g.c. sander; p.b. hairsine; c.w. rose; d. cassidy; j.-y. parlang -- unsteady soil erosion model

8/9/2019 G.C. Sander; P.B. Hairsine; C.W. Rose; D. Cassidy; J.-y. Parlang -- Unsteady Soil Erosion Model

1/17

ournal

of

ydrology

E L S E V I E R Jou r na l o f H ydr o l ogy 178 ( 1996) 351- 367

U nste ad y soil erosion mod el analytical solutions and

comparison with experimental results

G . C . S a n d e r a *, P . B . H a i r s i n e b , C . W . R o s e c , D . C a s s i d y a , J . -Y . P a r l a n g e d ,

W . L . H o g a r t h c , I .G . i s l e c

aFaculty o f Science and Technology Grif fi th University Nathan Qld . 4111 Australia

bCS IRO D ivision o f Soi ls P.O. Bo x 639 Canberra AC T Austral ia

CFaculty o f Environmental Sciences Grif fi th University Nathan Qld. 4111 Australia

dDepartmen t o f Agricultural and Biological Engineering Cornell University Ithaca N Y 14853 USA

Received 9 M arch 1995; accepted 31 M arch 1995

Abstract

H a i r s i n e a n d R o s e d e v e l o p e d a s o i l e r o s i o n m o d e l w h i c h d e s c r i b e d t h e e r o s io n t r a n s p o r t o f

t h e m u l t i p a r t ic l e s iz e s i n s e d i m e n t f o r r a i n - i m p a c t e d f l o w s i n t h e a b s e n c e o f e n t r a i n m e n t i n

o v e r l a n d f lo w . I n t h i s p a p e r w e e x t e n d t h e i r s te a d y - s t a t e s o l u t i o n s to a c c o u n t f o r t h e t i m e

v a r i a t i o n o f s u s p e n d e d s e d i m e n t c o n c e n t r a t i o n d u r i n g a n e r o s i o n e v e n t. A v e r y s im p l e a p p r o x i -

m a t e a n a l y t i c a l s o l u t i o n is f o u n d w h i c h a g r e e s e x t r e m e l y w e l l w i t h e x p e r i m e n t a l d a t a o b t a i n e d

f r o m n i n e e x p e r i m e n t s . W e a r e a b l e to r e p r o d u c e t h e r a p i d i n i ti a l i n c r e a s e t o a p e a k i n t h e t o t a l

s e d i m e n t c o n c e n t r a t i o n , w h i c h o c c u r s a b o u t 3 - 5 m i n a f t e r t h e c o m m e n c e m e n t o f ra i n f a ll , a s

w e l l a s t h e s u b s e q u e n t d e c l in i n g e x p o n e n t i a l t a il to w a r d s s t e a d y - s t a t e c o n d i t i o n s . W e a r e a l s o

a b l e t o s h o w t h a t t h e f r a c t i o n o f s h i e ld i n g o f t h e o r i g i n a l so i l b e d r e s u l ti n g f r o m d e p o s i t i n g

s e d i m e n t r e a c h e s i ts e q u i l ib r i u m v a l u e o n a b o u t t h e s a m e t i m e - s c a le a s t h e to t a l p e a k s u s p e n d e d

s e d i m e n t c o n c e n t r a t i o n . I n t e r e s t in g l y , w e f i n d t h a t t h e m a s s e s o f t h e i n d i v i d u a l p a r t i c le s w h i c h

f o r m t h is d e p o s i t e d l a y e r a r e f a r f r o m e q u i l i b r iu m , a n d t h a t t h e r e i s a g r e a t d e a l o f c o n t i n u o u s

r e w o r k i n g a n d s o r t i n g o f t h i s m a t e r i a l d u r i n g t h e e r o s i o n e v e n t . F i n a l l y , o u r s o l u t i o n s h o w s

t h a t t h e i n i t i a l p e a k i n t h e t o t a l s e d i m e n t c o n c e n t r a t i o n i s d u e t o t h e e n r i c h m e n t o f th i s s e d i m e n t

b y t h e f i n e r si ze c la s s e s a n d t h a t a s t h e e v e n t c o n t i n u e s t h e i r p e r c e n t a g e c o n t r i b u t i o n d i m i n i s h e s .

1 I n t r o d u c t i o n

I n a r e c e n t p a p e r b y H a i r s i n e a n d R o s e ( 1 99 1 ) a s o i l e r o s i o n m o d e l h a s b e e n

d e v e l o p e d t o d e s c r i b e t h e e r o s i o n a n d t r a n s p o r t o f s e d i m e n t i n r a i n - i m p a c t e d f lo w s

* Cor responding au thor .

0022-1694/96/ 15.00 1996 - Elsevier Science B.V. A ll r ights reserved

S S D I 0 0 2 2 - 1 6 9 4 ( 9 5 ) 0 2 8 1 0 - 2


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35 G.C. Sander et al . / Journal o f Hydrology 178 1996) 35 1- 36 7

in the absence of entrainment by shallow overland flow. Soils are naturally composed

of a range of particle sizes and aggregates, and in the past, soil erosion models have

tended to lump or average over all size classes, resulting in a single conservation

equation for the total suspended sediment concentration, c, as a function of distance

x and time t Govindaraju and Kavvas, 1991; Laguna and Giraldez, 1993). In

contrast, the approach o f Hairsine and Rose 1991) is to consider the contributions

of the individual size classes to the total suspended sediment concentration separately.

This leads to separate coupled conservation equations for the suspended sediment

concentration c i x , t ) for each particle size class i. While this does increase the number

of equations and complicates their solution, there are sound physical grounds for

justifying this approach.

Once particles have become suspended, they will naturally begin to settle back

towards the soil surface owing to their immersed weight. The fine particles, having

relatively small settling velocities, can be carried significant distances in the direction

of the surface water flow before depositing back to the soil surface. The larger

particles however, will return very quickly to the soil surface from suspension and

would make their way to the end of the plane by saltation. Therefore, at the beginning

of an erosion event, most of the contribution to the total suspended sediment

concentration c will be coming from the finer particles, but, as time increases and

most o f the finer material has been removed, the larger particles will begin to increase

their contribution to c.

Prediction of size distributions of eroded particles then enables estimates to be

made of quantities of sorbed pollutants, e.g. Palis et al., 1990). An understanding

of the dynamics of the suspended sediment concentrations ci of the individual

particles, as well as the overall total concentration c, therefore has significant

implications on the understanding of the supply of non-point source pollutants to

waterways.

In this paper we will only be considering erosion from situations where raindrop

impact is the only erosive agent. Under these conditions the role of overland flow is

simply to transport the sediment eroded under the action of raindrops. The fluid flow

velocity is low enough so that no sediment is entrained into the fluid by the shear

stresses acting between the soil surface and the fluid. This allows us to compare the

predictions of the model with the experimental results of Proffitt et al. 1991) and

considerably simplifies the mathematical solutions. As the development of the model

is discussed in detail in Hairsine and Rose 1991) and Hairsine et al. 1995) we will

only give a brief description o f it here.

