m i n i s t e r i e van v e r k e e r en w a t e r s t a a t r i j k s w a t e r s t a a t
d i e n s t weg- en waterbouwkunde a f d e l i n g a d v i s e r i n g waterbouw
Comett course on c o a s t a l morphology D e l f t , a p r i l 1990
D u n e e r o s i o n
K.J. Verhagen, r i j k s w a t e r s t a a t , c i v i l e n g i n e e r i n g d i v i s i o n , D e l f t
1 . I n t r o d u c t i o n
1.1 The d i f f e r e n c e between a c u t e e r o s i o n and c h r o a i c e r o s i o n
I n t h e N e t h e r l a n d s t h e dune c o a s t i s a f l e x i b l e sea-defence a g a i n s t t h e N o r t h Sea, C h a r a c t e r i s t i c i s t h e c o n t i n u o u s movement of sand i n t h e c o a s t a l Eone, There i s an exchange of sand between t h e s u b a e r i a l and s u b a q u a t i c p a r t of t h e beach. C u r r e n t s and waves move t h e sand on t h e shore i n c r o s s s h o r e and l o n g -shore d i r e c t i o n . T h i s p r o c e s s may course a l o s s of sand from t h e sea-defence zone t o a d j a c e n t c o a s t a l s e c t i o n s or t o n e i g h b o r i n g i n l e t s . Because of t h e s e p r o c e s s e s t h e r e i s a c o n t i n u o u s movement of t h e b o r d e r l i n e between l a n d and w a t e r . E r o s i o n and a c c r e t i o n a l t e r n a t e each o t h e r . E r o s i o n n e a r l y always causes p r o b l e m s .
There are two t y p e s of c o a s t a l e r o s i o n :
* A f a s t , b i g e r o s i o n caused by q u i c k e r o s i o n of dunes d u r i n g s t o r m surges * A s l o w l y , c h r o n i c e r o s i o n , which i s n o t so s t r i k i n g , caused by s e a - l e v e l
r i s e and o t h e r m o r p h o l o g i c a l phenomena. The c h r o n i c e r o s i o n moves sand o u t o f t h e c o a s t a l defence zone. An i n c r e a s e s e a l e v e l r i s e may cause an i n -c r e a s e d -c h r o n i -c e r o s i o n . I n t h a t -case a l s o t h e -c o a s t a l p r o f i l e w i l l adapt t o t h e new w a t e r l e v e l by moving i n a landward d i r e c t i o n .
I n t h e p a s t c h r o n i c e r o s i o n was o f t e n not p r e v e n t e d . On l o n g e r terms c h r o n i c e r o s i o n weakened t h e dunes, so t h a t one had t o f e a r t h a t d u r i n g a s t o r m t h e s t o r m surge e r o s i o n c o u l d wipe away t h e r e m a i n i n g dune. T h i s would cause f l o o d -i n g o f t h e p o l d e r s . T h e r e f o r e t h e P o l d e r Boards, r e s p o n s -i b l e f o r s e a - d e f e n c e , improved t h e dunes by p l a c i n g more sand b e h i n d t h e dunes on t h e l a n d w a r d s i d e . T h i s caused c o n s i d e r a b l e damage t o n a t u r e ,
1.2 cause and e f f e c t
I t i s assumed t h a t f o r a g i v e n w a t e r l e v e l and a g i v e n wave c o n d i t i o n t h e r e i s an e q u i l i b r i u m p r o f i l e . U n f o r t u n a t e l y t h i s c o n d i t i o n o c c u r s o n l y i n l a b o r a t o -r i e s , and c o n s e q u e n t l y we d o n ' t f i n d e q u i l i b -r i u m p -r o f i l e s a l o n g t h e c o a s t . C o n s t a n t l y t h e w a t e r l e v e l and t h e wave changes, and t h u s t h e p r o f i l e changes. E s p e c i a l l y d u r i n g s t o r m surges (much h i g h e r w a t e r , much h i g h e r waves) t h e p r o f i l e has t o a d a p t . T h i s causes dune e r o s i o n . Sand i s moved f r o m t h e dune t o -wards t h e beach. On s t a b l e c o a s t s , one may expect t h a t t h e dune r e c o v e r s com-p l e t e l y a f t e r t h e s t o r m . F i g u r e 1 shows t h e l o c a t i o n of t h e d u n e - f o o t as a
s h o a l s o r t r e n c h e s a r e p r e s e n t near t h e c o a s t . These problems w i l l n o t be d i s -cussed h e r e , some of t h e problems and t h e suggested s o l u t i o n s a r e g i v e n i n t h e " g u i d e " [TAW 1 9 8 9 ] . I t i s c l e a r t h a t t h e shape o f t h e p r o f i l e b e f o r e t h e s t o r m i n f l u e n c e s t h e p o s i t i o n of t h e r e s u l t i n g X - c o o r d i n a t e Xe.
T h i s model i s c a l l e d t h e Duros-model (DUne eROSion) o r V e l l i n g a - m o d e l , The wave h e i g h t (H) t o be a p p l i e d i n t h e e q u a t i o n i s t h e s i g n i f i c a n t wave h e i g h t , on "deep" w a t e r . The o t h e r parameter i s t h e s e t t l i n g v e l o c i t y o f t h e s e d i m e n t . T h i s parameter can be computed f o r seawater and f o r a t e m p e r a t u r e o f 5 as f o l l o w s : ^ " l o g (1/w) = 0.476 ( ^ " l o g D )2 + 2.180 ^ " l o g D + 3.226 i n w h i c h D i s t h e median g r a i n s i z e o f t h e e r o d i n g dune m a t e r i a l . The c a l c u l a t i o n i t s e l f i s an i t e r a t i v e p r o c e s s , because o f t h e f a c t t h a t t h e e r o d i n g q u a n t i t y has t o be e q u a l t o t h e a c c r e t i n g q u a n t i t y . V a r i o u s computer programs a r e a v a i l a b l e t o c a r r y o u t t h e c o m p u t a t i o n . 3.2 p r o b a b i l i s m
A c c o r d i n g t o t h e p r e s e n t Dutch r e g u l a t i o n a sea defence has s u r v i v e w i t h o u t problems a s t o r m surge w i t h a p r o b a b i l i t y o f o c c u r r e n c e of 1/10,000 per y e a r . For a d i k e t h i s g e n e r a l l y means t h a t i t has t o w i t h s t a n d a w a t e r l e v e l w i t h a p r o b a b i l i t y o f 1/10,000 per y e a r . Dune e r o s i o n i s a more c o m p l i c a t e d p r o c e s s , and n o t o n l y t h e w a t e r l e v e l i s o f i m p o r t a n c e . A l s o t h e wave h e i g h t and t h e d u r a t i o n o f t h e s t o r m surge a r e i m p o r t a n t . T h e r e f o r e i t i s b e t t e r t o d e s i g n a dune i n such a way t h a t i t has a g i v e n p r o b a b i l i t y of f a i l u r e .
T h i s p r o b a b i l i t y of f a i l u r e has t o be l e s s t h a n 1/10,000 per y e a r . A good d e s i g n e d d i k e which can s u r v i v e a 1/10,000 s t o r m w i l l n o t f a i l . Research has i n d i c a t e d t h a t t h e p r o b a b i l i t y o f f a i l u r e of a good d e s i g n e d d i k e i s approx. 1/100,000. The d i f f e r e n c e between s u r v i v a l and f a i l u r e i s t h e r e f o r e a f a c t o r 10.
