DSpace at VNU: A three-dimentional simulation of the tidally modulated plume in the rever entrance region

VNU JOURNAL OF SCIENCE. Nat . S c i. & Tech . T XIX. N01. 2003

A T H R E E - D I M E N T I O N A L S IM Ư L A T IO N O F T H E T I D A L L Y
M O D Ư L A T E D P L Ư M E IN T H E RIVKR E N T R A N C E R E G I O N
N g u y ê n Minh H u a n
D e p a r tm e n t o f H y d r o -M e te o r o lo g y a n d O c e a n o g r a p h y
C o llcg e o f S c ie n c e , V N U

A s t r a c t . A three dim cntionaỉ mathcmatical model ùH presented to computc the
ivatcr levcl, velocity and salinity distributions in stratified Coastal Lvaters a n d
tid a lly m o d u la te d p lu m c o f th c rivcr en tra n cc region T h e m o d c ỉ sy s tc m c o n s is ts
of hydrodynamìc, transport and turbulence closure modcls In the hydrodyn am ìc
m o d e l co m p o n cn t, th e N a v ic r -S to k c s cq u a tio n s a re so lved iv ith th e h y d r o s ta tíc
a s s u m p tio n (Itìd th e B o u ssin csq a p p ro xim a tio n . T kc tr a n s p o r t m o d cl coruĩists o f
th c ivatcr tc m p c ra tu rc a n d s a lin ity tra n sp o rt mocleLs. T h e v a r ia tio n s in th e Lưatcr
te m p c r a tu r e a n d s a lin ity in fĩu en ce th e Uỉater d e n sity , a n d in rc tu r n th e velo c ity
fic ld . T h e e q u a tio n s o f m o m e n tu m a n d c o n tin u ity arc soỉvccl n u m e r ic a lly u s in g
th e m o d c -s p littin g tcch n iq u e. A s th e tu rb u len ce m odel, a o n e-eq u a tio n k -e p siỉo n
tu rb u le n c e m o d e l is a p p lied . ỉn th e tra n sp o rt rnodel th e th r c e -d im c n tio n a l
tíd v c c tiv c d iffu 8 Ìo n e q u a tio n are solved. The m o d el is a p p lie d to a rec ta n g le
b a s in en clo sed by a Coastal b o u n d a ry a n d 'th ree opcn sea b o u n d a ries, tid a l
fo r c ỉn g is im p o se d in th c fo rm o f CI /ric tio n less K elvin w a ve w ith o Ị fr c q u e n c y
c n te r in g a t th e ivestcrn b o u n d a rỵ, frc sh w a te r lo a d in g w a s ta k c n in to a c c o u n t a t
lo cu tio n o f one river m o u th , Uỉhich rcached a to ta l o f lOOOm1 s 1

1. I n t r o d u c t i o n
A n e stu a r y is an a r ea o f in ter a c tio n betvveen s a lt a n d ír e s h vvater. T ho nnơ)§t
c o m m o n d e fin itio n u sed t h a t s t a t e s "an estu a ry is a s e m i-e n c lo s e d C oastal b o d y

of

vvater vvhich h a s a free c o n n e c tio n w ith the open sea and vvithin vvhich s e a w a t e r is
m e a s u r a b ly d ilu te d w ith fresh w a te r derived from ia n d d r a in a g e ”. T h e e s t u a r i m e

inHuence may extend to nearshore Coastal waters vvhere seavvater is diluted by Ịaind
d r a in a g e but b ey o n d th e c o n íìn e s o f e m e r g e n t land - m a s s e s .
T h e c la ss ic d e fin itio n of an estu a r y in c lu d e s t h e s e th r e e c h a r a c t e r is t u c s :
s e m ie n c lo s e d , free c o n n e c tio n vvith th e open sea , and ír e s h w a te r d e r iv ed from laind
d r a in a g e .

T hese

th r e e

c h a r a c te r is tic s

govern

th e

c o n c e n tr a tio n

of

s e a w a t; e r ,

therefore, s a lin ity is th e key to e s tu a r in e c la ss iíìc a tio n . T h e m ixing o f fresh V, a \ter
and

sea w ater

p rodu ces

d e n s ity

g r a d ie n ts

th a t

d r iv e

d is tin c t iv e

e stu a r in e

( g r a v ita tio n a l) c irc u la tio n p a tte r n s.
T h e s e c irc u la tio n ancỉ s h o a lin g p a tte rn s differ w ith e a ch e s t u a r y sy stc e m
accorcỉing to th e d ep th , tid a l a m p litu d e and p h a se at t h e m o u th , and th e a m o u n lt of
fresh w a te r flo w in g in to th e basin.

30


A t h r c c - d i m c n t i o n a l s ir n n la t ÌOĨI o f th c.