2.

heory

Once sediment has become suspended in the overland flow, this sediment will begin

to settle or deposit back towards the soil surface. The rate o f deposition di is directly

dependent on the size of the sediment and can vary by orders of magnitude from fine

sediment to large aggregates. The deposition rate d i of a given sediment size class i is


3/17


4/17

54 G.C. Sander et al. / Journal of Hydrology 178 1996) 351-367

R t ) , c o n s e r v a t i o n o f m a s s o f s e d i m e n t o f siz e c la s s i r e q u i re s t h a t

O D c i ) O q c i )

~ - - - - e i + e d i - - d i i = 1 , 2 , . . . , I

(6)

O t a x

w h e r e

D x , t )

a n d

q x , t )

a r e t h e d e p t h o f f lo w a n d t h e v o l u m e t r i c f lu x p e r u n i t w i d t h

o f p l a n e , r e s p e c ti v e ly . I n t h e c a s e o f s h a l lo w k i n e m a t i c o v e r l a n d f l o w , D a n d q a r e

r e l a t e d b y

01)0t ~- -~xOq= R t )

(7)

a n d

q = K D m (8)

w h e r e K a n d m a r e c o n s t a n t s d e p e n d i n g o n t h e fl o w r e g i m e a n d

R t )

i s t h e r a i n f a l l

e x c e s s . I n p a r t i c u l a r K = s l / 2 / n w h e r e s is t h e s u r f a c e s l o p e , n i s a m e a s u r e o f t h e

s u r f a c e r o u g h n e s s a n d m i s a p p r o x i m a t e l y 5 / 3 f o r t u r b u l e n t f l o w a n d 3 f o r l a m i n a r

f lo w . W i t h D a n d q k n o w n E q s . ( 6 ) a n d ( 7) c a n a l s o b e c o m b i n e d t o g iv e

O c i q O c i 1

O t + D O x - 1 ) e i + e d i - d i - R c i ) i = 1 , 2 , . . ., I (9)

F i n a l l y , w e n e e d a c o n s e r v a t i o n e q u a t i o n w h i c h g o v e r n s t h e v a r i a t i o n o f

M d i

i n t h e

d e p o s i t e d l a y e r . T h e o n l y p r o c e s s e s w h i c h e f f e c t t h e d e p o s i t e d l a y e r a r e

d i

an d edi, t h u s

t h e c h a n g e i n m a s s M d i i n t h e d e p o s i te d l a y e r f r o m t h e c o m m e n c e m e n t o f ra i nf a ll

d e t a c h m e n t is s im p l y t h e d i f f e re n c e b e t w e e n d i a n d e d i o r

OMdi

- - = d i - e d i i -- - - 1 , 2 , . . . , I ( 1 0)

O t

T h e r e f o r e E q s . ( 9 ) a n d ( 10 ) f o r m a s y s t e m o f N N - - 2 1 ) p a r t i a l d i f f e r e n t i a l e q u a t i o n s

f o r d e t e r m i n i n g th e N u n k n o w n s c i x , t ) a n d M d i X , t ) .

3 Approximate analytical solution

I n t h is s e c t i o n w e s e e k a n a p p r o x i m a t e a n a l y t ic a l s o l u t i o n t o E q s . ( 9) a n d ( 10 ) w i t h

a v i e w t o r e p r o d u c i n g t h e e x p e r i m e n t a l d a t a o f P r o f f i t t e t a l. ( 19 9 1) . T h e i r e x p e r i m e n t s

w e r e p e r f o r m e d f o r c o n s t a n t r a i n fa l l r a te s P , l o w s l o p es ( ~< 1 % ) , t h r e e s h a l lo w m e a n

f lo w d e p t h s o f 2 , 5 a n d 10 m m , a n d n o l o s s e s o f r a i n fa l l b y i n fi lt r a ti o n , h e n c e

R t )

= P = c o n s t a n t . F o r P r o f f i tt e t a l. (1 9 9 1 ) t o b e a b l e t o m a i n t a i n t h e s e m e a n

f l o w d e p t h s f o r t/ > 0 t h r o u g h o u t t h e i r e x p e r i m e n t s . . . a b a r r i e r w a s t e m p o r a r i l y

i n s ta l le d a t t h e e n d o f th e f l u m e a n d c l e a r w a t e r i n t r o d u c e d g e n t l y s o t h a t t h e r e w a s n o

t u r b u l e n c e . O n c e t h e c o r r e c t m e a n d e p t h o f p o n d i n g h a d b e e n a c h i e v e d , r a i n fa l l w a s

s t a r t e d a n d t h e b a r r i e r r e m o v e d s i m u l t a n e o u s l y . T r a n s i e n t u n s t e a d y [ w a te r] fl o w

o c c u r r i n g p r i o r t o a c h i e v in g a s t e a d y - s t a t e a p p e a r e d t o b e n e g l ig ib l e a f te r a p p r o x i -

m a t e l y o n e m i n u t e . I f w e a r e p r e p a r e d t o i g n o r e th e in i ti a l t r a n s i e n t h y d r o l o g i c a l

e f f e c t s o n c i , w e c a n t h e r e f o r e s i m p l i f y E q . ( 9 ) b y r e p l a c i n g D a n d q w i t h t h e i r

a v e r a g e d q u a n t it i e s w h i c h a re i n d e p e n d e n t o f x a n d t, w i t h o u t i n t r o d u c i n g a n y


5/17

G . C . S a n d e r e t a l. / J o u r n a l o f H y d r o l o g y 1 7 8 ( 1 9 9 6 ) 3 5 1 - 3 6 7

355

s i g n i f ic a n t e r r o r s . S o , f o r t h e e x p e r i m e n t a l c o n d i t i o n s o f P r o f f i t t e t a l. 1 9 9 1 ) t h e

f o l l o w i n g d i m e n s i o n l e s s v a r i a b l e s c a n b e d e f i n e d

P t

T = -= - l l a )

D

x

z = - 1 l b )

L

c i

C = - - l l c )

a

M d i 1 l d )

m d i - ~ aD

I

gd t

E m d i

l l e )

m d r a y e )

i=1

a n d d i m e n s i o n l e s s p a r a m e t e r s

v i

ui = ~ 1 l f )

e = - - l l g )

P L

a n d

adZ)

o~ -- - - 1 l h )

a d

= - - 1 l i )

a

w h e r e L i s l e n g t h o f t h e f l o w d o m a i n . C o m b i n i n g E q s . 1 ), 2 ) , 4 ), 5 ), 9 ), 1 0 ) a n d

1 1 ) g i v e s

OCi~_eOC

1 a )

Or Oz = I 1 - ~ m d ~ + a m d i - - l + u i )C i i = 1 , 2 , . . . , I 1 2 )

a n d

O m d i

O F ~ - v i C i - - a m d i i =

1 , 2 , . . . , I 1 3 )

