DSpace at VNU: Extremum sea levels in VietNam coast

YVhen com pute vvith astronom ical param eters of diurnal tide choose f for.

EXTKEMƯM SEA LEVELS IN V1ETNAM COAST

Pham Van ỉỉu a n

D epartm ent o f Hydro Meteorology a n d Oceanology

CoIIegc o f Science, V N U

A b s t r a c t : A revievv ()f th o in v e s tig a tio n s on t.hcì sea le vol changcỉS in S o u th

r h in a soa is presenUỉíl and th e methods o f approxim ate e a U u la tio n o f

th e o retica l tid a l oxtrem cs w crc oxplaineil in (letail

Tho soa level changes ncar V ietnam coast (ỉuo to global w a rm in g and o lh e r

<ìffccts is evaluated to be from 1 to 3 mm per ycar

Kor sevon sta tio n s vvith fu ll sot o f harm onic constants (le tc rm in o tl thc

th e o retica l extrem o heights o f tiíỉa l level by p re d ictin g h o u rly t i do h e ig h ts in a

2 0 -year ptỉriod lf,or o th e r ninctecn stations w ith 1 1 h arm onic constants o f m ain

tid a l co n s titu e n ls the theoretical astronom ical cxt.rome hìvols wortĩ calculatod

by the ito ra tio n mcìthod Thtĩ coniparison showeđ a good agrcỉomen! l)c(wcM n l.wo methods

T he e n ip iric a l oxtrem e analysis was carried out for 25 tid e gaugcs along

V ie tn a n ì coast to e v a lu a tc t h r design values <)f sca lcĩvel <)f ( ỉiffo ro n t ra re fr(?qucncies

T h o a n a lysis a lso show ed th a t th o tiđ a l e xtre in e s and đ csig n le vo l valu o s ()f 20-

y ea r r c tu rn poriod arc of the samc rango 'rhc levcl valuos ()f longer rcĩturn

p e rio d arcì affect.o(J m a in ly hy Hoods and surges

1 In tr o d u c tio n

T h e e x tre m e sea le v e ls a re s tu d y s u b je c t o f m a n v p u rp o s e s T h e m a x im a l a n d

m in im a l v a lu e s o f sea le v e ls a n d t h e ir o c c u rre n c e p r o b a b ilitie s a re ta k e n in to

a c c o n n t in d e s ig n in g h v d ro te c h n ic a l s tru c tu re s

T h e th e o ry o f e x tre m e a n a ly s is o f s ta tis tic a l m a th e m a tic s is a p p lie d to th e

h v d ro m e te o ro lo g y w it h d if fe r e n t d is tr ib u tio n s o f th e o b s e rv e d s e ric s o f c lim a tic a n d

h y d ro lo g ic a l p a ra m e te rs [5 t7 ] T h e m a in c o n c c p ts o f th e s e m e th o d s w ill be

p re s e n te d in s e c tio n 2 1

In th e case t h a t o b s e rv e d se rie s o f sea le v e l a rc n o t lo n g e n o u g h to a p p ly th e

p ro c e d u re s o f e x tre m e a n a ly s is th e o ry , t h a t u s ụ a lly h a p p e n in th e d e s ig n

in v e s tig a tio n s in t h c Coastal zone a n d e s tu a rie s , one m a y use th e o r e tic a l e x tre m e

v a lu e s o f p u re ly t id a l le v e ls

In m a n y p r a c tic a l p ro b le m s th e m in im a l th e o r e tic a l le v e l is a s s u m e d to be th e zero d e p th in t id a l seas T h is le v e l can be c a lc u la te d by s u b tr a c tin g m a x im a l lo w

h e ig h t o f tid e due to a s tro n o m ic a l c o n d itio n s fro m m e a n sea le v e l III so m e c o u n trie s

th is v a lu e is d e te rm in e d b v a n a ly z in g a p re d ic te d s e rie s o f tic ỉa l h e ig h ts 1 9 -y o a r