31

T ho tid e t h a t a p p r o a c h e s t h e m ou th of th e e s tu a r y is th e r e s u lt o f all th e
a str o n o m ic a l, m e te ọ r o lo g ic a l, se is m ic , and m a n -m a d e íactors a ffe c tin g am plit.ude
and (Yequency o f t h e vvave. A s th e tide e n te r s th e e s tu a r y , it is g r ea tly in flu e n c e d bv
th e river d e p th , w id th , an d d isc h a r g e.

S u p e r im p o s e d on t h is tid a l action is th e fr e s h w a t e r /s a ltw a te r in ter a c tio n . S a lt
w ater w ill a d v a n c e up a s y s t e m u n til th e tidal flơw can no longer o v e rc o m e th e
riverflow. D e p e n d in g on th e r ela tio n s h ip betvveen tid al flow a n d river flow, th e
estu a r y c a n be c la s s if ie d by its s a lin ity s tr u c tu r e a n d r e s u ltin g c irc u la tio n p a tte r n s.

2. T h e o r e t i c a l c o n s i d e r a t i o n s
To s im u lt e w ind d r iv e n circ u la tio n and d e n s ity c u r r e n ts th a t occur in Coastal
w a ters e s p e c ia lly in e s t u a r y s tr a tifie d bv s a lin itv and te m p e r a tu r e la y e r s c a u s in g
s ig n iíìc a n t la te r a l d e n s ity g r a d ie n ts , th r e e -d im e n tio n a l m a th e m a tic a l m odel are
n ecessa ry . T he d e v e lo p e đ th r e e -d im e n tio n a l

m a th e m a tic a l

model is c a p a b le of

co m p u tin g th e w a te r lev el a n d vvater particle v e lo city clistribution in th r e e princip al
directio n s

by

s o lv in g

ap p ro x im a tio n

and

th e

th e

N a v ie r -S to k e s

a s s u m p tio n

of

e q u a tio n s

ve rtica l

u s in g

h y d ro sta tic

th e

B o u s s in e s q

e q u ilib r iu m ,

th e

c o n tin u itv e q u a tio n an d e q u a tio n s of te m p e r a tu r e and s a lin ity .

2.1 G o v e r n in g eiỊ uations o f the m odel
T h e b a sic e q u a t io n s in th e th ree-cỉim en sion a l c a r te s ia n co o rd in a te s y s t e m are:
du

cu
— + // —

ct

dx

cu

du

d/ 1 y\ ỉ

ri ỵ

r

õp

0

p tí dx

õ

+ V — + Ví' — - A' = —

Đz

dv
«h'
cv

—- + // — + V
dí
dx
dy

õv
d2

1

——
-+

1 dp

.
~

(1
dz

p(] õ y

cu
T ortz

d_

du

dz

( 2 . 1)

ơy

dx

+~

rx

r*y +

õy

(2 .2 )

yy

<ỉ

(2.3)

~\

(z
du
dx

dv
ôy

=0

(2.4)

1

dJ

d

ÕT
?T
cT
dĩ
— + tỉ——+ V ——+ U’---oí cx (>V
02

--- --------- - + ---

ds

PS

d.S

ơ (

cỉ

rx

Õz

Õz \

(7s

-----+
5 . I I -----■> + V’ —

+ H' -—

A) c p d z

dz

dĩ

ar
õx

■H

(~ :x

~ <• \
ô (

ÕS)
ò
ÕS)
+—
-T- -t- ---+
s
11 X "
d x { 11 ~õ x j õ y
5z

ar
ày

7/

(2.5)

ôy

(2 . 6 )

vvhere (u ,v tw ) a r e t h e c o m p o n e n ts of th e c u r r e n t , T d e n o te s t h e t e m p e r a t u r e , s th e
s a l i i i t y , f = 2 0 s i n 0 t h e C oriolis íreq u en cy, ũ

=271/86164 racl/s th e r o ta tio n


N guy en Minh Hu an

32

(Yequencv of t h e E a rth , g t h e a cceleratio n o f grav ity, p t h e p r e s s u r e , VT and ẢT th e
v e r tic a l

ed d y

v is c o s ity

and

diffusion co effic ien ts,

Ản th e

h o r iz o n ta l

diffusion

c o e ffic ie n t for s a ỉin it y a n d te m p e r a tu r e , p th e d e n s ity , Po a r e fe r e n c e d e n s ity , cft th e
s p e c iíic h e a t o f s e a w a t e r at c o n s ta n t p r e ssu r e and 1 (x, y, z, t) so la r irracliance.
T h e h o r iz o n ta l c o m p o n c n ts of th e s tr e s s te n s o r are d e fin e d by
■5

-2 v

yx

XV'

= V',

du

(2.7)

11 c x

du

ôv

dV

õx

( 2 .8 )

(2.9)
ỡy
w here

VH

is t h e h o r iz o n ta l diffusion coefficien t for m o m e n tu m .

