## Thermodynamics of phase equilibria in snow

Bulk equilibrium temperature in a pure water system

Thermodynamic relationships among the three water phases determine the grain growth and metamorphism of snow. Whether the compound H2O exists as ice, water, or water vapor depends on its temperature (T) and pressure (p), as shown in the phase diagram in Fig. 2.5. The curves for evaporation (A), sublimation (B), and melting (C) trace the points where two bulk-water phases coexist in thermodynamic equilibrium. All three phases coexist at the triple point, where the temperature is 0.01 °C and the vapor pressure is 6.1112 hPa. This is not the same as the ordinary melting point of ice, which occurs at a temperature of 0 °C and at 1 atmosphere of pressure (1013.25 hPa). Vapor is in metastable equilibrium with respect to supercooled water along the dotted line A', which extends the evaporation curve below 0.01 °C.

Air filled with vapor at equilibrium is saturated and the addition of more vapor will cause condensation. Below the triple point, the saturation vapor pressure at equilibrium with respect to water always exceeds that with respect to ice and thus favors the growth of snow crystals, as discussed in Section 2.1.2. Excess saturation vapor pressure reaches a maximum at -11.8 °C (see Fig. 2.5). Supersaturation relative to ice for vapor at equilbrium with respect to water increases from 0% at 0 °C to about 46% at -40 °C.

The curves in the phase diagram are determined by integrating the Clausius-Clapeyron equation dp Lji

where i and j are the two phases, T is in kelvin, V is the volume per unit mass (or reciprocal density) of the phase, and L is the latent heat. At 0 °C the latent heats of fusion (Lie), evaporation (L¿v), and sublimation (Liv) are, respectively, 3.335 x 105, 2.505 x 106, and 2.838 x 106 J kg-1. L depends moderately on temperature (see Pruppacher and Klett, 1997, p. 97). For an ideal gas and constant Lkv, integration of Equation (2.1) provides approximate expressions for the evaporation and sublimation curves of the form

where T is in °C. Here, pvsat,k is the saturation vapor pressure (hPa) with respect to ice or water, and the coefficients a, b, and c are empirical fits for given temperature ranges. For the range between —40 and 0 °C, Buck (1981) recommends the expression

pv.sau = [1.0007 + (3.46 x 10—6pa)]6.1121 expf ^ \ (2.3)

pv,sat,i = [1.0003 + (4.18 x 10—6pa)]6.1115exp (--) (2.4)

for water saturation and between —50 and 0 °C

for ice saturation, where pa is atmospheric pressure (hPa). The initial coefficient in both expressions is a slight correction factor used when the gas phase is moist air rather than pure water vapor.

The effect of curvature on phase equilibrium in snow The T-p curves derived from the Clausius-Clapeyron equation are for phases separated by plane surfaces. These equilibrium conditions change when the surface is curved, as is the case with ice grains or water menisci in wet snow (see Fig. 2.4). Because work is needed to extend interfacial films, the phases behind convex interfaces experience a higher energy and pressure. The pressure difference (pij = pi — pj) across a curved surface is given by Laplace's equation as pj = -1, (2.5)

where a j is the interfacial surface tension and Z ij is the mean radius of curvature (Defay et al., 1966, p. 6 Dullien, 1992, pp. 119-122). The surface tensions a^, aiv,

Table 2.3 Ratio of saturation pressure over ice spheres to that over plane surfaces.

Table 2.3 Ratio of saturation pressure over ice spheres to that over plane surfaces.

 rg (1 x 10-3 m) 10-6 10-5 10-4 10-3 10-2 10-1 pv ,sat,i/pv,sat,i 0 °C 6.05 1.20 1.018 1.002 1.0002 1.000 02 -20 °C 6.97 1.21 1.020 1.002 1.0002 1.000 02

and oag are, on average, 0.028, 0.104, and 0.076 Nm—1 at 0 °C and vary strongly with temperature (Pruppacher, 1995). Higher surface tension and tighter curvature result in larger pressure differentials.

For curved surfaces, the Clausius-Clapeyron equation generalizes to the Gibbs-Duhem equation (Defay et al., 1966, Colbeck, 1980)

Vrdp, Vjdpj

Relating the phase pressure through Laplace's equation, Gibbs-Duhem's equation integrates to Kelvin's expression for the equilibrium vapor pressure p'vsat k over a curved surface

'2okv 1

Zkv RvPkT

Here pv sat k is the equilibrium pressure over a flat surface, pk is the density of water or ice, Rv is the gas constant for water vapor (= 461.50 J kg-1 K—1), and T is in kelvin. Equation (2.7) predicts that the vapor pressure will be higher over smaller ice particles or fine-structure with higher curvature. Table 2.3 shows p!v sat k/pv sat k for ice spheres of varying radii (Z¡v = rg) at temperatures of 0 and —20 °C.

The Gibbs-Duhem equation also predicts the melting temperature of snow. In the case of a two-phase water-ice system, such as water-saturated snow or pockets of water-saturated snow within the pack (grain A in Fig. 2.6a), pressure in the water computes from capillary pressure, pat = pa — pt, and that in ice from Laplace's equation. We discuss capillary pressure in Section 2.4.2. The melting point depression of water-saturated snow thus decreases directly with capillary pressure and inversely with grain radius as

273.15

2ou Vi'

Accordingly, larger ice grains have higher melting temperatures and will grow at the expense of smaller grains. In a totally saturated snowpack, the air/water interface is flat and the first term in Equation (2.8) drops out.

At lower water contents, liquid is usually configured as pendular rings about two-grain contacts or as veins in three-grain clusters, as in Figs. 2.6b and c r g

Ca« Pa

Figure 2.6. Pressure relationships and shape of water inclusions in wet snow, where pk is the pressure and Z ai is the radius of curvature for the air/water interface. (a) Saturated zones in snow with high water content (not to scale). (b) Two-grain contact with pendular inclusion in snow with low water content (reprinted with modifications from Colbeck, 1979; copyright 1979, with permission from Elsevier). (c) Three-grain cluster with vein-fillet inclusion in snow with low water content (reprinted from Colbeck, 1979, as in b).

(Colbeck, 1979). With air occupying much of the pore space, pressure in the ice computes from Laplace's equation applied to the ice/air interface and the melting point depression becomes (Defay et al., 1966; Colbeck, 1979)

273.15

2ctw Vi

Because the second term in (2.9) is small, Td depends mainly on capillary pressure, which increases with decreasing liquid water.

Melting temperatures predicted by Equations (2.8) and (2.9) are depressed only slightly below the bulk value of 0 °C and thus, for energy balance computations, the effects of surface tension and curvature on temperature are insignificant. They are, however, of prime importance to snow metamorphism and bond strength. At low water contents, the melting temperature of grain surfaces is below that of the ice bonds, directing heat away from the bonds and causing them to grow and strengthen (Colbeck, 1979). When snow grains become totally surrounded by water, there is a thermodynamic reversal and heat flows towards the bonds, causing them to melt. Saturated or "slushy" zones in snow are therefore cohesionless and have little r g strength. Impurities in the meltwater further depress the melting temperature by KfM, where KF (=1.855 K kg mol-1) is the cryoscopic constant for water and M is the molality of the species (moles of solute per kg of solvent).