Fundamentals of the Theory of Snowdrifting

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Fundamentals of the Theory of Snowdrifting Dyunin, A. K.

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Snowdrifts can be a major source of disruption in the operation of transportation services and a general nuisance in the normal wintertime activity of a community. Such drifts are formed whenever a wind, strong enough to transport horizontally a significant amount of snow, encounters an obstacle which forces it to deposit some of this snow. The usual approach taken in defending an area or structure against snowdrifting has been to locate the structure properly so that the drift problem will be a minimum and to erect obstacles, such as snowfences, to control where the snow will be deposited. The approach taken in the development of these defences has been

largely empirical. Attention has been directed primarily to the character of the air flow with little attention being given to the material transported. In some

circumstances, it would be an advantage to have a more complete defence against snow drifting than is now

available. In their attempts to develop this defence, engineers are giving more consideration to the

theoretical aspects of the problem and in particular to the relationships between the air flow and the snow being transported.

It is one of the responsibilities of the Snow and Ice Section of the Division of Building Research to collect and make available information required for the solution of snow and ice problems. The present paper, translated from the Russian, 1s a contribution to the theory of snowdrifting. This paper will give to the reader an appreciation of some of the factors to be con-sidered in the theoretical description of blowing snow and its deposition as snOWdrifts.

The paper was translated by Mr. G. Belkov of the Translations Section of the National Research Council Library, to whom the Division of Building Research wishes to record its thanks.

Ottawa, April 1961

R.F. Legget, Director

Technical Translation 952

Title: Fundamentals of the theory of snowdrifting (Osnovy teorii metelei)

Author: A.K. Dyunin

Reference: Izvest. Slbirsk. Otdel. Akad. Nauk SSSR, (12): 11-24, 1959

The theory of two-phase flow, one phase being a granular solid, is being developed at the present time mainly in relation to the movement of river deposits. Snowdrifting, as a unique variation of this type of flow, has received less attention although the drifting of snow has long attracted the attention of specialists interested in combatting snowdrifting in transport and in industry.

The first serious investigation of snowdrifting, made by the engineer E.D. Zlotnltskii, was published more than 80 years ago(l). Since that time the theoretical aspect of this problem has received little attention and only some aspects of the problem have been considered. Field observations are very few and their accuracy is not high. There have been no laboratory experiments. Therefore there is no ordered theory of snowdrifting. The quantitative characteristics of snow transfer are in the initial stage where a multitude of empirical formulae is being accumulated. At times even the very possibility of making an analytical evaluation of snow

transfer is questioned.

Differential Equation for Snowdrifting

By the terms "metel'" or ..pozemka" (snowdri ftLng ) we understand the process of transfer of previously fallen snow by the wind. Snowdrifting differs essentially from snowfall accompanied by wind called "verkhovaia metel'" (snowstorm). In a "verkhovaia metel,n only those snowflakes are involved which have not yet touched the

gr-ound ,

Let V

s be the vector of the mean velocity of snow particles in relation to the earth's surface, s is the volumetric concentration of the snow in the flow. Then the flux of snow particles j through an area normal to V_s can be written as

Its vertical component has the form sv •

S3

Assuming that outside the boundary layer the turbulence of the wind flow is locally isotropic one can consider the value of ｶｳｾ as being near the mean hydraulic falling veloci t:'l of the snow

particles. Consequently the solid matter from a snowfall with wind and without wind is distributed uniformly over a horizontal surface,

i.e. in both cases the snow will be distributed over the surface of the ground evenly.

When wind-borne falling snow encounters an obstacle it is possible to have a deviation from the even distribution of snow

but this deviation cannot be large because of the very small inertia of the snow particles.

The transfer by the wind of snow already fallen is an incomparably more complex phenomenon.

Referring to Barenblatt(2) and Frankl' (3), a.ssuming that the tensor of instantaneous stresses in the solid particles and the medium varies continuously, and neglecting the dependence of the density of the atmosphere on its dynamic state and humidity, we obtain the following fundamental '\Fector equations for the insepara-bility and the conservation of momentum of the gaseous and solid

phases of the wind-snow flow. For the gaseous phase (air)

a(l - s)

-p

at

+ p'V • (1 - s) v == 0,

a

{l -

s>'V'

- -

-

-p

at

+ p [ 'V • (1 - s) v

l

v

=

p(1 - s) (g + e)

-(1 - s)\7· IT - \7 • T + E.

