Mechanical Conditions of Snow Erosion

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Mechanical Conditions of Snow Erosion

Dyunin, A. K.

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PREFACE

S n o w d r i f t s can be a major s o u r c e of d i s r u p t i o n i n t h e o p e r a t i o n o f t r a n s p o r t a t i o n s e r v i c e s and a genera 1 n u i s a n c e i n t h e norma 1 w i n t e r t i m e a c t i v i t y of a col~imunity. Such d r i f t s a r e formed whenever a wind, s t r o n g enough t o t r a n s p o r t h o r i z o n t a l l y a s i g n i f i c a n t amount of snow, e n c o u n t e r s a n o b s t a c l e which f o r c e s i t t o d e p o s i t some of t h i s snow. The u s u a l a p p r o a c h t a k e n i n d e f e n d i n g a n a r e a o r s t r u c - t u r e a g a i n s t s n o w d r i f t i n g h a s becn t o l o c a t e t h e s t r u c t u r e p r o p e r l y s o t h a t t h e d r i f t problem w i l l be a n~inirnun and t o e r e c t o b s t a c l e s , s u c h a s snow f e n c e s , t o c o n t r o l where t h e snow w i l l be d e p o s i t e d . The approach t a k e n I n t h e d c v e l o p n ~ e n t o f t h e s e d e f e n c e s h a s been

l a r g e l y e m p i r i c a l . A t t e n t i o n h a s been d i r e c t e d p r l m a r i l y t o t h e c h a r a c t e r o f t h e

air

f l o w w i t h l i t . t l e a t t e n t l o n b e i n g g i v e n t o t h e m a t e r i a l t r a n s p o r t e d . I n some cir~cutr:stances, i t would b e a n advan- t a g e t o have a more complete d e f e n c e a g a i n s t s n o w d r i f t i n g t h a n i s now a v a i l a b l e . In t h e i r a t t e m p t s t o d e v e l o p t h i s d e f e n c e , e n g i n e e r s a r e g i v i n g more c o n s i d e r a t i o n t o t h e t h e o r e t i c a l a s p e c t s o f t h e problem and i n p a r t i c u l a r t o t h e r e l a t i o n s h i p s between t h e a i r f l o w and t h e snow b e i n g t r a n s p o r t e d .

It i s one o f t h e r e s p o n s i b i l i t i e s g f t h e Snow and I c e S e c t i o n of t h e D i v i s i o n of B u l l d i n g Research t o c o l l e c t and niake a v a i l a b l e i n f o r m a t i o n r e q u i r e d f o r t h e s o l u t i o n of' snow and i c e problems. The p r e s e n t p a p e r , t r a n s l a t e d from t h e R u s s l a n , i s a c o n t r i b u t i o n t o t h e Lheory of s n o t : d r i f t i n g . T h i s p a p e r w i l l g i v e t o t h e r e a d e r a n

a p p r e c i a t i o n of some of t h e f a c t o r s t o be c o n s i d e r e d i n t h e t h e o r e t - i c a l d e s c r i p t i o n o f blowing snow and I t s d e p o s i t i o n a s s n o w d r i f t s .

The p a p e r was t r a n s l a t e d by M r . D . A . S i n c l a i r o f t h e T r a n s l a t i o n s S e c t i o n of t h e N a t i o n a l Research Councll L i b r a r y , t o whom t h e D i v i s i o n of B u i l d i n g Research wishes t o r e c o r d i t s t h a n k s .

Ottawa

November

1963

R.F. Legget

NATIONAL RESEARCH COUNCIL OF CANADA Technical Translation 1101

Title : The mechanical conditions of snow erosion

(0 mekhanicheskikh ueloviiakh erozii snega)

Author: A.K. Dyunin

Reference: Trudy

Transportno-Energeticheskogo

Instltuta,

(4):

59-69, 1954

TIU PIUCIIAN I C A L CONDITIONS OF

SNOW

EROSION

Surnrnary

Formulae a r e developed f o r t h e c r i t i c a l wind speed f o r t h e s t a r t of snow e r o s i o n , u s i n g t h e t h e o r y o f dimensions.

