5.1.1. Salinity and temperature

The macroporosity decrease is related to the increase in the block microporosity (Eq. 8) and thus depends on the salinity and temperature of ice blocks that form the ridge. The salinity range of 6–10 in Fig. 6a corresponds to observations of the salinity of young ice with an ice thickness of 0.3 m (e.g., Cox and Weeks, Reference Cox and Weeks1974; Kovacs, Reference Kovacs1996). With a freezing point of $-1.9^{\circ}\mathrm{C}$ and seawater salinity of $S_\text{w} = 34.8$, this corresponds to a relative salinity range of $0.17 \lt S_i/S_\text{w} \lt 0.29$. To produce Fig. 6b for Baltic Sea brackish water ( $T_\text{f} = -0.19^{\circ}\mathrm{C}$, $S_\text{w} = 3.5$), the same relative salinity range has been chosen, which is reasonable for the Bay of Bothnia (e.g., Leppäranta and others, Reference Leppäranta, Lensu, Kosloff and Veitch1995). The temperatures in Fig. 6a reflect warm ( $-5^{\circ}\mathrm{C}$) and cold ( $-15^{\circ}\mathrm{C}$) ice surface temperatures in the Arctic, although more extreme values are possible. For the Baltic Sea, a more moderate ice surface temperature range (−2 to $-10^{\circ}\mathrm{C}$) was chosen.

The higher end of the assumed ice salinity range 8–10 has, while realistic for thin ice of 0.2–0.5 m thickness, rarely been reported for ridge keel ice blocks. This may be explained as follows for the different parts of the keel. First, for the consolidated part, the freezing front has passed a second time through the ice and will likely create a new equilibrium salinity for slower freezing (than for the blocks). Second, in terms of the ice blocks below the consolidated layer, the proposed thermodynamic adjustment will transform them into blocks of high microporosity. From such ice, one can expect high brine drainage during sampling and thus an underestimate of the in situ salinity. Observations of ice block density in the range 860–890 $\mathrm{kgm}^{-3}$ (e.g., Leppäranta and Hakala, Reference Leppäranta and Hakala1992; Kankaanpää, Reference Kankaanpää1997; Hoyland, Reference Hoyland2007) support such salinity loss. Leppäranta and others Reference Leppäranta, Lensu, Kosloff and Veitch(1995) even noted that air bubbles due to diver’s breath could go trough several ice blocks in the keel. Laboratory experiments with submerged ice blocks, where the salinity decreased while air porosity increased (Bailey and others, Reference Bailey, Taylor and Croasdale2015), also support this view.

An interesting result of Fig. 6 is that the model predicts a somewhat stronger change in the macroporosity for the Baltic Sea compared to the Arctic, even though the temperatures are assumed higher. This is explained by the fact that microporosity scales reciprocally with temperature (Eq. 8), which implies a larger change in the Baltic Sea with a 10 times higher freezing point. The salinity thus plays a more important role in the macroporosity change in the Baltic Sea—the warm and cold temperature regimes overlap in Fig. 6b. It should be noted that, when considering Arctic Seas with intermediate seawater salinities, one should apply the same relative ice salinity range (e.g., for a sea with $S_\text{w} \approx 28$ the ice salinity range from 5 to 8 rather than from 6 to 10), resulting in a similar macroporosity range as in Fig. 6a.

In the Arctic field data, there are two observations of high macroporosity close to the loose packing prediction. One of these is a ridge for which Hoyland Reference Hoyland(2007) has reported that it probably formed from very warm ice with a temperature close to the freezing point of seawater. The second data point is from the study by Bonath and others Reference Bonath, Petrich, Sand, Fransson and Cwirzen(2018). However, the latter authors have estimated the age of the high porosity ridge by analysis of ice drift tracks, temperature and wind forcing, suggesting rather cold formation conditions, not consistent with the high porosity point in Fig. 6a. The only Arctic data point for which the ridge formation conditions are reasonably known is from the study by Coon and others Reference Coon, Echert and Knoke(1995). These authors report an ice temperature of $-9.4^{\circ}\mathrm{C}$ at a depth of 15 cm in the consolidated part of a young ridge, which appears to be consistent with the cold formation condition expected from the location of their data point in Fig. 6a. For the Baltic sea, the spread is larger. The highest porosities refer to a shallow ridge from Kankaanpää Reference Kankaanpää(1997) and are the most uncertain because these were corrected for a consolidated layer that reached half of the keel depth. One ridge from Leppäranta and others Reference Leppäranta, Lensu, Kosloff and Veitch(1995), visited several times during winter, shown with three diamonds at the same aspect ratio, formed under cold conditions, which is consistent with Fig. 6b. Given the unknown formation conditions of most ridges, the overall agreement between model and observations is reasonable.

In Fig. 7, the model approach is summarized by choosing settings considered typical for young ridged Arctic sea ice (aspect ratio of 4, ice salinity $S_i = 8$, $T_\text{f} = -1.9^{\circ}\mathrm{C}$, $S_\text{w} = 34.8$). The figure highlights how the changes in micro- and macroporosity depend on the ice surface temperature during ridging, as well as the total porosity. The microposity is computed as a porosity average over an ice block with a linear vertical temperature gradient and constant salinity. Right after ridge formation, ϕ ₀ does not depend on temperature, yet v _b0 and ϕ_t do so. After thermodynamic adjustment (about a day later), v _b1 does not depend on initial temperature, yet ϕ ₀ and ϕ ₁ are lower the lower the ice temperature initially had been. For warm ice, when the ice temperature is close to the freezing point, no thermodynamic adjustment takes place, the total porosity of the ridge keel is above 0.5 and all porosity metrics remain unchanged. Note that for cold ice, the total porosity is smaller, both initially and after thermodynamic adjustment, with its decrease given by the specific heat term in Eq. 11.

Figure 7.

Predicted average macro- and microporosity ϕ and v_b and total porosity ϕ_t change during thermodynamic adjustment, for a pressure ridge keel forming from ice blocks with a length to thickness ratio of 4 and salinity $S_i = 8$ in water at the freezing point of $T_f = -1.9^{\circ}\mathrm{C}$ ( $S_w = 34.8$). The initial macroporosity from random loose packing is $\phi_0 \approx 0.44$, shown as upper solid line. The macroporosity ϕ ₁ after thermodynamic adjustment (after the ridge has reached the freezing point $-1.9^{\circ}\mathrm{C}$) is shown as the solid curve (lower bound of the dark grey shaded area) in dependence on initial ice surface temperature. The corresponding increase in the microporosity of ice blocks from $v_{b0}$ to $v_{b1}$ is shown as the lower hatched area between stippled curves, reaching $v_{b1} = 0.21$ after thermodynamic adjustment. Note that $v_{b0}$ is computed by averaging over the brine volume profile, assuming a linear temperature gradient in the ice blocks prior to ridging. The total porosity, shown with dashed curves, is computed as $\phi_{t} = \phi + (1-\phi)v_{b}$.

