Bull. l{al. Insl. Ocn. & Frsh., A.R,E, 1993^ il9): 1-19 I{UI{BRICAL II{VESTICATIOI( OF I{2.TIDE8 II{ THE HEDITERRII{EM AS A CLOSED EASII{. 8y G.F. SOLI}IAII*; I.N. AEDALLAH; F.l{. EID; S.H. SHARAT EL-DII{ AIID I.I. NAIYZA, t l{atronal Institute of 0ceanography and Prsherres, Kayet-Bey . Alerandria, ECYPT. Key rords: Tides, Hediterranean. iFsT&AqT A ilumencal model for calculali.ng X2 - Independenl Trde rn the Medrlerranean (as a closed basin) has been developed. The sea Has consrdered, frrsLly, as conposed of ltro comrnunicaled reclangular areas of constanl deplh of 500 r, ni.i.h a rescluLron of lo X Io , secondly, as bounded nearly Hr.th tls naiural coaslal and botlon confrgurattons and lested twrce rrlh resoiutions of lo X 1o and l5'X l5'respectrvely, The sea ls assumed to be drrven by the polenLral lidal torces. The conpuled anrplrludes and phases obtarned from lhe proposed cases irere compared srlh each olher and also rtth observallons. The deCrced vaiues oblained from rectangular basins lype vrlh constant depth ot 500 m are relatt vely greater than thal calculaled by lhe reai bounCarres lype" lleanrrhr"le, the results of the model rtlth frner grrd resoiulron 15' X I5' are more stgnrfrcanl than that wilh resolutrcn of grid lo X Io and more confirnatrve tlith observat tons. The boundari"es have nrnor rnfluences rhile Lhe d:.rnensions of the basrns have signifrcanL irrpacls on lhe tidal pattern. _IIT!_0-W_q1"l_o! Th* HeCrterranean rs a deep, elongated, irregular depressron str*lched fron 6' H io ahout 3Lo E and *xtended tn uidlh itom l0o l{ tc *b,:iit {lu H. it Is i'i)rrrrrr,l jrated ',trl!l the Allantrc 0r*an thrcugh the 5l rarl of Cibralt;*l ihrth rs it3i.r c:{ ri rlt 1y ?0 kr, wide an'.l llfl :n deplh ai ihe stli. The t{editertalean ls matll:. iir']tr!-lrrritlil ci inl ba::lr;s iEasirr.ii anil lit"-'irln). ghrcll rirt rititllPt'teC tilt0ilqi; iire 5r.r; j;.ur rlii;nnel 'riti a srll 'ri tir dveragt ritpli: i5C m arri:i the verl neItlil It.ia:t af Hes.rlnia rylth al avet,ige depth of 300 il1. The eastrtn basrn ts ;'.rniiirfed urlir the Arirl,rir': Sea thtough the $l:rait ';r Ctranto anC prth lht A*ge;l X',a ihiouqh ir,lr stteits iu^ilted tr"*est atid easi ,:f ilreat islan'j (Fr{t.i)
t l{atronal Institute of 0ceanography and Prsherres, Kayet-Bey .
Alerandria, ECYPT.
Key rords: Tides, Hediterranean.
iFsT&AqT
A ilumencal model for calculali.ng X2 - Independenl Trde rn theMedrlerranean (as a closed basin) has been developed. The sea Has
consrdered, frrsLly, as conposed of ltro comrnunicaled reclangular areasof constanl deplh of 500 r, ni.i.h a rescluLron of lo X Io , secondly, as
bounded nearly Hr.th tls naiural coaslal and botlon confrgurattons and
lested twrce rrlh resoiutions of lo X 1o and l5'X l5'respectrvely,The sea ls assumed to be drrven by the polenLral lidal torces.
The conpuled anrplrludes and phases obtarned from lhe proposed cases
irere compared srlh each olher and also rtth observallons. The deCrced
vaiues oblained from rectangular basins lype vrlh constant depth ot500 m are relatt vely greater than thal calculaled by lhe reaibounCarres lype" lleanrrhr"le, the results of the model rtlth frner grrdresoiulron 15' X I5' are more stgnrfrcanl than that wilh resolutrcnof grid lo X Io and more confirnatrve tlith observat tons. The
boundari"es have nrnor rnfluences rhile Lhe d:.rnensions of the basrnshave signifrcanL irrpacls on lhe tidal pattern.
_IIT!_0-W_q1"l_o!
