the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 08 Dec 2025
Nondimensional parameter regimes of Arctic ice keel-ocean flow interactions
Abstract. Sea ice keels modulate upper-ocean momentum and mixing through internal wave (IW) generation, yet their effects are difficult to represent in climate models because their spatio-temporal scales are smaller than those of climate models and difficult to study in idealized simulations because geometry, forcing, and stratification span a large parameter space. We construct a compact description of the idealized representation of this problem by deriving five nondimensional parameters: lee-wave radiation potential (χ), IW nonlinearity (J), keel steepness (ζ), mixed-layer depth relative to keel draft (η), and pycnocline strength (Ri). We then calculate these five nondimensional parameters over the Arctic Ocean using monthly data from NEMO–CICE model output over the 2000−2017 time period. After extracting only the data points that fall within the lee wave radiating range (0 < χ < 1) and time-averaging, we apply the unsupervised Gausian Mixture Model (GMM) clustering to find regions with similar nondimensional parameter distributions. GMM reveals mechanically distinct, geographically coherent regions: boundary and marginal seas (Clusters 0−2) versus open-ocean regions that span from the central basin toward shelves (Clusters 3−5). The parameter regimes differ systematically in η and Ri: large η near boundaries implies weak keel–pycnocline coupling, whereas smaller η and steeper keels characterize the central Arctic regions. To diagnose dynamics, we run idealized two-dimensional nonhydrostatic numerical simulations with Boussinesq approximation with nondimensional parameters associated the mean values of each GMM cluster and quantify turbulent kinetic energy dissipation above, within, and below the pycnocline. The boundary regions (Clusters 0−1; η ≈ 27–55) show negligible IW and turbulence response below the pycnocline. The central Arctic regions with larger ζ and J (Clusters 3−5) exhibit enhanced near-pycnocline turbulence, but downward energy propagation is limited where Ri is large (∼290–500) and increases in regions closer to shelves with a smaller Ri value (∼130). Recasting previous IW drag parameterization to a nondimensional form shows it is most sensitive to η, increasing sharply as η →0 and weakly to Ri at fixed η. However, the results of our numerical simulations suggest that there may be some deviations from this parameterization that need to be further explored. Together, the nondimensional framework and clustering bound the physically relevant parameter space, identify where mixed-layer IW-drag parameterizations are credible, and provide concrete target ranges of nondimensional values to use in numerical simulations for calibration of the parameterizations.
-
Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
-
Preprint
(0 KB)
-
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
Journal article(s) based on this preprint
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2025-5284', Anonymous Referee #1, 08 Jan 2026
-
AC1: 'Reply on RC1', Varvara Zemskova, 22 Mar 2026
We thank the reviewer for their insightful comments and a thorough list of suggested observational literature. Please find our point-by-point responses regarding the implementation of these comments along with the revised and new figures at the end of the document.
-
AC1: 'Reply on RC1', Varvara Zemskova, 22 Mar 2026
-
RC2: 'Comment on egusphere-2025-5284', Anonymous Referee #2, 14 Jan 2026
General comments
This manuscript is concerned with ice-ocean drag and ocean dissipation that results from moving sea ice keels, a topic worthy of study. I find that the main novel conclusions are quantification of the parameter space that exists in the Arctic. The behavior of lee waves in various parameter regimes appears to be known from previous work (but correct me if I am wrong on this point). This conclusion regarding what parameter space is applicable for the Arctic could be better supported in a number of ways, including providing details of how the NEMO model is validated against the parameters of interest (keel geometry, mixed layer depth, and ice-ocean shear in particular). This conclusion could also be expanded to more robustly consider how that parameter space is related to ice conditions, ocean conditions, or both ice and ocean conditions (see comments below).