5.1.2. Aspect ratio and ice block geometry

In Fig. 5a, the numerical simulations with GeoDict were compared with the empirical formula from Zou and Yu Reference Zou and Yu(1996). While both show a similar trend, the simulated porosities are 0.05–0.08 smaller. This is not unexpected, as discussed by Seckendorff and Hinrichsen Reference Seckendorff and Hinrichsen(2021), because true random loose packing is usually not realized in particle drop experiments. Geodict, however, appears to simulate this idealized state. The simulated solid fraction at an aspect ratio of 1 is consistent with the random loose packing of spheres noted above ( $\phi_{p} \approx 0.056$ or ϕ ≈ 0.044). While the empirical formula from Zou and Yu Reference Zou and Yu(1996) agrees better with laboratory packing experiments, including sea ice rubble, the numerical simulations allow us to study some details, like the effects of particle shape and boundaries. In Fig. 5a, the difference between square plates and disks is shown for the empirical and numerical model. The formula from Zou and Yu Reference Zou and Yu(1996) only predicts a difference of 0.01 for the loose packing of plates and disks, due to the difference in their sphericity. The numerical simulations show that one can expect a larger difference between the disks and square plates and that it increases with aspect ratio, where the difference is approximately 0.03 for ϵ = 4.

The blocks in a sea ice ridge are not idealized square plates of a single length to thickness ratio, but may have different shapes and consist of blocks of different size. For example, several authors have described that blocks in ridges have a length (L _b) to width (W _b) ratio close to 3/2 (Veitch and others, Reference Veitch, Kujala, Kosloff and Leppäranta1991; Kankaanpää, Reference Kankaanpää1997; Strub-Klein and Sudom, Reference Strub-Klein and Sudom2012; Guzenko and others, Reference Guzenko, Mironov, IMay and Porubaev2022), while Kankaanpää (Reference Kankaanpää1997) noted that block shapes are not simply rectangular. To study these differences, additional numerical simulations have been performed and are compared in Table 2. Also shown is the (small) effect of the empirical prediction due to sphericity differences. The numerical simulations show (i) a higher porosity for packing of quadratic blocks compared to disks with the same aspect ratio; (ii) a slightly higher porosity for a uniform distribution of aspect ratios and (iii) a smaller porosity for rectangular blocks with a horizontal aspect of 3:2. Accounting for such details will change the porosity prediction by a few percent, falling in the range 0.49–0.53 for an aspect ratio of 4. For natural sea ice ridges, effects (ii) and (iii) will likely cancel to some degree. The problem can thus be reasonably approximated by square plates of a fixed aspect ratio.

Table 2.

Random loose packing porosities predicted by Eq. 3 from Zou and Yu Reference Zou and Yu(1996), based on average particle sphericity, as well as the presented numerical simulations with GrainGeo (2021). The repeatability of the numerical simulations is ≈ 0.004.

While the porosity observations are limited, they show some consistency with the derived increase in macroporosity with aspect ratio, in particular when focusing on three studies. The data from Guzenko and others (Reference Guzenko, Mironov, IMay and Porubaev2022, Reference Guzenko2023) for more then 100 Arctic sea ice ridges follow, for 4 of 5 regions, the predicted increase with aspect ratio. In addition, detailed observations of the Baltic Sea ice ridge by Leppäranta and Hakala Reference Leppäranta and Hakala(1992) and Leppäranta and others Reference Leppäranta, Lensu, Kosloff and Veitch(1995) indicate a slight increase in macroporosity with aspect. Due to the effects of temperature and salinity, more observations would be needed for a proper validation.

5.1.3. Timescale of thermodynamic adjustment

The proposed model predicts the initial macroporosity after a ridge has piled up, followed by a decrease when it reaches a new thermodynamic equilibrium. The time scale of the thermodynamic adjustment will then depend on the congelation rate V of the cold ice blocks. A crude estimate may be obtained by assuming a similar freezing rate as of level ice from which the ridge forms, $V = (K_\text{i} \Delta T)/(H_{b} L_\text{v})$, with ice thermal conductivity K _i, ice-seawater temperature difference $\Delta T$, ice block thickness H _b and the volumetric latent heat of fusion L _v. I assume thin ice with $\Delta T = 10$ K, $H_{b}=0.2$ m and taking $K_\text{i} = 2.2\ \mathrm{W~m}^{-1}\,\mathrm{K}^{-1}$ and $L_\text{v} = 3.0 \times 10^8$ J m ${^3}$. With half the thickness H _b, as heat loss takes place in both directions normal to the horizontal ice block planes, the effective temperature gradient is $\Delta T = 10$ K over $H_{b}/2=0.1$ m, giving a growth velocity of V ≈ 2.6 mm/h on both block surfaces. Suppose an increase (see Fig. 6) in the solid fraction from 0.55 to 0.65, by approximately one sixth, the timescale to increase the thickness of the block by that fraction is $H_{b}/(2V)/6 \approx 0.2/(2 \times 0.0026)/6 \approx 6.4$ h. This approximation neglects the 3-dimensional nature of blocks, the initial temperature distribution and convection within the blocks and the void space of the ridge. However, it indicates that ridges would have to be sampled very rapidly after their formation to monitor the thermodynamic adjustment.

Shestov and Marchenko Reference Shestov and Marchenko(2016a) performed laboratory experiments on sea ice blocks submerged in water at a higher freezing point Shestov and Marchenko Reference Shestov and Marchenko(2016a), to study the growth of the ice block and the corresponding decrease in macroporosity. Ice samples were thinner (7 cm ) than in the field, and the example above had higher salinity (≈ 10 ppt), smaller temperature differences resembling summer conditions ( $2^{\circ}\mathrm{C}$) and a geometry different from a random ice rubble field. The experiments also focused on changes in salinity during cyclic exposure of ice blocks to saline and fresh water. However, the relative increase in ice mass during the submersion in freshwater was fast, typically 0.15 within only 3 h, comparable to my estimate above. Another estimate of the adjustment time can be obtained from ice rubble laboratory experiments reported by Bailey and others Reference Bailey, Taylor and Croasdale(2015). In that study, ice cylinders (9 cm diameter x 20 cm height) with a salinity of 6–7 ppt and a temperature of $-10^{\circ}\mathrm{C}$ were submerged in a 32 ppt NaCl salinity solution at freezing temperature ( $-1.6^{\circ}\mathrm{C}$). Most of the temperature adjustment appeared to take place within 3–7 h. From these considerations, one can expect that the adjustment time is proportional to the block ice thickness and should be less than a day even for moderately (0.7 m) thick ice blocks.

