INTER THE

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ON THE NONLINEAR EVOLUTION OF INTERNAL GRAVITY WAVES

By

@LirenYan, B.Sc.

Athesissubmitted to the Sch oolofGradu a t e Studi es inpnr tinlfulfillmentof the

requirementsfor thedeg re eof Master ofScience

Dcpn rtm cn t ofPhysics Memoria lUniversity of Newfoundl and

Octo ber,19 93

51.John's Newfoundla nd Ca na d a

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1+1

NallooalUbrafY

of Canada Bibliothequenaeonaie

duCanada Acquisitionsand Orecton des acquisitionset Bibliographic Services Branch des servces bibliographiqoos 39SWellingtooSI'eel 395.rveWellinglon

=40nlano ~:~~Onlar"'l

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Can ada to reproduce, loan,

distr ibute or sell copies of his/herthesis

by

any means and in any form or format,making thisthesis available tointerested persons.

The authorretains ownership of the copyright in his/herthesis.

Neither the thesisnor substantial extracts fromitmay be printed or otherwise reproduced without his/herpermission.

l'auteur a accorde una

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a

la Bibllotheque nationale du Canad a de reproduire,preter, distribue rau vendre des copiesde sa these de quelque menlere et sous quelque formeque ce soit pour meltre des exemplairesde celt e these it la disposit ion des personnes lnteressees,

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Abstj-a c t

TIll'purpose oft1li~1,IIt'~i~is1.0{'111l11Jilrt'1Ill'IIOIlIlilll'ar"I'Ullll,itJll"f iul,'tllal gravitywnvcs ohlai nl'dfl"01I1aIullynoulinourprim iLi\'l' "qnalillllUUIl!.'!withIIII' evolution predicted hyweakly1I0111iUI'n1'tlll'lI1'\I'S[i.o.,till' !\,IV"IIIII\II';.IV1'llml l,iulI~), Inorder to focusonUl('noullm-ar1'\'ulu1.ioll,we-nmsidl'rtill'id,'a1i;;'l'dnl.~"ufall invlseid, iucompresslblcnC)lIssim's'lIlnldoft:tmsta llt'!c'llt hwlrhsimp[,'stl'iIt.llindi"IIS, Insuch <IU environment,tIll','volllt io nofinternalW,lVl'Sis 1.11t'tm'l.kall.l''lllll l.v~,.. 1 upto second order illamplitude hy 1.11"llll't,hud or 'lsyml'1.lI1.kI'Sllill1~i"lIr"UIIWill/!;

LeeN.Beardsley\l97'IJ.TIlt'n'sultill~~{)Vl'rllin~l"(III,.1.i" lIistilt'Ild';,IV .'tllli,li"Il, MI'lII1Whilc,tlll~evclutlonoftIll'Hllm,'HYH~t'111illshnul a k,1I,ylLfully l1lmlillm l', invisd'[

numericalmodel.TIlt'modelishelk-vr-dtuI",rt'!iilIJ1I'(1..11 11 [' [l!I!I:111,10])"1111L1111 .~nlll he usedto quantitatively ver ifytl1l'(1,~ri\'l',1t.ln'oty,TIlt'llll'01'yis ,'utlll'iln', 1ap;ilills1.

the modelresults. Au initia ld''Prt's.~ionIs gCIll'ratl,<1nn.lthonlJnlilt,',.rH1.''t 'I '' 'llirl~uf thewave fronl and its snbscqueut I'volnllol[into1111nudnl..r 1m1'\'isluvl's1.i$';al"',1fill' differentstratificationsandWaveIl111plitltd"s, Itisrlll llt,1tllid, til!'Lluxn-yi.~ill vr-ry good quuntlturiveagrccnu-utwiLhlhl~mudd1"I'S1l[lH,TIlt'l11[{d V {'<IllatiunW'lJ,'rally imlHOV(.'Sthe flrst-ordcr](tiV resultsIorwaveswilh Illlllllillll'IIS1CJlI,'[ 1I1I11'Ii1I1 d4"111'1.0 0.07,Forsmallwaveswith,<0.02,til"s,,,,ljlld()rdt~rrltmlilll,itrll,yis lilli,nlldal. 'I'll"

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IIlI\dV1'llUillioliisIIUtl,p IJl"upri lt ll'for waves with(I;l r~"rr.hanll.Oi.Thesoresult s

»rt>Vi,I(·[urf.hr-r('vi,l('un't111111.lwfullyuouliurnrprunitive ('f1,w.1. inll1I1011('!is rI,liahl{'

;'Il.1~iVl'sum"ir" ljnrt.i"" orwls-nlilt' l\dV<11111rnl\ 11V '·qlla t.ionscun( ·,·tlypr<~di dl.l w W<I\'I ' l' volllliuli.ItisIlls"showntha tarlt'rlilt' nndulnrbon'bcgius10form,rotntien fillSIlminordfl"'lunibsrrl.sl'ljlll'1l1.evolution.

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Acknowledgement.s

duringtill'COlIN'ofthis tlu-slswork.

l.Dr,1\.G.Lamb,mySUIll'r\'i~ol',illt rodlll'I'll nw."1.lll'lil'ldofuouliucarinn-mal jl;Til\·ityWll\'l'S,111' 0ll1.lilll'lltilt'rl'SI'<l rl"1,idl';},Iwlpl'lInil'L..1I11l1,'r~l illld,1IId 1.11 analyze thephpic~ofllll'prtOSl'lll.I,Up;I·.IIisrontluunl1-!:lIi,I,ltln ' fOllLril>lllrd ;1 gn'f1ldoal to tilt' "lIl11p](,t iou oflilyt11l'si ~work,Dr.Lalll hHlsH kil:dlypro"i ,ll',l nu-with lillilllt"ial slll'l' Ul'ltlrnJll1-!:lrolilL1wr't>lll' Sl'Ill'I.lIiswork(a[uu-t."fl.1w

') Dr.H..J,Gn'atl ml rhulf,.t!..1runny1'1lIi1-!:hl,lmillJ.(mlvin' swhen11'II....lI111.l'r..drlif- Iicultics , and,l;1' Ill.rolls lyslippor t",IIII" duduJ.(till'lirs lLwuy"il LS(il1',1I'l"I' l.Ill' suppo r tis FromNSEH( : 0lllOratilll',/II"st' lllTl.(;rillll,s;lwa n l,..1I.t,II.,J,(:1'..;,1·

hatr-h},

eonstructl voSIl~I's l i(JlI Sill maki ng thist,lll'siscll'an'r alllinnu'r-"'mds,·.

'I.A.Gtluldingund I',1....mlygilV ,'Ill"itluI.ofIwl"with lolli','u IIlIJlI1.,'!' sySI."III,

II.lIa lilll'SIIllI·,..1wlrhm,'Iri.'(,'xI'Nil'IW "

o r

lISill.!!;Lill,'X,Xli,!!;,..r.,', iii

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!i.Sdu,u luf<:ra.lllllk SLllllil'~il!l.1DI'llilrlllH'lIlofl'hysles pr(l\'idl'(lg<'lwmlls li- lJiUlI':aI1l.ssi.,ta lll:l'furl,hisworklhroll ~hGraduall'FdlolYshipandDopnrtnn-nt al

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2. a

Hunuingtllf'Model.

12 12 l!i

zn

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:~ Noui lucnr Theor yof LongInternalGravityWaves

a.1

In1.l"Ud lldiuli .

:1.2 Ikrivilt.i"utlfModilil',ll\dVr;;'I',atioll .. :l.~.r TIIl' O{I)prcbk-tu.

Tlw O(( ) IJroh!f'llI .

;1.~.:1 TIlt'O(fl)pruhhuu•. :I.~A TIll' O(c'l )1'1'()IJlP1l1 ••

:1.2..'\ SllllImary .

;1.:1 Ap plka ti t!lIlo our~]lt't,ialn~~t'. .•. . • • • . . •• • •. • •. TIS

;1.:1.1 Sillll,lili" alioliof ",._,'\'·Xll1"l·~sillllS.

;I .;l.~ COIIVl'rlliulIlo dilllt' llsimml Fcnn..

:1.'1 SUllll iu llforvcrtirulmodrs.

:L"i Nunu-rlculsolutionor 1Ill\ dVuquation NlIllU'rindllwlliOlI.. . . . .. . . •.

Slllhili1.y..

.. Theo ryvs. N uruer-ica!Model -l.I EXl fiwtioll or wa w profilo

uo

65

n n

76 80

82

,,'

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.1.1.1 About1.1I1'W'II'"prolih-lI( I..,'}.,.

·1.1.2 ])"pthim]"]WIl'!.-llf't- uf l,]lt'lI'<lw'proflk-. ,1.2 Ilt-solu tionh'~l

'1.:\.1 RUllSwlthNI(=)

'1.:t2 HunswithSIIHMll hN{(=) .

'1.:1.:1 With,I sl,,'pwis,'Nf (;:; ).

·I.:U I~m~dsofrotution. 4.'1 Discllssion.

5 Su mmaryandCon cl usion

vii

Sl,

lUI

lOi

mu

1:1:1

1,1~l

Ill!

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Li st of Fig ures

OJ'I <:l'i,1sd u' l1I('wit hquadrilnterulgridrt-llxill(.1:,.:)S]Jiln~. \:1

~.:I ,[,11l~skddl nrI' xl ri, pu);lliollFromti;j 1!1

:lA TIll'd.'II.,itystra1.ifkll l,jolis 'l5

'l.!'". 1l"1I1<lly('ull t o ll l'llforsll!outhNl(:;)nndmoderatefoe'dug. :m

:!.H lklls;ly "oll1.o11rsfor SlIlllUI,hNn:)and strongf01'cll1g . :10

'l.i Ikns H,yrontcurs furNit:)awlstrong forci ng :11

:t~ D"lls;tyl"lmtuu rs[urNJ(;;)andsl.rull~fo rl';ug,with1'Ol ilt;ullillC!lllhl :12

;1.I ZC'l'Ir U" ,!l' I"vl' rl ,kal 11lUl!I'r,6(.::)undib,It,rival;vl' (it!

:l.'l First-order vvn.irulmodr-q,l.u,(; )anditsl!t'rivatiV(' 1m :1.:1 First- or der wr lil' allll UlIt·".,0,1,(=)and itsderivativt' 70

;l..l Sl't"llml-un !l'l'wr1.iri,l l11ml,'0/11.11.(:;)amiilsdcrlvarl vo . 71

viii

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:1....' 'l'l«-plul

or

Funr-Liuuf(/J.k,\'./J) "'~l :I.(j 'Iesr.s uftIll' lH11l1l'ric'i1lltll\d VII, cIV~ullil illils. :.;1

.1.I ('0111.imU', :';~l

'I.~ Vilrillliolluf11(1,:r )with dilr"1"l'lIt,,1,11. !l:!

,1.:1 Ill'pl,11ilUll'pl'lIl h'l1n~

"r

11prulill'wit.llNf (: )nurl".~."'" -U.lJ077I ~17

'1..l Sll"lldlln'or 1( ;\ V1lIIil ll1l11ll'! 11m

,1.7 Ttnn-l"p!mlllt iu ll !I'stor (CWsil1l. IUli

·1.8f1.II(.:)wilitNf ( :)

ror

dil rl'l"l'nl,,,'."• 111I

·UJ Dept him!c'lll 'lIdl'lIn'ofHIor1.11<'TImwithNf(:: ),mo,I" "llll'11Jl"r ill}!;

eud0'·11=-O.O(JXli7

,1.10COlllpilrisl!1lfortill'runwithNi (:).InurINH1.I'rOI"l'ill~;11 ,,1 "I.":- -O.OOXli7•.

