List of Symbols

• t time coordinate

• x _k space coordinate

• y mean weight concentration of bearing phase, g/cm³

• y_sol mean weight concentration of solid phase, g/cm³

• y_a concentration of falling snow particles, Mg/m³

• y_s mean density of snow, Mg/m³

• q_i,q_k components of mean weight flux of bearing (air) phase, g/cm ² s

• q_si,q_sk components of mean weight flux of solid (snow) phase, g/cm² s

• Q_max maximum solid flux of snow-storm, g/m s

• Q_s total solid flux of snow-storm, g/m s

• Q eight of blown snow, Mg/m³

• v_i,v_k components of mean velocities of bearing phase, cm/s

• v_si,v_sk components of mean velocities of solid phase, cm/s

• v₁(x₃) wind speed at x ₃ -level, m/s

• v_n wind speed at the height h from the snow surface, m/s

• v₀ wind speed at beginning of snow transport, m/s

• v_0.2 wind speed at 0.2 m height, m/s

• v₁₀ wind speed at 10 m height, m/s

• v_g group speed of the snow particles, m/s

• v_m mean speed of snow-storm current, m/s

• U_* friction velocity, cm/s

• w falling velocity of snowflakes, m/s

• g, gⁱ magnitude and i-component of the gravitational acceleration, m,s²

• M’_k sum of correlation moments between weight concentration pulsations and velocities of both bearing and solid phases, g/cm² s

• M"_ki sum of correlation moments between the weight-flux pulsation and velocities of bearing and solid phases, g/cm² s

• IIki tensor of molecular stresses in a two-phase medium, g/cm²

• T_ki generalized tensor of stresses, g/cm²

• s dimensionless volumetric concentration of snow particles in mixture “snow + air”

• p mass density of air, g s²/cm⁴

• p_s mass density of snow, g s²/cm⁴

• T_s3 quasi-static pressure difference, g/cm²

• μ₃ diffusion coefficient, cm²/s

• K Kármán’s constant

• Y criterion of stability of saltation layer

• x₀ linear index of roughness, cm

• h thickness of the snow cover, cm

• d size of snow particle, cm

• δ_s size of surface snow particles, cm

• α slope angle, grade

• L_n limit length of transporting uniform snow, m

• x_p growth length of snow-storm, m

1. Utilitarian Problems

The larger part of the Soviet Union lies in the zone of snow-storm activity that greatly interferes with the normal functioning of traffic, industrial and power units. In economic development of high-latitude areas of the Northern Hemisphere, man in his practical activity will have to face the effect of especially strong snow-storms on engineering structures.

The scientific and applied significance of such natural two-phase flow as snow-storms is enormous. The formation of glaciers and snow avalanches, the regime of water-flow from snow melt, water conservation on agricultural land, snow-drifts on settlements, roads and enterprises—this is not a complete list of the phenomena greatly dependent on snow-storms.

The regularities of mountain snow-storms and snow-drift during the strong blizzard conditions are not sufficiently studied. Methods of control of heavy snow-storms are not fully worked out. AH these utilitarian problems are consequences of one important theoretical problem—the mechanism of deflation in suspended streams. Therefore interest in the theory of snow-storms is quite natural.

Any generally adopted concept of the main causes of solid-particle movement in two-phase flow is until now missing. Reference ShulyakShulyak (1971) supposes that deflation is a consequence of a development of the initial disturbance of the surface of the loose medium. Reference RadokRadok (1970) considers that the main cause of deflation is turbulent diffusion of the suspended particles. We are convinced of the decisive role of the quasi-static pressure field in the boundary layer of two-phase How.

Theoretical, field and experimental researches on snow-storms (Reference DyuninDyunin, 1963; Reference DyuninDyunin and others, 1973) show that the main zone of deflated snow-storm action is a comparatively thin boundary layer 2-3 m thick; more than 90% of the total drifting snow being the lowest sublayer about 10-20 cm thick.

In the strongest winds some small snowflakes can rise to a considerable height, but they constitute only an insignificant part of the total solid flux of the snow-wind flow. The deflated particles move over the ground unevenly by means of “saltation”. This essentially facilitates the study of such complex phenomena as up-snow-storms in hilly and mountainous regions. Up-snow-storm is the atmospheric snow-flake drift until the moment they touch the Earth’s surface. Deflation and up-snow-storms have strictly defined zones of action and do not essentially ’’hamper" each other.

