b y
W i l l e m T.Bakker
T h i s r e p o r t has t h e f o l l o w i n g c o n s t i t u a n t s :
I . A p a p e r p r e s e n t e d on t h e Symposium a t Haren,
o r g a n i z e d by t h e E s t u a r i n e and C o a s t a l S c i e n c e s A s s o c i -a t i o n :
ECSA-23; A u g u s t 30- S e p t . 3 , 1993. The Symposium was c a l l e d :
" P a r t i c l e s i n E s t u a r i e s and C o a s t a l W a t e r s " )
The p a p e r has been s u b m i t t e d f o r p u b l i c a t i o n t o t h e N e t h e r l a n d s J o u r n a l o f A q u a t i c E c o l o g y .
I I . A c o m p i l a t i o n o f u n d e r l y i n g c o m p u t a t i o n s .
These a r e c o m p r i s e d i n t h e Supplements A t o E, w h i c h have t h e f o l l o w i n g c o n t e n t : Supplement A: C a l c u l a t i o n o f t i m e - a v e r a g e d p r o d u c t s o f h a r m o n i c a l l y v a r y i n g v a r i a b l e s : H a l f - t i d e a v e r a g e d v a l u e s . Supplement B: D e r i v a t i o n o f e q u a t i o n ( 2 ) . Supplement C: The t i d a l - a v e r a g e d v e l o c i t y b e l o w t h e l e v e l o f LW; D e r i v a t i o n o f e q u a t i o n ( 5 ) .
Supplement D: C a l c u l a t i o n o f d e r i v a t i v e s o f module and argument.
Supplement E: C a l c u l a t i o n s , c o n c e r n i n g a m p l i t u d e and phase i n a c l o s e d c h a n n e l .
by W.T.BAKKER On secondment f r o m R i j k s w a t e r s t a a t a t t h e N e t h e r l a n d s C e n t r e f o r C o a s t a l R e s e r a r c h P.O.Box 5048, 2600 GA D e l f t R u n n i n g head:
Areas l i k e t h e Western Wadden area have a l e n g t h o f a p p r o x i -m a t e l y 1/4 o f a t i d a l wave l e n g t h . T h e r e f o r e t i d a l -m o t i o n i s s e n s i b l e f o r resonance. I t i s i n v e s t i g a t e d , under w h i c h c i r c u m s t a n c e s m o r p h o l o g i c a l i n s t a b i l i t i e s may o c c u r . I n s t a b i l i t i e s a r e d e f i n e d i n t h i s way, t h a t s m a l l d e p t h i n c r e m e n t s i n t e n s i f i e s t h e v e r t i c a l t i d a l a m p l i t u d e i n t h e b a s i n , w h i c h i n t u r n t r i g g e r s i n c r e a s e d e r o s i o n o f t h e b a s i n . For p r i s m a t i c a l c h a n n e l s , o c c u r r e n c e o f i n s t a b i l i t i e s depend on t h e l e n g t h o f t h e b a s i n and t h e b o t t o m r o u g h n e s s . Dimen-s i o n l e Dimen-s Dimen-s c r i t e r i a a r e g i v e n . E f f e c t Dimen-s o f Dimen-s h o a l Dimen-s a r e d i Dimen-s c u Dimen-s Dimen-s e d . T r i g g e r words:
1
EFFECT OF TIDAL RESONANCE
ON THE MORPHOLOGY OF WADDEN AND ESTUARIES.
by
W.T.BAKKER 1 INTRODUCTION.
Since t h e S t a t e Commission Zuyderzee [ 1 ] i t i s known, t h a t t h e Western Wadden area has a l e n g t h o f a p p r o x i m a t e l y 1/4 o f a ^ t i d a l wave l e n g t h \ and t h e r e f o r e t i d a l m o t i o n i s s e n s i b l e f o r r e s o n a n c e . V e r t i c a l t i d a l d i f f e r e n c e s a r e l a r g e r near H a r l -i n g e n t h a n a t Den H e l d e r . I n t h i s paper t h e e f f e c t o f r e s o n a n c e on t h e morphology o f t i d a l b a s i n s l i k e t h e Western Wadden i s i n v e s t i g a t e d . S t a r t i n g f r o m a power-law sand t r a n s p o r t f o r m u l a , a n a l y t i c a l e x p r e s s i o n s can be d e r i v e d f o r t h e t i d e a v e r a g e d l o c a l e r o -s i o n , w h i c h a g a i n can be a v e r a g e d o v e r t h e c h a n n e l l e n g t h . I n c h . 2 t o 3, t h e area i s s c h e m a t i z e d as a p r i s m a t i c t i d a l c h a n n e l , c l o s e d on one s i d e and c o n n e c t e d t o t h e sea a t t h e o t h e r s i d e . I t i s shown, t h a t a n e a r l y - r e s o n a n t t i d a l b a s i n
( w i t h a l e n g t h , somewhat l a r g e r t h a n K/4 does n o t need t o be s t a b l e , i . e . : a s m a l l i n c r e a s e i n d e p t h can i n d u c e a l a r g e r d e p t h i n c r e m e n t .
T h i s h a s t h e f o l l o w i n g r e a s o n . Maximum r e s o n a n c e o c c u r s , when t h e b a s i n has a l e n g t h o f A./4. R o u g h l y , t h e t i d a l wave l e n g t h
i s p r o p o r t i o n a l t o t h e s q u a r e r o o t o f t h e w a t e r d e p t h . Thus, i f a c h a n n e l o r i g i n a l l y i s somewhat l a r g e r t h a n K/4, a s m a l l i n c r e a s e i n d e p t h c o u l d f a v o u r r e s o n a n c e c o n d i t i o n s , as t h e f r a c t i o n " c h a n n e l l e n g t h / w a v e l e n g t h " d e c r e a s e s . The r e s u l t i n g l a r g e r t i d a l a m p l i t u d e a t t h e c l o s e d s i d e o f t h e b a s i n s t i m u -l a t e s t h e r e t u r n c u r r e n t and t h e e r o s i o n p r o c e s s . I n ch.4 i s d i s c u s s e d , how t h e t h e o r y i s a f f e c t e d , i f a d j a c e n t ( t o t h e t i d a l c h a n n e l t h e r e e x i s t s a s h o a l as a k i n d o f p a r a l -l e -l c h a n n e -l ( f i g . 1 ) .
2 HALF-TIDE- AND TIDE-AVERAGED EROSION OF A TIDAL CHANNEL. I n t h i s c h a p t e r , a p r i s m a t i c t i d a l c h a n n e l i s c o n s i d e r e d , c l o s e d on one s i d e and c o n n e c t e d t o t h e sea a t t h e o t h e r s i d e . W i d t h and i n i t i a l d e p t h a r e u n i f o r m i n l o n g i t u d i n a l d i r e c t i o n . The v e l o c i t y v i n t h i s c h a n n e l i s assumed t o have a h a r m o n i c a l component ( w i t h a m p l i t u d e v) and a t i d e - a v e r a g e d component v. We s h a l l r e f e r t o m a g n i t u d e o f v and v and t o t h e cause o f v. ( The wanted r e s u l t o f t h i s c h a p t e r , t h e h a l f t i d e and t i d e a v e r a g e d e r o s i o n o f t h e c h a n n e l , r e s u l t s f r o m a n a l y t i c a l a v e r a g -i n g , s u p p o s -i n g t h a t t h e sand t r a n s p o r t S -i s an m'^ power o f t h e momentaneous w a t e r v e l o c i t y v:
s = s , \ i r - ' \ v ( 1 ) where S j i s t h e t r a n s p o r t ( p e r u n i t o f w i d t h ) i f t h e v e l o c
3 ( i . e . : a f t e r s e d i m e n t a t i o n . For f i n d i n g t h e s o l i d volume, one s h o u l d m u l t i p l y by 1 - p o r e c o n t e n t ) . I n t h e f o l l o w i n g , as an example m w i l l be t a k e n e q u a l t o 3; e x t e n s i o n t o o t h e r v a l u e s o f m can be c a r r i e d o u t i n t h e f u t u r e and i s r a t h e r s t r a i g h t - f o r w a r d . W i t h r e s p e c t t o e m p i r i c a l v a l u e s o f S i and m i s r e f e r r e d t o [ 2 ] . The momentaneous b o t t o m e r o s i o n -Zt, e q u a l s t h e l o n g i t u d i n a l g r a d i e n t o f S, w h i c h can be e x p r e s s e d ( u s i n g ( 1 ) ) i n p r o d u c t s o f v e l o c i t y and l o n g i t u d i n a l v e l o c i t y g r a d i e n t . C o n t i n u i t y o f w a t e r r e l a t e s p e r i o d i c a l v e l o c i t y g r a d i e n t t o v e r t i c a l t i d e . Thus, when a v e r a g i n g p r o d u c t s o f v e l o c i t i e s and v e l o c i t y g r a d i e n t s , t h e phase a n g l e (j) between v e r t i c a l and h o r i z o n t a l t i d e comes i n t o t h e p i c t u r e . L e t t h e n o t a t i o n " ( ) ' " i n d i c a t e a v e r a g i n g p e r h a l f t i d e , between two s u c c e s s i v e t u r n i n g s o f t h e t i d e . T h i s t i m e l a p s has been assumed t o be T/2 ( t h e e f f e c t o f Ö, d e f i n e d below, on t h i s t i m e l a p s has been n e g l e c t e d (Ö « 1) ) .
For m=3, a n a l y t i c a l l y can be d e r i v e d [ 4 ] , t h a t a h a r m o n i c a l v e r t i c a l t i d e and ( i n t h e zone below LW) a t i d e a v e r a g e d v e l -o c i t y V , r e s u l t i n a l o c a l s e d i m e n t a t i o n v e l o c i t y ( Z b ) '
d e p e n d i n g on t h e phase s h i f t (j) between v e r t i c a l and h o r i z o n t a l t i d e and on t h e t i d e - a v e r a g e d l o c a l i n p u t o f w a t e r w i n t h i s
zone: (Zö) = . 3S,v .-.2 1 -10 ö^ + -ö + - +uj5sin(t) -Ö "^0 + — n ( 2 ) i n w h i c h h i s t h e ( u n i f o r m ) d e p t h o f t h e c h a n n e l , z and v a r e t h e a m p l i t u d e s o f v e r t i c a l t i d e and h o r i z o n t a l t i d e r e s p e c t -i v e l y and where uo i s t h e a n g u l a r v e l o c i t y o f t h e ( M 2 ) - t i d e . The meaning o f 5 and w i s r e s p e c t i v e l y :
5 = v/v w = h — ( 3 )
I n t h e case o f one p r i s m a t i c c h a n n e l , w r e s u l t s f r o m t h e t i d e - a v e r a g e d t r a n s p o r t o f w a t e r i n t h e zone between LW and HW; when a d j a c e n t t o t h i s c h a n n e l a s h o a l i s f o u n d ( f i g . 1 ) w
i s a f f e c t e d as w e l l by t h e t i d e - a v e r a g e d t r a n s p o r t o v e r t h e s h o a l .