22

VMU JOURNAL QF SCIENCE, Nat., Sci., & Tech., T XIX NọỊ, 2003 _

E x tv e m u m s c a lc v e ls i n 23

long, o n e ch o ose th e lovvest h e ig h t n m o n g a ll lo w vvatồrs in th e s e rie s III R n s s ia th o

m ỉin m a l th e o r e tic a l le v e l is c le te rm in e d bv k n o w n m e th o d o f V la đ im ir.s k y

V la d im ir s k y m e th ơ d g iv e s an a n a lv tic a l s o lu tio n o f th e p ro b le m w ith

h a rm o n ic c o n s ta n ts o f 8 m a in tid a l c o n s titu e n ts T h o r r s t tid a l c o n s titu c n ts nre

ta k c n in t o a c c o u n t a p p r o x im a te ly lỉe c e n tly th e c a lc u la tio n s can be p e ríb rm e d

ra p id ly in c o m p u t r r s , e v a lu a tin g e x tro m e h e ig h ts o f tid e can be c a r r ie d o u t by m ore íie ta ile d sch e m e s a n d th e a c c u ra c y is im p ro v e d by w ith c lr a w in g a n o n - r e s tr ic te d

n u m b e r o f t id e c o n s titu e n t s ìn to c o n s id e ra tio n [G] S e c tio n 2.2 vvill e x p la in in d e ta ils

a schom e to im p le m e n t th is m e th o d in p ra c tic e a n d in s e c tio n 3 vvill p re s e n te d th e

n p p lic a tio n r e s u lts to o b ta in m a x im a l c h a r a c te r is tic s o f sea le v e l in so m e re g io n o f

V ie tn n m c o a s t

T h e o b s e rv a tio n o f sea le v c l a lo n g V ie tn a m co a s t is m a in ly c a r r ie d o u t by a System o f t id a l g a u g e s o f th e V ie tn a m H v d ro m e te o ro lo g ic a l S e rv ic e G e n e r a lly

s p e a k in g u p to novv t h e n u m b e r o f t iđ a l gauges th a t b e lo n g s to V ie tn a m w a te rs is not m a n v a n d th e n u m b e r o f o b s c rv a tio n y e a rs is n o t lo n g e n o u g h So th e r e is no niuch d e a l w it h th o b e h a v io r o f sea le v e l in g e n e ra l a n d th e e m p ir ic a l e a lc u la tic m s ơf leve l e x tre m e s in s p e c ia l

In so m e r a r e w o r k s th c r e re p o rte d th e r e s u lts o f a n a ly z in g c h a n g e a b le n e s s o f sea le v e l a n d th e e s tim a tin g th e tr e n đ o f.s e a le v e l ris e in th e base o f a n a ly s is o f observed s e rie s o f sea le v e l so m e y e a rs lon g T h e s p e c tru m a n a ly s is [2] s h o w e d t h a t besiđes th e s e m ia n n u a l a n d a n n u a l p e rio d s , in th e a lm o s t o f t i d a l gauges

o á c illa tio n s o f p e rio d o f G to lO y e a r s an d lo n g e r e x is t (íìg u re 1 )

8 < p /

0.0»

(k*.'ỉ

o.c»«

0 0 5 6

txtrb

0.0*0

n t X ‘ 2

ao:-*

mo 4

ii.ođ

Hon 0*1.

\ p‘«ni>d irtiìntt)

0.(1»

0.030

•>.0 'i?

