Internal wave drag coefficient induced by moving ice keels, CIW, provides a metric to quantify how sea ice interacts with and impacts the stratified upper ocean. An existing analytically-derived parameterization for CIW by considers a two-dimensional problem with (x,z) representing horizontal and vertical directions, respectively. A schematic view for this problem is shown in Fig. . Keel geometry is set by its maximum depth h0 and horizontal spacing between keels L. This definition of horizontal spacing follows previous theoretical works on flow induced by rough topography e.g.,, where L=2π/k0 for k0 being the horizontal wavenumber of a sinusoidal topographic feature. considers sea ice underside roughness to be represented as infinitely many sinusoidal ice keels. Also, importantly, their parameterizations are derived for small keel aspect ratios, i.e., k0h0≪1, and subsequently, small disturbances. Note that the keel is depicted in Fig.  as a single Versoria shape not a collection of continuous sinusoidal features, which will be addressed in Sect. .

The forcing arises from the horizontal velocity of the drifting ice relative to the underlying ocean current, with components (u,v)=(uice-uocean,vice-vocean). Combining the two velocity components, we define relative forcing magnitude to be u0=u2+v2 (see Fig. a). The vertical stratification of the ocean is characterized as two layers: well-mixed layer of depth z0 with zero buoyancy frequency N=0 separated by a sharp pycnocline with buoyancy jump Δb from the lower weakly stratified layer with buoyancy frequency N0 (see Fig. b).

derives the IW drag coefficient CIW by finding analytical expressions for horizontal velocity ũ and vertical velocity w̃ at the pycnocline and computing the wave radiation Reynolds stress: 3τ=ρũw̃=ρCIWu02. Expressions for the velocities at the pycnocline ũ and w̃ are derived by matching wave solutions in the mixed layer and the stratified interior at the interface. CIW can then be expressed as the product of a drag coefficient for a fully stratified water column, CDNW, and a damping factor Γ that depends on the buoyancy jump and mixed layer depth: 4CIW=ΓCDNW. The damping factor Γ accounts for the reduction of drag due to finite mixed-layer depth. The formulations for CDNW and Γ given in and are written in terms of the six dimensional sea ice- and ocean-related variables: 5Γdim(u0,N0,k0,z0,Δb)=(1+N02u02k02+Δbk0u022sinh⁡2(k0z0)-Δbk0u02sinh⁡(2k0z0))-1, while the drag coefficient in a fully stratified (deep, non–mixed-layer) ocean is 6CDNWdim(u0,N0,k0,h0)=k0N0h022u01-u0k0N02. This formulation, implemented in a recent coupled ice–ocean model , reflects how variations in keel geometry, current speed, stratification, and mixed layer depth combine to regulate the efficiency of momentum transfer from drifting ice keels into the ocean interior.

Since these six dimensional quantities (u0, h0, k0, z0, Δb, and N0) span different units and scales, it is more effective to describe the system in terms of four nondimensional ratios that capture the relative importance of keel geometry, velocity forcing, and ocean stratification. These nondimensional parameters reduce the number of free variables and provide a compact framework to identify dynamical regimes.

The first two nondimensional parameters following the classical flow-topography interaction theory are the lee wave radiation parameter χ defined in Eq. () and keel criticality parameter J defined in Eq. (). The combination of these two nondimensional parameters is another nondimensional parameter that measures keel steepness: 7ζ=h0L/2=h0k0π=χJπ. Greater ζ corresponds steeper keel sides, i.e., to deeper and/or less wide keels. However, as it is not independent of the other nondimensional parameters, we will not explicitly use it in our analysis.

The third parameter is the depth ratio η, which quantifies how far the keel protrudes below the mixed layer: 8η=z0h0, The larger η is, the smaller the keel is compared to the mixed layer depth, and the larger the distance from the keel to the pycnocline. This would potentially limit the keel's ability to generate disturbance at the pycnocline and radiate IWs into the stratified interior.

The final parameter is the Froude number that compares kinetic energy of the forcing to the potential energy barrier of the pycnocline and is related to the Richardson number Ri in the parameterization: 9Fr2=u02k0Δb=1Ri Larger Fr (smaller Ri) corresponds to a weaker pycnocline and/or stronger forcing and has been previously found to result in more unstable, supercritical flow conditions .

We can then re-write the drag coefficient in the stratified interior CDNWdim (Eq. ) and the attenuation factor Γdim (Eq. ) using these four nondimensional parameters, which will simplify our exploration and understanding of the parameter regimes: 10Γ(χ,J,η,Fr)=(1+1χ2+1Fr4sinh⁡2(χJη)-1Fr2sinh⁡(2χJη))-1,11CDNW(χ,J)=χJ221-χ2.

In order to illustrate the differences in the dynamical regimes for each cluster identified by the GMM, we perform numerical simulations of the idealized problem shown in Fig. . Similar to previous studies e.g.,, our simulations are two-dimensional in (x,z). Technically, the domain is two-and-a-half dimensional with just one grid cell in the y-direction, and the velocity in the y-direction can be non-zero but all derivatives with respect to y are zero. Specifically, we solve the following non-hydrostatic rotating Navier-Stokes equations with the Boussinesq approximation: 12∂u∂t+u⋅∇u-f×u=-∇pρ0+ν∇2u+bk+fu0j,13∂b∂t+u⋅∇b=κ∇2b,14∇⋅u=0, where u=(u,v,w) is the velocity in (x,y,z) directions, p is pressure, b=-g(ρ-ρ0)/ρ0 is buoyancy for density ρ(x,z,t) and constant reference density ρ0, f=fj for local Coriolis parameter f, j and k are unit vectors in y and z directions, respectively, and ν and κ are kinematic viscosity and diffusivity, respectively. For steady velocity forcing, the term fu0 is added to the y-momentum equation analogous to the simulations by and .

