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Three-Dimensional Model of Currents in the Bay of Seine
427
THREE-DIMENSIONAL MODEL OF CURRENTS I N THE BAY OF SEINE J.C. SALOMON, 6. THOUVENIN and P. LE H I R IFREMER, B.P. 337, 29273 BREST CEDEX
ABSTRACT Water movement and m i x i n g processes i n t h e Bay o f Seine a r e e s s e n t i a l l y i n duced by t i d e s ; winds and d e n s i t y d i f f e r e n c e s due t o r i v e r i n f l o w and temperat u r e g r a d i e n t s near t h e coast. To study t i d a l c u r r e n t s alone, i t i s s u f f i c i e n t t o use a two dimensional model. Howeveryfor wind and d e n s i t y e f f e c t s , t h e t h r e e s p a t i a l dimensions must be taken i n t o account. For t h a t purpose a three-dimensional model o f c u r r e n t s has been developed which solves t h e n o n l i n e a r Navier-Stokes equations i n a s t r a i g h t f o r w a r d manner through a f i n i t e d i f f e r e n c e technique. The model has been t e s t e d i n various schematic cases f o r which a n a l y t i c a l s o l u t i o n s o r r e s u l t s from previous models were a v a i l a b l e . It was then a p p l i e d t o t h e Bay o f Seine and proved t o be an e f f i c i e n t t o o l f o r coastal studies.
1 .REGIONAL SETTING The Bay o f Seine, as d e f i n e d by t h e c o a s t l i n e , i s shaped more o r l e s s l i k e a r e g u l a r q u a d r i l a t e r a l about one hundred k i l o m e t e r s long and f o r t y t o f i f t y kilometers wide, opening onto t h e E n g l i s h Channel ( F i g . 1).
The bay i s q u i t e
shallow (twenty f i v e meters deep) except f o r t h e a n c i e n t submerged v a l l e y o f the Seine which runs d i a g o n a l l y across t h e bay and reaches depths o f f o r t y met e r s i n t h e northwestern p a r t o f t h e bay. With an average flow o f 400 m 3 / s , t h e Seine i s a modest r i v e r . i t i s the o n l y s i g n i f i c a n t f l u v i a l discharge i n t h e Channel.
Nevertheless,
The estuary, which
used t o be upstream of t h e c i t y o f Le Havre, has now a s m a l l e r area, because o f the improvement o f t h e waterway. Over t h e l a s t hundred and f i f t y years, the estuarine area has decreased by a f a c t o r of two, and a t t h e same time t h e locat i o n has s h i f t e d towards t h e bay. downstream o f Le Havre.
Today t h e d e n s i t y nodal p o i n t i s s i t u a t e d
The c h a r a c t e r i s t i c s of t h e bay a r e q u i t e complicated. Indeed, t h e present s i t u a t i o n includes almost a l l t h e p o s s i b l e causes f o r water movements i n t h e coastal zone : The main p a r t of t h e dynamics i s due t o t h e t i d e . The t i d a l range can exceed
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e i g h t meters i n s p r i n g t i d e s .
The
c u r r e n t s increase r e g u l a r l y from e a s t t o
west, from one t o t h r e e knots. The t i d a l wave and t h e associated c u r r e n t s are s t r o n g l y d i s t o r d e d w h i l e propagating i n t h e shallow areas, so t h a t .the n o n l i near terms i n t h e hydrodynamicequations a r e i m p o r t a n t .
428
F i g . 1. Location map and bathymetry o f the Bay o f Seine.
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The wind i s another d r i v i n g f o r c e o f water movements i n the bay, a l l the
more important as i t i s r a t h e r intense and as water depths are small.
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As mentionned before, d e n s i t y c i r c u l a t i o n does e x i s t too, c l o s e l y related
t o the f l u v i a l r u n o f f i n the eastern p a r t o f the bay. It i s almost o f estuarine type (converging v e l o c i t i e s near the bottom, nodal p o i n t ) w i t h some geostrophic features ( r o t a t i o n o f surface v e l o c i t i e s t o the r i g h t ) .
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F i n a l l y water movements i n the bay i s p a r t l y due t o the general c i r c u l a t i o n
i n the Channel.
2 MODEL SPECIFICATIONS The t i d a l components o f t h e c u r r e n t s i n t h e bay can be studied, f o r t h e i r essential p a r t , w i t h two-dimensional (2D) models, b u t those which are l i n k e d t o the d e n s i t y f i e l d o r t o the wind do r e q u i r e a three-dimensional ( 3 D ) s l c u l a t i o n .
429
A few years ago we made a f i r s t attempt a t modelling currents in three dimensions in t h i s area (Salomon and Le Hir, 1980). We proceeded i n a quite s i milar way t o t h a t used by Nihoul (1977) a t the time, by developing along the vertical coordinate the r e s u l t s of a 2D model a t specific points. The results revealed some interesting apects of the currentology, b u t such calculations are not really three dimensional :the f r i c t i o n a l s t r e s s i s considered to be in the same direction as the vertically averaged velocity. Moreover, the nonlinear advective terms are d i f f i c u l t to take into account, since the calculations are local. The technical specifications of the present 30 model are that the model should: - solve the complete nonlinear equations; take into account the precise form of the sea bed (very important in coastal problems) , which eliminates " s t a i r step" models; - have the same behaviour a t a l l times of the computation and over the whole bay. Due t o the depth differences in our zone, and due t o the tidal amplitude, we discarded the use of reduced coordinates. As different grids a c t l i k e d i f ferent media, waves and dynamic structures are partly reflected and distorted. On the other hand, the model could have modest performances as f a r as horizontal turbulent fluxes calculations are concerned because f o r the time scales we are interested in, these terms are small compared t o the advective terms. Simple considerations on the dynamics of the bay show t h a t due tx i t s restricted dimensions and the intensity of f r i c t i o n forces, i t s "memory" towards barotropic waves and even baroclinic ones i s of a few hours. I t s memory towards phenomena of transport of matter ( l i k e s a l t ) i s much higher. Calculations based on the lack of s a l t in the bay give a residence time of about a month. For financial reasons (among others) i t was impossible t o measure boundary conditions during such a long period (except f o r the t i d e ) and the project of such simulations was given u p . Model indications, in terms of density circulation, will essentially r e s u l t from the i n i t i a l s a l t and temperature distribution, and n o t from modifications of the boundary conditions d u r i n g the calculations. Barotropic movements, on the other hand, are driven through the boundary condi tioos.
