lu GO?3

lu GO?3

. '

"

,AVIS

THISDISSERTATION HAS BEEN MICROFILMED

EXI\CTLYAS RECEIVED P;ovlousiycopyrightedmaleriei~ O~l.Irnal.artic'les, publi shed tesl$,etc.jarenO '-':I~,e~.'

The qualitylhequality'ofthior the or(ginalthesls submitted lor mlcr?liIm-smicroficheIsheavilydependentupon . :qual lt6de la

qualitil ci~Celte·~lcrOlictle~ilPend

thes.soumlseavmlcrofilmage.

~rtmdeme"t

NO,:,s8V0I15^de

^{I I J '"}

' :.; lng.Everyeuc rt hasbeen made10 ensureIhehighest 'to u t'aJIpour assurerunequalitllsuper ieure derepro- -~ '

~, -...;.qUalily~lleprodr~lon pOSsible. ,~ ^{dUe,tlo.n..." "} .'~ ^':', ,i,"' ,.." )':

~~U^{pageslifemissing,contacttheuniversitywhich". 5'11manque des}pages. v8uUlelcom munlque r avee<0 . ,

granled't~edegree.. , ' . l'univorslt6 QUIacontt,.Iegradi. . . . , ::. Some pages may 1'18\10 Indistinctprint especiallyII "

;~'

^qualitil'd'lmpressionde

cellairi~s pag~s

^paul-'.::'

tho originalpages weretyped witha poortypewriter' 'lalsser,a desirer,surt~ut^{,I !espages}OrlginaleSO"Vt6 .~ : ribbon or ilthe ul'I;iversily senl us apoor photocopy, daetylographieesaraided'un rubanuse ou s1l'unlversitlr__ ,;

nou s ala!~pa ...onlrun~Pllot~C~piecl~mauv~;se^qualillt. '--;f~

Losdocumentsquiiont dcjill'cbjot d'un droi td'au- '~~

to,",r (ilrticl, "

" ,e

r~..,ue.exemenspubillts:ete.)ne sonlpas ',1 i,',

mi.crolilm~,S.,, · ^{/ . .,:-}

" ,:: ; '1

Lareprod uctlon-;mllma partielle,de cemicrolllmest .f soumise/lia~canadlennesurIe dro itd·auleu r.SAC \ 1970"c.c-30.Veuillez prendr8connaiS$8ncedesfor.

.-~ulesd~aulo riuli o nquiacco mpagnenlcenelh~se.

:'~"

^{' "}

" ",LA THESEA

ETE

MICROFILMEETELLEQUE NOUS L'AVONSRECUE

." , , ' -,,,

.

--...

,

f -,

'by

.t~:~~~~~: n.~~:~~;:~~ ~~~ ~.~.~,I!~~;~,g~~~~

1~:', PleaS-El readthe authorization forms'Which accompany

~. thisthes! s. ' "

~t..

. . ]

by

~I~ I TE~'ELI"'E~,FORMULATI ON OF VISCOUSSE A,ICEMODEL

':oi'the '~eq~1 rellleDtsfortbe'd~greeof',

._M.aster'ofEDg'~Deer1D£

h \

~ l

. .~:..

De-partmen~

^of

ED~i p~rlDg ~d A~pl1~d Se1eti~~':

.i~Orl~l :Ull1~e, ~sitY .bf Ne~~OUD~a~d

^{. ' \}

:J..~uar,.1917_.

:•• 1 I

;. Th~\steadY^drift o~s~~^{.r ce in}·the,Arct i c,Basi~'·bas

bee~calc~l1at,ed

^{byusdo g}

tile'

Finite

El~ent Meth~d .

^assuming

~h8;~^{t? e'}ib~^{is a}viSC,OU,~'inaterili.l : The,.A;Ct i C.Ba~io ^{is ,:' .} discretized into,9x13squ8. r~elemen t s Of,si z e~50kmby250~.,

~be'co~st1ttit1ve:.rela t i,onS proposedbyGl~n(197.0) bave,been, used in tbemodeling. Tbe'mo'me,\tum and continui ty equations

'ar e sol ve d for.tbede e velocity due towind's~r~ss o.c e~llcurrent

.and o,cean t~lt. The,.driftbas beenCalCUla~ed~,or'f~U~ ~easoDs

ina,spa noftwo.ye a r s'by,using.proper viscositiesio,summ_er.

\_,~~ winte~. Tb~v:e c t.o r.add i tion,~f~ceV~locU,i,es^{due to}~in'4.

woce a ecurre nt a~dtil t are'p l o tte d,a scombi ne dve l o~i t1 eg,fo r

\ 11 10'.!r sea sons Summe r'1,Winter1,

~er

^2.

an~

Wi'n,ter ';

~.

'''~ , ^{. . ..}". ' ,.. " . ^'.

.. Theresultsof the'FiliiteElement Metbod 01~nalysi s are coepared'Wit h that

.of

't b e

Fo,urierTr~n~form Hetbo~ ( lfib~~r ,

1976)and i t is,found,t hat

th,~ r~~ults ~btiL.iDedbYb~tb '~e'th~ds

^-.

·eereewithea~l:iother in tlu;:middle'of' t,be Basin. A,comparison .

hetween slip'and,no slipboundar y·-condit.1;on.is'a l s opeeeent.ed.. Boundll.~yeaeeeecear eueee,a.n d an1terat1.ve·procedurels

llowedtolnc or por'atetbe'nonlinear.boundaryCOlld i t1oDs.

.

" " " ',

no n- ete.ady'dr if t ofpack ice

.r n-

eneStrait,ofB.e-lle:

Isle.,wbic ea-beckand fortb,d u e',to-t-i-da.i-streams-and .~~'-c-'- .cur-e e nta\b a ve bee:nc~lcuia'ted'

i;y

the'Fi n i t e Elemen't 'ldetbod

fo r mula tio n. It'1s ob s e r ved that··theP~Ckicevelocities .:arQ.~·barnlonic^asare"tbewater velocitiesb~'t'w1tb^a ti~e^{la g'}

·:which depends upon viscosityparamet~rBOf,i,~e. '

.

ya~iclu~' ;~elo~~,lty ,·~leld,s .~ave " bee~' :p,io,'t'ie~ :a~d -aU : : . . .

·patterDs,'Q~;.ce rcuret e a v e r c crties areobservedtobe,realistic..

't'io:;' ,

MyS'lnCere,fhank's

, ar~

^{eo eti/}

Professo~

^D.S•

S~dhl,., Fac~ltY. O~ E~~ -,~'~g 'a~d APPU~~" S.C-ie-~~e . M~or-i·il·_u~~verslt~: "

'of

•

Ne~fOU~d~j1:

.'- '.

'

•^{J thout}"'-"

whoS~guid

.- J

a~ce

' ^and"

,~upervlsi~n~'

' ^thil'i. .'\

wor k.WOU;l1 not-hav e'beenpossible..