01.11Asill01.10

mlfl",1,0=-O.U:.!~H7

ix

III

II~

11<1

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'1.1:1<:utlll'ilti~ullfurtill'run wlthNi(=),modcrnteflm~ill~and.,.1.U= -O.fJ1Xli7.

,1.14 Asill,1.1:1

awl,.1.11=-1.UmW7

'l.!tiColtlparison fortlu-rimwit hNl(::).modl'ra tl'forc ing and0-',0

=

[I!)

116

[17

-1.mIS!i7. 118

. . . . .. . •.. IHJ ,1.11'1D" llt hhnk-poudoncc

,.1

Iifortil<'timwithNl (::),weak fordllg111111

n',lJ =-O.Oll li!i7.

·1.1!ICompilrisullfor lItt' runwithN](.::),weak foruluguud(tl,O

=

-0.00SI\711,1

·1.11lD" I'U,illdt'pt"ltlt' lw,' ofIifurlilt'nmwithNl(=),stro ng forcingi~nd

«!"=-O.IlIlSli7 12!j

-1.21COlllpllrislIllfor 1hl'runwithNj(=),strong forcingand01,(/

=

-U.II0867126

·1.22 MIllI,'\1\'a\'t'lt'll~lhaut! sp,.,'dfor tilt'TtlIIwithNI(::), 128

,1.1:\ [Ii-tuih-rlr-omp aris ou ofwil\'('lt'ngt hfortilt"runwithNi(::) 12!J

·1.2,1 1J.'1.llill.,lnunp llrison ofSIII'{·t1fortlterunwithNf{:::) lao

·1.2!",H^I•lI( : )withOI,U:=n;Jl, la:l

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,1.26Dl'p Lh indt' pt'll tlt'lln'I>f

n

forI,ll<'rimwit.h~l11l1lltllN~~(::).weakf"IOt' iug and01.0

=

-0.1)077.1

40:a

ComparisonfurLIlI'ru nwith~l lwu~ 1iN;(::).wonkfunoin).!;,m,l l1l,n= -0.0077-1.

4.2~Depthindt'pl'lIt!t'Ilc('IlfIIfort.lu-run withsmoothN;(.:).111" ,11-1"11[,' forcingendOI,U=-0.OU77'1

,1.29Comparison fortlu~runwith smoothN;(::),mudt·r"t,t'fon·ill).!;IIl1,l

01

.0=-0.0077-1..

,I.a o

Dt.'pt hindependem-eof

n

forth,' runwith~nltlllthN; l.: ),~t.l'UIlP;fo rd ng and('(1.0

=

-000077'[

Il.alC()111[lal'i~()11fo rtit"runwith~l1lo(Jl,hN;'(::),~lorollP;fUl'l';Ilp;0111,1nL"=

-0.007711 .

I:m

[·1Il

HI ,1.:12 Mudd wav,.]t·l1gthandsp"t~lfertheruns with~1H"olJ.N;(::). J,11 11.:1:1Dt'lll ilt' drourparisonofW1tVt'I,' up;lllforlIlt'runs wit hl;lllUtJll1N;{ ::) 14:\

,I.:HDetailedrumpnrisunofSpt~'{[fer tilt'runs withsml/oLI.Nf(::) ItH 1.:15Depthindt-pnudcno-ofBfurll l(~filiiwith~l('I'W;~"Nt(::),III"d(,l"alt'

ford nganti('fLU

=

-O.00791i

11.:16Uompnrisonfortil(' nm wilh~t(,pwis,'N;(::),llI"d, ' ra tl'fl)rt~ill,l!;atn]

01,O= -O.0079Ii..

l'1fi

1<7

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thl' olll<'rwlrh1I1C'slc'lJwil'C'N:( z).

1.:lilCOlllflarisullhdwI·t~l1rutat ingalltllIon-rutatingres uhs '1.:I!JCUlIIl'ariscJIIIIt~twt"( 'ntl1C'theory11,1111tile'rotatingrc>sull forth«slrong

[orr.ilLl!:

-tAO Al'<inf'igll~4.:mhutforthemodera tefordng. 4.'11mKclVsol itolls •.••__._.

xii

i48

1M 1'>1 158

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List of Tables

4.1 Theli~loftilllSanda.!I.<;oda.ll"\pa.ranwll'J)I• . . • . • . . . • ..••• III...

xiii

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Chapter 1 Introduction

Thl'~ "wnvcaplaya Ilip;nilin lnt TO],'inlrallsporliugmomentum11111lt>1ll'rgy withi ntill"

,W,'lmlI1ul canIll' auil1l1'uttitntsource of mixing.IthasbeenrealizedthatIGWsand thd rskh-('m'ds(1('s! 'r V(~rarofulattention,forthey can significantly influence ora-au ("lIrrt'n t ltU'il:<lI rC'llll'lll,nndorwaternavigation,antisubmarinewarfareop(~mtioIl Hand

"\"'111,111'fl't·tlillgofmadill'animals(SI'POsh orlle,itBurdi[[980]).

(Iw-rtilt'pilsl two dC'nlll,-s,the re havebeen manyreportsof obsorvaticns of 1;1l"~"internalwavestJlTlIrinp;illlIlt'ocean, Thesewaves

',W

usu allyinthe formof inlt'rf nd;llundulnrhorr-»,llu]itOlisorsolitarywaves,typifiedbylarge,amplitudenud

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or,Il'!"withtill'IHrgl'stII' ilVl'1"H , li ll~till'~r"'ll',11'111'1'1"11[11171 11,1,I,ll<1l1ry.-ful.[1 ~ '7~11 and Chen-skin[l!llnlwllOrtl'<!l<l1 dl WiI\','r"rIIwl,i,,"suhs,'r\'t'clill1\1;,ssa ,.}mss,'ltsIla.\', Far mer&.Smith[l!lROa.h]ill I\lIip;lltlulot, 01'1101'111' .\'.Hurl'l l[1 !I~lIJilltill'Audnutnn SpaamiApI'! dol.[IIJK."i)illtill'Sull1SmandSlllld.4wlll,~:Elli"u[I!IKlJ"II1,1)(' SnitillllSlllM,dlo!Js,'r v,',I ,simil llrwlIv"I'" u.,' rn s,'vul vingrrullll'it lll'r;,I,ll1'rllll",1illl' depresslou01';11,'1'wnvetra in r"nrll'dloysll1Jnit it"all,i,li,l lIowm'nras, 'a sill.'1'111'1'"nrt- alsolebo ra tc ryexperimentswhichrt-prudm-cd similar Wi\V,'r"rmal.iulls ill "1,\\,,,,1;,,\',·1' lluhlSYStl'l11, I~,P;,l\'li ,xwo rt ll,Y[1!'7!l),I\()()[l&-Huth-r[I!'HI], S"g'II'.t"11;"1Il1l;...k11~1,'i~I . La ViolettorIai,[HlHO],1i sc lIs.'It~,1phuttl l;f/l ph sluk"11 OV1'1"tilt'(l,,<..,~,,sHauk(rullltI", ILS.span'ShllUI"H,whid.d1'arlyrevvnltl,,'slldiWI'siAlIiltur l'""rilll"Tllillwa y,'I'n,.[«'t"

pt"op agatillg away fru mtilt'bank\"lgI'.

At thesam!'tilill'.theoreticalillVI'stiAatiolisofI( ;\Vshnvr-],1'1'11 t'lU"ril'f!0111.

withvariousrO(:IISI'S. IIItIll'rrallll'wo rkurweakly1I01lIill"I.rshallow wl,t!'rU"'lI'y, Benney [1966).Lee&Be-ardsley [l!J71].IJnu[1!J7!"I],1\ lIbull' [1!J7K]d,'riv" ducnliu-

eartheoretica lmodelsforlOW cvolntlon illstrit lil ic~1illili/or slll'ilrI,,1lluidorshl.l·

low,finite amiilllini t"depths,respcrtlvr-ly.Tllt'sl'1,1j(~Jri,'ssay tll<,ttil"(,Y"llIti"llur weaklynonlinea rIG Ws :s~Ilvernl'llI,yl'II"" tio IiSofJ{dV t.YI'"(I(dV sl.awlsril l'lll"

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l<urll·WI·,i!;.dl ~ Vl' i (~~('(jll1tt iOIl),illwhic htheasso cia ted nonliuoari ryanddisp e rsion are totallyd('knllilll~1hyth«cnvlronmcnral equllihrium stale . Lee&,Beardsley(1974)

1.l'a i"sl ,I.sl'l'Vl',1illM' lSSi\I:!lllssd lsHay.Max wort hy(1979],F'anl1t1r&Smit h(19S0] and Il il,iy l' [ImW] illYl'st igatl'dth(~gl'IU'rati unmech anismofinter nalwave trains through 1.illill-tol'o gral'hkiuteructlou.1<001'&Bu tler(1!J81],Scg ur&Ham ma ck[1982],and CUlllmi ns&Ld ilulld(HJ8tllksle d1I1(~runge ofvalillil yof theKdV,Benjamin -Ono nnrl J<lIlml.a solita rysolutionsaga instex per ime ntsandobservatio nsof intern a l soli- tllryWilVl~S ,forusiugOil1I1l~waveshepoand amplitudl'. wavelc nglh scalingrelation- shi p.TIII'Yfoulltl th llt llwKdVtheory hasil la rger rangl"ofvaliditythanthetheory lI1i~l ltimply;in plu'l kill ar,itisnotstrictly lim itedtoshallow water .Liur:Ial. [1985) IIllulilll'dKubota'smodelby incor por atingspr ead ing anddissipati on,"lfedsintolilt' 1ll,,,ld,thusf"rmlll'l1.,'dt.11I1 evolu tion oftheinte rn a l par kl,ts obser ved ill Sulu SI·a.

TImcompnrisonshows thatLIll'sim ulat edpropa gationspeed oftilefront ,thenumber uf sulit oliJol,tln-irnmplitudr-sandIl'Ilgthsare ingoodqualitative agreem ent wit hthe datu.

Ih pointed outhy Ost rovs ky&Stcpanyaut s[HJ~9l ,athorough theoretica lde- sr-r- ipf.inn orIGWevolution prorx'ssraillrealoceansisitrath erdiffie.ultproblem.The 1\IIV-t,}'11I'tlll'uril'SIISi'l1nrctwodimen siona l andincludeonly fil'liLor puslliLlys(...~-

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endorder nouliuoar,IIHIdispl 'rsi\'l'l'Ir~,t'll'(audpOl'l'ihly par'lI1l1'l.t·ri;((·tlnllliali'lI\ and

alred"-llhy many hYllrOllynmninllrnd(Jr"~,sllchasLIII~\"llri~lli (ln sHr sln-nr,.~I,ral.ilj,.a lion, topography,radiation,hllUol1l!I'Ild ytlisl<il'at,ioll,I.ith·,"ll',It.il< all«',lim"lIlt1.0 followthe spnCt'·t inwevolution or a WiLVl' rur l'u'llJli lril<lJllwilli llwHl'il'S.TI,il<milk'>l<

p;uodquantitativecomparison sdilflcult. Thus,lilt' nl11lpill'iSUIlSMI'uulytilla lit.aliv, ·.