For the solid phase (snow particles)

as

p_s -

_at

+ P_s'V • sv_s

=

0,

(1 )

asv

Ps at s + _{ps(v • svs)vs}

=

P_ss (g + ｾ _s)

-(4)

S"i7 • IT -

v·

T

s -

E ,

where v is the vector of the mean velocity of particles in the medium,

Ps'P are the mass densities of the particles and the medium, g is the vector of acceleration due to gravity,

ES,E are the vectors of acceleration of non-gravitational mass forces,

I I is the averaged tensor of the instantaneous stresses inside the solid particles and the medium,

Ts,T are the averaged tensors of additional stresses caused by

mixing and collisions of solid and gaseous particles,

E is the vector of the averaged forces of interaction between the medium and the solid particles at a given point.

If there are interphase transitions (sublimation and crystal-lization from a vapour) then (1), (3) and (2), (4) cannot be

separated. It should be noted that these equations were obtained from integral relations with arbitrary forms for the fields of

integration. Therefore putting together (1) and (3) and (2) and (4)

D

and denoting by dt the operator of the substantial derivative and taking the expression

Pa

=

sp s + (1 - s) P ,

for the average density of the dispersoid P

_a

we get an equation of the S.G. Teletov type(4)

( a s - )

ｾＮｮＭｾﾷｔ _s ＭｾＮｔＬ

where

K

is a vector due to changes in momentum resulting from phase transitions, and called by us sublimation vector.

The magnitude of the sublimation vector

K

=

(V'5 -

v)

ps

Ｈｧｾ

+

ｾ

• SVa ) •

Having obtained the system of equations (5), (6) we will estimate the order of magnitude of some of their terms.

Non-gravitational Mass Forces

(7)

Electrically neutral particles when carried by the wind are not affected by any other fields of mass forces except the gravity field force. However snow particles, as a rule, carry an electric charge and in the atmosphere there is an e Lect.r-Lc field of ponderomotive forces. This latter factor was referred to by B.N. Vedenisov in defending the theoretical possibility of aerial transfer of snow by the wind(5).

Leaving aside the analysis of the value of

e

which is difficult to take into account and is apparently not very essential, let us consider the order of magnitude of the accel:ration

sa

in m/sec2

under the influence of ponderomotive forces F in mgm acting on a particle with charge e in SGSE units being in an electric field with

intensi ty grad ¢ in vim where ¢ is t he field potential in volts. If g is the modulus of acceleration due to gravity in m/sec2 _,

then conserving dimensionality we get

F

=

e • grad fA mg.

Let a be the weight of a particle in mg. Then

e grad p / :2

=

a • 3 . 103 m sec. (9)

Tables I and II give a summation of the available data on determination of maximum values of

I

grad

pi

during electrification

(6-9)

in strong blizzards and the maximum charges of snow

particles{lO,ll). In the wintertime the values of

I

grad

pi

without wind have the order of tens - hundreds VOlts/m.

According to

ａｾ

Shteger(12), under laboratory conditions the value of e/a for particles of sugar dust could be brought to 0.66 SGSE/mg and for crystals of methacetaldehyde it was 0.33 SGSE/mg.

For snow particles with minimum radii of surface curvature 0.01 - 0.1 mm the absolute maxima of e/a should be less than that of the experimental limit if one assumes the absence of corona diScharge(ll) •

When

I

grad

pi

ｾ 10,000 vim in the summertime thunderstorms occur. It is hardly likely that in windblown snow the gradient of electric potential ｳｨｯｵｬｾ reach this value which is apparently in this case an absolute maximum. The same value of the maximum of

Igrad

pi

is mentioned in the paper of D. Pearce and B. Currie(13). Table III shows the calculated values of

le

_s

I.

Even in

max

experimental cases they make up only a small portion of the gravita-tional acceleration g and cannot be considered as a serious factor causing suspension of snOw particles during a blizzard.

However, the absolute magnitude of the maxima of grad

¢

is sufficiently large to explain the observed radio interference, light effects, etc., during blizzards.

Sublimation Vector

In an air-snow stream the snow particles are divided and

dispersed, and at low temperature there is an evident predominance of sublimation of the solid phase over the crystallization from the vapour.