It is not particularly clear what new conclusions result from other portions of the analysis. 1) The paper claims to conclude that existing parameterizations are not sufficient and should be re-examined. I do not find this to be a robust conclusion, it is more of a suggestion or possibility to consider. 2) This paper additionally uses 2D numerical simulations, and it is not clear what new conclusions are derived from these simulations. They are useful illustrative examples, and can be retained in the paper, but should be presented as illustrative examples instead of new conclusions.
Overall, I think the novel and most useful aspects of this paper are the quantification of the five parameters for the Arctic environment. I think this paper could be shortened significantly to focus on this narrower set of novel conclusions.
Specific comments
It is difficult to follow this paper because of its use of variables, e.g, chi, J, eta, instead of their more dynamical names / descriptions, along with the use of cluster numbers, instead of more physical descriptions of these regions. For example, cluster 3 could be referred to as Cluster 3 (central arctic) or Cluster 3 (perennial sea ice zone) as the authors see fit. Lines 125-130: parameters like chi and J that have names should have their names stated here as well as in the abstract. In general, use of these names, e.g., internal wave nonlinearity, in addition to their symbols throughout the text would make this paper easier to read and understand.
Lines 71-80 are not necessary. These are not examples of GMM relevant to this paper.
Line 96-97: More details of the model validation and whether it is realistic or not need to be given in this paper. This study still relies on a model representation of keel characteristics (height and steepness). How do we know this is valid? Line 317-323: does the model accurately represent observed ocean stratification for the Arctic? Many arctic models fail to do so. Based on the information given in this paper, the claim that this model is a representative sample is not backed up. Figure 2: Can any of these parameters be compared to observational estimates? Is the geographic distribution of these parameters accurate?
Figure 2, line 230, and elsewhere: Three of these five parameters rely on u0. How is u0 estimated from the model? Is ocean velocity at a specific depth used, or is it averaged over a certain depth range? What drag coefficient formation is used in the model from which u0 derives? Is the drag coefficient constant or variable? Do the details of u0 varying with depth in the turbulent boundary layer and Ekman layer matter to this problem? How has u0 been validated? In general, I am also confused about when u0 is allowed to vary, and when a constant value of u0=0.1 m/s is used (2D simulations I think). Variability in u0 is potentially very important to this problem.
Figure 2: How often do the parameters satisfy the lee wave radiation condition?
Line 153-155: Is there seasonality in any of these parameters? Are these parameters only within the lee wave radiation range during winter? These are important details to convey. I realize that seasonality is addressed later in the paper, but it is important in this context.
Line 161-162: I’m confused about time averaging in the GMM setup. This sentence seems to state that data are NOT averaged ('at each timestep'), but also that time averages are used ('after fitlering and time-averaging').
How sensitive are the main conclusions of this paper to some of the choices and assumptions made? For example:
Figure 3: Do the conclusions of this paper change if only 5 clusters are used?
Line 197: A pycnocline width of 0.5 m is likely fine. But do the results depend on this parameter? A comment might help.
Line 211: If u0 were a factor of 2 larger or smaller, would this affect the conclusions?
Figure 8: The winter surface layer depths in the Arctic shown here are approximately 50 m depth on average, and deeper for cluster 5. There can be a range of surface layer depths across the Arctic, with wintertime values of 40 m or even shallower. How much difference would a 10 m change in MLD make to diagnosed mixing, or any of the other conclusions?I would interpret Figure 4-5 as having Cluster 4 representative of marginal ice zone / seasonal ice zone conditions. Is it the ice keel dimensions and not the ocean conditions that distinguish this cluster from 3 and 5? This is more relevant than listing the names of the seas that clusters occupy (line 267).
This paper would benefit from some context about ocean mixing in the mixed layer, pycnocline region, and interior. Is diagnosed mixing weak, average, or strong? Are lee wave processes a dominant source of mixing? The diagnosed dissipation rate ranges from order 10^-15 to 10^-7, which is a wide range of values. The largest values for the ocean interior are 10^-10, which I believe is small to average compared with other studies (e.g., Fine and Cole, 2022, Decadal observations of internal wave energy, shear, and mixing in the Western Arctic Ocean, J. Geophys. Res., https://doi.org/10.1029/2021JC018056).