Alt ho ug hitispossible now ttl uhta inagl' lll'ralp~LU.t·rI1ofproPllJ.(~It.in,l!;IUWsrnJlIl ail'ernftandsatd lites,rcliahk-Sil1111llanl'llllShytlrolup;iraldlll,auru:<l.illr1Ir,'lyavail~Lhlt·

forthe Formulationofthccrotir-almodels. Alloft.ht~sl ·prt'St'lll SiJ.(lIilkiUlt.ohs L;,,-lt'Slu rho unders tandingort.henature ortlhst'rvt~ll1C:\Vmotions.

NllllWrit~ll1modelling oflewt·vo lnt.iullillallitl.·nli~.t·dlIt"( ';UIl"ll1lldI,,·apH'- liminurysolu tionofthopruhll·l11.Bytuklugintot"I!IIl'i.lt~rnt.illllimp ort.ant fad"rs

"II"

byone,their rolesill IG\Vevolutiont-auheIll't.l('flllldc'rst.ourl.1\ndinhh-mutll·1nl.ll h,..usedto quantitatively veri fytilt'mdstingUwo ril'swhicha1'l' usually d,'fivl'tl1111·

dcr idealizedconditions.Incout rust, Iield1I1t~aSllr'~1ll1·nt.sal"l'!.1·1I1porlilly1I1" spatially sparse and ilrrl,:dedhy~1numberurphysicalpror.l'SSI'S.lIt'"e:t~,tln-yMt'lIulyn.~,·flll for qualitativer-omparlsons. Tlwrl'amalso SOIIIl'aclvallla~I 'SorlJlllm'ri c~alrrllltlt~l lillA over laboratoryexp eriment sfo rt.llt~(~IL~t 'St:(Jllsidl'r( ~,1h,'n ,:.Fur(~X1U llpl(~.II.nllnll~ril~ill modelcan he run in an lnvlscidmcd«ur at lilrW',n·albtit:lu~yn"ltb11I1I1I1,,·r:<.It;,Isu

(24)

allllw.~~n'att'rcoutrclofthe phys ical precessesthatcaninfluence thewaveevolu t io n, aUtI callilll:(j rpomt( ~1I10 rt~rt'alist irback gro und ccndltlons.Inaddition,n l:Olll lllett' da tasdnUlheobl,ai,u,t!.

IIIthistllO'siswork,WI"!investigate thn uonliucarevolut ion ofanisopynnal de- Ilrt'Ssio li. Tile depn 'Ssi'm steepensbeca use ofnonlinearit y andthenevolves intonil 1Jrlflll liLrlllln~tln'cughdisl)ersiv(~ellects.Thisprocessissimulat ed withitfullynon - linearprimitive equutioumode l (Lamb (1993 a,b])whic hsolvestheiuviscid,incom- prt'Ssihlt ,lIo11ssiue s(1t~(I"I~tiolls.TheKdV andml\ dV eq uat ions appropriate 1,0tll{' S(' ('fl"id,ionsarethenderivedfollowingBenney [19(j{j]andLee&.Bea rdsley[19741. The evolutionprcdicu-d hytilt" KdVamlml\ dVequatio nsis thencompa red withtill' evoluf iun~i Vt'nbytlll~FullynOlllin ('I~rprimitiv eequatio nmodel.Any differ en ces ill 1,11l' evolut.iuna enuon lylit· tim' toth (~npproxim at jonsmade in derivingthe KdV ll1ltI1l11\(IV('(Illa tion s,to11Iullt'ricalerr-or inmnncrlcallysolvingthe[( <IVandm!(dV equathms, urto1111llwrkalerror-illthemodel.Thenumericalerrorsuanbereduced by sllrrid(' l1lly higllresolution.TIll'theo ret ica levolut ionsarecom pa red withthemo d el rosnltsforsovr-ralhydrologicalbackgr ou nds andwavemuplitudes,am! alsoforacase

withrot auo uluchnh-d.Inun le rloS(~·th eim p ro vementmadeby mKdV ,com pa riso ns

hd \\'I'f'11lh('1l1tJ(1t'1and tht'first-order1<<lV th eoryarc also given.

Inuhuptr-r 2,the11l1nw rin(1modelused isbrieflyint roduc ed ,audthe nth (~

(25)

domaingeometry 111111 tlu-hydrul{)gil'illr-nvirouun-nt,inwhichtln-muddis1'1111IIl1d rho theoryisdt'riVt'd,aft'given. Ind',lptl'r:I,alterahridn-vh-wof lOll,!!;\\'ill't ' theo ries,WI~given tldnilt'!1dcrivaf.iou

or

tilt'1l,[' c!V!'tlllal,i"",IU,I!;l't.lwrwitll1,111'

tosolvetilt,l\d V ;\lltl ml\lIV1"1ll1 lltilll1.~is,11'Sl'rll,,'c!.'1'1".1.I1l~lryis"'lI11l'ill"',[wit h lhemodel illchapter<I,,1,loll~within-situ millp;!'I1l'ral,l is'~llssil!lIs.'I'ln-1'''1l''lll.~illlls art' in Chapl('l'.'i .

(26)

Chapter 2 Numerical Simulation

Inlhis {"hapll'"WI'briclly introducetileuuutc ricalmodelwhichhas been usedillL1lb sl lllly tllsimlila te'IC: W.~.Wl'ak lynonlinearthcoret.ic ul resultswillbe compared to llwfullynonlinear Tt'sultsobtainedwiththismodel.Thismodel wasdevdoped hy Kr-vlu Lamb [Umaa.h]andwillbereferred toliSIGW sim inthisthesis (standingfor lnternalGravityWave Simulation).The modelisrun withinIIfixedphysicaldomuiu and (ur Ilirfl'Tt'l1lbackgroundlltal~.Someof theresultswillhe give nla lerin this rhOlVl l'rl~~oxampk-s.

(27)

2.1 The Numerical Model

IG\Vsi m 801v(';Ithe-tim edependent,iuvisr -id, illn'1I1prt's sih l, 'lltlllssil ws' l1" lllill.iollS

01+(0.9)0 _-~II-_P!i, (:!.I)

PI +O.9p

0, (:!.:!)

v ·1i =

^O. ^(1.:1)

Here

a

^{is the}^velocityvector withhorizontalami\'t'rtin dnomr",w'u!.11 (u,m). (.r,;:;) arethecorrespondingro ol"dilla Lt'Swith ;:;l1I"nsllrl,,1l'0si1.j\/l'111'1'0';11'.1111111(7istln- gradient operato r(iJ/fh,iJID::).TIll'Iluld densityis~iVl'lIby(III(J

+

^p)ilwl/llI(!!:+11)

theBoussine sqappmxi1llatiollr,'rnuv{'stil,'rl,dul'orI

+

p'J11tIll'ldtsi,I,·of(<!.I).

giventopograp hyandabove hY '1rigi,l lidasshownillllgum :!.1.Attl ,. ~1l1 111,~riUlll lower boundariesiuvlscld hOlllltlarycondl tlens [nolIurnmllluw)arl'IIs( ..I.TIII~now isforcedbyspecifying

0,

at1I1(~leftbeuudnry, AlloutflowIHllllldary "IJlldi liulIis specifiedat theright boundary(~l~~BdlkMarcus[1!J!l21,Lamb[1!1!I:1a,b)),

(28)

' L

x

Figlln, 2. 1:TIll'sketchorphysicaldomaln.

jl·t~li(JlImethodtlevdopetlbyBdlrIaf.[1989 11,1,]andBell&Milfl'llSIID9::!].TIlt' llll'u ryof1.I1l'un-tho.lis111ln('(1 onthefollo wing:

i.Ho dgedecom posit.ion: AllYvecto r£ii'llI

II

callIll' decomposedintoadivl' rp;Plln1 fr PI ' l' tl l1l PUIII 'III,;\11<1ap;ratli"lltpart,i.o.

(2.4)

wln-n-

\7.

(id ::::0and 'Pis asCHI,~rfield.This decompositionis uniquegiven appro- Itria!.!· Imlillthll'yromlitions for\i.'^[e.g.,

Vo( .

Ii

=

0).

Byn,wril.ingtil('momentum equation(2.1)as

0, +

^'Qp=

V;::;; -( 0 .'(7)0-

pg, _(2.5J our- s,...,:!thnt,wlthslIit ahlt'bouuduryconditions1111I1bCCllU1<C ofequa tion (2.:J,

tit

is

(29)

ccrrcspoudiug gradit'liLpart,

ii. Or th o g onali t y : For nny\'('('lur

P

,111(1sn dilr<.p,WI'11<1\'1'

where11 lsthoeompututi onnldUlllain,iJIIilshlllllillilry tun]fistilt'(·j,'l1WU1.1"lIl-:lll ofthebonndury,IIIpart ic ula r,fur11c1iVt'rp;(·IIt'I,. fn~ ,iio/,WI'!Jav,·

llll~do main.

boundar y conditionissub tr actedfro m lh,'mouu-utum1"I"a1.iuII,(1.!i) rl'dll<"t'S I,ll

where

0/ =Or - O rB

has zeroDirid rldImllll,larynUl<litiOll.A"...mli ll~^tu^ti^ll'al ", vI' ,

ill

is orthogonallo

V ",

iii.Proje cti on : Theprcjcctlon is(ldim'l!I,ys[lI'dfyillp;itha~is"fdiVl·rW'IJI:I,·fn·(· vectors

iii"

withzeroDirichlctboundary,Thenl~"isorllrllWllIalto

V,I.

I'rum(:l.K)

10

(30)

11'"hl'V!'

''Ullrlil,iutis fur1I11'PI'l 'S,~lI rl'ure not rr'flllil'l·dillsolvingthisproblem.

(1. JO)

SIlI,sl,itu l.ill,l!;int ..(1.!1)p;iws alil11'lIl'SyS1.I·1l1 with~,'·(lllilt.iollsandI.'unknown(l',S,

i.e.,

withIIp,..iu,l!;InunIl,uLTllPr(oforr',(urilll)'known

VI,

IVI'nl11tlptl" l'lllilU'llw(IN,' sHnd rllnsl,(!'lr'lIllytln-Ili\'('r,l!;I'lU'l'(tH'l'( 111 )JUIII' IlL

ti/.

Thisis whatIlwSQrnlh-d projor-rlon Ld."rl"lw1c'this I'l"Uj(·"t ilJllOpl'l'1lt.ilJ1L,TIll' 1I'I'P('XlJl'('SStill'llwjl'dilJ!Ias

(2.12 )

OUl'iutcrlordi\'t'rp;I'lIr''' frr'(·velocityflckl

0/

is ulJta illl'(lh)'Pl"Ojt'(~lill~

Ol-lir.

II

(31)

lu tlu- follu\\"inp;,il],1"il'fnl'p1"\"il'\\'

"r

I,ll<' mdhudis .l!;in·n.Ollt' slllluldnlllsllh the above mcutlcuedpapPI's furmol'!',It,tails,

2.1.1 Gl'id scheme

the'mostnatural sdtilll;f01" 1I11'h'HlCllin!l:efI,ht, nml"'t,til'C'u-nusintilt's,l'sl"Ill,

tl'fl

nne!

