JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B4, 2222, doi:10.1029/2002JB001935, 2003

2. Assumed Physics of Basal Freeze-On

[6]   Direct investigations of basal freeze-on are rare because of the logistical challenges associated with studying such processes in their modern subglacial environment [Lawson et al., 1998]. Although basic theoretical treatments of this phenomenon have been introduced into glaciology several decades ago [Weertman, 1961], there is a paucity of theoretical and empirical investigations of heat, water, and solute flow during basal freeze-on. To develop our model we have assumed that basal freeze-on resembles the phenomenon of frost heave that has been studied extensively by permafrost engineers during the last several decades [Fowler and Krantz, 1994; Konrad and Duquennoi, 1993; Miyata, 1998; O'Neill and Miller, 1985]. This assumption is supported by the general macroscopic similarity of ice formed by basal freeze-on and by frost heaving, as seen in Figure 2. This figure contains images of basal ice from Ice Stream C (Figures 2a and 2b) obtained from a borehole camera system [Carsey et al., 2003] as well as the outcome of an experimental frost heave study (Figure 2c) [Watanabe et al., 2001]. Although there are some stratigraphic differences between sub-ice sheet freeze-on and ice segregation generated in an idealized porous medium, a fundamental macroscopic similarity clearly exist. Secondary differences in interlayering may be caused by very large differences (~orders of magnitude) in freeze rate and temperature gradients between the sub-ice stream conditions and the laboratory experiments of Watanabe et al. [2001].

[7]   Frost heaving occurs when soil freezing induces water flow and volumetric expansion beyond that caused by the mere expansion of water on freezing [O'Neill and Miller, 1985]. Whether given sediments are susceptible to frost heaving and segregation ice growth depends on grain-size distribution [Everett, 1961; Hohmann, 1997; O'Neill, 1983; Tester and Gaskin, 1996]. Whereas fundamental physics of basal freeze-on and frost heave may be similar, the physical setting of the subglacial environment is different from a typical permafrost setting. Frost heave is a seasonal surface process in which the overburden pressure is typically small (~10–100 kPa) and the vertical temperature gradients are typically large (~1–10°C m^-1). The overburden ice pressure acting during subglacial freezing is very large (~1–10 MPa), and vertical temperature gradients in basal ice are relatively small (~0.01–0.1°C m^-1). Hence, subglacial freezing rates will be slow compared to the freezing rates observed in near-surface permafrost processes. However, we expect that the long timescale over which subglacial sediments may be exposed to freezing (~100s–1000s years) provides basis for significant freeze-induced pressure changes in the basal environment.

[8]   We do not include a basal water system in our numerical simulations. Instead, we use the end-member assumption of entirely local hydrological balance, with no loss or gain of water from long-distance transport in a basal water system. We are not claiming conclusively that long-distance basal water transport is indeed negligible beneath the West Antarctic ice streams. The problem of existence and physical nature of such long-distance transport is still open to interpretation. Much previous research has emphasized the importance of a distributed, throughgoing basal water system beneath ice streams, e.g. in lubricating their beds and supplying latent heat to areas of basal freezing [e.g., Alley et al., 1994]. Theoretical analysis [Weertman and Birchfield, 1982; Walder and Fowler, 1994] and scaled physical models [Catania and Paola, 2001] indicate that any such water system, if it exists, should remain widespread. Recent numerical modeling has addressed the question whether a regional basal water system must exist beneath the West Antarctic ice streams by estimating the regional net balance of basal melting and freezing. Constraints on the magnitude of basal shear heating and geothermal flux are, however, so insufficient that it is possible to calculate either large net melting or net freezing rates for the same parts of the ice stream system [Parizek et al., 2002; Joughin et al., 2003; Vogel et al., 2003].

