy please discuss about it?

Response 5: I thank the Reviewer for highlighting the importance of boundary‑condition treatment on an Arakawa C‑type grid in complex coastal domains. In the work presented, I introduce the first numerical implementation of the analytical model using a robust finite‑volume solver on a C‑grid, which delivers excellent stability and accuracy for gently varying bathymetries. I recognize, however, that real shorelines often feature rapidly changing or highly irregular solid boundaries. To address this in future model developments, I plan to implement a cut‑cell approach for cells intersecting irregular shorelines, use ghost‑cell extrapolation adjacent to cut cells to populate velocity and pressure fields consistent with desired wall conditions along complex geometries or even incorporate boundary‑fitted meshes for smoothly curving coastlines.

Comments 6: Figure 3: Labels are too small

Response 6: I thank the Reviewer for pointing this out. In the revised manuscript, the labels in all subpanels of Figure 3 have been enlarged to improve legibility. I have also increased the label sizes in the other figures as much as possible, while preserving the compact layout of the results presentation.

Comments 7: Section 4.2: This section should not stay in the Results and discussion section.

Response 7: I thank the Reviewer for this suggestion. In the revised manuscript, the subsection 4.2 Input Data has been moved from the Results and Discussion section to the Methods section. It now appears as Section 3.5 Input Data, immediately following the data acquisition and preprocessing descriptions.

Reviewer 2 Report

Comments and Suggestions for Authors

General comments

The manuscript addresses three-dimensional numerical modelling of coastal zone hydrodynamics. The investigations analyse differences between 2D and 3D model solutions, and theoretical calculations based on analytical solutions. They demonstrate the necessity of incorporating three-dimensional bathymetric variability in models describing zone hydrodynamics to enhance accuracy in coastal engineering and environmental management applications.

The manuscript is generally well structured, and it is well written. It includes a throughout critical literature review, detail description of the model and adopted assumptions. A credit should be given to use of field data in the study.

The findings are of importance for coastal engineering and environmental management applications, although as the Author writes they are preliminary and have illustrative character. Further investigations still are needed to reach more conclusive results.

The manuscript is well suited for the publication in the Water Journal. However, I have some comments which considered by the Author prior to publication.  

Specific comments

1. Page 2, Lines 67-76, Introduction – the assumptions adopted in the model should rather be mentioned in the model description, later in the text, and not in Introduction.
2. Page 14, Lines 431-434 – Data from two Lubiatowo experiments (2002, 2006), are used. How this impact consistency of the presented analysis. This should be commented.
3. Page 15, Line 474 – (e.g., [? ]), the reference number is missing.
4. Page 17, Line 502 – Figures 8 & 9 represent measurements averaged over a 30-minute period. Some comments should be given why the 30-min averaged was adopted.
5. Page 23, Line 591, Conclusions – Written: “sufficient agreement». What does it mean? This statement is an unclear. It could be written, e.g., “generally well agree”.
6. Page 23, Conclusions – The analysis carried out is advanced and very complex, based on several assumptions. I miss recommendations in Conclusions which of these assumptions are critical for the presented results, to which future investigations should put particular attention.

Author Response

I thank the Reviewer for their positive assessment and constructive suggestions. Below we address each point in turn.

Comments 1: Page 2, Lines 67-76, Introduction – the assumptions adopted in the model should rather be mentioned in the model description, later in the text, and not in Introduction.

Response 1: Thank you for the suggestion. I agree that the model assumptions belong in the technical description rather than in the general Introduction. Accordingly, the corresponding lines have been moved from the Introduction to Section 2, “Analytical Model,” where the discussion has been expanded.

Comments 2: Page 14, Lines 431-434 – Data from two Lubiatowo experiments (2002, 2006), are used. How this impact consistency of the presented analysis. This should be commented.

Response 2: Thank you for raising this point. I have added a brief explanatory paragraph to Section 4, “Results and Discussion,” (highlighted in purple) that clarifies why both Lubiatowo data sets are used and explains how consistency between them is ensured.

Comments 3: Page 15, Line 474 – (e.g., [? ]), the reference number is missing.

Response 3: Thank you for catching that omission. I have now inserted the correct citation (now reads “(e.g., [44])”).

Comments 4: Page 17, Line 502 – Figures 8 & 9 represent measurements averaged over a 30-minute period. Some comments should be given why the 30-min averaged was adopted.

Response 4: Thank you for this suggestion. I  have added the following justification in the revised manuscript (Page 19, Lines 462-464): “all ADCP profiles were binned into 30‑minute averages to suppress high‑frequency wave‑orbital velocities and instrument noise while retaining wave‑driven mean flow”.

Comments 5: Page 23, Line 591, Conclusions – Written: “sufficient agreement». What does it mean? This statement is an unclear. It could be written, e.g., “generally well agree”.

Response 5: Thank you for pointing this out. We have revised the sentence in the Conclusions to read: The results obtained from the presented three-dimensional model, specifically the averaged free surface elevation and velocity field distributions,   are generally in good agreement with both theoretical values (for particular conditions) and measured data in natural conditions (in the coastal zone of the Coastal Research Laboratory, Institute of Hydro-Engineering of the Polish Academy of Sciences, Lubiatowo). (Line 552)

Comments 6: Page 23, Conclusions – The analysis carried out is advanced and very complex, based on several assumptions. I miss recommendations in Conclusions which of these assumptions are critical for the presented results, to which future investigations should put particular attention.

Response 6: Thank you for this suggestion. I have now added a new “Critical assumptions and recommendations for future work” paragraph to the Conclusions, in which I (1) identify which model assumptions—linear wave theory and roller omission, zero‐equation turbulence closure, and linear bottom‐stress formulation—are most critical to our results, and (2) outline targeted studies (e.g. higher‑order wave kinematics, advanced turbulence closures and  fully quadratic drag laws) that will help validate and improve the model’s predictive skill.