9"

il1""tl.'lilwdattill'\'{'t't u l'/o\ri,1I'" inh 1I'11it'11in<'llld,·tlll'illl,'ri"r vectorgrid points !ot'lIl"dat ""lIn 'lIl re'smullli,'IItJlllltlar,\,\'{','lor1!:1'id lu,inl,s ut.Iln- midpoints ofcelll'(Ip;(':IIyill,!!;,clO1l1;tIll' hounduries. '1'111'lutu-rIItt' liSt'",I,ll spl't'ify bo unda rycoudirions .'I'll(' srnlnr p;ritlpuillts ,II'"10calc',1al."..II C'tJrllt'I'S1I'11t'1'"s,',)I"r Ik-lds(Sill'llliS

9 'tl

andtllC' \"dd ill"d lat"r)MI'ddi,Il"1.'1'111'"X'·"l'l.itJlI tot.Ilis is "

whirllisp;i\'l'lI at llw Vc'l'lt!r p;riclp<lillls,

2.1. 2 Transforma tion

do mainill(;r,::)Sp,I('"i.~IllilP]Jc,d0111.0ilrc·'~1.a llp;ll la1"1'll1lllJlll.atilHl'l1 rluruniu ill(~,ll) fip" t'"I'll)thntunyp;i\'l'lIIIIHHlrilal l'ra[ p;l'i,1I'dlill(x,::)lW"ollll's«uultsquan-1'1,11in (e,ll),Thismakl'stIll' 11\(j,11'I1I10r('IlI'xi!>lc'11l1ll1ll it c:i1l1ta kc' any/.!,rid ulI'sh Jl;iv"11ill

I~

(32)

. . _. _r-

. " 1"--' .1"-- .1'--

" . ^{1'\ •}

,wL ^-, ^. ^i"-

x.u

\

<>lnlel lor veclor gr ld polnls

\"

• bound ary vector grId polnl!ll

•IcalargrIdpolntll

Figll1'l' 2,2;Grid sehr-nu- wit h quadrilatera l gridcellsin(x, z)spare.

C.I',z)uud,Itlthe-cak-ulatlouintill'slimeway011 theHllitsquawgridmeshill(!,I/).

Thisis,1,,"1'hyitCl>Urdillilll'transformation

X((,'/ ),

'1'111'mrr t·sp nmling.Jilf nhillllistld ilWdas

(2.1:') (2.1-')

(2,1')

(33)

From (2.1:1)illld(2.1-1)\\"(·ha\·l·

( ~

~,

^"

'I<

^' ⁾ ⁼ ^:7 ⁽

-

^,.

,1'"

^-")

,1'(

⁼ ⁵ ^.

whc'w7'i~tll'llllc..1as

T=

(~:, -~. ) .

J(:"II(

+

,1'(1'. -r"ll"-=( "( )

J ^V. ^TO ^.

{:Ulil

(1.17)

(:!.IX) (1.1!1)

ii,

+ }(l/.

^v)ii -VI'-

P!f,

f' ,+J, J.v,.

0,

v .pi

0,

w!n'fl'V E( iJ/ iJ(, iJ/ihl);\I111

{j= '1'ii .

1:J..1U)

IlnOSSlIfC grmlil'lIlisn."(~11'l.1inlht~(Jc]r:lllalioll.

(34)

Only tlwl'lIurllilllll l'Surthe dualgrillin(x , z)space need tohe provided.The 1~l'llro]Jrilll,I ' dirrl~rt'llfl"~uftIll' dunlgrilll~ullf(l il1aL('surt'used1,0evaluate theclements

"r1.II"T I1I1,lrixllll .ltlu'.e;rilll' " inlrllor.lilllLlCsor«(,I/) . Thill cllsllf('l;thlLt ll )('ll wt hod IIn's"rVl'SItunifnrtu flow(Sl 'f 'Hi'llrId. [i!)89b]).

2.1.3 Nu me rical Algor ithm

T"s"l l/"t.llt~llhoVI~SYllt"I1I, III" rewrit eLhe'momentum {'<illation(2.20)inthe' 111'1'0 111 - positlonformliliin (:.J..."i):

wlll'rt'

AITlltding 1.0till'theory, (:.J..:.J.!'I)means that the uon-dlvcrgeuce-Ireevector

V-

or

isch-composr-dhy UIl'prujt'd ionintoIIdivergen ce-free compone nt

0

1 -

Ot

^Jwhich has1.l'ro J)irid lldboundaryrondltlou, anditscalargradi"ll tpart

VI ',

Thedesired uptla1.ing l'l'll/dtylil·ldis1Ill'1Irr-oovcredhy addingbarkthebound aryfield

lilH .

The urlh og<>111l1propvrf.yortI...projcct lonPeliminatesthe pressure entlrcly fromtile

15

(35)

syste m,andthecquntlons<ttl'iI\ Lt'~ralt',1ilS IIwh...il.,Vnvoluf.iuu s,I'SI"lllwithi nIIIl' spaceofadiwrgellc ,·· fn'l' V('!'lurfh-ld.Thist,h''Ill1'l,inlll.l','xpl .,i ll,' wIlYII"!lm ll ltla!')' cond itio nis 1I('I,t1t'd furt.1ll'ptC'l!su rl' .

Dlrecrlydis r.tdi:-.ing(2.2!i)IISi ll~'lsoroud-onk-rdilf,'n'llt'in~bl'o"...i1,lc'.lollt.lilt' l'f'l'llllti llglinearayst cruispoor lyroudlthun-d(:<1'1'Ilc'lId nl.[HJ:\!)IJ, 'l'lwn-ful'l',till' following il.pproa chisnsodLutn'altill'nnnlim-arfOIlVt'l'livI' b'n lls,

111ordertilhave as.,Jll'Il 11'whk-his s"" o nd·tlnl"rinlillll'WI'di"ITt'Lhl'(:!,:!r,) llsingUcenteredLillII'Ililfc·rt' IWillP;,TIlt'uouliueart'(>II\'t 't'livl ~1,1'I'I1IS attlw hillfHuH' stop11

+t

an~cakulat.t'tlll sili gill!explir-itsl't~lm,I·(J1'tI.,retlllll lltlVprot'l'tln n'falIllH ] ' HiedPil'ce wis ('Pa r abolicMt'Lhud)dl'scril ll't!bylit'll"/Ill,[1!IK!I Il],VuluesiltLln- mid pointsofthorelledp;t~at t!lt· llllirthnok-vol111'1'I'Vahliltt'tlhy 'I';'yl"r C'Xllil.lISiuliS

or

lhe knownva l uesilttlwcellrt~lltrt'Satt.imt' II.Th«l.il!1I' '!c't iva li vl' s IIp l' .."ri n~

illtheTaylor expansions arethen dilll ill,ttetlbyIlsin gtlu-go wr lli llg('IIUitl,iu lI,thus timelaggl'llprc ~~Slltcgrudif'lIhilrt~intnl1ll1n'dIIl~I1'(thisistlll~onlypl1ll'\'wln-n-till' pressur egriltlimlt isllsl~,I).TWf!kindsof xpafiu l,It"'iviltiw silPlll'HrilltlU'l'I'lilllling expressions:normala.ml transverset11'ri v;~tivl ~S(wit hl'l'SJ)(,d to llll'n,111"I.l:c·s). TJ,,' nor malderiva tives arcevaluatedIIsill~mouutonieity-Hnriu-dt'l'lIl raldiJf('n 'lIn',~which preve n tintroducingIW\Vmax ima orminimaililu tIll'Howlicld,whilt ~1I1\'tr;lllSVt'rSf~

deriv a tiv esarecwdlla tet! via amodiliodIlilwiutlSd Wllll'.For ..;wl! "I'll"II/.('·L1l1'rt~MI' lfj

(36)

two.,dl!~!~lltfl'sto !'xt rapulat!'from.TII~llj)pro priawchoiceisthecellcent reupwind

tr,i.~prun'dll rn isshownlwluwfurtimterm[0,VlI)u+i (t hetra nsfonu att onisdropp ed 11" ro',sill!'!'itintroducesonlyroelliciouf.sandmakesnodifference10thealgorithm,It sluml.1alsoIll'nutI'dthatri"+i ,which istheextrapolation from the divergence-free vd l).:il.yfid.l'lt tillll'IIandinvolveslaggedpressure, isnotdi vl'rgl'll~e- frce),

(Ii 'VII)"+ ~

4(u:~tJ ⁺ u :':L> lI~l.j ^:;r.U;':!J ⁺ ⁴ ^(u,;::1! ⁺ III;~:!)1I:~}~~Z":;-\2':l(i)

Not.,I,hutl.ll ( ~!'!'ntf/' d.liffe rellringlls,'d hereindicates thesecondorderaccuracyin sparl',

EXlrapulati ngFrom

ti ::

j(Rl.,('flguro 2,3)givesfour mid-edged,halftimeval-

ill'Sfor1'1I1'h

0: ;;,

^which^an,^denoted^by^L,

n,

B,T(forleft,right ,botto mandtop,

ro'sl' !'rli Vt'ly),Fur illstallt'(', extrapolat ingfromthelefl,

ll; :!jL

isgivenby

II I; + ~1I~,;j ^+TIt;:;j

IIi ; + 7 "~,ij + -¥( -

^It^{ij t':}^,;j^-

w;jtl~,ij - II ;, :} )

17

(37)

TIl('governing equationforIIhils1J{'(~ 11used to l'1i11lhHlt"Llu-Lilllt'IIt'rivOiLiw. This introduc es thei.inwlaggrdpressuregl'1uli"lll.

rrcatodin dilfcrcutwaysasmcnt.loucd bofon-.D"tails willnutIll'giVl'1IILI'I'l',1tc-lld,'l's shouldrefer1.0HI'IIrl aI,11!1~fJa,hl,Thopurpnseofilll'1.rt'lllllH'lll!<isttlpreve-nt introducingspuriousoscillatio nstil'iustahllities,evr-n fur,l isnJll l iliIlUl I.~data .'rllt, normaltll'l'ivai iwsinvolvecl'nlmlllirf,'rt'lwillg,which1ll!;iliuIt',lIlslu lilt,St~'U1l tltJ1'l[t'r accurucyinspan',

extrapolatedmid-edgevalueshyil ll ~following:

lI:~t:t ~

⁰^all,1

ll::tt;, + 1I::l:t

^2:^II

lI::fj"

^<⁰ami

ll:: tj

^u^>ⁿ ^(l.:!X)

ot herwise

w;~l:/'

,,+!,II

wi+tJ 1{w~:lf +rJ!:~lr)

if

1I:~t > 1l

if

1<~ttJ

^<⁰ if

n:~ttJ=1l

(l .:.!!!)

(38)

"' /:1.l ..Ir..~

U;.lJti U,.lJ2,i

.. '/:I.T U",'/:1 ..,/:1."

U",'/2

li';'11

o

.. ,n.l ..1/:1.~

U" 1/:IJ U;"'i:lJ 0t;"' .Il

Fip;nrl' 2.:J:

I. t!l' IIUL!'Htllt~l~x trnpol/ll;ol1of(Ilii .

vi i)

to theId tslde or edge(i

+

t,j) ; B,It' lllIh 'Htill'extrupolnt.ionof(lli~',"1i)tothe right sideorNtg!." (i-t,j );

IItl"ll ol('l'til\!ext.rupolntlonof(1l:~ ,"1i)tcthebottomside of edge(i,j

+

t);

T,J,'l1n lc'lI1I11'c'xtr;lpo!llt iollof(1I1j, 1!;j) 10tiletopsidt,or f'dge(i,j

-!).