One o f t h e most i m p o r t a n t q u e s t i o n s i n t h e t h e o r y o f snow d e p o s i t s i s t h e e x p l a n a t i o n o f t h e c o n d i t i o n s u n d e r l y i n g t h e s t a r t o f snow e r o s i o n . We e x p l a i n t h e s e c o n d i t i o n s u s i n g t h e methods of t h e t h e o r y o f dimensions. Let us c o n s t r u c t t h e d i f f e r e n t i a l e q u a t i o n f o r t h e motion o f a p a r t i c l e i n a l i q u i d , c o n s i d e r i n g t h e p a r t i c l e a s a m a t e r i a l p o i n t : where m

-

mass of t h e p a r t i c l e ; v

-

s c a l a r v a l u e o f i t s speed; f

-

f o r c e e x e r t e d on t h e p a r t i c l e by t h e l i q u i d ; R

-

r a d i u s o f c u r v a t u r e of t h e p a r t i c l e motion; T

-

u n i t v e c t o r s of t a n g e n t and normal f o r c e ; b

-

u n i t v e r t i c a l v e c t o r ; i

-

a r b i t r a r y u n i t v e c t o r ; g

-

a c c e l e r a t i o n due t o g r a v i t y ; p

-

mass d e n s i t y o f l i q u i d ; A

-

mass d e n s i t y o f p a r t i c l e s . C o n v e r t i n g t o t h e non-dimensional form ( r e f .

4,

e q u a t i o n l ) , we o b t a i n :

where 1

-

a g i v e n l i n e a r dimension o f t h e flow; d

-

l i n e a r dimension of t h e p a r t i c l e . D i v i d i n g t h e two s i d e s o f t h e e q u a t i o n by d 3 A g ( l

-

f),

we o b t a i n i n t h e r i g h t - h a n d p a r t o f t h e non-dimensional conibination: where t h e f o r c e f

i s

an unknown q u a n t i t y . F o r a p r e l i n ~ l n a r y e s t i m a t e o f t h e v a l u e f we make u s e o f

N.E.

( 3

1

Z h u k o v s k i i l s t h e o r y o f e l l i p t i c a l v o r t i c e s

.

C o n s i d e r i n g t h e p a r t o f t h e flow n e a r t h e ground, N.E. Zhulcovskii assumed t h e e x i s t e n c e i n t h i s s e c t i o n of e l l i p t i c a l v o r t i c e s p l a y i n g a

p r ' i n c i p a l r o l e i n t h e p ~ ~ o c c s s o f creation of t h c t u r b u l e n t p u l s a t i o n s r e s u l t - i n g i n t h e suspension of t h t : g r a n u l a r n i a t c l > i a l s . T h i s was confirmed I n t h e l a t e r cxperirr~erlts o f G . D z h i l l b e r t , L. P r a n d t l and o t h e r s ( r e f . 2, page

18).

O n t h e b a s i s of t h c s e expcrlnicnts t h e p a t t e r n of t h l s proctfss. is r e p r e s e n t e d

i n t h e followinp; form ( s e e F i g . I ) * .

An e l l i p t i c a l v o r t e x A i s forrr~ed i n t h e p a r t of' t h e t u r b u l e n t flow n e a r t h e ground and suclts i n p a r t l c l c s s i t u a t e d a l o n g t h e s u r f a c e 1

-

1. These v o r t e x f o r ~ ~ l a t i o n s a r e u n s t a b l e . They forni, i n c r e a s e i n s i z e and d i s i n t e g r a t e g i v i n g p l a c e t o new o n e s .

On disintegration t h e ground ("bottorn") v o r t i c e s e j e c t a mass of l i q u i d M s a t u r a t e d w i t h p a r t i c l e s from t h e i r hind p a r t .