0.024

0.071

0.019 0.015

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F igure 1. S p e c tru m o f sea level a t tid a l gauges H on D a u a n d Q u y N h o n

T a b le 1 lis t s th e r e s u lts o f e s tim a tio n o f th e sea le v e l ris e by tr e n d a n a ly s is vv.th m o n th ly m ean le v e l [2 - 4 ] It is fo llo w e d t h a t th e s u m m a ry e ffe c t b y th e g lo b a l vv.irm in g a n d o s c illa tio n s o f sea bed in re g io n o f V ie tn a m c o a s t causes a r a t e o f lc v e l riíe a b o u t 14-3 m tn p e r y e a r

24 P h a m V a n H u a n

T a b le 1 R a te o f sea le v e l ris e a t som e p o in ts a lo n g V ie tn a m c o a s t

y e a rs

T re n d (m m / y e a r)

A f u ll c u m b e rs o m e c a lc u la tio n o f le v e l e x tre m e s vvas p e ríb rm e d in [1 ] I n th is

r e p o r t í ir s t ly lis te d s e rie s o f m o n th ly a ve ra g e , m a x im a l a n d m in im a l le v e ls fo r a ll gauges a lo n g V ie tn a m co a st u p to m id d le o f n in e tie th T h e e x tre m e a n a ly s is w as

c a r r ie d o u t by an a s y m p to tic G u m b e l fu n c tio n o f p r o b a b ility d is t r ib u t io n o f th e

e x tre m e s

2 T h e m e th o d o f s t u d y

2 1 E x t r e m e s a n a l y s i s i v i t h e m p i r i c a l d a t a

A s s u m e VỊ t h e v a lu e s o f in c ic ỉe n ta l v a r ia b le V a t t im e / a n d

A'<m) = m a x {F 1 ) F2 ) )Fm} ; x (m)= m in { F , ,F 2 , ,Vm

O ne is o fte n in t e r e s t in e s tim a tio n th e p r o b a b ility w it h w h ic h m a x im a l o r

m in im aỉ value exceeds a th re s h o ld , Iì{ Xim > x} or P{X(m) < x } If t h e o b serv atio n s

on th e h y d ro m e te o ro lo g ic a l p a ra m e te rs a re in d e p e n d e n t a n d d is t r ib u t e d if fe r e n t ly

due to d istrib u tio n fu n ctio n F{x) - P{Vị < x ) , t h e p recise d is tr ib u tio n of m ax im u m

a n d m in im u m ca n be cxp re sse d :

/ ’ { A '('n) < x } = | / - - ( v ) f a n d < x ) = 1 - |1 - /<-(x)|m (1)

The extrem es a n a ly sis theory says th at w ith th e enough length of sam ple m ,

t h e p r o b a b i l i t y d i s t r i b u t i o n o f t h e n o r m a l i z e d m a x i m u m Y (m)= ( X ' m

bm > 0 c a n b e a p p r o x i m a t e d by o n e o f t h e t h r e e íơ ll ov vi ng f o r m s o f a s y m p t o t i c

function

0*3(v) = c x p |- ( - v ) 1 k K v < 0 k > 0 (Wcibull function)

E x t rem u m SCO ie v e ỉs ỉ n 25

a n d s i m i l a r for t h e m i n i m a l v a l u e

//,(>•) = l - c x p ( - i ' v)

/ / : ( v ) = I - C \ p | - ( - v ) 1 A) V < 0 Ả < 0 (3)

/ /,( > ’) = i - c x p ( - v 1;*), y > 0 * > 0

T hese d if fe r e n t form s o f a s y m p to tic fu n c tio n s a re d e p e n d e n t to th e sh a pe o f

th e t r a i l o f p r o b a b ilìty d is t r ib u t io n F(x) (th e r ig h t sicỉe fo r th e m a x im a a n d th e le ft

s id e fo r th e m in in ia ) I n p ra c tic c th e s a m p le c o n c ỉitio n s (th e h o m o g e n e ity , th e incỉe p en đ o n cc a n d th e d im e n s io n ) in ílu e n c e on th e p re c is io n o f th e a p p r o x im a tio n by

th e above a s y m p to tic íu n c tio n s

A s y m p to tic e x tr e m e c lis tr ib u tio n s in c lu d e th re e p a ra m e te rs : k - s h a p e