Th* HeCrterranean rs a deep, elongated, irregular depressron str*lched fron 6'H io ahout 3Lo E and *xtended tn uidlh itom l0o l{ tc *b,:iit {lu H. it Isi'i)rrrrrr,l jrated ',trl!l the Allantrc 0r*an thrcugh the 5l rarl of Cibralt;*l ihrth rs
it3i.r c:{ ri rlt 1y ?0 kr, wide an'.l llfl :n deplh ai ihe stli. The t{editertalean lsmatll:. iir']tr!-lrrritlil ci inl ba::lr;s iEasirr.ii anil lit"-'irln). ghrcll rirt rititllPt'teC tilt0ilqi;iire 5r.r; j;.ur rlii;nnel 'riti a srll 'ri tir dveragt ritpli: i5C m arri:i the verl neItlilIt.ia:t af Hes.rlnia rylth al avet,ige depth of 300 il1. The eastrtn basrn ts;'.rniiirfed urlir the Arirl,rir': Sea thtough the $l:rait ';r Ctranto anC prth lhtA*ge;l X',a ihiouqh ir,lr stteits iu^ilted tr"*est atid easi ,:f ilreat islan'j (Fr{t.i)
Solrman, G.F. et, ai.
Ii';t:
t: '.. ai'
-2-: - -: -
'.a a:-
-aa
----r: -
t. --.
. - :-
:.
'El€EEfc
-|'ar.D
!3(Da.
(D
c,Eoe,Egr(DqJ
The l{edrterranean as a closed system
The restern bastn ts more deeper than lhe easlern basrn' In addrlr-on. the shelf
rs relatrvely wrde rn the eastern basrn, particularly to the east of Tunrs and
along the Egypl:.an coasL, rhLle rt is nearly absent rn the HesLern basln'
Although the Lrdai range In the Hedrterranean rs snall, not erceeding I m, few
tgBT; iosettr and purga, I9B5) have attenpted fo erplarn lhe trdal phenonena of
lhe Hedlterranean. Slerneck (i9I5) used fhe coastal observaLrons of Lhe waler
level in the t{estern and EasLern basins to investigate the tidal motion rn the
Mediterranean, He found that three nodal lrnes are exl"stLng along the basLn, one
in the middle of each basin, rhile the third one is located belween lhe tgo
basins in the Straits of Hessi.ni.a and Tunis. He also added that lhe rolall'on of
the earth causes sfrong transverse osci.llations rhich transforn lhese nodatr
I ines into anphi.droml.es riLh anticlockrise sense of rotati.on. l'loreover, he
applied lhe hidrodmamical equations to calcuiate the tidai componenls in the
Hldi.lerranean. He eranined fi.rsl lhe d:"recl effecl of lhe tide generalrng
forces for each of the tro basins, assuming lhat these basins are cornplelely
closed. He found lhat lhe co-oscrllatrng lidal rnotron of lhe easlern basrn Hl'Lh
lhe other adjacenl seas is nearly negiigible compared r:.th the independenl L:-dat
conponenls. Horeover, Lhe Hesfern basin has co-osci.llafrng Lides tfrlh both lhe
Atlantrc 0cean and the eastern basin.
Actually the prevlous resuils obLained by the lheoreLrcal calculalrons as velias thal obtained fror, observations are nol suffici.ent to inlerpolate lhe
co:range and the co-ti.dal Irnes over the Hediferranean Sea parliculartry along
lhe Afrrcan coasl and in the offshore rater.
Later, nunerical soluti.ons of the hydrodynanrcai equatrons developed by many
inveslrgalors to conpute lhe tides and storm surges rn the oceans and adjacent
seas (Hansen. 1956 e I952; P:.scher, i959 ; Bretschner.der, I958; Krauss, I973;
Flather and Heaps, I9?5; Sharaf Ei Din I t{orsy, 1977; Soliman, 1979; Parke, ]982
and Sch$i.derski, I983). The paltern of the trde rn lhe [edilerranean could be
investi.gated nore deeply by applying these nunerical melhods after sone
developing Lo conpute bhe natural tides or the ti.des in closed areas.
The present study deals Hl.th the mathenalicai model and lhe difference scheme'
The model ras appli.ed to investrgate fhe semr-diurnal H2-independenL trde rn the
ilediterranean, rhich ras assumed to be formed by one of the following shapes:
t- ?uo rectangular basrns of constant deplh (500 m), rhi.ch are connecled
together lhrough an opentng of 2o width, having a grid system of mesh size Io flo (Fis. 2a),
2- The basin rs nearly represented by its natural boundarres wrth lhe folioutngresolutrons:
lo * ]o (Fiq. 2b).15' * 15' ( Pig. 2c ) .
a)b)
Sol rman, G. F. et . al ,
:l
t.