T h e n u m e r ic a l s o lu tio n s o f th e m odel e q u a tio n s are g r e a t ly sim p liíled by
in tr o d u c in g a n e w v e rtica l coo rd in a te th a t tr a n s íb r m s both th e su rface and th e
b o tto m into c o o r d in a te o f su r fa c e s (P h illip s 19Õ7).

1

Suríàce ơ

Bottoni o

0

u

F igure 1.1. T he a-coorcỉinate transfom ation in th e vertical
T h e fo llo w in g c o o r d in a te tra n sfo rm a tio n is applied:
(t*. X*, y*, z*) = ự, X, y, Lf(a)),

(2 .10 )

vvhere
ơ =

z +h

z +h

c +h =1 T

( 2 . 11 )

is t h e c o m m o n ly usecỉ a -c o o r d in a te v a r y in g b e tw e e n 0 at th e bottom and 1 at the
s u r ía c e . T a k in g /(0 ) = 0 a n d / ’( 1 ) = 1 th e e q u a tio n o f th e b otto m ta k e s th e sim ple

A t hrcc-dirtient iotial simulation o f the.

33

form z* = 0 w h ile t h e m o v in g su rface tr a n sío r m s in to z* = L. T h is is íu r th e r
illu str a te d in F ig u r e 1 . 1 .
The

t r a n s ío r m e d

v e r s io n s

of

th e

e q u a tio n s

of

h o rizo n ta l

m o m e n tu m ,

h y d ro sta tic e q u iỉib r iu m , te m p e r a tu r e , s a lin ity and c o n tin u ity are g iv e n by

I c
>
I C
(:

—7 Ụu) + ^7 - ^ ụ i r
7 (Ci't
J ũx
..
=- » -p r r X

\

*/*« „
l £ ' Vj
t - 4 + Q\ + ,
~r

I
pu

õx

ũ , ,

./

x ì

c X

lí*

J

J

1

. .

-7 — r ( »

+ 7

J rí

0

,

V

1

r
T
1 õ
- ~ ự v ) + -7
J cv
J (2

7

(-/w v ) + — -

1 r‘/ >,

^

1

p í I^r r v

« - p ĩ - - - 7 - * r - f + c>; + -7 —

(■>'

( 2 . 12)

J õz*

v

0
7

J cx

cC

D

cz

du

/>i, (>>'

J (í:

♦
v) +

fu

(2.13)

VJ d z ,

L Ũ S l = />

(2.14)

(V

J

+ -■!/7 ;ạ
d ...
V

+ 77 7 7 ^

J dt

_L -ẼL I g
Jp íi

dz’

c p

L A
7 / H r> •
./ d.v
^ >

/í

~ G / .Ĩ ) + -ị—

1 ổ íJki

1

(^.S-) +

J ôx

./ rV
í)

J cy

Xi

(2.15)

dĩ'

Ỡ7' '

L_L
./ í V

J dt

JÀT

J dz*

+ - 7/ #-<■*•*•>
^7

.\

r-s
ỡr

1

-Ạ r(JvS ) + 1 ị ( V S )

J dy

a

J dx'

J dz

, \

JẢ II

ÕS
,

dx

fl.v

(2.16)


Nguyên Minh Hucunì

34

2.2. T u r b u l e n c e s c h e m e s
O n e of t h e m o s t in t r ic a t e p r o b le m s in o c e a n o g r a p h ic m o d ellin g is a n a đ e q u a tee
p a r a m e te r is a tio n o f v e r tic a l e x c h a n g e p rocesses. In th e p r e s e n t m od el th ey

a.ree

r e p r e s e n te d

tvvco

th r o u g h

th e

ed d y

co effic ien ts

VT

and

Ả r.

V a lu e s

for

th e s e

p a r a m e te r s a re to b e p r o v id e d by a tu r b u le n c e s c h e m e .
A large v a r ie t y o f tu r b u le n c e p a r a m e te r is a tio n s w ith a s u b s t a n t ia l r a n g e o:>f
c o m p le x ity h a v e b e e n p r o p o se d a n d v a lid a te d in t h e lite r a tu r e . T he se le c tio n o f ía
s u it a b le s c h e m e is o fte n a d iffic u lt ta s k sin ce it d e p e n d s o n t h e ty p e o f p h y s i c a i l
p r o c e s s e s specific for th e s im u a t e d area (e.g.tid es, t h e r m o c lin e s , river íronts,...).
In a n a lo g y w ith m o le c u la r d iffu sion w h ere t h e eddy v is c o s ity an d d if f u s io m

c o e ffic ie n ts are p r o p o r tio n a l to th e m e a n velocity t im e s and th e m ean free p a t h cof
t h e m o le c u le s, th e e d d y c o e ffc ie n ts

