List of symbols and units
Introduction
Most of the important material properties of the porous system snow depend on liquid-water saturation and liquid distribution. As the water content increases, granular porous material typically shows a sudden transition from the pendular into the funicular mode of water distribution. This transition causes rapid changes in the physical properties.
Based on a model of face-centered spherical grains, Reference ColbeckColbeck (1973) suggested that the transition occurred at a liquid saturation of about 14% of the pore volume. From measurements of the dielectric constant of wet snow, a transitional zone in the range of 11 to 15% liquid saturation was found experimentally (Reference DenothDenoth, 1980). Additional information as to the upper limit of the pendular distribution was derived from measurements of the drainage-flow of free water through aged snow with rounded ice grains (Reference Denoth and DenothDenoth and others, 1979).
It is obvious that the snow structure, and therefore the stage of snow metamorphism, influences the way free water is distributed around contact points of ice grains. In this paper, results of measurements of the pendular–funicular liquid transition at different stages of snow metamorphism are reported.
Snow classification
In order to account for the influence of snow metamorphism on the pendular–funicular transition, the snow samples have been classified according to the size and the shape of the ice grains or clusters. Assuming that the ice grains can be represented by spheroids, the shape is characterized by the ratio m of the two principal axes a 1 and a 2 of the ellipsoids: m = a 1/a 2. The axial ratio m was determined experimentally using two different methods: by analyzing photographs of the surface of the snow samples or by an analysis of the static dielectric constant (Reference DenothDenoth, in press) based on the mixing formula of Polder and van Santen.
Reference Polder and SantenPolder and van Santen (1946) developed a general theory to calculate the relative permittivity of a mixture containing ellipsoidal inclusions. In an adapted form for the system of snow, this formula reads as follows:
The theory of Polder and van Santen is of particular importance for modelling the dielectric constant of snow, because it allows us to account for the snow structure by the shape factors g i, j and to account for the structure and distribution of liquid inclusions by the shape factors gl,k . Consequently, the stage of snow metamorphism may be taken into account in this theory. The general applicability of this theory to snow has been shown by Reference Denoth and SchittelkopfDenoth and Schittelkopf (1978). A detailed analysis and an application to three different types of wet snow were given by Reference ColbeckColbeck (1980).
The shape factor g i, j required in Equation (1) can be calculated from the axial ratio m of the ice grains (Reference StonerStoner, 1945):
To account for the influence of snow metamorphism on liquid distribution, the snow samples have been grouped into three types according to their shape factor g i and the mean grain-size d m: Snow type I is characterized by d m < 0.5 mm and g i ≤0.06; this corresponds widely to new snow or to snow which has not undergone appreciable metamorphosis. Snow type II is characterized by 0.5 <d m ≤ 1 mm and 0.09 ≤ gi ≤ 0.16; this corresponds widely to aged. Alpine snow, some days to weeks old. Snow type III is characterized by d m > 1 mm and 0.18 ≤ gi ≤ 0.33; this corresponds to old, coarse-grained snow or firn. A compilation of these characteristic parameters for the classification of snow samples is given in Table I.
Table I. Snow classification characteristics
Experimental procedure
Based on the mixing formula of Polder and van Santen (Equation (1)), the shape factor g l of the liquid inclusions was determined experimentally by measuring the high-frequency relative permittivity ε ∞ of wet snow, liquid saturation, and porosity. The liquid content was measured with a freezing calorimeter; porosity and saturation were calculated from the density ρ and the water content W of the snow sample:
The shape factor g i of the ice grains was calculated from the static dielectric constant ε s of the snow sample using Equation (1) with the appropriate values for the static dielectric constants of the components. In addition, for some snow samples, g i was calculated from the photographically determined axial ratio of the ice grains using Equation (2).
In order to determine with high accuracy ε s and ε ∞ using Equation (1), it was necessary to measure the relative permittivity in the frequency range of 100 Hz up to 100 MHz. In the frequency range of 100 Hz up to 20 MHz a capacitance bridge and a twin-T-bridge were used; in the range of 500 kHz up to 100 MHz the relative permittivity was measured with a network analyser. The limiting relative permittivities ε s and ε ∞ were obtained by a least-square-fit using the model of Cole and Reference Cole and ColeCole (1941). The measurements were carried out in natural snow cover in the Stubai Alps at 3000 m a.s.l. (Schaufelferner/Daunferner).
Liquid distribution
For the three different types of snow characterized in Table I, the shape factor g l is shown against liquid saturation in the Figures 1 to 3. The solid line shown in the figures represents a least-square approximation using Chebyshev polynomials. The weight function was chosen according to the relative importance of the errors of g l in the different regimes of liquid saturation. The error E(g l) of g l is mainly determined by the errors in measuring liquid saturation and snow porosity. Typically it is E(g l) <0.1 for saturations S ˃2%; it increases to about E(g l) ≈ 0.7 for lower saturations.
Fig. 1. The shape factor gl for type I snow as a function of water saturation. Sc is the critical saturation: Sc ≈ 8%. The solid line represents a least-square approximation by a fourth-degree polynomial.
Fig. 2. The shape factor gl for type II snow as a function of water saturation. SC is the critical saturation: Sc ≈ 5%. The solid line represents a least-square approximation by a fourth-degree polynomial.
Fig. 3. The shape factor gl for type III snow as a function of water saturation. Sc is the critical saturation: Sc ≈ 3%. The solid line represents a least-square approximation by a fourth-degree polynomial.
