Ice crystallization from brine

As crystals of ice form in a brine solution, nearly all of the solute (salt, air, etc.) remains in solution (Reference Weeks, and AssurWeeks and Ackley, 1982, p. 4). As the crystals grow into the melt they form a series of parallel blade-like platelets about 0.5 mm thick. As growth proceeds, observers (Reference Anderson, and Weeks,Anderson and Weeks, 1958: Reference Weeks, and AssurWeeks and Ackley, 1982: Reference Grenfell,Grenfell, 1983) report that the platelets thicken and bridges begin to form approximately 2.5 cm above the lower tip of the platelets. The bridging process between platelets develops around vertical columns of brine which might extend vertically through the entire crystal. As the ice cools further, the columns of brine collapse into arrays of nearly spherical or ellipsoidal brine cells. As the brine cells age, they have an increased tendency to lake on an ellipsoidal geometry.

Utilizing this outline of crystal development, we assume that the ice crystal and trapped brine are stress-free immediately after the brine columns collapse into arrays of nearly spherical brine cells. Then the driving question becomes, what happens as the ice and brine are cooled further?

Here we will focus on the role of the phase transformation that brine, which is trapped in these cells, undergoes as the crystal temperature drops either during the growth stage or while in storage.

Phase changes

To quantify the phase changes, we will define the salinity of the brine (s^B) by the non-dimensional form

To understand the relationship between the phases of brine and ice, consider the phase-diagram cartoon (Fig. 2) from Reference Weeks, and AssurWeeks and Ackley (1982). For a brine solution with a salinity of 0.035, when the temperature drops to −2°C, water precipitates out of the brine in the form of ice. If the brine is contained in a small closed system, then the transformation from water to ice implies that the mass of brine decreases significantly while the mass of salt in solution remains “constant”; thus the salinity of the brine increases. The increase in brine salinity requires a decrease in temperature before more ice can precipitate out. This process will continue until the brine reaches the eutectic point, at a salinity of 0.233 and a corresponding temperature of −21.2°C. Before the temperature of the brine can drop below the eutectic point, the brine solution must assume the solid form of sodium chloride dihydrate (NaCl.2H₂O).

Fig. 2. Phase diagram for H₂O and NaCl.

Phase transformations are dependent on changes in pressure and temperature (Reference LaChapelle,LaChapelle, 1968). That is, as pressure increases, the phase curve typically depresses. To keep the mathematical development relatively simple, we will assume that the phase change is independent of pressure. Reference Cox, and Weeks,Cox and Weeks (1975, Table I) use the following third-order polynomial to model the phase relationship between the brine salinity (s^B ) and the freezing/melting temperature (T):

where the transformation temperature (T) is in °C. Their equation is based on a statistical fit of experimental data with a correlation coefficient of 0.999927 and a standard error equal to 0.4334.

Phase densities

For brine density, we assume that pressure and temperature are independent variables and that temperature and salinity are directly correlated. For the stress-free density of brine (ρ^B ). we use the approximation for sea brine given by Reference Cox. and Weeks,Cox and Weeks (1983, equation 16).

where

and ρ^W is the density of water at 0°C. For the density of non-saline ice (ρ^I ), we use the approximation given by Reference Pounder,Pounder (1965, equation 29).

where the temperature T is in °C and ρ0^I represents the density of ice at 0°C with the value:

When the temperature of the brine drops below the eutectic point, and the brine takes on the solid form of sodium chloride dihydrate (NaCl.2H₂O), the density of the solid is approximately 1630 k gm⁻³ (Reference Weeks, and AssurWeeks and Ackley, 1982, Table I); comparing the density of this solid form with the density of brine (Equation (4)) at the eutectic point ρ_e ^B = 1186 k gm⁻³ implies that the brine contracts as it transforms into solid form. Given the-nature of our query, we will not concern ourselves with temperatures below the eutectic point.

Notation

For state variables, m is mass, s is salinity, ρ is density, and V is volume. For the components, the superscripts “S”, “B”, “I”, and “W” designate salt, brine, precipitated ice, and non-saline water, respectively, Superscript “O” will identify combined components of the “original” brine system. Subscripts index the suite; in particular; “o” denotes the initial state, and “T” denotes variations due to the phase transformation from brine to ice.