E v e n t h o u g h E q s . 1 2 ) a n d 1 3 ) f o r m a s y s t e m o f l in e a r p a r t ia l d i ff e r e n ti a l e q u a t i o n s

t h e r e i s n o e x a c t a n a l y t i c a l s o l u t i o n a v a i l a b l e . C o n s e q u e n t l y w e s e e k a si m p l if ie d

a p p r o x i m a t e a n a l y t i c a l s o l u t i o n w h i c h w i ll s ti ll c a p t u r e t h e u n d e r l y i n g p h y s i c s o f

t h e e r o s i o n p r o c e s s e s . E x p e r i m e n t a l d a t a f ie ld o r l a b o r a t o r y ) a r e ty p i c a l l y c o l l ec t e d

a t th e e n d o f t h e fi el d s l o p e e .g . H u d s o n 1 9 8 1 ), o r a t t h e e n d o f t h e t i m e , e .g . P r o f f it t

e t a l. 1 9 9 1 ). S o i n o r d e r t o c o m p a r e w i t h t h is d a t a w e o n l y n e e d s o l u t i o n s o f E q s . 1 2 )

a n d 1 3 ) a t z = L . F o r l o w s lo p e s a n d l o w w a t e r f lo w v e lo c i ti e s i t i s r e a s o n a b l e t o


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3 5 6 G.C. Sander e t aL / Journal of Hydrology 178 1996) 35 1-3 67

a s s u m e t h a t t h e se d i m e n t c o n c e n t r a t io n d o e s n o t v a r y g r e a tl y w i t h z n e a r z = L ,

c e r t a i n l y f o r e a r l y t i m e s i n a n e r o s i o n e v e n t t h e g r e a t e s t c h a n g e s i n C i n e a r z = L

a r e t i m e d e p e n d e n t . N o w i f w e a s s u m e t h a t t h i s is t r u e f o r a l l t i m e i. e.

OC i /t~ F

>>

E(OCi /OZ)z= l . ,

t h e n t h e s o l u t io n f o r

C i L , 7 -)

c a n b e a p p r o x i m a t e d t h r o u g h

t h e m u c h s i m p l er s e t o f l in e a r a u t o n o m o u s o r d i n a r y d i f f e re n t ia l e q u a t i o n s

d r = 1 - - ~ m d . r + a m d i - - l + v i ) C i i = 1 , 2 , . . . , I 1 4 )

a n d

d m d i

d = l / i C i - a m d i i = 1 , 2 , . . . , I 1 5)

s u b j e c t t o t h e i n i t i a l c o n d i t i o n s

7- = 0, C i -~ 0 , m d i = 0 i = 1 ,2 , . . . , I 16 )

E q s . 1 4 ) a n d 1 5 ) a r e n o w e a s i ly s o l v e d b y s t a n d a r d t e c h n i q u e s . F i r s t w r i t e E q s . 1 4)

a n d 1 5) i n v e c t o r n o t a t i o n a s

d w

d r A w + b 17)

w h e r e

W = C l , C 2 , . . . ,

e l ,

m d l ,

md2,

. . . , r o d / ) T

18)

19)

} l l ) T

b = , j , . . . ? , o , o , . . . , a o

a n d A is a b l o c k N x N ) s q u a r e m a t r i x

[ D I S ] 2 0)

A = D2 D3

T h e m a t r i c e s D 1, D E a n d D 3 a r e d i a g o n a l I x I ) m a t r ic e s a n d S i s a s y m m e t r i c I x I )

m a t r i x . D 1 h a s d i a g o n a l e l e m e n t s - 1

+ v i ) , i = 1 , 2 , . . . , I , 1 )2

h a s d i a g o n a l e l e m e n t s

v i , i = 1 , 2 , . . ., I , a n d D 3 h a s d i a g o n a l e l e m e n t s a , i = 1 , 2, .. ., I . T h e m a t r i x S i s f u ll ,

w i t h d i a g o n a l e l e m e n t s [ a - a / t ~ I ) ] , i = 1 , 2 , .. . , I a n d a ll th e o f f - d i a g o n a l e l e m e n t s

a r e g i v e n b y

- a / t ~ I ) .

T h e s o l u t i o n o f E q . 1 7 ) i s t h e n g i v e n b y

W ~l~.gl e x p -- A lr ) + ~2~Lg2 xp -- A 2r ) + ... +

~ N ~ I N

exp - -A N r) + Ws 21)

w h e re A j an d t ~ j , j = 1 , 2 , .. ., N , a r e t h e e i g en v a l u e s an d e i g en v ec t o r s o f A r e s p ec t i v e l y .

T h e r / j ,j = 1 , 2 , ..., N a r e t h e c o n s t a n t s o f in t e g r a t i o n f o u n d f r o m s a t i s f y i n g t h e i n it i a l

co n d i t i o n r = 0 , w = 0 , C i = 0 an d m a i = 0 , o r f r o m s o l v i n g

E n = - w s 2 2 )

w h e r e E i s t h e m a t r i x o f e ig e n v e c to r s t~j w i t h t h e e i g e n v e e to r t~ j f o r m i n g c o l u n m j , a n d

Ws i s t h e s t e ad y - s t a t e s o l u t i o n o f Eq . 1 7) .


7/17

G.C. Sander et aL / J ou r n a l o f H y dr o l o gy 178 1996) 351- 367

3 5 7

4 . S t e a d y - s t a t e o r e q u i li b r iu m s o l u t i o n

T h e s te a d y - s ta t e o r e q u i l ib r i u m s o l u t i o n o f E q . 1 7 ) is g i v e n w h e n b o t h d C i / d r a n d

dmd i / d r

a r e z e r o o r

l i 1 - ~ m d ~ ) +

amdi - -

1 +

v i ) C i

= 0 2 3 )

v i - otmdi

= 0 24)

t h e s o l u t io n o f w h i c h i s g i v e n b y H a i r s i n e a n d R o s e 1 9 9 1 ) a s

C i

= 1 + i = 1 , 2 , . . . , I 2 5 )

a n d

V i j . = ~ l ~ ) ] - - 1

m d ~ = - 1 + i = 1 , 2 , . . . , I 2 6 )

W e n o t e a t e q u il ib r i u m

C i

s in d e p e n d e n t o f i, t h a t i s,

C 1 = C 2 . . . =

b u t

md i

d o e s

v a r y w i t h i .