19

(39)

trarer. If

lI;~t.j

^>0,

w ::l

Jisunl)'alf,'r ll'dbyUIUs\' vnhn-s cu lIu'Il'fI.

~i.h' or

1.11l' edg es.Thus, itisna t ura ltosdit equaltow;~f:/'whb-his,'xLr;!JlUlilh'dfrnm 1I1l' leftside.

Bythesubstit ut ionofthooluaim-dmid-,'d,l!phnlf-tlnu-vaILit'si1l1.H(1.:W),Lilt'

similar mnnncr.

Thevel ocity,densityand pressure art' then1IpdaL('11bysulviu).(1.l1t'fultowill).(

tlmeforward ing"<jllations p"'+l_p"

---;s:t 0"+ 1_0,,

- ,,-, .-

- J{I ;. _~p)"+~

(<!.:lll)

I'(V"+~

- 0/

¹)

+O r

p

(..!.( {] . ~{])"+~

^_^{p" +1-}^(I"^Ii^_^/ill)

+

/ill (:!.:ll)

J 2 . I I

(1-f,)(v"+t-or). (2.:12)

Since cen t red timedilf('rcllcingis used,till'sdlt'lIll'isills"st'," JIld urtll'l"HlTIlI·i\l..in time .Theflowfieldlnflueuccs thedl!lIsily flektthrllll).(hlul wditlll.unrltil,'ill flm'n",',l tll't1sityfieldfl'Cdsbackintolll.. vd ucityfield. Thisf"OllpJill).(is rd b:1.I'11in (l.:Ul) ami(2.:11).Sillwthehydrost atic upproxjural.iunhll." !lot101'('11 rlHtlJ", 1Ill' I'rI'SSllrt'is affectedbyboththevelocityanddensityilS(~Xprt~sS('din(:!.:tl).

Theprojectionis1.11f'1Idiscrt'lizl'd,whidlluvolv us solv ill).(IIlim'al'(!flllatiw i

(40)

systemIor1I11'IT".'s,1'ifiru',1 ill(2. [I).TIll'asso dall,dIl1lsis of dlscretiz cd dive rgence-

whereIt~0l'SIrom[ to1(.1-1),:,<" isth,~basisor diserctlzedscalar fieldswithzero Dirh.hlel.IIulintiary,whichhl~~a constantvalue on0111'scalarpointinde xed by11and

'I'hus,lIw disr.rdizt'llprcjcctloucanbe rewritten as

l'iJ= .'i ,

wlu-reI'isUlI'projectionmat rix given hy1'",,,=(,pll',,pll)",ifrepresentsl(.l- I) 111lknUIVIIS,1I1111

R

is(,pOl,

V,,+t - Ot')",

(Ikatlers shouldr('re rto Bellrl al.[1989 a,h]

I'isaposif.i ve, symmetr ir- matrix whichisblockteidiagoual. (2.34)is solved ll.~il\~11 st all'[iu't!hlm'ktridi agonelsolver(C olub&VanLoan(19H91l.Notethatthe ['matrixis fixl,t1 once enlculated,siuecthebasis vec to rs arc Independentortime.

TIll'lly stl'mis1I11{[lI.t('dhy(2. 30) 10 (2,32) ,Fortheexplicit Goduuov scheme IIsl',I,allnear cunstuut-coelliciont analysis sho ws thatfersta bility,thetime ste pmust

21

(41)

satisfy aCFtronditlon[Courunt-Ftiedrichs-Lewy eondulon},i.e-.

where,~isitsafetyfactortakon11~O,."i.TI Il~li1.ilflll·SS orUll',I1;l"'lvil,nl,iolllllfon'inp;1," 1'111 requiresa mornresrrlc rivetimestep,IIIpr111'tin ',lor uuruonlim-ar 1110d,·1,./Ii,~tnkx-u considerablysmallerthantheV11hll'givenhytill'sl,melllniCFL,'slima1.I'.

2.2 D omain geometry a n d b a ck gro u nd state

2.2.1 Domaingeometry

Thetwodimeusloualphysk ill domai n,,llIshownill lip;lll"l'1.1.is,'ulllin,·,I I,yill'ip;iol upperboun dary at ::

=

/Iand 1.han kI'd,!?;,· topup;raphy at ::

=

h{:r).'I'lu-l.auk ",lA"

i:tshuulatedbyIthyperbolicflludionp;iv,'lI by

(1.:l/i)

where(IIishalfthe height of11whanktop,atis a slupI' parll11 w1.I'I· alllifl:1islilt'""IlLI'I' ofthebank I·dge.::~is llil'Il<'iglitahuv'lwhichtill'Iluldissip;lIilif'lll11.ly,~lrnlili,,,I,

ForSll('11a system,U1(~nmlmllingIlltrll1rwkr,~I,.n~ 1111~gl'lIllll'lril'l"1l1il'fl l l". rho bank slope(It,amitherlyuamlealpllrill111't" 1"

r:

=1I,1J/ fl 1(0 is11ll'S'IIHrrt·uf 1.111'

(42)

;nVl ~r~c~ Fnmdl ~numberforSGW~. It liM no diroct rolevancoto supcrcrlt.iealityor suhcl'i tkalityoflOWs),whichwllluppnarinthenon-dimcnsioualequerionsystem (~l"'~dmlllt' r a),wherelJistilt'velocity sealo.Fe r fixedrati os and ditTen' lItUlind 11, if(fis tlwsault',Un'flowpattnruwillbtllilt' S1U1W butonII.differenttimescale,

IIIour runs, tlwgt"Jl1lcl ricpnrumetcrs thems elves areusuall yfixedasH= :lOUm , [.

=

HOlull,z~

=

:l60m,III

=

~20rn,fl1

=

0.000375andQ;)

=

0.25 ..L.The onlyparumeter controllin gthoflowis thedynamical

a,

whichdepends011Uonly (Ili n n~Ifi.~Hxcd} .Therefore,ditTc f('nlUwillres ultindiffer e nt flowpatte rns .

Ourprimar ypurpcae ill~pl·t~iryinga bank geometryis togenurateau initialwave throug h1I0w-topo~r<q)hyiutornctlon. Ther efore,whenthebank can notproduce a tlt'sin,1I init inlWUVl~,wewilllise aflalbottomandartificiallyspeciry an init ia l WIWt1.

2.2. 2 Backgr ound state

Twokillt[s orcll' lIsitypro lif.,sart'usedillthepresentstudy.POl'convonieucc,instead or$!;ivi ll!!:tilt' dl'llsityprofiles ,weprt~I' l1Lbelowtheassociatedslrat ificat ionsdefined by

N'(z)= -p,g. (2.37)

(43)

The firstlsgivenby1~hypcrholir-fUlld iull<If1.1u·Iorm

where1/is1\slopeparanu-tcruudNl~istIlt'cliaral'tl'ristir\'Hllil'oflllt'slwl,ilit"lt.ioll whichis setto0.002/.~:l .:~istlw Ill'ip;htuhovewhn-htill'Iluhlis t<i,l(uilin lllt.lyst.-,ll.ilil·t!.

Whl>nq--+00,Nl(:)lLJlIHOitdwsit stcpwbn- fundi"ll.Fip;IIw2..l (<t,)sl",\\·,;Nf(: ) witha~lIIlLlIIf=0.007 umla,l,lIW''1:010.T1It·llIUlI.,1 it< r-au f"r1",1.11'/ I'llhll'.~.Tilt·

1Il0l1l'-OllP'harodinkph,ISI'sp,,,·,]ilsSt>l~;at,'. 1with Nf(:)is;,Io"ul, '.!..rltll/sf"r1",UI'I va lues,whilethemode-twospl ..~dis11lmnLn.!l.'lm/ s.

Secondly,weconsider aIll~nt<i lysLralilimlillll ofL111' fo rm

{

O.OO;H :;-11+ 10 1I.!I(: -I/+XIl}

!1 ----:ifi'·XII --:1.'l-

-;-ri ⁺

1<_:':~~~lI)'ll

+

0,0' (, -/I

+ ~OO ) }

II + l:-':u";,IKll' r '

('.!..:J!l)

whichis showninfigllTl~2.'1(h).TIll' mOlII',ull(' ],'ITodilik"hllsc'spl'l'lli1.~stldl,tt't l withNi(:)is abo utI mis,nndlIwmod c·-twuph,~..r-sl'c'C'c1is ,,1">lItll..'lill/s.

TheselwosLralificatiOlIprulilc·s al' ]Jruxilllat,·O],SC'I'I'I',J.1'·l1sit.yp.-uril,·s fill UIt·

SeotlanSluM(N l ( z) withsma llq)aml atthr-''lIW'of<:t'orw~Hank (Nf(::J) wln-n-

LodC1T d1/1.(1992111.", 1 LaVi()ldl.~rlnt.(l!J!IO]).

(44)

U.(JO:\r--.---,---,---r-~-~--,..,

~lJ 'J' wj...

'{.,=

10.0 ) -

".u,,," l h{'I= O.IJ7),

,100 .lr,O ::(111)

'00

,, 0

_ll.IlII 1 1

L,-~-~-~~-~-~-~-~

rsu 2UO asu ,IOU :li,O

::(m) 100

.(l. 0I1[L-_~_~_~_~_~_~_ _LJ

u

Fi,l\II1"t,~."1 :TIlt'dl'II,.Hoys1.riltilic/ll,iuIISused. (n), St,'pWiSt,Hil t!smoothNl(::).till'

\";,1II\'s..f'l arr-iudicat e'II Il,\'k.·ys.[h].N](::) .

(45)

Bt'lii,lt'liIht·tlt ·lIsity sltlllifkil linlls.l l1t'lIui,llIn 'tlsIIIlu-drivenfrom I"<",tlit'tl1111 IG\V"rnuh,'r;('1I1'ralNItlLmllJ!;hIII.,illl"n" 'li""1,,·1\\·'...·11lilt'[lowi1Iull" I''' ,I!;fi11,1,.I'.

Thisis/U'lIit'\,I'tl ll)'illilin]ii\ill~till'mo.h-lwh.h11unlronnl'iJ!;h lll'i1nlIh,l\'1/in1111' shallowIInln-giun.TIll'1'11I]llilll,I,·ortill',I!;'·I11'r'It!·.1IUW",lqll'l1Ilsuu1/,,,,,IL]u- ha nk~I()p{'gjvcnhy"l '{lll...r~1'''lIld ri,'l'i1tiltlwlt·1'll im'I"' \'('rdliln~....1.

TIll' lll<'oryll..riv,'t! in1111'III'xldlil]ll<·t is ...'>;I·lIli" llyfutl(a VswilhslIIill;I..