FIG. 1
( The second t e r m i n s i d e t h e b r a c k e t s o f (2) shows t h e e f f e c t on ( Z b ) ' o f t h e phase c o u p l i n g between v e r t i c a l t i d e z and h o r i
-z o n t a l t i d e V ( l a r g e f o r a s t a n d i n g wave, where z and v a r e o u t o f phase and s m a l l when t h e s e a r e i n p h a s e ) .
The f i r s t t e r m shows t h e e f f e c t o f t h e p e r i o d a v e r a g e d v e l o c -i t y g r a d -i e n t .
5
a r e i n p o s i t i v e x - d i r e c t i o n ; 6 has been assumed s m a l l w i t h r e s p e c t t o 1. Asymmetry o f t h e v e r t i c a l t i d e ( w i t h r e s p e c t t o t h e h o r i z o n t a l ( t i m e - ) a x i s ) can be i m p l e m e n t e d i n (2) i n a more p r a c t i c a l t h a n m a t h e m a t i c a l l y e l e g a n t way, by a s s i g n i n g d i f f e r e n t v a l u e s t o z and v f o r t h e e b b - t i d e and f o r t h e f l o o d t i d e . ( From (2) , a t i d e - a v e r a g e d f i g u r e z^, f o r t h e s e d i m e n t a t i o n v e l -o c i t y f -o r t h e w h -o l e t i d a l c y c l e i n t h e case -o f a s y m m e t r i c a l t i d e i n a p r i s m a t i c c h a n n e l i s f o u n d t o be [ 4 ] ( i n f i r s t o r d e r o f Ö; f o r m = 3) : — -1^2}L(^-iu^2büözs\x\^) ( 4 ) 2 h Now t h e a t t e n t i o n w i l l be f o c u s s e d on t h e m a g n i t u d e o f ö and w, o c c u r r i n g i n ( 2 ) . These v a l u e s depend as w e l l on t h e phase
d i f f e r e n c e (}) between h o r i z o n t a l and v e r t i c a l t i d e . For t h e ( c l o s e d p r i s m a t i c c h a n n e l under c o n s i d e r a t i o n , l o c a l a m p l i t u d e
and phase o f z and v a r e known (see A p p e n d i x ) , i n l i n e a r a p p r o x i m a t i o n , when t h e v e r t i c a l t i d e Z Q a t t h e c l o s e d end i s known; t h a t means: g i v e n t h e c h a r a c t e r i s t i c s t i d a l p e r i o d 7, t h e w a t e r d e p t h ti and t h e b o t t o m f r i c t i o n a n g l e 0, d e f i n e d i n
[ 1 ] and [ 3 ] , where 6 = 0 f o r an i n f i n i t e l y f l a t b o t t o m and 9=45° f o r an i n f i n i t e l y l a r g e f r i c t i o n .
H y d r a u l i c m o t i o n b e i n g known, v a r i a b l e s u;, 6,4) i n (2) can be c a l c u l a t e d . I n t h e f o l l o w i n g , x w i l l be assumed i n seaward c h a n n e l d i r e c -t i o n w i -t h i -t s o r i g i n a -t -t h e c l o s e d end o f -t h e c h a n n e l . C o n c e n t r a t i n g on ö f i r s t : f o r a s i n u s o i d a l wave, t h e a v e r a g e f l u x o f w a t e r p e r t i d e above t h e LW- l e v e l can be c a l c u l a t e d by i n t e g r a t i o n o f t h e v e l o c i t y a t a c e r t a i n l e v e l d u r i n g t h e ( t i m e o f submergence and i n t e g r a t i n g t h i s amount o v e r a l l
l e v e l s between HW and LW. Because o f c o n t i n u i t y , t h i s f l u x s h o u l d pass ( i n o p p o s i t e d i r e c t i o n ) as w e l l between t h e L W - l e v e l and t h e b o t t o m . I t i s assumed, t h a t t h e l a t t e r f l u x i s s t a t i o n a r y and u n i f o r m o v e r t h e d e p t h . A p p r o x i m a t i n g f u r t h e r t h i s s e p a r a t i o n p l a n e between seaward and l a n d w a r d f l u x as h o r i z o n t a l , i . e . e v e r y w h e r e e q u a l t o t h e L W - l e v e l a t t h e c l o s e d end o f t h e c h a n n e l , one f i n d s f o r Ö: 0 = (z/Zo)cos(l) ( 5 ) Here z i n d i c a t e s t h e l o c a l t i d a l a m p l i t u d e and ZQ t h e a m p l i -t u d e a -t -t h e c l o s e d end o f -t h e c h a n n e l . For w i s f o u n d f r o m ( 3 ) : ( 6 )
7
As t h e X - a x i s i n seaward d i r e c t i o n , t h e phase d i f f e r e n c e <\> i s
180° f o r a p r o p a g a t i n g wave and 90° f o r a s t a n d i n g wave.
r i g . 2 F i g . 2 shows ( f o r v a r i o u s v a l u e s o f t h e b o t t o m f r i c t i o n ) t h e change o f <^ a l o n g t h e c h a n n e l i n r e l a t i o n t o t h e c l o s e d end o f t h e c h a n n e l . R e f l e c t i o n o f t h e t i d a l wave a g a i n s t t h e c l o s e d end o f t h e c h a n n e l causes v a r i a t i o n o f (f; t h e v a r i a t i o n ^ r e a c h e s f u r t h e r i n seaward d i r e c t i o n , when t h e b o t t o m f r i c t i o n i s l e s s .
A l s o t h e r e l a t i o n between v and ~ depends on t h e s i t e o f t h e c o n s i d e r e d p o i n t , w i t h r e f e r e n c e t o t h e c l o s e d end o f t h e c h a n n e l . The same h o l d s f o r t h e v a l u e s o f w and 5, g i v e n i n
(6) .
Remembering t h a t ( f o r m = 3 ) g S , i s d i m e n s i o n l e s s (g= a c c e l e r -a t i o n o f g r -a v i t y ) one m-ay w r i t e (4) i n -a d i m e n s i o n l e s s w-ay. D i m e n s i o n l e s s v a r i a b l e s w i l l be d e n o t e d w i t h a s t a r . Thus h' ( e q u a l s /I/ZQ; r e m i n d i n g t h a t e q u a l s -dh/dt, one f i n d s
-dh'/^iuot) f o r I^/UÜZQ. As t h e f o l l o w i n g c o n s i d e r a t i o n s gen-e r a l l y d gen-e a l w i t h a t i m gen-e s c a l gen-e , l a r g gen-e compargen-ed t o t h gen-e t i d a l p e r i o d , i n t h e f o r m e r s e n t e n c e t h e b a r above h has been o m i t t e d , f o r s h o r t n e s s .
Thus (4) g e t s t h e d i m e n s i o n l e s s shape:
= / ( x' . e ) ( 7 )
where x* = x / T { g h and where 9 depends on t h e f r i c t i o n . F u n c t i o n / ( x' , e ) ( o f w h i c h t h e f o r m u l a i s g i v e n i n t h e a p p e n d i x ) f o l l o w s
f r o m (4) t o (6) and t h e h a r m o n i c a l t i d e t h e o r y ( [ 1 ] , [ 3 ] ) . I t i s d e p i c t e d i n f i g . 3 .
9
3 INSTABILITY OF TIDAL CHANNELS
W i t h t h e a i d o f ( 7 ) , t h e m o r p h o l o g i c a l i n s t a b i l i t y o f a c h a n -n e l ca-n be t r a c e d . Here i -n s t a b i l i t y w i l l be d e f i -n e d as a-n a c c e l e r a t i o n o f e r o s i o n {d^h/dt^>0).
Mechanism w i l l be: i n c r e a s e d h g i v e s i n c r e a s e d t i d a l wave l e n g t h ; t h e r a t i o " c h a n n e l l e n g t h / wave l e n g t h " becomes l e s s . When o r i g i n a l l y t h e c h a n n e l was somewhat l a r g e r t h a n 1/4 o f a wave l e n g t h , now a d e c r e a s e o f t h e r e l a t i v e c h a n n e l l e n g t h ( enhances t h e wave a m p l i t u d e Zo a t t h e c l o s e d end o f t h e
chan-n e l . T h i s may echan-nhachan-nce t h e e r o s i o chan-n p r o c e s s i chan-n t h e c h a chan-n chan-n e l , e t c . I n t h e f o l l o w i n g some s i m p l i f y i n g a s s u m p t i o n s w i l l be made: t h e e f f e c t o f changes i n f r i c t i o n ( i n a n g l e 9 and t h u s i n / ( . \ : ' , e ) ) w i l l be n e g l e c t e d . As 9 w i l l d e c r e a s e d u r i n g t h e p r o c e s s d e s c r i b e d , t h e p r o b a b i l i t y o f i n s t a b i l i t y w i l l be u n d e r -e s t i m a t -e d . L e t x's be t h e d i m e n s i o n l e s s c h a n n e l l e n g t h and l e t ƒ be t h e mean v a l u e o f f between 0 and . V j .
( I f 7 > 0 , no i n s t a b i l i t y has t o be e x p e c t e d . I f 7<0, t a k e as t i m e s c a l e toi
(„ = - 7 g S , 7 / 2 n
Thus to w i l l be l a r g e r t h a n z e r o .
I n s t a b i l i t y b e i n g i n d u c e d by t h e change o f Zo when h changes, a dependent v a r i a b l e h' = h/z, ( i n w h i c h z, i s t h e a m p l i t u d e a t
t h e seaward b o u n d a r y ) , o n l y c h a n g i n g i n c o u r s e o f t i m e because h changes, has t o be p r e f e r r e d above h'. D e n o t i n g t/to by t \ and
z,/Zo by zl, t h e m u t a t i o n s i n d i c a t e d above t r a n s f e r (7) t o :
'JL=[^h'z:)\h-z:-\)]" ( 9 )
dt
Thus (9) shows, t h a t i n s t a b i l i t y o c c u r s , when h'z] d e c r e a s e s i n t h e c o u r s e o f t i m e . T h i s means ( a s dh'/dt' i s p o s i t i v e ) t h a t d{h'z])/dh' s h o u l d be n e g a t i v e . Thus t h e s t a b i l i t y c r i t e r i o n i s : dz' 1 + — > 0 ^stable ( 1 0 ) dh
Here one f i n d s dzl/dh as:
dz s _ d Z s dx ( 1 1 ) Th^J^P' dh D i f f e r e n t i a t i o n o f x' = x/T gh to h g i v e s -x'/(2h) as r e s u l t . Thus t h e s t a b i l i t y c r i t e r i o n becomes: l i ^ < ^ ^stable ( 1 2 ) d X X Here A ' d e n o t e s t h e c h a n n e l l e n g t h , e x p r e s s e d i n T^Jgh, w h i c h s h o u l d be t h e wave l e n g t h , when f r i c t i o n was n e g l e c t e d (see
( A 8 b ) ) .