A p r o p e r dune-design t h e r e f o r e r e q u i r e s a p r o b a b i l i s t i c c a l c u l a t i o n . I n t h i s course t h e t h e o r y o f p r o b a b i l i s t i c c o m p u t a t i o n w i l l n o t be d i s c u s s e d . For back-ground i n f o r m a t i o n can be r e f e r r e d t o Ang & Tang [ 1 9 8 4 ] . For a s i m p l e case ( f o r example t h e s t r e n g t h of a t h r e a d ) on can say t h a t t h e t h e p r o b a b i l i t y o f
( s t r e n g t h l o a d ) i s n e g a t i v e , s h o u l d be s m a l l e r t h a n a g i v e n v a l u e . I n p r o b a b i l i s t i c c a l c u l a t i o n s i t i s customary t o d e f i n e a s o c a l l e d r e l i a b i l i t y f u n c t i o n Z. T h i s f u n c t i o n s h o u l d be such t h a t Z ) 0 r e p r e s e n t s f a i l u r e o f t h e s y s -tem. I n t h i s case, as an example:
P r { Z < 0 1 = 1/100,000.
I n case o f t h e s t r e n g t h o f a t h r e a d Z = S - F, i n w h i c h S= s t r e n g t h of t h e t h r e a d and F i s t h e l o a d on t h e t h r e a d .
For dunes one s h o u l d ask t h e f o l l o w i n g q u e s t i o n : "given a certain
x-coordi-nate Xg, what is the probability that an erosion profile occurs such that the erosion point Xe is situated more landward than Xg ?" o r :
figure 3: definition of Xe and Xg
So t h e Z " f u n c t i o n s h o u l d meet t h e f o l l o w i n g r e q u i r e m e n t : Z = Xg - Xe.
Xg i s a p r e s c r i b e d v a l u e f o r t h e X - c o o r d i n a t e . However Xe depends on t h e parameters c a u s i n g t h e e r o s i o n p r o f i l e . So t h e problem i s t o c a l c u l a t e Pr {Z<0!. where Z i s a f u n c t i o n o f t h e parameters i n v o l v e d . S t a n d a r d methods a r e a v a i l a b l e f o r c a l c u l a t i o n of Pr iZ<0!, p r o v i d e d t h a t t h e i n v o l v e d parameters are known and moreover: t h e s t a t i s t i c a l d i s t r i b u t i o n of t h e v a r i a b l e s must be known, as w e l l as t h e s t a t i s t i c a l d e p e n d e n c i e s .
The main p a r a m e t e r s i n f l u e n c i n g t h e shape of t h e e r o s i o n p r o f i l e a f t e r a s t o r m a r e :
- t h e w a t e r l e v e l d u r i n g t h e s t o r m surge - t h e wave h e i g h t d u r i n g t h e s t o r m surge - t h e s t o r m d u r a t i o n
- t h e g r a i n - s i z e of t h e dune sand.
The shape of t h e e r o s i o n p r o f i l e can be c a l c u l a t e d w i t h t h e above mentioned Duros-model f o r a g i v e n s e t of p a r a m e t e r s .
A r e a l e r o s i o n p r o f i l e a f t e r a s t o r m surge w i l l c e r t a i n l y d i f f e r f o r m t h e c a l c u l a t e d e x p e c t e d e r o s i o n p r o f i l e . The r e a s o n f o r t h i s d i s c r e p a n c y might be:
- The a c c u r a c y of t h e model f o r t h e e x p e c t e d p r o f i l e . - The v a r i a n c e of t h e sediment d i a m e t e r D50.
The i n i t i a l p r o f i l e j u s t b e f o r e t h e s t o r m i s never known, u s u a l l y a p r o -f i l e i s used t h a t i s measured some t i m e b e -f o r e t h e s t o r m o c c u r s .
- The r e a l v a l u e s f o r t h e p a r a m e t e r s t h a t c h a r a c t e r i z e t h e s t o r m d i f f e r f r o m the v a l u e s assumed f o r d e t e r m i n a t i o n of t h e t h e o r e t i c a l e r o s i o n p r o f i l e a f t e r a s t o r m s u r g e . These a r e : t h e w a t e r l e v e l , t h e w a v e - h e i g h t , t h e s t o r m d u r a t i o n and a l s o t h e p o s s i b l e o c c u r r e n c e of g u s t bumps d u r i n g maximum w a t e r l e v e l .
Van de G r a a f f [1983, 1986] g i v e s f u l l d e t a i l s on how p r o b a b i l i s t i c c a l c u l a -t i o n s can be made f o r -t h e e r o s i o n of dunes.
I n a g e n e r a l purpose p r o b a b i l i s t i c program t h e z - f u n c t i o n (= t h e Duros model) can be b u i l d i n , as w e l l as mean v a l u e s , s t a n d a r d d e v i a t i o n s and s t a t i s t i c a l d i s t r i b u t i o n of t h e f o l l o w i n g 7 s t o c h a s t i c v a r i a b l e s :
1 hmax t h e maximum w a t e r l e v e l d u r i n g a s t o r m surge
2 Hs t h e s i g n i f i c a n t wave h e i g h t d u r i n g t h e s t o r m surge 3 T s t t h e t o t a l s t o r m d u r a t i o n
4 D50 t h e median g r a i n s i z e of t h e dune sand 5 Bpr f l u c t u a t i o n s of t h e beach p r o f i l e
L e t A be t h e c a l c u l a t e d e r o s i o n (m /ra) above t h e s t o r m surge l e v e l a c c o r d i n g t o t h e Duros model f o r a g i v e n c o m b i n a t i o n o f hmax, Hs, D50 and Bpr. The e f f e c t of t h e o t h e r t h r e e s t o c h a s t i c v a r i a b l e s T s t , Bump and Mod can be s c h e m a t i z e d t o an a d d i t i o n a l e r o s i o n above s t o r m surge l e v e l : A e x t r a , h a v i n g a normal d i s t r i -b u t i o n and t h e f o l l o w i n g c h a r a c t e r i z i n g p a r a m e t e r s :
mean o f A - e x t r a = A/20
v a r i a n c e of A - e x t r a = 0.02 * sqr (A) + 4*A + 400
figure 4: A-extra
The p r o b a b i l i t y d e n s i t y f u n c t i o n s f o r hmax and Hs must be known. For example: the maximum w a t e r l e v e l d u r i n g s t o r m surge can have an e x p o n e n t i a l d i s t r i b u t i o n :
Pr {hmax>h! = exp [ - { h - a ) / b ] ,
where a and b a r e parameters c h a r a c t e r i z i n g t h e d i s t r i b u t i o n .
3.0 4.0 5.0 6.0 7.0
~ storm surge level Im.NAP)
figure 4: Expected value of the significant wave height as a function of the storm surge level at a number of locations along the Dutch coast.
Up u n t i l now a l l c o n c e r n i n g v a r i a b l e s a r e c o n s i d e r e d t o be s t o c h a s t i c i n d e pendent (however t h i s i s a r g u a b l e ) , An e x c e p t i o n must be made f o r t h e s i g n i f i -cant wave h e i g h t d u r i n g s t o r m c o n d i t i o n s and t h e maximum w a t e r l e v e l : t h e h i g h e r the w a t e r l e v e l , t h e h i g h e r t h e waves w i l l be ( b o t h a r e s t r o n g l y i n f l u e n c e d by the wind) . A l s o two storms h a v i n g t h e same maximum w a t e r l e v e l , w i l l n o t have the same s i g n i f i c a n t wave h e i g h t s . For t h e Dutch c o a s t t h e s t a t i s t i c a l depen-dency o f Hs as f u n c t i o n o f hmax has been p r o v e n t o be as i n f i g u r e 5.