While the parameterization in was derived for a sinusoidal ice keel shape, in more recent numerical studies e.g.,, it has been more common to model keel shape h(x) using a Versoria function 15h(x)=h0σ2σ2+4x2, with width σ, which we also use in our numerical simulations as shown in the schematic in Fig. . However, the CICEv5.1 model output reports L rather than σ and theoretical nondimensional numbers are typically expressed in terms of k0 (wavenumber of a sinusouidal keel), so we make the approximate connection that σ=π/(2k0).

The equations are solved using Oceananigans.jl to take advantage of the enhanced computational speeds by GPUs . In its current implementation, an immersed boundary grid to model an obstacle (e.g., bottom topography or an ice keel) can only be specified along the bottom boundary. However, we can apply the property of the Boussinesq flows in that the flow is symmetric when flipped vertically, assuming that the buoyancy is also flipped in sign. That is, for example, in the Boussinesq approximation, cool dense water sinking and warm light water rising appear as vertically-flipped mirror images. Therefore, we model a flipped version of Fig.  by imposing a Versoria-shaped (Eq. ) immersed boundary along the bottom of the domain and initializing the buoyancy profile as 16b0(z)=12Δb-N02-H-z0-z1-tanh⁡-H-z0-zμ, where H is the maximum depth of the domain and μ is the pycnocline width taken to be 0.5 m for all simulations. The width μ is taken to be a small finite value to represent a sharp pycnocline yet to maintain numerical stability of the simulations. However, it can also be varied after examining observational measurements of the stratification profiles in the Arctic, thus creating a fifth nondimensional parameter.

In order to avoid reflections off the top rigid-lid surface, we implement an exponential sponge layer e-z2/2δ2 with δ=-H/20, which corresponds to the sponge layer being applied within approximately the top 20 m. Within the sponge layer, the flow is relaxed with a damping rate of 1/(20Δt) to the initial conditions: b0(z) for buoyancy, u0 for u, and zero for w. In order to maintain numerical stability of the simulations, we take values for ν, κ, and Δt similar to those of , namely, ν=κ=10-3 m² s⁻¹ and Δt=6×10-3 s. However, unlike , we do not apply sponge layers along the left and right boundaries, as these sponge layers were found to trigger artificial disturbances that travel downstream generating flow instabilities. Instead, we set the horizontal boundaries to be periodic and run the simulations for 6 h, which we found to be enough time for the flow to reach a quasi-steady state, but not enough time for the instabilities re-entering the domain through the periodic boundaries to reach the topographic obstacle. Finally, we set f=1.36×10-4 s⁻¹ corresponding to 70° N, though rotation is not likely to significantly influence the simulations as the total length of the simulation time is less than one inertial period (≈12.8 h).

The domain for all numerical simulations is x∈[-Lx/2,Lx/2] for Lx=1200 m and z∈[-H,0] with H varying depending on the mixed layer depth for the simulation (see Table ). For all simulations, we set u0=0.1 m s⁻¹ and width of the Versoria-shaped obstacle (i.e., ice keel) as in Eq. () to be σ=40 m. Recall that k0=π/2σ. We then compute all other dimensional parameters using the nondimensional parameter values as: 17N0=u0k0χ,Δb=k0u02Fr2,h0=χJk0,andz0=-ηh0.

We perform nine numerical simulations: one for each of the annually-averaged clusters A0–5 and additionally for two summer clusters (S0 and S2) and one winter cluster (W2). The additional clusters are selected based on relatively large predicted CIW from the parameterizations as will be shown in Sect. . These additional clusters also allow us to investigate the parameter regime were the ice keel protrudes below the mixed layer depth, i.e., η<1 (h0>z0), or the keel depth is close to the mixed layer depth, i.e., η∼1 (h0∼z0), which is not represented by the mean values of clusters A0–5. The numerical simulations are set up taking the mean nondimensional parameter values χ, J, η, and Fr for each of the GMM clusters. These values are summarized in Table  and are further discussed in Sect. . The horizontal resolution is the same for all simulations (Nx=4096 grid points), but the vertical resolution varies to allow approximately the same number of points within keel height h0 as shown in Table .

To minimize the influence of the flow re-entering through periodic boundary conditions on the interpretation of our results, we limit the horizontal extent of the region of analysis for numerical simulations to x∈[-200,200] m. All horizontal averages and integrals are performed only within these bounds. Also, in order to be consistent with the orientation of ice keel being at the surface (rather than along the bottom as in the numerical simulation set-up), all of the subsequent equations and figures will be shown in terms of z^=-H-z.

In order to compare the flow dynamics across the numerical simulations with different parameter regimes, we compute the fluctuating kinetic energy 18EK(x,z^)=12u′(x,z^)2+w(x,z^)2 and fluctuating kinetic energy dissipation 19ϵK(x,z^)=ν∂u′∂x2+∂u′∂z^2+∂w∂x2+∂w∂z^2, where u′(x,z^)=u(x,z^)-u0 is the velocity of fluctuations defined as the deviation of horizontal velocity u from the background u0. This is a common way to define perturbations and compute fluctuating (or turbulent) kinetic energy budget terms in numerical simulations involving internal waves e.g.,. Note that in this definition of fluctuating kinetic energy budget terms, we include both the internal waves and smaller scale motions.