My,si n ce r e tb anks are al sodue:to Prof .'R .T.Dempster •

•

-~~aD '~f ;~'cu'l'tYOf

Engine l!l,ri'ngand

AP'Pli~d' Sc'i~nce, ~d '

,Prot .

F.A.Aldri~b

^{-.D" ,;;;..}

·SOb~oi

^ot

Grodu."St~die,;

^t;r

the i r gen erous.heip·~i.ndassistan~ed~rin~;~istu~U~si.J?-d .stay'hfthiscountry .

F{~anaialassista ncefr om theNati onal Resear chcouncrj,

of. Ca~ad.a

^'apd

toh; . ~~mori3:~_U~_iver~·~ty· O'{New~~Un~landl

_~--c~~.

J;S '

'gra._t~fully acknowledged~

23.

.

^{,' "}

4.2HyPOTHETICALATIWSPBERICPRESSUREVARIATIONS

4.1

BOUNDARYELEI4ENTS

"''4.5STRAIT-OF BELLE<dISLE

a.a

THE,FI'NITEELEM,E~T"FO"n.YULAT:[ON-., CHAPTER 4: RESULTSOFCOMPUTATION"

.4.1 INT RODUCT I ON

REFEim~~S 'AP PENDI X A

56

64 .Page

' : "

;': .':, '

The pattern'o~Be~ufo;'t^{Sea gyre'an d}

t~~~'spoi~~ drif~' ~tr~.~·' ~n

^tbe'

~r'ct~c ':,'oc~

4.1 Wi ndve locity,li e.ld due"to hypothetica l

.

,atmo~Phe;i:c,

^_preeerrre

vari~tlon

^{",\ .57'.}

4..

2a ';~'ck

^tee

V~~~City fi,~ld

^{lor'I; "'ri'};',.2 . 0Xl 0^1·lkg'

s~l

^,,5&

~~.2b

^Pack

'tc'e ve IO~ity·;

^{fie l d,for'C}

~

^n.=-,

4:0~101O'k~'s-i

.4 . 5

'4 . 8 Packice velocity'fiel d due tow l nd "

curre n tsandocea n til t illWint er1. ;

4 .~

^{.PaCk ic e}

~~'~~'cit~ ,

^fhld

'~ue t~wind ,~

^oc,ea:n

aur re n ts and ocean t llt,in~ummer2

4.io

Pack ice,velocity fiel d dueto'wlnd . oce ~

D:iscr~t l'za"tion

^{of theArc ti c '}

Ba~in

I

"', ;. ' . ' " ,

4.4~,'~a"ck:,ice velocity,~ield'd'ue to wind.·stresseso~n .

",in

Summe~

^{1 • . .. ". 61}

.P,ok ice vel oc ity,

f,iel~

^{.doe,to'ocean}

,"~~~nt

^•..

~nl

^{-.' 62}

.4.6.Pack,ice.,,~e.locit,:P,~ld,d~e^{to ocean} ~ilt^{'o nl:'.,','.....63}

4.7 Pack lC~fe.l~c1ty ,fiel d duetO,wind, ocean.cur nts

and oceantiltin!Summer1

current~'^{and oceantilt>}:i~W~nter2 67

··4. ~1

^A,compa r is.dri

'betwee~

tbe ' r esui t s

ot

fini te elemen t methodan d Fouri erTransform,meth od' ..,4.~2,Efrect.ci!.~l1~and^no'slipb~UDdary^{condit ions}

onthesol u t ion wit hin',:t bedomain 4.13:Di~criti~at1onof tbe~:rcticBa sin'w~th

.bo.un.da.rY.111 em,eDts

69,

, " \ .U , \

~~\ ^{- - -m- - - - \}

4. 14 'P~Ck

^ice

v~i'OCitY

^field

d~e ^{'t; '} WiJ

^eteeeeesonly

.:

ID\~UIlIller.l witbth~ . inc~rpor at~?n ~t 'no,lJ-li~ea~

bO~dar;conditions <71

., • \ 72

4.1 5 Stral\of Belle Isle _ .

4.16Discre tization'of-straitof BelleIs l e 73

\ , "

4 .17ve toct

ty

profilesat the narrowestsection'of the st r a it

f~r d;f'fer~nt

_~' ^{valuesoftheviscos ity}

' ,

paz-aeeter-\.

4.18piot'ot parii{ ice and'waterve to ci.treewt.tb'r e spect

\': ' , ' : '

~to.timefor'~ij'feren t'visco s ity parameters,

.4.19S1mPl1:r:ie~ ui\del A;"' l Tringularfinite element

, . '

\

A-2 Bo=dory,S"".S.\ " ,'

\ : \

wate r,system..'

Unitve~tor.pe:':'PEmdicu.~ar-'to'I'[- yp~aDe.

Ll~'~2 ,L3.N~tural-co-olrdiDat~~ fo~l1tr~ an~~larelement.

l i j Lengtb paramet er.

t Wavelen gtb:of fo r cing,funct i on.

. ."

[Hi: "'as s mat ri x•

.p

'{Q}.',Loa dvect o r. . (qJ.

s \

Bo~ndary li~e.

~1ablelen gtb

! ..

Ocea n

til t :

Velo city~mponen-tin Velocity -C<?mpo-n.e n t..·in

,....

,.. _ -. . . , . . ^pr:·~·'U· . ,

/ \ . "ntor

S""8

;'.IOe ,·

l ·

, T:'.~~1r 8tre8~n le~. : ^{. .}

e Ekmanangl e..~nWAt er. ._.

',

~(~

^':

E~an ' angle.

^{_I n.:atr.. •}

..". ~ ' :;J ~:::;::IO·:(:::~:o~po ue. : I : i e ,

. £~j . St'~l~':'r~te co~~neD~~ ... ! .: '. ' -;

Y~~·.·',~~~~.~ -r"t~: d~e.·to . ·Sb~~r_. _.;;

._

'~~:.'

:.~ot~tionof,eartb inrad~ aD8.,p~ ,:.eeeccd.

peec ake.

,

Sb?reS'aD.~ " P~~k

^...