I n t h e model t h e f o l l o w i n g s c h e m a t i z a t i o n was made f o r t h e s t a t i s t i c a l d i s -t r i b u -t i o n o f -t h e wave h e i g h -t as f u n c -t i o n o f -t h e maximum w a -t e r l e v e l d u r i n g s -t o r m s u r g e :
For a c e r t a i n v a l u e o f hmax, t h e wave h e i g h t i s normal d i s t r i b u t e d w i t h a mean o f Hs = c * hmax" ( w i t h a p r e s c r i b e d maximum Hmax) s t a n d a r d dev. of Hs = 0.60 m
and where c and d a r e l o c a t i o n dependent c o e f f i c i e n t s , as does t h e maximum v a l u e o f Hs: Hmax.
3.3 example o f a p r o b a b i l i s t i c c a l c u l a t i o n
P r o b a b i l i s t i c c a l c u l a t i o n s can be made w i t h v a r i o u s t y p e s o f p r o b a b i l i s t i c models. As an example a c a l c u l a t i o n f o r a Dutch dune i s made w i t h a L e v e l I I method, implemented on a p e r s o n a l computer. See example 1 i n t h e a p p e n d i x . 3.4 t h e Dutch g u i d e t o t h e assessment o f t h e s a f e t y o f dunes as a sea defence
A few y e a r s ago no g e n e r a l purpose p r o b a b i l i s t i c programs were a v a i l a b l e which c o u l d be used e a s i l y . T h e r e f o r e i t was d e c i d e d n o t t o p r e s c r i b e t h e f u l l p r o b a b i l i s t i c c o m p u t a t i o n i n t h e Guide, b u t an a p p r o x i m a t i o n o n l y . For normal dunes a l o n g t h e Dutch c o a s t t h e r e s u l t s o f t h e a p p r o x i m a t i o n a r e n e a r l y e q u a l t o those o f a complete c o m p u t a t i o n . For d e t a i l s can be r e f e r r e d t o Van de G r a a f f , 1 9 8 3 , 1 9 8 6 . As i n d i c a t e d above, now such programs a r e a v a i l a b l e . I t i s expected t h e r e f o r e t h a t i n a n e x t e d i t i o n of t h e G u i d e l i n e s a p r o b a b i l i s t i c c o m p u t a t i o n w i l l be p r e s c r i b e d .
The r e s u l t o f t h e suggested a p p r o x i m a t i o n i s as f o l l o w s :
C a l c u l a t i o n s s h o u l d be be made, n o t u s i n g t h e s t o r m surge l e v e l and t h e g r a i n s i z e o f t h e dune sand, b u t u s i n g a c o m p u t a t i o n a l l e v e l and a c o m p u t a t i o n a l g r a i n s i z e . The c o m p u t a t i o n a l l e v e l i s t h e d e s i g n l e v e l + 2 / 3 d e c i m a t i o n h e i g h t . The d e c i m a t i o n h e i g h t i s t h e d i f f e r e n c e i n h e i g h t between t h e w a t e r l e v e l w i t h a p r o b a b i l i t y o f exceedance 1 0 t i m e s s m a l l e r t h a n t h a t o f t h e d e s i g n l e v e l , and t h e d e s i g n l e v e l . I n t h e N e t h e r l a n d s t h e d e c i m a t i o n h e i g h t i s ap-p r o x . 7 0 cm. The c o m ap-p u t a t i o n a l g r a i n s i z e i s c a l c u l a t e d as f o l l o w s " ^comp = MD5 0 - 5 I (C 7 D5 0 ) ' / MD5 0 '
i n which Mn^o t h e e x p e c t e d v a l u e o f t h e median g r a i n s i z e and Oncn i s t h e s t a n d a r d d e v i a t i o n .
With t h e s e v a l u e s an e r o s i o n a n a l y s i s i s made f o r each y e a r , u s i n g t h e Duros-model. A s u r c h a r g e i s added t o t a k e account o f t h e i n f l u e n c e s o f t h e i n a c c u r a c y of t h e c o m p u t a t i o n a l model, t h e g u s t o s c i l l a t i o n s and t h e g u s t surges and t h e u n c e r t a i n t y about t h e t i m e d u r i n g which t h e w a t e r l e v e l remains a t about m a x i -mum l e v e l . The e f f e c t o f t h i s s u r c h a r g e i s expressed i n an a d d i t i o n a l r e c e s s i o n of t h e s t e e p dune f r o n t . The s u r c h a r g e T i s computed as f o l l o w s :
T = 0 . 2 5 A + 2 0
P o i n t F i s t h e i n t e r s e c t i o n o f t h i s s h i f t e d dune f r o n t w i t h t h e c o m p u t a t i o n a l l e v e l . See f i g . 6. The changes i n t h e p o s i t i o n o f P a r e a measure f o r t h e c h r o -n i c c o a s t a l e r o s i o -n from t h e f o l l o w i -n g p o i -n t s o f v i e w :
- s a f e t y f o r t h e p o l d e r s b e h i n d t h e dunes - o n l y f o r a r e l a t i v e l y s h o r t p e r i o d .
figure 6: Definition sketch
The above c a l c u l a t i o n s y i e l d a t i m e s e r i e s f o r t h e p o s i t i o n o f p o i n t P. These p o s i t i o n s can be p l o t t e d i n a d i a g r a m as a f u n c t i o n o f t i m e (see f i g . 7) . I t can be e a s i l y i n d u c e d from t h e p o s i t i o n whether t h e r e i s q u e s t i o n o f a s t a b l e , e r o d i n g o r e a c c r e t i n g c o a s t . The t r e n d of t h e p o s i t i o n of p o i n t P as a f u n c t i o n of t i m e can be e s t i m a t e d by means of r e g r e s s i o n a n a l y s i s . A l i n e a r a p p r o x i m a -t i o n w i l l u s u a l l y do ( u s i n g approx. 10 y e a r s o f d a -t a ) . The p r o f i l e f l u c -t u a -t i o n s are e x p r e s s e d i n t h e s c a t t e r e d p o s i t i o n o f t h e p o i n t s P around t h i s r e g r e s s i o n l i n e . time (yearl 1960 SEASIDE 1970 1980 d*g= distance over which the regression line is
shifted landwards so as to include : - the processing of profile fluctuations (d)
- the influence of the gradient in longshore transport ig)
extrapolation
critical position
expected point in time when the| safety standard
LIMIT PROFILE
The found r e g r e s s i o n l i n e has t o be moved landward over a d i s t a n c e d+g. The v a l u e d i s t o t a k e i n t o account t h e p r o f i l e v a r i a t i o n s d u r i n g t h e y e a r , and i s d e f i n e d as: _
d = z / 275
i n w h i c h a 2 i s t h e s t a n d a r d d e v i a t i o n o f p o i n t P around t h e r e g r e s s i o n l i n e and z i s t h e average h e i g h t of t h e p r o f i l e . The v a l u e g i s f o r t a k i n g i n t o account t h e c u r v a t u r e of t h e c o a s t l i n e : g = G / z G = A*/300 (H/7.6)^-''2 („/o.0268) ^•'^^ G A = A + T G^= f ( c o a s t a l c u r v a t u r e ) A dune i s r e g a r d e d as s a f e as l o n g as l a n d w a r d of p o i n t P t h e r e i s a t l e a s t a so c a l l e d " l i m i t p r o f i l e " i s p r e s e n t . T h i s l i m i t p r o f i l e (see f i g . 8) has a c r e s t h e i g h t of a t l e a s t 2.5 m above c o m p u t a t i o n a l l e v e l . I n case of h i g h and l o n g d e s i g n waves the c r e s t of t h e l i m i t p r o f i l e may be h i g h e r .
figure 8: The limit profile
The method o u t l i n e d i n t h e above p a r a g r a p h i s d e s c r i b e d i n d e t a i l i n t h e " G u i d e l i n e s " [TAW, 1989] .