We then compute the area-averaged integrals of EK and ϵK in three different vertical regions to understand the effects of the different parameter regimes on the flow. The first region denoted by subscript pyc is around the pycnocline, which we define to be z0±10 m, and the integral is notationally expressed as 20⋅pyc=1Apyc∫-200200∫z0-10z0+10⋅dz^dx. The second region denoted by subscript above is above the pycnocline, that is between the pycnocline and the ice keel, i.e., 21⋅above=1Aabove∫-200200∫z0+10-h0/3⋅dz^dx. The upper bound is taken to be z^=-h0/3 to exclude numerical boundary layer effects due to the immersed grid. The third region denoted by subscript below is below the pycnocline, i.e., 22⋅below=1Abelow∫-200200∫-H+50z0-10⋅dz^dx. The lower bound is taken to be z^=-H+50 m to exclude the sponge layer. In Eqs. ()–(), Apyc, Aabove, and Abelow are areas of each respective region. We report values that are time-averaged over the last hour of the simulation to account for any small-scale temporal fluctuations.

We expect that most of the influence of the ice keel on the flow will be confined within the mixed layer, i.e., above the pycnocline. As we aim to quantify the relative influence of the ice keel on the pycnocline and the stratified interior of the ocean below the pycnocline, for each simulation we also compute the ratios: 23EKaboveEKbelow,ϵKaboveϵKbelow,EKaboveEKpyc,ϵKaboveϵKpyc. The smaller magnitudes of these ratios indicate greater effects of the keel on the energy propagation and dissipation within the pycnocline region and below the pycnocline.

We first focus on the GMM clusters identified based on the annually-averaged data identifying their geographical extents and discussing differences in flow characteristics below the sea ice based on the illustrative numerical simulation results. Figure a shows these six clusters within the Arctic region identified by applying GMM to four time-averaged nondimensional parameters. Each cluster reflects different oceanographic and sea ice conditions as will be discussed below. These clusters exhibit coherent geographic patterns despite latitude and longitude not being used as input variables for the clustering.

Before discussing the nondimensional parameter regimes for each of the clusters, we first put their geographical distributions in perspective using the Arctic clusters derived by based on the spatio-temporal patterns of sea ice concentration (SIC) observations over the 1979–2023 time period. This previous study categorized the Arctic based on the seasonal SIC cycle into three categories: permanent sea ice cover (fully ice covered in winter and minimum of 70 % SIC in the summer), full winter cover (mostly ice-free in the summer), and partial winter cover (maximum 70 % SIC in the winter and ice-free in the summer). Cluster A0, which is the largest cluster (Fig. a), occupies the central Arctic basin. This cluster mostly corresponds to the region in characterized by permanent sea ice cover consistently throughout their time period of consideration. Cluster A1 (Fig. d) extends outward from Cluster A0. Comparing its spatial extent with the analysis by , this region is still mostly within parts of the Arctic that are permanently covered by sea ice but undergoing temporal shifts in the seasonal SIC cycle. Clusters A2 (Fig. g) and A3 (Fig. j) predominantly correspond to the full winter sea ice cover regions , with A2 covering mostly outer parts of the Eurasian basin of the central Arctic and A3 occupying coastal regions. Finally, the smallest clusters A4 and A5 (Fig. m, p) occupy boundary regions falling predominantly within the partial winter sea ice cover regions identified by .

Now that we have identified the geographic patterns of each of the clusters, we will compare the nondimensional parameter regimes associated with each cluster. The distributions of each of the nondimensional parameters separated by the GMM cluster are shown in the left column of Fig. . In this subsection, we will focus on the mean values of the nondimensional parameters for each cluster (Table ) and discuss the dynamics of the different parameter regimes supported by the numerical simulations results. Snapshots of the flow fields for the numerical simulations are shown in Figs. – and the energetics metrics are summarized in Tables -.

Broadly speaking, the central Arctic clusters A0 and A1 that fall within the permanent sea ice cover regions are characterized by relatively small values of depth ratio η (Fig. g) and Froude number Fr (Fig. j) and large values of keel criticality J (Fig. d). Large values of J and small values of η indicate that keels in this region possess strong potential to overcome stratification and drive vertical mixing. In the numerical simulations, we find large turbulent velocities (Fig. a, e) and dissipation rates (Fig. c, g), in particular above the pycnocline. Out of the six annual clusters, we find the largest amount of kinetic energy and dissipation rates in all parts of the domain (above, below, and within the pycnocline) for these two clusters (Table ). The stratification in the vicinity of the pycnocline is also perturbed with evidence of small-scale turbulent motions, and the deviation from the initial stratification Δb‾ is the largest across all simulations (Fig. b, d, f, h). However, the small Fr values reflect strong density stratification, which may suppress sustained turbulence and limit mixing to localized, shear-driven interfaces. We find energy propagation below the pycnocline into the stratified interior to be small (large values of EKabove/EKbelow and ϵKabove/ϵKbelow in Table ). This suggests that the large value of density jump across the pycnocline (small Fr) can inhibit the effect of the ice keel despite the small value of η (so keel depth is the largest relative to mixed layer depth in comparison to other clusters). Comparing the two clusters, cluster A1 has smaller J and larger η than cluster A0. As a result, despite having larger value of Fr and thus smaller potential energy barrier at the pycnocline, the keel is less likely to reach or perturb the pycnocline, reducing mechanical coupling compared with cluster A0, thus smaller amount of kinetic energy and dissipation rates. However, because of a smaller density jump across the pycnocline (larger Fr), the lee-wave signature below the pycnocline is more coherent and the relative energy propagation below the pycnocline is larger (smaller values of EKabove/EKbelow and ϵKabove/ϵKbelow) compared to cluster A0.