-
3 MODEL CHARACTERISTICS 3.1 Equations The model solves the Navier-Stokes equations i n t h e i r usual form, using the hydrostatic approximati on , and assuming Fi cki an diffusion:
430
p = PO ( 1 t as)
as ~ t
L
as ax e t
v
as ay - t
asw az
as
a(KZ E) az
F, = o
: h o r i z o n t a l coordinates
: v e r t i c a l coordinate w) : v e l o c i t y components : surface e l e v a t i o n 4 H : t o t a l depth o f water : mass per u n i t volume P f : Coriolis factor : salinity S : Fx, FY h o r i z o n t a l d i f f u s i o n o f momentum Fs : horizontal diffusion o f s a l t Nx, NY : v e r t i c a l eddy v i s c o s i t y c o e f f i c i e n t s Kz : c o e f f i c i e n t f o r s a l t d i f f u s i o n along the v e r t i c a l
The h o r i z o n t a l d i f f u s i o n terms Fx, Fy, Fs are p h y s i c a l l y important i n vert i c a l l y i n t e g r a t e d models where they are c a l l e d dispersion because they include a t the same time the v e r t i c a l v e l o c i t y gradients and t h e v e r t i c a l 'diffusion. Here they are e s s e n t i a l l y r e q u i r e d t o damp small scale computational modes. It would be d i f f e r e n t i f long term
simulations were t o be done.
They are expres-
sed i n t h e f o l l o w i n g usual way : FX = NXX FY = NYX FS = KX
a2u
a2u + NXY 7 aY
a2v a2v v t NYY aY
a2s t ax.
7
KY
a2s
-2-
aY I n t h e present a p p l i c a t i o n t o the Bay o f Seine, which we know t o be rather
well-mixed, t h e pressure gradients are expressed i n the f o l l o w i n g simple form:
431 3.2 C o e f f i c i e n t s f o r v e r t i c a l exchange The model makes use of t h e concept o f t u r b u l e n t exchange c o e f f i c i e n t s parameterized i n a simple form.
The other, more sophisticated, p o s s i b i l i t y , which
would c o n s i s t s i n solving, a f t e r closure, the turbulence equations, has been rejected because: no i n i t i a l o r boundary c o n d i t i o n s are a v a i l a b l e w i t h enough accuracy,
-
i t has been established by d i f f e r e n t i n v e s t i g a t o r s t h a t i n the case o f t i d a l
currents i n shallow waters, a r e l a t i o n between the d i f f u s i o n c o e f f i c i e n t s and the v e l o c i t y gradients ( o r t h e concentration g r a d i e n t ) through a mixing length, gives reasonable r e s u l t s . point
.
No a d d i t i o n a l complication has been seeked on t h i s
The e f f e c t o f buoyancy forces, which h i g h l y modify t h e mixing length, i s introduced by means o f a global Richardson number:
AS: S a l i n i t y d i f f e r e n c e between bottom and surface U : v e r t i c a l l y averaged v e l o c i t y .
Several expression ( g e n e r a l l y empirical ones) have been t e s t e d i n t h e model. The f o l l o w i n g r e l a t i o n s appear t o g i v e the best r e s u l t s :
K, =
3.3.
t
5.10-2U log(O.l z 3 t 1 ) ( 1 t R i ) -
7
7
Method o f r e s o l u t i o n
The numerical procedure and t h e d i s c r e t i z a t i o n g r i d are extrapolated from the two dimensional v e r t i c a l model t h a t we already used i n several estuaries: Gironde, S t Lawrence (de Borne de Grandpre e t a l . , 1979, 1981), Seine (Salomon, 1981). It i s a mixed i m p l i c i t - e x p l i c i t f i n i t e d i f f e r e n c e technique. The o r i g i n a l i t y o f t h e method l i e s e s s e n t i a l l y i n the f a c t t h a t the locat i o n s o f the upper and lower f r o n t i e r s of t h e i n t e g r a t i o n area change during the c a l c u l a t i o n s . I n order t o achieve t h a t , use i s made o f v i r t u a l p o i n t s 10cated above t h e surface, and o t h e r s s i t u a t e d under t h e bottom (Hamilton, 1975). D e t a i l s o f t h i s technique have been exposed by Thouvenin and Salomon (1984).
432
HORIZONTAL
VERTICAL
j+l
j
j-1
Ax i-1
i
Ax
i +1
i -1
i
i +1
Fig. 2. Computational g r i d s B r i e f l y stated, a t ev.ery time step a f i n i t e d i f f e r e n c e method i s used t o go over the usual c a l c u l a t i o n sequence i n s i d e the area o f computations. The results are then e x t r a p o l a t e d t o external v i r t u a l p o i n t s by a n a l y t i c a l f u n c t i o n s which take i n t o account t h e upper and lower boundary conditions : -+ -+ a t the surface -+
av = k V1, Nz az -+
/Vim/
and
as
= o
a t t h e bottom
+ 1 ',
: t h e v e l o c i t y vector one meter above t h e bottom
T~
: wind s t r e s s
k
: f r i c t i o n coefficient
+
The f u n c t i o n s used a r e u s u a l l y polynomials o f t h i r d degree (splines), except f o r the v e l o c i t y near the bottom. To ensure a c o r r e c t v e l o c i t y p r o f i l e j u s t above t h e bottom where i t i s known t o be logarithmic, we use e i t h e r a l o g a r i t h m i c
curve o r an exponential one, obeying the c o n t i n u i t y o f
the v e l o c i t y curve and o f i t s f i r s t d e r i v a t i v e . 3.4 Numerical scheme The numerical scheme i s mixed i m p l i c i t / e x p l i c i t . A l l s p a t i a l d e r i v a t i v e s are centered except advective terms i n t h e s a l t equation. The pressure gradient term, p a r t i c u l a r l y important f o r the s t a b i l i t y o f t h e scheme, i s t r e a t e d as implicit.
433 Considering t h e equations i n t h e sequence described i n 3.1.,
i t i s possible t o
solve the complete system i n an e x p l i c i t way, i . e . w i t h o u t s o l v i n g any m a t r i x . The complete numerical scheme i s described and discussed i n Thouvenin (1983). 4. MODEL APPLICATIONS
4.1 Schematic t e s t s Before applying t h e model t o t h e Bay o f Seine, we t e s t e d i t s performances i n sane simple and r a t h e r schematic cases f o r which aspects o f t h e s o l u t i o n are known. I n p a r t i c u l a r , we have studied (Thouvenin, 1983):
- The propagation o f a sinusoidal wave i n a rectangular - The f r e e o s c i l l a t i o n s i n a closed basin. - The v e l o c i t y f i e l d i n a channel crossed by a trench. - The e f f e c t s o f t h e wind blowing over t h e open sea. - Convective movements along isopycnal surfaces.
basin.