3.5 p r a c t i c a l a p p l i c a t i o n i n t h e N e t h e r l a n d s
The Dutch v e r s i o n of these g u i d e was p u b l i s h e d i n 1984. I n t h e f o l l o w i n g y e a r s t h e whole Dutch c o a s t was t e s t e d upon s a f e t y .
The r e s u l t of t h e c o m p u t a t i o n s i s p r e s e n t e d i n f i g u r e 9. As can be seen most of t h e dunes were s a f e enough. At some p l a c e s o n l y t h e seadunes d i d n o t f u l f i l l t h e r e q u i r e m e n t s , b u t t h e dune as a whole were s t r o n g enough. I f t h e s a f e -t y was n o -t s a -t i s f a c -t o r y , dune improvemen-t works were e x e c u -t e d . G e n e r a l l y -t h e s e improvement works were c a r r i e d o u t by p l a c i n g sand l a n d w a r d of t h e weak dunes. C h r o n i c e r o s i o n was t a k e n i n t o c o n s i d e r a t i o n . G e n e r a l l y t h e amount of sand was s u f f i c i e n t t o g u a r a n t e e s a f e sea-defences u n t i l t h e y e a r 2000. Of c o u r s e t h i s g i v e s s e r i o u s problems i n t h e year 2001.
figure 9: Safety of the Dutch dunes in 1984
V l Q G n d e r e n
• »•• heoyy erosie»
I t i s c l e a r t h a t i t i s n o t s u f f i c i e n t t o t e s t t h e dunes o n l y once. T h i s s h o u l d be done c o n t i n u o u s l y . At t h i s moment l e g i s l a t i o n i s b e i n g p r e p a r e d f o r a g e n e r a l t e s t i n g of a l l t h e s e a - d e f e n c e s , r i v e r - and l a k e d i k e s . The new B i l l on Waterdefences w i l l p r o b a b l y be a c c e p t e d by p a r l i a m e n t i n 1990. A f t e r p a s s i n g the B i l l , e v e r y w a t e r d e f e n c e system i n t h e N e t h e r l a n d s has t o be t e s t e d a t l e a s t once i n f i v e y e a r s . A r e p o r t has t o be made by t h e manager of t h e seade-fence system, and p r e s e n t e d t o t h e n a t i o n a l government [Verhagen and V o l k e r , 1989] .
For t e s t i n g t h e dunes t h e manager i s o b l i g e d t o use t h e method d e s c r i b e d i n the " g u i d e " .
3.6 examples
As an example a p r o f i l e a l o n g t h e Dutch coast i s t e s t e d upon s a f e t y a c c o r d i n g t o t h e " g u i d e " . The chosen p r o f i l e i s a s t r o n g l y e r o d i n g s e c t i o n of t h e Dutch c o a s t , where t h e s a f e t y i s r a t h e r m a r g i n a l . I s i s n e c e s s a r y t o improve t h e coast h e r e r e g u l a r l y . See example 2 i n t h e appendix.
4. T h e D u r o s t a m o d e l
4.1 f o r m u l a t i o n o f t h e n o d e l
general
The f a c t t h a t Duros i s not t i m e - d e p e n d e n t , but o n l y g i v e s t h e f i n a l p r o f i l e a f t e r t h e s t o r m surge i s n o t a problem. T h e r e f o r e i t i s p r i n c i p a l l y n o t neces-s a r y t o d e v e l o p a time-dependent dune e r o neces-s i o n model.
However, t h e r e are a few s i t u a t i o n s w h i c h cannot be s o l v e d u s i n g t h e Duros-model . These s i t u a t i o n s a r e :
* e r o s i o n on v e r y c u r v e d c o a s t l i n e s ; d u r i n g e r o s i o n t h e d e p o s i t e d sand d i -r e c t l y -removed f-rom t h e beach by a g -r a d i e n t i n t h e l o n g s h o -r e c u -r -r e n t ; t h i s w i l l cause e x t r a dune e r o s i o n . Duros i s n o t a b l e t o p r e d i c t t h e e x t r a e r o s i o n .
* I f t h e r e i s a d u n e f o o t p r o t e c t i o n , Duros cannot c a l c u l a t e t h e f i n a l p r o -f i l e .
* On p l a c e s were dunes t r a n s f o r m t o d i k e s , t h r e e - d i m e n s i o n a l a s p e c t s a r e i m p o r t a n t . Duros cannot cope w i t h them.
* I f t h e r e i s a v e r y s t e e p f o r e s h o r e , sand w i l l n o t o n l y be d e p o s i t e d on t h e beach, b u t a l s o d i s a p p e a r i n t o t h e t i d a l c h a n n e l . I t i s q u i t e d i f f i c u l t t o f i n d an e q u i l i b r i u m - p r o f i l e i n these s i t u a t i o n s .
A time-dependent dune e r o s i o n model i s a b l e t o s o l v e t h e above m e n t i o n e d p r o b l e m s . T h e r e f o r e i t was d e c i d e d by t h e T e c h n i c a l A d v i s o r y committee on Wa-t e r d e f e n c e s (TAW) Wa-t o have Wa-t h e D u r o s Wa-t a model d e v e l o p e d . The b a s i s of Wa-t h e model i s v e r y s i m p l e :
1 C a l c u l a t e t h e t i m e - a v e r a g e d w a t e r movement i n t h e b r e a k e r z o n e . 2 C a l c u l a t e t h e t i m e - a v e r a g e d c o n c e n t r a t i o n .
3 M u l t i p l y w a t e r movement and c o n c e n t r a t i o n i n o r d e r t o f i n d t h e sediment t r a n s p o r t
4 C a l c u l a t e t h e e r o s i o n and s e d i m e n t a t i o n from t h e d i f f e r e n c e s i n sediment t r a n s p o r t
D u r i n g de e l a b o r a t i o n of t h e model, i t proved a l i t t l e b i t more c o m p l i c a t e d t h a n s c h e t c h e d above.
The b a s i c f o r m u l a f o r t i m e averaged sediment t r a n s p o r t i s : S = U.C + u ' ( t ) . C ' ( t )
I n D u r o s t a i t i s assumed t h a t u ' ( t ) . C ' ( t ) << u.C
Data a n a l y s i s has i n d i c a t e d t h a t t h i s a s s u m p t i o n i s c o r r e c t . I n t h i s c o u r s e t h i s w i l l n o t be worked o u t . For d e t a i l s see S t e e t z e l [ 1 9 9 0 ] . I t i s c o n c l u d e d i n t h a t r e p o r t t h a t under c i r c u m s t a n c e s where much sediment i s p r e s e n t i n t h e v e r t i c a l p r o f i l e , t r a n s p o r t i s m a i n l y d e t e r m i n e d by t h e average v e l o c i t y . The c o n t r i b u t i o n of f l u c t u a t i n g v e l o c i t i e s t o t h e t r a n s p o r t , t h e s o c a l l e d c o r r e l a -t i o n c o n -t r i b u -t i o n , i s r e l a -t i v e l y s m a l l ( o r d e r 5 % ) .
waves and water movement
The b a s i s f o r t h e d e s c r i p t i o n o f t h e v e l o c i t y p r o f i l e i s t h e v e l o c i t y near t h e b o t t o m . Based upon s t u d i e s from S t i v e and Wind and o t h e r s t h e shape o f t h e p r o f i l e i s g i v e n as:
u ( z ) = u j ^ + P/ E Q + a / s^ . z 2
I n t h i s f o r m u l a a,p a r e f u n c t i o n s o f t h e shear s t r e s s , t i s a m i x i n g c o e f f i c i -e n t .
