Change in Thickness of a Snow Layer by Viscous Compression

In order to compute the evolution of total snow depth with time or the depth–density profile of the snow cover by viscous compression, it is necessary to know the relationship between the compressive viscosity η and the density ρ of snow.

Reference Kojima.Kojima (1957, Reference Kojima1958, Reference Kojima and Oura.1967) has found that the relation between these factors for dry fine-grained snow within a density range 100–500 kg m ⁻¹ can be expressed as

where η ₀ and k are constants. But, it has been pointed out recently by Reference NakamuraNakamura (1987) and Reference Kajikawa and Quo.Kajikawa and Ono (1990) that the values η and k in the density range of less than about 100 kg m⁻³ are differenl from those in other ranges. This is inconvenient for the computation of snow depth or snow density.

Then, Reference Endo, Ohzeki and Niwano.Endo and others 1990 re-investigated this relation and indicated that η could be expressed by the following power function of ρ for dry snow with a dcnsiiy range 40–300 kg m⁻³ and a snow temperature of 0° to −5°C:

(1)

where C ≈ M 0.392 Pa s (kg m⁻³)−a and a ≈ 4 (Eudo and others, 1990). The value C is considered to be dependent on snow temperature and the shape of the snow crystals. Reference EndoEndo (1992, 1993) has shown that an approximation of snow depth or snow density is possible by using this equation. It has been used to predict direct-action avalanches.

On the other hand, the compressive viscosity of wet snow is known generally to be less than that of dry snow, when the other conditions of snow, for instance, wet-snow density, grain shape and others are the same. Reference Kojima and Oura.Kojima (1967) has found that, if η is related to dry-snow density ρ _dry which is obtained by subtracting the contribution of free water from the wet-snow density, the relation for wet snow with a free-water content of less than 5% is almost the same as that for dry snow of 0°C. However, the relation for very wet snow is unknown.

Thus, we have assumed the following equation as the relation between η and ρ _dry (cf. Rndo and others, 1990):

(2)

where C and a are constants as described previously.

Now, we consider a thin snow layer called the i layer which was deposited from time t _i–1 to t_i . Let the thickness of the i layer at time t(> t_i ) be h_i (t), and let the wet-snow mass of the i layer per unit horizontal area and mass of free water contained in it at time t respectively, be w_i (t) and q_i (^t). Then, the dry density of the i layer at time t is

(3)

where α _i , (t) is the free-water content by mass, of the i layer at time t and this is given by α _i (t) = q_i ,(t)/w_i (t). If the i layer of thickness h_i (t) at time t is compressed by dh _i (t) in a short time interval dt by the normal stress σ _i ,(t) exerted on the i layer, the compressive viscosity η/(t) is denoted by

(4)

Substituting Equations (2) and (3) into the above equation and, integrating both sides from the previous timet _n−1 to the present time t _n after separating the variables, the thickness h′_i (t) of the i layer at the present time t_n due to viscous compression is given by

(5)

We assume that the mass of the ice part, except for the water in the i layer, is unchanged from time t _n−1 to t _n.

Here, h_i (t_n−1 ), w_i (t _n−1, and α _i (t _n−1) are the thickness, total mass and free-water content of the i layer at time t _n−1. As shown in Figure 1, Q_i (t _n−1), t _n is the time integration of normal stress σ _i (^t) exerted on the i layer from time t _n−1 to t_n :

(6)

Fig. 1. Weight of each snow layer at time t_n−1 and t_n and the stresses exerted on a column of i layer

Here, taking an interval Δt between time steps 0, t ₁,..., t_n−1 and t_n to be short enough, we assume that changes in thickness, total mass and free-water content in each snow layer, due to snow melting and percolation of mcltwater, can be calculated separately using the change in thickness due to the viscous compression given by Equation (6).

Then, Q_i ((t_n−1 , t_n ) is approximated by

(7)

where p(t_n) is the amount of precipitation deposited in the form of snow or rain from time t_n−1 to t_n , and g is the gravitational acceleration.

Equations (5) and (7) show that the thickness h′_i (t _n ) of each layer at the present time t _n, resulting from viscous compression, is given by the precipitation amount p(t _n ) at the present time-step and the height h_i (t _n−1), the total mass w_i (t _n−1) and the free-water content α_i (t _n−1) of each layer at the previous time-step t _n−1 .

Calculation of New Snow Depth

Total snow depth changes by snow accumulation, ablation and viscous compression. As shown in Figure 2, we assume that snow accumulation and ablation lake place, after the snow cover at time t _n−1 changes its height solely by viscous compression and settling to a sum of all h′_i (t_n ). Also, we assume that the wind is so calm that it hardly Iranspnris snow particles on the snow surface such as at our site (average wind speed of 1.1 m s⁻¹). In this case, snow accumulation and ablat ion occur respectively, by snowfall and snow melting. Then, we consider the following index D(t_n ):

(8)

where H(t_n ) is the total snow depth measured at time t_n , and ∑h′ _i(t_n ) is the total snow cover estimated as a result of viscous compression. And g(t_n ) is the thickness of snow melted on the ground from time t _n−1 to t_n . This melt thickness has been neglected because measured g(t_n ), using a snow lysimeter, is less than 0.1 cm d⁻¹ of snow.