p a ra m e te r u m - lo ca l p a r a m e te r a n d hn - scale p a ra m e te r

O fte n , in s te a d o f e s tim a t in g th e d is t r ib u t io n o f m a x im a (o r m i n i m a) t one

e xecutes a d iv e rs e p ro b le m : d e te rm in e a d e s ig n v a lu e , i e a v a lu e x fnii such as

probability o f th e d esig n v a lu e Xp to r e t u r n period T = 1/(1 - p ) , w h e re 7 - th e tinie

to b e e x p e c t e d t h a t t h r e s h o l d X r is e x c e e d e d f o r t h e f i r s t t i m e , o r t h e a v e r a g e t i m e

betvveen tvvo above th r e s h o ld e v e n ts

U s in g th e a s y m p tọ tic e x tre m c d is t r ib u t io n th e d c s ig n v a lu e s can be e a s ily

e xp resse đ K o re x a m p le , v v itlì G u m b e l d is t r ib u t io n , one has:

e x tr e m e variable X m a y b e c a l c u l a t e d k n o v v in g p a r a m e te r s u a n d h :

w here y is also called f,norm alized design value".

A q u e s tio n o f p r in c ip le in th e a p p lic a tio n o f e x tre m e s a n a ly s is th e o ry is th e

p re c is io n o f th c a p p r o x im a tio n (2) 01* (3), i e th e q u e s tio n on th e ra te o f

convergence of p recise d i s t r ib u t i o n of e x tre m e s i ,r’ to th e a sy m p to tic one, in

p rac tic al aspect, th e p re c isio n of design v alu e x p e s tim a te d by asym ptotic

d i s t r i b u t i o n i n c o m p a r i s o n vvith ií's r e a l v a l u e ( b u t o f t e n u n k n o w n ) x ('"].

26 P h a t n V a tì H u n u

T h e m e th o cls o f e s t im a tio n o f e x tre m e d is t r ib u t io n a im a t s e ttle m e n t th e

q u e s tio n on th o i n i t i a l s e rie s , th e r e la t iv e ly s h o r t le n g th o f i n i t i a l s e rie s T ib o r

K a ra g o a n d R ic h a rd w K a ts [5 ] e x p la in d if fe r e n t m e th o d s to e s tin ia te t h e e x tre m e

p a ra m e te rs a n d d e te r m in e d e s ig n v a lu e s a n d t h e ir e s tim a te a c c u ra c y S e c tio n 3.3

p re s e n ts th e r e s u lts o b ta in e đ b y a p p ly in g th e s e m e th o d s to s e rie s o f a n n u a lly

m a x im a l a n d m in im a l le v e ls o f som e t id a l g a u g cs a lo n g V ie tn a m coast

2.2 M e t h o d o f c o ì ì i p u t i n f í e x t r e m e v a lu e s o f t i d c

T h e t id a l h e ig h t a b o ve th e m e a n le v e l m a y be e x p re s s e d by th e fo llo w in g

íb r m u la

I

vvhere f t - th e re d u c e c o e ffic ie n ts d e p e n d e d o n lo n g itu d e o f th e r is in g k n o t o f lu n a r

o r b it ; / / t - th e a v e ra g e a m p litu d e s a n d (pt - th e p h a s e o f t id a l c o n s titu e n ts

D e p e n d in g on t h e t i d a l íe a tu r e , th e h e ig h t o f tid e m a y a c h ie v e th e e x tre m e s

w h c n lo n g itu d e o f th e r is in g k n o t o f lu n a r o r b it N = 0 (fo r d iu r n a l tid e ) o r N = 1800 (fo r s e m id iu r n a l tid e ) I n th e s e c o n d itio n s (7V = 0 ,1 8 0 °) th e phases o f t id a l

c o n s titu e n ts a re e x p re s s e d th r o u g h a s tro n o m ic a l p a ra m e te rs i n ta b le 2