11
above t h e l i n e 2/x* ( d e n o t i n g a c e r t a i n d i m e n s i o n l e s s c h a n n e l l e n g t h and a c e r t a i n f r i c t i o n a n g l e 9 ) show p o t e n t i a l i n s t -a b l e c h -a n n e l s , -as l o n g -as ƒ ( c f . f i g 3) r e m -a i n s n e g -a t i v e . When t h e c h a n n e l l e n g t h i s l e s s t h a n h a l f a wave l e n g t h and t h e f r i c t i o n i s v e r y s m a l l , no i n s t a b i l i t i e s a r e f o u n d . How-e v How-e r , a l s o t h How-e n , f o r a c h a n n How-e l l How-e n g t h o f a b o u t .3 X., t h How-e chan-n e l i s ochan-n t h e v e r g e o f s t a b i l i t y . C FIG.4 4 EFFECT OF SHOALS I n t h i s s e c t i o n t h e s i t u a t i o n w i l l be d i s c u s s e d , t h a t a d j a c e n t t o t h e t i d a l c h a n n e l t h e r e e x i s t s a s h o a l , as a k i n d o f p a r a l -l e -l c h a n n e -l ( f i g . 1 ) . S t a r t i n g f r o m t h e L o r e n t z t i d a l t h e o r y , as b e f o r e , t h e f o l l o w i n g e f f e c t s o f t h o s e s h o a l s can be d i s t i n g u i s h e d . The c e l e r i t y o f t h e v e r t i c a l t i d e w i l l d e c r e a s e . I n L o r e n t z s c h e d u l e [ 1 ] r e f r a c t i o n e f f e c t s a r e n e g l e c t e d : i t ^ i s assumed t h a t t h e c r e s t o f t h e t i d a l wave i s p e r p e n d i c u -l a r t o t h e c h a n n e -l a x i s , i n d e p e n d e n t o f d e p t h changes i n t r a n s v e r s a l d i r e c t i o n . A way o f c a l c u l a t i o n o f t h e wave c e l e r i t y i n such a c h a n n e l w i t h c o m p l i c a t e d p r o f i l e i s g i v e n i n [ 3 ] ( i t i s l e s s t h a n t h e one i n t h e c h a n n e l , l a r g e r t h a n t h e one on t h e s h o a l ) . By t h i s e f f e c t , wave l e n g t h w i l l be s h o r t e n e d . Bakker
t h e h o r i z o n t a l t i d e on t h e s h o a l w i l l d i f f e r ( i n phase and a m p l i t u d e ) f r o m t h e one i n t h e a d j a c e n t c h a n n e l . Thus t h e phase a n g l e (f) between h o r i z o n t a l and v e r t i c a l t i d e w i l l be d i f f e r e n t on t h e s h o a l , compared t o t h e a d j a c e n t c h a n n e l . The v a l u e o f w i n t h e t i d a l c h a n n e l w i l l become ( o f t e n much) l a r g e r t h a n c a l c u l a t e d i n ch,2, because o f t h e e f f e c t , d e p i c t e d i n f i g 1 ( d i s c h a r g e o f w a t e r t h r o u g h t h e ^ c h a n n e l d u r i n g ebb t i d e , w h i c h i s s u p p l i e d o v e r t h e s h o a l d u r i n g f l o o d t i d e ) .
5 DISCUSSION AND FUTURE RESEARCH
The p r e s e n t c o m p u t a t i o n s do n o t p r o v i d e more t h a n a p i l o t s t u d y . I t i s t h e i n t e n t i o n t o e x t e n d t h e c o m p u t a t i o n s i n t h e f u t u r e t o n e t w o r k c o m p u t a t i o n s a c c o r d i n g t o t h e L o r e n t z model. Then e f f e c t s o f s h o a l s and o f c h a n n e l s , becoming s h a l l o w e r i n
l a n d w a r d d i r e c t i o n can be i n c l u d e d q u a n t a t i v e l y i n s t e a d o f q u a l i t a t i v e l y .
( 6 SUMMARY
T i d a l b a s i n s l i k e t h e Western Wadden a r e a ( h a v i n g a l e n g t h o f a p p r o x i m a t e l y 1/4 o f a t i d a l wave l e n g t h ) can become m o r p h o l -o g i c a l l y i n s t a b l e , because t h e t i d a l m -o t i -o n i s s e n s i b l e f -o r r e s o n a n c e .
T h i s s t a t e m e n t has been e v a l u a t e d i n t h e p r e s e n t p a p e r o n l y q u a n t i t a t i v e l y f o r an on one s i d e c l o s e d p r i s m a t i c c h a n n e l ;
f o r c h a n n e l s w i t h a d j a c e n t s h o a l s q u a l i t a t i v e c o n s i d e r a t i o n s a r e g i v e n . Under some c i r c u m s t a n c e s ( f o r w h i c h d i m e n s i o n l e s s c r i t e r i a a r e g i v e n , f i g . 4 ) s m a l l i n c r e m e n t s i n d e p t h i n t e n s i f y t h e v e r t i c a l t i d a l a m p l i t u d e a t t h e l a n d w a r d s i d e o f t h e b a s i n , w h i c h i n t u r n t r i g g e r s i n c r e a s e d e r o s i o n o f t h e b a s i n . For p r i s m a t i c a l c h a n n e l s , o c c u r r e n c e o f i n s t a b i l i t i e s depends on t h e l e n g t h o f t h e b a s i n and on t h e b o t t o m r o u g h n e s s . 7 ACKNOWLEDGEMENTS C o l l a b o r a t i o n w i t h p r o f . d r . i r H.J. de V r i e n d i s g r a t e f u l l y a c k n o w l e d g e d .
8 APPENDIX. THE FUNCTION /(.v'.G)
The f u n c t i o n f ( x ' . B ) has been d e r i v e d f r o m ( 4 ) . The e q u a t i o n has been made d i m e n s i o n l e s s by d i v i d i n g by -COZQ; t h u s emerges t h e l e f t h a n d s i d e o f ( 7 ) .
D i m e n s i o n l e s s v a r i a b l e s have been i n t r o d u c e d :
2' = 2/Zo Xj* - V / Z oigTh (/lla.ö) iA2a,b)
A l s o Ö i s r e p l a c e d by a more c o n c i s e v a r i a b l e 6*:
14
and t h u s , a c c o r d i n g t o ( 5 ) , 5 = ö*/(/i'- 1). Thus (6) r e a d s , i n a d i m e n s i o n l e s s way: 1 f . , d b ' w .30' ouZo 2 n ( / i ' - l ) V ^x' d x ' j So t h e d i m e n s i o n l e s s shape o f (4) i s Tl gS 1 (/14) cjuZo h ' \ h ' - \ ) 2 3 - * 7 .-V
I
V
:+5 : +2Ö s i n c f )
(/15) _ 2 i i ^ ax a.YU s i n g (7) and ( A 5 ) , and a f t e r a f t e r s u b s t i t u t i o n o f ( A 3 ) , one can e x p r e s s /(x'.B) i n v a l u e s , t o be d e r i v e d f r o m t h e L o r e n t z t h e o r y [ 1 ] , [ 3 ] : - 1 r . . dz* — V z sin(t) + v coscp-—: 4nV dx dx COS(j) dv dx 'cos(J)sin (j) (.46) T h i s f u n c t i o n has been d e p i c t e d i n f i g . 3 . I t i s memorized, t h a t i n t h e L o r e n t z t h e o r y v e r t i c a l and h o r i z o n t a l t i d e a r e f o u n d as t h e r e a l p a r t o f t h e complex v a l u e s and v' (made d i m e n s i o n l e s s a c c o r d i n g t o ( A l a , b ) and t o be m u l t i p l i e d w i t h exp(icuO) '
z* = cos kx V =vcos29.i slnkx {A7)
w i t h : (JO /c = - ( l - t t a n e ) c c = vg/^( 1 -tan-^e) (/18a,Ö) Bakker
The d i f f e r e n c e i n argument between =' and v' i s d e p i c t e d i n f i g . 2 ; i) and z a r e f o u n d as t h e modules o f z' and v .
9 REFERENCES
[ 1 ] LORENTZ, H.A, e t a l . 1926. R e p o r t o f t h e S t a t e Commission Z u i d e r z e e . The Hague, A l g . L a n d s d r u k k e r i j .