The problems of sublimation of dispersed particles is presented in detail in an earlier report(14).

When atmospheric humidity is low, which is possible during a blizzard, the duration of complete sublimation of snow particles is 10 - 20 minutes, which can be considered a minimum for a majority of solid particles. Let us assume that the mass concentration of particles at the surface of the earth is 600 gm sec2/m 4 _which

corresponds approximately to a wind velocity of 20 m/sec at a level of 1 m above the surface of the earth. Let us assume that the

difference

Iv

_s - vi at the surface of the earth is of the order of the hydraulic velocity of a falling particle, i.e. 2 - 3 m/sec. Then in formula (7),

- 600 / 3

K

=

3 • 10 • 60

=

3 gm m •

Comparing K with the term Pag , the value of which is greater than 7,000 gm/m3_{, it can be seen that the sublimation vector is}

small in comparison with the basic terms of equation (7).

Consequently for this investigation one can consider that the use of the system of separate equations (1 - 4) is well founded.

For prolonged blizzards the sublimation factor of snow carried by the wind is very important since it determines the loss of snow

in the places where the snow is deposited and the distance over which the snow is carried.

Stress Tensors

The tensor of instantaneous stresses I I can be written in the following form

IT

=

I • p - _ITB, (10)

where p is the average normal pressure, I is a unitary tensor,

ITa

is a magnitude which in meaning is analogous to the viscosity tensor of tension in a single-phase liquid (reference 15, page 66).

The tensors T_s and T are not considered in their complete form but only their "viscous" parts T

Bs and TB which depend on internal deformations in a two-phase flow.

Then the generalized tensor ｾｩｫ of stress tangents is written thus:

ｾｩｫ

=

_{IIBi k} + _{TBs i k} + _{TBi k,} i ｾ k, i, k

=

1, 2, 3.

The mean normal pressure p will receive, apparently, an addition owing to the normal components Ts and T.

For actual wind velocities the gradients of normal pressures p, as is ｫｮｯｾｭ (reference 16, page 65) are so small that they ｣ｾｾ be neglected.

It has been shown(17) that the probable formula for

ｾｩｫ

for a single-phase liquid is the following:

(11)

where Y is the coefficient of molecular kinematic viscosity, T

Yi k is a coefficient of turbulent kinematic viscosity having a tensor character.

T 0

Here we established the order of magnitude of Yiko

(12)

where t is the characteristic dimension of the flow,

ｾｶ is the difference between the mean velocity at a given point of the flow and the characteristic velocity corresponding to the boundary or initial conditions of motion.

Since the tensor T_s is not well known we must use as a first approximation the expression (11) for a two-phase snow-air flow. Here one must keep in mind that, as shown previously(17), the

expressed in the form of an apparent increase in the dimensions of the obstructions on the surface. This has been observed also by G. Liljequist in many experiments in the Antarctic(lS).

The forces of interact10n of the particle and the medium for low concentrations is estimated using the usual expressions for the resistance to motion of bodies in a fluid

Us E "" Cs -₅₂ • s p(v - v )2 S 2

where C is the dimensionless resistance coefficient depending on the shape of the particle and the Reynolds number Be,

U

s,S2s is the mean area of the air-snow cross-section and mean volume of particles,

b is a unitary directional vector. Fundamental Criteria

Taking into account the observations made above equation (4) for a plane flow is written thus:

(14)

where q is the solid flux in the snow-wind flow in gm/m:! sec over a cross-section normal to the wind.

We have an obvious equality

It is assumed that the direction of the wind coincides with the axis x₁ •

The second term in the right-hand portion of equation (14)

gives the momentum force of the particle counteracting the force of gravity. In the notations adopted here, as a transfer condition,

the following ordinal relation will be valid:

Further

p(v - v ) 2

S

2 • (15)

where ｾｳ is the characteristic dimensions of the particle. Directly from (14) we get a dimensional relation

P ｧｾＨｶ - v ) 2 ,

S S

(16)

(17)

where ｾ is the characteristic dimension of the flow.