I
I
F:" i :.1 .'^t:tr " .'-]L ,;+
-!--
II
t-j
iI
l-.lllI
t
Figur* ?: Yhe s*a Hi t.h di i trr eni $h":Fr'J :llC diif :t"
Solrman, C.l'. et" ai
HATHEHATICAI COI{SI DIRAT IOI{S :
The rnlegrated hydrodynamrcal drfferentraicolumn have been t.aken as folior,:
equat rons over t he whol e Hat er
AJ
6t
6 fHrrl+_+
6x: lt"l6y
6u
6t
Ta sol ve
t ) iniiral
The trdal
ir ) Boundary conditrons:
At the bourrdary, thel.e. u 0 along thetrcrrrrd.rr y. or generrl ly.
6ffv+ku+q--l--X
6x
6{+fu+kv+g--"---y
6v
('l )
t)t
(3)
f orm,
tn easl and north
6v--'-
ot
rhere:xry
tt
U'Vn
fk
Y
x,Y
:cartesi.rn co-ordrnates rn lhe easl and norlh drrectron respeclrvely,: lime,: yater elevation of lhe sea surface,:cornponenls of the depLh mean current rn r E y directrons respec[rveiy,
total rater depth,Corrolrs parameter,coeffrcrent of bottosr frictron in the lrnearacceleratron of fhe earth's gravi.ty,the conponents of the lide - producing force
directions respeclivel y,
such equatrans, the inr.tral and boundary ('rJndrtr0ris urre tntrcdu.,...i
condrtions:
motion ras suggesled to start irom rest;
u : v : 0 at t. 0
normal component of the drpt[ *..r,longr t lrdrn.ri bouli,i.tr I an,l v 0
In order to damp groning of raves r.ithaecount whrch uright be consrdered either as
tine, bottom frietron was laken rnloguadraLic or lr.near forn , r,e,:
rn Lhe r-drrectron,. .(5)
in the y-direction
6.k = I0 sec-'
.. '(8)
The grri syslem Has const.rucLed such thal each grrd has a u- prrni 0n rhe yd:,rected sides and denoted by a (r; srgn, a v- pornl on lhe r- directed srdesand denoted by a (x) srgn and a J - pornl at the rn:.ddle of the grrd. The ralerelevalt-on as rell as the conponenLs of the depth mean currenl u and v were thencompuled slep by sLep at each of J -, u- and v- points respecLively tFlg 3),
ii) fhe di.fference schene:
I{UI{ERICAL SOLUTIOI{ :
Tc solve the hydrodynanicaifrnrIe dlffertnce scherne Hilh a
i) 6rid systen:
r u (u2 r u2 lt'' lH or ku
rv(u2 rn2 lt" lH orkv
rith r= l0 3
and
* u(t,j,t) I X(r,t)l6
ano ccnttnurr-y equations (l 3, n,.un,rr,i:allygrrd syst-em as gtven belou lias used:
. A relatively srmple explrcrt fi.nile drfference scheme is employed tr evaluateJir,:), u(I,j) and v(i,:) al lime l+rt from the knonn values at.i.ay computetial llne t ililh forward llne differences and central drfferarces tor ai I spacederrvatrves as rndicated below:
J (i,j,t*at= J (r,j,r) - 0.5 ratr[ (( H(i,j)
+H(i,j+l)) i u(i,j,t)-(H(r,j)+ fi(i,j-I))tu(r, j-l,t)lax + ((H(r,l)+ H(r+l,j))r v(i, j,t)
-(H(i,i)+ H(i-l,i))t v(i-I,l,t)ra! I .....(?)
u(r,j,t*at)- u(i,l,t) +at t [o.zst f (r)t(v(i,j+t,t)
+ v(i,l,t) + v(r-I,l+l,t)+ v(r-l,l,t))- g/or
* (J (i,j+I.t.trt)- J (t,r,t.rt - h
The Hediterranean as a closed sysLem
5q,
ttbran
-) 'E
ttC)aF
a.)q)
,an
.dlr
o
;;
i
I;I0 -t--irL-- -ri*x
I
x*xi
X- 6a
.l(\l .n C\l ,C)
-,F € Oo,oc{ io6'€
c!r rr)ea, $
o l' o
oro
The l,ledrterranean as a closed system
v(L,J,t+at): v(l,l,t),at[ 0.25 t t(r] r1u(r+t,t,rr
at each v-pornt on a r-drrected boundary . . . . . [] )
At (ar / ( zgttmar )t/l
andat < k I f.,i'i',e::e ielar,:-rns were applred to estlm,atef,:rnsiiereci lurtnq the pre:rent studl;
stabiirty of a
for l=Fa, 5r vt I
t frs t Lme
Sllnpi e r-aVe t: gi vt'l by
f = 0 (i2)
I o (13)
tnf.er',,ai for r.cih case
i The fr"rst case, rrth grrd slslem of mesh srte Io r Ioar = 90.6 krn
H =500m and
tn lunar ttme
2- The second case, nrth grrd system of nesh srce:
il l::t: :
nx 90.5 km ,
H mar : {200 m and
!)