[9]   Borehole studies of sub-ice stream hydrology yielded many important observations that are, nonetheless, often difficult to interpret or even contradictory [Kamb, 2001a]. In his overview of borehole observations from West Antarctica, Kamb [2001a, 2001b, sections 9.2 and 9.3] provided a detailed discussion of the undrained-bed model and concluded that it offers a useful framework for understanding ice stream dynamics in general, and for explaining the stoppage of Ice Stream C in particular. A key limitation of borehole studies is that they necessarily sample small spatial areas. In our opinion, the most complete view of sub-ice streambeds comes from a large quantity of geophysical and sedimentological data acquired in the Ross Sea. The data have widespread regional coverage [Shipp et al., 1999, Figure 2] with high horizontal resolution (down to ~1 m; ibid. p. 1512) and include areas over which the West Antarctic ice streams extended during the Last Glacial Maximum (LGM). Anderson [1999, p.72] has summarized relevant observations in the following passage: “Notably absent in the … records from the Ross Seafloor are tunnel valleys, subglacial braided channels, outwash fans/deltas, and eskers, which would imply channeled subglacial meltwater. In addition, hundreds of piston cores … have only on rare occasions recovered graded sands and gravels that might be associated with subglacial meltwater systems. The few exceptions … are cores acquired near the termini of valley and outlet glaciers. The virtual absence of meltwater features and deposits … is perhaps the most important difference in geomorphic character between the Antarctic continental shelf and Northern Hemisphere glacial terrains.”

[10]   Notwithstanding the controversy regarding the existence and nature of sub-ice stream water drainage, we adhere to the undrained assumption. We do so to generate an end-member view of ice streamflow. We consider it possible that the undrained model is the best description of the ice streams. Further, we note from the work of Parizek et al. [2002] that an undrained model likely was even more applicable in the past. Regardless, the reader should bear in mind that we are working on an end-member of possible ice stream behavior.

2.1. Ice-Water Interface Curvature and Surface Tension Effects

[11]   Phase changes in an ice-water system are typically thought of as being controlled mainly by temperature and pressure. In the glaciological literature, the concept of the pressure-melting point is commonly utilized as a synonym for the freezing point. However, there are other less commonly considered factors that influence the temperature at which the ice-water phase transition occur. One such factor is the presence of solutes in the liquid water. An increase in solute concentration has the same effect as an increase in fluid pressure as it depresses the freezing point. The influence of solute concentration can be expressed formally through a pressure term, which is referred to as the osmotic pressure [Padilla and Villeneuve, 1992]. Another factor, which is often overlooked, is ice-water interfacial effects, especially surface tension arising from interface curvature. This factor becomes important when ice crystal growth is restricted to fine inter-crystalline veins [Harrison, 1972; Raymond and Harrison, 1975] or micron-sized pore spaces of fine-grained subglacial sediments [Tulaczyk, 1999]. The surface tension effect is paramount in setting up a hydraulic gradient that drives water flow in a freezing porous media [Everett, 1961]. In general, the finer grained a sediment is, the higher is the curvature of ice-water interfaces, and the greater is the depression of the ice-water phase change temperature, i.e. the freezing point [Hohmann, 1997]. Unfrozen water has been observed in clays at temperatures lower than -10°C [O'Neill, 1983].

[12]   When liquid water and ice co-exist in a curved interface configuration, there is a pressure jump between the two phases due to the interfacial effects [Fowler and Krantz, 1994; O'Neill and Miller, 1985]. The size of the pressure jump depends on the curvature of the ice-water interface as proposed by Gold [1957]. The most simplified assumption for the interfacial pressure jump between p_i and p_w is [Everett, 1961; Hopke, 1980; Tulaczyk, 1999]:

where p_i is the ice pressure, p_w is the pore water pressure, _iw is the ice-water surface energy, dA/dV is the curvature of the ice-water interface. At the ice-till interface, dA/dV is a function of the effective pressure. At zero effective stress, dA/dV = 0 and the ice base is planar. If the effective pressure reaches a critical value, the ice-water interface complies with the particle surfaces and its structure should be numerically equal to the specific surface area of the sediments, SSA. The ice-water curvature is in this case dA/dV = SSA [Tulaczyk, 1999]. The specific surface area is small if the sediment is coarse-grained and ice may freely intrude the pore spaces. If the sediment is fine-grained, the specific surface area is high and the ice-water interface has a high curvature, which impedes ice formation. This effect can be ascribed, at least formally, to an interfacial effective pressure [Tulaczyk, 1999]:

where subscript ‘_iw’ refers to the ice-water interface, SSA is specific surface area, and r_p is the characteristic particle radius.