5 . C o m p a r i s o n w i t h t h e e x p e r i m e n t a l d a t a o f P r o f l i t t e t a l . 1 9 9 1 )

T h e e x p e r i m e n t s p e r f o r m e d b y P r o f fi tt e t a l. 1 9 9 1 ) w e r e o n t w o b a r e s o il s o f

4 0 . 0 0

30 00

20 00

d

c -

O

. . ) 1 0 0 0

O

o 0 . 0 o o 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

o l o o o 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

T i m e m i n s )

F i g . 1 . S e d i m e n t c o n c e n t r a t i o n f o r t h e S o l o n c h a k a s a f u n c t i o n o f t i m e w h e n P = 1 0 0 m m h - 1 a n d

D = 2 n u n . P a r a m e t e r v a l u e s a r e a = 1 0 0 0 , ~ = 2 0 a n d a = 1 2 3 3. E x p e r i m e n t a l d a t a p o i n t s g i v e n b y t h e

stars


8/17

358

G.C. Sander et al. /Journ al of Hydrology 178 1996) 351-367

E

x 3 0 . 0 0

~'~20.00

6

c-

1 0 . 0 0

0

L D

~D

- 4 . - -

0

F

0 . 0 0 i 1 , , 1 , , 1 i 1 , 1 1 , , , , 1 i i , i i i i i i i i i i i i i i i i i i , , , i i i i I

o o o l o o o 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

Time mins)

Fig. 2. Sediment concentration for the Solonchak as a function of time when P = 100 mm h -1 and D =

5 nun. Parameter values are a = 1200, ~ = 20 and a = 718. Experimental data points given by the stars.

different type (a vertisol referred to as Black Earth and an aridisol referred to as

Solonchak ) at slopes between 0.1 and 1.0 . Both sediment conc entr ati on and

settl ing velocity characteristics were measured through time under two different

cons tant rainfall rates of 56 and 100 mm h -1 . The experiments were also performed

for three different average constant depths of water of 2, 5 and 10 ram.

Figs. 1-3 give the experimental

c t )

(stars) for the Solonch ak for depths of 2 mm,

E

1 0 0 0

t2n

5.o0

t-

O

0

b -

0 . 0 0

I ; i , l i , , , l , 1 , , , l , , , l , , , , , , , , , l , , a , , , , , i l

o o o l o o o 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0

T i m e m i n s )

F i g . 3 . Sediment concentration for the Solonchak a s a function of time when P = 100 nun h -1 and D =

I 0 m m . P a r a m e t e r v a l u e s a r e c~ = 1 6 0 0 ~ = 2 0 a n d a = 4 1 2 . E x p e r i m e n t a l d a t a p o i n t s g i v e n b y t h e s t a rs .


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G.C. Sander et al . / Journal o f Hydrology 178 1996) 35 1-3 67

359

5 m m a n d 1 0 m m , r e s p e c t iv e l y , w h e n P = 1 00 m m h - 1 . T h e s o l id li ne s i n Fi g s. 1 - 3 a r e

t h e p r e d i c t e d s o l u t i o n s o b t a i n e d f r o m E q s . ( 2 1) a n d ( 22 ) f o r te n s i z e c l as s e s, i. e.

I = 1 0. T h e c o r r e s p o n d i n g f a ll v e l o c i ti e s

v i ,

i = 1 , 2 , . .. , 1 0 u s e d , w e r e o b t a i n e d f r o m

t h e s o i l cu r v e i n F i g . 2 o f P r o f f i t t e t a l. ( 1 9 9 1 ) l ab e l l ed o r i g i n a l s o i l ( w e t t ed p l u s r a i n f a l l

a t 1 00 m m h - l f o r 4 0 m i n ). T h e t e n v e l o ci ti e s w e r e t h e n s i m p l y r e a d o f f a t e q u a l

i n t e r v a l s a l o n g t h e i r y - a x is . T h e f i n e s t s e d i m e n t c l a s s s iz e i s g iv e n b y i = 1 ( s m a l l e s t

v e l o c i t y ) w i t h t h e l a r g e s t c l a s s s iz e c o r r e s p o n d i n g t o i = 1 0 ( l a r g e s t v e l o c i ty ) . T h e

p a r a m e t e r v a l u e s o f c~, ~; a n d a a r e o b t a i n e d b y m a t c h i n g t h e p r e d i c t e d c o n c e n t r a t i o n

c u r v e t o t h e e x p e r i m e n t a l d a t a p o i n t s .

T h e l ev e l o f a g r e e m e n t b e t w e e n t h e e x p e r i m e n t a l d a t a a n d o u r p r e d i c t e d c u rv e s

is e x c e ll e n t, w e l l w i t h i n t h e e x p e r i m e n t a l e r r o r o f + 1 5 . T h e m o s t i n t e r e s ti n g

f e a t u r e o f t h e s o l u t io n s i s th e v e r y r a p i d i n c r e a se i n th e c o n c e n t r a t i o n f r o m 0 a t

t = 0 , to i ts p e a k c o n c e n t r a t i o n w h i c h o c c u r s a r o u n d 1 r a in , f o l l o w e d b y a c o n -

t i n u o u s d e c l i n e t o w a r d s i t s s t e a d y - s t a t e o r e q u i l i b r i u m c o n c e n t r a t i o n . T h e p e a k

c o n c e n t r a t i o n p r e d i c t e d b y t h e t h e o r y is n o t a b s o l u t e i n th a t b y m a k i n g s li g ht

v a r i a t i o n s t o t h e v a l u e s o f a , n a n d a , w e c a n e i t h e r i n c r e a se o r d e c r e a s e t h e

p e a k p r e d i c t e d c o n c e n t r a t i o n w h i l e st ill m a i n t a i n i n g t h e e x c e l l e n t a g r e e m e n t w i t h

t h e e x p e r i m e n t a l d a t a . T h i s i s d i s c u s s e d i n m o r e d e t a i l i n a l a t e r s e c t i o n . W h a t i s

n e e d e d i s t o o b t a i n m o r e m e a s u r e d s e d i m e n t c o n c e n t r a t i o n s a t t i m e s e a r l i e r t h a n

1 .5 m i n i n o r d e r t o f ix t h e p e a k c o n c e n t r a t i o n . I t w a s a l s o f o u n d t h a t w h e n P = 5 6

m m h - ] e x a c t ly th e s a m e le v el o f a g r e e m e n t w a s o b t a i n e d b e t w e e n t h e o r y a n d

e x p e r i m e n t a l d a t a a s t h a t d i s p la y e d f o r P = 1 00 m m h - ] . N o t e f r o m n o w o n t h e

b a r h a s b e e n d r o p p e d f r o m t h e d e p t h D t h o u g h r e f e re n c e s t o D w i l l s till i m p l y

a v e r a g e d d e p t h s .

F i g . 4 s h o w s t h e d e v e l o p m e n t o f th e d e p o s i t e d l a y e r d u r i n g t h e e r o s i o n e v e n t. W h i l e

t h e s e d i m e n t c o n c e n t r a t i o n t a k e s q u i t e a w h i l e t o o b t a i n e q u i l i b r i u m , t h e l e v e l o f

s h i e l d i n g o b t a i n e d b y t h e d e p o s i t e d l a y e r a t t a i n s e q u i l i b r i u m v e r y r a p i d l y . T h e

e q u i l i b r i u m sh i e ld i n g v a l u e o f a p p r o x i m a t e l y 9 8 s e em s t o b e a c h i e v e d o n th e

1 0 0

0.75

~- 0.50

0.25

0.00 I I J l l l l l l lS l l l J l l l l I l j L l~ l l l l *~h j l l j l l j l l l l l

0.00 10.00 20.00 30.00 40.00 50.00

TIME mlns)

Fig. 4. Developmentof the fractionalcoverageH as a function of time for P = 100 mm h -I and D = ram.


10/17

360

G.C. Sander e t al . Journal o f Hydrology 178 1996) 35 1-3 67

0 030

E

w.