T a b le 2 E x p re s s io n s o f p h a s e s a n d re d u c e c o e ffic ie n ts o f t id a l c o n s titu e n ts [ 6 ]

Tidal

;V = ()• N = 180'

2r + 2h - 3 s + p - g V; 0,963 1,037

Ỡ t + h - 3.V + p - 90° - g Qí 1,183 0,806

Sa

SSa

E x tv c t ỉiu tt i sca l c v c ls iti 2 7

III ta b le 2 Ị - a v e ra g e zone tim e fro m m ic ln ig h t, lì a v e ra g e lo n g itu d e o f th e

S u n ; V a v e ra g e lo n g itu d e o f th e M o o n ; p - a v e ra g e lo n g itu d e o f lu n a r o r b it

p e rig ro ; fỉ s p e c ia l i n i t i a l p h a s c r c la te d to th o G re e n vvich lo n g itu d e

T h o e x tre m e h e ig h ts o f tic le m a y be c o m p u te d fro m (7) i f th e v a lu e s o f

a s tro n o m ic a l p a r a m e te r s í lì. V a n d />, vvhich fo rm a c o m b in a tio n c o r re s p o n d in g to

an e x tI ^ m e c o n d itio n , a re k n o w n I n v e s tig a tin g on e x tre m e s th e íu n c tio n z ( í h s p )

fro m (7 ), vve o b ta in a s y s ti m o f fo u r e q u a tio n s w it h fo u r u n k n o w n s í h s a n d /?

w hose v a lu e s d e te r m in e th e e x tr e m e c o n d itio n o f th e t id a l h c ig h t:

I f th e a p p r o x im a te v a lu e s o f a s tro n o m ic a l p a r a m e te r s c o rre s p o n d in g to

e x tr c m e c o n d itio n ( t \ h \ s é % p f) a r c knovvn, w e m a y Ie a d e q u a tio n s ( 8 ) to a lin e a r

fo r m by T a v lo r e x p a n s io n W h c n a p p r o x im a te v a lu o s o f th e u n k n o v v n are

s u ffic ie n t ly close to th e €?xact v a h ie s ụ /? ,.V />,,) th e c x p a n s io n ca n be r e s tr ic te d

in f ir s t o rd e r ite m s

VVith (ie s ig n a tio n s o f c o r r c c tio n s to th e a p p r o x im a te v a lu e s o f a s tro n o m ic a l

p a r a m e te rs as fo llo w in g

th o r e s u lt o f th e e x p a n s io n is a s y s te m o f fo u r lin e a r e q u a tio n s vvith d ia g o n a lỉv

s y m m c tr ic c o e ffic ie n t m a t r ix :

I M 2 sin <pu ^ 4* 2.v: sin (pSs + 2N 2 sin <PX' + 2 K2 sin (pKy +

Kị SIn<pK 4‘ (ỉ ị SIIU/9,, + ì \ s i i í ặ ? , , s i n +

4À /4 sin (pSỊ + 4 M S Ằ sin <PSỊS + 6 M 6 sin <PSỊ = 0

2 M ■ > sin <psr + 2;V, sin <p < % + 2Ả \ sin (pK% + KI sm (pK 4*

(), sin<pa + /* sin + 0 | sinv?£> + 4iV/ 4 sin$?A/ + >

2 A /: sin (pSỊy -f 3 iV , S1I1 (ps% + 2 0 Ị sin (p(ỉ + 3 (/ị sin ^ + 4A/ 4 sin <PA/ 4- 2A/.V4 sin (pSỊSị + 6A /Ố sin - 0

A,r: sin <Pv: + 0 , sin <PC,( = 0,

(8)

vvhcre

P h a m V a n ỉ ỉ u a n

/4

a ) - 4 M 2 c o s<p'm + 4S 2 cos (p's + 4 N 2 cos (p's + 4K 2 co s<p'Kĩ +

+ K | c o s ^ í + 0 ị COSỰ)'0 +P\ COS{?Ị» + ( í| COSộ?£ + + 1 6 A /4 cos +16M S 4 cos <p'KiS + 3 6 A /6 cos •