1Il001,·ra l '· llllll'l il l1l1... W,· wanll..k"uwIIUWf/ltlIM'llll....ry nl1l lo.·.·xLo·l"l,'t!fOlrna vs wit hlat~l·tiu uplitll.lt·.TllC'rdufI',lilt' 11111.11·1isrllnfur lion...·.Iilfl·n 'nln \VIIii'll

'.m.]...·

Slll111l.m'HI,·rllll·i111111i1tp"· U;W,,. Sl','t,ilir'ully,II i"l...ll"1l.·1t7I11j".1l$ l:IIl,j,,"", I n.flfl!imj s,amltill'flJ1Tl'Slll lI"lillP;runswilli",n'fl'rrt,.I I,lli1S1,]11'\\",,,1.:,1I1",I.·r1l1.l'a l"]

strong fon'inp;l'ilSl'S,tl'SI,,·,'livo-ly,

2.2. 3 Runningst ra te gy

In onh,tto uhlai"tILl'd.'Sin'tlIlowl'all<·tI1sintill'fix' '11,!ulrli,illwilililla "Iw.-ili,'tl t'VOllllitlll lin.... !<IJlII" I'afal1l1'l....1<SOlO·" i1Sllll'1>iI,-k" n" ,,"! v"ltol·jlyIf.II,,'l'i1l1k"IOIH' Ill .,ll'l"(·lt·t illjUIIliulI',"ll"."It \' ( ·tilIH'it.lljll~ tt'll.

Wlu-niij~latp;l', lh,' lIt/wUll til l' lUI'!If1.111'ImukllIi~111.11l''''>l llf''~"f" ,,..'til.it'l".

(46)

t',Ip;I~.S'lfll11.situatiollwille1l.11SI~st ro ngoVI,rlu flii ngwhiehresultsill largedropsill llwtlun- sl.t·JIawlIIllilllitt d y in large1Ill1llCritnl errors , To avoid tIlis, wedl~r,dt~mlt·

1.11<'IIl1i<lllflt'rlIlt'lO Wsnn~l;elll~ rl\tt', 1andlliewnvefrollthas becometll'l achedIrom tI",LC>II"Ar~l phy. ThisIins110dfl'd ontill'sllhsl>qllClltevolution of the wave fro nt.

TIlt'It'r1lli'm lvnlill'sufIIforthothree forcing casesnrc listedinTable'1.1.

Furlilt'slrflll~fOI'l:ill,t?;rilse,since the /?:cllera tcd initia l depre ssionlias11large amplil.urk-,tIll'neulinenreffectwillquicklysteepenthewave front,th usresu lt ing in L111~tlt ~vd oPIlll'lltof annndulnr horn befon' thefro ntranIlrop ilgillt·away fromthe hank.Wha tWI'IWI'(Ii,~ilsmo ot hlnitialdeprrsslcn with onlya mode-onewave which i,s c1dlldu'(!fromthehanksotllnt th..theor y mil beapplied toitandits theoretical

"v"l lItiu lit'IUII...SI'(' 11eh-nrly.This('lUI],e doneby using a sm aller bunksl o p(~111' TIIC'II 1Ill'AI·IU'rit1.l'ddt'pn 'S.sio ll will hav easmootherfrontaudpropag a t eawayIrom llll·I ,;lllklu.fun'llt' vI'lupin grlppb-s.

2.3 Running t he Model

Usually,t.lu-modelisrun fo r a timeleng th ofabo utT=16000s [app roximatelyfour und ha lftla}'s) .1.11l'uutput(t.Ilt'velocityfield,LIII' lll'nliityfieldandthepressu refield]

is lil,un 'tl nl. un inl,t'1'\'11 1

or

1(j00.~ami lslabeled hyoutput number [ to10.Figur n 2.5-

27

(47)

2.8areS01l1l'uftill'f1111r('snltswithbackgroundlit.ilt.C·S iIHlirllh',jill!.Ill"fapl,iuTls.,\

11111lf('.~s ionin isopycnulsintIlt' ,lc't'!l na tl'c'p;ioll I'vulv,' ililu 1\UIIn,llIla r hurt, al,un1 100008la ter.Noteth1ltonlytill'donsitylil'lllof tilt'uul,pllli1< l'rt'sc'II1c',I,1<;11...·tlu- theoryde rived later willh,'rurnp an-dwithtIll'rh-uslty1ll'l"llll'h<l 1.iolisIIllly,

The cede I(lWsim wasoriginallywrittenltl ;lIws1.i,!!;att· til,·W1lWSP;C'lU'l'illt'd 1,.1"

tidal nowacross 11ha nkI't.lp;c'.BI'("lIlSl'of th;1<itW1l1<st1'1li,l!;IMu1'wanltu!(c'IIl'ra l" H {lc'pl't'1<S;OUvia thismochaulmn.La t,,1' 011, for runs withtlll'sll'iltirin ll.iollNil::),il wasfo unt!thatitWllShardto p;c'lIl'l·al,'all iuit.inl ,I"pr,'SsilJut.hal.W<lSs!.'~· 1'"11" 111-\11 to formIlIInndularhOl'ehc fn rt~itIdllhc~ c~"ll1plll.llliulilll.huuuiu.This,lilli" ull.yWill' partially,hll~to 81rcng u\,c'rlllrlli up;at.till'hankc·,I,!!;I·,'I'ln-n-Inn-,rurtill'nilIS with Ni(::),!.I1einitial ,]"prl'lisiuuis uul p;c'III'ral.,',1byHowOV"l"11 IOitllk ",I,!!;".Ins\.,·,ul,wr simply illitill.li;wtill'model witha I1lmlc'·ollt, lin"ilr IVilVI'in alI'd.hu ll ulil.knnuiuloy thl'111"1'oftilt' first 01"<1,,1'lH'rturhatiou'!l'fi v" dlater.

(48)

2" 10

• em)

Co/'lI"y, r."m-O.O'O .Ofl2 1,, 1l.IlJ7 57m-O,[,b y O.OO1J &717

Figll rl~2.!i:Densitycontours for smoothN?(z )andmoderate forcing. Theupper one

iNlltout put " undt111~lowerisat output10.

29

(49)

2X10

•(m)

Contoucl rom _0.0 2040112 to0.000 00 hyO.0 0IJ60r.4

~x10 .(",j

Contour"0'"_0.0 2040112 to 0.00000 hyOlI01360!'>~

Figure 2.6: Density conto ursforsmoot hNf(z)uudSt.roll~rurr_iIlK.Tl lf~uppr-rflllf~is at outpu t'" and theloweris atoutput !J.

:10

(50)

. (

...

⁾

Co"' o~ rlro...O.1I22 ~1116Ie>11.11212665by0.0003130611

:E11101---~V

_{...l

Conl"",'..,."O.012 ll71161e>O.0212665DyO.000313Cl60

I·'igllrt'2,7:Densi tycontoursforN:f(::)andstrongforcing.The upperoneis at output B 1I11, I 1.lit, lowerisat output 13.

31

(51)

o Con!"","tI;m e1? f100 .0

- -- - -

"--~~."

~.:~

_, .~\;;. ",...-~~-:--~~~,;:o.,;,. "

..

~~,;;;;c~...J

~(m)

Con'o'""Om1I.0n~706'oO.0l'n66!1h10,1XI0Jl.IOIIO

figu re2.8: Densitycont o ursforN.:(z )iLndsttull .!!;fut dllP;,witl lruln tiult il1o:1l1, I",1.

The up peroneis al output8and lll.,low, ~risILl.outplltla,

(52)

Chap ter 3

Nonlinear Theory of Long Internal Gravity Waves

3.1 I n tro duc t ion

Tltt,nonlinear tllt'ury of IGWsisall I'X1I'I1Si0I1ofthesllrract~gravity wave(denoted

;(1';S( :Wshl'I"('"fh-r)Ihcuty,IIItlw dassk<tlprobl emsccuceming SGWs,the Iluld is

hUIlIU/-;I'lIt'Ullliandinccmpresslhlcundthe motionilltakeutobeirrot a ton al.011these hasi.' assumptlons,two distilldtheorieshave beendeveloped:

(A).Allinflnitcshualamplitudetheorywhich leadstothelinearization of Euler's 1'<1l1i1liuliSandresults in1i111',~rdispersiveSGWs.Thistheoryisthezerothterm (or

(53)

the zeroth order solution)inanilSy ml>tot i,'r-xpunsionillPUII'I'r"ufLjn-p.'rLllr],;diull parnmo tcrf

=

(111-1withf1and1/(!<'Ilotillj!;LIll'1\',1\' "umpliuuk -,m,lUlI'lIIuli,;1.url" ',1 dept h oftill'Iluld.TI1l'theory WilS alsut'xtc'ud",1ttllinit" alllp lil l1t!<'I\'a\'l',;loy!.Ill' perturba tionmethod. Tilt!ba•sloidrnisL1111 t. tIll'!l i~llt'runl,'rl.t'rI11S ill1,11l','xl,a llsiuli can be regarded ascorwetionL,' rm sto til";wruth onk-r lim-ursU!lltiul1 fora"il.l1ill ill ll when-finiteillllplit lltl" 1\'aVt'S tllTIIl"andtill'1II1I1Iilll'<ll"it)'ruullu1.lll'111'~h ,." l t 't1.

(B),1\long-wave(01"shallowwater]tlu-oryfurIV1lVt'Swhk-hun-IUILJ':nlHljlllrt' t l 1,0 theundisturbeddep th oftill'fluid.Thlstheo ry is Ionst',1upoutilt'SHlilll lU'ssuf tin' contr ollingscale pa rnmct er11=IPII}wlll' l'l'IJis tilt'l.Ylli,..dwiIWlt'lIp;l h.111rlu-limit II--t0thonow becomeshyd rusta tir.alldl.ht~pt'rt Hrh 'l1.iulitl'l,w titil's an ';ntlt'IU'II,lt'nt ofdepth,'I'heresultisa nonlinea rshallowwat er ('(jll"ti tlllSyst.'III. In this tlu-ury,it isroundthat allinitialdistllrha ll t't~ll'lillsto 1)l'tlis1.(Jr LI ~ 1andIm'l,k ,aIlllt'lIlJlll t'lItlll absent in the1il/CMWiWP.theory.

Thetwotheoriesca n be combined.This !I'a tlstnuuuliuour,,lisllt 'rsivl: waws rOI'ahOlllOgC IJ('O IlSlluid.

Itiswellknown,throughtilt'r)itJ1lI~'ri u~wor k of}ko1.1. Illlss"JI [Pl:17,INtHl, n,Lyieigh[18761,Stokes[HH7,IIlHOj,and J(o rl l'wt,p;all.1 r!"Vri,Oj;IIN!J.'ij,t11i11 lUll,!!;

nonlinearSG \ Vl;illahOlllogl'IINJIlS sha llowfluidrl('pt~I!,1nut:ial1y(Ill twoplU·;IIIl, 'l. ~rs ,

(54)

,lumina ll'silll,l tlll~waVI~Stl'ndtu suepon andbrea k, wh ile iff//I

«:

I,thedispursl vo lolfl'd ,11IIlIilJa lf'S lin dtill!WilVllS111ndtH1)(1 linear dispersive nues ,Whcll

fill -

Iti lt, twoUI'II<I.~itf'drrds tf'lldtilcmwdl~adilltlll'r.Inthisr.a.~wIId.~~ sofnnitpamplitllde wuveI'rufilt\SufPI'TIlH\l Il~nLfonncanhr-ohtai llC'd,l'.g.,solita ryorcnoidalwaves.