E

.2

- 0 . 0 2 0

/ 3

5

0 .010

~J3

. > a, confirming the greater detachability of the deposited layer over the original

T a ~ e l

L i s to f pa ra m e te rv a lu e s ~ v i n g b e s t a ~ m e n t w i t h t h e e x p e f i m e n t ~ d a t a

S o l o n c h a k B l a c k E a r t h

P D a ad M~t P D a a d M~t

m m h - i ) m m ) k g m - 3) k g r n - 3) k g m - 2) m m h - 1) m m ) k g r n - 3) k g m - 3) k g m -E )

56 2 1113 2226 0 0.074 56

5 358 7160 0.18

10 319 6380 0.071

100 2 1233 24 660 0.05 100

5 718 14360 0.06

10 412 8240 0.051

2 3910 7429 0.15

2 3738 7102 0.18

5 1950 3705 0.23


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36 G.C. Sander e t al . / Journal o f Hydrology 178 1996 ) 35 1-3 67

s o il . B e in g a w a r e o f th i s b e h a v i o u r h e l p s r e s tr i c t t h e r a n g e o f p o s s ib l e p a r a m e t e r

v a l u e s a a n d t~ w h i c h y i e ld g o o d a g r e e m e n t w i t h t h e e x p e r i m e n t a l c t ) d a t a .

W e n o t e t h a t t h e v a l u e s o f t h e d e t a c h a b i l i t y a f o u n d i n T a b l e 1 a r e m u c h h i g h e r

t h a n t h o s e r e p o r t e d e x p e r i m e n t a l l y in P r o f f i t t e t a l. ( 1 99 1 ). T h i s d i f fe r e n c e a r is e s

b e c a u s e t h e e x p e r i m e n t a l v a l u e s o f a w e r e b a s e d o n a n e q u i l i b r i u m v a l u e o f

H = 0 .9 o b t a i n e d b y v is u a l o b s e r v a t i o n . I n c o n t r a s t F i g. 4 s h o w s t h a t H s h o u l d b e

m u c h c l o s e r t o o n e , o r t o b e p r e c is e 0 .9 8 . S i n c e t h e d e t a c h a b i l i t y in c r e a s e s v e r y r a p i d l y

a s a f u n c t i o n o f H f o r H n e a r o n e ( s e e F i g . 4 o f P r o f f i t t e t a l. , 1 99 I ) , t h e n s m a l l e r r o r s

i n e s ti m a t i n g H n e a r o n e a r e m a g n i fi e d , a n d c a u s e v e r y la r g e e r r o r s in d e t e r m i n i n g a .

T h u s i t is n o t r e a ll y a p p r o p r i a t e t o c o m p a r e t h e e x p e r i m e n t a l a n d a n a l y t i c a l v a l u e s o f

a e x c e p t t o n o t e t h a t t h e s ig n o f t h e d i f fe r e n c e is c o r r e c t . O n t h e o t h e r h a n d , a d

d e p e n d s o n l y m i l d ly o n H f o r H n e a r o n e a n d w e fi nd g o o d a g r e e m e n t b e t w e e n th e

e x p e r i m e n t a l a n d a n a l y t i c a l v a l u e s . F i g . 4 o f P r o f i t t e t a l. ( 1 9 9 1) g iv e s e x p e r i m e n t a l

v a l u e s o f a d f o r D = 2 , 5 a n d 1 0 m m o f th e o r d e r o f 2 3 0 0 0 k g m - 3 , 1 2 0 0 0 k g m - a a n d

7 5 0 0 k g m - 3 , r e s p e c t iv e l y , w h i c h a l l a g r e e w i t h t h e v a l u e s r e p o r t e d i n T a b l e 1.

H a i r s i n e a n d R o s e ( 1 9 9 1 ) d e v e l o p e d t h e o r y f o r t h e g e n e r a l c a s e w h e r e D i s a

f u n c t i o n o f d i s ta n c e x a n d t h e d e t a c h a b i li ti e s a r e f u n c t i o n s o f d e p t h . T h e y c o n s i d e r e d

e x p r e s s io n s f o r a a n d a d o f th e f o r m

( 2 7 a )

= a o D D o

a = a o D / D o ) - b D > . D o

a d = ado D < D o

a s = a d o Z ) / n o ) - b D >1 D o

a n d

( 2 7 b )

28a)

( 2 8 b )

I n E q s . ( 27 ) a n d ( 28 ) D o is u su a l ly t a k e n t o b e o f t h e o r d e r o f t w o t o t h r e e r a i n d r o p

d i a m e t e r s b e i n g a b o u t 2 m m f o r t h e e x p e r i m e n t s o f P r o ff i tt e t a l. ( 1 99 1 ). U s i n g t h e

d a t a a v a i la b l e in T a b l e 1 w e c a n e s t i m a t e t h e p a r a m e t e r s i n E q . ( 27 ) b y p e r f o r m i n g a

l e a st s q u a r e s c u r v e f it . T h e p a r a m e t e r a do in E q . ( 28 ) th e n f o ll o w s a u t o m a t i c a l l y f r o m

Eq s . (1 l i ), ( 2 7 ) an d ( 2 8 ) f o r

D >~ D o

a / a d = a o / a d o = n (29)

F o r t h e p u r p o s e s o f t h e l e a s t s q u a r e s f i t E q . ( 2 7 ) c a n b e s im p l i f ie d t o a = w D - b w h e r e

w = aoDbo a n d r e s u l t s i n

a = 1 7 3 0 D - 8 P = 5 6 m m h - 1 ( 3 0 a )

a n d

a = 2 0 1 7 D - 7 P = 1 00 m m h - 1 ( 3 0 b )

T a k i n g D o a s a p p r o x i m a t e l y 2 m m g iv es

ao

= 9 9 4 k g m - 3 ,

ado

= 1 9 8 8 0 k g m - 3 f o r

P = 5 6 m m h - 1 a n d

ao

= 1 2 4 2 k g m - 3 ,

ado

= 2 4 8 4 0 k g m - 3 f o r P = 1 00 m m h - 1 . T h e

a g r e e m e n t b e t w e e n E q . ( 5 .1 ) a n d t h e d a t a i n T a b l e 1 , i s s h o w n i n F i g . 7 . T h e r e s u l ts f o r

P = 1 0 0 a r e e x c e l le n t , h o w e v e r t h o s e f o r P = 5 6 a r e n o t a s g o o d . T h i s i s d u e m a i n l y t o


13/17

G.C. San~re tal. /JournalofHy~olo gy178 1996) 351-367

1400.0

363

1200.0

1 0 0 0 0

8 0 0 0

25

6 0 0 0

Q

4 0 0 0

2 0 0 0

A

Q

A

=

0 O 0 0 2 0 0 1

q q i T ~ i t i i ~ i ~ L ~ i i i i ~ i i i i i i i i i i i J i

4.00 6.00 8.00 10.00

Depth mm)

F i g . 7 . D e t a c h a b i l i t y a s a f u n c t i o n o f d e p t h f o r d i f f e re n t r a i n fa l l ra t e s . S o l i d l i ne s a r e f ro m E q . 2 7 ) b a s e d o n

a l e a s t s q u a r e s r e g r e s s i o n f i t t o t h e d a t a p o i n t s s h o w n i n T a b l e 1 .

the data point at D = 5 for P = 56 being too low. We also note that the values of b

derived from the least squares procedure are in general agreement with those found by

Proffitt et al. 1991).