Cluster A2 continues this trend moving away from the Central Arctic Ocean towards parts of the Arctic that are only fully ice-covered in the winter with smaller keel criticality J and larger depth ratio η and Froude number Fr, pointing to a weaker pycnocline and greater susceptibility to intermittent IW activity below the pycnocline. It also has larger values of lee wave radiation parameter χ in comparison to the other Central Arctic Ocean clusters. The numerical simulations show overall smaller magnitude EK and ϵK and less turbulent motions around the pycnocline (Fig. i–l) compared to central Arctic clusters A0–1 simulations, most likely due to smaller J (reduced nonlinearity of the flow) and larger η (small keel compared to mixed layer depth). However, because of larger Fr, wave energy propagation into the stratified interior below the pycnocline is larger (smaller EKabove/EKbelow) in comparison to clusters A0–1.

Cluster A3 exhibits a blend of characteristics: keel criticality J larger than that of A2 suggesting some nonlinear flow motions, and intermediate stratification strength with Fr smaller than that of A2 but larger than those of A0–1, and depth ratio η larger in comparison to the central Arctic clusters A0–2 and smaller in comparison to marginal ice clusters A4–5. These conditions suggest lee wave generation and intermittent mixing, likely governed by the interplay between moderate mechanical forcing and stratification. Indeed, we find lee waves radiating below the pycnocline in the numerical simulations for this regime (Fig. i–l). While EK and ϵK are smaller in magnitude in comparison with those for clusters A0–2, kinetic energy around the pycnocline EKpyc and above the pycnocline EKabove is relatively large (Table ). The dissipation within the pycnocline region is relatively small (larger value of ϵKabove/ϵKpyc) possibly due to smaller Froude number (or stronger pycnocline).

Clusters A4–5 represent boundary or transitional regimes characterized by large values of η, i.e., a much deeper mixed layer in comparison to the keel depth (Fig. g). Data points within cluster A4 have more extreme or distinctive parameter values (e.g., largest mean Fr and χ and smallest mean J) and the largest variability within the cluster. We find that the ice keel has negligible effect on the flow for these parameter regimes (Fig. e–l). Both EK and ϵK are smaller by at least 2–3 orders of magnitude in the numerical simulations for these clusters compared with the other clusters (Table ) as wave propagation is getting suppressed by the deep pycnocline. The only notable exception is that EK above the pycnocline is larger cluster A5 simulation than that of cluster A4 simulation and comparable to that of cluster A2, most likely due to a steeper keel (larger keel criticality J).

The numerical simulations presented here do not exhaustively capture the variability in the dynamical regimes and are primarily for illustrative purposes. However, through these simulations, we can already observe how differences in the nondimensional parameters change the flow characteristics, IW generation, propagation, and dissipation. For example, in general, we find that the amount of fluctuating KE and KE dissipation increases with increasing J (steeper keels, increased nonlinearity of the flow) and decreases with increasing η (keel further away from the pycnocline) with more complicated correlations with χ and Fr, though these relationships will need to be investigated thoroughly with a more comprehensive parameter sweep.

While in the previous section we explored specific combinations of parameter values associated with each cluster in order to qualitatively assess their effects on the flow, in this section, we consider the spatial and seasonal variability of the nondimensional parameters in order to identify the ranges of values that are relevant to the sea ice keels. We then explore how this variability translates to the variability of IW drag values estimated from the parameterization in Eqs. ()–().

Seasonal differences between the summer- and winter-averaged distributions of the nondimensional parameters are shown in Fig. . Winter months are characterized by larger values of lee wave radiation parameter χ due to larger relative ice keel speeds u0, especially in the Eurasian basin (Fig. S1b, c in the Supplement), and smaller buoyancy frequency N0 in the stratified interior throughout most of the Arctic (Fig. S1h, i). Smaller N0 and larger u0 in the winter than in the summer also yields smaller keel criticality values: over most of the Arctic, J<1 in the winter and J>1 in the summer (Fig. e, f). The seasonal differences in the depth ratio η are mostly due to the differences in the mixed layer depth z0 (Fig. S1n, o) rather than the keep depth h0 (Fig. S1k, l). From these estimates, we find η<1 (mixed layer depth shallower than the keel depth) for most of the Arctic in the summer (Fig. h) and η>1 (z0>h0) in the winter (Fig. i). Froude number Fr is larger during the winter months in comparison to the summer (Fig. k, l), with particularly larger in the Eurasian Basin in the winter. This is due to smaller buoyancy jump across the pycnocline Δb in the winter (Fig. S1q, r) and larger relative keel speeds in the winter (Fig. S1b, c). Interestingly, the seasonal differences in the keel horizontal wavenumber k0 (Fig. S1e, f) are not strongly reflected in the seasonal differences of the nondimensional parameters. Overall, k0 is smaller in the winter, which would make χ and Fr smaller in the winter, but we do not find that to be the case.