Using equations (15) and (16) it is easy to separate from (17) the four criteria which are of essential significance in the theory of two-phase flow, p ｾｯ =

p'

s V ｾＱ］ＭＭｬＬ V_s 9 (Ie) (19) (20) (21) An important role is also played by the symplexes C and S. Expression (17) is a complex of the form

ｾｾ

ｷｾ

1 W₂

ｾＳＮ

Criteria (19) and (20) were used by us for the analysis of average blizzard observations. A criterion formula for solid flux has been found with their

ｨｾｬｐＨＱＹＩ

which is valid within accuracy limits of the experiments, for snow-air, air-sand and water-sand

flows.

Assuming that the initial and boundary flow velocities are equal to zero, from equation (2) taking into account (11) and (12), we find the generalized Reynolds criterion

where a is the dimensionless coefficient characterizing the extent of turbulence of the flow,

Re 1s the Reynolds criterion depending only on molecular viscosity.

Under ordinary conditions, for the air flow around obstacles under winter conditions according to our observations a

ｾ

0.05(20). Consequently when Re » 200, as seen from (22) , He' tends towards a constant value of ｾ｡Ｒ and does not depend on velocity and dimensions of flow which correspond to the region of self-modelling. Therefore the velocity field of ｡ｩｾｳｮｯｷ flow at a sufficient distance from the boundary of motion and with an invariable Re' number can be

investigated with the use of reduced-scale mOdels which is confirmed by many years of experiments at the Transportation Problems Division of the Transportation-Power Institute, SOAN, 5SSR (Siberian Division of the Academy of Sclences USSR).

In considering the motion of solid particles one should in (22) take the difference v - V

s instead of v and put t ,.., .e, •

s

Calculation shows that in this case when the size of blizzard particles is 10-4 - 5 • ＱＰＭｾ m and relative velocities from 0 to

5 m/sec He < 200 and self-modelling no longer occurs. It is

particularly difficult to maintain the fundamental criterion ｾＳ (21) since one can decrease the absolute dimensions of the particles, retaining the suitable connection between them, only to a specific limit having the order of 10- ｾ m,

The static picture of particle accumulation depends on the

kinematic field of flow and, according to what has been stated above, the general contour of the deposition on the model should be similar to the natural picture, as was obtained by I.V. Smirnov(20).

However it is exceedingly difficult to bring about an approximation of dynamic deposition when the scale is greatly reduced.

Some investigators, neglecting criteria (21) and (22) admit an error. For example, A.I. znamenskii(2l) considered that most of the eolian sand would be harmlessly carried over the Karakum Canal, which was shown experimentally in an aerodynamic tube using natural

sand in a greatly reduced mOdel, where, of course, similarity in dynamic trensfer did not hold.

The Transporting Abilitx of Wind

The specific load-lifting capacity is the most important

characteristic of a blizzard as in the case of any two-phased flow containing a heavy solid phase.

The load-lifting capacity of a blizzard should be determined from a simultaneous solution of equations (1 - 4) and the gammas of energy equations, however the theoretical difficulties of such a solution have not been overcome.

An approximate estimation can ｢ｾ given either from the energy balance of a two-phase flow, introducing the concept of the "work of

suspension", as was done by M.A. velikanov(23), or from the balance of forces using equation (4).

Assume that the process is quasi-stationary and on the average does not change in directions x₁ and x₂ • Then construct both parts

of equation (4) on the x₃ axis and integrate over x3 from the level

of the surface of the ground up to a specific height of suspension x₃

=

h. Assuming that the x₃ axis goes in the same direction as the vector g, we get

where f

c is the "plane" concentration of snow in the flOW, i.e. the gravimetric quantity of snow lifted by the wind, taken for a unit of surface of the snow cover

fa are the suspension forces converted to a unit of horizontal area H fa

=

J

[(

fN • II +

E) • :

+ f (x_{3 ) ]} dx_{3 ,}

o

where f(x 3 ) is a certain function of x3 •

Let us introduce a hypothesis according to which the main suspension factor is considered to be the decrease in pressure at the centres of small but highly intense eddies formed directly at the surface of the earth. The drawing pressure fa at the axis of the centTe of the eddy is, as is known (reference 24, pp. 292-293),

. 2

ｾ

" ( aV

1 )

fa

=

P 4

ax

₃

'

where D is the diameter of the centre.

aV

₁

The gradient

ax-

is determined on the basis of reference 17

3

where with the use of expressions (11) and (12) the following

the earth·:

.J

In

_u.