Here, we s h a l l b r i e f l y comment on the l a s t two experiments.
N
WIND
Point 12 (170 Point 10 (130 m.) Point 6 ( 5 0 m . )
Fig.3. Water p a r t i c l e s t r a j e c t o r i e s under wind a c t i o n (Distance from the surface i n d i c a t e d i n brackets).
434
Fig. 3 shows the t r a j e c t o r i e s o f water p a r t i c l e s induced by a constant wind supposed t o blow during 33 hours from t h e south, and then t o stop.
According t o t h e known a n a l y t i c a l s o l u t i o n , we observe t h e superposition o f t h e Ekman d r i f t s p i r a l and o f i n e r t i a l o s c i l l a t i o n s .
When t h e wind stops t h e i n e r t i a l
r o t a t i o n , stronger near t h e surface. diminishes.
Some energy i s transferred
through viscous s t r e s s t o the lower l a y e r s which are thus accelerated. Comparing t h e thickness o f the Ekman l a y e r produced by t h e model w i t h the t h e o r e t i c a l s o l u t i o n , we f i n d f o r Nz a d i f f e r e n c e o f 20 %. This d i f f e r e n c e which sums up several numerical e r r o r s seems q u i t e reasonable. The movement associated t o a s a l t wedge i n a rectangular tank i s i n t e r e s t i n g Here, we observe the formation o f a d e n s i t y d r i v e n c i r c u l a as w e l l ( f i g . 4 ) . t i o n along the i n t e r f a c e and t h e appearance o f a b a r o t r o p i c wave propagating very q u i c k l y f a r from t h e s a l t wedge.
Also t o be noted i s
the polarisation
o f v e r t i c a l movements, more intense on t h e r i g h t side, due t o t h e C o r i o l i s f o r ce
.
Fig. 4 S a l t wedge c i r c u l a t i o n a) l a t e r a l views a f t e r 30 and 300 minutes
b) view from above a f t e r 5 hours
435 4.2 A p p l i c a t i o n t o the Bay o f Seine ( i ) I n i t i a l and boundary conditions.
Because o f computer l i m i t a t i o n s , t h e
application t o the Bay o f Seine i s done w i t h a r e l a t i v e l y coarse g r i d s i z e ( 4 km. 4 m).
It must be acknowledged t h a t due t o t h e l a c k o f i n f o r m a t i o n on
boundary conditions, a g r i d refinement would n o t have been completely j u s t i f i e d . I n i t i a l conditions f o r s a l i n i t y are taken from i n s i t u measurements made during the year 78. Boundary conditions along the open boundary (water l e v e l s and v e l o c i t i e s ) are taken from a 2D model (Salomon, 1985).
The v e r t i c a l s t r u c t u r e o f the h o r i -
zontal v e l o c i t i e s i s supposed t o obey a l o g a r i t h m i c p r o f i l e .
The v e r t i c a l com-
ponent o f the v e l o c i t y i s l i n e a r l y i n t e r p o l a t e d between t h e bottom and the surface where i t i s . c a l c u l a t e d through the f o l l o w i n g r e l a t i o n s : 'bottom
=
( i i ) T i d a l currents. The t i d a l component o f t h e currents i s the e s s e n t i a l part o f the dynamics b u t as i t seemed already w e l l known through measurements (Le H i r and l'Yavanc, 1985) o r 2D modeling (Salomon, 1985), l i t t l e a d d i t i o n a l information was expected. I n f a c t , we observe (see F i g . 5): important v e r t i c a l movements which, o f course, could n o t have been measured. They are t h e r e s u l t o f t h e bottom shape and h o r i z o n t a l convergences.
-
-
phase d i f f e r e n c e s and r o t a t i o n s along t h e v e r t i c a l which come from the d i f f e r e n t r a t i o o f the i n e r t i a l f o r c e and the d r i v i n g f o r c e (surface slope) as well as from c h a r a c t e r i s t i c s o f bottom Ekman l a y e r . Thus we are more aware o f the e r r o r we make i n a 2D model assuming t h e f r i c t i o n s t r e s s on the bottom t o be i n the same d i r e c t i o n as the average velocity. ( i i i ) Wind induced movements. The wind induced dynamics i s t y p i c a l l y threedimensional and cannot be approached by 2D simulations. This component o f the c i r c u l a t i o n i s l i n k e d t o others and e s p e c i a l l y t o the tide.
The r e s u l t s may be very complex and d i f f i c u l t t o understand from i n s i t u
measurements.
The model here i s o f g r e a t assistance, even i f i t s i n d i c a t i o n s
are n o t very accurate. Theoretically, c a l c u l a t i o n s should take i n t o account the response o f the e n t i r e Channel t o wind f o r c i n g and l a r g e scale atmospheric pressure gradients. However, some i n t e r e s t i n g i n f o r m a t i o n can be obtained from l o c a l simulations about the response o f t h e bay t o wind f o r c i n g , a t l e a s t q u a l i t a t i v e l y .
64
-40 16
96Km
32 Km
-
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8 -8
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32
m 1 16
80
-24
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32
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-24
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-32
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64
16
32
Km
80
16
96Km
32
16
48
32
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.--+--*__-
.\-&-&-b*D*Q*
BOTTOM
16
80
64
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80
%Km
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437
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F i g . 6. Wind induced v e l o c i t y f i e l d ( h o r i z o n t a l sections near the surface and the bottom ; N-S v e r t i c a l sections ; E-W v e r t i c a l s e c t i o n s ) . Wind blowing from the west.
438 So we have c a r r i e d o u t some s i m u l a t i o n s o f t h e d i r e c t a c t i o n o f t h e wind on t h e bay alone. The boundary c o n d i t i o n corresponds t o a h y p o t h e t i c a l steady regime.
I n the
dynamical equation along t h e n o r t h e r n boundary a balance i s assumed between t h e surface slope, f r i c t i o n stresses on t h e bottom and a t t h e surface, and the C o r i o l i s force. The r e s u l t s presented i n Figs. 6 and 7 are f o r u n i f o r m wind f i e l d s o f 15 m/s.
A permanent regime i s g e n e r a l l y observed a f t e r seven hours. The h o r i z o n t a l s e c t i o n s c l e a r l y show t h e s t e e r i n g o f t h e c u r r e n t s by bathym e t r i c features, p a r t i c u l a r l y near t h e bottom ( f i g . 6 ) . Cross c u r r e n t s a r e h a r d l y ever observed i n t h e Parfond ( a n c i e n t Seine valley) even close t o t h e surface, where c u r r e n t s a r e l e s s r e l a t e d t o t h e bottom shape. V e l o c i t i e s t h e r e o f t e n r o t a t e a few tens o f degrees.