·1~~

^-.-The

~; C,tiC .'~·~~k

^{ic e}

~~~~~r ~

^.•.

i.~~~ ~a~.::·~~, .- ..

·oftheArcticOc.ea n"" T h e,~h~1Ckl}ess:o ~ice',~r1 e s'fr,om,z e :r:o .

· ::~:::d~,::r::te::';::::,:;:~t :;:; ·;;~,~::;::~~~:':.·. L " do;tr , .

open.water-orthi nsheetso.f<>ice.~n"tne.A.rcti c re gion. Pack.ic e

is'.founddurillg

t~e ~'~e-a~er par.{'~~ '~be

^{}le ar}

l~':'~ost p or:l,Oris"

"

'.

.

.- .

- ,- . " .' -

of'^the',

,m~rgln~l,~

,s eas^I

Baf~{n

_Bay

:~nd,;th~ w~:~,te,z;,~·.~a~.~ ',,~i;,t

be'

r n

^Sea,the Ca n a dianArCbl~elago: nll,v is.S.trait and

~ tGreenland Coast. • "

~.ck lc~ ^{mo;'" on th e} sUr':ce ~~f s~ ^{• • • ie/ du e} :t"'''f;~. ·

oCE!a.n

· ~~rrJD~S

^{•.bOdY"forc,es}

·an~ .~;terac~tion ~f~ ";elgb,::,~~i~g ,... . ~

Ice,_

A s'

i tmoves.

it m~y

^beap'UP.

i,ead~ ·1.~i. ~~~~ I»-es~~re"

^{.' ,}

rid ge san db' _d ckBintb"e' ,f Orm.ot,rubiHe':.t ueddf t.cr-Lce-

."

^{, \ '" .' "}

^.

^\',,'

^{. . ..}

^{'. ', . '.}

^.

·-te.no ~gener~p:'¥n~~lc.cable.ec apex sonstan~1Il g:.9D

thou gh be migbt ha ve'rooved'huJidred~

o f

ki;lomet~rs._'_.,._.- ,.

mont bs .

Nansen'( 19 0 2 )·a.n d Sver-drup-(1928) ,: and it eeebeeu an·1.mpOr-

" .;';,'

. "

',',

..

^",

'. '

ta u t re se'arcbfi eld in recentyearsdue to the.r e c en t, dis c overy

' . ...' " ;'~ . '' , " '.

~fhydrocarbon,deposit.a intheBeaufortSe a, Antarc ti c ice' d.r1; t_....

'w~~\

_:')

s tudled

^by_{,.' ,}

Bren~'~,cke

_{' . :"'}^{(1 9 21 )}_{, ,,'}

' a~d't~'e'

_.,.

dev

_.,

i~tion

_.^{an; l e}_{0' '}

,bet~~E;ndr 1t ~of'ice,and,w~ddirecti onwas foundtobe34.,'

..'t.t~the l eH, T~~(Nans~nan,d.,sver dr~ S~~d~.e·softhedev:at~.

,. a~~le.between t :e:win~directionand tbe~rc~iC ~cedrift': .~.,.

direction is30 to the,right. Nansen expLa Lned thi5 dev ia-

/::-> :-

t ron,was:d uet.c't hecor{olisrcr-ce• t.

-,'j

" ,

. 'There!

are permaiJent ddf,t~t'~acks^{In'the'Arcti'c}B~sii;i .which'

bav~ ~~!,nObserve~ 7e~~~~. ·;eats'

^{io'r e}

' C?l~~,,:iy

^;han

. .. .

^'

.

..'....:.

' ~ the 'sUggeSt!On 'Of N'ilnse~ ',

,Ekma n (ieo5)f6und tbat

U:it:: e:'t,~ct

JOf windwould be/ p'

to:, ~ome

depth in,.seawate r .

?'

Heobserved-tba t:tbe.velocitiesof: 'd i f f e r e nt.layers would' '.,:,dev 1li.te· more a,~d/~ore

a s

w~go:deeper- and:,deeperi~,tbe, water,but tbq. magnitudes'ot .verccrtv of different layers_-,

,~~d~'e ~xpo~ent,i~ilY ,w~-{h ~E'P1;~

^{• .:' , . .}

..

...

I,

. :' ': , . ' -,' ,.'.~ 3

Ma'ny

-s~~~n-'t~'~1;S' h~~~'''p'r~POSed , ~itte~rent ~o~e'is '

^{'t o\.:}

::::::' .:::.::::~.::.m::~::e~r;::::::i:~.~::;I" ^;

·"rea ; .!s tt c'now"'

~lel'i1s'

^{,c oul d}

b~ ~a.lc~ated

^..'

_~~ ' mor~,.1~P?r~n·t.l~',

^.

. t

_in

o ."".ee

te rpre ted

W . . b.'b~r

sensibly i

'b .e" ~;'ess.ure

n the light

. . . t1~

^..

~w

^..,

'. bi..

wn

. . t b e

lC~tion'

. . iC~ . '

^{cov,e. rcaD b..}1n

~

^.

~iff'e~?,ntare~s Of .'the-.~a·si~..

^{. : ,-.," , . ..:. ,'}

': , '. • . ,'. . . c'.": ",

The re are.two aspects of seaic e modelingwhichare

n~ t:we llunderstoo"d~ anddU feren~ the or i e s on lcedri'f~-h a ve"

. been,~roposed'due t"'o11.~lted~nOWledge,anddata: The'first' aspect

de~s

^{with the}

'lnechaa1~al propei'tie~ 't,~

^ice

i'Il , B~~ll ' ~~~le: ^and

^'-€I'

se~ond as~ec~ :-~e~ls w1t~ ~he beha~ior

^{of pack'}

. ~c:~

^{00' :0.. ar"}

·scd~

^{. .}

c.:nS~derab~e l~';er~iure~X'i~~S ~~.

^the-'.

· research in tb e s e twofleid~~ .

. , ' &

Ice'is a crystall i nemat~rialwi th·hex a goo a l synunet r y .

.it s

·~ha'r9,cte ~isUcs.

^s'u.c.pas.

-ib~.ck~e~s;

^qual1'ty,

a~d

^t'ype

·of gr a.iost~u~ture;,nre s~bject-:to varinbil i ty~it b wea tb~r. .\

Elastici ty,plastici t yandal lmechanica l prop e rti e s are ( , sensitiveto temper a t ure

jy

~rinevol ume1.nseaic e bas a noticeab le influenceon itspropehi~s~":'Sla'tis tiC,l?and·proba~llity

""

.~In. ~'i,e".

^of.

tbe. ~bo:Ve ~ent10~'ed 1f8.~tors.~ ' se~

^{ice,has ,}

been,modeie d a s avLscous mater,lal ~ym'anyBciEint1sts'(Ca~Pbel,i:^'

. '"

" ,

^"

. " ....

. I

.

;

...

.. .

\ ,

-,

withtho seobtainedbyot hermC1;bods.·The advant ageof the driven'

by

w.i nd,a'ce a rl~u~.J;~n~andoc e ant"Ut-,~Tbe^results ob ta ine dby'the"fi nite elementmethod',are.i"o go od agreement metersaresolv,ed to ob t a invelocities of pac k'i c ewhi cb").s

, . \ '"

.'

...^,

"

Afini teel e ment fOl"lllUla t ion'of the vis co u s.sea ice-_

. \

mode l.ispr

.

e-sented

.

bere^. . Gle n^{. .}(197 0)pro pos ed the

\ . . . .

constitutive

..

' 'ie laU o ns, andHibl er.l!-0dTucker (1 976 )PfOp\?se d tbatthevrecc-

. , . \ .. .

sityparame tersin tbe....vis co usse a ice'mceei.vary acco r di ng·to"

" , " .'..., .

' .

.

theseasons: In.thetioiteel~~n t.~ alysi s" a setof two

.-as'a!lo;inv l s:cid.Ll u ic;lby Rotbrock.,(l9 73)-:

"(19 72 ). ~~ ~laS\;·ic.,.plaSticlIlaterial by,Coon, et

ar

(~974).~and

.bo·unda ry'. Theres.u.lts of cOl!!pu.tatioD8are pre s ent ed tor·tbe

.:st eady

s~ate

^{icedr i f t in}

, th~

Arctic Basin andthe transie nt '

.'boUndary:ele me'nt s .bos~^proper;:ies cban ge accordi~g^{to tbe}

.~:r~ven'byst rong t,i,dalstre ams.

' . "

"

'. .1965~i·Gl~n.-1970;·,.H ible'r'.1974), as aplasU c:ma terlalby Coon

Co'linear>-dl,ff~rcnti al~-uat ions',with va riable viscosity p,a ra-

··,i..

,.

'

.. .

^~

WECHANICALBEHAVIOROFPACK ICE

..2.1 KINEMAT ICS

'The

'velO? i t y..11eld of pack· ic e.v~ries^{in space}~d

t»,

"-.-

.CHAPTER ~

.

^{. .}

.

^.

Butveloc,lt l~salonearepoorIDd1co.t o rsof ice ba h &. -

v~o'r 'beca'us~ t'he "mechani cal

:b e h a'v i o ; ofpack ice:i 6

rei~i~d

t~'~~lO~l~ydi tterenc~s'between,'.no~s,·o~.'t hev.~loei.ty,gra.,... \.. ..~

d.en't., thatIsvo r t1 n 1tynodstra'nrate. V.'OCity.grad'e.ta .\

may be wellde fi n e d ina contin uwn. bu t in a real ice cover

~

^]

tbey.af e Dot••Abrup t cha n g es ma.y tak epla c e at floe bo.u n da r ie 8,

Tomeasurethe re spon s:e of icetoreal weather'systelllS,.wbieh . mayvaryover aperiodsofab o ut aday . tbe..easu rellle nt~,