Another assumption for the proposed thermodynamic adjustment is that the brine, rejected during freezing of the macroscopic void space, is rapidly draining and replaced by less saline seawater. Otherwise, the seawater in the void space would increase its salt concentration and retard freezing. Thus, the void space must be permeable enough to allow for rapid gravity-driven brine drainage. The problem may be compared to desalination of young level sea ice, where brine convection drives large changes in the solid fraction during hours to a few days. With the length scales of young ice brine convection being millimeters to centimeters, while void sizes are proportional to ice block sizes (decimeter to meter), the assumption of rapid brine release is reasonable. However, while the salt fluxes probably do not play a major role in the timescale of thermodynamic adjustment, they could be relevant for the salinity evolution of a ridge.

5.1.4. Oceanic heat fluxes

In the presented calculations, it is assumed that the internal redistribution of latent heat takes place in the absence of oceanic and atmospheric heat fluxes. What role may oceanic heat fluxes play during the initial phase of ridge formation? Amundrud Reference Amundrud(2006) have both observed and modeled keel melt rates during summer and shown that melt rates depend on keel draft. They found typical melt rates in the range 1–2 centimeters/day per meter keel depth, a result that is consistent with more recent results obtained during the MOSAiC expedition (Salganik and others, Reference Salganik2023b). As an example, for an unconsolidated keel of 4 m thickness (e.g., Guzenko and others, Reference Guzenko, Mironov, IMay and Porubaev2022) with a solid fraction of 0.7, this corresponds to a heat flux of 100 to $200\,\mathrm{W/m}^{2}$. While such oceanic heat fluxes have been observed during the summer season, they are an order of magnitude smaller for most winter conditions (e.g., Salganik and others, Reference Salganik2023b). With regard to the present work, these fluxes need to be compared to the internal redistribution of latent heat by thermodynamic adjustment. With a typical change in ϕ by 0.05–0.15 (see Fig. 3), the equivalent ice thickness change in a 4-m keel is 0.2–0.6 m. Using an adjustment time scale of one day (due to the estimates from the previous section), this corresponds to a heat flux of 700 to $2100\,\mathrm{W/m}^{2}$. The estimate thus shows that oceanic heat fluxes during winter are unlikely to contribute considerably to the energy budget during initial ridge formation. Over time, however, oceanic heat fluxes may play some role. For example, an oceanic heat flux of $5\,\mathrm{W/m}^{2}$ (Lei and others, Reference Lei2022; Salganik and others, Reference Salganik2023b) over 4 months would, if distributed equally over the depth of a 4-m ridge keel, increase the total porosity by 0.05.

Hoyland and Liferov Reference Hoyland and Liferov(2005) investigated the early consolidation of artificially created sea ice ridges and estimated exceptionally high oceanic heat fluxes of up to $235\,\mathrm{W/m}^{2}$ during winter. They budgeted the latent heat consumption in the ice blocks (corresponding to the second term on the right hand side of Eq. 9) with the latent heat release from the growth of the consolidated layer (thickness change times initial macroporosity ϕ ₀) and atmospheric heat loss to obtain the oceanic heat flux as a residual (their equation 5). However, as in their equation 3 (p. 52) for the ice block heat term the factor $(1-\phi_0)$ is lacking, the heat fluxes appear overestimated. They also had to use the macroporosity (0.34) of a third ridge for their calculations, noting that the ice of the investigated ridges was too soft. A higher initial macroporosity would have further reduced their derived oceanic heat fluxes. A revised heat budget based on the present study (with Eq. 9 and a typical initial $\phi_0 = 0.45$) gives an order of magnitude smaller oceanic heat fluxes, much more consistent with other observations. Hence, it seems unlikely that oceanic heat fluxes were relevant during the initial phase of ridge formation described by Hoyland and Liferov Reference Hoyland and Liferov(2005). Instead The heat transfer during ridge formation is dominated by latent heat exchange between macro- and microporosity inside the ridge.

5.2.1. Drilling

Only a few investigators have reported uncertainty estimates for their reported macroporosities. Leppäranta and others Reference Leppäranta, Lensu, Kosloff and Veitch(1995) mentioned a standard deviation of 0.05 for an average porosity ϕ ≈ 0.30 obtained from 19 to 27 drill holes. Kharitanov Reference Kharitanov(2019) has tabulated results of the unconsolidated keel macroporosity of an Arctic pressure ridge based on 881 drillholes along 8 lines. The average and standard deviation for these 8 lines was $0.28 \pm 0.04$. Salganik and others Reference Salganik(2023a) provided, for two transects of a ridge during MOSAiC a macroporosity of $0.29 \pm 0.15$ based on overall 16 drillholes. These results indicate the variability of porosity measurements from single drill holes and transects. Assuming a normal distribution of porosity values, one can estimate the uncertainty of the mean values by dividing the standard deviations by the square root of the number of measurements. The results from Leppäranta and others Reference Leppäranta, Lensu, Kosloff and Veitch(1995) then correspond to an uncertainty of 0.010 of the mean, while those of Salganik and others Reference Salganik(2023a) to an uncertainty of 0.04. The value of 0.014 for the results from Kharitanov Reference Kharitanov(2019) reflects the mean of different transects.

5.2.2. Slush and snow

Several investigators have characterised ridge keels not only by ice and voids yet also in terms of slush or soft ice. Coon and others Reference Coon, Echert and Knoke(1995) noted a macroporosity of 0.35 that included 0.1 of mushy ice. Leppäranta and Hakala Reference Leppäranta and Hakala(1992) mention the presence of slush and present a ridge section for which a slush fraction of 0.09 can be estimated for the keel. Kankaanpää Reference Kankaanpää(1997) report slush fractions between 0 and 0.12 for Baltic sea ice ridges, with an average of 0.05. With a solid ice fraction in slush of $\approx 0.2-0.3$ (Maus and la Rosa, Reference Maus and la Rosa2012), a slush macroporosity of 0.05 to 0.1 would correspond to a keel-averaged solid ice fraction of 0.01 to 0.03 in the slush. A potential candidate for the slush in ridges is snow on ice blocks. For an estimate a snow layer of $1/5$ to $1/10$ of ice thickness is assumed (e.g., Kacimi and Kwok, Reference Kacimi and Kwok2022) and a snow density of $1/3$ of solid ice. For a macroporosity of 0.4 this leads to ( $1/5$ to $1/10) \times 1/3 \times (1-0.4) \approx 0.01$ to 0.02 for the snow-derived solid ice fraction in a ridge. Hence, the presence of slush in ridge keels may be well related to snow on ice. The contribution of slush to the total latent heat budget of a ridge is likely a few percent.