A t t h e surface, t h e v e l o c i t i e s are l a r g e r i n t h e shallow zones, a consequence o f t h e f a c t t h a t t h e d r i v i n g f o r c e r e l a t e d t o t h e u n i t o f volume o r u n i t o f mass i s i n v e r s e l y p r o p o r t i o n a l t o t h e water depth.
This i s t h e basic reason
f o r t h e general dissymmetry between t h e eastern and western p a r t s o f t h e bay.
Fig. 7 H o r i z o n t a l v e l o c i t y f i e l d s f o r d i f f e r e n t wind d i r e c t i o n s .
439 For the same reason,the wind induced v e l o c i t i e s above t h e Parfond are weaker, When the conditions are such t h a t a d e f i c i t o f water appears i n t h e southern p a r t o f the bay, an adjustment c u r r e n t appears i n the Parfond, opposite t o the surface v e l o c i t i e s .
This i s a w e l l known f e a t u r e o f wind a c t i o n i n l i t t o r a l
zones and lakes : a back c u r r e n t appears i n the deepest parts, g e n e r a l l y the waterway.
I t i s a l s o i n close agreement w i t h measurements made during t h i s
study. The v e r t i c a l sections a l s o show very obvious v e r t i c a l s t r u c t u r e s corresponding t o v e r t i c a l movements near the banks and h e l i c o i d a l movements i n s i d e the trench o f the Parfond, which measurements would n o t have revealed. These simulations, as w e l l as the measurements, i n d i c a t e t h a t t h e wind i n duced v e l o c i t i e s are o f t e n o f an order o f magnitude o f 10 t o 30 an/s, which i s much higher than t i d a l r e s i d u a l currents.
This mechanism i s thus an essential
vector o f t h e long term movements o f waters i n t h e bay.
5 CONCLUSION . The model presented i n t h i s paper i s b u t one element o f a general study o f the bay. The study l a s t e d t h r e e years, during which a considerable e f f o r t has been made on measurements : t h r e e cruises o f a few months each have been c a r r i e d out, i n v o l v i n g d i f f e r e n t measurement l o c a t i o n s (up t o twelve), sometimes a t t h e surface, a t t h e bottom and a t mid-depth. Important human and technical resources have thus been involved.
Fig.8 Diagram showing t h e r e s i d u a l c i r c u l a t i o n i n t h e Bay o f Seine
440
The measurements produced a l o t o f i n t e r e s t i n q i n d i c a t i o n s but,finallv,when t came t o e l a b o r a t i n g a comprehensive synthesis o f thedynamical regime o f the bay, the main r o l e was devoted t o the model. This way we were able t o understand what occured and we could separate t h e proper e f f e c t s o f each o f the physical mechanisms involved : wind, t i d e , density, C o r i o l i s ... This way we understood what measurements where showing. The diagram f o r the residual c a l c u l a t i o n presented i n fig.8,
was e s s e n t i a l l y
deduced from model r e s u l t s . So, i t proved t o be an e f f i c i e n t t o o l f o r conducting coastal studies.
Some advantages o f t h i s model have been already mentioned, e s p e c i a l l y the p o s s i b i l i t y o f c o n c i l i a t i n g a f i x e d l o c a t i o n o f computational p o i n t s and a continuous representation o f the bottom and t h e surface, i n s p i t e o f a great t i d a l range. It i s h i g h l y p e r f e c t i b l e from the numerical p o i n t o f view as w e l l as f o r
the t u r b u l e n t exchange simulations, b u t before complicating i t , i t i s worth weighting t h e pros and cons. Our opinion i s t h a t f o r r e s t r i c t e d coastal s i t e s under t h e i n f l u e n c e o f the t i d e and the wind, preventing high s t r a t i f i c a t i o n t o occur, and f o r which i n i t i a l and boundary conditions are h a r d l y a v a i l a b l e , such a model i s an i n t e r e s t i n g compromise and an e f f i c i e n t t o o l . 6 REFERENCES De Borne de Grandpre, C., 1979. Modele bidimensionnel en temps r e e l de l a c i r c u l a t i o n v e r t i c a l e estuarienne. A p p l i c a t i o n I l a Gironde. Oecanologica Acta, 2, 1, 61-68. De Borne de Grandpre, C., E l Sabh, M . I . and Salomon, J.C., 1981. A twodimensional numerical model o f t h e v e r t i c a l c i r c u l a t i o n o f t i d e s i n the S t Lawrence Estuary. Estuarine, Coastal S h e l f Sci , 12, 375-387. Hamilton, P., 1975. A numerical model of t h e v e r t i c a l c i r c u l a t i o n o f t i d a l estuaries and i t s a p p l i c a t i o n t o t h e Rotterdam Waterway, Geophys. J.R. Astron. SOC., 40, 1-21. Le H i r , P. and l'Yavanc, J., 1985. Obssrvations de courant en b a i e de Seine. Rapport Scient. CNRS/Greco Manche, 7-14. 1977. Three-dimensional model o f t i d e s and storm surges i n a Nihoul, J.C.J., shallow well-mixed c o n t i n e n t a l sea. Dynamics o f Atmosphere and Oceans, 2,29-47. Salomon, J.C. ana Le H i r , P., 1980. Etude de l ' e s t u a i r e de l a Seine. Modelisation numerique des phenomenes physiques. Rapport Scient. CNEXO/UBO, 286 p. Salomon, J.C., 1981. Modelling t u r b i d i t y maximum i n t h e Seine Estuary. I n : J.C.J. Nihoul ( e d i t o r ) , Ecohydrodynamics. Elsevier, Amsterdam, 285-317. Salomon, J.C., 1985. Courantologie calculee en b a i e de Seine. Rapport Scient. CNRS/Greco Manche, 15-22. Thouvenin, B. , 1983. Modele tridimensionnel de c i r c u l a t i o n e t de dispersion pour des regions c b t i e r e s a maree. These 3e c y c l e U.B.O., 269 p. Thouvenin, B. and Salomon, J.C. , 1984. Modele tridimensionnel de c i r c u l a t i o n e t de dispersion en zone c b t i e r e 1 maree. Premiers essais : cas schematique e t baie de Seine. Oceanologica Acta, 7,4, 417-429.