VT

a nd

ẢT

a re c o n s id e r e d a s th e product off :a

t u r b u le n t v e lo city s c a le a n d a le n g th scale / u s u a lly d e n o ted by th e K olm o go rov /P r a n d tl “m ix in g l e n g t h ”. A c o m m o n ly u se d v e lo city s c a le is th e sq u a re root

ooĩ

t h e tu r b u le n t k in e tic e n e r g y . T h is p a r a m e te r c a n b e o b ta in e d by s o lv in g a tr a n s p o r r t
e q u a tio n . T h e m o s t g e n e r a l form o f th is e q u a tio n s , as u s e d in th e program , iis
vvritten as

w h e r e th e tim e d e r iv a tiv e , t h e h o rizo n ta l and v e r tic a l a d v e c tio n as th e d iffu sico n
o p e r a to r s are d e fin e d by
1 d
T(k) = ^ ( J k )
J õt

( 2 . 1 8?a)

(2 .1 8 íb )

A Jk)= \~ (Jw k)

J dz

(2 .1 8 k )

(2.18^d)

N 2 and Af2 ar e s q u a r e d b u o v a n c v and sh ea r fr e q u e n c ie s g iv e n by


A thrce-dimcntional simulơtion of thc.

35

and /; d e n o t e s th e d is s ip a tio n r a te o f tu r b u le n c e e n e rg y . T h e d is s ip a tio n r a te is
p a r a m e te r is e d according to
k' 2
e = e „ = -ị-

>2 . 2 1 )

w h e r e cu is a c o n s ta n t d e te r m in e d by L\, = 0 .1 8 8 .
AU
d iffu sio n

tu r b u le n c e tr a n s p o r t
c o effic ien t

e q u a tio n s

are s o lv e d

vvith th e

vvhich is th e s a m e as t h e o n e u s e d

ẢfỊ

sam e

h o r ỉz o n ta l

in t h e e q u a t io n s o f

te m p e r a tu r e an d s a lin ity .
T h e ed d y c o effic ien ts ar e th e n e x p r e s s e d as

(2 .22 )

>*T = s b k2/c + xb,

VT = S Mk 2/c + vbt

w h e r e vht Ảh are prescribed b a c k g r o u n d c o e f fic ie n ts f Vi = / 0 7 [ m 2!s]; Ảfj = 7 0 5 Ị m 2/ s j
a n d s m, S), a r e u su a lly reffed a s th e s ta b ilitv íu n c t io n s . T h e ir e x p lic it íò rm s are
0 l()X ^ ()0 2 2 9 q v
‘

" ” 1+ 0.47l a V + 0 .0 2 7 5 a ỉ
0.177

•v

s

(2.23)

U 0 4()3av

k*

where ữ N = — Ari?

(2.24)

o

for th c

O n e -e q u a tio n k -ep silo n tu r b u le n c e m odel is u s e d for p a r a m e t e r i s a t i o n
m ix in g le n g th and

d is s ip a tio n rate. W hen o n e -e q u a tio n

m o d el is c h o s e n , th e k-

eq u a tio n is s till so lv ed vvith c m o d elled a c co rd in g to (2 .21 ) w h ile / is d e te r m in e d
u sin g th e ío r m u la tio n , in itia lly proposed by B la c k a d a r (1 96 2), h a s th e form
—

= —

/

/>

+ — +

Ạ

—

.

(2 .2 5 )

/a

H o r iz o n ta l diffusion te r m s are m e a n t to p a r a m e t e r iz e s u b g r id s c a le p r o c esses,
in practice th e h o rizo n ta l d iffu s iv itv
scale c o m p u ta tio n a l

n o is e th e y

sp a cin g s an d th e m a g n itu d e

V ịị

and

are ta k e n

Ả ịị

a re u s u a lly r e q u ir e d to d am p sm a ll

p ro p o rtio n a l to

o f th e velo citv d e íb r m a tio n

t h e h o r iz o n ta l

te n s o r

in

griđ

a n a lo g y w ith

S m a g o rin 3 ky*s (1963) p a r a m e te r is a tio n

V'// = C m0Ax Ay D T a nd

/.,/ = c .o Ax Ay D T.

(2.26)

2.3 B o u n d a r V a n d i n i t i a l c o n d i t i o n s
C o a sta l b o u n d a r ie s are c o n s id e r e d as im p r e g n a b le w a lls . T h is m e a n s
currrents, a d v e c tiv e and d iffu s iv e flu x es are s e t to zero

h a t all


Nguy en Minh Hucềìn

u = 0,

w = 0.

y = Q'

V= 0 ,

Juiị/ =0, ẢH— = 0

(2.27?)

Jvtỵ = 0 , ẢH— = 0 .

