Comments on:

Two-dimensional Numerical Simulations of Mixing under Ice Keels, https://doi.org/10.5194/egusphere-2023-1756, by De Abreu et al.

I enjoyed reading the manuscript. The topic is timely and is of interest. The idealized experiments are of course highly limited in their applicability to nature. The authors do a good job of emphasizing the limitations. I also do not mind that the authors opt for visual determination of regimes, differing from categorizing using formal thresholds based on evaluation of the Froude number.  In this study, I actually think the visual approach is a better choice. There are sufficient results from this study to qualify a scientific paper. More advanced and complex simulations can be left for a future study. I recommend publication after addressing my comments below. My recommendation should be considered keeping in mind that I am not a numerical modeler and cannot comment on the technical choices made about the numerical model setup and its solution.

Please consider (editor and authors): As a reader and editor for Ocean Science, I found the topic and content of this submission to be more suitable for Ocean Science. If you agree, the submission could be transferred from TC to Ocean Science and retain the reviews and discussion.

Major comments:

The introduction may be improved we reviewing the relevant literature (see details below).

Li 146: definition of the buoyancy difference, DeltaB. There is some inconsistency with using the summer and winter DeltaS bounds and the mixed layer depth from Peralta-Ferriz and Woodgate. They used a density step threshold of 0.1 kg/m3 which roughly translates to a buoyancy difference of 1e-3 m/s2. But your range is from about three times to 75 times that value. I certainly do not ask for new simulations. However, you should discuss the implications of this.

Referring to Figure 8, if the vertical reach of mixing is roughly two times z0, i.e., one additional mixed layer depth below the mixed layer depth of z0, for a relatively thick pycnocline layer (in real ocean) below a shallow mixed layer (say, z0=10 m and the diffuse pycnocline thickness is 20 m), mixing will not penetrate below the pycnocline and will not contribute to entrainment into the mixed layer. I would like to see some discussion about this.

Discussion includes "Implications" (actually, climatological and trend estimates), and "Limitations". I would like to see some discussion of the results too, on the findings in general but also including perhaps a discussion on the context/applicability of other studies on flow over sills etc, on the excluded interfacial/internal wave drag and related processes.

Minor comments (including comments on the literature review):

Opening paragraph: the narrative suggests the issue is a misrepresentation of ocean mixing under ice-covered waters. But this is only part of the story of the poor performance of large-scale models.

Second paragraph: studies diverge on the effect of decreased sea ice cover on potentially increasing wind-induced mixing. The literature review on this is not up-to-date. There are several studies that attempted to quantify the change in the near-inertial energy field in the Arctic in recent decades and how this is influenced by the sea ice cover.

Third paragraph: I am not a sea-ice expert, but I suspect the cited literature on changes in sea ice thickness and age may be outdated (newest 2018). Given that this is a submission to TC, the state-of-the-art can be improved.

Li 46-48: Agreed, but please also include some seminal papers from McPhee on the effects of under-ice roughness. (Actually, the only McPhee reference cited is from 1976.)

Li 51-52: Although not directly an ice-keel study, laboratory experiments in cases where the ice floe protrudes into the pycnocline reported in Carr et al (2019) can also be insightful. [Carr, M., et al. (2019). Laboratory experiments on internal solitary waves in ice-covered waters. Geophysical Research Letters, 46, https://doi.org/10.1029/2019GL084710]

Li 67: one of three and one of four stirring regimes can be confusing for the reader. Perhaps simply “we categorize the different stirring regimes in the upstream and downstream of the keel for each simulation.”

Li 84: I generally agree to ignore Coriolis in this study, but note that you do not need to go far from the boundary layer before the effect of rotation has a significant influence on the mixing length (so-called outer layer, see the McPhee book or book chapters).

Fig 1 caption can also define phi, sigma and h or refer to text.

Throughout, please use Roman Fr for the Froude number and Re for the Reynolds number.

Li 169: I’m not sure how to interpret this Re when the viscosity is replaced with a large value that mimics turbulent viscosity. I guess it is common practice in modeling. One of its implications, in mixing through low buoyancy Re is discussed later. Perhaps here a comment is also needed, about this implication and others if any, for the non-modeler reader.

Eq.10: Why is the sorted density not a function of the horizontal distance, x?