Reviewer 3 Report

Comments and Suggestions for Authors

This is an interesting paper with new results, presenting a 3D numerical ocean model with a focus on radiation-stress parameterizations in the coastal environment. Several numerical tests are shown to illustrate the model's performance, including comparison with field data. This work is commendable for a single author. I would recommend publication after the author has addressed the following few (minor) points:

• page 2: assumptions and novel features of this model are clearly acknowledged. However, it is not clear how these stand as compared to existing ocean circulation models such as MITgcm, Princeton Ocean Model, ROMS. I'm not asking that you add comparison with results from these other models, but it would be suitable to add a short discussion to highlight any differences.
• page 2: you mention that linear Airy theory is used for the wave motion representation. Can you point out more clearly where exactly in the parameterizations described in Secs. 2 and 3 this linear wave theory comes into play?
• page 3 and onwards: the double vertical bar notation is used to denote tensors or matrices (I guess), which is unusual. Double bars are usually used to denote the norm (or magnitude) of a vector. Can you clarify this notation or use a more standard notation for tensors?
• page 8: staggered grids are widespread in CFD and are important in the present area of application. For the broad readership, it would be suitable to cite other references where such grids play a key role, e.g. Gilbert et al. (J. Comput. Acoust. 21, 1250017, 2013).
• page 8: what is the typical order of accuracy in space and time for the numerical scheme that solves the model's equations?
• page 14: values of the spatial grid size are given for these computations. Can you also provide information on the temporal grid size?
• page 15: some figures seem too small (e.g. Fig. 7 or 16) and may need to be enlarged, as details are not clearly visible (e.g. legend or labels).

Author Response

I thank the Reviewer for their positive comments and have addressed each point as follows:

Comments 1: page 2: assumptions and novel features of this model are clearly acknowledged. However, it is not clear how these stand as compared to existing ocean circulation models such as MITgcm, Princeton Ocean Model, ROMS. I'm not asking that you add comparison with results from these other models, but it would be suitable to add a short discussion to highlight any differences.

Response 1: Thank you for this comment. A concise discussion comparing our model’s assumptions and innovations with those of widely used circulation models (MITgcm, POM, and ROMS) has been added to the second paragraph of the Introduction and is highlighted in green.

Comments 2: page 2: you mention that linear Airy theory is used for the wave motion representation. Can you point out more clearly where exactly in the parameterizations described in Secs. 2 and 3 this linear wave theory comes into play?

Response 2: A clarifying sentence has been inserted in Section 2, immediately after the bullet list (highlighted in green).

Comments 3: page 3 and onwards: the double vertical bar notation is used to denote tensors or matrices (I guess), which is unusual. Double bars are usually used to denote the norm (or magnitude) of a vector. Can you clarify this notation or use a more standard notation for tensors?

Response 3: Thank you for drawing attention to our use of the double vertical bars ‖ · ‖ to delimit tensors. I fully agree that, in many texts, ‖ · ‖ denotes the norm (magnitude) of a vector or matrix. In the analytical formulation introduced in my earlier paper—Gic‑Grusza, G. The Three‑Dimensional Wave‑Induced Current Field: An Analytical Model. Water 16 (2024) 1165, doi:10.3390/w16081165—I adopted the pair of vertical bars to mark tensors, while reserving single bars | · | for absolute values, square brackets [ · ] for commutator–type operators and component arrays, and bold italic symbols for vectors. I retained that scheme here to keep the present article fully consistent with the analytical groundwork laid in the 2024 paper and.

Comments 4: page 8: staggered grids are widespread in CFD and are important in the present area of application. For the broad readership, it would be suitable to cite other references where such grids play a key role, e.g. Gilbert et al. (J. Comput. Acoust. 21, 1250017, 2013).

Response 4:  I appreciate the Reviewer’s suggestion to elaborate on the use of staggered grids. In the revised manuscript I have added a new paragraph (highlighted in green) that explains our use of an Arakawa C-type staggered grid in the numerical model, and we have included citations to the broader CFD literature as requested. We have marked this new text in green in the manuscript for easy reference.

Comments 5: page 8: what is the typical order of accuracy in space and time for the numerical scheme that solves the model's equations?

Response 5: Thank you for pointing this out. The manuscript now explicitly states the formal accuracy of the numerical discretisation. Spatial accuracy: second‑order, owing to the centred finite‑difference stencils used on the Arakawa‑C staggered grid. Temporal accuracy: first‑order, because an explicit forward‑Euler (single‑step) time integration is employed. This sentence has been inserted immediately after the paragraph that introduces the staggered grid in the Numerical grid section and is highlighted in green for easy reference. 

Comments 6: page 14: values of the spatial grid size are given for these computations. Can you also provide information on the temporal grid size?

Response 6: Thank you for noticing this omission. I've  added the missing information to Section 4.2:  “The simulations were advanced with a constant time step of Δt = 2 s, which satisfies the Courant–Friedrichs–Lewy (CFL) condition (maximum Courant number ≈ 0.55) for the entire domain. Sensitivity tests with Δt = 1 s and 4 s altered the peak significant‑wave‑height results by < 0.5 %, indicating that the chosen time step is sufficiently small for temporal convergence.”

Comments 7: page 15: some figures seem too small (e.g. Fig. 7 or 16) and may need to be enlarged, as details are not clearly visible (e.g. legend or labels).

Response 7: I have enlarged most of the figures to improve the readability of the labels. If you would like further adjustments, please let me know. I can scale all the figures proportionally, but doing so would change the overall layout of the paper—rows with four figures would need to be rearranged into two‑figure rows to fit the column width.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

After the edits the manuscript very much improved, therefore I suggest acceptance