In I t s g e n e r a l f o r n ~ t h e Slow f u n c t i o n o f t h e e l l i p t i c a l v o r t e x f i e l d , a c c o r d i n g t o V.N. Goncharov ( r e f . 2 , page 2 0 ) , i s :

where a , b

-

maximum semiaxes of t h e v o r t e x c o r e ( ~ l g . 1 ) ;

v,

-

v e l o c i t y a t t h e p o i n t B.

n

-

a f a c t o r which a c c o r d i n g t o N.E. Zhukovskil i s e q u a l t o one and a c c o r d i n g t o V.N. Goncharov e q u a l t o two.

We d e t e r m i n e t h e p r e s s u r e of t h e i n t e r n a l c o n t o u r

AIBAB1

due t o t h e s u c t i o n of p a r t i c l e s e n t e r i n g t h e zone of i n f l u e n c e of t h e v o r t e x .

Let u s t a k e t h e Navier-Stokes e q u a t i o n :

where i , k

-

1 , 2 , 3

...

a r e i n d i c e s o v e r which sunmations a r e c a r r i e d o u t ; Sl

-

co~nponents of t h e e x t e r n a l f o r c e s ; p

-

mass d e n s i t y of l i q u i d ; P

-

p r e s s u r e ; % i k

-

v i s c o u s s t r e s s t e n s o r . F o r t h e s a k e of s i n l p l i c i t y we assume t h a t i n t h e g i v e n i n t e r v a l of time 0

-

t t h e f i e l d i s s t a t i o n a r y . For t h e two-dln!ensional s t a t i o n a r y v o r t e x e q u a t i o n

( 5 )

h a s t h e form: where v

-

k i n e m a t i c c o e f f i c i e n t o f v i 6 ~ 0 S i t : r .

*

T h i s schernc must be regarded a s o n l y a n approximate n ~ e c h a n i c a l model. A

n o r e a c c u r a t e r e p r e s e n t a t i o n of t h c p r o c e s s i s g i v e n by t h e methods of s t a t i s t i c a l p h y s i c s .

According t o e q u a t i o n ( 1 1 ) we have: d ~ k

4

vx = - - = - - v 3 - - dx, an @ _Ivx _-n (n - I ) x,"-

'

s,; d" S u b s t i t u t i n g t h e s e e x p r e s s i o n s i n ( 6 ) , m u l t i p l y i n g t h e f i r s t e q u a t i o n ( 6 ) by d x , , and t h e second e q u a t i o n

( 6 )

by dx, and combining then] we g e t :

1 n (n- 1) x:-' sn (n

-

1) xy -

'

- -

d P + v V ,

[-dh-

d x , - --

P ',n

+

I

A f t e r i n t e g r a t i o n w e have:

A t t h e p o i n t A ( a , 0 ) ( ~ i g . 1) l e t t h e p r e s s u r e be P o . The p r e s s u r e d i f f e r e n c e ( s u c t i o n ) w i t h i n t h e contour ABA'B i s then w r i t t e n :

Hence t h e s u c t i o n i s equal t o t h e sum of t h r e e terms, one of which

depends on t h e square of t h e forward v e l o c i t y , t h e second i s proportioned t o t h e same v e l o c i t y and t h e t h i r d ( p i e z o m e t r i c term) i s n o t dependent on t h e v e l o c i t y .

The r e s u l t obtained s e r v e s a s a b a s i s f o r t h e adoption of t h e f o l l o w i n g working h y p o t h e s i s :

Force f

i n

e x p r e s s i o n

( 3 )

i s e q u a l t o t h e sum of t h r e e terms, one of which i s p r o p o r t i o n a l t o t h e square of t h e forward v e l o c i t y of t h e flow n e a r t h e around, t h e second d i r e c t l y p r o p o r t i o n a l t o t h e same v e l o c i t y and t h e t h i r d i s independent of t h e v e l o c i t y .