I m p o r t a n t i s t h a t c o a s t w a r d sediment t r a n s p o r t i s n o t t a k e n i n t o r e g a r d , Coastward sediment t r a n s p o r t , caused by wave asym.m.etry can be n e g l e c t e d d u r i n g s t o r m surge c o n d i t i o n s . T h i s makes i n d e e d t h e D u r o s t a model o n l y v a l i d f o r s t o r m c o n d i t i o n s .
concentrations
The average c o n c e n t r a t i o n v e r t i c a l can be d e s c r i b e d by C(x) = CQ.f^(z) i s t h e c o n c e n t r a t i o n near t h e b o t t o m , f ( z ) i s a d i s t r i b u t i o n - f u n c t i o n over the v e r t i c a l . From a n a l y s i s f o l l o w s t h a t z f (z) = exp [-W . dz ] c S oJ £(Z)
i n w h i c h Wg i s t h e s e t t l i n g v e l o c i t y o f t h e sediment and t i s a m i x i n g parame-t e r .
A l l t h e d e r i v a t i o n s i n o r d e r t o a c h i e v e t h e s e r e s u l t s w i l l n o t be p r e s e n t e d h e r e . Reference i s g i v e n t o S t e e t z e l [1990] .
sediment transport and sand balances
When a d e s c r i p t i o n i s a v a i l a b l e f o r t h e v e l o c i t y and f o r t h e c o n c e n t r a t i o n , t h e r e s t o f t h e model i s q u i t e s i m p l e . Sediment t r a n s p o r t i s found by m u l t i p l i -c a t i o n o f -c o n -c e n t r a t i o n and v e l o -c i t y . S e d i m e n t a t i o n and e r o s i o n i s -c a l -c u l a t e d from t h e d i f f e r e n c e s i n sediment t r a n s p o r t . T h i s i s a l l b u i l t i n a a m a t h e m a t i -c a l t r a n s p o r t . Changes i n t h e p r o f i l e d u r i n g a s t o r m -can be p l o t t e d on t h e computer s c r e e n , and t h e r e s u l t s l o o k v e r y r e a l i s t i c .
The model has been c a l i b r a t e d upon t e s t r e s u l t s f r o m dune e r o s i o n t e s t s on a v e r y l a r g e s c a l e i n t h e D e l t a f l u m e o f D e l f t H y d r a u l i c s (De V o o r s t ) . F i g u r e 10a and 10b shows a comparison o f model r e s u l t s and computed r e s u l t s .
examples
D u r o s t a w i l l be used f o r t h e d e s i g n and t e s t i n g o f v a r i o u s p r o b l e m s . At t h i s moment t h e use f o r t e s t i n g e x i s t i n g dune f o o t p r o t e c t i o n s i s most u r g e n t . As an example a c a l c u l a t i o n i s p r e s e n t e d f o r t h e d u n e - f o o t p r o t e c t i o n n e a r Den H e l d e r . See example 3 i n t h e a p p e n d i x .
figure 10a: comparison measured and calculated erosion for a fixe waterlevel
figure 10b: comparison of measured and calculated erosion for a varying waterlevel
5. C o c i s t a l d a t a base^s a n d t h e i r u s e 5.1 o l d databases
Because a sandy coast i s a v e r y dynamic c o a s t l i n e , and because s t a b i l i t y o f t h e c o a s t l i n e i s v i t a l f o r t h e s a f e t y of t h e N e t h e r l a n d s a g a i n s t i n u n d a t i o n by s t o r m surges (dunes p r o t e c t t h e p o l d e r a r e a , which i s s i t u a t e d below sea l e -v e l ) , a l r e a d y i n f o r m e r c e n t u r i e s i t was n e c e s s a r y t o g e t some knowledge about t h e movements of t h e c o a s t l i n e . I n t h e m i d d l e of t h e 1 9 t h c e n t u r y s y s t e m a t i c c o a s t a l measurements s t a r t e d . On f i x e d i n t e r v a l s ( a p p r o x . 250 m) t h e p o s i t i o n of t h e d u n e - f o o t , t h e h i g h w a t e r l i n e and the low w a t e r l i n e was measured e v e r y y e a r . I n o r d e r t o have a f i x e d r e f e r e n c e , monuments were p l a c e d on t h e beach.
A l l t h e g a t h e r e d data a r e now s t o r e d i n a c o m p u t e r i z e d database and a r e a v a i -l a b -l e f o r v a r i o u s types of c o a s t a -l r e s e a r c h . Because t h e database c o v e r s a -l o n g p e r i o d and a l o n g c o a s t l i n e , i t i s v e r y u s e f u l f o r t h e s t u d y of l o n g p e r i o d i c phenomena l i k e moving c o a s t a l sand waves.
From t h i s database i t became c l e a r t h a t the L W - l i n e , and a l s o t h e HW-line shows b i g f l u c t u a t i o n s . F i l t e r i n g of d a t a i s n e c e s s a r y . That makes t h a t t h e d a t a cannot be used f o r d e t e r m i n i n g s h o r t term c o a s t a l movement (< 10 y e a r s ) . Besides t h e above mentioned sandwaves, a l s o c l i m a t i c changes have a b i g i n f l u -ence on t h e c h r o n i c e r o s i o n [Verhagen, 1989] .
5.2 J a r k u s
Around 1960 i t became c l e a r t h a t more d e t a i l e d i n f o r m a t i o n was n e c e s s a r y . T h e r e f o r e t h e system of monuments and base l i n e s was somewhat u p d a t e d , and a d e t a i l e d program f o r c o a s t a l measurements was s e t up.
A l o n g t h e whole N o r t h Sea c o a s t l i n e f r o m Cadzand i n t h e s o u t h t o Rottumeroog a f i x e d s e t of measuring l i n e s was d e f i n e d (see f i g . 1 1 ) . I n t o t a l approx. 3000 l i n e s a r e d e f i n e d t h i s way. The l i n e s have a i n t e r m e d i a t e d i s t a n c e of 200 m and are p e r p e n d i c u l a r t o t h e base l i n e . The base l i n e i s p a r a l l e l t o t h e c o a s t -l i n e , and i s p o s i t i o n e d a p p r o x . near t h e h i g h w a t e r -l i n e . On c o a s t a -l s e c t i o n s w i t h g r o i n s , t h e measuring l i n e s are p l a c e d i n t h e m i d d l e between t h e g r o i n s . P r o f i l e s are measured e v e r y y e a r from 200 m l a n d w a r d o f t h e base l i n e t o 800 m seaward of t h e base l i n e ( t h a t means t o an average d e p t h o f 8 m below mean sea l e v e l ) . Every f i v e y e a r s e v e r y k i l o m e t e r a l i n e i s measured u n t i l 2500 m f r o m t h e base l i n e . A l l measurements are made i n t h e p e r i o d between a p r i l and sep-tember.