5.2.3. Boundary effects

The agreement of the model with laboratory experiments is promising, yet raises the question of how boundary layers affect the porosities of the ridge obtained in the field. Indeed, the case is different then. Bottom and side boundary layers lead to a larger average porosity when filling a fixed volume container. However, for field observations, when drilling downward, the bottom of the keel is always defined by the last ice block. One would then rather expect an opposite boundary layer effect, and that field observations based on drilling give a porosity that is smaller than the porosity bound for an infinite volume. To estimate this effect on the keel-averaged macroporosity consider that the maximum sail height and keel depth of typical first-year ridges have been found to scale with the square root of the block thickness as

(15)\begin{equation} H_{max} = a_{max} H_b^{1/2}, \end{equation}

where typical values for a_max for the sail height are in the range $3.7\,\mathrm{m}^{1/2}$ (Tucker and others, Reference Tucker, Sodhi and Govoni1984b; Strub-Klein and Sudom, Reference Strub-Klein and Sudom2012) to $4.3\,\mathrm{m}^{1/2}$ (Guzenko and others, Reference Guzenko, Mironov, IMay and Porubaev2022). Melling and Riedel Reference Melling and Riedel(1996) and Amundrud and others Reference Amundrud, Melling and Ingram(2004) have, based on moored sonar observations, estimated upper bounds for the maximum keel depth in the range 16 to $20\,\mathrm{m}^{1/2}$. The average keel depth maximum was not given by the latter authors, yet according to their data plots, appears to be in the range 13 to $15\,\mathrm{m}^{1/2}$, while Guzenko and others Reference Guzenko, Mironov, IMay and Porubaev(2022) determined $13.4\,\mathrm{m}^{1/2}$. Reasonable numbers for a_max are thus $4\,\mathrm{m}^{1/2}$ for the sail and $14\,\mathrm{m}^{1/2}$ for the keel. Corresponding numbers for the average depth of triangular ridges would be half these values. However, ridges often are better approximated by a trapezoidal shape (Timco and Burden, Reference Timco and Burden1997; Strub-Klein and Sudom, Reference Strub-Klein and Sudom2012). The statistics provided by Guzenko and others Reference Guzenko, Mironov, IMay and Porubaev(2022) give average sail heights and keel depths that are 10 to 20% larger than a triangular shape approximation. Based on these results I will use an average sail height of $H_{s, av} \approx 2 H_b^{1/2}$ and an average keel depth of $H_{k,av} \approx 8 H_b^{1/2}$ in the following calculation.

To estimate the effect of the corresponding boundary layer on macroporosity, I divide the average depth of the ridge keel $H_{k,av}$ into a part $(H_{k,av} - H_b)$, where the solid fraction takes the loose packing limit $\phi_{p0}$ and H_b, where the solid fraction is one (representing the bottom ice blocks in the keel). The average solid fraction ϕ_p in the ridge keel may then be approximated as the sum of the major keel volume term $\phi_{p0}(H_{k,av} - H_b)/H_{k,av}$ and the block thickness term $H_b/H_{k,av}$, leading to

(16)\begin{equation} \phi_{p} = \phi_{p0} + (1-\phi_{p0})\frac{H_b}{H_{k,av}}. \end{equation}

Inserting the average keel depth $H_{k,av} \approx 8 H_b^{1/2}$ as well as the macroporosity $\phi_0 = (1-\phi_{p0})$ one obtains

(17)\begin{equation} \phi_{p} - \phi_{p0} = \phi_{0} - \phi \approx \phi_{0}\frac{H_b^{1/2}}{8}. \end{equation}

for the average solid fraction increase (macroporosity decrease) due to the bottom ice block in a keel. Interestingly, the effect increases with ice block thickness. Assuming $\phi_0 = 0.45$ and an ice block thickness range $0.15 \lt H_b \lt 0.7$ m, one finds that the solid fraction increases (macroporosity decreases) by 0.02 to 0.05. Making a similar estimate for the sail one obtains Eq. 17 with a four times smaller denominator of about $2\,\mathrm{m}^{1/2}$. The resulting effect on the sail porosity is thus much larger, being in the range 0.09 to 0.19 for the same block thickness range. Before discussing these estimates, as well as differences between keel and sail, it is noted that a low porosity regime has been reported at the very bottom of the keel by many authors (Leppäranta and Hakala, Reference Leppäranta and Hakala1992; Kharitanov, Reference Kharitanov2019; Reference Kharitanov2021b). Such a regime is the logical consequence of interpreting downward drilling results, where a solid ice sheet defines the lower keel boundary.

5.2.4. Differences between keel and sail

The analysis indicates two major differences between sail and keel. The first is the thermodynamic adjustment, that only takes place in the keel, implying larger sail porosities. The second is the top and bottom ice block effect, that leads to a lower average macroporosity, and is primarily expected in the sail. Both effects combined have the potential to lower the macroporosity by up to 0.2 in the sail and the keel, depending on ice thermodynamics and thickness. Observations show on average a lower porosity in the sail compared to the keel. For Baltic sea ice ridges (small block thickness), Kankaanpää Reference Kankaanpää(1997) reported a keel porosity of 0.30 versus 0.20 for the sail. A similar difference has been found by Guzenko and others Reference Guzenko(2023) for 104 Arctic ridges, with average sail porosities of 0.17 and 0.23 versus 0.26 and 0.31 in the keel. The smaller values are for larger ridges, a trend that is consistent with Eq. 17, as thicker ice blocks tend to form larger ridges.