The spatial distribution of GMM clusters also highlights the seasonal differences in nondimensional parameters (Figs. –). For the annually-averaged data, the largest cluster A0 (40 % of data points, Fig. a) encompasses most of the central Arctic. In contrast, for the summer- and winter-averaged data, the central Arctic is more evenly and clearly divided into the Amerasian (S0 in Fig. b and W0 in Fig. c) and Eurasian (S1 in Fig. h and W1 in Fig. f) basins. In both basins, the values of χ, η, and Fr are overall smaller and the values of J are larger in the summer than in the winter. The nondimensional parameter ranges are different between these clusters and the inter-cluster differences vary across seasons. For example, in the winter months, we find larger values of J, η, and Fr (Fig. f, i, l) in the Eurasian basin than in the Amerasian basin, but the opposite to be the case in the summer months.

As captured by the IW drag parameterization in Eqs. ()–() and observed based on the kinetic energy metrics in Sect. , the effects of the nondimensional parameters on these quantities is nonlinear. To explore whether IWs might play an important role in certain parts of the Arctic, we plot in Figure  the pair-wise distributions of nondimensional parameters χ, J, η, and Fr calculated based on ice keel and ocean variables averaged annually (first column from the left), over the summer months (second column from the left), and over the winter months (third column from the left). Ultimately, we wish to perform numerical simulations with parameter sweeps over these nondimensional parameters to test parameterization schemes included in regional and large-scale models, so it is important to limit the range of values for such sweeps. The pairwise distributions help us identify combinations of parameter values that might not need to be as thoroughly explored, e.g., combinations of (i) large χ and large J (Fig. a–c), (ii) large J and large η, and (iii) large J and large Fr. Furthermore, the breakdown of the pairwise distributions into the GMM clusters helps us identify combinations of nondimensional parameter values that, while present in the Arctic, might be less ubiquitous. For example, clusters A4, A5, and W4 show a high variability with respect to values of η and Fr (Fig. g, i, j, l), but each of them contains less that 15 % of all the data points in the Arctic (Fig. m, o, p). Therefore, even though the pairwise distribution shows values over the full range of η-Fr space (Fig. u, w), we find that most of the points (possibly over 85 %) have smaller values of η (≲10) and Fr (≲0.5).

These pairwise distributions are compared with the distribution of the IW drag values predicted from the parameterization CIW (Fig. , right column), where red (blue) colors indicate CIW values larger (smaller) than the canonical form drag coefficient value of CD=5.5×10-3. For example, in the χ-J space (Fig. a–d), we find that most points in the Arctic for all time averages fall within the range of nondimensional parameter values that produce CIW>CD. The joint distributions also illustrate the temporal variability of the nondimensional parameter regimes. For example, CIW is large for large values of J and small values of η (Fig. h, p). This pocket is not well-populated with data in the annual and winter distributions (Fig. e, g, m, o) but become pronounced in summer (Fig. f, n), when the distributions shift toward larger J and smaller η, thereby intersecting the regions of large CIW values. In the J-Fr space (Fig. q–t), larger CIW are found for Fr≳0.5 and J≲3. While some points in the annual- and winter-averaged fall within this space, many of the the summer-averaged points have smaller Fr values, and subsequently smaller predicted CIW.

While subplots in the right column of Fig.  shows the dependence of CIW for pairwise distributions of the nondimensional parameters, it still does not capture the full variability in the four-dimensional space. We now plot the distribution of values of CIW for each cluster (annual, summer, and winter clusters) computed using Eqs. () and ()–() based on the distribution of all four nondimensional numbers for each cluster (Fig. a–c). For the clusters using the annually-averaged values, unsurprisingly, clusters A4–5 with larger η (i.e., smaller ice keel depth relative to the mixed layer depth) have smaller CIW, which is consistent with the energetics metrics discussed in Sect. . The parameterization also predicts clusters A0–1 to also have smaller IW drag, possibly because of smaller values of Froude number Fr (≲0.1), despite having relatively large keel criticality J and small depth ratio η values (Fig. d, g, j). When considering seasonally-averaged data, we find the parameterization predicts larger values of CIW in some parts of the Arctic in comparison to the CIW estimated based on the annually-averaged data. For instance, some of the marginal ice clusters (S2–4, W2–3) and even Central Arctic Ocean clusters (summer Amerasian basin S0 and winter Eurasian basin W1) have CIW values larger than those for any of the annual clusters.

Now that we have explored the spatio-temporal variability of the nondimensional parameters and the subsequent variability in the predicted values of the IW drag, we put these findings into a broader context in the next section.

In this study, we perform GMM clustering over four nondimensional parameters (lee wave radiation parameter χ, keel criticality J, depth ratio η, and Froude number Fr) to identify parts of the Artic that have similar distirbutions of these parameters. Our clustering is performed both on the annually-averaged data and data averaged over the summer and winter months separately to deduce any seasonal patterns. We find that GMM clustering generally performs well to separate the spatial patterns of perennial multi-year ice (A0–1, S0–1, and W0–1) and of seasonal first-year ice zones (A2–5, S2–5, W2–5) identified based on maps from and . The seasonal clusters also spatially separate the Eurasian (S1, W1) and the Amerasian (S0, W0) basins for both seasons. This agrees with the current understanding of the noticeable differences between the two basins, for example, in terms of the seasonal stratification . Our estimates of KE dissipation rates from the idealized numerical simulations and the estimates of CIW from the current parameterization both indicate a differences between the regions of perennial sea ice and seasonal sea ice. This can be important as the proportion of perennial sea ice in the Arctic has been decreasing in the past three decades .