( 2 .2 8 Ỉ )

ôx

õy

O pen s e a (or river) b o u n d a r y co n d ition for the 2-1) m o d e n eed to be s u p p ỉie c d

eastern b o u n d a r ie s a n d for V a t S o u t h e r n a n d n o r t h e r r n
boundaries. A selection can be made between different types of open b o u n d a r -y

for

a t western a n d

u

conditions. T h e y h a v e th e form o f a ra d ia tio n co n d itio n d e r iv e d u s in g th e m e t h o d cof
c h a r a c te r is tic s [H e d str o m
[Rancỉall J. L eV e q u e

1979),

[Roed

and

Cooper,

1987],

[R uddick,

19 95Ị].

1997]. T h is is b a se d on the in te g r a tio n o f th e e q u a t io n s for t h ie

incoming and o u tg o in g R ie m a n n v a r ia b le s

) = ( Ũ ± cỉ; . V t c C )

.

(2.2 Í9 )

3. N u m e r i c a l s i m u l a t i o n
T h e ai 111 of th e t e s t is to s im u la t e th e ev o lu tio n o f a tid a lly m o d u la te d riveer
plum c u s in g th e fo llo w in g c o n d itio n s o f a b a s in w ith vvater d e p th r a n g in g from 3im
in the sh o r e lin e to 2 0 m in th e offshore boundary. T h e co m j)u ta tio n a l d o m a in , h a is
the form o f a r e c ta n g le b a s in en c lo se d by a Coastal (solicỉ) b o u n d a r y and th r e e opoĩn
sea b o u n d aries. For c o n v e n ie n c e , th e Coastal b o u n d ary w ill be d e n o te d b y

thìe

Southern b ou n d ary, th e la t t e r by th e vvestern, e a s te r n c r o ss-sh o r e b o u n d a r ie s anul
the nortbern a lo n g s h o r e b o u n d a r y . T he b a s in h a s a le n g th of 120 km, a w idth o f 410
kin, in th e Southern b o u n d a r y th e r e is tho river mouth s it u a t e d in th e d is ta n c e o f fí)0
km from th e vvestern b o u n d a r y and d isc h a r g e vvator to b a sin insid o ono h a b ío r
constructed by 2 g ro in s. T h e h o r iz o n ta l r eso lu tio n of grid is 5 0 0 m and 2 0 le v e ỉ s aire
used in th e v e rtica l. T h e a r ea is rillecl in itia lly w ith seavvater h a v in g a unifor*m
sa lin ity o f 30 PSƯ.
ỉ ỉ iittsei I

Km
40

30

20

10

Hi ve ì n u m ỉ h

F i g u r e 3. ỉ:

The computational domain


A threc-dirncntional sirntỉlntion of thc.

37

Ticlal ĩo rcin g is im p o se d in th e form of a f r ic tio n le s s

K elvin w a v e with

ír eq u e n c v of Oj e n t e r in g at th e vvcstern b ou n d ary a n d p r o p a g a t in g a lo n g th e const
[V a n Rijn,

1 98 9 a n d

Rudciick et. al.,

1995]. T h e in c o m in g

R ic m a n n varial>le,

s p e c ia liz e d at t h e vvestern bou n d a rv , th e n ta k e s th e form
(3 .1 )

R. = u +cỊ = 2 c F |Mr = 2cA e ,v/c coso)~t ,

w h e r e th e C oriolis fr e q u e n c y is e v a lu a te d at a l a t i t u d e o f 2 0 , (0 is th e 0 . tiđal
ír eq u e n c y , A = 0 .8 m a n d ư , r, c are th e d e p t h -in te g r a t e d a lo n g s h o r e cu r re n t, th e
barotropic

vvave s p e e d

and

th e s u r ía c e e le v a t io n .T h e

a m p ỉit u d e

o f th o w a v e

d e c r e a s e s e x p o n e n t ia llv vvith d is ta n c e to th e c o a s t vvith a d e c a v s c a le g iv e n hy th e
barotropic R o ssb y r a d iu s c / f -

120 km. T h e a m p lit u d e A e '

o f th e harm onic

fu n ction Flíflr is storecỉ for e a ch o p e n boundary node.
A zero n o r m a l graciient co n d itio n is s e le c te d at th e e a s t e r n a n d northorn
bouncỉaries, i.e.
~ ự /
<\x

-c£) =0

(3.2)

4 -(V -c O
ry

=0

T he la te r c o n d itio n is ju s tifíe d bv th e fact th a t th e vviđth o f t h e b a s in is much

sm aller than th e external Rossby radius c / f .
Sin ce th e v a lu e o f Q is unknovvn at th e r iv er m o u th , t h e o p e n boundary
concỉition a t t h e in le t is n o longer d e íìn e d in t e r m s o f th e in c o m in g R iem an n
va riab le

but by s p e c if y in g th e c r o ss-sh o r e c o m p o n e n t o f t h e clepth

current.