[13]   Freezing of liquid water takes place when the pressure components and the temperature satisfies the generalized form of the Clapeyron equation [Fowler and Krantz, 1994; Miyata, 1998; O'Neill and Miller, 1985]. When solutes are present in the liquid water, the generalized Clapeyron equation becomes [Padilla and Villeneuve, 1992]:

where p_w is the water pressure, p_i is the ice pressure, p_o is the osmotic pressure, _w is the density of water, _i is the density of ice, L is the coefficient of latent heat of fusion and T is the temperature in °C. This form of the Clapeyron equation represents a general thermodynamic relation whose validity is not limited to our specific purpose. Its validity has been verified experimentally [Biermans et al., 1978; Konrad and Duquennoi, 1993; Krantz and Adams, 1996; Miyata and Akagawa, 1998] and it provides the fundamental basis for frost heave models [Fowler and Krantz, 1994; Miyata, 1998; O'Neill and Miller, 1985]. The Clapeyron equation provides a mean for coupling pressure terms and temperature in a freezing porous medium. An examination of this equation demonstrates that a significant pressure jump can develop across the ice-water interface when supercooled liquid pore water is present beneath the freezing interface.

[14]   The effect of interface curvature, equation (1a) or (1b), and the Clapeyron equation (2), can be linked together. Solving the former for ice pressure gives p_i = 1/r_p + p_w, and insertion into the latter yields:

This equation is the same expression used by Raymond and Harrison [1975] to treat freezing of water in micron-sized veins between ice crystals. It is also the fundamental equation for the temperature of ice-water phase transition given in Hooke [1998, p. 5]. The first term of equation (3) specifies the effect of water pressure on phase transition. The commonly utilized pressure-melting point is an approximation of the phase transition temperature based only on this term. The second and third terms of equation (3) include the effects of interfacial pressure and osmotic pressure in a complete treatment of the ice-water phase transition. These additional factors are fundamental in our treatment of the response of subglacial sediments to basal freezing. The term supercooling is here used to refer to any liquid water present at temperatures below the pressure-melting point.

2.2. Coupled Vertical Transport of Water, Heat, and Solutes

[15]   From force equilibrium it follows that the ice pressure at the ice base equals the gravitational load of the overlying ice, p_i = p_n. Hence, supercooling of a freezing ice base overlying a fine-grained till will be associated with lowering of the pore water pressure (equation (3)). This localized drop in pore water pressure gives rise to a hydraulic gradient that drives water flow toward the freezing ice base as outlined conceptually in Figure 3.

[16]   The upward flow of water (toward a freezing interface) is the characteristic process that we see as being common to both, the frost-heave phenomenon and subglacial freeze-on. The freeze-induced water flow in porous media with water present in solid and liquid phases is driven by cryostatic suction, which is analog to capillary suction in sediments with liquid and gas phases of water in the pore spaces [Fowler and Krantz, 1994]. When freeze-driven water flow becomes significant, one must assess the thermodynamics of the freezing process in the context of coupled flows of water, heat and solutes. Important feedbacks arise because the flow of water initialized by cooling may carry sufficient heat and solutes to significantly influence the freezing conditions at the ice base.

[17]   Vertical water flow through unfractured, porous media occurs in response to a gradient in excess water pressure. The excess water pressure, u, is defined as the water pressure component in excess of an initial hydrostatic pressure, p_h. The total water pressure is thus p_w = p_h + u [Domenico and Schwartz, 1990, equation (4.50)]. From Darcy's law, vertical water flow velocity is assumed to be proportional to the excess water pressure gradient, u/z [Domenico and Schwartz, 1990, equation (4.53)]:

where _w is the water flow velocity, K_h is the coefficient of hydraulic conductivity, _w is the density of water, and g is the acceleration of gravity. Vertical gradients in excess pore pressure that build up in a freezing till can be obtained by solving a one-dimensional diffusion equation [Mitchell, 1993, equation (13.19)]:

where, t is time, c_v is the hydraulic diffusion coefficient, and z is the depth coordinate (taken here to be zero at the ice base). Once water flow rates are determined, vertical transport of heat can be derived from a diffusion-advection equation [Domenico and Schwartz, 1990, equation (9.21)]:

where T is temperature, _t is the thermal diffusion coefficient, and _w is the velocity of water flow. Vertical transport of solutes in subglacial sediments underlying a freezing ice base can be determined from an analogous diffusion-advection equation [Domenico and Schwartz, 1990, equation (13.9)]:

where C is the concentration of solutes and _c is the chemical diffusion coefficient. We neglect the influence of convection on heat and solute redistribution because the Rayleigh number for the considered problem is several orders of magnitude smaller than the usual convection threshold [Domenico and Schwartz, 1990, equation (9.25)].