Surkov and others Reference Surkov, Astafyev and Polomoshnov(1997) reported an interesting result for three of their box-filling experiments (macroporosity in Fig. 3b). After determining the porosity the authors poured out the water, let the ice blocks freeze together, and cut sections on which they determined linear porosity estimates to resemble drilling. The linear macroporosity was on average a 0.09 smaller than obtained by the box filling (0.33 compared to 0.42). These results are consistent with the present estimates of a boundary effect and underline the challenges to compare porosity estimates from the lab and field. It is further noted that Eq. 17 is based on simple geometry and averaging, and that the effect for a sail may be more complex. The build up of the sail may involve different fracture modes and forces, as submersion of ice blocks takes place under ten times smaller buoyancy. More rafting in the sail may result in lower porosities and also a different block to thickness aspect ratio in the sail could make some difference. The two main points that follow from this brief discussion are that sail porosities may be much more affected by the geometry of a few individual blocks, and thus do not necessarily resemble the porosity of the major volume of a ridge - which is in the keel.

To close the sail-keel comparison it is noted that some investigators have estimated keel porosity ϕ in dependence of sail porosity ϕ_s assuming isostatic equilibrium between keel and sail (Bowen and Topham, Reference Bowen and Topham1996; Timco and Burden, Reference Timco and Burden1997). Neglecting effects of snow, the keel porosity may be estimated as

(18)\begin{equation} \phi = 1 - \left(1 - \phi_{s}\right)\frac{A_s}{A_k}\frac{\rho_{is}}{\left(\rho_w-\rho_{ik}\right)}. \end{equation}

This method requires observations of the cross-sectional sail and keel areas, A_s and A_k and the ice densities in the sail (ρ_is) and keel (ρ_ik), while the seawater density ρ_w is often reasonably known. Assuming a sail porosity of $\phi_s =0.30$, Bowen and Topham Reference Bowen and Topham(1996) estimated ϕ = 0.50 and Timco and Burden Reference Timco and Burden(1997) arrived at ϕ = 0.38. While these estimates appear high, they are not unreasonable, considering that this method of calculation of ϕ includes the microporosity v_b of the ice blocks in the keel. Hence, iso-static estimates of keel porosity may have some potential, when combined to other observations, and may be of relevance for remote-sensing based approaches. Again, such estimates will depend on several observational variables, in particular the macroporosity in the sail. The approach may be compared to other indirect, non-destructive methods to measure macroporosity discussed next.

5.2.5. Indirect, non-destructive methods

Several indirect methods have been applied to estimate the macroporosity of pressure ridges. The first approach was proposed by Leppäranta and Hakala Reference Leppäranta and Hakala(1992) based on comparison of the thickness of the consolidated layer in a ridge and the thickness to which the level ice has grown since ridge formation. It can be illustrated by a simplified form of Stefan’s growth law

(19)\begin{equation}H^2-H_0^2=\frac{a_q(t-t_0)}{L_v}\end{equation}

for ice thickness H, where H ₀ is the ice thickness at ridge formation, a_q reflects the growth conditions and physical properties, L_v the latent heat of fusion, and $t-t_0$ is the time since ridge formation. Assuming an effective latent heat $\phi L_v$ for the ridge keel, as part of the ice is already frozen, and similar a_q, one can estimate ϕ from the thickness of level ice H, the consolidated layer H_c and the initial H ₀. Leppäranta and others Reference Leppäranta, Lensu, Kosloff and Veitch(1995) first used the approach to estimate the consolidated layer thickness consistent with the observed ϕ. The method has several limitations (Leppäranta and others, Reference Leppäranta, Lensu, Kosloff and Veitch1995; Salganik and others, Reference Salganik, Hoyland and Maus2020): the initial thickness H ₀ in the keel is often unknown, the effective thermal properties (e.g. snow cover) may be different for ridges and level ice, and the growth law of thin ice is affected by a linear term. In line with the current study one also needs to take into account that both the consolidated ice in the keel and the level ice are porous, and that the estimate should be interpreted as total porosity. Hence, while comparing the thickness of level ice and consolidated layer is the easiest one to estimate porosity, it is also subject to the largest uncertainty, reflected in the range in observed thickness ratios (e.g., Hoyland, Reference Hoyland2007; Kharitanov, Reference Kharitanov2008; Salganik and others, Reference Salganik, Hoyland and Maus2020). However, applying this approach Salganik and others Reference Salganik(2023a) estimated a porosity of 0.53 for a ridge keel during the MOSAiC expedition. Furthermore, they derived another porosity estimate ( $0.43-0.46$) based on comparison of the change in consolidated layer thickness with directly observed (by temperature strings) conductive heat fluxes. The macroporosity obtained by drilling was $0.29+-0.15$ and much smaller. Interpreting the higher estimates as total porosity (0.43-0.53) is consistent with the model results in Fig. 7.

The macroporosity of a sea ice pressure ridge has recently also been determined by nuclear magnetic resonance (Nuber and others, Reference Nuber2013) allowing for a 3D integrated porosity estimate. For the upper part of the keel a porosity of $0.30 \pm 0.07$ was found, for the lower part a slightly higher value of $0.40 \pm 0.10$. The corresponding macroporosity estimates based on drilling were only 0.10 and 0.27. As Nuclear Magnetic Resonance (NMR) is suitable to derive total water content, these values and differences are also consistent with the predictions in Fig. 7. So far this method has only been applied during one field campaign.

Last but not least, Komorovskii Reference Komorovskii(1984) monitored a pressure ridge that formed during the closure of a lead. From budgeting of lead and ridged ice volumes he estimated an initial ridge macroporosity ϕ ₀ of 0.52. The blocks in the sail had an aspect ratio in the range 2.5 to 10, for which Eq. 3 gives the range 0.42 to 0.52, in reasonable agreement. Although rarely achieved in the field, this type of budgeting underscores the importance of properly quantifying the redistribution of lead ice to ridges.

5.3.1. Thermodynamic modeling

A few studies have focused on the thermodynamics involved during the early phase of ridge formation. Leppäranta and others Reference Leppäranta, Lensu, Kosloff and Veitch(1995) only considered the effect of the specific heat term in Eq. 8 on the macroporosity. While the contribution of the microporosity to the total porosity has been discussed in some studies (Hoyland and Liferov, Reference Hoyland and Liferov2005; Nuber and others, Reference Nuber2013; Salganik and others, Reference Salganik, Hoyland and Shestov2021), these have not considered the thermodynamic adjustment of submerged ice blocks and resulting micro- and macroporosity change. For example, as discussed above, Hoyland and Liferov Reference Hoyland and Liferov(2005) associated the microporosity change (the second term on the right hand side of my Eq. 8) with an oceanic heat flux, in contrast to budgeting it with a corresponding macroporosity change.