THREE-DIMENSIONAL MODEL OF CURRENTS I N THE BAY OF SEINE J.C. SALOMON, 6. THOUVENIN and P. LE H I R IFREMER, B.P. 337, 29273 BREST CEDEX
ABSTRACT Water movement and m i x i n g processes i n t h e Bay o f Seine a r e e s s e n t i a l l y i n duced by t i d e s ; winds and d e n s i t y d i f f e r e n c e s due t o r i v e r i n f l o w and temperat u r e g r a d i e n t s near t h e coast. To study t i d a l c u r r e n t s alone, i t i s s u f f i c i e n t t o use a two dimensional model. Howeveryfor wind and d e n s i t y e f f e c t s , t h e t h r e e s p a t i a l dimensions must be taken i n t o account. For t h a t purpose a three-dimensional model o f c u r r e n t s has been developed which solves t h e n o n l i n e a r Navier-Stokes equations i n a s t r a i g h t f o r w a r d manner through a f i n i t e d i f f e r e n c e technique. The model has been t e s t e d i n various schematic cases f o r which a n a l y t i c a l s o l u t i o n s o r r e s u l t s from previous models were a v a i l a b l e . It was then a p p l i e d t o t h e Bay o f Seine and proved t o be an e f f i c i e n t t o o l f o r coastal studies.
1 .REGIONAL SETTING The Bay o f Seine, as d e f i n e d by t h e c o a s t l i n e , i s shaped more o r l e s s l i k e a r e g u l a r q u a d r i l a t e r a l about one hundred k i l o m e t e r s long and f o r t y t o f i f t y kilometers wide, opening onto t h e E n g l i s h Channel ( F i g . 1).
The bay i s q u i t e
shallow (twenty f i v e meters deep) except f o r t h e a n c i e n t submerged v a l l e y o f the Seine which runs d i a g o n a l l y across t h e bay and reaches depths o f f o r t y met e r s i n t h e northwestern p a r t o f t h e bay. With an average flow o f 400 m 3 / s , t h e Seine i s a modest r i v e r . i t i s the o n l y s i g n i f i c a n t f l u v i a l discharge i n t h e Channel.
Nevertheless,
The estuary, which
used t o be upstream of t h e c i t y o f Le Havre, has now a s m a l l e r area, because o f the improvement o f t h e waterway. Over t h e l a s t hundred and f i f t y years, the estuarine area has decreased by a f a c t o r of two, and a t t h e same time t h e locat i o n has s h i f t e d towards t h e bay. downstream o f Le Havre.
Today t h e d e n s i t y nodal p o i n t i s s i t u a t e d
The c h a r a c t e r i s t i c s of t h e bay a r e q u i t e complicated. Indeed, t h e present s i t u a t i o n includes almost a l l t h e p o s s i b l e causes f o r water movements i n t h e coastal zone : The main p a r t of t h e dynamics i s due t o t h e t i d e . The t i d a l range can exceed
-
e i g h t meters i n s p r i n g t i d e s .
The
c u r r e n t s increase r e g u l a r l y from e a s t t o
west, from one t o t h r e e knots. The t i d a l wave and t h e associated c u r r e n t s are s t r o n g l y d i s t o r d e d w h i l e propagating i n t h e shallow areas, so t h a t .the n o n l i near terms i n t h e hydrodynamicequations a r e i m p o r t a n t .
428
F i g . 1. Location map and bathymetry o f the Bay o f Seine.
-
The wind i s another d r i v i n g f o r c e o f water movements i n the bay, a l l the
more important as i t i s r a t h e r intense and as water depths are small.
-
As mentionned before, d e n s i t y c i r c u l a t i o n does e x i s t too, c l o s e l y related
t o the f l u v i a l r u n o f f i n the eastern p a r t o f the bay. It i s almost o f estuarine type (converging v e l o c i t i e s near the bottom, nodal p o i n t ) w i t h some geostrophic features ( r o t a t i o n o f surface v e l o c i t i e s t o the r i g h t ) .
-
F i n a l l y water movements i n the bay i s p a r t l y due t o the general c i r c u l a t i o n
i n the Channel.
2 MODEL SPECIFICATIONS The t i d a l components o f t h e c u r r e n t s i n t h e bay can be studied, f o r t h e i r essential p a r t , w i t h two-dimensional (2D) models, b u t those which are l i n k e d t o the d e n s i t y f i e l d o r t o the wind do r e q u i r e a three-dimensional ( 3 D ) s l c u l a t i o n .
429
A few years ago we made a f i r s t attempt a t modelling currents in three dimensions in t h i s area (Salomon and Le Hir, 1980). We proceeded i n a quite s i milar way t o t h a t used by Nihoul (1977) a t the time, by developing along the vertical coordinate the r e s u l t s of a 2D model a t specific points. The results revealed some interesting apects of the currentology, b u t such calculations are not really three dimensional :the f r i c t i o n a l s t r e s s i s considered to be in the same direction as the vertically averaged velocity. Moreover, the nonlinear advective terms are d i f f i c u l t to take into account, since the calculations are local. The technical specifications of the present 30 model are that the model should: - solve the complete nonlinear equations; take into account the precise form of the sea bed (very important in coastal problems) , which eliminates " s t a i r step" models; - have the same behaviour a t a l l times of the computation and over the whole bay. Due t o the depth differences in our zone, and due t o the tidal amplitude, we discarded the use of reduced coordinates. As different grids a c t l i k e d i f ferent media, waves and dynamic structures are partly reflected and distorted. On the other hand, the model could have modest performances as f a r as horizontal turbulent fluxes calculations are concerned because f o r the time scales we are interested in, these terms are small compared t o the advective terms. Simple considerations on the dynamics of the bay show t h a t due tx i t s restricted dimensions and the intensity of f r i c t i o n forces, i t s "memory" towards barotropic waves and even baroclinic ones i s of a few hours. I t s memory towards phenomena of transport of matter ( l i k e s a l t ) i s much higher. Calculations based on the lack of s a l t in the bay give a residence time of about a month. For financial reasons (among others) i t was impossible t o measure boundary conditions during such a long period (except f o r the t i d e ) and the project of such simulations was given u p . Model indications, in terms of density circulation, will essentially r e s u l t from the i n i t i a l s a l t and temperature distribution, and n o t from modifications of the boundary conditions d u r i n g the calculations. Barotropic movements, on the other hand, are driven through the boundary condi tioos.