~':-':~ 6~OUld

^be

arraDI~d

^{ove r perio ds ofle s s tbanaday. Tempo :al ."}

average s are of no t much use . If sp ac ial averag e s areto be I

. .

_ uS~f ul. theremust be a continuum len g th of sc~leIe...bic b 1s

':",la r ge.,c o m! aredto floe sh:~ l( ~dsmallcompa r e d.tothe. ,.

smal~est ':~pa~e' ~cale

^If

of . ib~ '

^dt:iVing,

f~r,ce8

^{(Wa y ku t/T h'orn'dike}

. ·.'aDdUnt erste i n e r,19 72 ) ,' that:..is".Ic:i8

to

;exifit~^SUCh,.'~~~t

- . \

..

>~ , ~ , lc · .-<\lf

^{(2.1).: ,}

",

.:

' . , ' . 'Thel.eng i,h·s,Ii'~d· 'lt ·are.~.~ t^....ell..det 1ne~.~:b~)in:th e..A~c~.i~·

Ocean.,.~..rigi d piecesot icelarger than 50kili,across'are.t-.

'rar~'> an~' ai~~'S~b~rlQ

~

.·preseure·

,

syet'emB

"

. i~

^{tbeArotic, ' .Oeelto,"}

, /

' elimi~'atiil.g

,v a ri a.t i o ns"s h o r t e r than

<'

^and

tb~n, d?!i~i~g

\s t r a i n rates. But t,hls'i de a is notp rac tLca.L. To.obta i,~

v_'.c c r rect size_{', "}

of

_.

' g;u~e

'l e n g t h 1_{'" g,}

. "t~o

_,1

m~tbods

_, ^{have been}_'.

a~~~ted;

In the first method,st rain ra t e s were simultaneously measured

. " I ' " , ' \'.

o~,a ,gaugeOf,10 0 k m and

{3

a gauge,o f 20~and'le.ss. If both,th e s e gauge',lengthsare within-t be:continuum scales ,the

v. , ' , '- :' , ", . .

~~a'~~rCd'strainrat~.S'WOUl~·c~rr~lat'e'perfectly. :These 'at measur~ments show~ dthat_t h e net dUatatio,n,over,a 25da~

period (Hib ler, eta j , 1974)was'stronglydependent.00.,the g~~ge^{length. 'I}t was clear,th~t^..one

..

^,did,not. 'obtal~ t~e.^{' same}

.

estimate

01

strain.r-ate bymeasiJr~ ngover these dift'ereIit

.

.. \ . .

....

. .

gauge lengtbs. In,i~e ~ecO~dmet,hod" Nye and,Th o ma s(1~74) I' showed that'th e variance of strainrate measurementsdecreases

. ,' , . ' . .

as the'g Q.ugele ng th19 Lnc reaeea , and this e'tfectwaso:t>,Berve~

up to aga uge len'gthof'100'~. From'ebeee.'two methods,'i \

I" ,, ' '" .

appear~.that the continuum le ng tb Bc'alE! is,lOCjkmor'l arg e r . F~~~diffe,~et'l.tda ta'(Col~.ny~;"dflash'.1975.Dunhar and: Wit tm an , ~,6 3 ).,we now have a goodidea of typical velocittes . vortic1tie~and strain ra t e s,i n·t h e central.Arc t i c Ocean.

I I ;

Pack.tee moves at speeds of 10, iuD/day ,_occasionally.reach Ing twlce"tha:tspeed...

:Tb~

actual

P~tb,fol 10wed

by a'p o i n t in the

'p e r c e n t. ,'per" d~Y~'bUt''tue:Lo ng-t e r m--' , -~v~rB.ge~are conside rab ly" , - -"

smaller_ Duri ngth e40 day period'documented

by

Thorndike

.., .\.- ' ,' , ': ,' . ":-- ',

.

._( 1~74 )'_themean-.rat e s of diverge nce and shear'(defined.as

.the sumandcliffe-renc,: of pri nC!p.!l.st r-atn ra t e s )_'we:reO.001 and 0.0 02 per day, re s pe c ti ve l y. These',riumbers re fl e c t the tende ncy"of ice tode f orm"morein'shea'r- tha nindiv e r gen c e or-:

.

^{" ' ,~.}

converg ence..Tbe vortiC1t~,(deti~e da~.thever-tLc aj,compone~",..

of'curl .!:!,''whe re~is'the velocity), behavesmuch,.th~sameas th~

. ra

t.eC!fshear.

Fr 6 m,mosto!'the se obsar'J'ations;one can,interthat

. ,,'_ . l J , ," ': ",.,

there.ismuchdif f eren c e be t ween 'in t r a' and "I nte r-fl oe ' .str ai n,and,tha t the

str~in , I'Ihi~b con't~ibutes

to:-

~be l~rge'

scalecl rcul a t iono f pa c kic ei'~c:aused-bytll~ rEHat~ve movemen t of pie,?eso f lc~d-not bythe cont i nuo us deforma ti on of pi e c e s -tbemselves.

,

(

_;.•..

19~3).

" .

~U:!!§.~[llL:!!!!ill!&§U!L..!E!L.!C~0!lCVE!!R· .:,,;

t.. , . ., ' , ._. ,_ pa~~. iCe

^{' ..}

complicated as.i t-de p en ds on chan g esin,we a ther . The-Lce th1'Jkne's s

in~y ,

ancr eas e due,toridg ing 'or, ra ftio g;or.tne rmc-

" "-dyna~iC ~h'an~es

ⁱⁿ

w~a~'he:r.

^{so'me.op en-}

l'e~s" ma~

^{fOrm due'}

t~

" .\

." ' . . .

' . ,

"r~ ~ glrg and~a fting.':t~llusthe co nti n uo us,ly v:arYlng'char a c te ris -

ticsI-sucha$'c ompactn e s s and thick nessof icemust be

~onSl~er~d ~n. th~ _m~del ^{0 , : '}

^{sea ac e. rnere:a,redifferent}

ap~r.Cl~Ches f01.10wed:~Y .di 1'ferent scie,~ t~~ti- t~

^{·i ncl ude}

.thes e,:e f fects.1ntBeirmodel s.

. Ino,oe ~p.pro a~h ,t.lle~o~.p actnes sClha~.been,definedas ' t.h~fre et.Lonof,thear ea cove r e d'b y rce . Th~remalni ~ gfrac- t ion(~-C)i S cove r e d by ope n.w a 'ter.. Droga Ltsev(,19 56) an d.

. . .

Nl kifo rov(19 5 7 ).usrsrumed tha t the ice covered ar e a C within

t~e to'ta~

^{ar e a,B}

is ,co~served

_.

~~

_,\ ^'"0)..

Tbis 'm~ans , ~h'at

^{ol;ll y}

'the'a r ea cove red by openwat er(B-C) canicbange, Doronin, ':.( 1970)"mod Ui~dth~"above equa tiOD by introducin g'a source

• 'j " -. ' ",' ,' : •

term'in the co mpact riess equationto a.llowforthermodynamic

~ha~geS

ⁱⁿ C. Intbe caseot'ridgin g, term'in

· tb~· ~ompa'~in~B'ri

equat'ion,

~' ./." , ' , .', , ,' , ..

A.dl ~f erent butIll?re c,omplete a p p r o a'Cbis 'toin t r o duc e terms of:i.ce thickn'e~~'d ist r i b u t i o n,.g(h ) ,'wh ieh'i s.defi n ed as't he

': f~acti(~~

^of

'~h~ '~rea:

cov e re d'b Y

'ic~ '~n t~e ~bickriess

,band..b I'

,.'~~' ^b

2 an d

i~, ~i ven

^{by :}

J~2::'g

^..

( ~)

^{'db .}

(}f ",~a.;kut·.'~D~

^'Thor.n d yk'e"

', ' ~1 ' .

!

~".,.,:",

....

^'

"'

b~ "

^the

b~i.~";~S<:·~Si

^{tymOduius"and}

:":",she'a I',v1scoS:itymodti.l ~8.

component sof c:~rrent'a nd airstre.ss'esaregiv~h·by emodifi~d

\

' " . . ,. ',-. ;. . " ,

.• "

Wb~re .~

is,""

le~:i.~lt~

^whose

.comp~n.~nts' ar~,"u, a~ d.,V:'ln

^'.)

the,x and'r directions,' 1:t-be Coriolis parameter wbichis

Y

~ :,', "','- -'-

' ..

^{. '}

equa l to twice_the product.of ear t h'8 rate of rotaUon-art ,.

:sJ o e .o f

·i,!l'~~tude

^•.!a

unlt'.vector~Orina.~·,: to ,X-Y '~la:n~ytri t~e .

'ice"ma s s pe. .r'unit\ area,!:'the"',forc e

-

-d~e^{.., ""}to variation's^{' . -}in^" t^"lie.

internal.icestr~ss'~i

..

^{crlj.j),'4,and ~at~e-.wai,er}

^and

^air,"

.s tre s s e s·o.n'tbe ice"and~-arid '!t.heeffect'of.longt~~

-r ."

8

(SiD~'U •..c c e e.V') •