[ 2 ] RUN, L.C.v. & A.J. de LEEUW. 1985. Sand t r a n s p o r t model W e s t e r n S c h e l d t ; s t o c h a s t i c a l r e s e a r c h on a sand t r a n s p o r t
f o r m u l a . D e l f t Hydr.Lab. rep.S 582 ( i n d u t c h )
[ 3 ] THIJSSE, J.Th. I 9 6 0 ? . Theory o f T i d e s . L e c t u r e n o t e s ; D e l f t U n i v e r s i t y o f Technology and NUFFIC
[ 4 ] BAKKER, W.T. 1993. F l i r t i n g w i t h Anna L y s e . N e t h e r l a n d s C e n t r e o f C o a s t a l Research; i n t e r n a l r e p o r t
16 i Legends of f i g u r e s F i g . 1 . D u r i n g t h e f l o o d w a t e r i s t r a n s p o r t e d o v e r t h e s h o a l . T h i s w a t e r has t o be d i s c h a r g e d a g a i n ( t h o u g h t h e c h a n n e l ) d u r i n g t h e ebb. F i g . 2. The change o f <^ a l o n g t h e c h a n n e l ( f o r v a r i o u s v a l u e s of t h e b o t t o m f r i c t i o n ) i n r e l a t i o n t o t h e c l o s e d end o f t h e
c h a n n e l . The h o r i z o n t a l s c a l e i s x / T i g h , where igh i s t h e p r o p a g a t i o n v e l o c i t y o f a p r o g r e s s i v e t i d a l wave, when f r i c -t i o n e f f e c -t s can be n e g l e c -t e d ; T d e n o -t e s -t h e -t i d a l p e r i o d F i g . 3 . D i m e n s i o n l e s s r a t e o f a c c r e t i o n /(.v'.G) as f u n c t i o n o f d i m e n s i o n l e s s d i s t a n c e f r o m t h e c l o s e d end o f t h e c h a n n e l . F i g . 4 . I n s t a b i l i t y o c c u r s , when t h e l o n g i t u d i n a l g r a d i e n t of t h e t i d a l a m p l i t u d e exceeds 2/x'. T h i s happens f o r l a r g e f r i c t i o n a n g l e 9 or f o r l a r g e c h a n n e l l e n g t h s . Bakker
HARNSUP .DOC November 1 1 , 1993 T a b l e o f C o n t e n t s Supplement A. C a l c u l a t i o n o f t i m e - a v e r a g e d p r o d u c t s o f h a r m o n i c a l l y v a r y i n g v a r i a b l e s . H a l f - t i d e a v e r a g e d v a l u e s s 1 Supplement B. D e r i v a t i o n o f e q u a t i o n (2) - s 3 Supplement C. The t i d a l a v e r a g e d w a t e r m o t i o n . D e r i v a -t i o n o f e q u a -t i o n (5) C a I n t r o d u c t i o n t o s u p p l e m e n t C - s 6 C.b T i d a l - a v e r a g e d c o n t i n u i t y e q u a t i o n - s 6 C.c The m a g n i t u d e o f w and ö t o be used f o r t h e
( c o m p u t a t i o n o f e b b - s e d i m e n t a t i o n i n a c l o s e d s i n g l e
p r i s m a t i c c h a n n e l - s 6
Supplement D. C a l c u l a t i o n o f d e r i v a t i v e s o f module and
- S 17 argument D.a Problem " s 12 D.b C a l c u l a t i o n o f t h e d e r i v a t i v e o f t h e module - s 12 D. c C a l c u l a t i o n o f t h e d e r i v a t i v e o f t h e a r g u m e n t . ... - s 14 Supplement E. C a l c u l a t i o n s c o n c e r n i n g a m p l i t u d e and phase i n a c l o s e d c h a n n e l " s 16 E. a I n t r o d u c t i o n t o Supplement E - s 16 E.b The a m p l i t u d e o f t h e v e r t i c a l t i d e - s 16 E.C The a m p l i t u d e o f t h e h o r i z o n t a l t i d e - s 17 E.d The phase d i f f e r e n c e between v e r t i c a l and h o r i z o n
-t a l -t i d e - s 18 E.e The h o r i z o n t a l g r a d i e n t o f t h e a m p l i t u d e o f t h e
v e r t i c a l t i d e s 19 E.f The h o r i z o n t a l g r a d i e n t o f t h e phase d i f f e r e n c e
between v e r t i c a l and h o r i z o n t a l t i d e s 19 i R e f e r e n c e s " ^ 20
Supplement A. C a l c u l a t i o n of time-averaged p r o d u c t s o f har-m o n i c a l l y v a r y i n g v a r i a b l e s . H a l f - t i d e averaged v a l u e s . I n t h i s supplement w i l l be c a l c u l a t e d : t h e t i m e - a v e r a g e d v a l u e o v e r a h a l f p e r i o d ( f r o m 0 t o T/2) o f : sin"oot s i n ( a j t +<()') Thus, i n t h e n o t a t i o n o f t h i s r e p o r t , t h i s e x p r e s s i o n w i l l be d e n o t e d as: { s i n " a ) t sin(cA)i +(|)')} S o l u t i o n :
sin"(jut sin((jüt+([)') = sin"*^ cose})' + sintj)' sin"cA^? cosoot ( A l ) The l a s t t e r m o f ( A l ) i s a n t i s y m m e t r i c a l between 0 a n d T/4 on
one hand and between T/4 and T/2 on t h e o t h e r hand; t h u s t h e l a s t t e r m c a n c e l s i t s e l f when a v e r a g i n g between 0 a n d T/2:
{sin"cjof sin(uof + (t)')>* = {sin""^coO*cos(t)' (A2)
I n t h i s way t h e p r o b l e m has been r e d u c e d t o f i n d i n g t h e mean v a l u e (between 0 and T/2) o f sin""^cot. C a l l : o j f ^ i p , t h e n t h e v a l u e o f t h e f o l l o w i n g e x p r e s s i o n w i l l be w a n t e d : (sin^^^uot)* = - f s i n " ' ^ ^ d^\) Tt »/ 0 F i r s t t h i s e x p r e s s i o n w i l l be c a l c u l a t e d f o r even v a l u e s o f n and t h e n f o r odd v a l u e s . Even v a l u e s o f n: It
sin""^ip dip = - r (1 - cos^\p)"^^d(cosa|;) J
0 II
sin^'^ip di|; = - I (1-cos^ip)"''^d(cosi|;) J
Sin" > d ^ = Z (-1) , x^^dx k = 0 k
HARNSUP1.DOC S 2 November 1 1 , 1993 i x ^ ' d x = ^ I 2 / c+ l - 1 2 - 1 + 1 x^'^dx = 2k+ 1 2 J i j Tlfcro" ~2k+ 1 n/2
s i n " ^ > dop = - Z (R even) (A3)
Thus one f i n d s f o r n = 2: n J s i n ^ i p di|j = 2 ( s i n .p) = -r 5 , ^ 4 2 16 j s m ^ dip = 2 g =
-^^^^ ^) = T ^
The f o l l o w i n g way o f c o m p u t a t i o n i s v a l i d as w e l l f o r odd as even v a l u e s o f n; however, t h e c o m p u t a t i o n above l e a d s ( f o r even v a l u e s o f n) q u i c k e r and s i m p l e r t o i t s g o a l .
n
n+ 1
sin"^^T|; d ^ = — ^ ( Y ""l^ e'^'^'-''-'\-l)''^'-'d^\) (A4)
( 2 0 J fc=oV k J
When 2 k - n - l = 0 , i . e . k = (n+ l)/2, t h e e-power i n (A4) becomes 1 ; t h e i n t e g r a l o f t h a t s p e c i f i c t e r m w i l l be /dap; t h u s t h e v a l u e Jt w i l l be t h e r e s u l t . W i t h r e s p e c t t o t h e o t h e r t e r m s u n d e r t h e i n t e g r a l : i t i s assumed, f o r t h e p r e s e n t c o m p u t a t i o n t h a t n i s odd; t h u s t h e f o l l o w i n g r e l a t i o n can be u s e d : j ^ i ( 2 k - n - l ) 2
Thus i t i s c l e a r , t h a t t h e t e r m s i n q u e s t i o n a r e z e r o when n i s odd and k^(n+ l)/2. Thus one f i n d s :
n+ 1 0 Summarizing t h e r e s u l t s : . • ^* 2 ( s m i p ) = -( s i n ^ i p ) ( s i n ^ i p ) " (sin'^ip) ( s i n ^ i p ) J l 1 2 3n 3 8 16 15JI _5_ 16 Supplement B. D e r i v a t i o n of e q u a t i o n (2) I n t h i s s u p p l e m e n t , e q u a t i o n (2) w i l l be d e r i v e d f r o m e q u a t i o n ( 1 ) . U s i n g t h e t r a n s p o r t e q u a t i o n ( 1 ) , one f i n d s f o r t h e l o c a l e r o s i o n : dx ^ dx T h i s u n d e r t h e a s s u m p t i o n , t h a t v i s p o s i t i v e . The t i m e l a p s t o be c o n s i d e r e d i n t h e f o l l o w i n g i s t h e t i m e between two s u c c e s s i v e t u r n i n g s o f t h e t i d e (a " h a l f t i d e " ) and t h e r e f o r e t h e e f f e c t o f t h e a b s o l u t e s i g n i n (1) can be d i s r e -g a r d e d . N e g l e c t i n g t h e e f f e c t o f b o t t o m changes, t h e e q u a t i o n f o r momentaneous c o n t i n u i t y o f w a t e r g i v e s : -h — +w = — ( ö 2 ) dx dt w
HARNSUP1.DOC S 4 November 1 1 , 1993 h — = w QB3) and s u b t i t u t i n g (B3) i n t o { B 2 ) : dx dt T i d a l a v e r a g e s a r e i n d i c a t e d b y a b a r and f i r s t h a r m o n i c s b y a t i l d e . S p l i t t i n g ( B l ) i n t o means and h a r m o n i c s g i v e s : dS „ - ..m-lf dv dv\ r n c • ^ — = mSi(ü + ö) — + — ( 5 5 ) dx \dX dx J I n t h e f o l l o w i n g , t h e symbol " ( ) * " w i l l be used f o r a v e r a g -i n g o v e r a h a l f t -i d e , s t a r t -i n g f r o m and f -i n -i s h -i n g w -i t h t u r n -i n g o f t h e t i d e 2. S p l i t t i n g (B5) i n t o t e r m s , t h e p e r i o d i c a l t e r m s w i l l be powers o f V, w h i c h may o r may n o t be m u l t i p l i e d w i t h a l i n e a r power o f
dv/dx. A c c o r d i n g t o (B4) dv/dx i s p r o p o r t i o n a l t o dz/dt, o f
w h i c h t h e phase i s 90° ahead w i t h r e s p e c t t o z. D e n o t i n g t h e phase d i f f e r e n c e between v and z b y ((), t h e phase d i f f e r e n c e between v and dv/dx w i l l be ^ + 9Q°. I n Supplement A t h e f o l l o w i n g r e l a t i o n i s d e r i v e d f o r t h e h a l f - p e r i o d a v e r a g e o f a p r o d u c t o f 2 h a r m o n i c a l f u n c t i o n s v and z', h a v i n g a phase d i f f e r e n c e (j)': (ö".z')' = X„.iö";z'cos(t)' ( 5 6 ) Here, Xn d e n o t e s : X,_ = (sin"(JoO* (^7) The c o e f f i c i e n t Xn, d e p e n d i n g on n, i s c a l c u l a t e d as w e l l i n Supplement A, f o r n=1..6. Denote 5 a s : b = v/v ( 5 8 )
Thus one c a n c a l c u l a t e (depending on t h e v a l u e o f m) t h e s e d i m e n t a t i o n o v e r a h a l f - p e r i o d ^ , a p p l y i n g (B3) t i l l ( B 8 ) . I n
2 T h i s t i m e l a p s w i l l be assumed t o be T/2. The e f f e c t o f v on t h i s t i m e l a p s w i l l be n e g l e c t e d .
