K. M. Golden, ^* S. F. Ackley, V. I. Lytle

Sea ice exhibits a marked transition in its fluid transport properties at a critical brine volume fraction p_c of about 5 percent, or temperature T_c of about -5°C for salinity of 5 parts per thousand. For temperatures warmer than T_c, brine carrying heat and nutrients can move through the ice, whereas for colder temperatures the ice is impermeable. This transition plays a key role in the geophysics, biology, and remote sensing of sea ice. Percolation theory can be used to understand this critical behavior of transport in sea ice. The similarity of sea ice microstructure to compressed powders is used to theoretically predict p_c of about 5 percent.

K. M. Golden, Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA. S. F. Ackley, U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, NH 03755, USA. V. I. Lytle, Antarctic Cooperative Research Centre and Australian Antarctic Division, University of Tasmania, Hobart, Tasmania 7001, Australia.
^*   To whom correspondence should be addressed. E-mail: golden{at}math.utah.edu

Brine transport is fundamental to such processes as sea ice production through freezing of flooded ice surfaces, sea ice heat fluxes, and nutrient replenishment for sea ice algal communities, as well as being an important factor for remote sensing. However, the basic transition controlling brine transport has received little attention. Percolation theory (5, 6) has been developed to analyze the properties of materials where connectedness of a given component determines the bulk behavior. We show that it provides a natural framework to understand the critical behavior of sea ice. In particular, we apply a compressed powder percolation model to sea ice microstructure that explains the law of fives, the observed behavior (4) of the fluid permeability in the critical temperature regime, as well as data on surface flooding collected recently on sea ice in the Weddell Sea and East Antarctic regions.

It was observed in the Arctic (7) that a snow storm and its resultant loading on a sea ice layer can induce a complete upward flushing of the brine network. In the Antarctic, it was observed that the freezing of a surface slush layer, with resultant brine drainage, induced convection within the ice, whereby rejected dense brine is replaced by nutrient-rich sea water from the upper ocean (8), fueling autumn blooms of algae in second-year ice (9). During the autumn freeze-up, this process provided about 70% of the salt flux into the upper ocean and increased the total heat flux through the overlying ice and snow cover. The proliferation and growth of sea ice organisms is favored by permeable ice, which allows nutrient replenishment (10, 11). For remote sensing, surface flooding and subsequent freezing can affect microwave backscatter from sea ice (12, 13), and connectedness of the brine inclusions affects the permittivity of sea ice (14, 15). As yet another example, it was observed in the Arctic that there was about a 20-day time lag between the start of the spring snow melt and the occurrence of freshwater input into the mixed layer (16, 17). Presumably, part of this lag was the time it took for the ice sheet to warm to above the critical temperature to allow drainage out of the ice (16).

Percolation theory (5, 6) has been used to successfully model a broad array of disordered materials and processes. The simplest form of the lattice percolation model (6) is defined as follows. Consider the d-dimensional integer lattice Z^d and the square (or cubic) network of bonds joining nearest neighbor lattice sites. To each bond, with probability p, 0  p  1, we assign a 1, meaning it is open, and with probability 1 - p we assign a 0, meaning it is closed. Groups of connected open bonds are called open clusters, and the size of a cluster is just the number of open bonds it contains. In the percolation model, there is a critical probability p_c, 0 < p_c < 1, called the percolation threshold, at which the average cluster size (p) diverges and an infinite cluster appears, so that the open bonds percolate. In two dimensions, p_c = 0.5; in three dimensions, p_c  0.25. For p > p_c, the infinite cluster density P(p) exhibits power law behavior near the threshold, P(p) ~ (p - p_c), where  is the percolation critical exponent,   1. This model deals only with the geometrical aspects of connectedness in disordered media, yet we are interested in the transport properties as well. Then we consider a random resistor network, where the bonds are assigned the conductivities 1 and h  0 with probabilities p and 1 - p. With h = 0, for p < p_c, the effective conductivity (p) = 0, whereas near the threshold with p > p_c, (p) exhibits power law behavior (p) ~ (p - p_c)^t, where t is the conductivity critical exponent, with 1  t  2 in d = 2,3 (18). Analogously, we may consider a random pipe network with effective fluid permeability (p) exhibiting similar behavior (p) ~ (p - p_c)^e, where e is the permeability critical exponent, with e = t. Critical exponents for lattice models are generally believed to exhibit universality, meaning that they depend only on dimension and not on the type of lattice, although continuum models can exhibit nonuniversal behavior, with exponent values different from the lattice case, t > 2 in d = 3, and e  t.