-
3 MODEL CHARACTERISTICS 3.1 Equations The model solves the Navier-Stokes equations i n t h e i r usual form, using the hydrostatic approximati on , and assuming Fi cki an diffusion:
430
p = PO ( 1 t as)
as ~ t
L
as ax e t
v
as ay - t
asw az
as
a(KZ E) az
F, = o
: h o r i z o n t a l coordinates
: v e r t i c a l coordinate w) : v e l o c i t y components : surface e l e v a t i o n 4 H : t o t a l depth o f water : mass per u n i t volume P f : Coriolis factor : salinity S : Fx, FY h o r i z o n t a l d i f f u s i o n o f momentum Fs : horizontal diffusion o f s a l t Nx, NY : v e r t i c a l eddy v i s c o s i t y c o e f f i c i e n t s Kz : c o e f f i c i e n t f o r s a l t d i f f u s i o n along the v e r t i c a l
The h o r i z o n t a l d i f f u s i o n terms Fx, Fy, Fs are p h y s i c a l l y important i n vert i c a l l y i n t e g r a t e d models where they are c a l l e d dispersion because they include a t the same time the v e r t i c a l v e l o c i t y gradients and t h e v e r t i c a l 'diffusion. Here they are e s s e n t i a l l y r e q u i r e d t o damp small scale computational modes. It would be d i f f e r e n t i f long term
simulations were t o be done.
They are expres-
sed i n t h e f o l l o w i n g usual way : FX = NXX FY = NYX FS = KX
a2u
a2u + NXY 7 aY
a2v a2v v t NYY aY
a2s t ax.
7
KY
a2s
-2-
aY I n t h e present a p p l i c a t i o n t o the Bay o f Seine, which we know t o be rather
well-mixed, t h e pressure gradients are expressed i n the f o l l o w i n g simple form:
431 3.2 C o e f f i c i e n t s f o r v e r t i c a l exchange The model makes use of t h e concept o f t u r b u l e n t exchange c o e f f i c i e n t s parameterized i n a simple form.
The other, more sophisticated, p o s s i b i l i t y , which
would c o n s i s t s i n solving, a f t e r closure, the turbulence equations, has been rejected because: no i n i t i a l o r boundary c o n d i t i o n s are a v a i l a b l e w i t h enough accuracy,
-
i t has been established by d i f f e r e n t i n v e s t i g a t o r s t h a t i n the case o f t i d a l
currents i n shallow waters, a r e l a t i o n between the d i f f u s i o n c o e f f i c i e n t s and the v e l o c i t y gradients ( o r t h e concentration g r a d i e n t ) through a mixing length, gives reasonable r e s u l t s . point
.
No a d d i t i o n a l complication has been seeked on t h i s
The e f f e c t o f buoyancy forces, which h i g h l y modify t h e mixing length, i s introduced by means o f a global Richardson number:
AS: S a l i n i t y d i f f e r e n c e between bottom and surface U : v e r t i c a l l y averaged v e l o c i t y .
Several expression ( g e n e r a l l y empirical ones) have been t e s t e d i n t h e model. The f o l l o w i n g r e l a t i o n s appear t o g i v e the best r e s u l t s :
K, =
3.3.
t
5.10-2U log(O.l z 3 t 1 ) ( 1 t R i ) -
7
7
Method o f r e s o l u t i o n
The numerical procedure and t h e d i s c r e t i z a t i o n g r i d are extrapolated from the two dimensional v e r t i c a l model t h a t we already used i n several estuaries: Gironde, S t Lawrence (de Borne de Grandpre e t a l . , 1979, 1981), Seine (Salomon, 1981). It i s a mixed i m p l i c i t - e x p l i c i t f i n i t e d i f f e r e n c e technique. The o r i g i n a l i t y o f t h e method l i e s e s s e n t i a l l y i n the f a c t t h a t the locat i o n s o f the upper and lower f r o n t i e r s of t h e i n t e g r a t i o n area change during the c a l c u l a t i o n s . I n order t o achieve t h a t , use i s made o f v i r t u a l p o i n t s 10cated above t h e surface, and o t h e r s s i t u a t e d under t h e bottom (Hamilton, 1975). D e t a i l s o f t h i s technique have been exposed by Thouvenin and Salomon (1984).
432
HORIZONTAL
VERTICAL
j+l
j
j-1
Ax i-1
i
Ax
i +1
i -1
i
i +1
Fig. 2. Computational g r i d s B r i e f l y stated, a t ev.ery time step a f i n i t e d i f f e r e n c e method i s used t o go over the usual c a l c u l a t i o n sequence i n s i d e the area o f computations. The results are then e x t r a p o l a t e d t o external v i r t u a l p o i n t s by a n a l y t i c a l f u n c t i o n s which take i n t o account t h e upper and lower boundary conditions : -+ -+ a t the surface -+
av = k V1, Nz az -+
/Vim/
and
as
= o
a t t h e bottom
+ 1 ',
: t h e v e l o c i t y vector one meter above t h e bottom
T~
: wind s t r e s s
k
: f r i c t i o n coefficient
+
The f u n c t i o n s used a r e u s u a l l y polynomials o f t h i r d degree (splines), except f o r the v e l o c i t y near the bottom. To ensure a c o r r e c t v e l o c i t y p r o f i l e j u s t above t h e bottom where i t i s known t o be logarithmic, we use e i t h e r a l o g a r i t h m i c
curve o r an exponential one, obeying the c o n t i n u i t y o f
the v e l o c i t y curve and o f i t s f i r s t d e r i v a t i v e . 3.4 Numerical scheme The numerical scheme i s mixed i m p l i c i t / e x p l i c i t . A l l s p a t i a l d e r i v a t i v e s are centered except advective terms i n t h e s a l t equation. The pressure gradient term, p a r t i c u l a r l y important f o r the s t a b i l i t y o f t h e scheme, i s t r e a t e d as implicit.
433 Considering t h e equations i n t h e sequence described i n 3.1.,
i t i s possible t o
solve the complete system i n an e x p l i c i t way, i . e . w i t h o u t s o l v i n g any m a t r i x . The complete numerical scheme i s described and discussed i n Thouvenin (1983). 4. MODEL APPLICATIONS
4.1 Schematic t e s t s Before applying t h e model t o t h e Bay o f Seine, we t e s t e d i t s performances i n sane simple and r a t h e r schematic cases f o r which aspects o f t h e s o l u t i o n are known. I n p a r t i c u l a r , we have studied (Thouvenin, 1983):
- The propagation o f a sinusoidal wave i n a rectangular - The f r e e o s c i l l a t i o n s i n a closed basin. - The v e l o c i t y f i e l d i n a channel crossed by a trench. - The e f f e c t s o f t h e wind blowing over t h e open sea. - Convective movements along isopycnal surfaces.
basin.
Here, we s h a l l b r i e f l y comment on the l a s t two experiments.
