Snow-ice

In near-shore areas, where snow blown off the Antarctic continent accumulates in large quantities, the resulting overburden pressure can lead to flooding of the ice surface and the formation of snow-ice. It has been observed in some coastal zones that snow-ice is the main constituent of the fast-ice sheet (Reference KozlovskiyKozlovskiy, 1981). Snow-ice has also been found in abundance in the winter pack-ice cover, where high oceanic heat fluxes restrict the annual ice growth to 0.6–0.8 m, and flooding is possible on a large scale even though the snow accumulations are less (Reference Wadhams, Lange and AckleyWadhams and others, 1987). If estimates of the snow and ice densities are available, then the equilibrium hydrostatic water level (H _w) can easily be calculated using

where ρ is density, h is thickness, and the subscripts i, s, and w refer to ice, snow, and water, respectively (Reference RøthlisbergerRöthlisberger, 1983). Determining exactly when, or if, flooding will occur is considerably more difficult. A reasonable assumption might be that surface flooding will take place as soon as H _w becomes positive. If, as observations indicate, flooding occurs in discrete events but eventually reaches equilibrium levels, then although the timing would be incorrect, the resulting error in the growth calculations would be small. However, the McMurdo Sound observations throw some doubt on the validity of this assumption. Throughout the winter, drill holes revealed consistently positive hydrostatic water levels, but there was a conspicuous absence of large-scale flooding. Because sea ice normally does not crack under thermally induced stress, in order for positive H _w values to be relieved flow must take place through interconnecting channels in the ice. Reference PounderPounder (1965) suggested that the transition between interconnected and discontinuous brine-drainage channels occurs at a temperature of roughly −15°C. This figure will be highly dependent upon the salinity and crystal structure of the ice, and measurements of ice-surface temperatures in McMurdo Sound during several flooding events showed that a value of −8°C is more appropriate in this specific case. We therefore have two criteria which must be satisfied before flooding can occur: positive hydrostatic water levels, and surface temperatures greater than or equal to −8°C. It is recognized that imposing a constant minimum temperature for flooding is a crude approximation, but very little is known about the relationships between the temperature, salinity, and crystal structure of sea ice and its permeability and, until such information becomes available, the threshold temperature criterion must suffice.

It is assumed that the transport of liquid to the surface has no effect on the thermodynamics of the ice sheet, apart from the obvious effects of changes in h _s and the thickness of the snow-ice layer (h _si). Spot measurements made on the ice immediately in front of Scott Base in March 1986 showed the flooding liquid to be high-salinity brine (≈125 ppt), at roughly the same temperature as the snow into which it was flowing (indicating that flooding takes place through the brine-drainage network rather than through stress fractures, and that the liquid in the brine pockets is pushed to the surface ahead of the ocean water below). The snow/liquid mixture produced is therefore highly saline (about 70 ppt), but the temperature profile through the ice remains virtually unchanged, and the amount of advected heat energy is quite small. We have also assumed that the amount of latent heat released as the flooded layer solidifies is negligible. This is of course not strictly true, but McMurdo Sound observations suggest that the amount of latent heat released from the flooded layer is small. Since the temperature and salinity of the flooding liquid do not change when mixed with the snow, the mixture will be in a state of chemical equilibrium for that temperature. The observed salinity of 70 ppt, and temperature of −8°C, indicates that the brine volume in the mixture was about 465 ppt (Reference Frankenstein and GarnerFrankenstein and Garner, 1967), and therefore the volume of ice was roughly 500 ppt, slightly greater than the 350 ppt occupied by the ice grains in the original snow layer. Since the temperature did not change, most of this increase must be due to compaction of the snow on flooding. In fact, in the absence of any significant decrease in temperature, most of the solidification of the snow-ice layer must be the result of compaction of the snow in the presence of the liquid.Although reductions in the temperature of the flooded layer will result in some freezing of the flooded liquid, the deep snow cover which remains over the “snow-ice” layer acts to moderate atmospheric temperature fluctuations, and in this case the observed brine-volume decrease resulted primarily from snow compaction and drainage of the interstitial fluid after the initial flooding event (the salinity of the snow-ice layer was 70 ppt on 1 March, 44 ppt on 30 March, and 15 ppt several months later when the sample shown in Figure 1 was taken). It is not known to what extent freezing of the flooding liquid contributes to snow-ice formation on a regional scale (this subject should be investigated further); however, we feel that the errors introduced into this simple type of ice-growth model by neglecting the latent heat released from the snow-ice layer will be small.