The b a s i c f a c t o r s ir1S7uer1cin~; t h c val-uc-1 o f f' a r e ( t h e dimensional

forrnulae a r e shown i n t h e squarbc b r a c k e t s : P

-

f o r c e , L - l e n g t h , T - t i r n e ) : Mass d e n s i t y of l i q u i d p [ ~ ~ - 4 ~ 2 ]

V i s c o s i t y of l i q u i d ~ [ F I , - ~ T ]

Plean forlward v e l o c i t y o f flow V[LT-

'

]

A c c e l e r a t i o n due t o g r a v i t y i n t h e l i q u i d ~ [ L T - ~ ] S i z e o f p a r t i c l e d [ L ] According t o t h e t h e o r y of dirnenslons: where a ,

-

non-dimensional c o n s t a n c e . We now c o n s t r u c t t h e dirr~ensiona 1 e q u a t l o n : 4 7'

L

1'

L

T-ZJZILFL-~T]' ( [ F ) ' = [ F L -

I t [

1

[ * LT-IJZI z , +r,= 1 -.It, + % 2 + ~ a - 2 Z ~ + ~ g = 0 whence : Hence : 1

Assuming z , =

H ( 3

-

I ) and z , = 2

-

1, w e o b t a i n t h e f i r s t t h r e e terms of t h e sum i n f u l l c o n f o r m i t y w i t h t h e working h y p o t h e s i s formulated above:

S u b s t i t u t i n g

( 7 )

I n

( 3 ) ,

we have:

K,

O r , a f t e r introduction o f t h e c o n s t a n t s K , =

0

Let v ' b e t h e mean v e l o c i t y f o r t h e l a y e r of t h e flow a t a h e i g h t e q u a l t o u n i t y (1

m).

Taking t h e d i s t r i b u t i o n of v e l o c i t i e s n e a r t h e ground

( 6 ) .

a c c o r d i n g t o S.A. Sapozhnlkov

.

where vy

-

wind speed a t t h e h e i g h t y v,

-

wind speed a t a h e i & t of 1 m

6

-

l i n e a r c h a r a c t e r i s t i c o f roughness.

We d e t e r m i n e t h e niean v e l o c i t y i n t h e l a y e r from 0 t o y:

Because of t h e sniall v a l u e of 6, y 1" y

-

6 . The r e f o r e :

( e

-

b a s i c N a p i e r l o g a r i t h m ( e = 2.718 . . . ) ) .

Employing formula ( 1 0 ) we determine, a f t e r a simple t r a n s f o r m a t i o n , Vl mean9 e x p r e s s i n g t h i s value i n terms o f v, and vy:

The i n i t i a l p e r i o d of motion o f heavy m a c r o p a r t i c l e s i n l i q u i d o r gaseous medium i s c h a r a c t e r i z e d by t h r e e v e l o c i t i e s :

( 1 ) v t

-

c r i t i c a l o r "non-moving" v e l o c i t y a t which complete absence o f p a r t i c l e motion

i s

a s s u r e d ;

( 2 ) v"

-

t h r e s h o l d v e l o c i t y , 1 . e . v e l o c i t y a t which s e p a r a t e p a r t i c l e s b e g i n t o b r e a k away from t h e s u r f a c e and t o be d i s p l a c e d ;

( 3 )

v"'

-

upper c r i t i c a l v e l o c i t y , 1 . e . v e l o c i t y a t which mass motion of p a r t i c l e s b e g i n s .

For t h e c a s e of sand i n w a t e r ("sand

+

w a t e r " ) t h e f o l l o w i n g r e l a t i o n s between t h e s e v e l o c i t i e s have been e s t a b l i s h e d ( r e f .

7,

page

443):

If

K,,

Po

and f3, have been determined from t h e c o n d i t i o n s o f non-motion o f s u r f a c e p a r t i c l e s , t h e n i n formula

( 8 )

we s h a l l have v = v l .

Let u s assume t h a t t h e p a r a m e t e r s

K,,

Po and f3, determined i n t h i s

manner a r e c o n s t a n t , i n d e p e n d e n t l y o f t h e c h a r a c t e r o f t h e medium i n which t h e p a r t i c l e s a r e moving. To v e r i f y t h i s s u p p o s i t i o n we may t a k e two v e r y

d i f f e r e n t media s u c h a3 w a t c r and a l r .