One of the significant contributions this study is identifying parts of the Arctic that potentially have elevated values of IW drag – but are these values large enough? We can estimate the relative importance of the IW drag by comparing the CIW values from parameterizations with skin Cs and ice-ocean drag coefficients Cio estimated from observations (Fig. a–c). We take the skin drag coefficient value of Cs=7×10-4 from the measurements under unridged summer ice by . The range of values for Cio is estimated from various observational studies: 1.3–12.3×10-3 Beaufort Sea, annual cycle by, 4–6×10-3 Amudsen and Nansen Basins, summer by, 1–10×10-3 Canada Basin, annual cycle by, and 1–10×10-3 average value of 3.4×10-3, Nansen Basin in July by. However, notably, measured that Cio can be as large as 0.13 at times. We also make comparisons to the canonical ice-ocean drag coefficient value of CD=5.5×10-3, which is approximately in the middle the Cio range (Fig. d–f). When the IW drag estimates are made with the annually-averaged data, we find that the CIW values are typically smaller than the measured Cs and Cio values and mostly less than 10 % of CD. In the regions of perennial sea ice (e.g., S0–1, W0–1), CIW is still smaller in comparison to the form and skin drag coefficients; we find that at most only 50 % of the points in those clusters have IW drag coefficient values larger than CD (Fig. d–f). However, in some marginal ice zones (e.g., S2–3, W2), CIW can be 10 %–20 % on average (or even larger for some points) of CD. The CIW values for cluster W3 (along Greenland and in the Chukchi Sea) can be as large as or exceeding CD. Combined, these clusters contain about a third of all data points that we examined over the Arctic for each season (cf. Fig. ). So, even though the IW drag may be relatively not as important in the pack ice regions in the Central Arctic Ocean, it could be important in the marginal zones, especially in the winter.

It is important to note that the values for the IW drag coefficient CIW presented here are calculated using the parameterization by . This parameterization was developed for a two-dimensional model assuming small keel height h0, i.e., small topographic criticality parameter J for a fixed stratification N0 and relative keel speed u0. A study by recently assessed the parameterizations of form drag for flow over seamounts. They found that the disagreement between numerical simulations and the two-dimensional parameterization also derived for small mount heights increased as J increased. Specifically, they found that the parameterization underestimated the form drag more in comparison to the numerical simulations for larger values of J: at J=1, the parameterized drag was only about one third of the value calculated from the simulations. Their results suggest that the current parameterization for CIW by might also be underestimating the drag at larger values of J. In this study, we find many points in across the Arctic with J≳1: 67 % of all grid points in the annual average, 91 % during the summer months, and 7 % during the winter months. Our findings indicate that this supercritical regime J≳1 might be an important parameter regime, especially during the summer, but it is not well-represented by the current parameterization and needs to be re-evaluated through future numerical studies.

Another assumption made in the model of ice keel-flow interaction is that the pycnocline lies below the ice keel, meaning that the depth ratio η is greater than unity. However, in the summer, the canonical mixed layer can be absent in parts of the Arctic, such that even the near-surface ocean layers are stratified . We also find that approximate 67 % of data points in the summer months (and 1.9 % for the annual and 8.4 % for the winter data) have the depth ratio η<1. For the idealized numerical simulations conducted to assess the parameterization of CIW, the absence of the mixed layer does not pose a problem, as it would be just the limiting case of setting the nondimensional parameter η to zero (as the mixed layer depth z0=0). In on our preliminary numerical simulations (clusters S0, S2, W2) with the mixed layer depth less than or about the keel depth (η≲1), we find kinetic energy and dissipation rates to be larger by at least an order of magnitude in comparison to the other parameter value combinations examined in the numerical simulations here (Table ). In particular, in cases with η<1 considered here (S0 and S2), we find an energetic flow field with many small-scale motions that are possibly enhancing the energy cascade and dissipation (Fig. a–c, e–g). Overall, we would expect that smaller η would enhance the IW generation and the IW drag as the ice keel would be in a more direct contact with the stratified layer, potentially without the buffer of the mixed layer. Therefore, the parameter regime of such smaller η values might not be well-captured in the current IW drag parameterization and needs to be further explored through numerical simulations.

With recent GPU-acceleration of computational fluid dynamics numerical codes (e.g., Oceananigans.jl), conducted a large numerical simulation sweep to test the existing parameterizations for the drag due to steady and tidal flows interacting with topographic obstacles along the ocean floor (i.e., seamounts). However, that problem has a different set of nondimensional parameters compared to the sea ice-flow interaction problem. Namely, additional nondimensional parameters to characterize the ratio between mixed layer depth and keel depth (η) and the ratio of the kinetic energy of the flow relative to the potential energy due to the buoyancy jump across the pycnocline (Fr) are relevant in the upper Arctic Ocean stratification, whereas constant stratification is assumed near the ocean bottom. Therefore, there is a need for similar studies with a consistent numerical set-up to test the existing sea ice drag parameterizations. Previous modeling efforts typically have only considered the variability of one or two of the relevant nondimensional parameters and only certain parameter regimes, e.g., only relatively deep ice keels (η=0.25–2 in and ) or in contrast, homogeneous fluid (η→∞ in and ). We find η to be in the range of [0,140] and concentrated in the η∈[2,4] range in the annually-averaged and winter-averaged data and in the η∈[0,1.5] range for the summer-averaged data. From our preliminary numerical simulations, we also find that the kinetic energy metrics might be enhanced for mid-range values of η∼5 depending the values of other nondimensional parameters. In the absence of well-distributed observational data of Arctic ice keel and near-surface ocean flow characteristics, our results provide a good starting point to consider for parameter sweeps in future numerical studies. Our results also suggest that certain joint ranges of parameter values might not need to be investigated in detail (e.g., large χ and large J combinations).