Table 1 also gives the variation of M~t with D for various rainfall rates. These are

the least favourable results, since M~t represents the mass per unit area kg m -2) of

the deposited layer which provides complete shielding, these values appear to be lower

than physically expected. There also does not appear to be any uniform relation

between M~t and depth. This could mean that the definition of

H t )

given by Eq.

7) is not quite correct and needs further development. However all other

comparisons between theory and experiment i.e. Figs. 1-6) are very encouraging

and lead us to believe that considering the individual class sizes as opposed to just a

total concentration per se is more useful in explaining soil erosion behaviour.

Certainly it is not possible to predict the sudden peak concentration of Figs. 1-3

by using less than three particle sizes I = 3). Even then these three size classes must

cover several orders of magnitude in their settling velocities vi or the peak will still not

appear.

5 .1 . Curve f i t t ing to exper imen ta l c t) d a t a

To curve fit the experimental data and determine the parameter values for a, a d and

M~tt we used the solution in dimensionless form as given by Eqs. 21) and 22) since


14/17

364

G.C. Sander et al . / Journal of H ydrology 178 (1996) 351-367

o n l y tw o u n k n o w n p a r a m e t e r s a a n d n a p p e a r r a t h e r t h a n t h re e . T h e s e tt li ng v e l o ci ty

d i s t r i b u t i o n v i i s o b t a i n e d d i r e c t l y f r o m s e t t li n g t u b e e x p e r i m e n t a l s ( a s i n F i g . 2 o f

P r o f f it t e t a l. , 1 99 1) o n c e t h e n u m b e r o f si ze c l a s s e s / h a s b e e n s e t. F o r a g i v e n r a in f a ll

r a t e e x p e r i m e n t t h e

v i s

a r e t h e n d e t e r m i n e d f r o m E q . (1 I f ) a s

v i = v i / P ,

i = 1 ,2 , . . ., I ,

w i t h

C i ( r )

a n d m d i(7 - t h e n c a l c u l a t e d f r o m E q s . ( 2 1) a n d ( 2 2) f o r a p r e s e t a a n d ~ .

T h e t o t a l d im e n s i o n le s s c o n c e n t r a t i o n C ( r ) is th e n f o u n d b y s u m m i n g t h e C i ( r ) f o r

i = 1 , 2 , . . . , I .

T o f i n d t h e d e t a c h a b i l i t y a w e m a t c h t h e e x p e r i m e n t a l s e d i m e n t c o n c e n t r a t i o n a t

t = 4 0 m i n , c e ( 4 0 ), w i t h t h e t h e o r e t i c a l s e d i m e n t c o n c e n t r a t i o n C a t 7- = 4 0 P / D a n d

t h e n u s e E q . ( 1 l c ) t o g i v e a =

c e ( 4 0 ) / C ( 4 O P / D ) .

T h e t i m e o f 4 0 m i n w a s c h o s e n s i nc e

t h is w a s t h e l a r g e s t ti m e f o r w h i c h e x p e r i m e n t a l c d a t a w e r e a v a i la b l e a n d a s s u c h t h is

d a t a p o i n t w o u l d b e t h e m o s t r e l i ab l e o w i n g t o a m i n i m u m i n fl u e n c e o f tr a n s i e n t fl o w

e f fe c ts . A c o n s e q u e n c e o f t h is m e t h o d f o r f i n d in g t h e d e t a c h a b i l i ty is t h a t t h e

t h e o r e t ic a l s o l u t io n w ill a lw a y s g o t h r o u g h t h e f in a l e x p e r i m e n t a l d a t a p o i n t . H a v i n g

a , t h e e n t i r e t h e o r e t i c a l c ( t ) s o l u t i o n c a n b e p l o t t e d a n d c o m p a r e d w i t h a l l t h e

e x p e r i m e n t a l d a t a . I f t h e le v e l o f a g r e e m e n t i s u n s a t i s f a c t o r y t h e n t h e w h o l e p r o c e s s

i s r e p e a t e d w i t h n e w v a l u e s o f a a n d n u n t i l t h e b e s t f it i s f o u n d . W i t h t h e f in a l v a l u e s

o f a , ~; an d a , Eq . (1 l i ) g i v e s a d an d Eq . (1 l h ) g i v e s M ~ t .

T h e e f f e ct o f t h e p a r a m e t e r s a a n d ~ o n t h e c u r v e f it ti n g o f t h e s o l u t i o n is b e st s e e n

t h r o u g h t h e ir in f lu e n c e o n t h e m a g n i t u d e a n d t im i n g o f t h e p e a k s e d im e n t c o n c e n t r a -

t io n . D e f i n e Cmax a s t h e p e a k o r m a x i m u m c o n c e n t r a t i o n r e a c h e d a n d tm ax a s t h e t i m e

a t w h i c h t h i s p e a k o c c u r s . I f w e r e r u n t h e m o d e l s e v e r a l t im e s d u r i n g w h i c h w e k e e p

f i x e d w h i l e a is d e c r e a s e d f o r e a c h a d d i t i o n a l r u n , t h e n w e f in d t h a t b o t h tm ax a n d Cmax

i n c r e a s e , t h a t i s t h e p e a k c o n c e n t r a t i o n g e t s l a r g e r a n d i s d e l a y e d , w h i l e th e a s s o c i a t e d

v a l u e s o f a a n d a d d e c r e a s e w i th M ~ t in c r e a si n g . T h i s o c c u r s b e c a u s e i f t h e d e t a c h -

a b i li ti e s a a n d a d d e c r e a s e , t h e n t o m a i n t a i n t h e s u p p l y o f so i l p a r t i c l e s i n o r d e r t o

m a t c h t h e m e a s u r e d e x p e r i m e n t a l s e d i m e n t c o n c e n t r a t i o n s , t h e f r a c t i o n H o f s h ie ld -

i n g m u s t d e c r e a se , I n p r a c t i c e , t h is o c c u r s t h r o u g h i n c r e a s in g nit i n E q . ( 5) so t h a t

t h e p r o d u c t a ( 1 - H ) , b e i n g t h e s o u r c e o f d e t a c h e d m a t e r i a l, r e m a i n s c o n s t a n t . I n

t e r m s o f t h e d i s t r i b u t i o n o f t h e s iz e c l a ss e s in t h e e r o d e d s e d i m e n t , a f i x e d ~; w i t h a

d e c r e a s i n g a w i l l d e c r e a s e t h e c o n t r i b u t i o n f r o m t h e l a r g e r p a r t i c l e s .