The first authors, to my knowledge, who pointed out the important role of the microporosity during ridge consolidation were Shestov and Marchenko Reference Shestov and Marchenko(2016b). They studied the case when summer melt water with higher freezing point drains into the void space of a ridge keel. In that situation, the same thermodynamic process as proposed in the present study can be expected, with microporosity increasing on the expense of macroporosity. In their theoretical analysis, Shestov and Marchenko Reference Shestov and Marchenko(2016b) used an initial ice block temperature of $T_{i0} =-2^{\circ}\mathrm{C}$, ice salinities of 3 and 6 and seawater temperatures (corresponding to $T_{i1}$) in the range $-0.2 \ {\text{to}}-2^{\circ}\mathrm{C}$ and compared the results for a closed volume and the case when seawater is washing out the void space. The latter case is comparable to the assumption that brine convection is rapidly exchanging enriched salinity in the void space. Shestov and Marchenko Reference Shestov and Marchenko(2016b) also report results for ice blocks of an average temperature of $-5^{\circ}\mathrm{C}$ and salinity 6 submerged in seawater at $-1.8^{\circ}\mathrm{C}$ (their Table 1). Their macroporosity reductions are within 0.01 of those predicted by Eq. 11. The small but significant difference may relate to a different freezing point relationship used. Another difference is that, while the present derivation assumes that newly accreting ice has the same porosity as the existing and transforming ice blocks, Shestov and Marchenko Reference Shestov and Marchenko(2016b) assume that all newly forming ice is fresh, which implies a smaller macroporosity decrease derived by Shestov and Marchenko Reference Shestov and Marchenko(2016b) compared to Eq. 11. Disregarding these minor differences, the present study shows that the concept proposed by Shestov and Marchenko Reference Shestov and Marchenko(2016b) is also highly relevant for winter conditions.

In their study of sea ice ridges from the MOSAiC expedition, Salganik and others Reference Salganik(2023a) also investigated the consolidation of ridge keels during spring and summer. Based on temperature profile data, they proposed that 2 m of the keel solidified during a period of 1 1/2 month from May to June. For a macroporosity of 0.3, this roughly corresponds to a heat loss of $50\,\mathrm{W/m}^{2}$ during a period when conductive heat transfer was an order of magnitude smaller. The authors suggested that refreezing slush and melt water, protruding into the ridge keel, were responsible for this rapid ridge consolidation. This hypothesis lacks a physical process for latent heat removal, as meltwater should not be able to attain high enough supercooling to freeze a void by more than a few percent. I also suggest that downward transport of partially frozen slush against gravity is unlikely. However, a consistent physical explanation may also be given here in terms of the process proposed in the present study and by Shestov and Marchenko Reference Shestov and Marchenko(2016b): meltwater with a higher freezing point, percolating into the keel, will lead to thermodynamic adjustment, also here resulting in larger micro- and smaller macroporosity. One may then suggest that the transition from macroscopic voids to microscopic pores would change the temperature field in the keel, which Salganik and others Reference Salganik(2023a) interpreted as rapid consolidation. With this interpretation, one could rather speak of redistribution of latent heat during the melt season, than of consolidation.

5.3.2. Packing versus compaction

Kharitanov (Reference Kharitanov2021a, Reference Kharitanov2021b) proposed that compaction by gravity leads to an exponential vertical porosity dependence in a sea ice ridge. As an example, he shows a predicted profile of macroporosity that decreases from 0.48 at the keel bottom to 0.14 about 10 m up in the keel (Fig. 3 in Kharitanov Reference Kharitanov(2021b)). First, this corresponds to a vertical macroporosity change that is 2–3 times larger than observed (Leppäranta and others, Reference Leppäranta, Lensu, Kosloff and Veitch1995; Kankaanpää, Reference Kankaanpää1997; Bonath and others, Reference Bonath, Petrich, Sand, Fransson and Cwirzen2018; Kharitanov, Reference Kharitanov2019, Reference Kharitanov2020b). For example, a representative average of thousands of thermal drilling profiles (Pavlov and others, Reference Pavlov, Kornishin, Efimov, Mironov, Guzenko and Kharitanov2016) shows a porosity of 0.30–0.40 near the bottom, decreasing to 0.25 at a level 8 m up in the keel. Second, the approach by Kharitanov Reference Kharitanov(2021b) predicts a much lower macroporosity than expected even for random dense packing. Third, the described behavior is not observed in general for random packing of other materials, where the vertical porosity gradient is generally small (Zou and Yu, Reference Zou and Yu1995; Philippe and Bideau, Reference Philippe and Bideau2002; Seckendorff and Hinrichsen, Reference Seckendorff and Hinrichsen2021). Last but not least, it appears to be more appropriate to consider compaction as a time-dependent transition from a loose to a denser packing state, driven by mechanisms like shaking/tapping (e.g., Knight and others, Reference Knight, Fandrich, Lau, Jaeger and Nagel1995; Richard and others, Reference Richard, Nicodemi, Delannay, Ribiere and Bideau2005), fluid flow (Gauthier and Gondret, Reference Gauthier and Gondret2019) or freeze-thaw cycling (Ludewig and others, Reference Ludewig, Vandewalle, Dorbolo, Pakpour and Lumay2015). The porosity change is then bounded by the difference between random dense and loose packing and thus not larger than ≈ 0.1 (see Fig. 3). One can conclude that there are fundamental differences between the compaction model proposed by Kharitanov (Reference Kharitanov2021a, Reference Kharitanov2021b) and the present approach to link the keel macroporosity to random loose packing. The solution presented here is more consistent with laboratory experiments, numerical simulations and existing theories of packing and compaction. However, it also has its limitations that will be summarized in the final discussion section.

5.4.1. Differences between keel and sail

Most block dimensions, also those summarized in this study, are from measurements in the sail. This means that the random loose packing model can only be validated by comparing block aspect ratios from the sail with macroporosities from the keel. So far there have been only a few comparisons of block dimensions in keel and sail. Kankaanpää Reference Kankaanpää(1997) has noted larger blocks in the keel compared to the sail, but also this was based on little data. Leppäranta and others Reference Leppäranta, Lensu, Kosloff and Veitch(1995) have compared block sizes in the sail with chord lengths from keel drilling—with little difference when comparing means, minima and maxima. Also Surkov Reference Surkov(2001) has computed ice chord lengths in the keels of ridges in the Sea of Okhotsk and the Baltic Sea and typically found values 2–3 times the block thickness. Such a chord length value can be expected when blocks of 4 times the ice thickness are randomly oriented in the keel. The existing data, while sparse, thus do not indicate a fundamental difference in block geometry and dimensions between keel and sail.