We can also compare the kinetic energy metrics from our numerical simulations with observations. Many observational studies estimate internal wave dissipation rates in the Arctic to be within the 10-10–10-9 m² s⁻³ range . These values are smaller than observational measurements of up to 10-8 m² s⁻³ in the upper parts in other global ocean regions . In our idealized numerical simulations, we find the dissipation rates below the pycnocline to be generally on the order of 10-9 m² s⁻³, which is within the range of observed values (Fig. 12b). However, larger values of ϵ have been found close to the sea ice, in particular when the mixed layer is thin . We also find larger values of kinetic energy dissipation within the pycnocline region and above the pycnocline in the case of numerical simulations with smaller η (shallower pycnocline) (e.g., simulations A0, S0, S2, and W2 in Fig. d, e). These results suggest that depending on the sea ice and flow characteristics, there could be a spatio-temporal variability in the dissipation and subsequently diapycnal mixing rates in the Arctic. Note that our estimates of the dissipation rates from the numerical simulations include energy loss from all motions that are deviations from the background flow (predominantly the generated IWs) and the model viscosity is larger than the molecular viscosity of sea water. Although some previous studies have made comparison of flow energetics and in particular dissipation rates with observations e.g.,, any direct comparisons with observational values should be cautious.

Using our numerical simulation results, we evaluate how well the CIW parameterization captures the variability in KE metrics. In the region below the pycnocline (Fig. c, f), parameterized CIW values generally capture the trend in the orders of magnitudes of both EK and ϵK well, though it might overestimate the KE and dissipation rates for large Fr (weak pycnocline, A4) and underestimate them for small η (shallow mixed layer, S0). Parameterized CIW values are not as well correlated with EK and ϵK above the pycnocline (Fig. a, d) and around the pycnocline (Fig. b, e), which might be expected as the parameterization was primarily developed for IW energy flux below the pycnocline into the stratified interior. Of course, there are many real-ocean processes that are missing in our idealized numerical simulations as will be discussed in Sect. , and in this study we only analyze nine specific cases. Therefore, we caution against overinterpreting the numerical values and trends of ϵK in our numerical simulations without a more thorough parameter sweep.

Finally, when discussing the role of ice keels in the Arctic, it is important to consider climatological changes and shifts in sea ice dynamical regimes. One potential change is sea ice smoothing. Measured from aircraft, above-sea surface ice ridges have significantly decreased in height and ridge density over the last 30 years. This loss is especially prominent in parts of the Arctic that are experiencing loss of multi-year ice like the Beaufort Sea and the Last Ice Area . Smoother ice leads to reduction of atmospheric surface drag coefficient on ice ridges. While this is not direct evidence for changes in the under-sea surface keel height and density, it is plausible that there is some correlation between the above- and under-sea surface ice properties. This implies that understanding the effects of ice keels on ocean mixing is important, as these effects could be reduced if the ice ridges become smoother. A large portion of these areas that found to have sea ice ridge smoothing are within Cluster A0 (and seasonally S0 and W0) in our dataset. This cluster is characterized by relatively large J and small η that enhance kinetic energy dissipation. Therefore, smoothing of the ice keels (a reduction in h0, so a reduction in J and η) can substantially change the ice-ocean dynamics in this region.

Arctic stratification has also changed in the last several decades. For example, through analysis of water column observations, found changes in both the pycnocline depth, approximated as the mixed-layer depth z0 in this study, and the change in buoyancy across the pycnocline (Δb in this study). Specifically, they found a pan-Arctic increase in z0, possibly due to surface-layer freshening or deepening of the mixed-layer due to intensification of wind-driven mixing . Combined with potentially smaller h0 from sea-ice smoothing, this would result in an increase in η and less ice keel-induced turbulent dissipation and mixing and reduced IW drag in the upper ocean. also found an increase in Δb in the Amerasian Basin (Cluster A0 and seasonally S0 and W0), which would correspond to a decrease in Fr. Based on our numerical simulations, this decrease in Fr would reduce KE and dissipation rates. However, the relationship between the IW drag and Fr, at least in the current parameterization, is nonlinear (Fig. (right)), so further analysis is needed to assess the ocean's response to these changes.

One of the main limitations of this study is that we use data from another model to estimate the distribution of dimensional and nondimensional parameters across the Arctic. While this is unavoidable as there are still no comprehensive pan-Arctic datasets for all the variables that we would need, we need to discuss how the differences in the values of the underlying dimensional parameters between the model output and the real ocean can affect our findings. For example, showed that their model in general overestimates the sea ice drift speed over most of the Arctic and across seasons in comparison to the observational data from the National Snow and Ice Data Center Polar Pathfinder dataset. This is consistent with climate models typically overestimating sea ice drift . From , the overestimation of the sea ice drift by the model is largest during the winter in the marginal sea ice areas. These regions are part of our clusters W2 and W3 (Fig. i, l) that have relatively large CIW predicted by the parameterization (Fig. c, f). For cluster W3, this is in part because of the larger Froude number Fr (Fig. f) that is proportional to the relative sea ice speed u0. Therefore, an overestimate of the ice drift by the model could, in turn, overestimate the IW drag CIW.