N
WIND
Point 12 (170 Point 10 (130 m.) Point 6 ( 5 0 m . )
Fig.3. Water p a r t i c l e s t r a j e c t o r i e s under wind a c t i o n (Distance from the surface i n d i c a t e d i n brackets).
434
Fig. 3 shows the t r a j e c t o r i e s o f water p a r t i c l e s induced by a constant wind supposed t o blow during 33 hours from t h e south, and then t o stop.
According t o t h e known a n a l y t i c a l s o l u t i o n , we observe t h e superposition o f t h e Ekman d r i f t s p i r a l and o f i n e r t i a l o s c i l l a t i o n s .
When t h e wind stops t h e i n e r t i a l
r o t a t i o n , stronger near t h e surface. diminishes.
Some energy i s transferred
through viscous s t r e s s t o the lower l a y e r s which are thus accelerated. Comparing t h e thickness o f the Ekman l a y e r produced by t h e model w i t h the t h e o r e t i c a l s o l u t i o n , we f i n d f o r Nz a d i f f e r e n c e o f 20 %. This d i f f e r e n c e which sums up several numerical e r r o r s seems q u i t e reasonable. The movement associated t o a s a l t wedge i n a rectangular tank i s i n t e r e s t i n g Here, we observe the formation o f a d e n s i t y d r i v e n c i r c u l a as w e l l ( f i g . 4 ) . t i o n along the i n t e r f a c e and t h e appearance o f a b a r o t r o p i c wave propagating very q u i c k l y f a r from t h e s a l t wedge.
Also t o be noted i s
the polarisation
o f v e r t i c a l movements, more intense on t h e r i g h t side, due t o t h e C o r i o l i s f o r ce
.
Fig. 4 S a l t wedge c i r c u l a t i o n a) l a t e r a l views a f t e r 30 and 300 minutes
b) view from above a f t e r 5 hours
435 4.2 A p p l i c a t i o n t o the Bay o f Seine ( i ) I n i t i a l and boundary conditions.
Because o f computer l i m i t a t i o n s , t h e
application t o the Bay o f Seine i s done w i t h a r e l a t i v e l y coarse g r i d s i z e ( 4 km. 4 m).
It must be acknowledged t h a t due t o t h e l a c k o f i n f o r m a t i o n on
boundary conditions, a g r i d refinement would n o t have been completely j u s t i f i e d . I n i t i a l conditions f o r s a l i n i t y are taken from i n s i t u measurements made during the year 78. Boundary conditions along the open boundary (water l e v e l s and v e l o c i t i e s ) are taken from a 2D model (Salomon, 1985).
The v e r t i c a l s t r u c t u r e o f the h o r i -
zontal v e l o c i t i e s i s supposed t o obey a l o g a r i t h m i c p r o f i l e .
The v e r t i c a l com-
ponent o f the v e l o c i t y i s l i n e a r l y i n t e r p o l a t e d between t h e bottom and the surface where i t i s . c a l c u l a t e d through the f o l l o w i n g r e l a t i o n s : 'bottom
=
( i i ) T i d a l currents. The t i d a l component o f t h e currents i s the e s s e n t i a l part o f the dynamics b u t as i t seemed already w e l l known through measurements (Le H i r and l'Yavanc, 1985) o r 2D modeling (Salomon, 1985), l i t t l e a d d i t i o n a l information was expected. I n f a c t , we observe (see F i g . 5): important v e r t i c a l movements which, o f course, could n o t have been measured. They are t h e r e s u l t o f t h e bottom shape and h o r i z o n t a l convergences.
-
-
phase d i f f e r e n c e s and r o t a t i o n s along t h e v e r t i c a l which come from the d i f f e r e n t r a t i o o f the i n e r t i a l f o r c e and the d r i v i n g f o r c e (surface slope) as well as from c h a r a c t e r i s t i c s o f bottom Ekman l a y e r . Thus we are more aware o f the e r r o r we make i n a 2D model assuming t h e f r i c t i o n s t r e s s on the bottom t o be i n the same d i r e c t i o n as the average velocity. ( i i i ) Wind induced movements. The wind induced dynamics i s t y p i c a l l y threedimensional and cannot be approached by 2D simulations. This component o f the c i r c u l a t i o n i s l i n k e d t o others and e s p e c i a l l y t o the tide.
The r e s u l t s may be very complex and d i f f i c u l t t o understand from i n s i t u
measurements.
The model here i s o f g r e a t assistance, even i f i t s i n d i c a t i o n s
are n o t very accurate. Theoretically, c a l c u l a t i o n s should take i n t o account the response o f the e n t i r e Channel t o wind f o r c i n g and l a r g e scale atmospheric pressure gradients. However, some i n t e r e s t i n g i n f o r m a t i o n can be obtained from l o c a l simulations about the response o f t h e bay t o wind f o r c i n g , a t l e a s t q u a l i t a t i v e l y .
64
-40 16
96Km
32 Km
-
\
-
8 -8
0
- 16
16
-40
32
m 1 16
80
-24
48
-32
32
-32
16
-24
I
-32
- 24 48
64
16
32
Km
80
16
96Km
32
16
48
32
64
48
.--+--*__-
.\-&-&-b*D*Q*
BOTTOM
16
80
64
32Km
80
%Km
96Km
I
437
1 R
. . . .! . . . . . . . . . . . l
,
t
?
a
X
'
l
'
~
I . . .
. . . . . .
. . . . . I
R S , . r ? ! l p ? ?
I . . . . . . . . . .
:ssnR.rcnnvnar
.'
F i g . 6. Wind induced v e l o c i t y f i e l d ( h o r i z o n t a l sections near the surface and the bottom ; N-S v e r t i c a l sections ; E-W v e r t i c a l s e c t i o n s ) . Wind blowing from the west.
438 So we have c a r r i e d o u t some s i m u l a t i o n s o f t h e d i r e c t a c t i o n o f t h e wind on t h e bay alone. The boundary c o n d i t i o n corresponds t o a h y p o t h e t i c a l steady regime.
I n the
dynamical equation along t h e n o r t h e r n boundary a balance i s assumed between t h e surface slope, f r i c t i o n stresses on t h e bottom and a t t h e surface, and the C o r i o l i s force. The r e s u l t s presented i n Figs. 6 and 7 are f o r u n i f o r m wind f i e l d s o f 15 m/s.
