WATER-MOVEMENT IN WET SNOW

The three phases of water co-exist at the triple-point temperature whi ch is very close to 273.1 K. Wet snow packs are therefore nearly isoth ermal. Weak temperature gradients result from the dependence of the tr iple-point on grain size and capillary pressure and these need to be c onsidered if metamorphism of the snow pack is modelled. The bulk energ y balance determines the amount of latent heat released by phase chang e that is necessary to maintain the triple-point temperature. Solar ra diation occurring on diurnal timescales generates significant amounts of melt water in a surface layer and the subsequent unsaturated flow, as the water percolates down through the pack under the action of grav ity, is governed by the theory of immiscible displacement. This introd uces a partitioning of the water into an immobile, or bound, phase, wh ich is trapped in the necks of the grain boundaries, and a mobile phas e. The theory of interacting continua (mixture theory) provides a natu ral framework in which to describe all these processes and the necessa ry assumptions are discussed in detail for a four constituent snow pac k consisting of ice, mobile water, bound water and air. Earlier work h as suggested that the capillary pressure is proportional to the recipr ocal of the mobile water saturation; however, the experimental data on which this is based is sparse and there are equally likely functional forms with more realistic behaviour. The most notable problem with th e existing capillary pressure is that it is unbounded in the limit as the saturation tends to zero. A physically realistic model can not sup port infinite pressures as these would produce infinite forces in the momentum balance and drive unbounded flows. Current water percolation models only work because the relative permeability fortuitously introd uces another singularity that controls the limit behaviour. While this does not present a problem when used in isolation, it will produce un bounded rates of mass supply and unbounded temperatures in wet snow me tamorphism models that include phase change and the triple-point depen dence on capillary pressure, which is clearly physically unsatisfactor y. A simple modification in which the capillary pressure remains bound ed is not sufficient to eradicate these problems and must be supplemen ted by a relative permeability which has a finite gradient at zero sat uration to obtain physically realistic results. This introduces a cree p state in which water continues to flow, however low the saturation b ecomes, not present with the power law relative permeability used in c urrent models. Complete drainage of the snow pack in finite time can o nly occur if the creep state model is used. Four combinations of the c apillary pressure and relative permeability functions are investigated to demonstrate that these changes have a significant effect on the na ture of the solution. Scaling arguments are used to draw out the balan ces in the equations and determine the appropriate magnitudes of the s aturation, velocity, and time and length scales for each class. At low water fluxes the length and timescales of the flows are those suggest ed by the diurnal forcing. However, for larger fluxes the nonlinear na ture of the equations cause a front to develop whose time and length s cales are ten times shorter than the corresponding diurnal scales. The water mass and momentum balances can be combined to obtain a nonlinea r diffusion equation for the saturation, whose diffusion coefficient i s given by the relative permeability multiplied by the capillary press ure gradient with respect to saturation. Travelling wave solutions are constructed for each of the four classes which provide a useful check on numerical methods. If the diffusion coefficient equals zero at zer o saturation, then the saturation equation has a degenerate form which admits the possibility of solutions with a discontinuous derivative. Numerical methods which can solve more general problems are developed for both non-degenerate and degenerate cases. Finally, numerical illus trations are presented for a simple forcing scenario in which the satu ration varies sinusoidally on the diurnal timescale at the surface.