~~~~~~~.I,l.y-'--- --

'it : '.

g

. I

;, I c-f- : ^{~ ' ~} .'/.

: _- \-!".~:~: -==

ccrepcner rta', ," " " ,.

- 1

r

^{Vw' In thecasewh en Qis}ne g l e c t e d , the sea ice 1saflee,tire,';'

j

^lycons i de re dtob,: moving,R,crossa e,tagna ntoce.an.,

.~h~.geo~.,;

, trophic cu;;~~t ^nO/l•. cOmput~d ^fromoc•• ri h.,g~t ^{". a. ·fOU ['.'.}

...•.. .

gf~i .!< \~

^J,"

^{.·. oJ· · · -. ' . ' .} (2.41 '

.' ,., i

•.1

I

.;

\ ", ".I

.::where"p'ae_tb~dens,ttyOf."a~r.

\::.~

_I"~,

:!i'

_<",

^the

_.:^{;e ar t H}_" ^{'r o}_{' , "}^{tat e s}_;^'

~~·~·~t'1.ts

_{"'" ', "}

·a~l's

_{' ,}

'(;:29Xio~5,;

_{.".: :.'"}

~'d:{ansl~~c)','"

_-,:,':;"

,a,'vBJOei ty ve c t or'o nit l s·con1J.lnuous l y cha ng in grel a i1'l.~·eto.

.:

:i,~pace

^...

~~b1~

^{cb ange}

liHl~!Q~"1Y p~~, -a~~~a:', 1~" ~b~ Cc;;10~,1S

. ~ ':~'C~~le~;lt'~oD " ~~,.·~'t .1~:·.~.~~'p~~~~i:.~~.';;~~) ·.~.otb·,:/~_e' .. ~~lO,Oi,t'~:, ~~c~;~

androtati on'vector attbe ear "tn. Aspack iee ba s'c onsiderab le

~aBs,/we ;e'a~llot."n~~'iee~.·tbe ·, ~r161'iB"'fciice:,

^w

'hie~

^.Ls.

gi·~~::~y:;:. '. .

. ti ~~'t "te~ :'ln; tbe~~e~~~ 'e;~~ti~n .

^{(2 '-2,; ,',""}

~

'~.

-:

w~s ' cl't~c~~~,~d ^, ~,~

^,Buyni

i~k,l'~:'

^,<

1~5;) ., ^{" ..'} Ro,~hroc~ : (~~7?)

't hat',

~he': ~cce'ler,a~lo~ ot' pa:~k

^{.r ce}

,i~'

^{o. f the}

'o'r~er ' ~f '

IC?-5c m/ ;; e <:2

;flS','ll,'-t:;'P1 Cal.'

~"l~e"

ⁱⁿ

' t~~ 'Arct~c ~as:i.n

^on'

'~he '

^time

:'~e'al~ Of-' ~~'e

d~Y, But.'ti,he'ri'two'floe~,~'ol1ide,th~tiiite'scaleis verY,nitich

'.' " . .' ', .'. .. '. ," .

Sh'or~.er th~n, '

^one..