Also in terms of context, I find it hard to visualize which regions have elevated mixing due to lee wave generation from the tables and discussion of the figures. Consider making a map showing pycnocline or interior mixing across the Arctic based on the clusters (obviously involving some assumptions). Line 446-455: The regurgitation in terms of relationships between different parameters is one view, but what is missing is the practical aspects. Overall, is lee wave drag or mixing significant for the Arctic? Where and when is it significant?
Line 422-429: These lines relate to one of the main conclusions of the paper, that the existing parameterizations are not quite correct. These lines do not give sufficient detail to support that conclusion. I do not think that the factor of 5 difference in internal wave drag at cluster 2 is particularly robust given the overlap in the 25-75th percentiles, and the general spread in the distributions of several orders of magnitude. The simulations that these are compared to also do not account for any variability in parameters, and are constructed with mean values. There might be a suggestion that the parameterization could be further investigated, but it is presented elsewhere as more of a conclusion.
Line 422-429: Again, context is needed. Can these values of the internal wave drag coefficient be compared with previous studies? Typical ice-ocean drag coefficients are order 10^-3, which is not mentioned in this paper and should be. Compared to that value, in all but the most extreme examples, the drag from lee waves is negligible.
Line 440: Are these sea ice regimes, ocean regimes, or sea-ice and ocean regimes? I believe it is the latter. The results section seems to focus more on their interpretation as ocean regimes more than ice regimes. Additional interpretation in terms of perennial ice cover (cluster 3) vs. seasonal ice cover (cluster 4-5) is also needed. Can arguments be made about ice regimes? What does this imply about a future arctic environment with increasing seasonal sea ice?
Lines 446-455: Please be explicit about what new results are revealed by these 2D simulations. It is my understanding that none of the behavior in different parameter-space regions is novel, it is simply that the quantification of those parameter-spaces for the Arctic has not been previously investigated. The 2D simulations should be presented more explicitly as an illustrative example instead of as new conclusions. I will also make a second point that I do not think conclusions about mixing or kinetic energy values from these 2D simulations are particularly robust. Not enough of the variability in parameter-space has been included to draw robust conclusions. I view these 2D simulation primarily as useful illustrative examples.
Technical comments
Abstract: Too many variables for an abstract. Use of the more common Ri might be okay, but I suggest removing the symbols for others.
Lines 2-4: Run on sentence with two uses of ‘because’
Line 57: Be more specific than ‘keel size’. Is this horizontal extent? Height?
Line 105: Horizontal spacing BETWEEN KEELS, L.
Line 227: Equation 13 has w without a prime, u with a prime. Which is correct?
Line 287: ‘small KEEP depth’
Figure 2d: log10=0 corresponds to eta=1, which is an important value for this parameter. The colorbar should be changed to emphasize this range of values, and the variations between eta=10 and eta=100 are less important.
Figure 5: cluster 0 and cluster 5 are hard to distinguish, especially when printed out.
Tables 1-3: These are hard to understand as just numbers on a table. I suggest plotting the values as a graph where shading or symbols can also be used to show standard deviations. Table 2 may need to be a logarithmic scale, but I think it is still useful to consider presenting it this way. Figure 6 is much better than Table 1! I think table 1 could be removed, and mean values could be added to Figure 6.
Tables 2 and 3 should make it clear in their description that these results derive from the idealized 2D model.
Figure 9: Text is too small. The color-scheme should be improved for readability, especially for summer. Panel (o) is also difficult to determine which clusters are which. Different seasons do not need to have different color schemes.
Figure 9 and 10: please adjust one figure so that the corresponding panels share the same axes. Example: Figure 9d uses a linear axis in eta and Figure 10d uses a logarithmic axes. Also consider adding dots for the mean values in each cluster and season to Figure 10.
