On The Use Of Mixture Theories To Model Heat And Mass Transport In Mountain Snowpack (Abstract only)
Authors: Brown R. L. , Adams E. E 
Published: February 2017

Snow is a mixture of ice, air, water vapor, and liquid water, and as such is a mechanically and thermodynamically very active material. In an attempt to gain a better understanding of the process of meta-morphism and the interaction of mechanical and thermo-dynamic processes within mountain snowpack, modern mixture theory was used. Only dry snowpack was considered, so that the liquid phase could be neglected.
Each constituent (ice, vapor, and air) was assumed to have its own density ρα, temperature Θα, partial stress Ta, particle velocity ζα, internal energy εα, entropy ηα, free energy tα, and heat flux qα. Each constituent was considered to have the ability of reacting with the other constituents, so that mass supplies tα, energy supplies êα, and momentum supplies pα can be defined to characterize the manner in which each constituent receives mass, energy, and momentum from the other constituents due to chemical and mechanical interactions. In addition, a concentration cα was defined in order to specify the extent to which the solid and vapor phases have exchanged mass.
The mass balance, momentum balance, and energy balance principles were then developed for each constituent, thereby giving a set of nine differential equations to solve for θα, ζα, and ρα for each constituent. In addition the second law was defined for the mixture and used to place restrictions on the allowable forms for the constitutive equations for Tα, εα, ηα, Ψα, qα, Pα, εα, and ĉα for each constituent. In this way the forms for these variables under non-equilibrium thermodynamic conditions could be specified.
For example the restrictions imposed by the second law require that, under nonequilibrium conditions, the stress, mass exchange rate, heat flux, and momentum supply may have the forms:

1

2

3

4

where Eα, ?Eα/?Z, Uα3, and ?θβ/?Z are respectively the strain, strain gradient, diffusion velocity relative to the air phase, and temperature gradient. An alternate form for ĉα which was also found to be consistent with the second law is

5

where ĉ2 is the rate of sublimation from the solid to vapor phase and μ1 and μ2 are the chemical potentials for the ice and vapor phases.
As a special example, densification due only to the presence of surface energy was calculated. In this case, the material is unloaded, and the free energy Ψ10 for ice is dominated by its surface energy. Therefore Ψ10 = σ1?1 where A1 is the surface area per unit volume and σ1 is the specific surface energy. If T1 and θ1 are set to zero and θ0, the solution to Equation (1) yields the result illustrated in Figure 1 for the density P1.C10 and γ10 _ are the initial elastic modulus and viscosity. This case corresponds to that of new, very fragile snow. Denser, stronger snows would densify much less, but cracking may still occur in the material if constrained froin densifying freely.

Fig. 1.

Densification of low-density snow due to effect of surface free energy.

Next, the added effect of equitemperature meta-morphism on densification was considered. This involves the process of transfer of water molecules from small grains to large grains to produce a more efficient distribution of ice and hence some densification. The solution was found, and preliminary results indicate densification is enhanced by this process. However, final quantitative results must await a better means of relating density, grain size, and grain shape to specific surface area A1. Current formulations for finding ?1 are of questionable accuracy.