F i x i n g t h e v a l u e o f a a n d l e t ti n g ~; d e c r e a s e f o r e a c h r u n , r e s u l ts i n a n i n c r e a s e d

p e a k c o n c e n t r a t i o n w h i c h o c c u r s a t a n e a r li er ti m e . H o w e v e r

~t

an d ad w i l l

r e m a i n e d f i x e d w h i le t h e d e t a c h a b i l i t y a i n cr e a se s . F u r t h e r m o r e , t h e t o ta l m a s s o f

s u s p e n d e d s e d i m e n t i n c re a s e s, b u t t h e r e is a te n d e n c y t o w a r d s a r e d u c t i o n i n t h e

p e r c e n t a g e o f s m a l l e r p a rt ic l e s c o n t r i b u t i n g t o t h is s e d i m e n t . T a k i n g t h e t w o p a r a -

m e t e r s i n c o m b i n a t i o n , t h e n Cmax c a n b e r e d u c e d b u t k e p t a t t h e s a m e t i m e t ma x b y

i n c r e a s i n g b o t h a a n d ~;, o r c o n v e r s e l y , t h e p e a k c o n c e n t r a t i o n c a n b e h e l d c o n s t a n t

a n d b e m a d e t o o c c u r e a rl ie r b y in c r e a s i n g a a n d d e c r e a s i n g ~;, o r t o o c c u r l a t e r b y

d e c r e a s i n g a a n d i n c r e a s i n g n.

W e a l s o n o t e t h a t w h e n n i s f ix e d a n d w e v a r y a , t h e d e t a c h a b i l i ty o n l y v ar ie s

s l i g h t l y . T h i s o c c u r s b e c a u s e w e c h o s e t o f i n d t h e d e t a c h a b i l i t y b y m a t c h i n g t h e

t h e o r e t i c a l C t o c e (4 0 ) w h i c h is n e a r t o t h e s t e a d y - s t a t e c o n c e n t r a t i o n . I f w e w e r e

a b l e t o u s e t h e t r u e s t e a d y - s t a te c o n c e n t r a t i o n t o f i n d a b y a =

C e (t ~ o o ) /

C ( r ~ c ~ ) t h e n a w o u l d r e m a i n i n d e p e n d e n t o f a a s E q . (2 5 ) s h o w s t h a t


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G.C. Sander e t aL / Journal of Hydrology 178 1996) 35 1-3 67

3 6 5

C r ~ o o )

is i n d e p e n d e n t o f a . T h u s t h e s m a l l v a r i a t io n s w e s ee i n c a lc u l a t in g t h e

d e t a c h a b i l i t y f o r f i x e d ~ s i m p l y is d u e t o t h e d i f f e r e n c e b e t w e e n c e 4 0 ) a n d

c e t ~ o o ) .

C o n v e r s e l y w h e n w e h o l d a f ix e d a n d v a r y ~ d u r i n g t h e c u r v e fi tt in g ,

t h e n l a r g e v a r i a t i o n s i n t h e c a l c u l a t e d d e t a c h a b i l i t i e s w i l l b e f o u n d w h i c h a r e

p r o p o r t i o n a l t o t h e c h a n g e s i n ~ .

5 . 2 . S e n s i t i v i t y t o n u m b e r o f s i z e c la s s e s

I t w a s d i s c u s se d e a r l i e r t h a t i t w a s p o s s i b l e t o o b t a i n a v e r y g o o d f it w i t h t h e

e x p e r i m e n t a l d a t a w i t h a r a n g e o f v a l u e s f o r a a n d ~;, a n d t h e r e f o r e a , ad a n d M ~ t .

I n t h i s s e c ti o n w e a r e i n t e re s t e d in h o w t h e se r a n g e s a r e a f f e c t e d b y t h e n u m b e r o f

p a r t i c l e s i ze cl a ss e s . S p e c i fi c a ll y , w e a r e l o o k i n g f o r a n o p t i m a l n u m b e r o f p a r t i c l e

c la s s si ze s lp a b o v e w h i c h t h e s u i ta b l e r a n g e s o f v a l u e s f o r t~ a n d ~ w h i c h r e p r o d u c e

t h e e x p e r i m e n t a l d a t a a r e e s s e n ti a ll y i n d e p e n d e n t o f L T h i s is p a r t i c u l a r l y i m p o r t a n t

w h e n s e e ki n g s o lu t io n s f o r t h e s p a t ia l d e p e n d e n c e o f t h e c o n c e n t r a t io n t h r o u g h t im e

f r o m E q s . 9 ) a n d 1 0 ). A n y r e d u c t i o n in th e n u m b e r o f p a r t ic l e si ze c la s se s le a d s to a

r e d u c t i o n o f t w i c e t h a t n u m b e r o f p a r t i a l d if f e re n t ia l e q u a t i o n s , a n d w o u l d t h e r e f o r e

h a v e a s i g n if ic a n t i m p a c t o n t h e c o m p u t a t i o n t i m e r e q u i r e d t o s o lv e n u m e r i c a l l y f o r

c x , t )

f r o m E q s . 9 ) a n d 1 0 ).

F o r t h e p u r p o s e s o f e x a m i n i n g t h e s en s it iv i t y o f th e m o d e l t o / , t h e e x p e r im e n t o n

t h e B l a c k E a r t h s o il f o r a 5 m m d e p t h a n d a 1 00 m m h - l r a i n fa l l ra t e is u s e d . T h e

r e s u l t s f o r I = 5 , 1 0 , 1 5 a n d 2 0 a r e s u m m a r i s e d i n T a b l e 2 . T h e p a r a m e t e r r a n g e s

g i v e n i n T a b l e 2 a r e b a s e d o n g o o d a g r e e m e n t b e t w e e n t h e t h e o r e t i c a l c t ) a n d t h e

e x p e r i m e n t a l c o n c e n t r a t i o n s , a n d a l so o n w h e t h e r t h e s e t tl in g v e l o c it y d i s t r ib u t i o n

c u r v e s a r e a c c e p t a b l e . F o r a l l p a r t i c l e s i ze c la s s e s i t w a s f o u n d t h a t ~c m u s t a l w a y s l ie in

t h e r a n g e 1 .9 ~< t e l 0 .0 e x c e p t f o r I = 2 0 w h e r e t h e u p p e r l i m i t o f n i s 4 a n d t h a t M ~ t

w a s i n d e p e n d e n t o f ~. F r o m T a b l e 2 w e s e e t h a t t h e r a n g e o f p o s s ib l e a v a l u e s

d e c r e a s e s s ig n i fi c a n tl y b o t h i n m a g n i t u d e a n d w i d t h a s I i n c re a s e s. T h e m a g n i t u d e

o f t h e d e t a c h a b i l i t y r a n g e a l s o d e c r e a s e s si g n i fi c a n t ly f r o m I = 5 t o I = 1 0 b u t t h e n

s ta b il is e s. T h e w i d t h o f t h e r a n g e i n M ~ t d o e s n o t s e e m t o s h o w a n y p a r t i c u l a r t r e n d

w h i l e it s m a g n i t u d e a p p e a r s t o b e i n c r e a s in g . B a s e d o n t h e r e s u l t s i n T a b l e 2 , I p = 1 5

w o u l d b e a n i d e a l n u m b e r o f p a r t ic l e s iz e c la s se s t o u s e b u t , o w i n g t o t h e c o m p u t a t i o n

r e s t r i c t i o n s i t w o u l d p la c e o n s o lv i n g E q s . 9 ) a n d 1 0 ) f o r c x , t ) w e fee l t h a t Ip = 10 i s

m o r e t h a n s u ff ic ie n t t o r e p r o d u c e t h e u n d e r l y i n g p h y s i c s o f t h e e r o s i o n p r o c e s s e s a n d

t h e s t r u c t u r e o f t h e e x p e r i m e n t a l d a t a .