Figure 1 represents type I snow, corresponding to new snow or snow which has not undergone appreciable metamorphosis. At low water saturations 0 ≤S≤S c the shape factor g l increases monotonically and takes an approximately constant value of g l ≈ 0.051 for saturations exceeding a critical saturation S c ≈ 8%. In the saturation range of 13%≤S <18% a transitional zone between the pendular and the funicular mode of liquid distribution is formed. This transition is characterized by a change in the shape factor from g l ≈ 0.053 in the pendular regime to g l ≈ 0.091 in the funicular regime. In the funicular regime, the shape factor g l shows only a small increase with increasing saturation.
Figure 2 represents the shape factor g l for type II snow. At low saturations, the shape factor shows a steep increase with increasing saturation. Compared to type I snow, the critical saturation S c above which the shape factor takes a nearly constant value of g l ≈ 0.054, is much lower: S c ≈ 5%. The transition from the pendular into the funicular regime occurs for this type of snow in the saturation range of 9%≤S≤14%; it is marked by an abrupt change in the shape factor of g l ≈ 0.054 to g l ≈ 0.080. A slight increase of g l in the funicular regime with increasing saturation is also observed.
The dependence of g l on water saturation for type III snow is given in Figure 3. The pendular–funicular transition occurs in the range of 7%≤S≤12%, whereas g l shows a flip-flop-like behaviour from g l ≈ 0.054 to g l ≈ 0.082. The critical saturation decreases to S c ≈ 2% to 3%. In the funicular regime, the shape factor shows also a gradual increase with increasing liquid saturation.
Discussion
Independent of the stage of metamorphism, liquid water in snow exists generally in four different types of distribution depending on the amount of liquid water present: the pendular regime with two different zones, the transitional zone, and the funicular regime.
Pendular regime
The pendular stage includes the saturation regime from the adsorbed-liquid limit to saturations at which some of the liquid menisci and fillets coalesce and form a more or less continuous water matrix. The solid-ice matrix is characterized by clusters of ice grains and by the formation of fillets and veins (Reference ColbeckColbeck, 1980). Therefore the average shape factor of the water inclusions is very sensitive to the liquid saturation: at very low saturations, g l is controlled by the shape of the water menisci surrounding contact points of ice grains; g l ≈ 0. Because of the small cross-sectional area of the menisci or fillets, capillary forces may be dominant in this saturation regime. With increasing liquid saturation, the veins become filled with water; g l increases. Since approximately two-thirds of the water is retained in the veins—only one-third in the menisci or fillets—the average shape factor g l is mainly controlled by the shape of the veins: this is the case for saturations exceeding the critical saturation S c. As the cross-sectional area of the veins is much larger than that of the menisci and fillets, gravitational forces may be dominant in the saturation regime S ˃ S c; this is the case for a freely draining seasonal snow cover with a liquid-water content W normally in the range of 3 up to 7% by volume.
The critical saturation S c separates therefore two significantly different zones within the pendular regime: a zone with saturations S < S c, characterized by a liquid shape factor which strongly depends on water saturation, and a zone with saturations S ˃ S c, characterized by an approximately constant shape factor g l ≈ 0.053. At saturations below the critical value S c, capillary forces may be dominant; at saturations exceeding the critical value S c, however, gravity forces may be dominant. It may be of interest in this connection that the irreducible liquid saturation Si derived from long-term drainage experiments (Reference Denoth and DenothDenoth and others, 1979; cf. also Reference ColbeckColbeck, 1974) compares favourably with the critical saturation discussed here. The irreducible saturation also decreases with wet-snow metamorphism. So, for comparison purposes, S i is shown in Table II together with the characteristic parameters g i, g l, S c, and the saturation S t at the pendular-funicular transition for the three types of snow defined in Table 1.
Table II. Compilation of the characteristic snow parameters
Transitional zone
The transitional zone, which is characterized by a more or less rapid change from the pendular into the funicular mode of liquid distribution, is highly influenced by the stage of snow metamorphism. As the ice grains round off, and therefore the shape factor g i of the solid-ice matrix increases, the transitional zone shifts to lower values of liquid saturation. In addition to that, the gradual transition from the pendular into the funicular regime for type I snow switches into an abrupt, flip-flop-like transition for type III snow. This change in liquid distribution at the transition also changes other physical properties such as. for example, the microwave reflectance and the water flow characteristics.
Funicular regime
The funicular regime is characterized by a more or less continuous water matrix; the snow grains are completely surrounded by water and the air exists in isolated bubbles trapped between ice grains. Therefore, the clusters of ice grains—a stable configuration in the pendular regime—break down into single, isolated ice grains. Changes in liquid saturation, therefore, cause only small changes in liquid distribution; this can also be concluded from the small changes of the shape factor g l.
As the natural snow cover is typically inhomogeneous, the funicular mode of water distribution is usually formed over impermeable boundaries and interfaces. Therefore, the funicular regime is of particular importance, as the comparably high saturation at these points cause rapid snow metamorphism (Reference ColbeckColbeck, [1975]) and leads also to the formation of unstable water channels which, in turn, significantly affect the flow field (Reference ColbeckColbeck, 1979). Here it must be noted, however, that in the funicular regime electric interactions between various water aggregates and also between ice grains can no longer be neglected. This limits the applicability of the basic theory of Polder and van Santen (cf. Equation (1)) to saturations below approximately 30%.
Acknowledgements
The research reported here has been sponsored by the Fonds zur Förderung der wissenschaftlichen Forschung through Project No. 3888. The Wintersport Tirol AG is thanked For supporting the field work in the Stubai Alps.