2a

(24)

Vo is the velocity of the flow at the boundary of motion, ｾｯ is the friction force at the surface of the earth. From (24) we get 1 ｾ 1 + '"

?t1J1 •

eljr • erf

4T

(25) (26) where

At the surface of the earth !:J.v =

o

and, as one should expect

av,

I

'to v2

•

aX

3

x

3= o

= _vp = _v •

Assume that at the level x₃

=

h the difference 6v is equal to !:J.vfi At some distance from the wall

!:J.vh

:*

J

In

ｾ

,

•

This formula is valid in the layer adjacent to the ground having a thickness of 40 - 50 em in the absence of any noticeable

temperature gradient. Wtth uniform roughness and non-uniformity of large dimensions the effect of (24) may be extended to even greater distances.

where () is the mean height of the protrusions of the rough surface which is approximately

() = ,f7C --L.. 2 av. Leaving out a in (26) and (27) we get

Consequently

Then the drawing pressure fa is defined as

(27)

(28)

Let D

=

a{) where a is the proportionality coefficient. It would be natural to assume that D and () are magnitudes of the same order. On the other hand it is obvious that a > 1, i.e. the

dimensions of the centre of eddies adjacent to the earth should be somewhat greater than the dimension of the heavy particle being suspended.

We obtain the inequality

(29)

H

Q

=

J

qdx3 gm/m sec

o

the total flux of snow.

In contrast to the flux q, convert9d to a unit of area of flow cross-section, the total flux Q represents the weight of snow trans-ferred by the wind in a unit of time over a unit of length of the flow front.

We have the obvious equality

(30) where v ｾｰ is the mean velocity of the motion of the suspended

ｳｾ

particles with respect to the surface of the earth.

Assume that v s cp is proportional to ｾｶｨＧ _{i.e. v s cp}

=

ＮｾＶｶｨＧ

where 13 is a certain coefficient depending on h. Then from (23),

(29) and (30) we find

ｾ

Q. > 64 • ｾ • (31 )

From (31) we get the fifth criterion for a two-phase flow which connects the total flux of solid matter ｾ with the characteristics of the medium

Let us make the following observation regarding the value of the boundary velocity of the flow v_{o •} As has been established by

experiment there are some critical velocities v' above which the transfer of the solid phase begins with a flux of Q > O. Repeated measurements of the velocity of wind in the layer adjacent to the earth show that for snow the critical velocity preceding the transfer

of snow under the ｾｯｳｴ favourable conditions of erosion can be

reliably taken to be 2 m/sec. When v_h ｾ v' ｾ ｾ 0 but in correspond-ence With formulae (23), (28) and (30) this can take place only under the condition D ｾ 0 or v_o ｾ VI. We assume the latter, i.e. V_o ｾ VI

-2 m/sec.

The author determined the value of ｾ experimentally. In

1958 - 1959 measurements were made in a special aerodynamic channel, having a cross-cection of 0.15 • 0.15 m2 and a length of 14.9 m, of the transit velocity of a cloud made up of snow particles. The results of these experiments are shown in Fig. 1. In the first approximation for temperatures below _5° they can be generalized empirically by the formula

v s cp ｾ 0.494(v_0.0:5 - 2) m/sec,

where V_{o ,}0 is the velocity of flow at the height of 0.05 m.

ｾ

Assuming that ｾ = 0.5, V_{o =} 2 m/sec, P = 130 gm sec2/m 4 h =

0.05 m, 6

=

0.0005 m(l?) , we have

Q > 0.686(v - 2)3 gm/m sec.

0.0:5 (34)

It is interesting to note that when v, = 1.4 v_O•

O:5 formula (34)

almost exactly coincides with out approximate criterion formula derived earlier(19).

Fig. 2 shows a curve constructed from formula (34) compared with experimental observations the author obtained of the total flux of snow in the aerodynamic channel of the cold laboratory.

One can check the theoretical formula (31) on other similar two-phase flows. For this purpose let us use the experimental data of V.N. Goncharov and G.N.