Till'ahflllf~tlll'ori('Sr'uuIwl'x ll~ll(l\'dto verticallytraJlJwdwuvrsinslmtin(1d .~lll'iLrHows.Ltlll,!!; [IWIG,Em:),1!172] liasshowntha tlOll;.:';non linearIC:W.~art'possible ill a10011l1l1l'dsl m W i('dfllI;11.BI'llj alllill[1906]fcuud thalpenuaneutIG Wsare also )l,~sih ll'illHhl'l~rHewswithslTlttil iC'al ;oll.Henney[196fi] extendedBenjamin'swork ttllneludr-LIII' lillll'(ll1Ilt' lldf'IIL IlTolll'rli,>s hyintroducing atwo-parruuet er(fand II) l'xpall sitlllIlwllitHl. BythismdllOd, hederivedtheI{,IVoquatlon whichP;0IlI' I"tLStill' 111111;ll"nl;Uf'iLr1t:\VllUll;sopyr n a k 1..1'11&Bcerdalcy(19741furtllcrdl'v e!o)ll'(1thi s Ilwtl l(ull,y;Udllll; lIp; inthe pl'rlllrhatio nexpansionRthi rdparameterIIlCILSUrillgthe UUlI-BOIlSS;lIt'lllll'lr('(·t,101111UIIiSohlai uediLI\dV typl'l'lllIat ionwhich ;nvolllf'sthree smllllpUmJlIt'LI'I1l.AIllUdWI'(II\(IVwitha soecndorder-lt~l'millf(acubickrill)was

;JI.'lt.1,It'ri w cl ;1It1lt'irpaper,inUlI'(:a.'>l'or IGWs wit h large amplitudes.

III1.lIi1lM'f1.iuJJ,IUlIgnonlinearle ws011lsopycnals in alllnviscid,stretificd, B(llI s.~illt'S( 1tluldwithshear1101\'confined by rigidboundariesare iuvestlga ted .By till'llll'LhtJrI of two-puramctcrperturbation expansion de ve lopedbyBenney,LeI'and Ik ilrds!,'y,amodlflr-dK,IV(Inl\IIV)('fl\lllt iun withthecubicwnuisrederived. TIJis

(55)

profiles with la rgeal1lplillllll'~'

3.2 Derivation of Modified KdV Equation

Considt'rlUI illt:ornpl"l'ssih lt"luviscid,1I1l1 11 lil rll Si \',~,sLra til i"t1lluidwith;~xlu-ar How illi\two-dimeusicnaldomaint~tJll 'llll'tlhyf1all" i ~il l IIlJllllllil1'il'~"

Thep;ov('rnill~t'Clllut.itJlls fertill'systr-mm'l'Eull'r':;equutluns,

fI, fI,

withuo-nonnal-IlowIlUl11I.liU"y t"(l1uliliUlI,

w(O)=lU(l1)=0,

when-f'ist!L<1density ,I'isLhl' pressun-,IIaudII!;ln~t.11l'!luri1.unlalawlVI'rlit"al vnlocitles, f{>speclivdy,.rJislilt'a("(~dl' rutitJlldlWI,ll p;rllvity,andtl ..•sllIJs,'ri pl,Htlt'nl/t, · llartialtlilrl~rt"ntiillioll.Tit" vl'rtil :a]'~ourtlillall'ZiHIlil"t'dl',l llpwar,1wlth1..'rtJnl,till' huLLum, andthl~horizontalwurdi llatt~;tls,li r('{~ll·,1lu111C~riKltl.

(56)

with

An(!xad suluriunof(:J.I)-(:l,5) isLIIl~hlU;1c sl ate:

,1;;=(i1('::),0,0), (:l.8)

(:l.9)

Wlll·l·l·f(.:: )istill'shearHow ,(i(::)1l1HIii(.::)an"the nudlstur boddensi tyandpn"SSIIf(' llisLrilllltiollS.

w(~IJIl Wconsidersm allpetturbetionsabout thisbasic state, Introducingper-

1,1lrhati ullsinf=(//11whichis as:mllU!(!sm all,

,i;;=I~+(Ii',p= ii+ (p',p=ji+ql,w=:ClIl, (:1.10)

HIllIsuhstit llt inp;thisinto(:1.1) through(3.'1)wehave

(fi+1,l )(II; +i"jlt~

+

u=w')

+

(Iii+(2p') (II'Il~+ttI'II~) -IJ" , (:l.ll)

u~

+

w~ =: 0, (3,14)

37

(57)

\VI'ucndhncuslonnlizcthisSy Sl 1't11h)'

.1'= L;, ::

=

1/=,I=

~i.,,' =

^I'"".

u'

= u«.

w'

=!!.f}li"/ / = l~lf1!11j,

^1/

=

(Ii,. (:l.Ir,) when-lilt'sCiLlill~IaetorIV

=

(Ill/I,f"llH'~Fnnntlil'I'lpwtioll ufl'ulIl,iullity(:1.:1).an,l

T

=

"Ill

ist111~ronvoc tio n time srule.'1'11('11 WI'havr-1.1l('nOll di ml'l1~ilJlr"I I" l'wl,iIlIlS (withthe hats droPPl'f.I):

-(,'Pro (:I.W)

-( "(,"

+ 1')'

^(:1.17)

II, (:I.l11 )

II, (:I.I!I)

wherer;

=

yl/liP istilt'squun-ofllll'illvl'rHl' Fnm, I,'IllrllllOl 't(s," <Hl~'liull~.~,~) audn=IJ~II}, Notetha t 1l0W allvnriuhloswitlroll lau (lVl·r!,M111"1 'jll·tl,llrl"'1.i"lI variables.

Wenow maketheHoussincsqupproximatjouwlridlIl·IH1.~I,u 1111'disill'l' l'ilrillr"I' ofthefactorfi

+

^((Iin11111 equntlou system(:I.lfi )1111,](:1.17),Tlli~isl,asl',1UIItin' fadtlra lthewaterdensityillth,~oem lllta.~only~Illallvaria1.jolls« fI,:\%)alllllJla co nstant uon-dlmensicn alvalueof1.,],III~irwlIl1II'H'ssif,ililyof Llw/lui,]allowsII.~til

(58)

illlr",I''''I'ilIJl'r tllrl'l llillllslt,'amf1Ilwtiulllf',SlIdlth.u

(:l.2O)

Aft,'rslll,Sl,il llt illp;UII'S"tWIJ1111I1dimina 1.inF:IJin(3.1fi)and(:J.17),WI'gdequal.ions for(IiH1l1

v ':

Equut.lon(:1.2:1)saysthat tIll'stroamfuucttonV'isconstantalongho undaries.This system

"r

(" Illa tiullslnvolvosI,h n'l ~pa rame ters

a,

fand II.

/,,1IlollP;willif,is small.

r;

isassumedtoII('0(1).Thus,W(1rail searchforan i1s.\'mpl u l,j,or-xpnnxionsolution oftheIorm

~I, = ~!.IJJI(.r , .::";I)

+

f",I,U(x,.::";I)

+

Ill/p,l(:r,=;l)

(:!.2.')

(59)

Suhslilillinglh, 's,~into (3.21),(a.:!:!)and(:1.2:1)~iVl's:

'1'1]('0(1)pr oblem:

(~ + fl{h)I/I~;U

-

fj" I/'~'o

_

(:p~.n

n,

(~+

[1!f;)pO.o -

p,if,~~·n

n,

TIH~Off)prohkuu:

(:I.2ti)

{~ + ftfh)if'~;U

^-11,,"';'u -

(,'(,~,u q,~.II'/J~;(:

^-

q'~'''if'~~~..

^(:1.211)

(~ +

fj/h)pl."-

ji" g " 1/,~·uf'~'"

^-

"'~'''('~'''.

^(:I,:IIJ)

(:<.:111

(£ ⁺ flf/;NJ~;l

^_

fl,.lfl~,l

^_

(,'p~.l {~+ il.-fh)p(J.I _ {J,t/)~'l

'"

II.

(60)

h'J~'()"':~

+

v!~;~V!;'O) -(V!~,oT/J:fr

+

t/,~;~v,:.o) (:I,31i) (V,~,op~,o

+

P~'0rJJ~'o) _( " '~'()p;.D

+

p~·[JV!:,lI) (:J.:l(j)

V,:,U(J',Ojl) =,gU(X,J;l) = 0 (a. :I7)

Inthe rollnwiull;,w,'willdiscussand solve' 1I1f'ahoveIlroh"'I11~one byoUt'lipto l.I11'Sf'f'uutlo rdl'l'i ll f,

3.2.1 The0(1)problem.

Asim pl"l11a tli llllln liuliuf (:I.:J.(i) and(:1.27)gi\'t's

(~ + fl-/h)1",~r - '1«(~ + 'l-#;.)ljJ~'D_ Gfi<rJ!~~

⁼^0, ^(:U8)

withl,11l'boundaryronditlonliSbefo re.

TI1l'f"llIlltil/ll(:l.:l8)has lin infinitenumber or solutions,most orwhicharc' ,liHi,'ult to lindlUll!1.0 df'lll with. It11'011111benleoifithad asep ara ble solution, 1lf'I'HllSI'lhilikind of solution iseasyto handle. Tosec.'whetherthe reexist such

(61)

solutions, We ilSSlIl11{'

(:I.:I!l)

andsubst it ute thisinto(a.aS).This~i\'I'l<

(:I.·W)

Thele ftsideof thisequation isItfunctionIJfzouly,so l1l1'rip;1itsid,'1I111 s1.I", as\\'\,11.

This l"cqui n'SLhat

(:1.-11)

whor e

I.

aIHI/1arc(ulietioll Surzonly.From(:lAI).itillul.violi llthatwr-IIIl1sl.Ili tV"

A"l=-cAnfor501111'cous tnutf'.Thislt'al ls to

Snbsuuruou

^{int o}(;H 2) showslhaldT(I)/ ,ll111l1stIIl~lLl"tlnstmlt..w(~111"1'nulinu-r- estedillsolutionswhichh''l 'orm'unhouurb-dasI,iIiCrt'I~WS,S!'WI'IlIlIs tsr-L'I'=:lJ.

Thus,(01" asup a rahlnsolution,AlIlllSt.sa1.i.~ fytil"lilll'ar rIlJlldisl"'rsiv,,WilV""'Ill at iu n

1:1.11)

(62)

11('::)f ('fur ,IllY '::[i.o. 110nil,klll lay!'I's),thereisIttlist:rd,..infilliLt· slw drlllllof t'il!:"'IVllllll-:l(',>('~>(':1> '" withr-orres pondiug"igl' nrlllicliollS<PI'1J,.'",TIll' solll1.ioll('" ,(p,,(.::)i~tilt'IIIOtI" -,,lim-arwave.TIlt'1l0lHlis pt ' TSi\'('uapcct,of till'Zt'1'U urd.' r solu tiollis111'('l'lIIL{IfII¢;J[i.e, the long wuveapproxiuuuiou].

FW1l1(:I,:n)Wt'iumuxlintr-ly~"l'tlmLpU,oalsohas tIll's-parabk - solution

(:1.'17)

wlIl'Tl'

(:JAR)

(63)

3.2.2 TheO{c- )problem,

Wt'againdIOOS" to s"l'kSI'pilr<lbl ,'SUIIlt.i tll IS forOtt)pruhl,'m(:I,:!!l)III (:1,:11).Tl ...u-.

fore,WI'ass utue

i11~'(.r,I)r/J"u(:), H^I,ll(.r,l) /) I.II(.:::).