t h e f o l l o w i n g , as an example, t h e r e s u l t s w i l l be g i v e n f o r m = 3 a n d m = 4 r e s p e c t i v e l y ^ . F u r t h e r m o r e , i t i s assumed: 5 > 0 . For m = 3 one f i n d s : = - 3 u ; ö ^ ( 5 ' + 2 X i 5 + X2) + 3 ^ ^ ö ' ' s i n K X i 5 + 2 X 2 Ö + X3) m = 3; ( 5 9 a ) S u b s t i t u t i o n o f t h e v a l u e s o f Xm c a l c u l a t e d i n Supplement A g i v e s as r e s u l t : — =-3wv^ 5^ + - ö + - + 3 ü o z ö ^ s i n ( l ) - 6 ^ + 5 + — Si\dxJ \ n 2J \n 3 n m = 3; ( 5 1 0 a ) As t h e t r a n s p o r t g r a d i e n t e q u a l s t h e l o c a l e r o s i o n , (BlOa) r e s u l t s i n t h e wanted e q u a t i o n (2) f o r t h e h a l f - t i d e a v e r a g e d a c c r e t i o n (Zt,) . For m = 4 t h e same k i n d o f c o m p u t a t i o n s r e s u l t s i n t o : _ A [ ^ = - 4 u ; ö ^ ( 6 ^ + 3 X i 5 ' + 3x25 + X3) + 4 a ) z ö ' s i n ( t ) ( x , ö ' + 3 x 2 ö ' + 3x38 + X4) m = 4; (59Ö) For Tn = 4 s u b s t i t u t i o n o f t h e v a l u e s o f Xn, c a l c u l a t e d i n Sup-p l e m e n t A g i v e s as r e s u l t : h f d S Y . . 3 r « 3 , 6 . . 3 _ 4 V , . . _ . , 3 „ . „ J 2 , 3 ^ 3 ^ 2 ^ 4 ^ ^ 3 s A ^ x J \ n 2 3nJ "^Kn 2 n 8 m = 4; ( 5 1 0 b ) E q u a t i o n (BIO) i l l u s t r a t e s , t h a t t h e s e d i m e n t a t i o n p e r h a l f -p e r i o d i s a f u n c t i o n o f t h e l o c a l w a t e r s u -p -p l y w ( f i r s t t e r m on t h e r i g h t h a n d s i d e ) and o f t h e l o c a l w a t e r s t o r a g e uoz.
The second t e r m d i s a p p e a r s f o r a p u r e l y p r o p a g a t i n g wave ( w i t h o u t f r i c t i o n ) .
Only when t h e r e a r e phase d i f f e r e n c e s between z and v (by r e f l e c t i o n o r b y f r i c t i o n ) s e d i m e n t a t i o n o c c u r s , w h i c h (as w e l l as t h e e r o s i o n caused by t h e w a t e r s u p p l y w) i s enhanced by t h e e f f e c t o f t h e s t a t i o n a r y c u r r e n t .
HARNSUP1.DOC S 6 November 1 1 , 199 3
Supplement C. The t i d a l - a v e r a g e d w a t e r motion. D e r i v a t i o n o f e q u a t i o n (5) C a I n t r o d u c t i o n to supplement C E q u a t i o n (2) g i v e s a f o r m u l a f o r t h e s e d i m e n t a t i o n p e r h a l f -t i d e , d e p e n d i n g on 5, b e i n g -t h e r a -t i o be-tween -t h e mean w a -t e r v e l o c i t y d u r i n g t h e h a l f - t i d e and t h e t i d a l a m p l i t u d e . I n t h e p r e s e n t s e c t i o n t h i s t i d e - a v e r a g e d w a t e r m o t i o n i n a t i d a l c h a n n e l w i l l be c a l c u l a t e d . F i r s t ( s e c t . C b ) a t i d a l - a v e r a g e d c o n t i n u i t y e q u a t i o n w i l l be d e r i v e d . I n o r d e r t o make t h e r e s u l t f i t f o r f u t u r e e x t e n s i o n s o f t h e t h e o r y , t h e p r o b l e m i s posed more g e n e r a l t h a n n e c e s s a r y f o r t h e p r e s e n t p a p e r . Thus, i n s t e a d o f t r e a t i n g o n l y one p r i s m a t i c c h a n n e l , a s y s t e m o f p a r a l l e l c h a n n e l s ( i n c l u d i n g s h o a l s above l o w w a t e r l e v e l ) w i l l be c o n s i d e r e d i n t h i s c o n t i n u i t y e q u a t i o n . Eqn. (C 12) b e l o w r e l a t e s t h e t o t a l mass o f w a t e r , t r a n s p o r t e d above l o w w a t e r l e v e l ( o n one hand) t o t h e t i d e - a v e r a g e d r e s u l t i n g d i s c h a r g e b e l o w t h e l o w - w a t e r l e v e l on t h e o t h e r hand. T h i s g i v e s a n i m p r e s s i o n o f t h e mean v e l o c i t y v t o be t a k e n i n t o a c c o u n t f o r t h e e b b - t i d e c o m p u t a t i o n ; f o r a s i m p l e p r i s m a t i c c h a n n e l t h i s v e l o c i t y w i l l be c a l c u l a t e d , t h u s g i v i n g v a l u e s o f w and 5, d e f i n e d i n eqn. (3) o f t h e p a p e r . T h i s g i v e s t h e p o s s i b i l i t y t o e l a b o r a t e t h e f o r m u l a e f o r t h e s e d i m e n t a t i o n v e l o c i t y (2) somewhat f u r t h e r ( s e c t . C 3 ) . S t i l l , t h i s o n l y h o l d s f o r a p r i s m a t i c c h a n n e l . C a l c u l a t i o n s c o n c e r n i n g t h e d i m e n s i o n l e s s t i d e a v e r a g e d v e l -o c i t y 5 i n a s e t -o f p a r a l l e l c h a n n e l s a r e p -o s t p -o n e d t -o a l a t e r d a t e . C b T i d a l - a v e r a g e d c o n t i n u i t y e q u a t i o n . I n t h i s s e c t i o n f i r s t ( e q n . ( C . 1 ) ) t h e t i d a l a v e r a g e d c o n t i -n u i t y e q u a t i o -n w i l l be d e r i v e d f o r a p r i s m a t i c a l c h a -n -n e l , i -n w h i c h t h e c h a n n e l d e p t h i s l a r g e r t h a n t h e t i d a l a m p l i t u d e . L a t e r o n (eqn (C2) e t c . ) t h e s c h e m a t i z a t i o n i s made more i n t r i c a t e : t h i s e q u a t i o n d e a l s on a s e t o f p a r a l l e l chan-n e l s , i chan-n c l u d i chan-n g a s h o a l . For a p r i s m a t i c a l c h a n n e l w i t h d e p t h h t h e t i d e - a v e r a g e d ^ c o n t i n u i t y e q u a t i o n r e a d s ^ :
Vf
I vdz dt = constant (.Cl) 0 -h 5 o r " t i d e - i n t e g r a t e d " ( w h i c h o n l y i m p l i e s m u l t i p l i c a t i o n o f t h e f o r m u l a w i t h T) 6 e v i d e n t l y i n r e a l ( n o t complex) n o t a t i o nwhere z(t) d e n o t e s t h e v e r t i c a l c o o r d i n a t e o f t h e w a t e r l e v e l , as u s u a l .
C o n s i d e r now a s e t o f p a r a l l e l c h a n n e l s , i n c l u d i n g as w e l l : a s h o a l , w h i c h emerge above w a t e r a t some t i d a l phase. Then eqn ( C l ) c a n be g e n e r a l i z e d t o : 0 -/i„ T B jVjdz dt = constant C C 2 ) j= 1 i n w h i c h J i s t h e number o f c h a n n e l s , i s t h e d e p t h o f t h e d e e p e s t o f t h e J c h a n n e l s , Bj i s t h e w i d t h o f t h e j'" c h a n n e l and Vj i t s momentaneous v e l o c i t y (assumed u n i f o r m o v e r t h e c r o s s - s e c t i o n o f t h e c h a n n e l ) . The s h o a l , a t l e v e l -h^h i s a s s i g n e d as " c h a n n e l number 1 " ; i t s w i d t h and v e l o c i t y w i l l be c a l l e d Band u.^ r e s p e c t i v e l y . Conform t o t h e L o r e n t z a s s u m p t i o n s ( [ 3 ] , [6] 7) z(t) w i l l be assumed e q u a l f o r a l l p a r a l l e l c h a n n e l s . E v i d e n t l y t h e r e w i l l be assumed: Vj=0 f o r
i n t e g r a t i o n l e v e l s l o w e r t h a n t h e t h e b o t t o m l e v e l -hj o f t h e 7'"' c h a n n e l . The phase d i f f e r e n c e b e t w e e n z( 0 and Vj w i l l be
c a l l e d ^ j . W i t h o u t l o s s o f g e n e r a l i t y a t ? = 0 t h e r e may be assumed, t h a t z( 0 ) = z : z ( t ) = zcosuot ( C 3 ) Thus (C2) t r a n s f e r s i n t o : cosooi zcoscof ^ "N ^ B,f^v^^dz+ j Y B jV j d z \ dt + T Y B j{h j~ zyiTj^ constant ( C 4 ) Here v~j i s t h e p e r i o d - a v e r a g e d v e l o c i t y i n t h e p a r t o f t h e c r o s s - s e c t i o n o f t h e c h a n n e l b e l o w LW; t h i s v e l o c i t y i s assumed u n i f o r m o v e r t h a t p a r t o f t h e c r o s s - s e c t i o n . For c o m p u t a t i o n o f t h e i n t e g r a l s i n (C4) t h e s u c c e s s i o n o f i n t e g r a t i o n ( f i r s t t o z and t h e n t o t ) i s r e v e r s e d . A c e r t a i n l e v e l "z"8 i s u n d e r w a t e r : f r o m t=0 t o t=t^ a n d f r o m t=T-t^ t o t = T, where: t., = —arccos{z/z) ( C S ) 00