ｌ｡ｾｳｨｩｮＨＲＵＩ

on the measurement of the transfer of sand in troughs. The size of particles in these experi-ments varied from 0.75 to 4.83 mm. This is the order of magnitude of the promontories of the rough surface 6. Let us assume that by analogy With the above, ｾ

=

0.5, a > 1. The mass density of the

medium p = 10200 gm sec2/m"'"_{• Then}

Q > _{596 (vo • o o -} v' ):3 gm/m sec when 6

₌

0.75 • 10-3 m,

Q. > 1073 (v - Vi )3 gm/m sec when _{6 = 4.83 •} 10-3 _m,

0.015

where Vi is the mean critical "non-transferring" velocity of the

flow which varied in experiments from 0.24 to 0.37 m/sec.

Fig. 3 gives curves constructed from these two formulae and the experimental points.

With the artificial blizzard in the aerodynaoic channel and in the artificial water channel the most favourable conditions for transfer of particles from the surface were created. Thus the flow had time to realize its entire transporting ability and the experi-mental points were close to the theoretical curve.

When the flow was completely saturated with the solid phase the criterion ｾＴ (32) in the water channel varied from 0.0084 to 0.049. From the data of our laboratory investigations of blizzard transfer this value varies from 0.015 to 0.034, 1.e. within the same range, which demonstrates that the criterion is universal.

The expression (28) is of interest from the point of view of phase interaction. In reference 17 it was stated that with satura-tion of the flow by solid particles there is an apparent increase in the roughness factor 6 by the factor of 10 and more. Then from

(28) it follows that With an unchanged diameter of the eddies adjacent to the earth the resultant suction pressure decreases by more than the factor of 50 - 100, i.e. in a saturated two-phase flow there is no possibility of further uptake of particles. Here the tangent surface pressure ｾｯ decreases as it should in correspondence with the physical sense of the phenomenon. G. Liljequist arrived at the erroneous conclusion that ｾｯ increases in time during a blizzard blizzard(lS) , which would be equivalent to maintaining that there was no limit to the transporting ability of ｾｩｮ､Ｎ

The author intends to devote a separate paper to a detailed description of experiments with artificial blizzards.

Transportation-PowGr Institute Siberian Division of the

Academy of Sciences USSR

References

Received

July 22, 1959

1. Zlotnitskii, E.D. Snowdrifts. Zhur. Min. putei soobshcheniya (unofficial section), 4 (1), 1878.

2. Barenblatt, G.I. The motion of suspended particles in a turbulent flow. Priklndnaya Metamatika i Mekhanika, 17 (3), 1953.

3. Frankl', F.r. On the theory of motion of suspended particles. Doklady Akad. Nauk SSSR, 92 (2),1953.

4. Teletov, S.G. Hydromechanics of two-phase mixtures. 1. Hydro-mechanical and energy equations. Vestnik MGU, sere matem., mekh , , a.st r-onom , , fiz., khLm , novz , 1958.

5. Vedenisov, B.N. On increasing the efficiency of snow removal from railroads. Tr. Mosk. in-ta inzh. zh.-d. transporta,

(71), 1948.

6. Pudovkina, I.B. Electric charge generation during a blizzard. II. Mezhduved. koordin. sov. po izucheniiu snega i

snezhnogo pokrova. Nal'chik, 1956.

7_ Gerasimenko, V.I. Atmospheric-electric investigations on Cape Cheliuskin in 1934 - 35. Sb. rabot po zemnomu magnetizmu i atmosphernomu elektrichestvu, no.2, Tr. Arkticheskogo in-ta, v.97. Leningrad, 1937.

8. Leushin, N.I. Results of recording the electric potential gradient on Dikson lsland. Tr. Arkticheskogo in-ta, v.78. Leningrad, 1937.

9. Cherniavskii, E.A. On the problem of the charge of the earth's surface. Izv. Akad. Nauk SSSB, ser. geofiz. i geogr. 12 (2), 1948.

10. Anikiev, I.S. Electric charge of rain drops and snowflakes. Meteorologiia i Gidrologiia, (4), 1951.

11. Norinder, H. and Siksna, R. air when blowing snow.

Electric charges measured in the Arkiv for geofysik. 2 (4), 1955. 12. Stager, A. Experimentaluntersuchungen uber

Kontaktelektris-ierung von Staub und wolkenformig zerteieten Korpern, speziell von Schnee als gewitterbildendem Factor. Ann. der Physik, v.76. Leipzig, 1925.

13. Pearce, D.C. and Currie, B.W. Some qualitative results in the electrification of snow. Canadian Journal of Research, v.27, no.l, Sec. A. Physical Sciences, 1949.

