Water movement in wet snow

Abstract

The three phases of water co-exist at the triple-point temperature which is very close to 273.1 K. Wet snow packs are therefore nearly isothermal. Weak temperature gradients result from the dependence of the triple-point on grain size and capillary pressure and these need to be considered if metamorphism of the snow pack is modelled. The bulk energy balance determines the amount of latent heat released by phase change that is necessary to maintain the triple-point temperature. Solar radiation 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 gravity, is governed by the theory of immiscible displacement. This introduces a partitioning of the water into an immobile, or bound, phase, which is trapped in the necks of the grain boundaries, and a mobile phase. The theory of interacting continua (mixture theory) provides a natural framework in which to describe all these processes and the necessary assumptions are discussed in detail for a four constituent snow pack consisting of ice, mobile water, bound water and air. Earlier work has suggested that the capillary pressure is proportional to the reciprocal 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 the existing capillary pressure is that it is unbounded in the limit as the saturation tends to zero. A physically realistic model can not support 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 introduces another singularity that controls the limit behaviour. While this does not present a problem when used in isolation, it will produce unbounded rates of mass supply and unbounded temperatures in wet snow metamorphism models that include phase change and the triple-point dependence on capillary pressure, which is clearly physically unsatisfactory. A simple modification in which the capillary pressure remains bounded is not sufficient to eradicate these problems and must be supplemented by a relative permeability which has a finite gradient at zero saturation to obtain physically realistic results. This introduces a creep state in which water continues to flow, however low the saturation becomes, not present with the power law relative permeability used in current models. Complete drainage of the snow pack in finite time can only occur if the creep state model is used. Four combinations of the capillary pressure and relative permeability functions are investigated to demonstrate that these changes have a significant effect on the nature of the solution. Scaling arguments are used to draw out the balances in the equations and determine the appropriate magnitudes of the saturation, velocity, and time and length scales for each class. At low water fluxes the length and timescales of the flows are those suggested by the diurnal forcing. However, for larger fluxes the nonlinear nature of the equations cause a front to develop whose time and length scales are ten times shorter than the corresponding diurnal scales. The water mass and momentum balances can be combined to obtain a nonlinear diffusion equation for the saturation, whose diffusion coefficient is given by the relative permeability multiplied by the capillary pressure 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 zero 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 illustrations are presented for a simple forcing scenario in which the saturation varies sinusoidally on the diurnal timescale at the surface.