/?ị = 4jV/2 cos<Pv/ + 4/V, cosợ?Y + 4 /v 2 c o s(p'K + ATj c o s(p’K +

+ ƠJ co s - 1\ co s ợ?), + Ọj cos <Pọ + 16;V/4 co s +

+ 8 A / S 4 c o s ọ 9 m s + 3 6 M ỗ cos<p'ví ;

Cị = - 4 A / , COS0>Ú - 6 ^ 2 cos (p'Ni - 2 0 ì cos <Pq - 3 Ộ | cosợ?£

- 16A-/, CCS «?;,4 - U í S A cos íỡ;,Í4 - 3 6 M 6 cos <?;,6 ,

í/, = 2 N 2 cos + 01 cos ;

/j = 2 M 2 s i n + 2 5 2 s i n (p's + 2 / V 2 s i n + 2AT2 s i n (p'K + + K x sin (p'K + 0 ] sin + l\ sin + (>! sin + + 4A* 4 sin ^ ; Í4 + 4M S 4 sin ^ + 6 M6 sin <p;,6 ;

b 2 = 4 A / : C0S(p's1 + 4 j V 2 cosợ?v + 4 / ^ c o s ộ \ + Ả 'j c o s p í +

+ ƠJ cosọ>'0 +!\ cos (p'j +Qị cos<Pq + ỉ 6A/ 4 cos<Pv/ +

+ 4MV4 cos + 36M ố cos + Sa cos <pýa + 4&SV7 cos <p'SSa;

c 2 = - 4 A / : c o s ọ h - 6 N 2 c o s ^ v - 2 ơ | cos <P o - 3 ( P ị C0SỘ?£

- 16A /4 cos - 4MV4 cos <P v ,v< - 3 6 A /6 cos <p v,A;

d 2 = 2 iV: cos{?v + (?! cosỌọ ;

/ 2 = 2 A / 2 sin + 2/V-, sin + 2/w 2 sin + Áf| sin +

+ ()] sin sin ự Ị, + Qx sin (pQ 4- 4A/ 4 sin + + 2M V4 sin + 6A/ 6 sin (pu + SV/ sin -f- 2 XVc7 sin ;

c 3 = 4 A / 2 c o s (p fM + 9 N : c a s ọ ' s + 4 Ơ , c o s t p ý + 9 Q ị c o s ( p ọ

+ 16A f4 cos + 4M V 4 cos^ /5 + 3 6 A /Ố cos ạ>Ệ Xi ,

J 3 = - 3 W 2 c o s (p \^ - 3 ( ^ J cos<pộ ;

/, = - 2 A / : sin^>v/j ” 3^2 s i n f v 2 - 20j s i n ^ -3Ợj s i n ^

- 4 A í4 sin - 2A/.V4 sin - 6jW() sin <p'M ;

J 4 = /v 2 cos V + ộ j cos ;

/ 4 = N z costpl/ + Q \ cos<Pọt ;

p h a s e o f th e t i đ a l c o n s t it u e n ts c o m p u te d t h r o u g h a p p r o x im a te v a lu e s o f th e

a s t r o n o m ic a l p a r a m e te r s í \ h \ s ' a n d p '

E x t r e m u m s c n lc v c ls i n . 2 9

I n o r d e r to c o m p u te th o v a lu e s o f a s tro n o m ic a l c o rre s p o n d in g to e x tre m c

concỉition (tt,,h0 s 0, pv) w ith a given accuracy t h e ite ra tio n m eth o d m ay be u sed If

m a g n itu d e a g iv e n v a lu e \ổ\ th e s o lu tio n w ill re p e a te d a n d th e n in o rd e r to c o m p u te

th e c o efficien ts of e q u a t io n s (9) vve will u se th e p h a se s (p' co m p u ted t h r o u g h th c

va lu e s c o r re c te d o f a s tr o n o m ic a l p a ra m e te rs :