The measurements are p e r f o r m e d i n two p a r t s (see f i g . 12) : * l e v e l i n g above t h e low w a t e r l i n e
* s o u n d i n g below approx. mean sea l e v e l l i n e
leveling
The l e v e l i n g s a r e done by t h e s u r v e y department o f R i j k s w a t e r s t a a t , u s i n g a e r i a l p h o t o g r a p h y (see f i g . 1 3 ) . I n t h e f i e l d o n l y some c a l i b r a t i o n p o i n t s a r e measured u s i n g c o n v e n t i o n a l equipment. I n f l i g h t d i r e c t i o n 60 % o v e r l a p i s used i n o r d e r t o g e t a good s t e r e o s c o p i c e f f e c t . I n more t h a n one s t r i p i s r e q u i r e d , an o v e r l a p of 20 - 30 % between t h e s t r i p s i s used. W i t h t h e use o f an a n a l y t i c p l o t t e r t h e h e i g h t i s measured i n t h e measuring l i n e s . I n t h i s p l o t t e r t h e o v e r l a p p i n g p h o t o g r a p h s are p r e s e n t e d a a t h r e e d i m e n s i o n a l image. The p o s i t i o n of t h e bends i n t h e p r o f i l e a r e read as d i g i t a l v a l u e s . The program i n t e r p o -l a t e d t h e d a t a , and e v e r y 5 m a h e i g h t - p o i n t of t h e p r o f i -l e i s s t o r e d i n memo¬ r y .
soundings
The soundings a r e done by s u r v e y v e s s e l s o f v a r i o u s R i j k s w a t e r s t a a t d e p a r t -ments and by s u r v e y v e s s e l s o f t h e w a t e r b o a r d s . P r o f i l e d a t a a r e a u t o m a t i c a l l y s t o r e d on board as x - y - z - v a l u e s . P o s t p r o c e s s i n g equipment c a l c u l a t e s t h e depths i n t h e p r o f i l e w i t h i n t e r m e d i a t e d i s t a n c e s o f 10 m.
L e v e l i n g d a t a and s o u n d i n g d a t a are c o u p l e d and s t o r e d on a mainframe compu-t e r o f R i j k s w a compu-t e r s compu-t a a compu-t . The d a compu-t a a r e a v a i l a b l e compu-t o c o a s compu-t a l managers, r e s e a r c h e r s and everyone e l s e who i s i n t e r e s t e d . A l l d a t a a r e o n - l i n e a v a i l a b l e s i n c e 1963 . T h i s data-base i s c a l l e d J a r k u s ( a b b r e v i a t i o n from J A a R l i j k s e KUStmetingen, y e a r l y c o a s t a l measurements).
Post p r o c e s s i n g s o f t w a r e i s a v a i l a b l e t o r e a d t h e d a t a , and t o p e r f o r m a d d i -t i o n a l c o m p u -t a -t i o n s . S-tandard P o s -t p r o c e s s i n g i s p l o -t -t i n g p r o f i l e s and making t h r e e d i m e n s i o n a l p l o t s ( f i g . 14) . A l s o v o l u m e t r i c c a l c u l a t i o n s can be made. T h i s can be done i n h o r i z o n t a l and i n v e r t i c a l s l i c e s ( f i g . 1 5 ) . The v e r t i c a l s e c t i o n s a r e used f o r g e n t l e c o a s t l i n e s , w h i l e t h e h o r i z o n t a l s e c t i o n s a r e used f o r c o a s t l i n e s f a c i n g r e l a t i v e l y s t e e p t i d a l c h a n n e l s .
figure 15: horizontal and vertical volumetric calculations
5.3 A p p l i c a t i o n o f c o a s t a l d a t a bases
The J a r k u s database i s used f o r v a r i o u s p u r p o s e s . I t i s used f o r s a f e t y a s -sessment o f dunes, as d e s c r i b e d above, b u t a l s o f o r o t h e r purposes r e l a t e d t o c o a s t a l movement. C h r o n i c c o a s t a l e r o s i o n can be d e t e r m i n e d q u i t e a c c u r a t e l y w i t h t h e database by s i m p l e v o l u m e t r i c i n t e g r a t i o n . V o l u m e t r i c i n t e g r a t i o n g i v e s t h e t o t a l volume o f sand i n a c o a s t a l s e c t i o n . Changes i n t h e volume i n d i c a t e e r o s i o n o r a c c r e t i o n o f t h e c o a s t l i n e . Recent a n a l y s i s o f changes i n t h e sand volume a l o n g t h e Dutch c o a s t i n d i c a t e e r o s i o n zones and a c c r e t i o n zones. These zones do n o t c o i n c i d e w i t h t h e e r o s i o n were a t t h i s moment f o r example t h e d u n e - f o o t i s e r o d i n g . V o l u m e t r i c a n a l y s i s shows t h e r e a l p r o b l e m , and n o t t h e s h o r t t e r m problem (Verhagen, 1989) .
F i g u r e 16 g i v e s t h e r e s u l t o f t h a t a n a l y s i s .
I n f a c t c o a s t a l e r o s i o n s h o u l d always be i n d i c a t e d as a l o s s o f sand ( i n m /year p e r m c o a s t l i n e ) . R e c e n t l y O e r t e l e t a l . [1989] suggested a s c a l e f o r q u a n t i f y i n g t h e e r o s i o n r a t e . A l t h o u g h i n t h e paper t h e i n t e g r a t i o n range n o t e x p l i c i t l y i s i n d i c a t e d , t h e y suggested a t t h e i r p r e s e n t a t i o n t o i n t e g r a t e o n l y between t h e low-water l i n e and t h e storm surge l e v e l . E x p e r i e n c e f r o m t h e N e t h e r l a n d s show t h a t e s p e c i a l l y below low w a t e r a l s o b i g v a r i a t i o n s may o c c u r . S t e e p e n i n g of t h e u n d e r w a t e r s h o r e l i n e i s i n f a c t a l s o c o a s t a l e r o s i o n . The proposed s c a l e i s a l o g a r i t h m i c one, u s i n g n - u n i t s .
n = 0.72 I n 5V - 1.8
i n which 5V i s t h e y e a r l y change i n beach volume ( m ^ / y e a r ) . T h i s g i v e s t h e f o l l o w i n g n o m e n c l a t u r e : volume change r a t e n - r a t e n o m e n c l a t u r e ( d u t c h d e s c r i p t i o n ) m V y e a r 768 3.0 192 — 2.0 96 ^^^-^^^^^^^^ 1.5 48 — - 1,0 24 0.5 12 •^•^^-^-^--^•^•^--^ 0.0 .„12 -^^^^^^ 0.0 „24 ^0.5 -48 -1.0 ^96 ^^^^^^^ „i_5 ^192 ..^-^^ ^2.0 ^76g ^3_o
P e r s o n a l l y I t h i n k t h e suggested e r o s i o n s c a l e can be used, b u t a v o l u m e t r i c a n a l y s i s s h o u l d be made f o r t h e whole a c t i v e p r o f i l e , i . e . from approx. 2 t i m e s t h e b r e a k e r d e p t h (under low w a t e r ) u n t i l t h e t o p o f t h e dune. When t h i s s c a l e i s a p p l i e d t o t h e N e t h e r l a n d s , we s u f f e r m a i n l y f r o m m i l d and moderate e r o s i o n . Only on few s p o t s we have s t r o n g e r o s i o n . Severe e r o s i o n does n o t o c c u r i n t h e N e t h e r l a n d s , i f one uses a y e a r l y b a s i s . F i g u r e 17 shows t h e d a t a o f t h e Dutch c o a s t (used e r o s i o n d a t a f r o m i n t e r n a l r i j k s w a t e r s t a a t r e p o r t GWAO-88-007; used p e r i o d 1965/1985) .