HARNSUP1.DOC S 8 - November 1 1 , 1993 I n t h i s way (C4) changes i n t o : Bsh f [ ^ s h d t d z + T Bj f f V j d t d z + T B j ( h j - z ) V j = constant ( C 6 ) y - 2 - z - t . J-2 For t h e " s t r o k e l e n g t h " ^ a t a c e r t a i n l e v e l z, t h e i n n e r i n t e g r a l o f t h e s e c o n d t e r m o f (C6), one f i n d s : 'z ö , c o s ( c o t - ( t ) , ) d i = — { s i n ( c » j f _ - ( ( ) , ) + s i n ( a ) ( ^ + ((),)> f ^ i J VjCos^uot- ^ j')dt = 2 — sin(A)^^cos(t)y ( C 7 ) where, a c c o r d i n g t o (C5) s i n ( A ) t ^ = s i n { a r c c o s ( z / 2 ) } = v l - ( z / z ) ' Thus t h e s t r o k e l e n g t h e q u a l s :
j V J cos ((JO t - 1^ j)dt = 2 —cos j\j 1 - ( z / z ) ' (C8)
F i g . C l shows t h e s t r o k e l e n g t h as f u n c t i o n o f z / z . z / t F i g . C l . S t r o k e l e n g t h ^ O as f u n c t i o n o f z / z 9 A b b r e v i a t i n g t h e e x p r e s s i o n " t h e i n n e r i n t e g r a l o f t h e sec-ond t e r m o f (C6)" b y " s t r o k e l e n g t h " i s o n l y c o r r e c t when L a g r a n g i a n e f f e c t s may be n e g l e c t e d (3u/3x = 0 ) . F u r t h e r m o r e , i t i s o n l y a r e s u l t i n g s t r o k e l e n g t h , t a k i n g r e v e r s a l s i n t o a c c o u n t ; i t i s n o t a h o r i z o n t a l a m p l i t u d e . Thus t h e e x p r e s s i o n o n l y s h o u l d be seen as a " m a t t e r o f s p e e c h " . 10 M i n d n o t e
As t o be e x p e c t e d , t h e maximum s t r o k e l e n g t h i s f o u n d a t s t i l l w a t e r l e v e l : above t h i s l i n e o n l y d u r i n g a s h o r t p e r i o d w a t e r i s t r a n s p o r t e d and b e l o w t h i s l i n e t h e w a t e r moves t o and f r o . I t shows f r o m (C8), t h a t t h e s t r o k e l e n g t h i s ( f i r s t ) a f u n c t i o n o f c u r r e n t c h a r a c t e r i s t i c s t i m e s (second) a f u n c -t i o n o f z / z , and t h e r e f o r e t h e r e s u l t a n t d i s c h a r g e p e r t i d e e q u a l s t h e f i r s t f u n c t i o n t i m e s t h e i n t e g r a l o f t h e second one o v e r t h e h e i g h t ; l a s t m e n t i o n e d i n t e g r a l e q u a l s : 2 ƒ • \ \ - ( z / z Y d z = z - z I n t r o d u c i n g (CB) f o r t h e i n n e r i n t e g r a l o f t h e second t e r m o f (C6) and (C9) f o r c a l c u l a t i o n o f t h e o u t e r i n t e g r a l , one f i n d s f o r t h e t o t a l second t e r m on t h e l e f t h a n d s i d e o f (C6) , r e p l a c i n g n/co b y T/ 2 : 7 z cos ooi Y B : \ v^dzdt = T T BjVj-cos(^j ( C I O ) ; = 2 J J j-2 2 ' 0 -z The o r d e r o f m a g n i t u d e can be r e a l i z e d b y t a k i n g , f o r a m i n u t e , ^j=0 ( p r o p a g a t i n g wave; no f r i c t i o n ) . Then i t i s t h e amount o f w a t e r , f l o w i n g ( c o n t i n o u s l y ) t h r o u g h t h e t i d a l p r i s m ( o v e r a h e i g h t 2z) w i t h a q u a r t e r o f t h e maximum ( l o c a l ) t i d a l v e l o c i t y i n t h e d i r e c t i o n o f t i d a l p r o p a g a -t i o n . 1 ^ X + -arcsin x 2 ~2 ( C 9 )
The amount o f w a t e r f l o w i n g o v e r t h e s h o a l can be c a l c u l a t e d i n t h e same way; t h e o n l y d i f f e r e n c e i s t h e l o w e r i n t e g r a -t i o n b o u n d a r y i n ( C 9 ) . Thus one f i n d s f o r -t h e f i r s -t -t e r m on t h e l e f t h a n d s i d e o f ( C 6 ) : C C z ^9/1 / hsh arcsin(hsh/z) \ Oh,, V V ( C l l ) Thus t h e t i d e - a v e r a g e d c o n t i n u i t y e q u a t i o n c a n be f o u n d b y s u b s t i t u t i o n o f (C 10) and (C 11) i n t o (C4), w h i l e d i v i d i n g by 7:
HARNSUP1.DOC S 10 November 1 1 , 1993
C.c The magnitude o f w and ö t o be used f o r the computa-t i o n of ebb-sedimencomputa-tacomputa-tion i n a c l o s e d s i n g l e p r i s m a computa-t i c chan-n e l . Purpose o f t h i s s e c t i o n C3 i s t o e x p r e s s ö and w , w h i c h d e t e r m i n e (see ( 2 ) ) t h e l o c a l s e d i m e n t a t i o n , i n t o : a. t h e l o c a t i o n i n t h e c h a n n e l ; b. t h e t i d a l a m p l i t u d e - t o d e p t h r a t i o ; c. t h e b o t t o m f r i c t i o n . Now f i r s t , t h e s e t h r e e v a r i a b l e s w i l l be s p e c i f i e d . ad a ; t h e l o c a t i o n i n t h e c h a n n e l : The f o l l o w i n g c o m p u t a t i o n s w i l l be c a r r i e d o u t f o r a p r i s m a t i c c h a n n e l , c l o s e d on one s i d e . The l e n g t h c o o r d i n a t e x w i l l be t a k e n w i t h r e s p e c t t o t h i s c l o s e d end.
However, i t must be remarked, t h a t t h e r e s u l t s o f t h e c o m p u t a t i o n s a r e q u i t e g e n e r a l and can be e a s i l y -e x t -e n d -e d t o c h a n n -e l s , op-en on b o t h s i d -e s . F u t u r -e r -e p o r t s w i l l e l u c i d a t e t h i s remark. For g e n e r a l i t y p u r p o s e s , t h e s i t e o f t h e l o c a t i o n w i l l be e x p r e s s e d i n a d i m e n s i o n l e s s x - c o o r d i n a t e . R e f e r e n c e l e n g t h w i l l b e ^ : L, = Tigh ( C 1 3 ) ad b: t h e t i d a l a m p l i t u d e - t o d e p t h r a t i o : as s t a n d a r d f o r t h e a m p l i t u d e o f t h e v e r t i c a l t i d e , t h e a m p l i t u d e Z Q a t t h e c l o s e d end o f t h e c h a n n e l w i l l be t a k e n . ad c: t h e b o t t o m f r i c t i o n : The b o t t o m f r i c t i o n i s d e t e r m i n e d b y f r i c t i o n a n g l e 9 (see [ 3 ] and [ 6 ] ) For a s i m p l e p r i s m a t i c c h a n n e l , c l o s e d on one s i d e , f r o m (C12) t h e v e l o c i t y v can be d e r i v e d : •Öcosè ( C 1 4 ) 2 ( / i - z )
Thus one f i n d s f o r 5, d e f i n e d i n (3) and u s e d i n ( 2 ) :
0 = cos(l) ( C I S )
11 I n f a c t , Lo w o u l d be t h e wave l e n g t h , i f f r i c t i o n c o u l d be n e g l e c t e d (9 = 0) I n t h e f i g u r e s 2 t o 4 t h e u n i t {x/Lo=l) has been d i v i d e d i n 36 p a r t s , t h u s g i v i n g t h e t h e h o r i z o n t a l a x i s t h e c h a r a c t e r o f an argument, w i t h 10 degrees as an i n t e r v a l
Wanted i s a d i m e n s i o n l e s s r e l a t i o n between ö,x/Lo, ZQ/IL and f r i c t i o n a n g l e 0.
For t h i s g o a l , t h e i n v e s t i g a t i o n s w i l l be c o n f i n e d t o chan-n e l s , f o r w h i c h i s v a l i d :
\ z - Z o \ « h ( C 1 6 )
Then one may a p p r o x i m a t e (C15) b y ( 5 ) .
The v a l u e o f ^ as f u n c t i o n o f X/LQ and 0 i s c a l c u l a t e d i n Supplement D4; i t i s d e p i c t e d i n f i g . 2 . The r a t i o Z/ZQ i s c a l c u l a t e d i n Supplement D2 and d e p i c t e d i n f i g . D l .
HARNSUP2.DOC - s 12 - November 1 1 , 1993
Supplement D. C a l c u l a t i o n o f d e r i v a t i v e s of module and argument.
D.a Problem.1
Suppose w i s a complex f u n c t i o n o f a complex argument u :
w = w(u) w complex ( T ' l ) u complex S p e c i f y f u r t h e r , t h a t u c o n s i s t s o f a r e a l f a c t o r a and a complex f a c t o r z: u=az a r e a l ( ^ 2 ) z complex Now c o n s i d e r a r e a l f u n c t i o n ƒ o f w, f o r i n s t a n c e a. t h e module o f w ; b. t h e argument o f w . Requested: An e x p r e s s i o n f o r d f / d a ( w h i c h e v i d e n t l y i s r e a l ) . A t f i r s t s i g h t , a p p l i c a t i o n o f t h e c h a i n r u l e seems l o g i c a l : d l ^ d l d w ^ ^ ( D 3 ) d a dw du However, ƒ i s n o t a p r i o r i an a n a l y t i c a l f u n c t i o n , a n d t h e r e f o r e t h e v a l u e o f d f / d w may depend on t h e d i r e c t i o n o f t h e p a t h i n t h e complex w-plane, used t o r e a c h a c e r t a i n v a l u e o f ƒ.