We ert~ploycd t h e c x i s t i n g experillrental v a l u e v ' f o r - t h e c a o c o f "sand

+

w a t e r " . C a l c u l a t i o n o showed t h a t t h c f o l l o w i n g riurner.lcal v a l u e s for. t h e p a r a ~ i l e t e r s of fornrllula

( 8 )

corr.c:.sponded c l o s e l y t o t h i s v a l u e : The l i n e a r c h a r a c t e r i s t i c of t h e r o u g h n e s s 6 was c a l c u l a t e d a c c o r d i n g t o V . N . G o n c h a r o v f s d a t a ( r e f . 2, p a g e s

46-48)

f o r t h e v e l o c i t y d i s t r i b u t i o n i n d e p t h a s i n d i c a t e d i n F i g . 2 . On t h e a v e r a g e l o g 6 =

-11.75"

( b r o k e n l i n e i n F i g . 2 ) . F o r sand w e have t a k e n A = 276

-

l o 3

e.

F o r w a t e r a t a t e m p e r a t u r e

.

.., A .

1 5 " C ,

p = 102 l 0 3 s c c 2

,

v = 1. 114

-

rn2/sec. A c c e l e r a t i o n due t o n14 g r a v i t y g =

9 . 8

nJsec2. I n F i g .

3

we g i v e t h e c u r v e v,,

,

= f ( d ) , c o n s t r u c t e d a c c o r l d i n g t o forniulae

( 8 )

and ( 1 2 ) . T h i s f i g u r e r e v e a l s t h e meaning o f t h e c r i t i c a l

v e l o c i t y d e t e r m i n e d by V . N . Goncharov ( r e f . 2 , page

158)

f o r q u a r t z p a r t i c l e s of a n a v e r a g e s i z e o f 0 . 3 5

-

39.3

and reduced t o a f l o w d e p t h o f 0 . 1 n . I n a d d i t i o n , we have p l o t t e d t h e d a t a o f D z h a l s t r o m ( r e f .

9,

page 1 6 0 ) on q u a r t z p a r t i c l e s o f v e r y s m a l l s i z e .

The v a l u e vo

.

,

mean f o r srnall p a r t i c l e s i s due t o t h e i n c r e a s e d

i n f l u e n c e o f t h e v i s c o s i t y o f t h e medium and a l s o t o t h e i n c r e a s e i n t h e i r s p e c i f i c a r e a s , which i n c r e a s e s t h e c o h e s i o n between minute p a r t i c l e s , i n h i b i t i n g t h e i r removal from t h e s u r f a c e o f t h e ground o r snow c o v e r .

S u b s t i t u t i n g t h e above c i t e d v a l u e s K,,

P,

and

Po

In f o r m u l a

( 8 )

we f lnd : where _{E, = 3,5g}

($-I

)

D i f f e r e n t i a t i n g t h e r i g h t - h a n d p a r t of ( 1 4 ) w i t h r e s p e c t t o d and p u t t i n g t h e r e s u l t e q u a l t o z e r o , we o b t a i n t h e c o n d i t i o n o f minimum vfmean. A t t h i s v a l u e o f d t h e c r i t i c a l v e l o c i t y i s a minimum. F o r v e l o c i t y s m a l l e r t h a n vfmin t h e r e w i l l be no removal of p a r t i c l e s ( e r o s i o n ) , a s though

*

S t r i c t l y s p e a k i n g t h e r o u g h n e s s 6 depends on t h e s i z e s d o f t h e p a r t i c l e s f o r m i n g t h e bed o f t h e f l o w and on l o g 6 , a s c a n be d e t e r m i n e d from t h e

t h c r e were no p a r t i c l e s of srnall dirncnsions.

Thc v a l u c s of E, and E, ( f o l b d e x p l ~ s s c d i n m l n ) a r e d c r i v c d f o r v a r i o u s

media and p a r t i c l e s and a r e g i v e n i n T a b l e I.

The v a l u c s of d and v ~ , c a l c u l a t e d a c c o r d i n g t o forlnul a e , ~ ~ ~ (111) and ( 1 5 ) a r e g i v e n i n T a b l e 11.