/ ' = /' + A/' v" = v' + Av', h n = h ' + A h '%p* = p ' + Ap'

step k o f system (9) becom e less th a n ổ :

T a b le 3 V a lu e s o f a s tr o n o m ic a l p a ra m e te rs a p p ro x im a te ly c o rre s p o n đ in g e x tre m e

c o n d itio n [ 6 ]

S e m id iu rn a l tid e

A s tr o n o m ic a l

p a ra m e te rs

M in im a l leve l

c o n d itio n

M a x im a l le ve l

c o n d itio n

r

I f i n i t i a l a p p r o x im a te v a lu e s ( í \ h \ s \ / / ) close to re a l v a lu e s (ía,h f),sa , pa ) th e ite r a tio n r a p id ly c o n v c rg e s T h e se v a lu e s o f a s tro n o m ic a l p a ra m e te rs c o r re s p o n d in g

to e x tre m e c o n d itio n m a y be c a lc u la te th ro u g h fo u r d iu r n a l o r s e m id iu r n a l t id a l

c o n s titu e n ts d e p e n d in g o n tid e fe a tu re T h e e x tre m e c o n d itio n fo r fo u r s e m iđ iu r n a l tid a l c o n s t itu e n t s a n d d iu r n a l c o n s titu e n ts is d e te rm in e d by th e íb llo v v in g

e x p re s s io n s :

- F o r s e m id iu r n a l tid e : <PKU = (ps = (pN =(pKĩ = <p

- F o r d a u rn a l tid e : ỌK = (pQ =<pr = Ọọ - (p,

w h e re (p = 180° + 2mi - fo r th e lo w c s t le v e l a n d (Ọ = 360° + 27U1 - fo r th e h ig h e s t le v e l

F ro m th e s c e x p re s s io n s fo llo w th e ío rm u la e fo r c o m p u tin g th e a p p r o x im a te

va lu e s o f a s tr o n o m ic a l p a ra m e te rs \ i \ h \ s \ p ) c o rre s p o n d in g th e e x tre m e

c c n d itio n s ( ta b le s 3 - 6 )

30 P h a m V a n H u a n

In o rd e r to c o m p u te a p p ro x im a te v a lu e o f a ve ra g e zone t im e / ' th e r e a re tw o

e x p re s s io n s fo r th e lovvest c o n d itio n a n d h ig h e s t c o n đ itio n s e p a r a te ly , s in c e fo r

s e m iđ iu r n a l tid e o n e day has tw o ' h ig h w a te rs a n d tw o lo w vvaters T h e ch o ic e o f

ío rm u la used in c o n c re te case m u s t be re fe re n c e to th e s ig n o f s u p p le m e n ta r y

c o e ffic ie n ts B a n d c (ta b le 4) C o e ffic ie n ts tì a n d c a re c o m p u te d b y following

fo rm u la e :

t ì - 0 J cosơị +1\ cos a 2 +Q\ COSƠ3 + KJ cosa4|

c = ƠJ sin ơ| + pI sin a 2 + Q\ sin a 3 + K ] sin ữ A J

= g Mỉ c, - g o , " 9 0 °; a ì= s.w , - ° ’58k, - g Ql - 9 0 ^ |

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« 2 = « 5, - 0 f5 g JCi - g p> - 9 0 ° , a 4 = 0 , 5 * ^ - g ^ + 9 0 c

T a b le 4 C o n d itio n s o f th e lo w e s t a n d h ig h e s t le v e l [6 ]

C o n d itio n s o f th e lovvest le ve l C o n d itio n s o f th e h ig h e s t le ve l

w h e n c > 0

t 2 w h e n c < 0

/j w h e n B < 0 /2 w h e n B > 0

T a b le 5 V a lu e s o f a s tro n o m ic a l p a ra m e te rs a p p r o x im a te ly c o rre s p o n d in g to

e x trc m e c o n d itio n [6 ]