T h is is g iv e n

in tcg r a te d

a s th e s u m o f a r e s id u a l v a lu e , r e p r e s e n t in g th e

river

cỉischarge, and a tid a l c o m p o n e n t
V = c F hnr = % + A tHcos(o) t - (pr)

w

(3.3)

w h e r e Q,i = 1000 m Vs is t h e r iv er d isc h a r g e, vv = 5 0 0 m t h e w id t h o f t h e in le t and Á,
= 0.6 m/s th e a m p litu d e o f th e tid a l c u rren t a t th e m o u th o f th e river. T h e p h a se tp,
is d eterm in cd bv

(pr = c

2

(3.4)

where D = 2 6 .0 k ĨTÌ so th a t D, l c r e p r e s e n ts th e t im e t r a v e lle d bv t h e K elvin w a v e
from the w e r s te r n b o u n d a r y to th e river m outh. O b s e r v a t io n s in tho river plum e
show th at th e a lo n g s h o r e a n d c ro ss-sh o re c o m p a n e n t a r e a n t i- p h a s e vvhich e x p la in s
th e use of th e factor 7t/ 2 [V an Rijn, 1989].


38

N gu y en Minh Ị ỉ Uan
III a d d itio n to thi' p r e v io u s c o n d itio n s for the 2 -D m ode, open b o u n d a ry

c o n d itio n s h a v e to be im p o se đ d u rin g th e final run for th e h o r iz o n ta l v e locity
d e v ia tio n s

(u\ V 9 a n d

the s a lin it y

s.

A t th e o p e n s e a b o u n d a r ie s a zero norm al g r a d ie n t co n d itio n is ta k en for all
q u a n t it ie s . In th e c a s e o f s a lin it y th is procedure is a r e a s o n a b le a p p ro x im a tio n
s in c e th e p lu m e n e v e r in t e r s e c t s th e vvestern and northorn b o u n d a r y w h ile the
cross-bounclary g r a d ie n t is m u ch sm n ller th an th e a lo n g b o u n d a ry g r a d ie n t a t the
e a s t e r n bou ndary.
T h e d e ía u lt c o n d it io n s a re no longer a p p lica b le a t th e river m o u th w h e r e u'
and

s

are s p e c iíìe d in th e form o f a tvvo-layer s tr a tific a tio n

s = 10 P S U ,
v ’ = - 0 .2 [m .s ‘]

v ’ = 0.6 [m.s !j

if

z> -5

i f - H < 2 < - 5,

( 3.5)

vvhere ổ = 5 m is th e s p e c iíìe d d ep th o f th e p lu m e la yer a t th e m ou th . In t h is vvay
fresh w a te r is r e le a s e d th r o u g h th e su rface la y e r vvhereas s a lt ic r s e a w a te r flows
into th e e s t u a r y th ro u g h th e bottom layer. A zero g r a d ie n t co n d itio n is a p p lie d for
s a lin it y in th e bottom la y er.

4. D i s c u s s i o n
A lth o u g h the d e v e lo p e d program is able to e x a m in e th e role of d ifferen t
p h y sic a l íorcin g n ie c h a n is m (b a th y m e tr y , tid es, w ind, w a ve) on th e p lu m e stru ctu re,
tho in te n tio n h e r e is to t e s t s o m e of th e a b o v e -m en tio n e d forcing a n d the role of the
S m a g o r iỉisk y ío r m u la tio n for h o r iz o n ta l diffusion and th e u p w in d sc h e m e for the
advort.ion o f m o m e n tu m .

F ỉg u res 4.1. Surface d istrib u tio n o f current and salin ity after õOh sim u la tion (fmal run)
F ig u r es 4.1 - 4 .5 c le a r ly sh o w how th e p lu m e e v o lv e s d u r in g a tidal cycle. At
tho tim e w h e n th e a lo n g s h o r e cu rren t r ev c r se s s ig n a n d tho o u tflo w roaches its


A three-dimentional simulation of the.
m a x im u m , a nevv blob o f írcsh w ater e n te r s th e b a s in . m o v in g seavvards (ỈMgure
á 1ì
xO.Skm

*°

20

10

4 0

5 0

6 0

7 0

8 0

9 0

1 0 0

1 1 0

x0.5km

F igures 4.2. S u rface distribution o fc u r r en t and sa lin ity after Õ2h sim u la tio n
A s th e e a s t w a r d d irected tidal w av e b e c o m e s s tr o n g e r , th e fresh vvater patch
is d e fle c te d to th e r ig h t (F igu re 4 .3 ).
x0.5km

20

40

50

60

70

80

90

100

110

0 5k

F igure 4 . 3 : S u ría ce distribution o fc u r r e n t and sa lin ity after 54h sim u la tio n
D u r in g th is p h a s e of th e tid e both the b u lg e an d th e C oas ta l p lu m e expancl
sea w a r d s . W h en th e tid a l c u r re n t r e v e r se s s ig n a g a in and t u r n s to th e w e s t, th e
c u r re n t in s id e th e p lu m e is first s o u t h e a s t w a r d s p u s h in g th e b u lg e to w a r d s th e
c o a st (F ỉg u re 4.4).