Brine transformation volume

Because the most significant volume changes hinge on the transformation from brine to ice, we will start by modelling the transformation volume of the brine (V_T ^B ).

Given the closed nature of a brine cell, the mass of salt contained in a cell remains constant. In addition, we will assume that all of the salt mass is retained within the brine to the extent that, upon freezing, ice rejects practically all of the salt. Based on experimental results (Reference Weeks, and AssurWeeks and Ackley, 1982, p. 4), this assumption is a good approximation for density calculations. Based on the preceding statements, we can write the following relation.

Rearranging terms, we can represent the transformation brine volume (V_T ^B ) by

Representing the brine density as a function of salinity (Equation (4)), we can write

If T ≤ T_o , then ζ^B = 1.25 > 0.233 ≥ s ^B ≥ s _o ^B implies the volume is approximately inversely proportional to the salinity and is related in a similar fashion to temperature.Footnote ³

The term indicates an approximation of the transformation brine volume (V_T ^B ); this approximation emphasizes the relationship between the brine volume and the brine salinity. In essence, as the salinity increases, there is less and less brine volume for ice to precipitate from; therefore, the most significant changes should occur immediately after a sealed brine cell begins to cool.

Differential changes for the total transformation volume

We will describe the stress-free (transformation) volume of the brine cell (V_T ^O ) as the sum of the reduced brine volume (V_T ^B ) and the precipitated ice volume (V_T ^I ):

As indicated earlier, the superscript “O” denotes the original brine system. Here, after the temperature is lowered, the original system consists of the remaining brine and the ice that has precipitated out of the brine. Differentiating the preceding expression yields

Considering the differential density change for brine, the stress-free brine density can be written as a function of temperature and salinity. We have chosen to express the density changes as a function of salinity, such that temperature effects are included implicitly (Equations (3) and (4)); thus

and differentiating yields

Considering the differentia] density change for ice, all but a very small amount of salt is rejected from ice so that salinity can be neglected. Also, since we are concerned only with temperature changes on the order of 20°C, density changes will be small so that the density differential may be neglected.Footnote ⁴

Returning to the expression for the differential volume change (Equation (13)), substituting for the brine density differential (Equation (15)), recalling V_T ^B = m^B/ρ^B, and allowing the density differential for the precipitated ice to vanish, we have

Focusing on the first two terms in the differential volume expression, the mass of the brine in the original system can be written as

because the mass of precipitated ice in the original system is zero (m_o ^I = 0). Differentiating the preceding mass relation, we have

Substituting into the volumetric differential (Equation (16)) and simplifying,

To make this equation more useful, we will express the first term in terms of salinity rather than mass. Using the definition of brine salinity (Equation (2)), the brine mass can be written as:

Recalling that the salt mass (m^Sr) is constant, differentiating, and substituting m^S = m^Bs^B and m^B = ρ^BV_T ^B , we have

Returning to the volumetric differential (Equation (19)) and substituting for dm^B (Equation (21)), we have

Substituting for ρ^B as a function of salinity (Equation (4)), we have

where

This differential relation gives us a measure of the increasing interference between the two phases, i.e. brine and ice.

Strain function

Consider the use of the preceding volumetric differential (Equation (23)) as a strain measure. The total volume differential (dV_T ^o ) gives us a measure of the total volume change of the original brine-cell contents under stress-free conditions. If we take the brine volume as a reference volume, then it appears that all of the expansion is taking place in the brine. Integrating this strain measure between the initial state and the current state yields a measure of the total strain (e_T ^B ) relative to the brine volume⁵, i.e.,

Making the change of variables from volumetric measures to salinity (Equation (23)), the strain can be described by

where ξ is defined in Equation (23).

To obtain the volumetric Strain measure as a function of temperature, replace the brine salinity s^B with a function of temperature, s^B(T) defined in Equation (3), so that:

То study the behavior of this strain function, recall ζ^B = 1.25 as defined in Equation (4), let the specific gravity of ice (ρ^I/ρ^W) equal 0.917, and let the initial temperature T₀ = −2°C (for the present, we assume the trapped brine starts with a salinity of 0.035, though it will probably be higher); the resulting volumetric strain profile is shown in Figure 4.

Fig. 4. Volumetric tansformation strain (e_T ^B).