2. Basic Theories of Snow-Storms

2.1. General theory of two-phase flow

The theoretical foundations of snow-storm mechanics are the common equations of mass, impulse and energy balances of the multicomponent continuous medium. The differential equations of continuity and balance of impulse flows for a two-phase medium averaged in space and time terms are (Reference DyuninDyunin and others, 1965):

(1)

(2)

where t,x_k are time and space coordinates respectively; _y, y_s mean weight concentrations bearing air and solid phases; (q_i q_k), (q_si,q_sk) components of mean weight flux of bearing(air) and solid (snow) phases respectively, (v_i,v_k), (v_si, v_sk) components of mean velocities of the bearing and solid phases respectively, g,g_i modulus and i-component of the gravity acceleration respectively, M’_k sum of correlation moments between weight concentration pulsations and of both phase velocities, M"_ki is the sum of correlation moments between the weight flux pulsation and velocities of both phases, Il_ki is the tensor of molecular stresses in a two-phase medium that is considered to be continuous.

Equation (2) supplies information on the analysis of the suspension of heavy impurities. Let T_KI be the generalized tensor of stresses defined as

(3)

It should be noted, that

(4)

where s is volumetric concentration of impurities in mixture, p, p_s are mass densities of air and snow respectively.

Let us consider a flat and stationary flow directed along the x₁-axis, over the horizontal plane x₃=0. Using the ergodics of the stationary process, let us choose a region of space for averaging in the form of a thin parallelopiped whose extension along the x₁-axis is large enough to neglect the dependence of averaged terms on x₁, and conform to the condition q_s=q_s3=0

Having a projection of Equation (2) on the x₃-axis directed vertically upwards and using Equations (3) and (4) we can easily find for the given case that

(5)

Equation (5) shows that the quasi-static pressure excess T_S3 ought to be considered the main factor in suspending impurities, since p ≪ p_s. Impurity suspension will only be possible when . The measurements show that this excess differs from zero only at the contact with the bed surface, i.e. there are no essential factors promoting the continuous soaring of impurities inside the snow-air flow (Reference DyuninDyunin and others, 1965; Reference DyuninDyunin, 1966).

The above theory is very consistent with the field and laboratory data.

2.2. Diffusion model of snow-storms

A simplified “diffusion" snow-storm model is possible, the most recent using only the continuity equations (1). Let us assume, that

(6)

where μ_s is the diffusion coefficient. Taking the previous assumption of a stationary regime we find y_s from Equation (1) with accuracy to a constant vector.

(7)

since yv_k, ≪ y_su_sk

Taking Equation (7) as the initial equation Reference RadokRadok (1970) supposed that

(8)

(9)

where v(x₃) is the wind speed at the x₃ level, K Kármán’s constant, x₀ the linear index of roughness, w the fall velocity of the snowflakes. The model of impurities diffusion is considered to be similar to the Kármán’s model of the turbulent diffusion of homogeneous flow. According to Equation (8) the diffusion coefficient is intensively increasing with height so that the greater part of snow mass must be transported as a suspension in the upper boundary layers of the atmosphere.

Analysing the diffusion equations, the Australian researchers (Reference BuddBudd and others, 1964) indicate that weight concentration of the impurities in some layer does not depend on wind speed. Therefore weight concentration in this layer is not increased with wind speed and an addition to drifting matter has been produced by diffusion of solid particles to the upper layers.

Reference Owen.Owen (1964) introduced the criterion for saltation layer stability

(10)

where u_* is the friction velocity, d is the particle size.

According to Budd and others (1964)

where v₁₀ is the wind speed at the 10 m height. Making the substitution u_* = 3.78 v₁₀ in Equation (10) gives

The zone of possible saltation is up to the inequality

(11)

When Y > 1.0 the shearing force exceeds the particle weight and all particles will be carried into suspension. Let us calculate the values of v₁₀ when Y = 0.01 or Y= 1.0 and p_s = 0.9 g/cm³ (the density of Antarctic snow particles) and d = 0.01 cm.

We have

Thus, saltation is possible under the range of wind speed from 2.21 to 22.1 m/s (at 10 m height) when d = 0.1 mm. Therefore when v_l0 > 22.1 m/s snow particles will be carried into suspension since Y > 1.0.

We shall call the boundary of such a transition the threshold of mass suspension. An experimental verification of the diffusion model would be reduced to detecting the deep qualitative changes in the mechanism of snow drifting when the wind speed reached the threshold of mass suspension. The vertical distribution of solid flux in a layer near the snow surface would be uniform all over the section. In other words when v₁₀ > 22.1 m/s the distribution of solid flux with height in a sectional wing-like trap of the kind used by R. Bagnold will be uniform.

This hypothetical statement was subjected to experimental test in a wind tunnel. The results of these experiments are given in section 3.

The verification of the theoretical models in a wind tunnel is possible when the following conditions are assured: (1) the vertical profile of wind speed is similar to the wind profile in Nature; (2) the turbulent two-phase boundary layer has time to stabilize, i.e. the growth length (acceleration length) of a snow-storm is confined to the working zone of the wind tunnel.

3. Experimental Results

All these conditions were satisfied in the wind tunnel designed by Reference IstrapilovichIstrapilovich (1972). The length of the working zone of the wind tunnel is 27 m, its cross-section 0.5 × 0.6 m². The fan ∐9-57 N 8 with its maximum rate of revolution (1000 r.p.m.) and with completely open throttle has provided a flow velocity up to 43 m/s. The cooling is natural.