Equation (8) is based on the assumption that the flooded layer becomes completely saturated. This is clearly not true, as close examination of the ice revealed numerous small air pockets trapped in the ice matrix (see Fig. 1). Qualitative observations of the snow-ice in McMurdo Sound show that its air-bubble content is similar to that of snow-ice formed on lakes. Reference Adams, Gray and MaleAdams (1981) has measured the density of lake snow-ice and we adopt his value for ρ_si of 820 kg m⁻³ (corresponding to an air-bubble content of 10%), and adjust the calculated hydrostatic water levels by increasing H _w uniformly by 10%. Also, because both brine and air are poorer conductors than pure ice, the low density and high salinity of the snow-ice layer will combine to reduce its thermal conductivity. To estimate the magnitude of this reduction, we use Reference SchwerdtfegerSchwerdtfeger’s (1963) model for the thermal conductivity of bubbly, saline ice as representative of the thermal conductivity of snow-ice (k _si). Then k _si can be calculated using

where S is the ice salinity in parts per thousand, T is the ice temperature in degrees Celsius, μ is a constant (−0.0182 C⁻¹), k _bis the thermal conductivity of brine

Fig. 1. Vertical crossed-polaroid photographs of (a) snow-ice, (b) columnar ice, and (c) ice formed by columnar growth into a sub-ice platelet layer. All three thin sections are from cores taken in McMurdo Sound during the winter of 1986; (a) and (c) from roughly 1.5 km offshore from Pram Point, and (b) about 100 m from shore in North Bay. Each section is 76 mm in diameter.

k _a is the thermal conductivity of air, V _a is the volume fraction of air, and k ₀ is the thermal conductivity of pure ice (Reference SchwerdtfegerSchwerdtfeger, 1963). At −10°C and a salinity of 20 ppt, Equation (9) gives k _s = 1.66 Wm⁻¹ K⁻¹ which is about 10% less than a value of k _i for “normal” sea ice calculated using Reference UntersteinerUntersteiner’s (1961) formula

in which β is a constant (0.117 Wm²kg⁻¹), and S and T are again in parts per thousand and degrees Celsius. For simplicity, we therefore set k _si to 0.9 × k throughout the calculations. An additional alteration must then be made to accommodate the three-layer (ice, snow-ice, and snow) system of series-connected conductors. This is accomplished by replacing Equations (2) and (3) with

see Reference LeppärantaLeppäranta, 1983).

Sub-ice platelets

A second modification which must be made to the model is to incorporate the contribution to growth of the sub-ice platelets; discoid, single crystals of ice formed when supercooled water comes in contact with the base of the ice sheet. They normally take the shape of plates roughly 5 mm thick and ranging in diameter from less than 10 mm to over 150 mm. Supercooled water and sub-ice platelets have been observed several times near ice-shelf margins in the Antarctic (Reference Kozlovskiy and CherepanovKozlovskiy and Cherepanov, 1976; Reference Foldvik, Gammelsrod, Torresen and JacobsFoldvik and others, 1985), and the general mechanism of formation has been described by Reference RobinRobin (1979). Although tidally modified, the currents in McMurdo Sound are principally from under the McMurdo Ice Shelf (Reference Lewis and PerkinLewis and Perkin, 1985) and therefore fit Robin’s explanation. The water flowing into McMurdo Sound originates in the Ross Sea and flows under the Ross Ice Shelf at roughly its mid-point (Reference Jacobs, Gordon and ArdaiJacobs and others, 1979; Reference Pillsbury, Jacobs and JacobsPillsbury and Jacobs, 1985). As the water moves southward, it is forced to greater depths by the overlying ice shelf, and may exceed the pressure-reduced melting point at the ice-shelf bottom even if its initial temperature was equal to the surface freezing point. The heat consumed in melting basal ice is drawn from the water, and reduces its temperature. If this heat loss is sufficient to lower the water temperature to below the surface freezing point, then on the return journey northward into McMurdo Sound it will become slightly supercooled as it rises to the surface and the pressure is released. The supercooling is relieved when the water comes in contact with the bottom of the ice sheet and nucleation results in the formation of a sub-ice platelet layer. The platelets will form under both sea and shelf ice, wherever a sufficient pressure reduction has occurred to produce super-cooling, but the greatest quantities will be produced where the pressure is lowest, that is at the base of the sea-ice sheet. It is also possible for sub-ice platelets to form at depth (Reference Dieckmann, Rohardt, Hellmer and KipfstuhlDieckmann and others, 1986), in which case they would simply float upward and accumulate on the underside of the ice, but there was no evidence to suggest that nucleation was occurring in the water column in McMurdo Sound during the winter of 1986.