~.ay

^..

a~d

^ebeee.

s~a,~l, 'S~lLle a'Cc:el~rlL~'iCmS,':,are :~

almost certai,nly,s i gn ific a nt.inw~ ch:case we,:,hflvetoincl~.de

~i!: 't~im" 'o~

the',rfgh'thand side

o'~

^{equation'( 2.2 );.: '}

'. ' .

Thedelormat1on behavi~r',\'~fpackca n be treat~das,

that

o.f

. '

^.

a'hi g h l y'.dampedItliit e r ial; The wi nd"drag s theic eover'~ rel~t1veiy

s:tat{:o~a~; ocea;/"oPp'~~e~ ~O!iltIY bY,' Wa~~r

^drag

a.nd"r~s~~ta~~e

^of

"the ice cover t~ de formation. Thewi ndu~~al1Y'Ch~ngeB.s o s'lowly

that',the

i~e_~ffers little ineiti~l resistanc~~t-o: 1t ~'

^"

.:The

~~rk

of the wi nd isd:L$s i pate cl

'p~rtly

'b'y.wo r k'against

i~tern'~l'

ne:~d c~~st~ tut'i~~ ·'rel~t1ims "~~'l'ch . r~~8"te -fbe .str~·ss'_·~c~ns·6r to

,

other' vir;abi~s .:~~ .th·~ ~r~~l~~~ :.~n '.~Il.Oer!at~'Dd.i~g. :i~~- '~y'~~les ;

tbis:has.b'eenthe gre;;'test-,si ngle s,tumbli ng'b io c k';-·DHferent.-'

," , - ,., ', . ' . :"',, ' .- - ' " ,",

S,c'i.entistshave used"~l,fferent.:cons t itut i ve,r e l a t 1.on s tr~at1ng,

pacit

·i~.e ~'~ .'a, ~~scous. ttia~er~~l , ,( C~Pb.ell·' :'19~.~) ., , ~n ,,~_~com·pr~~~.ib; ~ a~d

^invlscld,"tiuid

( Rothro~k';" 1 9;3)"

s:

~~vltadng.ilu.id. (.Niki~-"

fO~O~ .

^'et-

a1; '" U)67, Dotonl n ; .19'10) . and

aILelas1:1c-plluiqe

" material :

(Coon, e t" ,1974) . ... " , . '

;.;.' : ' . .

.Glen':

⁽¹⁹⁷⁰⁾

d:i~CU8~'ed _ varlg const~'iut1v~

're t it.i l ol;; s for"

.'f~cous :

model .nd proposed.

c~nsu iut've r", 'Uo~ ::,:::r" '

.

." ::::s'::' .:::e:::~ t::e~;::::.~,:~ .::':::d ;::.:::::~p;~:~:7 -.

..~~,-~le-~:. " .:.-.' ',' ..: -. .'::.": ':":.t '.,.~~-~S··. '

"t)~'i\: '

~f' ~'

~f:;

~!-

t,

~}

l

_'t'

·,:;t~} "; . ." '.; :;J.J.';:; ~t·/~;~ ' .ci').( '2;>%;;~ · i." :i::, ~';~·<;:' ;:>J·

·

^·:drift'

:r~:i::,:%~.~:::

rat e",u(x;y )< .odv( x ,y.).

~:~.

;;-

;.:Oi~.~.

.

P ^{. .} ~ £: ^{. .. •} ,~

•

1~:r. ^{. •.. .}

^·1.,....,·"^,·.·..•"

"" " 1 ." " , " . ,

.Bi bler'

'~~"'TuCke~ (1976) ~~stl~at~ .t.b~ V~l;e'~' 9t

^"bulk"

, '&nd-.:b~a:r_ vlsc&~lt1e8.

^by

" ~orr~liti~g tb~,-obse~ved, and ~r~dlc~ed

d~l;'~' ra'~~s ;'

,'"and-

i~~~:, f.OU~~· ~~~t.i·t~e

^:

~18e'~~1~/-~~i~.~s:g~~lng ~ ~.

, >tbe,be'st:"-.ilt'-bet~~eD"t~e:~b;~er;'e-d:~'dpr~dl~te~>val";~s<~b~~' •."

..

:';;~~n~wic:.e~.' se~so~~l

..

var'iatl~~" ~ba.t.' ~.or~~iate9· "~~11::.";'1 th,; '~'c:.~

'.

,.. • rgi~~t~",:rate ^{.... .} T~~Y, ^{: iu gg;ste ' d,'} t~e ':fOl~OW1~~ ,e'~P1rlc~;;' ,~~ne~r

^{,,-'.',: '"}

',;~~a~l~~~bi~:.,b~t·~~~n ~~~C:.~Sl~Y\~;~ '~.~~ i~~.~~b'

^reate

~~.~~,h_,."{~.: · ·\ ::....

Yle l ~,~_predi'c:~ lon6'Inre.as onable. agree!Dentwltb both:'long

,>.

/'-:

.. .

^'

va:dab'i~

^.(;hether'

' t't is ~~~~u~~': ,, ~empera~'ur'loJ. : '

.ve l o 'd.ty,

• .;:'.', '. : ..,:" \ . ,' . ' . '.":,'•. '""'., ,.:".' . ,v'"

~. " :.: : ::m:",:::::~::,~::i:~:::::::sp::.~:~::~::: ::r:r~':~::::::: '; "

~eg'~'~'n :'

^:',

'conS~q~(tti~lY' tbe'pr~blem

^{,i s'one}

w~"t·b. '.~n ~:inf1ri1te: D~be~

oj ~nk~OWJ;l:~. '

.

In,::,t~~ ,

^['!n'!te

-'~i~~~t: ~e't~:o~':'

the

;soi'ut~o~ r~gion

...,',:'-,",' " '. ' '" ~. " " ".\ ''''., .',...'

. .

"

:~s,di.6~~e~.~zedint<?ele~~t's.· .~ le~ent.,boy ndarie Sare conne<:t e d Wi.tbSP~.CHied· ~o!nt~ ~a.~l e.~n~s':or^nOjal~O~Dt,S . ^NQw,WE7: .' ex p r ess,tbeunknown.,fi eld:v a,r i ab l e,.1nte rms of,assumed"app'ro,-

'~lm~te," functlo~S C~l'le:d'.

^{'i ntEn}·pol at·lon.

·iu·i:c~~~ii's' '.Wi,~bi~· :~a~6

~lem~n't; 'T~ese lrite~polatioD.f~?Cti~Si^a:r:o'deiine~','in

,

'o~.the'val u~s'oftbe-f i e l d'va~i abl esa.t'·the" nod.;"i

:'

~hU~~:' tb~~p~~'lem i~ '~educ~dy" ~~ , 0De ~f

^',a

flni~'e -. ...

^{" ", ' , ; . ' ", ....,;" " ,." ..}

:

.~ ownsof.tbe !i,eldva r iab l e.at.t b e~oda~paints.

.'

~

~~d~Y1~,Uallel~::nt~~r~p~r: 1~~ 6~~ ~e

^..