Polynomial approximations

For numerical models, polynomials are much easier to work with. By choosing an initial temperature (T₀), corresponding to the initial salinity of the brine (s₀ ^B ) immediately after the cell is sealed, we can approximate the volumetric strain using a fifth-order polynomial. For the initial temperature, we chose to let T₀ = −2°C.

To model the entire range between the initial temperature and the eutectic temperature (T_e = −21.2°C), we used Chebychev polynomials as a basisFootnote ⁶ , obtaining a polynomial approximation with a maximum offset of 1.24 × 10⁻³, which occurred at T_o.

The polynomial approximation takes the following form

with the coefficients listed in Table 1, where T is in °C.

In some cases, it is advantageous to model the transformation expansion as thermal expansion. If large (logarithmic) strain components are used, the Volumetric strain can be represented by

where α is the coefficient of thermal transformation expansion and ∆T = T − T_o is the temperature difference between the initial stress-free temperature (T_o ) and the brine temperature (T). To approximate the expansion coefficient, recall the expansion is “zero” at the initial temperature (T_o). Coincidentally, for the polynomial approximation (Equation (28)), there is a root near T_o; therefore we can factor out of the polynomial model , so that

for T_e≤T≤T_o with the parameters given in Table 1, where T is in °C.Footnote ⁷

Discussion

Notice that the volumetric strain (e_T ^Β ) is larger than might be expected, if we considered a simple transition from water to iceFootnote ⁸ . This discrepancy is due to the fact that we are considering the apparent strain in the brine, measuring the volumetric strain relative to the brine volume rather than the the total cell volume. Because the strain represented by e_T ^Β is not small, engineering strain measures do not make good approximations.

Notation

In addition to the notation of the previous section, the Subscript “C” denotes components due to material constraints, and superscript “i” denotes parameters related to the brine/ice interface.

In particular, we let rⁱ represent the radius of the brine/ice interface and P^B represent the hydrostatic pressure in the brine and hence the pressure at the interface.

Compatibility

We start by enforcing compatibility at the interlace between the precipitated ice (I) and the brine (B). The displacement for each side of the interface is divided into two components. The first expresses the displacement of the interface due to phase transformations, denoted by subscript “T”. The second expresses the displacement of the interface due to the pressure at the interface, denoted by subscript “C”. To satisfy compatibility, the total displacement differential must be the same for each side of the interface:

Using an elastic model for concentric spheres (Reference Love.Love, 1944, §98) and assuming the outer radius of the ice mass is very large with no external pressure, we arrive al the relation:

for the radial displacement of the inner radius of the surrounding ice mass due to a brine pressure increment, dP^B .

For the stress-free displacement of the ice interface.

Because we are concerned with transformation displacement and the original cell is constrained by the surrounding ice. we assume the precipitation of ice results in displacement of the interface rather than changing the outer boundary , so that

For the differential displacement of the brine interface resulting from the constraining forces of the surrounding ice

and in terms of the spherical geometry of our conceptual model, we can write

then

where K^B is the bulk modulus of the brine.

For the stress-free displacement of the interface viewed from the brine,

implies

and solving for the differential displacement, we have

This gives us expressions for all of the displacement components.

Pressure model

To finish the development of the pressure model, we can substitute the expressions for the differential displacements (Equations (32), (34), (37) and (40)) into the compatibility Equation (31), to get

Combining terms and solving for dP^B ,

Recognizing that and that the brine volume (V_T ^B ) is a reasonable approximation for the volume enclosed by the interface (Vⁱ ), we can write

Integrating,

where the volumetric strain (e_T ^B ) was defined previously (Equation (26)). This model seems fairly intuitive.Footnote ¹⁰

Example

To study the behavior of the resulting pressure function, let the bulk modulus of the brine (K^B ) equal 2.0 GPa and let the shear modulus for the ice (μ^I ) equal 3.6 GPa. Then using the volumetric strain function (Equation (26)) with initial temperatures of T_o = −0.5°, −1°, −2°, −3° and −4°C, we obtain the pressure profiles shown in Figure 6.

Fig. 6. Brine pressure as a function of salinity.

Notice the pressure becomes significant immediately after the brine cell is sealed and begins to cool, particularly for cases with low initial salinity. For the cell model with an initial temperature T_o = 2°C, by the time the temperature drops 1°С the brine pressure (P^B ) approaches 50 MPa, and it reaches a maximum of 285 MPa at the eutectic point (T_e = −21.2°C).

