Benson (1960) (see also Reference Anderson, Benson and KingeryAnderson and Benson, 1963) studied the problem of densification of dry snow and derived a simple formula to describe the density of dry snow as a function of depth. Their formula is very accurate when compared with observation. However, to use it one usually needs to know two coefficients of proportionality for each location, in addition to given values of surface snow density and maximum attainable snow density. (The number of coefficients can be reduced, however, in the case of new snow.) Reference Bader and KingeryBader (1963) developed a theory which also describes very accurately the density distribution of snow, but one needs to know six parameters for each location.
This note describes a simple mathematical expression for the density distribution of dry snow in a way that requires only one free coefficient or none, at the expense of a little accuracy. We assume that, at any specific time, the change of density dρ in the vertical direction is related to the change of pressure dρ and ρ m – ρ (Reference RobinRobin, 1958; Reference Herron and LangwayHerron and Chester, 1980), where ρ m is the maximum attainable density for the dry snow at a certain location and ρ is the density at a certain depth, by
where c is a proportionality constant and the change of pressure is
where z is the distance from the snow surface. However, here we assume a more general relationship
where the parameters n and c are determined by comparison with the field data. Here n is a measure of stiffness of the snow and c is a proportionality constant. When n = 2, the integration of Equations (3) and (2) gives
where ρ 0 is the surface density of snow.
Letting λ = 1/L where L is a characteristic length scale, we have
or
Figures 1 through 4 compare Equation (4) with measurements in dry snow at four locations from Greenland, the Colorado Rockies, and Antarctica, and a combined dimensionless plot is shown in Figure 5. In the case of Antarctic and Greenland snow, the knowledge of a length scale would be enough to calculate the density, whereas for shallow new snow, L may be set equal to one-third the total depth of the snow. Integration of Equation (4) throughout the depth gives the total mass of dry snow per unit area as
and for shallow new snow, this expression reduces to
The curve represented by Equation (4) corresponds quite well with snow data as shown in Figures 1 through 5. It is concluded that the non-linear relationship between the change of density and the change of pressure represented by Equation (3) would be a good candidate for use in the density distribution of dry snow.
Fig. 1. Snow density versus depth for Station 2, Greenland.
Fig. 2. Snow density versus depth for Byrd Station, Antarctica.
Fig. 3. Snow density versus depth for South Pole.
Fig. 4. Snow density versus depth for Tower Station. Colorado; data collected during the winter of 1980; personal communication from E.G. Josberger, 1984.
Fig. 5. Snow density versus depth.
Acknowledgements
I want to thank E.G. Josberger and S. Hodge for their help in providing recent snow data they collected, and I want to thank W.J. Campbell for his suggestions. I also want to thank Professor C. Benson for his encouragement and help in this work.