The c u r v c s of v t , = f ( d ) f o r t h o c a s e s " a i r

+

sand" and " a i r .

+

s o i l p a r t i c l e s " ( s e e F i g . 4 ) a r e d e r i v e d fro111 t h e d a t a of Table I and I1 and

forr,iulae (11) and ( 1 1 1 ) . They aC;ree w e l l w i t h t h e e x p e r i r r ~ e n t a l Cata of

M.A.

Sol<olov ( I . 1, page 20'1) and C h e p i l and ~ a l l n ( ' ~ ) , c o n f i r m i n g o u r assump- t i o n a b o u t t h e c o n s t a n c y o f t h e p a r a ~ ~ e t c r s

K 2

and

B , .

S i l n i l a r cul-ves have been c o n s t r u c t e d a l s o f o r " a i r

+

snow".

According: t o G . D . R i k h t e r l s d a t a ( r e f .

5,

page 29) snow i s n o t blown away a t wind s p e e d s of l e s s t h a n 2 nJsec, r e g a r d l e s s of t h e s t a t e o f t h e snow c o v e r o r t h e s i z e o f t h e snow p a r t i c l e s . T h i s c o r r e s p o n d s t o t h e v a l u e of

V: n i n =

1.93

nJsec i n T a b l e 11.

According t o t h e d a t a of A . A . Komarov, a c o l l a b o r a t o l - of TEI ZSFAN*, a t a mean snow p a r t i c l e s i z e of a p p r o x i m a t e l y 0 . 5 nun snow i s n o t t r a n s p o r t e d a t s p e e d s v of l e s s t h a n

3

rifsec, c o r r e s p o n d i n g t o t h e broken l i n e c u r v e s I n F i g . 4 .

The mass t r a n s p o r t of d r i f t i n g snow p a r t i c l e s of a p p r o x i m a t e l y

0.5

mm

i n d i a m e t e r i n t h e absence of p r e v i o u s thawing b e g i n s , a c c o r d i n g t o

formulae ( 1 7 , ) and ( 1 4 ) ) a t u p p e r wind s p e e d s v"' = 2 . 6 8

.

1 . 5 6

=

4 . 2 n / s e c , assuming t h a t e q u a t i o n ( 1 7 ) a p l i e s t o t h e snow and t h e wind f l o w . According t o t h e d a t a o f A.W. Khrgian(8B, TsNII PIPS** and T G I ZSFAN, t h e t r a n s p o r t of snow r e a l l y b e i n g s a t wind s p e e d s o f v , = LC

-

4 . 5 n / s e c .

A f t e r thawing and r e f r e e z i n g o f t h e s u r f a c e c r y s t a l s ( " r e g e l a t i o n " ) and wind p a c k i n g of t h e s u r f a c e of t h e snow c o v e r , t h e l a t t e r becomes covered by a s o - c a l l e d " c r u s t " which p o s s e s s e s g r e a t e r r e s i s t a n c e t o e r o s i o n . The v a l u e

( 5 ) v, I t ' u n d e r t h e s e c o n d i t i o n s i n c r e a s e s t o 1 0 nJsec and more

.

The d e g r e e of r e s i s t a n c e t o e r o s i o n depends on t h e v a l u e of t h e snow c o h e s i o n .

I f t h e snow i s somewhat b r i t t l e , t h e n t h e v a l u e of t h e a d h e s i o n depends, a s was i n d i c a t e d above, on t h e s i z e o f t h e snow p a r t i c l e s . T h i s can be

d e t e r m i n e d b y r e p r e s e n t i n g t h e pheno~nenon a s f o l l o w s ( s e e F i g . 5 ) . L e t u s c o n s i d e r t h r e e c u b i c p a r t i c l e s a , , a , and a , o f s i d e d .

+ Thermal Power I n s t i t u t e of West S i b e r i a n Branch, Academy of S c i e n c e s

USSR.