As we would expect, given the inverse relationship between brine volume and salinity (Relation (11)), the rate of change in the pressure for a change in temperature is dependent on the salinity of the brine. Brine cells which start with a lower initial salinity display a much sleeper pressure profile initially, leveling off to display slopes parallel to those which started with a higher initial salinity.

Discussion

As can be ascertained, the elastic model would not be valid after inelastic deformation begins and, as we can see (Fig. 6), the pressure modeled at the brine/ice interface approaches the theoretical maximum strength for ice, µ^I/10 = 360 MPa; hence, there is little doubt that the stresses are sufficient to induce inelastic deformation. Though inaccurate for large temperature changes, the model may be useful for estimating the temperature change required to commence yielding as well as the extent of possible yielding.

Example

Let the initial interface radius r_o ⁱ = 1 for an initial temperature T_o = −2°C. Then the stress profiles for temperatures T = −2.5°, −3°, −8° and −21.2°C are shown in Figure 7 with corresponding interface radii rⁱ = 0.93. 0.88. 0.64 and 0.50.

Fig. 7. Stress profiles for the brine cell: (a) principal components, (b) maximum shear.

Notice that the stresses transferred to the region outside of the original brine cell change relatively little as the temperature drops. As the brine pressure increases, the pressure increase is borne by the precipitating ice that generates the pressure increase.

Discussion

The predominant concern regards the extent of inelastic deformation that might occur in the neighboring ice due to the pressures generated in the brine. As discussed in the previous section, the stresses resulting from our elastic model are well beyond the yield stress for ice. Higashi (1964) shows yield in non-saline single crystals of ice for basal plane shear stresses ranging from 0.06 to 0.44 MPa with strain rates in the range of 0.13–2.7 × 10⁻⁶s⁻¹ and temperatures in the range of −21° to −15°C. At warmer temperatures and lower rates, where we would expect the phase transformations to have a more significant impact, we would expect yield at even lower stresses.

Comparing this observation with the results in Figure 7, one can conclude that inelastic deformation takes place almost immediately after the brine pocket has sealed off and the temperature begins to drop. Therefore the pressure/stress curves shown in Figure 7 are unrealistic for saline ice and serve only to show the significance of inelastic deformation that might occur during crystal growth, if the ice surrounding the cell does not fracture immediately.

Reference Wakahama, and Òura,Wakahama (1967) deduced that for temperatures near −10°C basal slip would occur in ice single crystals for basal plane shear stresses of 0.02–2 MPa. For shear stresses above 2 MPa, the ice crystals would fracture. For the sake of argument, assume the shear stress at the interface is near the maximum for basal slip. Using the elastic model (Equation (47)), we can estimate brine pressure , as well as the radial location at which shear stresses have decayed to the minimum “yield” point , i.e.

indicating that there is the potential for inelastic deformation at distances up to five times the interface radius (rⁱ ) away from the brine cell. Because inelastic deformation would occur over the range (rⁱ≤r≤5rⁱ ), this estimate, from our elastic model, of the range of inelastic deformation is only a rough approximation.

Set-up

To model the stress field that develops in the ice, we used the ANSYS 5.2 FE model (Swanson Analysis Systems Inc., Houston, Pennsylvania) shown in Figure 8.

Fig. 8. Inner section of the finite-element model of a spherical ice cell.

Due to the rapid decrease in the stress field near the ice brine interface (Fig. 7), the shape functions in an FE model will tend to underestimate the stresses at the interface. To determine the accuracy of the FE model, we ran a test case where we applied an internal pressure (P^B = 100 MPa) and then compared the results with the equilibrium stress relations for concentric spheres (Equations (45)–(47)). We found that with the curved boundaries and somewhat distorted elements, we achieved better results using elements with quadratic shape functions (i.e. ANSYS, SOLID95). Additionally, to obtain results that were accurate at the inner radius, we had to use a sub-modeling (mesh-refinement) technique. For our coarse model, we started with an outer radius that was 20 times greater than the inner radius. We then used this “coarse” mesh to define displacement boundary conditions for the outer radius of a much finer model where the outer radius was only four times as large as the inner radius. By using this sub-modeling technique, we were able to reduce our maximum discrepancy, which occurred at the interface radius, rⁱ , to less than 3%. If we ran the analysis with only the coarse mesh, the stresses at the inner radius were underestimated by more than 10%.