The distribution of solid flux with height is found by means of a Bagnold wing-like trap. The effect of tunnel dimensions (its cross-section) on wind-speed profile was studied on several wind tunnels with different cross-sections. A scale factor was defined on tunnels with sections from 0.15 × 0.5 m² to 0.5 × 0.6 m². Some of the experimental results arc published in Reference DyuninDyunin (1974).

At present a wind tunnel of length 27 m and cross-section 1.0 × 1.5 m² is being used. This tunnel will permit an approach to natural conditions on the scale 1 : 1 and allow us to simulate flow around the small models of the mountainous regions.

The experimental results are given in Figure 1, which also shows theoretical curves of maximum solid flux calculated from the equations

(12)

(13)

Fig. 1.

Graph of dependence of maximum solid flux on wind speeds.

As Figure 1 shows, the measurements lie far beyond the limits of the threshold but significant deviation from the “pre-threshold" tendency was not detected.

Figure 2 shows the snow and sand contents in the trap sections (in %) from the total weight of the material collected by the Bagnold trap. The distribution of solid flux with height appeared to be similar both at weak and strong wind speeds. The main part of the snow is transported near the snow surface.

Fig. 2.

Vertical distribution of solid flux at different wind speeds.

Nevertheless the boundary-layer structure is essentially reconstructed when the Owen Radok criterion Y > I is satisfied. When Y > I at the beginning of deflation, the zone of negative quasi-static pressure differential is more strongly marked. Pressure differential is also observed inside snow cover, where it inevitably influences the snow metamorphism. These internal pressure differentials play a decisive role in the formation of cones of erosion around obstacles. In the literature the role of vortex rollers in the formation of erosion cones is grossly exaggerated although they are only secondary phenomena.

4. Influence of Snow-Storms on Snow Cover in Mountains

In mountainous regions, snow-storms influence the redistribution of snow on the slopes and in the valleys. They also greatly influence the avalanche regime.

Investigations on mountainous snow-storms are very difficult because of the extreme variety of mountainous microclimate in small space and time intervals.

Let us consider the plane movement of a snow-storm along a slope inclined to the horizon at an angle α (Fig. 3).

Fig. 3.

Theoretical model of snow-storm on a mountainous slope.

Having calculated the solid flux in the parallelopiped abcd we find (Reference DyuninDyunin and others, 1973):

(14)

where h is the thickness of the snow cover on the slope measured vertically, y_a the concentration of falling snow particles, w the falling speed of these particles, y_s the mean density of snow, Q the weight of snow blown through the edges of the parallelopiped abcd, L_n the limit length of transporting uniform snow.

With a constant falling speed of the snow particles w, and average snow-cover thickness H we have:

(15)

Where θ= v_g/v_m is constant and where Q_s is the total solid flux of snow-storm, v_m the mean speed of snow-storm current, and v_g the group speed of the snow particles.

The total solid flux Q_s, according to our field data and modelling the snow-storms in a specially inclined wind tunnel, depends slightly on the slope angle at α < 30-40°.

In general form:

(16)

where v_n is the wind speed at the height h from the snow surface, v₀ the wind speed at the beginning of snow transport, p and p_s the mass densities of air and snow respectively, and S_s the size of the surface snow particles.

Φ is the compound periodic function of x₁ given in Figure 4. The distance x_p called as a growth length and the wave length depend on the spectrum of turbulent wind pulsations, its gusts. In the mountains, wind gusts are extremely marked.

Fig. 4.

Graph of approximate dependence of Φ on limiting length of snow transport x₁.

Equations (14)–(16) give the following important parameters when the influence of snowstorm on the snow cover must be taken into consideration: the values of Φ and x_p, the absolute value of wind speed v_n and its direction, the gradient ∂ν_h/∂x ₃ (vector pulsations of wind speed), the gradient ∂Q_s/∂x ₁ evaporation intensity during snow-storms, growth length. L_n, the gradient ∂ν_m/∂x ₁ of blowing solid precipitation.

The role of up-snow-storm in snow accumulation in the mountains differs from the influence of deflated snow-storm. Snow transport during deflated snow-storm is carried along mainly by saltation. During up-snow-storm a simple falling of snow particles is observed, which is however sharply distorted near large obstacles. Deflated and up-snow-storms have strictly defined action zones.

5. Conclusions

• 1. Drifting snow is transported mainly by saltation at all conceivable wind speeds.

• 2. The gravitation field, the kinematic field of the wind speed, its pulsations, and the quasi-static pressure field in the snow-storm stream and the loose-medium influence the formation of deflational snow-storm and snow cover.

• 3. Up-snow-storm and deflational snow-storm can be considered as independent factors when the distribution of snow in the mountains is estimated.