It is interesting to see that the thickest platelet layers measured during two transects across McMurdo Sound on 4 August and 17 October correspond to the areas of maximum surface supercooling recorded by Reference Lewis and PerkinLewis and Perkin (1985). The timing of the platelets' first appearance and growth is also significant. The thickness of the platelet layer was observed to be increasing from early July, when it first appeared, until about mid-September, after which it remained fairly constant. Reference Wright and PriestleyWright and Priestley (1922) found that platelets grew on their manilla lines between 5 August and 12 October 1911, and between 28 August and 28 September 1912. The later dates of first appearance of the crystals may be the result of the greater distance of their measurement site (which was roughly half-way between Cape Evans and Inaccessible Island (Reference DeaconDeacon, 1975)) from the ice-shelf edge; nevertheless, their observations suggest that the phenomenon occurs at about the same time each year. This can be readily explained by referring to Reference RobinRobin’s (1979) theory. The water rising out from under the McMurdo Ice Shelf will be supercooled only if it were initially near its freezing point as it flowed under the Ross Ice Shelf. For supercooling (as opposed to merely cooling) to occur in the return flow, the water must lose enough heat through basal melting to have its temperature reduced to below the surface freezing point. The absence of platelet growth until mid-winter, and a reasonable estimate of the residence time of water under the ice shelf of 2–3 months, suggests that the water in the Ross Sea is not sufficiently cooled until early winter (roughly the month of May). There is no evidence to indicate that, as Reference Wright and PriestleyWright and Priestley (1922) hypothesized, changes in the currents themselves are responsible for the timing of the phenomenon, although it cannot be ruled out as a possible contributing factor.

Considering the thermodynamics of platelet growth on a regional scale, if we assume that a platelet layer 1.0 m thick, consisting of 50% ice and 50% sea-water, forms under the ice cover over a 60 d period, and that the layer extends horizontally away from the ice-shelf edge a distance of 20 km (figures based on measurements during the 1986 season), then 9 × 10⁶ kg of ice will be formed per metre width of ice shelf. Assuming that the “active” water column extends to a depth of 100 m (Reference Lewis and PerkinLewis and Perkin, 1985) and the average rate of flow out from under the ice shelf is 0.10 ms⁻¹ (Reference Gilmour, MacDonald and van der HoevenGilmour and others, 1962; Reference Tressler and OmmundsenTressler and Ommundsen, 1962; Reference Heath and DunbarHeath, 1977), then 5 × 10⁷ m³ of water per unit width will be acting to form the platelet layer over the 60 d period. Given that L _i ≃ 2.5 × 10⁵ J kg⁻¹, and c _p = 4.2 × 10⁶Jm^−3°C⁻¹, then the 2.25 × 10¹²Jm⁻¹ released on freezing corresponds to a mean supercooling in the active layer of 0.011°C, which is of the same order as, though slightly lower than, the values of surface supercooling of 0.03°C observed by Reference Lewis and PerkinLewis and Perkin (1985). Using the same assumptions, the ice-growth rate would be 9.6 × 10⁻⁵ kgs⁻¹m⁻², giving a mean oceanic heat flux of −24 W m⁻². However, because only a part of the ice formed in this manner actually becomes incorporated into the ice sheet (see Fig. 1), the remainder forming an unconsolidated mass of platelets beneath the ice which melts when the water column warms in spring, we cannot simply set F _b to −24 W m⁻². The influence of the platelets on the growth of the solid ice sheet is therefore as dependent on the conductive flux of heat through the slab, and the columnar growth which results, as it is on the growth of the platelets themselves. A more realistic solution to the platelet problem might be simply to increase the columnar ice-growth rate during the period that the platelets are present by an amount proportional to the volume of the platelet layer occupied by ice. For example, if the platelet layer is 50% ice, then Δh _Β would be multipled by two each day the platelet layer is present. This is of course a great over-simplification because it ignores the effects of energy and salt transfers due to the formation of the platelets, and the physical changes to the ice bottom due to their presence. It has the advantages of being both simple and requiring no detailed information about the rate of platelet growth.