,,,, - --
The l,ledrterranean as a closed syslenr
t ( 320 sec
accordrngly t lras taken as 207 sec
hr l( t lqpt
x , 22.65 Ln'. ,
H rnax : 4200 m andl( 80 sec,
l0/ 3' rn I urrar t inre.
I ' rn ,l unar I lme . Thelc make the perrod as
62.i2 sec 'I unar hour
lr-ne rncremenLtrnteger nuinDer
accordrngl y, t ras lalr.en as
was Laken:.n fractr,on of at,rme steps.
The model reached the:cmpiete L:.dal cycles.
:g
slab:.1 rty condrt ron afler ruJinr.ng lhe prcgram fr';t
RESUL?$ Aff D DI SC1JSSlO}i
The xejiirrrjrral;'s a ccnpi:,a"e,l sea 1:i ris lr-reqular b;.urle1-,.r:e: and seili;l,.::n!;;itr ti,Jj-,, Ar;oriinql;. :he'rr ccrre:p0;id;;tJ r ipar i rn il,e t-:.dei ru.ctr Jl ai,,lje:era: r i cn l-ie ua"tr lirc,...3r,r0l t:1 iiie l,{edi"erriiir6l l5 a.;sujiei t0 br s;::iill?,
l:,e pi'rssl: e€j'rejcplij ri,.:i*" lo;tiit is :ra:.rl;, dri,;eri Ly the pcieiilra: lrilarir'i.:es $it"-s pl]!,Jtl:i *[.ttinrl 'ri ;alcu]alr r"lre i{2 lrrcepe:rdenl :}"de t.n l.,iii rli;:r"r'.lit i:;,sl::s 1 ;i mesii ;: za I r x 1a i c0nnrlct ed t,l eaci clher thl u,r.g;i ;i,:r:4"{;ri;aif.. 5"rcn rvp::l:trLCl.rir is nearj}'sl.ilnl:r ln i0n:iguraltcn fc thetleci;erranrai:. The r,rcdel ua:, lhelr useo t.o r:aicuiate the Hl lnde;ender,t. r.; de lr,I iie sea ui iii neai. : y r ls r"eai bcundarres ani dept hs rrs;ng qrrd srsl erns ci meshsr?PSi js X l" atd 1.5'X l5'respectrvely. The bottcn frrcr:.ori $as assuiTrfd as.LC -l sec-i .
t) Tro rectanguiar basins:
The later depths r.n the llo basrns flere assurned to be constant (500 m) and theopening belgeen lhe tro basrns as 2o nrdth. Figure ({) shors the co-tidal Irnesof equal amplr.t.udes rn centrmeters and the co-range lrnes rn hours as the Hoontransr.l,s the equator at Creenrrch.
The patterns tndrcate th* presence of three nodal lrrres. cne of Hhrch istian:icrned by the ri:latron oi the earth rnto a ccntra solem amphrdromyianttclockiltse-Clrectlon) rhtch occuptes the Heslern basrn. The co-ranges arertearly ellt'pses arcund lheaurphrdrcmlcp0r.ntHhlchrs nearly locatedatthecpntrr rf tire basrn. The o-crrliat icls tit {irc r,re.;tern l,;l;rn appelr as d ::landing[.ive'.]srliiating:.n the ]ongtttr,jtilal rlrrei'llrin lq'ifh est:bltshment:;,:f jh.lnd lh.
Irr lhe eastern basrn tHo nodal i tnes uere produced nhrch descrlbed a
i ongiLudrnal rnotron propagatrng along the basrn. The cc-rarrges lu tiie uesternbasrn are relatrvely grealer than lhal in the eastern basLn.
ii- a- Real boundaries and nesh srze Iq X lo :
in lhrs case. lhe Ceplhs *ere averaged over each grrd and aliccaled at lticenler. it ts clear frcm Frg. 5 t.iiat t.he iledrlerranean;-s cc;islstirrq oI nicrr:than sne basl.n. Each basin has rls oyn length, width anC depth rhrch nrai
rnfluence separalely rts t.rdal characterrstrcs.