5.4.2. Boundary and finite size effects

The present study has shown that one needs to correct the proposed loose packing bounds (or observations) for boundary/finite size effects. For a fixed volume, as the filling of a container, observations will give a porosity that is biased high, which can be corrected for by Eqs. 5 and 6. When drilling data are considered, bottom (and top, for the sail) blocks will lead to lower porosity, for which Eq. 17 has been suggested as a correction. The latter is an approximation and should be validated by further observations.

5.4.3. Details of block interaction—friction and fracture

A principle question is if random loose packing is the dominant packing mode for sea ice blocks. According to Kankaanpää Reference Kankaanpää(1988) divers have reported a disordered distribution of ice blocks in keels, which supports the random packing assumption. A more thorough statistical data analysis would be needed to prove this (e.g., by comparing statistics from numerical simulations to drilling data).

A material with high particle friction is known to tend to looser packing, with a gradual transition from dense to loose packing with increasing friction (Salerno and others, Reference Salerno2018). The present GeoDict packing algorithm does not allow us to specify a coefficient of friction, yet produces results close to theoretical random loose packing bounds (Scott and Kilgour, Reference Scott and Kilgour1969; Jia and others, Reference Jia, Gan, Williams and Rhodes2007; Song and others, Reference Song, Wang and Makse2008). Other discrete element simulations, which allow to specify lower friction, predict porosities close to dense random packing (Jia and others, Reference Jia, Gan, Williams and Rhodes2007; Gan and Yu, Reference Gan and Yu2020). Intuitively, one would expect that sea ice packs as a high friction material: its static coefficient of friction is typically 0.5 (Schulson, Reference Schulson2018), and it has a rough surface and facilitates formation of freeze bonds between ice blocks. The argument to use Eq. 3 from Zou and Yu Reference Zou and Yu(1996) in the model proposed here is based on empirical data, while the random packing achieved by numerical simulations is a theoretical limit (Seckendorff and Hinrichsen, Reference Seckendorff and Hinrichsen2021). Support comes from the finding that laboratory-based macroporosity observations agree better with empirical Eq. 3 than with the numerical simulations. Other aspects that could play a role in block interaction during the packing process are hydrodynamic effects and the ice surface properties (e.g. the low strength of the sea ice bottom skeletal layer and/or snow on the upper ice surface). It may thus well be that an optimum empirical relation for sea ice will differ from Eq. 3 and depend on ice properties.

While Eq. 14 is a scaling law based on data for many materials and particle shapes, for sea ice, fracture processes are complex and may lead to a variety in aspect ratio modes (e.g., Tucker and others, Reference Tucker, Sodhi and Govoni1984b; Sayed and Frederking, Reference Sayed and Frederking1989; Lau and others, Reference Lau, Wang and Seo2012; Ranta and others, Reference Ranta, Polojaervi and Tuhkuri2018) that affect the packing density of ice blocks. For example, fracture may lead to a non-uniform vertical distribution of ice block length to thickness ratio. Strong shear and compression may also lead to block fragmentation and create small pieces that float up in the keel and increase the porosity locally. Palosuo Reference Palosuo(1975) shows cross-sections of ridges obtained by diving—in some of these, the block size is decreasing downward, in others not. Surkov Reference Surkov(2001) analyzed the ice chord length in ridges and found that it was independent of depth, indicating that fracture processes do not depend on depth. As observations are sparse, and packing simulations with fracture would be much more challenging, it remains an open question, which effect fracture inside the ridge may have on its macroporosity. It should also be noted that ridging may lead to a mixture of ice of very different thicknesses, e.g. when a ridge is formed from thin lead ice within thick pack ice. In that case, smaller macroporosity is expected, as thinner ice blocks may fill voids between thicker ones, and random packing equation 3 would have to be modified. However, the dependence of macroporosity on aspect ratio is weak, and for many purposes, it may suffice to choose a constant $\epsilon_b \approx 4$ and proceed with predictions as in Fig. 7.

5.4.4. Temporal porosity changes—heat transfer and compaction

To validate the present packing model, observations of macroporosity are essential. Such observations are seldom available right after formation of a ridge. One critical question is thus if observations for older ridges still reflect the formation conditions. However, observations of the temporal evolution of ridge macroporosity are sparse. Leppäranta and others Reference Leppäranta, Lensu, Kosloff and Veitch(1995) observed almost no temporal changes in the unconsolidated part of the keel of a Baltic Sea ridge, with the macroporosity remaining in the range of $\approx 0.29-0.32$ over 3 months. Zemluk and others Reference Zemluk, Astafyev, Surkov and Polomoshnov(1999) reported a decreasing average macroporosity with time, yet their study appears to include the effect of an increasing consolidated layer thickness and thus does not reflect the unconsolidated part of the keel. Beketsky and others Reference Beketsky, Astafyev and Truskov1996 reported an age dependence for three ridges offshore of northern Sakhalin, with macroporosities of $0.28, 0.23$ and 0.15 for ridges that were $0.5, 1$ and 2 months old, yet these ridges were from different areas. Recently, Guzenko and others Reference Guzenko(2023) have reported macroporosities and consolidated layer thickness for a large number of ridges in the Arctic (also shown in Fig. 6 as averages for 5 regions). These data show little dependence of macroporosity on consolidated layer thickness (that reflects the ridge age), yet indicate a decreasing macroporosity with ridge size. Although this might indicate the effect of compaction discussed above, one may also argue for a finite-size effect due to bottom ice blocks, Equation 17 predicting a lower macroporosity for larger ridges.

The main drivers of vertical and temporal macroporosity changes suggested so far are mechanical compaction and heat transfer. Above I have argued that compaction, if it is relevant, should lead to a transition from random loose to random dense packing and a porosity change of order 0.1, much smaller than the range proposed by Kharitanov (Reference Kharitanov2021a, Reference Kharitanov2021b). Also, Sazonov Reference Sazonov(2021) questioned Kharitanov’s compaction hypothesis, proposing that keel macroporosity changes are mostly linked to thermodynamics and heat transfer. In general, one would then expect an increasing macroporosity with depth in the keel, due to (conductive) heat loss close to the consolidated layer and (oceanic) heat gain near the keel bottom. For example, Blanchet Reference Blanchet(1998) have, based on mapping of a large rubble field, reported a partially consolidated layer of more than 1 m thickness. Such a layer develops, as the freezing front that passes through unconsolidated rubble is not sharp, ice blocks conducting heat faster than voids. The process has been studied for geometrically idealized fresh ice ridges (Salganik and others, Reference Salganik, Hoyland and Maus2020; Reference Salganik, Hoyland and Shestov2021). Although the macroporosity observations compiled in Fig. 6 are nominally from the unconsolidated part of the ridge, some of them may be affected by partially consolidated rubble and biased toward lower macroporosity. Another important aspect is that oceanic heat flux not only decreases the macroporosity near the keel bottom. The near-bottom ice melt implies freshening, a higher freezing point and the potential thermodynamic adjustment discussed above (Shestov and Marchenko, Reference Shestov and Marchenko2016a, Reference Shestov and Marchenko2016b). With meltwater moving upward, this will lead to lower macroporosity further up in the keel. Marchenko Reference Marchenko(2022) has recently modeled this process and its role for the evolution of ridge keel depth, consolidated layer thickness and macroporosity. The model accounts for thermodynamics, heat and salt fluxes and predicts that the macroporosity in the unconsolidated part of the ridge may increase or decrease over time, depending on the initial porosity. Marchenko reported another interesting model result for the case of high oceanic heat flux (of the order of $100\,\mathrm{Wm}^{-2}$). The macroporosity of the unconsolidated rubble then could increase strongly (equivalent to melting), remain at its initial level, or drop to low values. The result was dependent on initial porosity and oceanic heat flux forcing and indicates threshold behavior of the model.