Nguyen Minh H u a m

40
x0.5km

40

3 0

20

40

50

60

70

80

90

100

110

x 0 Skm

Kigure 4.4: Su rface đ istrib u tio n of current and salin ity after 56h sim u lation
A nd fin ally s o u t h w e s t w a r đ s red u cin g the e x t e n t o f th e b u lg e and th e c o a stía l
p lu m e (F ig u r e 4.5). T h e m a in fe a tu r e here is th at th e b u lg e a n d th e C o a s t a l p lu m ie
o s c illa te vvith th e tid e.

x0.5km

40

30

20

10

4 0

5 0

60

70

80

90

100

110

x 0 .5 k n i

Figure 4 .5 : Surface distribution of current and salinity after 62h simulation
T ho current a n d s a l i n i t y field s along tho tr a n s e c ts sh o w th e p r e se n e e of an
ọ s tu a r in e -t y p e c ir c u la iio n (P ig u r e 4.5). In tho c r o ss-sh o r è tr a n s e c t u p w e llin g tak
place at the coast vvhile cỉownwelling occurs at the eđge of tlìG plume by tòhe
c o n v e r g e n c e of tho s u r f a c e o u tflo w current. A s im ila r p h e n o m e n o n is s e e n in Ithe
Coastal jet whc»re d o w n w e llin g motions are c rea ted by th(* con vorgonce o f th e coasttal
jet. III th e c a s e o f a n o n -tiđ a l p lu m e the p lu m e la y e r is shallovver and th e ừ on ita l
gradients

are

stronger

comparccỉ

to t h e

tidal

(Nìse w h e r o

turbulent

diffusrion

in c r o a se s tho d ep th o f th e su r fa c e la y e r and red u ces th e v ertica l s tr a tiĩic a tio n .


A t h r e c - d i m e n t i o n a l s ỉ m u l a t ỉ o n o f th e .

Lớpơ

20

15

10

5

5

to

15

20

* 0 . 5k m

F igu re 4.6: Cross-sectional distribution o f cu rren t and sa lin ity

after 66h simulation in the a-coordinate
5. C o n c l u d i n g r e m a r k s

This paper presented a three dimensional model which consists of a
circu la tio n m od el, a tr a n sp o r t m odel, and a o n e e q u a tio n k -e p silo n tu r b u le n c e
m odel. T h e u se o f th r e e -d im e n s io n a l m od els is u n a v o id a b le in all c a s e s w h e r e th e
in flu e n c e o f d e n s it y d istr ib u tio n cannot be n o g le e d or a n d in w in d d riv en flow s,
w hich h a ve ty p ic a lly th r e e -d im e n sio n a l ch a ra cter. T h e d e v e lo p e d m odel a ls o m u st
to he vvell c a lib r a te d an d v erificated vvith a n o t h e r n u m e r ic a l e x p e r im e n t a l and

prototype data.

Questions about this article and source code in PORTRAN of prograni can be
a d d re sse d to N g u y e n M inh H uan , Paculty o f H y d r o -M e te o ro lo g y a n d O c ea n o g ra p h y ,

of Natural Sciences.

(í

A c k n o w le d g m e n ts. This work has been supported by the Hanoi University

((2001-2003) project coded 7317001.

1

REFERENCES
1.

B lack ad ar A.K., T h e vertical d istr ib u tio n o f w in d a n d tu r b u le n t e x c h a n g e in a
neutral a tm o s p h e r e , Journcil o f G e o p h y sic a l R e s e a r c h ? 6 7 ( 1 9 6 2 ) , 3 0 9 5 - 3 1 0 2 .

2.

B lu m b er g A .F .a n d M ellor G .L., A d escr ip tio n o f a th r e e c ỉim e n sio n a l Coastal

ocean circulation model. In: N.S.Heaps (Editor), Three-dimensional Coastal
Octean M odels, C o a s ta l a n d K stu n r in e S c i e n c e s , Vo].4, A m e r ic a n G e o p h y s ic a l

Union,Washington D.C., 1987pp.l -16.


42

Nguyen Minh H u a n

3.

D a v ie s A .M ., A b o tto m b o u n d a r y la y e r -r e s o lv in g t h r e e - d im e n s io n a l tid a l m od el:
A s e n s it iv it y s tu d y o f eddy v is c o s ity
O c e a n o g r a p h y , 2 3 ( 1 9 9 3 ) 1 437 - 1 4 5 3 .