~o~u~:~ted'~ ~Y, ,on~

^of-

t.~e'

^,;

!ol~,?~~ngr~lIr,appr-oaches~-:th ed1,re,C::,appr-oach,.,th~ yar~ tl:t.i~"<' Dlilapproach "~he.weighted'res:f:duals"appr.o a chan'(ltheene'rgy

~~l~nce", a~p;o~~h.

^The,

Ch?1·~e.

^Of'

i~e~e-~~pro~che:~'-'depen'ds ^{.o n ,}

'th e.t ype of.tl1~'prO~.l em.

;'~~:' t~e:,pr~~e'nt' P~Ob~~in ,

^";e

U:s~·'1ihe. va_~i:a ti.on"ii, aY~~~'ac.h: ·'

.In.th·e'..CbS.steal:,

V arl ati~~al ' fO!mU~,at1~n , t;be~'prObl~m

.js,t o.

'j'

'.; ^{> '", ' . -,«: one'of 'the,}

:~~~ri~~t

~d~~:tage5f({f 't~e' var1atiOnalapp~oach

^{.1s 'that it'}:1m; l.1c:L

tly'''~~.'·

. , " , ~" 'I·,,~" , ' ,: ' .: . ' ", : ' • "

..impo,:es..,t,~enat u r albpund~,ry.,COnd,it ~Ons . ,~n',t~e.fin;tteet e-:t

'me~t

_"

tDetbo~

_"

',

_,'

~~

_{'' ,}

CXPli~1tl~'

_'

tm~os~"geometric

_{,' 1 ·' . ..}

~OU~darYCOndi'~

_:,'.

'

tio ns, 'Thu s ,'dn't hi s method both types ol:bo unda ry-c.ond1tions

a~e sa;h~lied . ' 'lil~'

^lin!te

'.ei.~e~t ~ethoii~an :

be appl'lcdto pro-

.

.

^{, ', ~ ,. " " ~..}

...

^'

.

"" blemswit hcomplicated- bo und a r y coudftions alitlirregular

~. • • ~ 0

shaped'boundaries . • ,:.:, ,,

...f.

17

(2.~a ).

.

; '....

where"

'm , . E. '

"i s-" the change.'

:

of,"',Ioom'en tum

..

^' 'p; r,un i^{"," ,} t'a'r e a-. .:

..

In'~ol ":

vl n g ttie IDOlOOutUlllequetLon,weJ;akeice velocit y comp o n e n ts

.

. ..

'. .

.

^{' .}

aswik~owns, . ^{'A cOPlP'atibi eveloc ity'}. ^Held-^{'isas sumed in}

.~r:der~t~_h ~~e.acontlnu~wb.icb.

does

Dotoverlap.01" bas.

. holesIn''i t,duringdeforma t ion . Strai nrat es are. obt~ln edby' .

: : ~~II,~'i al- d1ff~r,e'ntiati~n

^of'veac c rty'

compo~·e·D1:S. ~t~es.s·

^{'c ompo-}

nents ca n be obt'aini d l nt~rmsof st ra i nrate.;compone"nt sby

i', . ' . . ..

.tnevu ee..of.constit utive.rela t io ns.y Th us , the set orvtwo eq u a-

t.lons(,~.2 a:!can be~x~ress~d'i n terms,Ofvelocitycomponen t~

u aD~.

v .: -,

Fort11.efinit eel~~e'nt,formulat1bll , we need'atUlle.,.

tionaI:as,explained'1n~~cti~n(3 . 1 ) ,'and tb1s'1,. ob t a i ne d'·

"' by

JnlJ,ltiPl;1ng

'~he 1lI0llle~tum -e~u:at1ons '('~ '''2a)\y ·~ar1at1ons·· ~f

~'"

^and

.;p ·'and'.·int';ig~~t·in~

^{the'.SUlll"Of'}

pr~dU~'~S,

^{over th e}

domal~

^.

.'v]..•.•.

[ l :2~S;D ' ;~SlD:] [J

.r:

,

' {

^.,"

'

"

.

J 9

1 "

".~.;' ';'-

+

r ^{CO~' ,}

^{D . . .,}

^{r~-'],~: ["}

^COB'

^{, C a ' i ·· I f] :}

. I:f:;:':'::j :i: ' .

_.

.:.:...::... ':it:i' ; .: >~;;':