If the above lattice model is applied to sea ice, where the open bonds represent brine and the closed bonds represent ice, then p_c would be about 25% in d = 3, which is much larger than the observed 5%. Even continuum models, such as ellipsoidal brine inclusions randomly distributed in an ice host (a commonly used model for sea ice), exhibit critical volume fractions in the 20 to 40% range (19). Instead, consider the critical behavior of composites made up of conducting particles suspended in an insulating matrix (20) and the problem of finding microstructures that reduce p_c. For some flexible polymer composites designed to be highly conducting, it was found that by compacting powders of large polymer particles with much smaller metal particles, low critical volume fractions of the (more expensive) metal particles are required to significantly lower the resistance of the composite (21, 22). The resulting microstructure of such compressed powders is remarkably similar to the cellular microstructure of columnar sea ice (Fig. 1). The key parameter in predicting the conduction threshold for compressed powders is the ratio  = R_p/R_m of the radii of the large polymer particles to the smaller metal particles (22, 23). For large , p_c is not very sensitive to the exact value of . For example, a range of 44 to 17 for  gives only a range of 3 to 7% for p_c. Using photomicrographs of sea ice microstructure and typical brine inclusion sizes (3), we measured the corresponding parameter for sea ice, obtaining an average of  24. Applying the compressed powder percolation model (23) yields a critical brine volume for columnar sea ice of about 5%. This result will vary with ice crystal structure. For example, the slightly higher values of p_c (lower ) observed in (4) are caused by the more random distributions of brine inclusions in granular ice as compared to columnar ice. The compressed powder model explains why sea ice exhibits such low values for p_c, as compared to the 20 to 40% range that might be expected, and provides reasonable estimates for p_c, which depends only on the geometry of the two phases. Observations show, however, that fluid transport then proceeds primarily through large-diameter brine channels (3, 8, 9). Dynamic models will likely be needed for analysis of brine-channel formation beyond the initial onset of percolation at p_c, although it is interesting to note the existence of the "independent crossings" of a sample in standard percolation models (24). They have large separations in comparison to the microstructural (or lattice spacing) scale for p near p_c and play much the same role for transport as the brine channels.

Comparison of the electrical conductivity of compressed powders and the fluid permeability of sea ice (Fig. 2) shows that near the critical temperature, sea ice permeability displays the same characteristic behavior exhibited, in general, by transport coefficients of composite media near a percolation threshold (19, 20). The data in (Fig. 2B) are thus best fit not with a straight line as in (4) but with an "S" curve that captures the actual behavior of (T) in the critical regime near the percolation threshold T_c^perc.

Compressed powders with low p_c exhibit large, nonuniversal values of t ranging between 2 and 7 (20). A rough estimate for the fluid permeability critical exponent e for sea ice based on the data in (Fig. 2B) is about 2.5, although there is significant uncertainty in this estimate (but e is probably between 2 and 4). Much more experimental work needs to be done to determine the actual range of e and its dependence on the type of sea ice and on other factors.