Li220: because of the division by [the molecular diffusivity] mu, …(to help the reader)

Li 257: cross-reference should be to section 2.1

Li 271: please clarify “ahead” of the keel, by using upstream or downstream

Fig 5 caption: the regime was defined without “Waves” in it [Unstable Subcritical regime]

Li 272: Fig3b shows the streamline not the vorticity. Perhaps use : “as we can see in the streamlines in Fig 3b… and in the spanwise vorticity field in Fig 5a.

Li 295: please comment on the presence or lack of mixing for this regime

Table 2 Caption: Missing “mixing” before depths. A missing closing bracket in the end.

Li 323: could insert: “… the largest mixing rate [in the upstream] does not …”

Fig 8 caption: could also mention overbar(Z) = 1 equals the mixed layer depth, z0.

Li 409: using a constant speed is an over simplification that is worth commenting

Li 413: typo in the ice speed trend. should be cm/s?

Li 478-485: On the positive side, your inferences can actually be representative of a floe. Your upstream and downstream control volumes are roughly (30-40)z0 long. For a 10 m MLD, this is roughly 300 m. One keel every 300 m should be typical (as you mention with reference to Wadhams). So effectively, your mixing calculations could be representative of the floe and not as local as you imply here.

Ilker Fer, University of Bergen

The paper is focused on the investigation of irreversible (diabatic) mixing of upper ocean layer by ice ridge keels. Effect of diabatic mixing is related to salt diffusion in conditions of complicated motion of stratified sea water.  Numerical simulations with created in a spectral partial differential equation solver Dedalus were used for the investigations. Although the calculation time is few tens of minutes, the effect has long-term consequences and could be applied to solve climate problems.  

There are following comments to the paper:

1. Nonlinear terms in momentum balance equations (1) and (2) are different from standard expressions .Viscous terms  and  in equations (1) and (2) are also different from standard form . Equations (1) and (2) are different from the momentum balance equations considered in the papers of Skyllingstad et al (2003) and Hester et al (2021) given in the reference list. More detailed explanation of equations (1) and (2) is necessary for improving of understanding of the problem statin.
2. Diabatic mixing is caused by salt diffusion in conditions of internal waves excited by the interaction of the ice keel with water flow leading to adiabatic stirring. Coefficient of salt diffusion is set to m²/s in numerical simulations. This value is much larger the molecular salt diffusion  m²/s. The large value of  is chosen to dissipate eddies smaller than the resolution of the grid (line 96). Further increasing  influence diabatic mixing according to formula (12). Please give more physical reasons for the choice of numerical value of .
3. Kinematic viscosity  m²/s is also larger molecular kinematic viscosity of  m²/s. Is it turbulent eddy viscosity? Please explain physical sense of .
4. Authors ignore thermal effects assuming water temperature equals -2 C. The water temperature is assumed depending on salinity (lines 91-92). Temperature at ice-water interface should be equal to the freezing point, and outside of the interface temperature is equal the freezing point or higher. Adiabatic mixing and diabatic stirring lead to increasing of water salinity and decreasing of the freezing point at ice-water interface. Decreasing of the freezing point influences ice melt leading to decreasing of water salinity and density near the interface. How strong this effect is in long term perspective?      
5. Estimates of ice drift speed using wind drag coefficient are not correct in the Barents Sea regions with relatively strong semidiurnal tide and influence of Spitsbergen, Franz Josef Land and Novaya Zemlya. Semidiurnal tide is stronger in the Barents Sea than in East Arctic regions. Speed of semidiurnal tidal current may exceed 1 m/s in the region between Bear and Hopen Islands. Also, water temperature below drift ice is frequently higher than -2 C in the Barents Sea. Depending on tidal phase and wind it varies from -1C to -1.9C.
6. All ice ridges in the Barents Sea are the first-year ridges. Shape of their keels is not no smooth as it is considered in the papers. Ridge keels are not completely consolidated, and macro porosity of their unconsolidated parts vary in the range 20-40%. Water can penetrate inside ridge keels, and boundary condition with zero normal velocity should be modified.

I recommend major revision of the paper to improve model description and thermodynamic justification of applicability of obtained results for the investigation of long-term processes in upper ocean layer.