v e r y severe a c c r e t i o n (zeer s t e r k e aanzanding) severe a c c r e t i o n ( s t e r k e aanzanding)
s t r o n g a c c r e t i o n ( a a n z i e n l i j k e aanzanding) moderate a c c r e t i o n (gematigde aanzanding) m i l d a c c r e t i o n ( e n i g e aanzanding) s t a b l e ( s t a b i e l ) m i l d e r o s i o n ( e n i g e e r o s i e ) moderate a c c r e t i o n (gematigde e r o s i e ) s t r o n g e r o s i o n ( a a n z i e n l i j k e e r o s i e ) severe e r o s i o n ( s t e r k e e r o s i e ) v e r y severe e r o s i o n (zeer s t e r k e e r o s i e )
P o l i c y makers have b i a problems w i t h t h e above s k e t c h e d d e f i n i t i o n of c o a s t a l e r o s i o n . The l o s s of m cannot be seen on t h e beach. I t i s a l s o i m p o s s i b l e t o draw a l i n e on a map, i n d i c a t i n g t h e c o a s t l i n e r e g r e s s i o n , i f one c a l c u l a t e s o n l y t h e l o s s i n ra^^. I n t h e N e t h e r l a n d s i s t h e a d d i t i o n a l problem t h a t t h e B i l l on Waterdefence says t h a t t h e low w a t e r l i n e i s an i n d i c a t o r f o r t h e c o a s t l i n e .
To overcome t h i s p r o b l e m i n t h e N e t h e r l a n d s t h e f o l l o w i n g d e f i n i t i o n o f " C o a s t l i n e " i s g i v e n : A v o l u m e t r i c i n t e g r a t i o n i s made f o r t h e c o a s t a l p r o f i l e f r o m t h e d u n e - f o o t ( a p p r o x . t h e h e i g h t o f a once-a-year s t o r m s u r g e ) u n t i l a d e p t h h under mean low w a t e r l i n e . The v a l u e h i s t h e h e i g h t d i f f e r e n c e between the d u n e - f o o t and t h e low w a t e r l i n e . The r e s u l t o f t h e v o l u m e t r i c a n a l y s i s i s d i v i d e d by 2 h. So t h e r e s u l t i s a d i s t a n c e , which i s g e n e r a l l y i n t h e n e i g h -borhood o f t h e low water l i n e . F i g u r e 18 shows t h e d e f i n i t i o n .
flSP-lijn
figure 18: Definition of coastline
The dune i t s e l f i s n o t p a r t o f t h e i n t e g r a t i o n , because t h e dune i s under j u r i s d i c t i o n of t h e Waterboard, w h i l e beach maintenance i s t h e r e s p o n s i b i l i t y of t h e n a t i o n a l government. A l a n d w a r d movement of t h e " c o a s t l i n e " i s e r o s i o n . The n a t i o n a l p o l i c y i s t o m a i n t a i n t h e c o a s t l i n e on i t s 1990 p o s i t i o n . So e v e r y y e a r t h e p o s i t i o n o f t h e c o a s t l i n e i s c a l c u l a t e d w i t h t h e above d e s c r i b e d me-t h o d , and a r e g r e s s i o n l i n e i s c a l c u l a me-t e d . I f me-t h e r e g r e s s i o n l i n e i n me-t e r c e d e s the 1990 p o s i t i o n , t h e c o a s t l i n e has t o be r e s t o r e d . G e n e r a l l y t h i s w i l l mean t h a t a beach n o u r i s h m e n t has t o be p e r f o r m e d . T h i s beach n o u r i s h m e n t moves t h e c o a s t l i n e t o a more seaward p o s i t i o n . E r o s i o n c o n t i n u e s , t h e c o a s t l i n e moves a g a i n i n a l a n d w a r d d i r e c t i o n u n t i l i t i n t e r c e d e s t h e 1990 p o s i t i o n , and t h e n e x t n o u r i s h m e n t i s due. See f i g u r e 19.
•J—I—I I 4—1—I—I—I—j—I—I—I—t—l—I—I—I—^-j—I—I—I—I—I—t—l—I—I—j—I—1—4.
1990 y e c i r i
figure .1.9: Determination of the required moment of beach nourishment
The a c t i v e h e i g h t i n t h e N e t h e r l a n d s i s g e n e r a l l y 2 0 m ( f r o m t h e t o p o f t h e dune u n t i l t h e d e p t h u n t i l one observes e r o s i o n d u r i n g s t o r m s ) . So g e n e r a l l y a c o a s t l i n e r e t r e a t o f 1 0 m/year causes a volume l o s s o f 200 m'Vyear ( n = 2 ; t h u s m i l d / m o d e r a t e e r o s i o n ) . I n a d e n s e l y p o p u l a t e d c o u n t r y t h i s e r o s i o n i s g e n e r a l -l y seen as a severe p r o b -l e m . So, i n t h e N e t h e r -l a n d s m i -l d e r o s i o n causes a se-v e r e p r o b l e m .
7. r e f e r e n c e s
Ang, A.H.S and Tang, W.N., 1984; P r o b a b i l i t y concepts i n e n g i n e e r i n g p l a n n i n g and d e s i g n , v o l 2. John W i l e y & Sons, New York.
G r a a f f , J . v . d . , 1983; p r o b a b i l i s t i c d e s i g n o f dunes, Proc. C o a s t a l S t r u c t u r e s ASCE, 1983.
G r a a f f , J. v . d . , 1986; p r o b a b i l i s t i c d e s i g n o f dunes; an example from t h e Ne-t h e r l a n d s . C o a s Ne-t a l e n g i n e e r i n g , v o l 9, 1986.
Kaufman, W. and P i l k e y , O.H., 1983; The beaches a r e moving, t h e d r o w n i n g o f America's s h o r e l i n e . Duke U n i v e r s i t y P r e s s , Durham, N.C., 336 pp.
O e r t e l , G.F., Ludwick, J.C. and O e r t e l , D.L.S., 1989; Sand a c c o u n t i n g methodo-l o g y f o r B a r r i e r i s methodo-l a n d sediment budget a n a methodo-l y s i s . Proc. C o a s t a methodo-l Zone '89 C h a r l e s t o n , ASCE, pp 43-61.
S t e e t z e l , H.J. [ 1 9 9 0 ] ; D u r o s t a , a time-dependent c r o s s - s h o r e model f o r extreme c o n d i t i o n s ; D e l f t H y d r a u l i c s , r e p o r t H298; T A W - p u b l i c a t i o n n r . CC90.xx ( i n Dutch)
TAW ( T e c h n i c a l A d v i s o r y committee on W a t e r d e f e n c e s ) . Guide t o t h e assessment o f the s a f e t y o f dunes as a sea d e f e n c e . Report 140. R i j k s w a t e r s t a a t / C U R p u b l i s h i n g f o u n d a t i o n , Gouda, The N e t h e r l a n d s , 1989.