2.3. Formation of Ice Lenses

[18]   Liquid water flows toward the freezing interface where it accretes into a layer of segregation ice. This layer forms as long as the surface-tension penalty for forming ice crystals in small pore spaces is large enough to suppress growth of ice within the till [Everett, 1961; Konrad and Duquennoi, 1993; Miyata, 1998; O'Neill and Miller, 1985]. When the distributions of water pressure, temperature and solute concentration are known, one can rearrange the Clapeyron equation (2), and solve it for ice pressure within the till pore spaces:

This pressure should exist inside any small ice crystal that may form in the confined pore space of fine-grained sediments. When the ice pressure exceeds the sum of gravitational overburden pressure and the ice-water interfacial pressure, nothing is left to keep the crystal from growing beyond the confines of the pore space in which it initially formed [O'Neill and Miller, 1985]. For ice lens initiation, we thus use the criterion [Hopke, 1980]:

where p_n is the vertical overburden pressure and p_iw is the ice-water interfacial pressure. An ice lens grows through accretion of segregation ice until a new lens is initiated. The complex dependence of ice pressure on temperature, water pressure, and osmotic pressure determines where within the till there will be a new, thermodynamically more favorable location for ice crystal growth (equation (9)). Through this process of progressive ice-lens formation, the freezing front moves into increasingly deep levels in the till. This mechanism produces the banded ice-sediment structure of freezing soils [Fowler and Krantz, 1994; Konrad, 1994], and it may play an equally important role in forming layered, debris-bearing basal ice [Gow et al., 1979; Herron and Langway, 1979; Koerner and Fisher, 1979; Lawson et al., 1998]. This concept of subglacial ice lens development is shown in Figure 4.

[19]   Ice lens initiation is a sensitive and particularly complicated part of frost heave simulations that typically involves an empirical treatment as in O'Neill and Miller [1985]. A micro-scale approach to frost heave analysis related to premelting of ice [Wettlaufer and Worster, 1995; Wilen and Dash, 1995] and thermomolecular force [Rempel et al., 2001b] may lead to a more complete thermodynamical treatment. However, macro-scale models can be tested and verified via experimental studies [Fowler and Krantz, 1994; Krantz and Adams, 1996; Miyata, 1998; Miyata and Akagawa, 1998] and they can also be compared to field observations [Hohmann, 1997; O'Neill, 1983]. There is thus a major advantage in the phenomenological use of the interfacial effects and phase equilibria [Michalowski, 1993]. The method proposed by O'Neill and Miller [1985], on which we base our work, is the most commonly used theoretical treatment of contemporary frost heave models.

[20]   Force balance associated with ice pressure acting in growing ice lenses and effective pressure and water pressure acting in the surrounding unfrozen till is shown conceptually in Figure 5. Incompletely frozen till can become trapped between two neighboring ice lenses and this may effectively isolate the till from further inflow/outflow of water and solutes. These inactive till layers separated by segregation ice should, after sufficient cooling, freeze completely and become fully incorporated into the basal ice.

2.4. Regelation

[21]   Previous models of ice intrusion into subglacial sediments have concentrated on the process of regelation, e.g., Iverson [1993] and Iverson and Semmens [1995]. Surface tension effects are negligible in coarse-grained sediments and the pressure that opposes intrusion of ice into pore spaces is the pore water pressure alone. For regelation into fine-grained sediments ice pressure must exceed the pore water pressure as well as the interfacial tension [Tulaczyk, 1999]. Modification of equation (1b) yields the critical subglacial water pressure below which ice may intrude into the pore spaces of the underlying till:

Once regelation starts, the velocity of regelation is finite and we use an experimentally based formulation given by Iverson and Semmens [1995]:

where K_r is the conductivity of the sediment to ice, and p = p_i - p_w is the driving stress for regelation, and z_r is the vertical penetration depth of ice.

Citation: Christoffersen, P., and S. Tulaczyk, Response of subglacial sediments to basal freeze-on, 1, Theory and comparison to observations from beneath the West Antarctic Ice Sheet, J. Geophys. Res., 108(B4), 2222, doi:10.1029/2002JB001935, 2003.

Copyright 2003 by the American Geophysical Union.