Citation: https://doi.org/10.5194/egusphere-2025-5284-RC2 -
AC2: 'Reply on RC2', Varvara Zemskova, 22 Mar 2026
We thank the reviewer for their comments and suggestions on the clarifications for our manuscript. We have addressed these comments point-by-point along with noting how we implemented these suggestions into the manuscript. Please also refer to the revised and new figures at the end of the document.
-
AC3: 'Reply on RC2', Varvara Zemskova, 22 Mar 2026
We thank the reviewer for their comments and suggestions on the clarifications for our manuscript. We have addressed these comments point-by-point along with noting how we implemented these suggestions into the manuscript. Please also refer to the revised and new figures at the end of the document.
-
AC2: 'Reply on RC2', Varvara Zemskova, 22 Mar 2026
Peer review completion
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2025-5284', Anonymous Referee #1, 08 Jan 2026
Please find my comments in the attached PDF.
-
AC1: 'Reply on RC1', Varvara Zemskova, 22 Mar 2026
We thank the reviewer for their insightful comments and a thorough list of suggested observational literature. Please find our point-by-point responses regarding the implementation of these comments along with the revised and new figures at the end of the document.
-
AC1: 'Reply on RC1', Varvara Zemskova, 22 Mar 2026
-
RC2: 'Comment on egusphere-2025-5284', Anonymous Referee #2, 14 Jan 2026
General comments
This manuscript is concerned with ice-ocean drag and ocean dissipation that results from moving sea ice keels, a topic worthy of study. I find that the main novel conclusions are quantification of the parameter space that exists in the Arctic. The behavior of lee waves in various parameter regimes appears to be known from previous work (but correct me if I am wrong on this point). This conclusion regarding what parameter space is applicable for the Arctic could be better supported in a number of ways, including providing details of how the NEMO model is validated against the parameters of interest (keel geometry, mixed layer depth, and ice-ocean shear in particular). This conclusion could also be expanded to more robustly consider how that parameter space is related to ice conditions, ocean conditions, or both ice and ocean conditions (see comments below).
It is not particularly clear what new conclusions result from other portions of the analysis. 1) The paper claims to conclude that existing parameterizations are not sufficient and should be re-examined. I do not find this to be a robust conclusion, it is more of a suggestion or possibility to consider. 2) This paper additionally uses 2D numerical simulations, and it is not clear what new conclusions are derived from these simulations. They are useful illustrative examples, and can be retained in the paper, but should be presented as illustrative examples instead of new conclusions.
Overall, I think the novel and most useful aspects of this paper are the quantification of the five parameters for the Arctic environment. I think this paper could be shortened significantly to focus on this narrower set of novel conclusions.
Specific comments
It is difficult to follow this paper because of its use of variables, e.g, chi, J, eta, instead of their more dynamical names / descriptions, along with the use of cluster numbers, instead of more physical descriptions of these regions. For example, cluster 3 could be referred to as Cluster 3 (central arctic) or Cluster 3 (perennial sea ice zone) as the authors see fit. Lines 125-130: parameters like chi and J that have names should have their names stated here as well as in the abstract. In general, use of these names, e.g., internal wave nonlinearity, in addition to their symbols throughout the text would make this paper easier to read and understand.
Lines 71-80 are not necessary. These are not examples of GMM relevant to this paper.
Line 96-97: More details of the model validation and whether it is realistic or not need to be given in this paper. This study still relies on a model representation of keel characteristics (height and steepness). How do we know this is valid? Line 317-323: does the model accurately represent observed ocean stratification for the Arctic? Many arctic models fail to do so. Based on the information given in this paper, the claim that this model is a representative sample is not backed up. Figure 2: Can any of these parameters be compared to observational estimates? Is the geographic distribution of these parameters accurate?