Turbulent fluxes

A third, but relatively minor, change from the Semtner model is the calculation, rather than explicit definition, of the turbulent fluxes F _s and F _l . This is achieved using the aerodynamic formulae

and

Here ρ _a is the density of air, c _pa is the specific heat of air, и is the mean wind speed, C _s and C _l are the sensible and latent energy-transfer coefficients (1.75 × 10⁻³; (Reference OverlandOverland, 1985)), f is the relative humidity, p ₀ is the surface pressure, and e _Saand e _S0 are the saturation vapour pressures of the air and at the surface, respectively. Reference MaykutMaykut’s (1978) fourth-order polynomial approximation is used to estimate e_sa and e _s0 from the air and surface temperatures, and f and p ₀ are assumed to be constant, with typical Antarctic values of 0.6 and 1013 mbar, respectively.

Predicted and observed ice growth

Figure 2 shows measured ice-core lengths from McMurdo Sound along with columnar and snow-ice thicknesses predicted by the original and modified versions of the Semtner model, after setting the initial conditions to observed values. Although many more ice-thickness measurements were made, difficulties in accurately determining the base of the ice sheet in the presence of a sub-ice platelet layer caused an unacceptable degree of uncertainty in the values of h _i, and only the core data can be considered sufficiently reliable for the present purpose. Input data for Т _а, u, F _r and F _Ld have been modelled using Fourier transforms of mean monthly data collected from a variety of sources. Air temperature, wind velocity, and incoming short-wave radiation are from Scott Base meteorological records, while incoming long-wave radiation is a synthesis of Antarctic measurements (Reference RusinRusin, 1961; Reference Thompson and MacDonaldThompson and MacDonald, 1962, Reference Zaitseva and ShlyakhovZaitseva and Shlyakhov, 1984) and values estimated from observed air temperature and cloud data, using empirical formulae (Оке, 1978). On the basis of observations, α, S, and ρ _S have been set at constant values of 0.75, 6 ppt, and 350 kg m⁻³, respectively. Changes in snow depth are represented by a linear increase of 0.35 m during the month of May, and held constant for the remainder of the year, while the thermal conductivity of the snow was calculated using the empirical formula of Reference Abel'sAbel’s (1893). The timing and quantity of sub-ice platelet growth were also based on measured values, and the density of the ice was fixed at 910 kg m⁻³ (Reference AndersonAnderson, 1960).