For t h i s r e a s o n t h e c a l c u l a t i o n o f d e r i v a t i v e o f module and argument ("a" and "b" above) w i l l each g e t a s e p a r a t e t r e a t m e n t . S e c t . D2 g i v e s t h e c a l c u l a t i o n o f t h e d e r i v a t i v e o f t h e mod-u l e and s e c t . D3 o f t h e argmod-ument. D.b C a l c u l a t i o n of t h e d e r i v a t i v e of t h e module. For any f u n c t i o n ƒ t h e e x p r e s s i o n h o l d s : d i p 1 d f ' da 2\fJ^ d.'^ ( T 4 )
The s q u a r e o f t h e module o f w may be w r i t t e n as t h e p r o d u c t o f w and i t s complex c o n j u g a t e w :
1 N o t a t i o n i n t h i s Supplement d i f f e r s f r o m t h e one i n t h e r e s t o f t h e r e p o r t
\w\^=ww (5'5) From (D4) and (D5) i t shows, t h a t t h e r e q u e s t e d d e r i v a t i v e
o f t h e module may be w r i t t e n a s : d\w\ 1 f dw —diu\ rnA.-\ da 2\w \ \ da da J I n t h e r e s t o f t h i s Supplement an a d d i t i o n a l r e s t r i c t i o n w i l l be made c o n c e r n i n g t h e f u n c t i o n w. F o l l o w i n g d e r i v a -t i o n s a r e o n l y v a l i d u n d e r -t h e f o l l o w i n g c o n d i -t i o n :
F u r t h e r m o r e , use w i l l be made o f some c o r o l l a r i e s c o n c e r n i n g t h e p r o d u c t o f t w o complex numbers a and b. These w i l l be a p p l i e d on (D6) a n d as (D6) o n l y d e a l s on r e a l p a r t s o f complex numbers, a l s o t h e c o r o l l a r i e s a r e f o c u s s e d on r e a l p a r t s . The p r o o f i s e v i d e n t , i f one keeps (D8) i n m i n d ^ :
From (DB) t h e c o r o l l a r i e s can be d e r i v e d : 3?Cab) = Ï R ( a b ) ( D 9 a ) : R ( a b ) = 3 ? ( a b ) ( D 9 b ) ï R ( a b ) = 3 ^ ( a b ) ( D 9 c ) U s i n g ( D 7 ) , one may w r i t e : d 11611 1 f -dw — dw = ——- wz — +wz—— da 2\w \ \ du du if (D7) is valid ( T I O ) and w i t h t h e a i d o f (D9a,c): d \ w \ 1 —dw^. .r^'7^ 1-^ r m ^ ^ •IR zw if ( T Z ) I S valid ( T U ) d a
I I
V
du A p p l i c a t i o n o f t h e g e n e r a l e q u a t i o n ( D l l ) , v a l i d f o r a l l f u n c t i o n s w, o b e y i n g ( D 7 ) , w i l l now be f o c u s s e d on t h e g o n i o m e t r i e f u n c t i o n s i/j = c o s ( a z ) , r e s p uj = s i n ( a z ) , d e n o t i n g . t h e a m p l i t u d e o f v e r t i c a l and h o r i z o n t a l t i d e i n t h e Appen-d i x . From e q n ( E l , 2 ) i t shows, t h a t (D7) a p p l i e s i n t h e s e cases. I n s e r t i n g f o r w t h e c o s - f u n c t i o n i n ( D l l ) :HARNSUP2.DOC - s 14 - November 1 1 , 1993 d | c o s ( a z ) | - 1 ^ / ' ^ ^ - • : R { z c o s ( a z ) s i n ( a z ) > d a I c o s ( a z ) d I c o s ( a z ) I - 1 d a 2 1 c o s ( a z ) d I c o s ( a z ) I - ( Z r S i n 2 a z , - Z i S i n h 2 a Z j ) 3 ? [ z [ s i n { a ( z + z ) > + s i n { a ( z - z ) > ] ] ( D 1 2 ) d a 2 I c o s ( a z ) I n t h e same way, i n s e r t i n g f o r w t h e s i n - f u n c t i o n i n ( D l l ) : d I s i n ( a z ) | 1 da | s i n ( a z ) | d 1 s i n ( a z ) | 1 3 ? { z s i n ( a z ) c o s ( a z ) } 3 ? [ z [ s i n { a ( z + z ) } - s i n { a ( z - z ) > ] ] ( Z ) 1 3 ) d a 2 | s i n ( a z ) d I s i n ( a z ) I _ ^ r ^ s i n 2 a z ^ + z ^ s i n h 2 a Z t d a 2 | s i n ( a z ) | D.c C a l c u l a t i o n of t h e d e r i v a t i v e of t h e argument. C o n s i d e r now t h e case, t h a t ƒ i s t h e argument o f u) :
/ = ^ ( l n u ; - l n w) ( ^ 1 4 ) U s i n g (D2), one t h e n f i n d s f o r t h e d e r i v a t i v e t o a : d{arg w) i f z d w z d w \ ... ^ n i ^ r ^ — i - = — = ^ = i j ( T 7 ) IS valid ( T I S ) d a 2i\iudu wdaj As t h e second t e r m o f (D15) i s t h e complex c o n j u g a t e d o f t h e f i r s t one, (D15) c a n be r e f o r m u l a t e d a s ^ : djarg w ) _ J z d w \ ( D 1 6 ) d a \ w d u ) Example: = t g ( a z ) A c c o r d i n g t o [ 8 ] , eqn.4.3.5.7 , t h e c o n d i t i o n (D7) i s v a l i d f o r t h e t a n g e n s . S u b s t i t u t i o n o f t h e t a n g e n s i n t o (D16) g i v e s : d [ a r g { t g ( a z ) > ] _ „ | ^ 2 z ^ ( D 1 7 ) d a V s i n ( 2 a z ) J For t h e i m a g i n a r y p a r t o f a q u o t i e n t a/ b can be w r i t t e n : 3 Here 3 d e n o t e s t h e i m a g i n a r y p a r t o f a f u n c t i o n .
^{ab/\b\'}= R e a l and i m a g i n a r y p a r t o f a s i n e - f u n c t i o n a r e g i v e n i n Thus one f i n d s : d j a r g i t g j a z ) } ] _ 2 Z i s i n ( 2 a z , ) c o s h ( 2 a z J - 2 z , c o s ( 2 a 2 , ) s i n h ( 2 a z , ) d a l s i n ( 2 a z ) | ^ ( D 1 8 )
HARNSUP2.DOC - S 16 - November 1 1 , 1993
Supplement E . C a l c u l a t i o n s c o n c e r n i n g amplitude and phase i n a c l o s e d c h a n n e l .
E.a I n t r o d u c t i o n t o Supplement E .
I n Supplement C.c t h e t i d e - a v e r a g e d v e l o c i t y i n a c l o s e d s i n g l e p r i s m a t i c t i d a l c h a n n e l i s e x p r e s s e d i n t h e t i d a l a m p l i t u d e and t h e t i d a l phase. These a m p l i t u d e and phase
( r e l a t e d t o t h e v a l u e s a t t h e end o f t h e c h a n n e l ) c a n be e x p r e s s e d i n t h e d i s t a n c e f r o m t h e c o n s i d e r e d s i t e t o t h e end o f t h e c h a n n e l , i n c o m p a r i s o n w i t h t h e t i d a l wave l e n g t h . The e x p r e s s i o n s f o r h o r i z o n t a l and v e r t i c a l a m p l i t u d e a r e g i v e n i n s e c t . E 2 a n d E3 o f t h e p r e s e n t Supplement.
The phase d i f f e r e n c e b e t w e e n h o r i z o n t a l and v e r t i c a l t i d e f o r t h e c o n s i d e r e d c l o s e d c h a n n e l i s f o r m u l a t e d i n s e c t . E 3 . However, f o r t h e c a l c u l a t i o n o f ö (eqn. (6)) n o t o n l y t h e a m p l i t u d e and phase i n a l o c a t i o n , b u t a l s o t h e h o r i z o n t a l g r a d i e n t o f t h e a m p l i t u d e a n d phase a r e o f i m p o r t a n c e . These a r e c a l c u l a t e d i n s e c t . E 4 and E5 r e s p e c t i v e l y . The c o o r d i n a t e s y s t e m o f Suppl.C.c w i l l be u s e d , i n w h i c h t h e seaward d i r e c t i o n i s p o s i t i v e . The o r i g i n w i l l be a t t h e c l o s e d s i d e o f t h e c h a n n e l . I n t h e f o l l o w i n g , t h e t i d e w i l l be i n d i c a t e d b y complex numbers; t h e p e r i o d i c a l f a c t o r exp{ioot) w i l l be o m i t t e d . F u r t h e r m o r e , i t w i l l be assumed t o be known, t h a t t h e magni-t u d e o f magni-t h e v a r i a b l e u n d e r c o n s i d e r a magni-t i o n i s magni-t h e r e a l p a r magni-t o f t h e complex number, w h i c h i s a s s i g n e d t o t h i s v a r i a b l e .
Some g e n e r a l f o r m u l a e c o n c e r n i n g t h e r e a l and i m a g i n a r y p a r t o f c i r c u l a r f u n c t i o n w i t h complex argument (ip) , ( w h i c h can be e a s i l y d e r i v e d ' l ) ^ w i l l be r e c a l l e d :
cosTp = c o s i] ; r C o s h ipj - isinip^sinhTpi ( T l )
s i n ip = sinip^coshTp; + i cos ip^ s i n h ip^ ( T 2 )
t a n op = s i n ( 2 i p r ) + i s i n h ( 2 i p i ) c o s ( 2 \ p r ) + icosh(2-\pi) ( T 3 ) I cosip 1 = -J cos^ipr + s i n h ^ i p i ( T 4 ) sinip 1 = i s i n ^ i p r + sinh^-ipi ( T 5 )
E.b The amplitude of t h e v e r t i c a l t i d e .
L e t Zo be t h e v e r t i c a l t i d e a t t h e c l o s e d end. Then App.,(A7) g i v e s f o r t h e v e r t i c a l t i d e z i n t h e c h a n n e l :
4 One c o u l d a l s o r e f e r t o , f o r i n s t a n c e , [ 8 ] , eqns. 4.3.55 t o 4.3.63
z = ZoCOs(/cx) ( T 6 )
The complex wave number k f o l l o w s f r o m [ 2 ] , (AG)^:
A: = - ( l - i t a n e ) ( E 7 ) c where c i s g i v e n i n t h e A p p e n d i x ; e q n . ( A 8 b ) . Thus t h e r e a l and i m a g i n a r y p a r t s o f k a r e r e s p e c t i v e l y : ^ c o s 6 _ ^/g7^^/cos(2e) O) s i n e r c • ö K ^ ki = - ^ = I ( T 8 5 ) Vg^Vcos(2e) Thus t h e r a t i o o f t h e a m p l i t u d e | z | a t a c e r t a i n s i t e , com-p a r e d t o t h e a m com-p l i t u d e ZQ c a n be f o u n d f r o m ( E 4 ) ; i n f i g . E l i t i s d i s p l a y e d as a f r a c t i o n o f t h e r e f e r e n c e l e n g t h ^ Lo, where: Lo = T4gh {E9) Here t h e f r i c t i o n a n g l e 9 i s i n c r e a s e d f o r t h e v a r i o u s l i n e s o f f i g . E l w i t h an i n t e r v a l o f 7.49 °, t h u s g i v i n g a maximum^ v a l u e , d i s p l a y e d o f 44.94°. One c l e a r l y o b s e r v e s t h e d e c r e a s e o f t h e wave l e n g t h f o r i n c r e a s i n g 0. F o r l a r g e f r i c t i o n , n e a r l y e v e r y w h e r e t h e wave h e i g h t d e c r e a s e s e x p o n e n t i a l l y ^ ; f o r s m a l l f r i c t i o n nodes and a n t i n o d e s become a p p a r e n t .