P a r t i c l e s a , and a , a r e on a g i v e n p l a n e while a, i s r a i s e d t o a p l a n e a t t h e h e i g h t

$.

Thc d i s t a n c e between t-he p a r 8 t i c l e s i s d .

'-

I f t h e r c were no r e s i s t a n c e t o t h e f a l l of p a r t i c l e a, i t would f a l l towards t h e s u r f a c e a t a r a t e vil =

4 3 .

Ilowever, s u r f a c e v i s c o u s f o r c e s

oppose t h i s motion.

S u b s t i t u t i n g vH = we o b t a i n

The t h i r d term of formula

( 8 )

i n t h e numerator of t h e r i g h t - h a n d s i d e can e a s i l y be expressed i n terms of .co

T h i s i s t h e p h y s i c a l i n t e r p r e t a t i o n of t h e t h i r d term of e q u a t i o n

8.

P,

I n t h e g e n e r a l c a s e t h i s term i s e q u a l t o

-

T , where T > -co f o r a packed o r P

f r o z e n snow c o v e r .

Comparisons of t h e c o n d i t i o n s a t t h e s t a r t of e r o s i o n of v a r i o u s g r a n u l a r m a t e r i a l s i n v a r i o u s media g i v e t h e b a s i s f o r d e r i v i n g a f i r s t approximation t o t h e c o r r e c t formula f c r d e t e r m i n i n g t h e c r i t i c a l speed v l ,

.

It w i l l have t h e form

From what has been s a i d t h e f o l l o w i n g c o n c l u s i o n s may be drawn:

( 1 ) The phenonienon of e r o s i o n of l o o s e snow i s a p a r t i c u l a r c a s e of t h e e r o s i o n of v a r i o u s g r a n u l a r m a t e r i a l s i n w a t e r and i n a i r and depends on t h e same p a r a m e t e r s .

( 2 ) The l a r g e r t h e v a l u e s of 6 , d and T t h e p;reater w i l l be t h e c r i t i c a l v e l o c i t y of t h e a i r flow r e q u i r e d t o i n i t i a t e snow e r o s i o n a t ground l e v e l . T h e r e f o r e t h e measures t o be t a k e n t o p r e v e n t e r o s i o n rnust c o n s i d e r t h e

follovring: ( a ) t h e roughness of t h e snow s u r f a c e , ( b ) t h e d i a m e t e r of t h e s u r f a c e p a r t i c l e s , ( c ) t h e a d h e s i o n between t h e s u r f a c e p a r t i c l e s .

In pursuance of t h e c o n s i d e r a t l o l l s p r c s c n t e d i i l t h i s p a p e r a d d i t i o n a l

e x p e r i m e n t a l m a t e r i a l i s r e q u i r e d f o r f u r t h e r developr,ient of t h e theoretical

Bodrov, V. I \ . I ~ > o t l a y a n~c 1 iol-a t s i y n ( ~ o r e o t iniprovcmcnta )

, 1952.

Goncharov, V.N

.

Dvizhcrllc nanosov (Thc I~love~iicnt o f a l . l u v i u ~ ~ l ) ,

1938.

Z h u k o v s k i l ,

N.K.

0 oneztlnylth zarlosakh i z a i l e n n i rclc ( o n snow d r i f t s and t h e s i l t i n g o f r i v e r s ) . S o b r . Soch. 111,

13!~3.

K i r p i c h e v , M.B. and Mlkhecv,

M.A.

M o d e l ~ r o v a n l e te p l o v y k h u s t r o i s t v ( ~ h c c o n s t r u c t i o n of therrrlal a p p a r a t u s ), C h a p t e r 1, 1936.

R i l d ~ t e r , G . D

.

Snezhnyi pokrov, ego f ormirbovanie i s v o i s t v a (The snow c o v e r , i t s f o r r ~ i a t i o n and p r o p e r t i c s ), 1945.

Sapozhniltova, S . A . M i ~ r o c l i n i a t i m e s t n y i k l i m a t ( ~ i c r o c l l m a t e and l o c a l c l i m a t e ) , 1950.