In Sect. , we noted the relative importance of the regime with smaller depth ratio η values. However, as noted in , the pycnocline or mixed layer depths shallower than 10 m cannot be accurately quantified in the NEMO model, hence, they are not present in our clustering results. This means that in addition to the ranges of values of η presented in this study, smaller values of η→0 would have to be considered when conducting numerical simulations to evaluate the parameterization of the IW drag. Accurate values of ice keel depths h0 are also necessary for estimating the depth ratio η, so we also compare the distribution of ice keel depths h0 with observed values. Figure S2 shows the distribution of h0 from the entire model output (not just our filtered data within the lee wave radiation regime) with the distribution from a large keel dataset by . The distributions are in overall good agreement, especially if we consider h0<9 m as one bin, as data for h0<6 m was not presented in . Notably, other observational studies e.g., have found such smaller keel depths (h0<6 m). These comparisons suggest that the distribution of the h0 values used in this study is in a reasonable agreement with the observations.

Other limitations of the study stem from the choices made in our numerical simulations that resulted in us not considering certain physical processes. One such limitation in the numerical set-up of this study is that we neglect the skin and form drag (i.e., the turbulent ice-ocean boundary layer) by imposing free-slip boundary conditions along the ice keel boundary. While we chose to implement this approach in order to separate the effects of IWs and because our results would change due to our subjective choices of the drag coefficients, in the real Arctic Ocean, all three of the drag components would have an effect on the ice keel speeds. Because the shape of the keel in our simulations and the conceptual model is not allowed to change, we neglect the effect of turbulent heat fluxes that can melt the sea ice. This process can be complicated. For example, found that while ice keels enhance turbulence, their effects on melting can depend on blocking of the flow and trapping of fresh water. This can alter the shape of the ice keel, though we do not expect it to take effect on the short timescales of our simulations. The choice of shape to represent the ice keel also plays an important role. A recent study examining submarine sonar data found that keels can have different shapes, e.g., triangular shapes, cusp shapes like Versoria used in this study, and trapezoidal shapes that have a flatter bottom. Their study also found that the keel shape is dependent upon the keel depth and processes that the keel undergoes throughout its lifecycle, e.g., it may form in a pointier shape but flatten along the bottom due to accelerated bottom melt in the summer. Such differences in keel shapes can affect IW generation and drag.

In this study, we also assume that the wind has already transferred momentum to the ice keel to move it, and hence neglect modelling the atmosphere-ice stress. This is a common assumption for an idealized process study e.g., such as the current one aimed to isolate the dynamics of internal wave generation by the ice keels, However, in the ocean, momentum transfer within the atmosphere-ocean-ice coupled system can be simultaneous and is a more complicated process . Another nondimensional parameter, the Nansen number, which measures the ratio between the atmosphere-ice and ice-ocean drag coefficients scaled by the ratio of air and sea water densities, can be useful to characterize this coupling. Better-constrained Nansen number, possibly through further observational campaigns of measuring atmosphere-ice and ice-ocean drag coefficients e.g.,, would be important for more accurately capturing the three-way coupling in climate models.

The two-dimensional set-up of this model also neglects certain physical mechanisms, e.g., three-dimensional turbulence and three-dimensional effects due to flow splitting around the keel . Because we only model a single keel, we neglect ice keel sheltering effects from upstream keels, which have been found to affect the dynamics of the flow and skin drag parameterizations in previous modelling studies . The parameterization in also omits the effects of rotation, so we also consider motions on time scales shorter than the inertial period in our numerical simulations. The effect of rotation is two-fold. First, it can shrink the range of lee wave radiation from 0<χ<1 (non-rotating case) to fN0<χ<1 (rotating case). However, in our dataset we find that less than 0.2 % of points have χ<fN0, so perhaps this is not a substantial limitation. However, in the presence of rotation, near-inertial waves are generated on the timescales of 2π/f (order of 11–17 h). Near-inertial waves that can interact with lee waves to enhance dissipation , which is not considered here or in the parameterization. This effect of the rotation via the near-inertial wave generation can be examined further in subsequent numerical studies running the simulations for longer periods of time.

Finally, in this study, we made a particular choice of six GMM clusters based on the statistical information from the BIC score and by considering the interpretability of our results. For the purpose of the discussion here, we focus on the summer-averaged data, though similar conclusions can be made for winter clusters. Based the BIC curve (Fig. ), we should have chosen a larger number of clusters (K≈10 clusters) in order to improve the GMM's ability to capture the variability in the data. Having more clusters would have allowed us have better-constrained clusters, i.e., reduce the standard deviation of the nondimensional variable ranges within each cluster and the overlap between clusters (Figs. , ). However, this would have been too many clusters to interpret in terms of physical regimes. On the opposite end of the spectrum, we can also divide the Arctic broadly into three regimes similarly to our discussion of the results in Sect. : (1) the central Arctic with perennial ice, (2) marginal ice regions with larger η (smaller keel depth, deeper pycnocline) and without substantial IW generation, and (3) marginal ice regions with intermediate η that support IW generation. However, such broad characterization would return a wide range of nondimensional parameter values for the central Arctic, which is perhaps not an insightful result. In order to examine the tradeoff between the accuracy of representing the statistical distribution of the nondimensional parameters (i.e., large K) and ease of interpretation (i.e., small K), we show the spatial distribution of GMM clusters for K=4, 5, and 7 in Fig. S3 for comparison with K=6 in Fig. b. Too few clusters (e.g., K=4–5) leaves the entire central Arctic region as a single cluster. However, for K=7, the GMM algorithm returns a relatively consistently clustered centered Arctic, separating the Amerasian and Eurasian basins, and the outer eastern Eurasian seas. Increasing the number of clusters K in this range (6–7) seems to predominantly break marginal regions into even smaller clusters. Based on this analysis, we choose K=6 to strike an overall balance, though recognizing that this choice is somewhat subjective.