Figure 2, line 230, and elsewhere: Three of these five parameters rely on u0. How is u0 estimated from the model? Is ocean velocity at a specific depth used, or is it averaged over a certain depth range? What drag coefficient formation is used in the model from which u0 derives? Is the drag coefficient constant or variable? Do the details of u0 varying with depth in the turbulent boundary layer and Ekman layer matter to this problem? How has u0 been validated? In general, I am also confused about when u0 is allowed to vary, and when a constant value of u0=0.1 m/s is used (2D simulations I think). Variability in u0 is potentially very important to this problem.
Figure 2: How often do the parameters satisfy the lee wave radiation condition?
Line 153-155: Is there seasonality in any of these parameters? Are these parameters only within the lee wave radiation range during winter? These are important details to convey. I realize that seasonality is addressed later in the paper, but it is important in this context.
Line 161-162: I’m confused about time averaging in the GMM setup. This sentence seems to state that data are NOT averaged ('at each timestep'), but also that time averages are used ('after fitlering and time-averaging').
How sensitive are the main conclusions of this paper to some of the choices and assumptions made? For example:
Figure 3: Do the conclusions of this paper change if only 5 clusters are used?
Line 197: A pycnocline width of 0.5 m is likely fine. But do the results depend on this parameter? A comment might help.
Line 211: If u0 were a factor of 2 larger or smaller, would this affect the conclusions?
Figure 8: The winter surface layer depths in the Arctic shown here are approximately 50 m depth on average, and deeper for cluster 5. There can be a range of surface layer depths across the Arctic, with wintertime values of 40 m or even shallower. How much difference would a 10 m change in MLD make to diagnosed mixing, or any of the other conclusions?I would interpret Figure 4-5 as having Cluster 4 representative of marginal ice zone / seasonal ice zone conditions. Is it the ice keel dimensions and not the ocean conditions that distinguish this cluster from 3 and 5? This is more relevant than listing the names of the seas that clusters occupy (line 267).
This paper would benefit from some context about ocean mixing in the mixed layer, pycnocline region, and interior. Is diagnosed mixing weak, average, or strong? Are lee wave processes a dominant source of mixing? The diagnosed dissipation rate ranges from order 10^-15 to 10^-7, which is a wide range of values. The largest values for the ocean interior are 10^-10, which I believe is small to average compared with other studies (e.g., Fine and Cole, 2022, Decadal observations of internal wave energy, shear, and mixing in the Western Arctic Ocean, J. Geophys. Res., https://doi.org/10.1029/2021JC018056).
Also in terms of context, I find it hard to visualize which regions have elevated mixing due to lee wave generation from the tables and discussion of the figures. Consider making a map showing pycnocline or interior mixing across the Arctic based on the clusters (obviously involving some assumptions). Line 446-455: The regurgitation in terms of relationships between different parameters is one view, but what is missing is the practical aspects. Overall, is lee wave drag or mixing significant for the Arctic? Where and when is it significant?
Line 422-429: These lines relate to one of the main conclusions of the paper, that the existing parameterizations are not quite correct. These lines do not give sufficient detail to support that conclusion. I do not think that the factor of 5 difference in internal wave drag at cluster 2 is particularly robust given the overlap in the 25-75th percentiles, and the general spread in the distributions of several orders of magnitude. The simulations that these are compared to also do not account for any variability in parameters, and are constructed with mean values. There might be a suggestion that the parameterization could be further investigated, but it is presented elsewhere as more of a conclusion.
Line 422-429: Again, context is needed. Can these values of the internal wave drag coefficient be compared with previous studies? Typical ice-ocean drag coefficients are order 10^-3, which is not mentioned in this paper and should be. Compared to that value, in all but the most extreme examples, the drag from lee waves is negligible.
Line 440: Are these sea ice regimes, ocean regimes, or sea-ice and ocean regimes? I believe it is the latter. The results section seems to focus more on their interpretation as ocean regimes more than ice regimes. Additional interpretation in terms of perennial ice cover (cluster 3) vs. seasonal ice cover (cluster 4-5) is also needed. Can arguments be made about ice regimes? What does this imply about a future arctic environment with increasing seasonal sea ice?