The lidal nolron rn the Albcran Sea rs very weak. the rater rrses and failsslorriy with e rarimun anrpirlude oi about {,0 cm. llore lhan Lhree nodai I:.nes aregenerally observed in the Hedrterranean, one amphidrony conLra solem rn theTyrrhenran Sea, a degeneraled cun sole amphrdrainy 0n the Trrnisran coast and a
nodai llne tn the Alboran Sea. The lr.dal rootton in the eastern basrn ls nearlysrmrlar to thal observed rn the preceding case.
Greal range values are observed in the Gult of Cabes (more Lhan 25,0 cml andalong lhe eastern boundary of Lhe t{edrterranean (nore than 10.0 cm). Suchpallern does nol appear in lhe case of reclangular basrns srlh consLant depth ct500 m, The trdal ranges rn the uestern basin are relatively louer than iilatobserveC rn Lhe above nentroned case. Across lhe Adri.alic, a nodal irne ais*extsls and the irater lei,el changes gradually nort.hwards. There rs a phasedt fference of about t hours betneen the occurrence of high tj.des at fhe moulh oilhe Adrralrc Sea and its norlhern end, The tidal ranges on the eastern srde aregeneraliy hrgher fhan lhal on lhe Hes[ern srde. A phase drfference of about 6
hours cas found betseen lhe occurrence of high water tn each of Lhe twcconsequenL seas: Levantine and Ionran; tevanline and Aegean; Levantr"ne andsouthern part. of lhe Adriat.i.c, Thrs indrcates that rrhrle lhe water falls rn theLevanline, norlhern part of the Adrj.atic and *estern basin of the Hedrterranean,rl rrses rn the lonran, Aegean, soulhern part of the ldrratic, and Lhe Gulf ofGabes. Generally, lhe results obtarned i.n thrs case rs much better lhan thatobtained wrlh constant depth and lhey are more or less in agreemenl w:.lhobservattons.
ii- b- Real boundaries and nesh srle l5'X 15' :
In order to get more confrlmattve resulls comparable rith observatr.ons. a grrdsystem rrth hrgh resolutron mesh srse of 15'X 15' tras used. The baslc arraycomprises I08I5 elenents tlrth a dimensron of I'54 and J. 159. The trme step oirteratton processes Has taken as 52.I seconds to fulfrl the stabilrty crrterra.
By usrng the new gtrd system nore preclse results trere obtatned everyrhere.Relatrvely hrgh ranges rere observed rn the eastern basrn, relatrve to that rnthe restern, partrcularly rn the Culf of Cabes (Lrbra) nrth a marrmum range ol
a:p.r.ir;iil.. iii:,i urti ,;aluei on the easlerrr s:.de 'rilich i: h;gier than i:,'I ;l'
:'r des':ei:. :*r*,,aiiy speaklng' lir* resujts cbiaihed l'I f irr lasi last al? n]r''
-:.i-i:"a!11'vrllh.riu1."tions iabit Lr' l'i'ire s1:udies art t::'ieiitlaiiv r:c:'-ill:;
- jer- rnore l.it0rr,si ior,, alc,ur t:he :.nriilence of hcuidarles;, lepr'f' f :ici;''r' aii-,:.;_ t.cr,a.:cr.,rf lhe ea;th cr, lhe s!gnLficance of the resulr-s"
_a
+i-i!
I---.i
_i
gq&IE!l$ c0t{cttjsl0{5
The l{eiiielraneaii lJds aSSUined tc be separ ai.ed f i.om the Atiant-tc c':ean b\'
:.:st:lg 1t al llibraltar, ils'*ar:er ras cnly drlven by the poleilrai tr'lai
::l:es . A t$nerlcal model ilt-th a f :.nite dr.f ierence scheme Has establ:'shed end
ii,en erarr\rfied i,o calculale Lhe H?- Independent trde in lhe sea' ''vi1i6l uas
:lns!dered, as conpos;d of l$o rectangular tasins urth a consfant depth cf 50C ti'
arc lhen Htr,h rts reat Uounaarres and depths. co-ranges and co-trdal Iines ';i
:he Crfferent cases have been presenled r"n charts'
The nrodei wrth reclangular basins gave a descrtptrve t"dea abcut the lype ot
l:dal notrons rn closed bas:.ns. Actually more informatlon are requrred to be
estirnaled lo dr.scuss the reasons of the abrupt changes in the tldal ranges rICn
i The uater riepth Cistrrbritluns i bott0m irrp0qraphyl has r qreat ttlllrt't"'':t':ir. trci.ll itiit]r-.i.i "s rrvrair,J lrom tiie constant tiepth pattelns tn fonpdll::Jll 'i