One can conclude that, with a lack in observations of ridges right after their formation, the present packing theory is best validated by macroporosity observations obtained during the freezing season and outside the (partially consolidated) transition regime below the consolidated layer. For a detailed validation, one would have to include a model that for each ridge predicts the time dependence in the macroporosity of unconsolidated rubble as proposed by Marchenko Reference Marchenko(2022). Such an approach will be the more important the higher the oceanic heat flux. However, while a concise validation remains a future need, the results of the present model in Fig. 6 are promising and provide a physically consistent estimate of the initial macroporosity of a ridge during the cold season, which may then be used in more sophisticated models.

5.4.5. Salinity and thermodynamics

In the present study, it was assumed that thermodynamic adjustment is accompanied by rapid drainage of brine rejected during freezing of the macroscopic void space. A more advanced approach would take into account the effect of salt fluxes and possible change in initial ice salinity on the redistribution from macro- to microporosity. For example, a delayed exchange of rejected brine against seawater would increase the water salinity in the voids, retard freezing and lead to higher macroporosity. Also brine convection from the ice blocks could decrease their salinity, implying a higher macroporosity after ridge formation. Detailed salinity modeling will also be relevant for the related melt-water driven summer consolidation of ridges described by Shestov and Marchenko Reference Shestov and Marchenko(2016a) and Salganik and others Reference Salganik(2023a). Some of these processes have been parameterized in a ridge consolidation model by Marchenko Reference Marchenko(2022) discussed in the previous paragraph. A more sophisticated model would account for vertically resolved thermodynamics, heat and salt fluxes and for heat and salt exchange between the macroscopic voids and the microscopic pore space. As indicated in the analysis by Marchenko Reference Marchenko(2022), such a system may exhibit threshold behavior and imply ridge consolidation or disintegration. An important aspect is the condition, when the macroporosity becomes very small and all liquid is present in the form of microporosity (given by setting the left hand side in Eq. 11 to zero). Here, the limit at which porous ice may be considered solid is of interest, which likely depends on growth conditions and microstructural details (Maus, Reference Maus2023, Reference Maus2025).

5.4.6. Unexplained observations

Having considered several model limitations, I close with a summary of those observations that may not be explained by the present model. In the present data compilation, Fig. 6 and Table 1, old ridges or ridges from the summer season are not included as these have, due to the reasons discussed above, notably lower macroporosity (e.g., Kharitanov and Morev, Reference Kharitanov and Morev2005; Shestov and others, Reference Shestov, Ervik and Hoyland2018; Marchenko, Reference Marchenko2022; Salganik and others, Reference Salganik2023a). An example is three ridges investigated during September 2004 in the central Arctic Ocean (Kharitanov and Morev, Reference Kharitanov and Morev2005), also reported by Strub-Klein and Sudom Reference Strub-Klein and Sudom(2012) in their ridge overview, that were characterized by macroporosities in the range of $0.12-0.14$. Other examples from the studies in Table 1 are a ridge with 0.01 macroporosity investigated by Bonath and others Reference Bonath, Petrich, Sand, Fransson and Cwirzen(2018), for which the authors reported that voids were not or hardly measurable. For a similar reason (soft ice and difficulty to detect voids), the 2003 ridges from Hoyland Reference Hoyland(2007) with a macroporosity of $0.10-0.11$ were not included. For these ridges, the authors reported soft ice, higher ice temperature and rainy weather, indicating that they were in a stage of melting when cored. As the ridge macroporosity is, as discussed above, likely to change from its initial value during the melt season, a direct comparison to the present initial porosity estimate does not make sense. The remaining data from Table 1, shown in Fig. 6, fall in the predicted macroporosity range when assuming realistic ice salinity and temperature. Although these datasets were chosen here due to the availability of block size measurements with macroporosity, other observations from Strub-Klein and Sudom Reference Strub-Klein and Sudom(2012) show similar ranges, except for one ridge from the Svalbard area. However, the macroporosity average for that ridge keel, as well as those for five more ridges around Svalbard reported by Sand and others Reference Sand, Bonath, Sudom and Petrich(2015), appears to include solid ice from the consolidated layer, which may explain the low macroporosity range of $0.04-0.16$.

As a final note, I like to point out that the observational points from Guzenko and others (Reference Guzenko, Mironov, IMay and Porubaev2022, Reference Guzenko2023) in Fig. 6 are of particular interest, as each represents the average of many (11–31) ridges from a specific region. Most of these points fall close to the model prediction with lowest ice surface temperature ( $-15^{\circ}\mathrm{C}$) and highest salinity (10). One could argue that these temperature/salinity values are a bit extreme and that more realistic model settings would give a higher initial macroporosity (e.g., for $-10^{\circ}\mathrm{C}$ surface temperature by ≈ 0.03). However, when correcting the observed macroporosity for the effect of a bottom ice block in a finite depth ridge keel, using Eq. 17 suggested above, one obtains a similar porosity increase for the observations ( $\approx 0.03-0.04$ for the ridge properties given by Guzenko and others). Hence, considering the model limitations (no time dependence, ocean heat fluxes and vertical resolution), the difference between model and observations is not large. It is anticipated that a more sophisticated comparison, based on the ridge statistics data from Guzenko and others (Reference Guzenko, Mironov, IMay and Porubaev2022, Reference Guzenko2023) and other environmental data, has the potential to validate the present model and further test time-dependent models in the lines of Marchenko Reference Marchenko(2022).