Data collected on Antarctic sea ice directly demonstrate the significance of the percolation threshold. During the winter ANZFLUX experiment (25) in the Eastern Weddell Sea, we encountered a thin ice pack, typically 20 to 60 cm thick. Unusually large vertical oceanic heat fluxes resulted in ice basal melt rates of up to 3 cm/day (average was 1 cm/day), which could have melted the ice in a short period. The persistence of the ice depended on flooding of the surface and the subseqent freezing of this slushy snow/brine mixture to form snow ice, which replaced the ice melting on the bottom (26). The surface flooding was controlled by upward brine percolation. Temperature profiles measured hourly during a 5-day drift camp over Maud Rise at about 4°E longitude and 65°S latitude (Fig. 3) indicated that while most of the sea ice layer remained above the critical temperature for percolation, the top 5 cm or so of the sea ice was impermeable, except during the typically warm storms. Sufficient loading led to surface flooding, which subsequently froze. This cyclic process occurred twice during the 5-day drift. The impermeable layer, defined by temperatures below a critical temperature of about -5.3°C, disappeared with the onset of the first storm during day 216 of the year (4 August 1994). For about a day, the entire ice sheet was permeable, and by noon of the next day, we observed a thick layer of slushy snow consisting of 30 to 50% liquid brine. Subsequently, a cold period set in, the slush froze, and the impermeable cap returned as frozen slush. Then another storm moved in, with resultant warming, flooding, and freezing. Late in the evening of day 219 during a warm storm, we observed large "boils" on the snow surface, which was apparently brine percolating up through the ice.

A similar type of flooding event was observed during the winter HIHO HIHO experiment (27) in the East Antarctic sea ice pack at about 145°E and 65°S. Brine volumes were calculated using measured salinity and temperature profiles (Fig. 4). The initial brine volume of the ice was near, yet below, the percolation threshold. As air temperatures increased, brine volume throughout the ice sheet was raised above the percolation threshold of p_c  5% (and in the upper 15 cm was even raised above a higher p_c for granular ice), resulting in surface flooding of the snow, which covered the sea ice. This saturated snow froze, forming snow ice. Ice and snow thickness measured before (day 230, 18 August 1995) and after (day 236) the flooding indicate that 9.5 cm of snow ice had formed. These data demonstrate that an air temperature increase alone can cause the permeability phase transition.

As a final example, consider the algal bloom observed in a porous sea ice layer at a depth of 10 to 30 cm during the autumn of 1992 in the western Weddell Sea (9). From day 60 (29 February) to 81 of the year, standing stocks of pigments in the ice increased at a rate of 0.8 mg m^-1 day^-1, yet after day 81, the algal growth rate was reduced to one-tenth the earlier value. Day 81 is when the downward advancing critical isotherm of T_c  -4°C passed through the bottom of the algal layer, effectively cutting off the community from significant nutrient replenishment, because ice above this isotherm was impermeable. The critical temperature higher than -5°C is understood by noting that the ice surrounding the algal layer was granular, and had a higher p_c, yet a salinity of only about 5 ppt.

It has been demonstrated that sea ice exhibits a percolation transition at a critical temperature. Recently, it has been found that this type of behavior in composite materials is mathematically analogous to a phase transition in statistical mechanics, like water at its freezing point or a ferromagnet at its Curie point (28). Thus, T_c may be viewed as a type of phase transition point, with sea ice at temperatures between T_c and -1.8°C (the freezing point for sea water) being a hybrid phase between liquid and solid.

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30. We thank the participants of the ANZFLUX and HIHO HIHO experiments and the crews of the R.V. Nathaniel B. Palmer and the R.S.V. Aurora Australis for contributing to the field work yielding the data we report. We also thank M. McPhee, D. Perovich, H. Eicken, and A. Efros for helpful discussions and P. Heil, A. Worby, and two anonymous reviewers for helpful comments on the manuscript. Supported by NSF grants OPP-97-25038, DMS-96-22367, and Office of Naval Research grant N00014-93-10141 to K.M.G. and NSF grant OPP-93-15934 to S.F.A.