To model the brine, we filled the spherical vacancy with a second mesh. We used the same element type for both the brine and the ice shell, though we altered the material parameters.

We assumed our model took the “stress-free” configuration at −2°C. To model the interference between the brine and ice due to the phase transformations, we used the ice interface as a frame of reference, so that the interference was due to the “apparent” transformation expansion of the brine. To model the “apparent” expansion of the brine in the FE model, we modeled the volumetric strain as a thermal expansion, using the large (logarithmic) strain capabilities of ANSYS. The components of thermal (transformation) strain were represented by

where ∆T = T − T_o is the temperature difference between the initial stress-free temperature (T_o) and the brine temperature (T), and α is the coefficient of thermal (transformation) expansion, defined in Equation (30).

For the elastic parameters of the brine, we used a bulk modulus of 2 GPa and a Poisson’s ratio of 0.498. For the surrounding ice, we used the coefficient of thermal expansion, α ^I ≈ 53 × 10⁻⁶, as given in Reference Fletcher,Fletcher (1970), though it is negligible relative to the transformation coefficient.

To validate the accuracy of the FE model relative to the preceding closed-form model, we ran the FE model, for T = −8°C, using the following isotropic elastic parameters for the ice (Reference Derradji-Aouat,Derradji-Aouat, 1992, table 4.3):

To evaluate the differences in the two isotropic models, we used the radial-siress data from the FE model and applied least-squares fit of the form , obtaining with a correlation coefficient of 0.9991 and a standard error of 1.02 MPa. Comparing this pressure estimate with the pressure estimate from the closed-form model (Equation (44)) yielded a difference of 2.5%, presumably due to the included thermal contraction of the ice and interpolation errors.

To determine the significance of the anisotropic material model relative to the isotropic material model, we used the following orthotropic material parameters,

Τo obtain the numerical values, we used the polynomial models of (Reference Dantl,Dantl l968, Reference Dantl,, Riehl., Bullemer and Engelhardt,1969) for the compliance parameters (S_ij) and evaluated them at a temperature of −8°C.

Analysis

Using the preceding isotropic properties, our FE model produced the tangential (σ_tt) and shear (τ_xz) stress fields shown in Figure 9a and c, respectively, The corresponding maximum stresses are shown in Table 2. Using the anisotropic properties, our FE model rendered the tangential (σ_tt) and shear (τ_xz) stress fields shown in Figure 9b and d, respectively. The corresponding maximum stresses are shown in Table 2.

The stress contours shown in Figure 9c and d represent the magnitude of the shear stress that would result on the basal plane of the ice crystal, if it remained elastic.

Fig. 9. Stress contours (T = —8°C): (а) σ_tt contours (isotropic model): (b)σ_tt contours (anisotropic model): (c) τ_xz contours (isotropic model): (d) τ_xz contours (anisotropic model).

Given that most of the anisotropic elastic parameters (Equations (53)) are within 30% of the isotropic components (Equations (52)), we would expect the stress results to reflect the same order of similarity. The contour plots in Figure 9 demonstrate such a case. The radial-stress contour plots displayed so much similarity that they were not deemed worth including.

Consider the stresses in the xy plane. Because the z axis in Figure 9 corresponds with the crystalline с axis, the elastic model is isotropic in the xy (basal) plane. In this plane, the only noticeable difference relative to the isotropic model is that the magnitudes of the radial and tangential-stress components decay slightly slower and faster, respectively, relative to R_i .

For stresses along the z axis, the behavior is again very similar to the isotropic case, except the rate of decay in the magnitude of the radial-stress component is significantly slower, yielding a magnitude 100% greater when R_i = 4.

Given the magnitude of the stresses predicted, this model is primarily useful for predicting the onset of yielding. Therefore, our primary concern is the magnitude of the shear components that would act on the basal plane. Comparing the shear-stress contour plots (Fig. 9c and d), it is clear that the change in the shear stress field due to anisotropic properties is small. The only noTable observation is that in the anisotropic ease the shear stress displays a maximum which is about 10% less, combined with a slightly faster rate of radial decay relative to the isotropic case.