ĩo r m u la tio n ,

Journal

of

P h ysic a l

4 . Phillips N.A., A coordinate system having some special advantages for
numerical íbrecasting, J o u r n a l o f M e tc o r o lo g y , 14(1957), 184-185.
5.

H e d str o m G .w . , N o n r e fle c tin g bou n d ary c o n d itio n s for n o n lin e a r h yp erb olic
s y s t e m s , ổ o u r n a l o f C o m p u t a t i o n a l P h y s i c s , 3 0 (1 9 7 9 ) 2 2 2 - 2 3 7 .

6.

Leo c . v a n Rjin, P r in c ip le s o f flu id flow and s u r fa c e in rivers, e s tu a r ie s , s e a s
and o c ea n s, A q u a P u b l i c a t i o n s , 1989, pp. 2 1 6 -2 2 6 .

7.

R oed L .p .a n d C ooper C .K., A s t u d y o f v a r io u s open b o u n d a r y c o n d itio n s f o r
u)ind-forccd

b a r o tr o p ic

B .M .J a m a r t (E d ito rs),

n u m e r ic a l

ocean

T h r e e -d im e n s io n a l

m o d e ls,
m o d els

In:

J .C .J .N ih o u l

o f m a r in e

and

and!

estu a rin eì

d y n a m ic s, E lse v ie r , A m s te r d a m , 1987, p p .3 0 5 - 3 3 5 .

8 . Randall J. LeVeque, N o n l i n e a r C o n s e r v a tio n L a w s a n d P i n ite V o lu m e M e t h o d l
fo r A s tr o p h y s ic a l F lu id F lo w , S p rin g er -V e r la g , W a s h in g t o n D . c , 1998.
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R uddick K.G., M o d e l l i n g o f C oastal p r o c e s s e s in ỷ ĩu en ce d by the fre sh u ja te rr
d is c h a r g e o f the R h i n e , U n iv . de L i'e g e, B e lg iu m , 1995, 2 4 7 pp.

TA P CHỈ KH O A HỌC O H Q G H N . KHTN & CN, T.X1X. Nọ1, 2003_________

K Ế T Q U Ả M Ồ P H Ỏ N G 3 C H I Ê U C H E Đ Ộ D Ò N G CHAY
V Ù N G CỬA S Ô N G C H Ị U TÁC Đ Ộ N G CỦA T H Ủ Y T R l Ể ư
N g u y ể n M in h H u â n
K h o a K h i tư ợ n g T h ủ y v ã n và H ả i d ư ơ n g h ọ c,
Đ ạ i h ọ c K h o a học T ự n h iê n , Đ H Q G H à N ộ i
Mô hìn h th ủ y đ ộ n g lực 3 c h iề u dược, sử d ụ n g tr o n g t ín h toán mô ph ỏng m ự íc
m íốc, v ậ n tốc d òn g c h ả y và p h â n bô độ m uỗi ở v ù n g nước cửa sô n g ph ân tầ n g c h ịiu
tá c đ ộn g của th ủ y tr iề u . M ô h ìn h bao gồm các phương tr ìn h th ủ y động lực, trưvểẩn
tả i v à được k h ép kín b ằ n g cá c sơ đồ rổì. Hệ ph ư ơng tr ìn h th ủ y động lực của mô h ìm h
là

hệ

phương tr ìn h

N a v ie r - S t o k e s sử

dụng

giả t h u y ế t th ủ y tìn h

và xấp

xỉ

B o u s sin e s q . Sự b iế n đ ộn g c ủ a n h iệ t độ và độ m uối s ẽ ảnh hưởng lên m ộ t độ cuảa

nước và mật độ biến đổi sẽ ảnh hưởng ngược lại lên trường dòng chảy. Hệ phươmg
tr ìn h dộng lượng và liê n tụ c dược giả i b ằ n g p h ư ơ ng p h áp p h â n tách th à n h phần, íá p
d u n g mò hình rôì k-ep silon m ột phương trình. Trong m ô hình tải, các phương tr ìm h
kh u y ẽch tán đôi lưu ba c h iể u được sử dụng. Mô h ìn h được áp dụ ng cho khu vực biêển
v e n bò có của sô n g vối một b iê n cứng và 3 biên lỏng, mực nước biến động ở biên lỏsng
phía tây do tác động củ a só n g triều 0 ,, lưu lượng nước sô n g chảy vào vùng tính có rmô
phỏng

khu vực

cả n g với giá tr ị là 1000m 3.s

Kết quả tín h toá n đã mô phỏng đượccch ế

độ đặc trưng của dòng c hảy 3 ch iểu v ù n g cửa sông chịu tác động của thủy triểu.