D iu r n a l tid e

A s tro n o m ic a l p a ra m e te rs C o n d itio n s o f th e lo w e st

leve l

C o n d itio n s o f th e

h ig h e s t le v e l

T h e choice o f re d u c e c o e ffỉc ie n ts to c o m p u te v a lu e s / / / is d e p e n d e d o n th e tid e

íe a tu re :

1) F or s e m id iu m a l tide, i f { h £ + H G ) ỵ H M < 0,5 th en f is chosen fo r iV = 180 ;

2) F o r d iu r n a l tic ỉc , i f [h K + / / ỡ ) / H K1 > 1,5 th e n f is chosen fo r N = 0*;

o f a s tro n o m ic a l p a ra m e te rs fo r b o th s e m id iu r n a l tid e (ta b le 3) a n d d iu r n a l tid e

(table 5) YVhen com pute vvith astronom ical param eters of diurnal tide choose f for

E x t r c r n u r n s e o l c v c l s ỉ tì 31

;V = 0 , vvhen c o m p u te w it h a s tro n o m ic a l p a ra m e te rs o f s e m id iu r n a l t id e choose /

fo r N 0 a n d N = 180 T h e h ig h o s t le v e l a n d th e lo w e s t le v e l o b ta in e d by th re e

v a r ia n ts vv ill bo a c c e p te d to be th e e xtre m e s

\V e also c o m p u tc th e a p p ro x im a te v a lu e s o f a s tro n o m ic a l p a ra m e te rs

c o rre s p o n d in g th e e x tr e m e c o n đ itio n s by V la c ỉim ir s k y m e th o d ; t h is m e th o d a p p lie d

fo r 8 t id e c o n s titu e n ts In V la d im ir s k y m e th o d th e e x tre m e h e ig h ts o f tid e is

d e te rm in e d by c o n s e q u e n tlv c h o o sin g v a lu e s (pK in th e in t e r v a l fro m 0° to 3G0°:

T h e choice o f re d u c e c o e ffic ie n ts to c o m p u te v a lu e s / / / is also macỉe as th e above re c o m m e n d a tio n s , i e w it h th e s e m id iu r n a l t id c / is chosen fo r yV ^lX O ,

w it h d iu r n a l tid e f is chosen fo r N = 0 W it h m ix e d t id e th e c o m p u ta tio n is

p e río rm e d w it h f f o r N = 180 a n d /V = 0 a n d th a n th e lovvest a n d h ig h e s t va ỉu e s

in tw o v a r ia n ts w il l b e th e e x tre m e lc v e ls

I f c o m p u te e x tr e m e le v e ls vvith 8 tid e c o n s titu e n ts th e n th e la s t re s u lts are

o b ta in e d d ir e c t ly fr o m th e e x p re s s io n s ( 1 2 ) In th e case o th e r c o n s titu e n ts a re ta k e n

in t o th e c o m p u ta tio n s , we m u s t re íe re n c e to v a lu e s (<pK )mm a n d ((pK )mxx fro m

a n a ly z in g ( 1 2 ) to c o m p u te th e a s tro n o m ic a l p a ra m e te rs c o rre s p o n d in g e x tre m e

c o n d itio n s í j ụ s \ p a n d u s c th e m as th e a p p ro x im a tio n s to c o m p u te th e c o e ffíc ie n ts

o f e q u a tio n s (9)

T h e c o n d itio n s o f th e lovvest le v e l:

H = A'j cosỌị + K c o s(2 Ọ K + a A ) +|/ỈJ 4- 7Ỉ, 4- / Ỉ 3Ị

/ = Ả' jC 0s ^ A + K : cos(2<pk + ơ 4 ) - | / { 1 + / ỉ 2 + / ỉ 5|

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w h e re

/í, =y[M Ỉ +()■' + 2 M , O xc o s r J ;