T a b l e 2

E f f ec t o f n u m b e r o f p a rt ic l e s iz e I o n p a r a m e t e r v a l u e s

I a ~ - 1 .9 ~ = 10 .0 M ~ t

5 1 6 0 ~ a ~ 2 2 0 2 5 4 7 ~< a ~ 2 7 3 5 4 8 5 ~ a ~ 5 0 0 0 . 1 2 ~< M ~ t ~ 0 . 1 5

1 0 6 0 ~ a ~ 1 2 0 1 85 1 ~ a ~ 2 0 9 7 3 5 2 ~ a ~ 4 0 0 0 . 1 7 ~ M ~ t ~ 0 . 2 9

15 40 ~ a ~ 80 1673 ~ a ~ 189 2 318 ~ a ~ 361 0 .23 ~ M ~t ~< 0 .4

20 40 ~< a ~< 60 16 74 ~< a ~< 18 09 79 5 ~< a


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366

G . C. Sander e t a l . / Jour nal o f H ydr o logy 178 1996) 351 -367

6 Conc lus ion

I n c o n c l u s i o n w e h a v e p r e s e n t e d a t i m e - d e p e n d e n t s o l u t i o n f o r t h e so i l e r o s i o n

m o d e l o f H a i r s i n e a n d R o s e 1 9 9 1) a n d s h o w n t h a t w e c a n r e p r o d u c e t h e e x p er i-

m e n t a l d a t a o f P r o f f it t e t a l 1 9 91 ) . T h i s s i m p l e a n a l y t i c a l s o l u t i o n i s b a s e d o n t h e

a s s u m p t i o n t h a t t h e s p at i al d e p e n d e n c e o f t h e s e d i m e n t c o n c e n t r a t i o n a t x = L is

s m a ll c o m p a r e d w i t h it s t im e d e p e n d e n c e . D a t a f r o m n i n e e x p e r im e n t s w e r e u se d t o

v e r i fy t h e r e li a b il i t y o f t h e t h e o r e t i c a l m o d e l w i t h e x c e ll e n t a g r e e m e n t f o u n d f o r a ll

n i n e . W e w e r e a b l e t o r e p r o d u c e t h e i n i t i a l r a p i d i n c r e a s e t o a p e a k i n s e d i m e n t

c o n c e n t r a t i o n w h i c h o c c u r r e d a p p r o x i m a t e l y 5 r a i n a f te r th e c o m m e n c e m e n t o f

r a i n fa l l , a s w e l l a s t h e t a i li n g e x p o n e n t i a l d e c l i n e f r o m t h i s p e a k t o s t e a d y - s t a t e

c o n d i t i o n s .

W e s h o w t h a t w h i le t h e t o ta l s e d i m e n t c o n c e n t r a t i o n a n d t h e f r a c t i o n o f s h ie l d i n g

r e s u l t i n g f r o m t h e d e p o s i t e d l a y e r a p p e a r t o r e a c h e q u i l i b r i u m f a i rl y q u i c k l y , th e

d i s t r i b u t i o n o f p a r ti c l e s iz e c la s se s w h i c h c o m p r i s e t h is s e d i m e n t i s f a r f r o m

e q u i l i b r i u m . T r u e s t e a d y s ta te f o r th e S o l o n c h a k e x p e r i m e n t s w a s a c h i e v e d b e t w e e n

1 5 00 a n d 2 0 0 0 m i n b e i n g o n t h e o r d e r o f o n e d a y . S e n s i ti v i t y t o th e n u m b e r o f p a r t ic l e

s iz e c l a s s e s w a s i n v e s t i g a t e d a n d i t w a s d e c i d e d t h a t t e n c l a s s e s w e r e s u f f ic i e n t t o

r e p r o d u c e t h e s t ru c t u r e a n d p h y s i c s u n d e r l y i n g t h e e x p e r i m e n t a l d a t a . A s it is o n l y

p o s s i b l e t o s o lv e f o r c x , t ) n u m e r i c a ll y , a n d t h a t a n y r e d u c t i o n i n t h e n u m b e r o f siz e

c l as s es l e a d s t o a r e d u c t i o n o f t w i c e as m a n y p a r t i a l d i ff e re n t ia l e q u a t i o n s , t h e n

k n o w i n g I p = 1 0 h a s s i g n if i ca n t p r a c t i c a l i m p l i c a t i o n s f o r d e t e r m i n i n g t h e s p a t i a l

a s w e ll as th e t e m p o r a l d e p e n d e n c e o f th e s e d i m e n t c o n c e n t r a t io n . T h i s p r o b l e m is

c o v e r e d i n f u t u r e p u b l i c a t i o n s .

References

Alberts, E.E., Mo ldenhauer, W .C. and Foster, G .R., 1980. So il aggregates and prim ary particles

transported in rill and interill flow. Soil Sci. So c. Am . J., 44: 590-595.

Alberts, E.E., W endt, R .C. and Prie st, R.F., 1983. Physical and chemical properties of eroded soil

aggregates. Trans. A SAE , 26: 465-471.

Gov indaraju, R.S . and Kavv as, M .L., 1991.Modelling the erosion process over steep slopes: approximate

analytical solutions. J. Hydrol., 127: 279-305.

Hairsine, P.B . and Rose, C.W ., 1991. Rainfall detachment an d deposition: sediment transport in the

absence o f flow-driven processes. Soil Sci. Sot:. Am . J., 55: 320-324.

Hairsine, P.B., Sander, G.C., Rose, C.W ., Parlange, J.-Y. an d H ogarth, W .L., 1995. Unsteady soil erosion

due to rainfall impact: I. Theo retical considerations and practical applications. Water Resour. Res.,

submitted.

Hud son, N., 1981. So il Conservation, 2nd edn. C ornell University, Ithaca, NY .

Laguna, A. and Giraldez, J.V ., 1993. The description o f soil erosion through a kinem atic wave model. J.

Hydrol., 145: 65-82.

M eyer, L.D., Foster, G .R. and Rom kens, M.J.M. 1975. Source o f soil eroded by water from upland slopes.

In: Present and Prospective Technology fo r Predicting Sediment Yields and Sources, Proc. Sediment

Yield W orkshop, Oxford, MS, 28-3 0 Novem ber 1972. USD A-ARS-40, pp. 177-189.

Palis, R.G., Okw ach, G., Rose, C .W . and Saffigna, P.G., 1990. Soil erosion processes and nutrient loss. I.

The interpretation of enrichment ratio and nitrogen loss in runoff sediment. Aust. J. Soil Res., 28:

623-639.


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