Fig. 2. Julian Day

Ice thicknesses predicted by the original (dotted line) and modified (solid line) version of the Semtner model, along with five measured ice-core lengths (large dots)

The agreement between the observed ice thicknesses and those predicted by the modified version of the model (solid line) is quite good, while the original model significantly under-predicts ice growth. Similar under-predictions resulted when the models of Reference MaykutMaykut (1978) and Reference Nakawo and SinhaNakawo and Sinha (1981) were run with the same input data (Reference CrockerCrocker, unpublished). As there was no flooding or snow-ice formation during the observation period (and none was predicted by the model), the improved predictions of the modified model are solely a result of including the influence of the sub-ice platelet layer on ice growth. Unfortunately, there are no data currently available which can be used to test the flooding and snow-ice growth parts of the model. Some isolated flooding events were observed at the end of the measurement period in mid-October, when the ice-surface temperature approached the prescribed threshold, but our treatment of the snow-ice growth process remains for the time being unproven. Bearing this in mind, it is nevertheless interesting to look at the patterns of ice growth and decay predicted by the model over the course of several seasonal cycles.

Multi-year projections

In order to extend the thickness predictions through the summer ablation period, we must first define a term accounting for basal melting. From measured summer oceanic heat fluxes (Reference Mitchell, Bye and JacobsMitchell and Bye, 1985), summer melting is simulated by setting F ₀ to a constant value of +70 W m⁻² through the months of November, December, January, and February. This heat energy is used first to melt the sub-ice platelet layer, and begins to reduce the thickness of the ice sheet only when all the platelets have been removed. An indication that this rather crude technique is at least generally representative of actual conditions can be found in the observation that both model predictions and measurements (Reference Paige and LeePaige and Lee, 1967) of the timing and magnitude of ice-sheet ablation coincide quite closely.

Figure 3 shows forecast growth and decay cycles over a 5 year period. Snow-ice does not form in appreciable quantities until the second summer, when a deep snow pack, the reduced winter growth accompanying it, and basal melt during the summer, combine to produce the necessary conditions of H _w and T ₀. There is a consistent reduction in the total ice thickness with time after the first winter, as the increased snow depth, and to a lesser extent the increased snow-ice depth, reduce the flow of heat through the slab. After 5 years, equilibrium conditions are reached, with snow-ice constituting just over 25% of the total thickness at the end of winter peak, and 100% of the total during the summer minimum. The predicted lack of surface melting, a product of the relatively strong winds and dry air prevalent during the summer (Reference Andreas and AckleyAndreas and Ackley, 1982), is consistent with observations, and has been found to be characteristic of Antarctic sea ice. The general pattern and magnitude of the thickness oscillations are also in agreement with observations from the multi-year fast ice in the southern end of McMurdo Sound, although it is difficult to assess the snow-ice predictions without corresponding snow-depth data. As a result, any assessment of the influence of different snow-accumulation regimes on the growth characteristics of the ice can only be considered tentative, but several features of the analysis are noteworthy.

Fig. 3. Time (years)

Predicted growth and decay cycles for snow, snow-ice, and columnar ice in McMurdo Sound over a 5 year period. It is assumed that freeze-up occurs on 1 March in year 1.

Since snow deposition is normally greatest near the coast, and decreases markedly with distance from shore, the range of snow conditions likely to be found in McMurdo Sound can be modelled by varying the snow accumulation from 0.35 m a⁻¹ to extremes of 1.00 and 0.05 m a⁻¹. With a deep snow-pack, true equilibrium conditions are not realized; rather a repeating 3 year cycle is established in which h _i progressively increases and h _si is progressively reduced, but on average snow-ice thicknesses are much greater than with 0.35 m of snow per year. Annual accumulations of 0.05 m, on the other hand, result in large initial columnar ice thicknesses (≈3.0 m) and a complete absence of snow-ice until the eighth winter, with an equilibrium not being established for over 20 years. When equilibrium conditions are reached, however, the total ice thickness at the end of winter maximum is only 1.85 m, 0.40 m of which is snow-ice.

Also, in the absence of surface melting, the equilibrium thickness is controlled by the summer oceanic heat flux. For example, reducing F _b by 75% produces a predicted equilibrium thickness (with 0.35 m a⁻¹ of snow) of over 10 m, providing a possible explanation for reports of extremely thick, old, sea ice near the edge of the McMurdo Ice Shelf (Reference Hoffman and StehleHoffman and Stehle, 1963). Unfortunately, crystallographic observations from this or other multi-year ice sheets are not available, and the predicted proportions of columnar and snow-ice cannot be verified.