A permanent regime i s g e n e r a l l y observed a f t e r seven hours. The h o r i z o n t a l s e c t i o n s c l e a r l y show t h e s t e e r i n g o f t h e c u r r e n t s by bathym e t r i c features, p a r t i c u l a r l y near t h e bottom ( f i g . 6 ) . Cross c u r r e n t s a r e h a r d l y ever observed i n t h e Parfond ( a n c i e n t Seine valley) even close t o t h e surface, where c u r r e n t s a r e l e s s r e l a t e d t o t h e bottom shape. V e l o c i t i e s t h e r e o f t e n r o t a t e a few tens o f degrees.
A t t h e surface, t h e v e l o c i t i e s are l a r g e r i n t h e shallow zones, a consequence o f t h e f a c t t h a t t h e d r i v i n g f o r c e r e l a t e d t o t h e u n i t o f volume o r u n i t o f mass i s i n v e r s e l y p r o p o r t i o n a l t o t h e water depth.
This i s t h e basic reason
f o r t h e general dissymmetry between t h e eastern and western p a r t s o f t h e bay.
Fig. 7 H o r i z o n t a l v e l o c i t y f i e l d s f o r d i f f e r e n t wind d i r e c t i o n s .
439 For the same reason,the wind induced v e l o c i t i e s above t h e Parfond are weaker, When the conditions are such t h a t a d e f i c i t o f water appears i n t h e southern p a r t o f the bay, an adjustment c u r r e n t appears i n the Parfond, opposite t o the surface v e l o c i t i e s .
This i s a w e l l known f e a t u r e o f wind a c t i o n i n l i t t o r a l
zones and lakes : a back c u r r e n t appears i n the deepest parts, g e n e r a l l y the waterway.
I t i s a l s o i n close agreement w i t h measurements made during t h i s
study. The v e r t i c a l sections a l s o show very obvious v e r t i c a l s t r u c t u r e s corresponding t o v e r t i c a l movements near the banks and h e l i c o i d a l movements i n s i d e the trench o f the Parfond, which measurements would n o t have revealed. These simulations, as w e l l as the measurements, i n d i c a t e t h a t t h e wind i n duced v e l o c i t i e s are o f t e n o f an order o f magnitude o f 10 t o 30 an/s, which i s much higher than t i d a l r e s i d u a l currents.
This mechanism i s thus an essential
vector o f t h e long term movements o f waters i n t h e bay.
5 CONCLUSION . The model presented i n t h i s paper i s b u t one element o f a general study o f the bay. The study l a s t e d t h r e e years, during which a considerable e f f o r t has been made on measurements : t h r e e cruises o f a few months each have been c a r r i e d out, i n v o l v i n g d i f f e r e n t measurement l o c a t i o n s (up t o twelve), sometimes a t t h e surface, a t t h e bottom and a t mid-depth. Important human and technical resources have thus been involved.
Fig.8 Diagram showing t h e r e s i d u a l c i r c u l a t i o n i n t h e Bay o f Seine
440
The measurements produced a l o t o f i n t e r e s t i n q i n d i c a t i o n s but,finallv,when t came t o e l a b o r a t i n g a comprehensive synthesis o f thedynamical regime o f the bay, the main r o l e was devoted t o the model. This way we were able t o understand what occured and we could separate t h e proper e f f e c t s o f each o f the physical mechanisms involved : wind, t i d e , density, C o r i o l i s ... This way we understood what measurements where showing. The diagram f o r the residual c a l c u l a t i o n presented i n fig.8,
was e s s e n t i a l l y
deduced from model r e s u l t s . So, i t proved t o be an e f f i c i e n t t o o l f o r conducting coastal studies.
Some advantages o f t h i s model have been already mentioned, e s p e c i a l l y the p o s s i b i l i t y o f c o n c i l i a t i n g a f i x e d l o c a t i o n o f computational p o i n t s and a continuous representation o f the bottom and t h e surface, i n s p i t e o f a great t i d a l range. It i s h i g h l y p e r f e c t i b l e from the numerical p o i n t o f view as w e l l as f o r
the t u r b u l e n t exchange simulations, b u t before complicating i t , i t i s worth weighting t h e pros and cons. Our opinion i s t h a t f o r r e s t r i c t e d coastal s i t e s under t h e i n f l u e n c e o f the t i d e and the wind, preventing high s t r a t i f i c a t i o n t o occur, and f o r which i n i t i a l and boundary conditions are h a r d l y a v a i l a b l e , such a model i s an i n t e r e s t i n g compromise and an e f f i c i e n t t o o l . 6 REFERENCES De Borne de Grandpre, C., 1979. Modele bidimensionnel en temps r e e l de l a c i r c u l a t i o n v e r t i c a l e estuarienne. A p p l i c a t i o n I l a Gironde. Oecanologica Acta, 2, 1, 61-68. De Borne de Grandpre, C., E l Sabh, M . I . and Salomon, J.C., 1981. A twodimensional numerical model o f t h e v e r t i c a l c i r c u l a t i o n o f t i d e s i n the S t Lawrence Estuary. Estuarine, Coastal S h e l f Sci , 12, 375-387. Hamilton, P., 1975. A numerical model of t h e v e r t i c a l c i r c u l a t i o n o f t i d a l estuaries and i t s a p p l i c a t i o n t o t h e Rotterdam Waterway, Geophys. J.R. Astron. SOC., 40, 1-21. Le H i r , P. and l'Yavanc, J., 1985. Obssrvations de courant en b a i e de Seine. Rapport Scient. CNRS/Greco Manche, 7-14. 1977. Three-dimensional model o f t i d e s and storm surges i n a Nihoul, J.C.J., shallow well-mixed c o n t i n e n t a l sea. Dynamics o f Atmosphere and Oceans, 2,29-47. Salomon, J.C. ana Le H i r , P., 1980. Etude de l ' e s t u a i r e de l a Seine. Modelisation numerique des phenomenes physiques. Rapport Scient. CNEXO/UBO, 286 p. Salomon, J.C., 1981. Modelling t u r b i d i t y maximum i n t h e Seine Estuary. I n : J.C.J. Nihoul ( e d i t o r ) , Ecohydrodynamics. Elsevier, Amsterdam, 285-317. Salomon, J.C., 1985. Courantologie calculee en b a i e de Seine. Rapport Scient. CNRS/Greco Manche, 15-22. Thouvenin, B. , 1983. Modele tridimensionnel de c i r c u l a t i o n e t de dispersion pour des regions c b t i e r e s a maree. These 3e c y c l e U.B.O., 269 p. Thouvenin, B. and Salomon, J.C. , 1984. Modele tridimensionnel de c i r c u l a t i o n e t de dispersion en zone c b t i e r e 1 maree. Premiers essais : cas schematique e t baie de Seine. Oceanologica Acta, 7,4, 417-429.