E.C The amplitude o f t h e h o r i z o n t a l t i d e .
From t h e A p p e n d i x , ( A 7 ) i t shows, t h a t t h e h o r i z o n t a l t i d e can be c h a r a c t e r i z e d b y :
5 The o r i g i n a l d e r i v a t i o n i s g i v e n i n [ 3 ] a n d [ 6 ] .
M i n d t h a t t h e p r e s e n t symbol "/c" d e n o t e s "ik" i n [2] .
6 I n f a c t , Lo w o u l d be t h e wave l e n g t h , i f f r i c t i o n c o u l d be
n e g l e c t e d (6 = 0 ) . I n annex .. t h e u n i t ( x / Z o = l has been d i v i d e d i n t o 3 6 p a r t s , t h u s g i v i n g t h e h o r i z o n t a l a x i s t h e c h a r a c t e r o f an argument, w i t h 1 0 ° as a n i n t e r v a l .
HARNSUP2.DOC - s 18 - November 1 1 , 1993 V = ^ c o s ( 2 e ) . ( - O e ' ' Z o S i n ( f c x ) ( T I O ) h Thus f o r t h e module i s f o u n d ; V = ^ c o s ( 2 e ) . |Zo | | s i n ( / c x ) I ( T i l ) h where t h e l a s t f a c t o r can be f o u n d f r o m ( E 5 ) . I n f i g . E l t h e v e l o c i t y a m p l i t u d e i s made d i m e n s i o n l e s s w i t h a r e f e r e n c e v e l o c i t y | Zq I -Jg/h. Thus t h e f u n c t i o n d i s p l a y e d i s : Z o l V g T ^ = V{sinh2(A:,x) + s i n 2 ( / c , x ) } c o s ( 2 e ) ( T 1 2 ) I t i s p r e s e n t e d i n t h e same way as t h e v e r t i c a l a m p l i t u d e . For l o w f r i c t i o n , h i g h e r v e l o c i t i e s a t t h e nodes can be p e r -c e i v e d . F o r h i g h f r i -c t i o n , v e l o -c i t y and v e r t i -c a l t i d e b o t h decay e x p o n e n t i a l l y (a wave p r o p a g a t i n g and d e c a y i n g i n n e g a t i v e x - d i r e c t i o n ) .
E.d The phase d i f f e r e n c e between v e r t i c a l and h o r i z o n t a l t i d e .
From (E6) and (ElO) one f i n d s f o r t h e phase d i f f e r e n c e between v e r t i c a l and h o r i z o n t a l t i d e :
({) = | - e + a r g { c o s ( / c x ) } - a r g { s i n ( / c x ) > ( T 13)
One may make u s e o f :
a r g { c o s ( / c x ) } - a r g { sin ( / c x ) } = - a r g { t a n (fcx)>
and t h u s r e s u l t s f o r (j), m a k i n g use o f (E3) :
n ^ . s i n h { 2 ( f c , x ) > ^ A^
è = --Q-atan — — — - (A 1 4 )
^ 2 s i n { 2 ( / c , x ) >
For s m a l l v a l u e s o f x, t h e v a l u e o f t h e s i n h and t h e s i n a r e a b o u t e q u a l t o t h e argument a n d t h u s (E7) makes c l e a r , t h a t f o r s m a l l x t h e v a l u e o f t h e a t a n i n t h e l a s t t e r m on t h e r i g h t h a n d s i d e o f (E14) i s a b o u t -9.Thus, a t t h e c l o s e d end of t h e c h a n n e l t h e phase d i f f e r e n c e b e t w e e n v e r t i c a l and h o r i z o n t a l t i d e i s J i/ 2 as e x p e c t e d . F o r l a r g e v a l u e s o f x t h e a t a n t e n d s ^ t o -n/2 and t h e r e t h e phase d i f f e r e n c e i s j t 9 . T h i s c o r r e s p o n d s w i t h t h e r e s u l t s , f o u n d f o r t h e p r o p a -g a t i n -g wave i n s e c t . 5 . 5 . R e s u l t s ( a c c o r d i n g t o (E14) a r e d i s p l a y e d i n f i g . 2 . On t h e 9 a c c o r d i n g t o (EBb) fc; i s n e g a t i v e
HARNSUP2.DOC s 19 November 1 1 , 1993
r i g h t h a n d s i d e o f t h e f i g u r e , a s m a l l h o r i z o n t a l b a r shows t h e l i m i t i n g v a l u e f o r i n f i n i t e l y l a r g e x . F o r s m a l l f r i c -t i o n and v a l u e s o f x, l a r g e r -t h a n Z . o / 4 even l a r g e r phase d i f f e r e n c e s t h a n it c a n o c c u r . The f i g u r e g i v e s a c l e a r p i c t u r e o f t h e t r a n s i t i o n f r o m p r o p a g a t i n g t o s t a n d i n g wave. E.e The h o r i z o n t a l g r a d i e n t o f t h e a m p l i t u d e of t h e v e r t i c a l t i d e . For f i n d i n g t h e h o r i z o n t a l g r a d i e n t o f t h e a m p l i t u d e , u s e i s made o f t h e t h e o r y , d e v e l o p e d i n Supplement D. The h o r i z o n t a l g r a d i e n t o f t h e a m p l i t u d e o f t h e v e r t i c a l t i d e can be f o u n d f r o m (D12) b y s u b s t i t u t i o n o f x i n s t e a d o f a and k i n s t e a d o f z: ' d I c o s ( f c x ) I _ - ( / c , / f c o ) s i n ( 2 / c , x ) + ( f c i / / C o ) s i n h ( 2 f c i X ) ( £ 1 5 ) d ( / C o X ) 2 | c o s ( f c x ) |
where fco i s t h e wave number f o r f r i c t i o n l e s s f l u i d :
ko = oó/{gh ( T 1 6 )
The r a t i o ' s kr/k^ and k^/ko can be f o u n d f r o m (E8a,b) . The r e s u l t (E15) i s u s e d f o r t h e c a l c u l a t i o n o f ö
( S u p p l . C . c ) .
E . f The h o r i z o n t a l g r a d i e n t of t h e phase d i f f e r e n c e between v e r t i c a l and h o r i z o n t a l t i d e . For Suppl.C.c, e s p e c i a l l y t h e g r a d i e n t a l o n g t h e c h a n n e l o f coscj) i s w a n t e d . T h i s g r a d i e n t w i l l be c a l c u l a t e d i n t h i s p a r t o f t h i s Supplement. A c c o r d i n g t o (E13) d ( c o s ( l ) ) / d x , i . e. ( s i n ( t ) ) d ( l ) / d x e q u a l s : — cos(t) = s m d x ^ - e- a r g { t g ( / c x ) } w h i c h can be w r i t t e n ( u s i n g (E13) a g a i n ) a s : — c o s è = - c o s [ 9 + a r g { t g ( / c x ) } ] . - ^ [ a r g { t g ( / c x ) } ] ( T 1 8 ) d x d x Use o f (DIB) g i v e s : d 2 ( f c | / f c o ) s i n ( 2/ C r X )c o s h ( 2 / c , x ) - 2(/Cr/A:o) c o s ( 2 A: r X )s i n h ( 2 / c , , ) _ c o s * = - c o s[ e. a r g < t g ( f c x ) > ] . | s i n ( 2 y t x) r ( £ • 1 9 ) The v a l u e s f o u n d a r e u s e d f o r i n s e r t i n g t h o s e i n eqn.(A6) o f t h e A p p e n d i x .
i R e f e r e n c e s [1] Bakker,W.T. I t a i n ' t n e c e s s a r i l y so [2] B a k k e r , W.T., M a t h e m a t i c a l s h a p i n g o f t i d a l i n l e t s I n t e r n a l r e p o r t ; v e r s i o n o f August 20, 1992 [3] L o r e n t z , H . A . , V e r s l a g v a n de S t a a t s c o m m i s s i e v o o r de Z u i d e r z e e ( R e p o r t o f t h e S t a t e Commission Z u i d e r z e e ) Den Haag, A l g . L a n d s d r u k k e r i j (1926) [4] Robaczewska, K.B., G e t i j d e s t r o m i n g i n de N e d e r l a n d s e Waddenzee, g e s i m u l e e r d met een m a t h e m a t i s c h model
( T i d a l c u r r e n t i n t h e D u t c h Waddensea, s i m u l a t e d w i t h a m a t h e m a t i c a l model ( i n d u t c h )
R i j k s w a t e r s t a a t , T i d a l Waters D i v i s i o n , GWAO-91.013 [5] R i j n L.C.v. & A.J.de Leeuw
Z a n d t r a n s p o r t model W e s t e r s c h e l d e ; s t o c h a s t i s c h o n d e r z o e k z a n d t r a n s p o r t f o r m u l e ( S a n d t r a n s p o r t model W e s t e r n S c h e l d t ; s t o c h a s t i c a l r e s e a r c h on a s a n d t r a n s p o r t f o r m u l a ; i n d u t c h ) D e l f t H y d r a u l i c s L a b o r a t o r y ; r e p o r t S 582 (19 85) [6] T h i j s s e , J.Th. T h e o r y o f T i d e s L e c t u r e n o t e s , D e l f t U n i v e r s i t y o f T e c h n o l o g y and NUFFIC [7] B a k k e r , W.T., C o n t r i b u t i o n ( A p p e n d i x A) i n : de Ras, M.A.M., Onderzoek n a a r de G r o o t t e v a n h e t L a n g s t r a n s p o r t l a n g s de k u s t b i j I J m u i d e n d o o r de c o m b i n a t i e v a n g e t i j s t r o o m en g o l f o r b i t a a l b e w e g i n g ( I n v e s t i g a t i o n o f t h e m a g n i t u d e o f t h e l o n g s h o r e t r a n s p o r t a t I J m u i d e n b y a c o m b i n a t i o n o f t i d a l c u r r e n t and o r b i t a l m o t i o n ) R i j k s w a t e r s t a a t , D i r . f o r Water Management a n d H y d r a u l i c Research; S t u d y R e p o r t WWK 71-11 (1971) [8] A b r a m o w i t z , . . and I . A . S t e g u n , Handbook o f M a t h e m a t i c a l F u n c t i o n s .