The shear stress contours (Fig. 9c and d) illustrate that basal-plane shear stresses will be significant only at specific points on the surface of the brine-pocket model. These are the points where the basal plane makes a 45° angle with direction perpendicular to the brine-pocket surface. As a consequence, one would expect to first see dislocations emerging from these preferred regions on the brine-pocket interface.

Discussion

In summary, based on comparisons between models with isotropic and anisotropic material parameters for single Crystals of ice, the isotropic elastic model yields approximations that are very reasonable relative to the anisotropic case. In particular, the isotropic model works quite well for predicting the onset of yield, though it may be a hit conservative.

Gas Inclusions

Notice our model does not include the effects of gas that will be coincidentally trapped along with the brine. When the brine is sealed in a cell, it will contain a certain amount of dissolved gas. As ice precipitates from the brine, the gas and salt in the brine will remain relatively constant. Hence, the remaining brine will become increasingly saturated with gas. At the same time, the remaining brine will become increasingly salty, and increasing brine salinity results in decreasing gas solubility in the brine, sometimes referred to as “salting-out”Footnote ¹¹ . To counter these two factors, both the brine-pressure increase and the temperature decrease which occur coincidentally with the reduction in brine volume will increase the gas solubility. In the situation modeled, the dissolved gas may never nucleate a bubble.

Whether a bubble forms or not, consider the amount of gas that might coexist in a brine cell. As a rough upper bound for the dissolved gas. we will use the closest solubility data provided by Reference Weiss,Weiss (1970), which is for sea water with a salinity of 0.035 and at a temperature of −1°C. In this case, the sum of the solubilities of N₂, O₂, and Ar yields a gas solubility of 0.02326 (1 of gas)/(1 of brine); so, up to 2.3% of the original cell volume might be occupied by dissolved gas. At the same time, it is possible that a bubble will float up from below and enter the cell before it is sealed off. In summary, the volume occupied by gas in a brine cell is likely to vary anywhere between 1% and 100%.

The more gas there is in the cell, whether dissolved or in bubble form, the less brine there is available to transform to ice, and, therefore, the less the cell contents will expand relative to their initial size. Coincidentally, the gas in the cell will compress much more readily, so that the pressure generated will be significantly reduced. As a lower bound, a cell filled Completely with gas will generate virtually no internal stress; our model gives an upper bound for brine cells containing gas.

The role that bubbles play in the transformation process is also dependent on the directional nature of the freezing process. Reference Nakaya,Nakaya (1956, plate 44), in his study of internal melt figures, demonstrated that typically when freezing is “isotropic”, the vapor bubble will shrink in size as the water/ice transformation progresses. In the case of bubbles generated by internal melt, the bubble will eventually vanish. If the freezing process is “orthotropic”, in particular if the rate of freezing is faster from the top down, the buoyancy of a bubble will typically result in its isolation as the precipitating ice freezes around it (Reference Nakaya,Nakaya, 1956, plates 45–47). Once a bubble has been isolated, the model proposed in this paper provides an approximation for the strain and pressure build-up, though with altered initial conditions.

Relevance to sea ice

As Reference Cox, and Weeks,Cox and Weeks (1975) point out, the phase transformations for brine in sea ice and saline ice are similar at temperatures above −8.2°C. At temperatures below −8.2°C, solid salts begin to precipitate out of the brine in sea ice and, according to Reference Peyton,Peyton 1966, §6.2.2), merge with the precipitating ice to form an inner shell of “salty” ice. For uniaxial specimens, Peyton found that the solid salts did not significantly alter the tensile behavior but had a Stiffening effect on the compressive behavior. Most likely the solid salts will have a reinforcing effect on The precipitated “salty” ice shell. We have not included any solid-salt effects, so our model will lack accuracy for temperatures below −8.2°C; however, we feel our model is accurate for sea ice at temperatures above −8.2°C. Of course, as in saline ice, the strain measures are relevant for all temperatures down to the eutectic point, while the pressure/stress measures developed in this paper are relevant only in the elastic range.

Natural sea ice does not normally experience rapid temperature changes. The bottom of an ice sheet is always at temperatures near the melting point, even though the top surfaces may experience temperatures as low as −50°C, particularly near the Antarctic coast. In this fashion, the heat capacity of the underlying sea water serves as a temperature moderator for the ice sheet above. With moderated temperature changes, less than 1°C, dislocations are likely to nucleate, indicating that the processes modeled here are quite relevant to typical sea-ice sheet in the polar regions.