Sllhs1.ilutil1gtIll' O(I)solut io1lintoLln-rip;1ttlmmlsi.t.,or(:I.~~ I).1111 1(:t:UI),weSl'l' that

AAAM'" ,-M ' ,,), Ai\A,~·I('I(~(.J, )"

Slnccthoil1hot11ogCIWOI1Stormshuvctil.·Sl.pill"illol,'i\i\...rill"t" l",weW<lutI/'~"",fl~·II,^I/'I1.^l!

iLlUl fll'otohi lVl'tl'I'f" d urilSwell.lIytilki l1~

SttlJs t il llling1/,1,0,/11,0iulll(:I.<!!l),,It.1(:L lfI),i1IlIldil11iltiLtill~/)1.,,(.:::),\'0'1'l'U'!

upwith<tilequationSYStl~lI1rurrbl ,lI:

(64)

~ [ (" - C Il"Om -O,.,,) H:.'(,,~,.)J

(;1." '1)

4,1'''(0)=4>1,0(1)= O. (:I..'")!;)

ro r1.111'O(I)pl"Uhl"1ll(:H!i).Thus(:l.rl~)is lUIiJl htll1l o~{' IIt"lJlISfo rm uftil!'eip;t'nvill n,' 1'.-,,1>1"111(a.'!!).Now 1I11'111tl'Htj,m iswhctl ll~reqnat.ion (:1..'"1'1) lliiNitsolut.iou which salis lil'sll lt~Aiv"11lmnudaryrondh.iou.

Ml1 rtipl yill~(:I.!iifl"y4>/(<< -f')~andthl'lIusing(:IA!i) w,'1';,-1

wlu-rr-

lh-n-WI'han - nssunu-d nonzoro(ii -I').

l ·/(: ),I.:=O. (:'.•8)

Thus,illunk-rfur thcr«to1)('11sc'par ahlpliUlul iollronditlon(:l.5S)111l1stI){'siltis fit'd.

{)1It',sppl"ii,l"ilS,'furwhichUrisistrueistIlt'ollt'with11=coustuntuud

P: =

^ronstant.

Bulill,I!;\'lwral,(:I.."i:S) isnot )i;ltislie d.

-15

(65)

byaddiuganO(()mrn' di"l1,,( tIl<'rUl'Ill

Wel1UII' mustreunu1.<1tl1C'0(1)problem.'l'lii stil1l1' \I"'slll ~ljlllt,'(:L!i!l).111sl.lwl"f (:1.-1'1).into(:lAO),T1iis~i\'('S

[(li-rl/)- ii,'~IAJ"=-d/(.r,I )/). (:Uil )

Till'S('0(1)l()1"1l1Slll'lungilllhe0(1)prolJll-lll.Tllllstill' 0(1)pnIIJI,-1Itis sl.ill~i v('l1I,y (:1..15)alll1(:lAG).Howevurint1w O(i )prul'!l'Il1, th«ril!:lllsid l'Sur p-rll) ,11111(:tr,~) Iwr.uIIll'

AAAM",-!>,",,)-(il"L)'I,,,

AAA~ (/~ J

^=-^l/(^J:,^llfJ,

,Hi

{:l.Ii:l j

(66)

I'IJIIslanlI,, ·:.!"su lh"t(:J..'ifl)101'l'UIlWS

Physin dly, 11H'('rrl' d (If1111'lJ(!W 1."1'111is to illLm d m '('^all0(1)cnrrr-etlon totheprop- ,,~a1.i(J11sl',·,',l,whirllisnnw(r-ttl'll),

N'IWl.1u·()(t)lll'o!III'IlII>,'corm's

01'uln-mutlvcly,

~(11",

-1,1,,)-"1",

~ Ct~ J~

^-^I'D,

0, (:1.67)

<Ii

(:l.68) (:I.m)

(67)

;111,1

\\'ilh 1.hl' I"lrrl'liIIUlIllillll:

r= ,

I,

O(f) prulol"ll1 r"r,p1)111111101)'_i~_~.."lvaJ.lI·.

3.2.3 TheO(/J)problem.

TIll'pruf l',IIlI'l'Illlopll',1lnthls1'1'0111" 111 istill' SHUll'liSillLilt'pre'villusI'fUlJl"III,S"

only/III,ml lim'ispr<"SI'lItl~IIII'Il"

·IX

(68)

WeIl~aillwantttlsoeki~scpaeabl«solutionforthesyst em (3.:l2)10(:l.:H ), so wc' s d

AO,I(;e,l),p°.I(Z), yo,I(:t,t)Do,I(:; ).

~llhsl. i tl1lingt{!o,lJinto theequations(:I.:l2)andp.:l3)suggests 1h at

'l'lwsnl!s litll1.iOl1

or

thls s('pllrahlnformandlltl 'olimlnation

or

DO,Igives (:1.75) (3.76)

(U7)

(U R) (UO)

1'0'111'1'('(lillytl)('1t'lulill~ordor term ofA,(namely-cAr )has beenused.Note again 1,lmt(:1.7H) is1111'illlwllIugl'lIl'UllSformuf(:1,45).BytheSli m eprocedureusedfortill' 0(1)I'whlelll,til"c'xis1.il1lC~I'ofilsolutionof (:1.78 )requ ires

l'

^.J(::)dz

⁼

^0,

\\'111'1'('

(3.RO)

(69)

ti,=-ctl ...

+

2{rtltlr

+

I, I/{J',l ), (:1.1'11)

With (:l.77),\Il,1(:J.lH)lIuhslil uLt..1 in(:I.:t~).L1111p.:!:I),lb,'r" 'l·ill.':I"flll"of

whichiluli"l\le1hl\1ror1\""p,nllhl.~""Iu1i..nWI'11111>111,1IHlV'~It

= ...

Au...wilh.~I...ill)l;

aconsuuu.Thillillrll('lilllplll'l1tlm1111" m-wter m illtill'.'v"!nLiu,,"'1'11I1;""rur II resultsindispersion.

Theilthe0(11)problem Iwmml'S

orah"rnalivdy,

0, (:UUi)

.'"

ⁿ

_ [~ +

^(il-c) ^{(li- cP}^(lP^. ^{] .'"}

oPU,I(O) =t/J°,I( I)

- rp-s

..f

r~ + ~,p]

fl-I') (lI - f:P • ^(:1.117)

0, (a.... )

(70)

.utllWll~('{IIl(~II11y,./(:)iNHOW

.1(,)=

-</'

^-^.<

[</,,</

_{l I -}_C

+

_{(II_C),1}

~P'</' J .

Thust1II,,'Ullll jtillll

L:

./rl=

=

0,;iVi~

(3."9)

(:1.90)

(:U II)

(:1.92)

illl,lI;~till'salll"asill(a.7'1). WithlIlisvalueofII /lud"wa wprofile"volvillp;

iU"I"tlT,lillp;lu

A,

=

-Cll"

+

:.!(rAA...

+

IMAu .., (:1.9:1)

tIl<' ()(/'ll'ruhl"lnfUT.U,Iallil [1J.1issolvable.eAllUltion(:1.g:l)iiitill'KdV equation whieh,ll~rillf'!lLh"tillll·,lt·vl·lopllll'lILor1011&Wtl.VNi uudertil(' combined drpcLorLlw lin;l uTll,'rIItmlim'i1rily and ,lisl'c·rsivily.

51

(71)

3.2.4 TheO(f²)problem.

Int1w last serlioll.W(·llt'rivl'tl llIC·l\ tIVt"llllaliullwhis-hi"lOrlil"l'l"nl.' r ill' all,I ";,",1 t1l1'rl'rOrt,i,.goodrnr!CaV,. withsmallnf1I11J1II'rillc'mlll'lil ll.I...1111\;11till"10...·,' 11.,IS

FOftlll'St'IG\ Vsl\N1't'lmtl-uflll'r,Hllillilnol,'dr'''I"tII1I1,.thi'i1ld,..Ir-d. TI,i"~1,·."1,,I"

tlu-t1er iva lio1lorllIutlili,'(1I\tlV"'Illalio ns

<i"

Wil"~oI"n,'lIy

I."" ,\',

1I"<lr,b ll'y[I!lj'l].III 1I1issection,rollO\vingLet'uudBI'ilnlsll',f,1'0'"Slllv\'

nil'

O(f1)I'r"IoI" 1ll,11101"I.liliniI

lllotliriNII{tl V 1'(!llalillll(will 1.1',1"lIull"lI ..,.lJl1\IIVlll'l'I'llrk r).

SlIhslillllillg the'Ilrt·viuus ft'snlts1/'";',I/,I,U.m,l",11,1jlllullll'rightsilll, ..rtill' 0(e²)probk-m(3.:I,'i)1\1lI1(:I.:l6), weS('(, t ha t

,,~»,,:;~

+

"~f:r~»

,,~.u"~f.,

+

",':f.,,,~;1

11¹A'(H~:~

+

1oP",,,I)').

A'A,(101>z.p:~1

+

1>".:;1), and so 011.Thus\V(~an- illlll)in'tl1.1.1sd Lhc~rulluwill,t!;IonurllritlIl'I"lfill,I,',;o;,llIli'J11

(:1.~Hil

(:I.~ 17)

(:I,a7),weobtaiutill'"fjlllll; Ullsysklllr..r1J¹,nlI11d L11l' "urr,,,,pulltlin/l,f'l>ll.liljllllfur

(72)

"lw"y ~ ~; , l i.~I *" r1,Hy1.1 ...~;LlIl<'I,r)!,rrull'rrlilSlu'fOrl',illlo th ,'r termlTlm "", rr t,ill~1111"\'( '11 sl" wl'l"..1111111;"intill'wave,~ll a p"isirrl.rotllll'(',1 intotIre'('(lI lat io ll 1?:01/(,1"llillgA

Nul'.' w,' slll,st.itlll.,'(:I.!JK)audulltho obtainedresultsoftilt'lowerorderpruhlems inl,"1,11t'full[w 1"1.rrrl,a1.io ll,'([rrill iullofti ll'r0 1'1ll

(a,!m)

n-ms.l'I"III"'I" I.riITrm l)!,ill)!, 1,1ll' 1,'nllSwi1.11tilt,sarrl"urrlor,WI'fiudHIl'('(lllil1.iol1s fur

:l:\tA...-l(1I_dIYl.lI _tl:~"I,UI=

[('IW~'"

+

;W=~l~'l-{'ltP::f)r.o

+

f)t/J~,,,_ .lrO I.^Ul]t11A...

-MO,

(:1.100)

(:1.101)

(73)

ro r a !'<'I>arilhll':oc,luli..n.Ld

(:l.IlI:! )

(ii _f'j/P ~1_P= <b~ ·1J

=

vVithD~)lelimiuau-d,1I1l'SI'ltrt'n-wrij.te-nlIS

<t>~~'

..

^- [~

(,i- r)

+ ~l

(.,- r F

.p~~'

⁼

:J(i1I

-

r)(~~~ +2~m4-I~1

^-

:1.1-:.p~:'

^-

~=:.'- -tr.'

^-^·\10^:=)

+:Jfli

( ~r)l(,p/)~~I ⁺

^'J.1J=^4P'^-^:1."'=^/)')'^-

/)".~)'

^-^,11'0^')'^-^.\11),^(:I.lIlIi)

l/IV'(Uj

=

^¢.'l,l'(I)⁼^11, ^l:l.I(7)