Lines 446-455: Please be explicit about what new results are revealed by these 2D simulations. It is my understanding that none of the behavior in different parameter-space regions is novel, it is simply that the quantification of those parameter-spaces for the Arctic has not been previously investigated. The 2D simulations should be presented more explicitly as an illustrative example instead of as new conclusions. I will also make a second point that I do not think conclusions about mixing or kinetic energy values from these 2D simulations are particularly robust. Not enough of the variability in parameter-space has been included to draw robust conclusions. I view these 2D simulation primarily as useful illustrative examples.
Technical comments
Abstract: Too many variables for an abstract. Use of the more common Ri might be okay, but I suggest removing the symbols for others.
Lines 2-4: Run on sentence with two uses of ‘because’
Line 57: Be more specific than ‘keel size’. Is this horizontal extent? Height?
Line 105: Horizontal spacing BETWEEN KEELS, L.
Line 227: Equation 13 has w without a prime, u with a prime. Which is correct?
Line 287: ‘small KEEP depth’
Figure 2d: log10=0 corresponds to eta=1, which is an important value for this parameter. The colorbar should be changed to emphasize this range of values, and the variations between eta=10 and eta=100 are less important.
Figure 5: cluster 0 and cluster 5 are hard to distinguish, especially when printed out.
Tables 1-3: These are hard to understand as just numbers on a table. I suggest plotting the values as a graph where shading or symbols can also be used to show standard deviations. Table 2 may need to be a logarithmic scale, but I think it is still useful to consider presenting it this way. Figure 6 is much better than Table 1! I think table 1 could be removed, and mean values could be added to Figure 6.
Tables 2 and 3 should make it clear in their description that these results derive from the idealized 2D model.
Figure 9: Text is too small. The color-scheme should be improved for readability, especially for summer. Panel (o) is also difficult to determine which clusters are which. Different seasons do not need to have different color schemes.
Figure 9 and 10: please adjust one figure so that the corresponding panels share the same axes. Example: Figure 9d uses a linear axis in eta and Figure 10d uses a logarithmic axes. Also consider adding dots for the mean values in each cluster and season to Figure 10.
Citation: https://doi.org/10.5194/egusphere-2025-5284-RC2 -
AC2: 'Reply on RC2', Varvara Zemskova, 22 Mar 2026
We thank the reviewer for their comments and suggestions on the clarifications for our manuscript. We have addressed these comments point-by-point along with noting how we implemented these suggestions into the manuscript. Please also refer to the revised and new figures at the end of the document.
-
AC3: 'Reply on RC2', Varvara Zemskova, 22 Mar 2026
We thank the reviewer for their comments and suggestions on the clarifications for our manuscript. We have addressed these comments point-by-point along with noting how we implemented these suggestions into the manuscript. Please also refer to the revised and new figures at the end of the document.
-
AC2: 'Reply on RC2', Varvara Zemskova, 22 Mar 2026
Peer review completion
Journal article(s) based on this preprint
Model code and software
2D_seaice_simulations Varvara E. Zemskova https://github.com/bzemskova/2D_seaice_simulations
Arctic_Clustering Fangchen Liu https://github.com/fcliu03/Arctic_Clustering
Viewed
Since the preprint corresponding to this journal article was posted outside of Copernicus Publications, the preprint-related metrics are limited to HTML views.
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 1,281 | 0 | 32 | 1,313 | 0 | 0 |
- HTML: 1,281
- PDF: 0
- XML: 32
- Total: 1,313
- BibTeX: 0
- EndNote: 0
Viewed (geographical distribution)
Since the preprint corresponding to this journal article was posted outside of Copernicus Publications, the preprint-related metrics are limited to HTML views.
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
Fangchen Liu
Varvara E. Zemskova
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
Please find my comments in the attached PDF.