S o k o l o v s k i i , D . L . Techno1 s t o k

h he

flow o f r i v e r s ) , 1952.

Khgian,

A.m.

F i z i c h e s k i e osnovaniya b o r ' b y s o snegorn na z h e l e z n y k h dorogakh

h he

p h y s i c a l b a s i s f o r c o m b a t t i n g snow on r a i l r o a d s ), I s s u e 1, T r . M G M I ,

1933.

Krumbeln, W.C. and S l o s s , L . L . " S t r a t i g r a p h y and s e d i m e n t a t i o n " , San F r a n c i s c o , 1951.

C h e p i l , M.S. and Milne, R . A . Cornparatlve s t u d y o f s o i l d r i f t i n g I n t h e f i e l d and i n a wind t u n n e l . S c i e n t i f i c A g r i c u l t u r a l , Ottawa,

Average and l a r g e p a r t l c l e s

....

Water

+

sand t o = 15O.

...

A i r

+

sand A i r

+

s o i l p a r t i c l e s , t o =

...

20°, p r e s s u r e 760 m i . . A i r

+

snovr, t o =

- l o 0 ,

...

p r e s s u r e 760 I I I I ~ . _I.I: s e c s e c 2 m

1

t114 Table

I1

I) D e n s i t y of f l a k e s r e l a t e d t o mean a c c o r d i n g t o Nakaiya ( g r a n u l a r and

d e n d r i t i c ) and a c c o r d i n g t o t h e d a t a of P r o f . A . D . Zamorskil. Medium and l a r g e p a r t i c l e s K a t e r

+

s a n d . .

...

...

A i r

+

s a n d . . A i r

+

s o i l p a r t i c l e s .

...

A i r

+

snow..

2)According t o Bagnold

.

See F i g . 2 and r e f . 1 0 . 3)According t o Chepll and Milne. F i g . 2 and r e f . 10.

4 S e e a u t h o r ' s p a p e r on

"Vertical

d i s t r i b u t i o n of s o l i d f l u x i n a snow-wind flow" i n t h i s c o l l e c t i o n . ( N R C TT-999). v

'

n i n v,., mean = 0.24 r d s e c trlln v' = 2.92 d s e c

'

min v 1 = 2.88 d s e c

'

min v I =

1.93

d s e c

'

min l o g 6

-4.75

-4

.682'

-3.12 ') -4.854) d t r n l 1 0.56 0.035 0.03 0 . 1

Y L 8 11 C' a .fo< o./n at2 a 1 a06 aofl P O I 0006 aoot5 mot F i g . 1 Diagrati~ o f a n a l y t i c a l v o r t e x a t t h e ground A. W d t ~ l .

Data o f V.P(

onc char-ov

A _- - B e d o f 5-7 mm yrd! hS u

-

,2.,8 ,r D e p t h s o f f o /II ~cm. + . 8, _{" 3 3 1 "} #, / - I 1 < : / L.' R - A i r . o-Data of Bsgnold f o r t h e s f a r f o f r n o t ~ o r ,

OF

sand. a n t h e sur/&ces o f s a n d d u n e s - -

I J S - ~ / D O I ) - ~ I P ( O M ~

-46d /.JS - 0.74 d r 15 Fig. 2 V e l o c i t y p r o f i l e of

a

f l o w o f w a t e r and wind

Graph o f t h e f u n ~ t i o n

v r

= f ( d ) for t h e c a s e of sand

+

water" ysec Graph v; =

f

(d) /I 0 10.0 .

- - -

- - - air

+

sand

---

a i r + S o 1 1 pattlcles a i r + s h o w $0

o -Data OF N-A . Soko lov

7J A - Data o f A A ( 0 l 8 f4 7J 10 -7I -1 I 0.0 I o. I a ~ r 05 ;.o Is dm Fig. 4 Fig.

5

Graph of t h e f u n c t i o n v; = f ( d ) f o r Diagram f o r t h e d e r i v a t i o n

t h e c a s e o f motion of heavy o f forinula ( 1 6 )

p a r t i c l e s i n a flow of a i r