Fig. 2. Estimation method for the depth of snow accumulation and ablation. We assume that D(tⁿ) indicates the depth of snow accumulation if D(t_n) > 0 and snow ablation if D(t_n) < 0

If new snow is deposited during the time interval from time t _n−1 i to t_n without melting of the old snow surface at time t _n−1, D(t_n ) is positive and the value of D(t_n ) gives thickness a h_n (t_n ) of the newly deposited n layer, as shown in Figure 2. In an opposite case, that snow melting takes place on the surface without snowfall, D(tⁿ )is negative, indicating the thickness of snow melted on the surface. Also, even in the case that both snowfall and snow melting take place during the interval, if snow melting Occurs after new snow is deposited, D(t_n ) shows the thickness of new snow in D(t_n ) > 0 and of melted snow in D(t_n ) < 0.

However, if new snow is deposited after the snow surface at time t _n−1 is melted, we cannot obtain the respective thickness of them from D(t_n ).

But, these cases are possible only at the beginning of a snowfall. If we take the time interval to be short enough, the ratio of occurrence of these cases compared to all becomes small.

Hence, taking the time interval Δt as 1 hour, wc assume thai D(t_n ) gives the thickness of new snow in the case of D(t_n ) > 0 and the thickness of snow melted on the surface in the case of D(t_n ) < 0. In the case of D(t_n ) > 0, the precipitation p(t_n ), considered as the amount of snow, gives a mass w_n (t_n ) of the n layer at a time t_n . Free-water content of the new snow is supposed to be 0, because it is unknown. Thus, height, mass and water content of the n layer are as follows:

In the other case of D(t_n ) < 0, we suppose thai the precipitation p(t_n ) is the same amount of rain as the surface is melting. In this case, the k layer located at depth D(t_n ) of snow melted from the snow surface satisfies the following equation:

(9)

As a result of snow melting, the thickness h_i (t_n )of each layer at time t_n becomes

(10)

Also, mass m(t_n ), containing free water, of wet snow melted on the surface is given by

(11)

We assume that this meltwater m(t_n ) is distributed with the precipitation p(t_n ) in the lower snow layers, according to such a simple tank model that free water contained in a snow layer can percolate into the lower only in the case when its free-water content α_i ,(t_n ) exceeds the maximum value α _max. Then, we can determine that mass w_i (t_n ) and free-water content α_i ,(t_n of each snow layer at time t_n .

If we compute the new snow depth from time-step 1 by such a method, we can obtain thickness, mass and free-water content of each snow layer at the present time t_n , using the values of total snow depth and precipitation measured at each time-step from time t ₁ to t_n . In this paper, as we take Δt as 1 hour, depth of hourly new snow at time t_n is given by thickness h_n (t_n ) of the t_n layer and that of daily new snow is given by

(12)

where D(t_n )_day is the depth of daily new snow at time t_n .

Comparison with Conventional Estimation Methods of New Snow Depth

Most of the local observation stations of the Japanese Meteorological Agency use two conventional methods, based on differences of total snow depth measured automatically, to estimate the depth of daily new snow; the first one is the method of substituting daily change of total snow depth for depth of daily new snow, and the second one is the method of estimating the depth of daily new snow by summing the positive increment of hourly snow-depth changes.

We investigated the error of estimation using these two methods and compared them with our estimate by the viscous compression model, using the same data of the three winters from 1992–93 to 1994–95. Figure 5 shows the relationship between daily change of total snow depth and the conventionally measured depth of daily new snow. As shown in this figure, the errors of daily difference of snow depth are usually negative and its maximum error is −23 cm; the standard deviation is 9.72 cm. Such large negative errors are mainly caused by viscous compression of the snow cover.

Fig. 5. Relationship between daily change in total snow depth and conventionally measured depth of daily new snow

The estimation method by summing positive change is considered to revise the negative errors due to viscous compression by removing the negative change of hourly snow depth. The relationship between the values estimated by this method and the conventionally measured depths of daily new snow is shown in Figure 6. Using this revision, the estimated values shown in Figure 6 are distributed equally around a solid line which indicates y = x. However, its maximum error is ±15 cm and standard deviation is 4.52 cm.

Fig. 6. Relationship between the daily sum of positive changes in hourly snow depth and the conventionally measured depth of daily new snow

These results show that estimates using the viscous compression model method are much more accurate than the other two conventional methods.

Relationship Between Compressive Viscosity and Dry-Snow Density

Taking C = 0.392 Pas (kg m⁻³)^−a, we determined the constant a = 3.6 in Equation (2) in order to minimize the errors of the estimated depth of daily new snow.

The result obtained is shown in Figure 7 as the plot of log η against the dry-snow density, together with the values measured for dry; fine-grained snow below 0°C by other researchers (Reference Kojima and Oura.Kojima, 1967; Reference Shinojhna and OuraShinojhna, 1967; Reference NakamuraNakamura, 1988; Reference EndoEndo and others, 1990; Reference Kajikawa and Quo.Kajikawa and Ono, 1990). The solid line represents the relation used in this paper and the broken lines those by the other authors. As shown in this figure, the solid line is not too far from the broken lines. Hence, the values of C and a used in this paper are considered to be appropriate values.

Fig. 7. Relationship between compressive viscosity and dry-snow density