V e i i i n g a , P. Beach and dune e r o s i o n d u r i n g s t o r m s u r g e s , C o a s t a l e n a i n e e r i n a v o l 6 , 1982
V e i i i n g a , P. Beach and dune e r o s i o n d u r i n g s t o r m s u r g e s . D e l f t H y d r a u l i c s Com-m u n i c a t i o n s n r . 372, 1986 ( P h . D . - t h e s i s D e l f t U n i v e r s i t y ) .
Verhagen, H.J. Sand waves a l o n g t h e Dutch c o a s t . C o a s t a l E n g i n e e r i n g , 1989 (13) pp 129-147.
Verhagen, H.J. and V o l k e r , W. S a f e t y a g a i n s t i n u n d a t i o n , t h e Dutch a p p r o a c h . C o a s t a l zone '89, C h a r l e s t o n , ASCE 1989.
AMELAND PGM NR d. n u m b e r o f v a r i a b l e s : 7 •""^ Name T y p T r A B C G E M „ H W D e t N 0.0 0.0 0, 0 Z E E S P D e t N 0.0 0.0 0. 0 D50 Nor N 1 6 4 E - 6 1 5 . 0 E ~ 6 0. 0 Hs_ONZEK Nor N 0. 0 0. 6 0 0. 0 Hh_WATERS E x p N 2 . 6 5 0 9 0 . 3 1 1 7 0. 0 Hh_ONZEK D e t N 0.0 0.0 0. 0 T O E S L A G F A Nor N 0.0 1.0 0. 0 n u m b e r o f c a l c u l a t i o n s : 13 AF-COORDINAAT : 2 8 0 3 0 0 3 2 0 3 4 0 3 6 0 3 3 0 -Fmmp)e. -4. D Mu S i X m i n Xmax 0. 0 0. 0 0. 0 0.0 0. 0 0. 0 0. 0 0. 0 0.0 0. 0 0. 0 1 6 4 E - 6 1 5 . O E - 6 l O ü E -6 5 0 0 E - 6 0,0 0. 0 0. 6 0 0.0 2. 0 0 0.0 4.0 0 . 3 1 1 7 3. 0 6.0 0.0 0. 0 0. 0 0.0 0.0 0. 0 0. 0 1, 0 0. 0 5.0 lO. --420 - 4 4 0 - 4 6 0 - 4 8 0 - 5 0 0 - 5 2 0 R E S U L T PROCEDURE P R O B A B I L , K o s t e r E n g . S o f t w a r e J o b I d e n t i f i c a t i o n = AMELAND PGM NR 4 R e s u l t s o f t h e M e a n - V a l u e a p p r o a c h f o r AF-COORDINAAT : = - 4 . 4 E + 0 0 0 2 P r o b a b i l i t y o f f a i l u r e = 1 . 6 E - 0 0 1 1 T h e Mean V a l u e o f Z i s . . . . = 2 . 1 E + 0 0 0 2 T h e S t a n a r d D e v i a t i o n o f t h e f u n c t i o n Z i s = 3. l E + 0 0 0 1 T h e v a l u e o f B e t a i s = 6 . 6 E + 0 0 0 0 D50 A l p h a ~ 2 [ 1 ] = 0 . 0 5 Hs_ONZEK A l p h a ~ 2 [ 2 ] = 0 . 0 0 Hh_WATERS A l p h a ' 2 C 3 ] = 0 . 8 2 T O E S L A G F A Alpha'"2 C 4 ] = 0 . 1 3 R e s u l t s o f t h e AFDA a p p r o a c h f o r AF-COORDINAAT : = - 4 . 4 E + 0 0 0 2 T h e n u m b e r o f i t e r a t i o n s w a s = T h e v a l u e o f t h e R e l i a b i l i t y -f u n c t i o n i n t h e c a l c u l a t e d d e s i g n p o i n t i s = T h e v a l u e o f B e t a i s = T h e P r o b a b i l i t y o f f a i l u r e i n d e c i m a l s i s T h e a c c u r a c y a c c o r d i n g t o t h e AFDA a p p r o a c h o f t h e P r o b a b i l i t y o f f a i l u r e i s ... = T h e J o i n t P r o b a b i l i t y F u n c t i o n i n t h e c o m p u t e d d e s i g n p o i n t = = 6 . 0 E - 0 0 0 5 = 5 . 1 E - 0 0 0 6 1 . 4 E + 0 0 0 0 T h e v a l u e s o f t h e d e t e r m i n i s t i c v a r i a b l e s G E M _ H U ) = O . O E + 0 0 0 0 Z E E S P = O . O E + O O O O Hh O N Z E K = O . O E+ 0 0 0 0 V a r i a b l e D 5 0 , T y p e N O R D e s g n P o i n t 1 .6 E - 0 0 0 4 A l p h a 0 . 0 1 Mu 1. 6 E - 0 0 0 4 S i g m a 1 . 5 E - 0 0 0 5
s i O ' " - 1 L i l j * N H P i . 5 i f - - O . i i i O - CrlJ S I G H . G O L F H O O G T E I H Ï n 3 G G L F P E K I Ü D E I H Ï S J K Ü K R E L D l H f l E T E R I H C l O ' - G H J - i .T •i . i O 1 2 . 0 0 i S 7 . 0 0 - 2 , 1 . 0 - 3 . 5 - ^ . 2 . D U I H P R D F Ï E L H F S L i i G P R Q F I E L 2 0 H D E K I Q E Ï L H G K O S T U S K H i i f i n J f t f i f t R i i H l ï ¥ P E D H T H D H T L H H E L H H D i S S S i ^ O O 0 i ^ O * i O s
imi 0.7 -0.0 - i - 2 , 1 5 I G N , GDLFHDÖGTE I N E n ] GOLFPEKIODE I N C S J K a R R E L D T H f l E T E R I N L i O " - F N J 3 , 10 1 2 . 0 0 1 5 5 , 0 0
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HHELiiHD iSU 1^20 0 H O ^ 103 ¥ 1 0 - 1 Efl^TNfipj i.i - 1,1 h 0.^ DUINPRÖFIEL HFSLiiGPRöFÏEL 2DNDEK ? Q E5L H G nFïLHGPRDFIEL NET T Ü E S L H G G K E N S P R D F I E L 0,6 A F S L A G H O E V E E L H E D E N A a n z a n d i n g ( e x c l , i n v l o e d l a n g s t r a n s p o r t ) A f s l a g ( e x c l , i n v l o e d l a n g s t r a n s p o r t ) A f s l a g b o v e n r e k e n p e i l ( e x c l . i n v l o e d l a n g s t r , ) T o e s l a g op a f s l a g b o v e n r e k e n p e i l AFSTANDEN A a n z a n d a f s t a n d (X [ A ] - X C B ] ) A f s l a g a f s t a n d ( X C B ] - X C E ] ) E x t r a v e r s c h u i f a f s t a n d t g v l a n g s t r a n s p o r t ( i n B C ) V e r s c h u i f a f s t a n d i n c l . l a n g s t r a n s p o r t (X C B ] - X [ C ] ) V e r s c h u i f a f s t a n d v o o r d e t o e s l a g ( X[ C] - X [ D ] ) V e r s c h u i f a f s t a n d i n c l . l a n g s t r a n s p o r t + t o e s l a g ( X C B] - X[ D 3 ) X c o o r d . v a n a f s l a g p r o f i e l op r e k e n p e i l ( X C C ] ) X c o o r d . a f s l a g + t o e s l a g p r o f i e l op r e k e n p e i l ( X [ D ] )
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