14. Dyunin, A.K. Sublima.tion of snow. Izv. SOAN SSSR, (2),1958. 15. Landau, L.D. a.nd Livshits, E.M. Mekhanika sploshnykh sred

(Mechanics of continuous media). GITTL, Moscow-Leningrad, 1954.

16. Yur'ev, B.N. Experimental aerodynamics, Part 1. Gl. red. aviats. l i t . Moscow-Leningrad, 1936.

17. Dyunin, A.K. On the semi-empirical theory of a turbulent boundary layer. Izv. SOAN SSSB, (5), 1959.

18. Liljequtst, G.H. Wind structure in the low layer. Norsk. Polarinstitut. Oslo, 1957.

19. Dyunin, A.K. Solid flux of a snow-wind flow. Tr. Transp.-energet. in-ta ZSFAN

sssa,

no.4 (Transport), Novosibirsk, 1954.

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Table I

Value of the maximum gradient of the electr1c field potent1al at the t1me of electrification during blizzards

Wind

veloc-Igrad

¢I

_max

Author Pl9.ce of 1ty

accord-observation 1ng to wind in volts/m

vane m/sec

V.I. Geraslmenko Cape Cheliuskin 13 2600

E.A. Cherniavskii The Iceberg Fedchenko 13 5400

V.I. Gerasimenko Cape Cheliuskln 20 6000

M.I. Leushln Dlkson Island 16 2000

M.I. Leush1n Dikson Island 18 2100

I.B. Pudovklna

si

'brus Slope 33 1050

Charges of snow particles

Characteristics Weight of Charge e,

«/c ,

Authors _{of particles} particles _saSE SGSE

in mg m2

Norinder and Siksna Blizzard particles in very high wind (assumed average

c 10-3

size 0.2 nun) ｾｏＮｏｏＳＳ 0.33 ｾＰＮＱ

I.S. Anikiev Blizzard particles

during snowfall

• 10-3

and strong wind 0.16 2.14 0.0134

I.S. Anikiev Snow particles

1

1•39 without wind 0.11 • 10-3 0.0126 , I N VI I

Table III

Values for maximum acceleration of snow particles caused by ponderomotive forces of the

atmospheric electric field

Hax i.num values Value of

Data

I

grad ¢

I ,

_{e SaSE}

I

e

s

I

characteristics in

_0'

according to vOlts/m mg formula (9 ) in m/sec2 Experimental data ••••• 6000 0.1 0.2 Assumed absolute maxima • • . . • . • . . • • . • • • ₁₀₀₀₀

I

0.3 1.0 0 - 1 - - - .-2 .01-3 )( - 4 ｾＭＵ x X-X )( X

•

fJ 8 10 12 II, 16 18 20 22 24 26 V_{o •}05' m/ sec Fig. 1

Average velocity of the blizzard particles in the layer near the ground

I - Velocity of the cloud of snow particles measured directly in an aerodynamic channel at a temperature above -5°C, 2 - the same at a temperature below _5°C, 3 - velocity of slipping particles measured in an aerodynamic channel, 4 - data of barometric measurements in an aerodynamic channel in a layer 0 - 2 em thick, 5 - data of batho-metric measurements under field conditions in a layer 0 - 10 em thick

" J 678110 V_o.0 - 2 m/sec ｾ J 2 1 ｾ \ 10 0- gm/m sec 1000 G «] ",-2 - J ｾ

:\\

h

ｾＬＧ

.

ｾｦ｢｜Ｉ IUO

«

Fig. 2

Solid deposition from a completely saturated snow-wind flow in an experimental channel

1 - Cross-section of a channel 0.15 • 0.15 m2

, 2 - cross-section of

Q. g!D/m sec 1600 100 10 0,' • - 1 - - Z - - - 3 ｾＳ O/f

4"

46 8,8 /.0 1,5 V - VI m/sec oＮｯｾ Fig. 3

Solid deposition from a water-sand flow in a hydraulic channel according to the data of V.N. Goncharov and G.N. Lapshin at an

average depth of flow of 10 cm

1 - Experimental data, 2 - Theoretical curve with the greatest diameter of particles being 4.83 mm, 3 - theoretical curve for a minimum diameter of particles of 0.75 mm