:

~~~~e

^.O··is

t'~'e:: ~~~th<~,:

^t:'llte

O~~, ,r~:~~_~·.~~,~ ',~

^,

18 "i~~ lll.tt,t~d.~!,. "

•.,Il Dd01j Le.thet~od1me.ns1~na:~tr~ss tens ~rwb<?~e',?.?._~poneDts.

"....

, '. , , " . ..' ' , .'

.:Il~~':~hevert l Cal1~.:~D tegrated·qUlln.~ ~ t ~ e8:aerolJs~:t~e",t,hlckneS&,

ot.pa Ck rce . ~he ~~1t'!ot 0.1J'areto:r:ce,~e~ u~.1t,leing_~b..

. I' , ·

I~ieg~at1Dg s~~~ ot " ~'be

^{term s}

' ~n -:the' il:b~ve~e~~a:tJ.o/ · ·

^,

.:.:(? 1.).~yp.ar ts.'....~·^{can.exp,re s s'}tbefu~C.~10Dll:l'^{i'D 'ttie'}toUOIf- -

... ·111 ·'

t(j<I"lII.by

USJ,I1&"' ~be'"'c_oIlS~1 ~Utive

^{.relatioDe'( 2.7)8.Ild.}

~be

eq uati o ll:(2'-9):

.

^.

.

'

::~ ;::0:::::::,::':.: : : d.:::: ::,::::::::~::::a,1:e!~U"" '"

.

;~.lace

^{'i n}

t~efoi-m ~f.d~forui~t1on:O!paCk -.1~e ; . ~rll.g ~:d~e

^{,'t o-,}

' :w~ter ' Il_~d

^..the;;

c~ah'ge : t ~iileU~" eri:ergy

^.,,:.,..'

• .

~?"he

^{'fini te}

~ment f~.rmUl'ti~"'

^the

,~not1oo~1~,.2)

ov~rthe whole'domaincan be

wrl

'tten as.t he-S Ulll-of',thi;.functlon·~l

~";~r-_fj'-n iie

^{'s i z e}

el~me.n~s

^{intowh Leh}

tl:ie :whoie --~~~l~'

^is

we bave

'.,:.", iC8.d:1t t.

(1972)

:··wa~··m~~~1i~d to ' c~~·~u·t~· v~iOCi't/ ~t,~~~k

^{'i ee/}

U8i~g

~l;1e' .' 1·0~Ul;"~iOIl 'gt.vI1, ~bOV~ ~ " ID 'tbl~"~r~'gn~: :, tour '~rla.ngul~~

al~lDe~,ts' a'r~ ~~s~~~b'ted " to' m:~a ,a-g~'~~ral q~~dr~i~~~-r:al :e~'~m~~t

~l1~l~'it..t~l1g ' tb~

^1,l1tarnai

:·Dod.~:,bY':'tb~' ,Bta,t:1C C~ndeDs~t~.c;in proce9~ (D~~al

a'nd,·Abe l. 1972 )..

Tbe , re~ul'tlng''''l~ebra.~c, ·s~~­

._:'t e m',

61'

~uation~^{ai,te r'}

assembiing

th'e~iem~D.t·pro~~rtie'~:,gl~e

·· 't~e'~· t~ ~ uD$y~etri~ ' baDd;d 'mair~x~

^I.'A

~~w, 's~~~t1~e,~~a~

. ,'. .',.

, . .

. .-" ,' , , . ". . ~- - '" .' . '.thema t r ix.i nt o.l ower and.uppe r.t r..!a ngula rma,t r~ces._ Sev~r a l··\.~

.ot her.{i:te pswere Lncorpora te d inthe"p'rogram in'or de r to com-

." .

' . ' . ' . ..

·p~t.e c,:,nSis~e~t g.en_er~llBe.dforces a<,the,n9d~S.^which~re^d,es;'

cr i bedi'n'''Appe ndi x'A.. .,. . .

•~ Rl:lnge-Ku t tame hto d:of1ntegration ofth e firstorder

. . ' .' .

equation(3.4)'iSlllsolncorporate: dintheprogram' inorder

~o .c~~pute

^time

de~endent, ve'l~~itY

^{of'packice. 'For t'h/'f1r s t}

.

or'd~'r SY5tem'in ''';'~1me, th~r'e

^{··1S a}

cbarac~er~st1c ti~~or ~he~ ::, 7

~ys~em' wn1chis_an'i~dicator

^of

tb~ res~onGe t'~me o~

^the

~s~em ~

^{"en di.t ,}

dep~ds.~~pon '~he

^inertial

mas~

^aridtbe

~nergy:: "~ ' ~-:":"

. , , . : . .. ' "'-;-""""''':'. :,

diS,SiPa ti D,gpra.p~.~.ty91the,

"?":

Wh~n t~e V1SCO.sltY~.~a~~\:~~,·' met.ersare large, as.t heyare.for the packd ce.1n theAro-t1c,'~ '"

Basin ,the

:r:espo~se ,t ime

.'

' is/small ;

^and

.-

^{we'may as'.}

we~l

^{' ,}

SOl~~

^'.

the'st eady state, veloc it y ofpack ic e us'i n g eQuj;,U on ,:( 3.5 ). For the,loos el y c.ompacte d 1<;e,the_v~scOS 1ty^{"par amet ers}Ill~Y

~e'small,{.an~.wemaY·,in t e gr.a :te theequ e.t.Lo n(3.4)"·us1n gil~lme~': \ st epeq.ual

<\,•

.

^~.

"

t~16 c~·ap~e.r

^'.'

·w~ .p~es.e~:t. ~~e. ';e~u~,t8 ~t~.tl~i'~ ' ~~~:.

^.....

ment.and ys b ot the'ic e..driftin the Arct:l,c:?Coanand in'tbe Strait.otBella'I'sle,us in g'vi s c ousse'areemodeldesc~lbedfn

.... . . .

/

. .. .

^'

. . .

previouscha pters.'

;1 0,0:r::d.er·.toc~eCk'the"f i ni te"eleme~ tcomputer'p~ogr~ . ,somehy~tbet1 cal a..tlllOs~he~icpre~8ure vat:iatlon.~are:a.ssume~-..

andtbe resulting wi ndst re s s(equa tio ns2.3and 2.6)isbpo-

sed' ~; t~e

^{pa ck'}

i'c~~""'. Th~ '~esu'lts ob~~in'~d

^by

~i~~te: elem~'D,t ':

,

~~tbod' ~ere ~IlPa.r~d ~i-tb tbe' , re8~lts' ObtaiD~~

^"

, b~ Fourie~

.

'fi'~~~ to~ ~~t~~d.

^'.

n~e' ·~~.~~~~t ~~ .~t.~~~.~:·t~.:~~ excel~:~~;.

^{; '"}

The program'WI.8 t.beousedtocompute,ve l ocity o'f'pack 10edue'

':'.,

....

- ..":' , .':-- "" " . ' '. " ',':.

.to rceeee caus ed'by.wl n d ...cur renteand·ocea ntilt.·,The,veto-

'. eiti~g of'pa~k'

^ice

Obt~'l~~d" f~~ ~~Cb ior~in'" ~u~~t:i~~ 'we~:·

^..

8~~e~·­

. ~S~d

^{·.t o}

obia;~' ,tb.~~o~~·r~ii'::'iC~··y~i~·~l·ty::.·,·,'~i>p~·~~~C~t., ~'h~ '~~~Ul~'S ;

o·f.

o~r

calcu lati'ons',foratwo.'-yearper'10d

'U~iDg ·d1ft~'r~[lt

^..

~aj'ue's .

"

'1~.

^{"t he}

~il;S~"s ".be"~"".t_be l ~e- "~~e~:l~ p_~~hed " ~ialD"st"" tbi "~~o:re."_~.""

.D~".be~:"_lt :^Is::r~~edln.g^tram:tbe.:sho re."":i.Spec;l~(bo~~ary- .~:...

":el~meDt: lsIneorp(;r'~ied^and

a n

:i.t~r~uve"s e h"e me

is"-followed,. ·

city,

.f1~1

^d-

~.~~U~d ~~~. c_enfre :o~_ 200~_.~'

by

':' 2 0.00'

.kmsQuare.

r:

'~$,

^..ebcwn

" ;~ Fl~ure ' ~.:. l.

^:.The

~eO',$~rop:~~~ ^{wihd. '} v~loF'H~ .

^coepo-.

Den t s areobtainedby,uai n g equ,l\.t ioD(2.6)

1 s

^{.101 1o ws}

. .

^.

.

where

p (den'shy,a:!

ai r >:·.'\.3

'k gm~3.^and_.

:~~rame.~e~) ^,-

^{1. 46 lC}

^{io- 4} ~~l.

^{- ,}

~b'''compon:ents

^o'f':'ap

a~d- w ater. 8tr~S13~~ ~re O'bt'dne~_

^:.

from.

eq,ua:U~ns , ( 2. ~)_ ue~n~

^,the

tf~1l6w~~'~::v~1ue~'_O:f

,dif f e r ent

r

'\,

l

u.:

'"' l

m,..,3.0 xiii

~'~

The'ex actsol u t ion ot:pack icedr Ht over,a stagnant

, t: -. -- ' ," ' .

ocean'due t othe atmosphe ricpressure variatio nsgivenin

7qU#~n '

(4.'1)- 'is

~V~il~bl~

^"'US-irig t,ne -pou;i.e r_

;;an ~form :

^{me t hod}

inwhi ch-.the

~olution . ~~~~pea:ted' e~ery"

^{4000'kmoveran iDfi -}

nleedomai~'in'tb~'x and'y direc.e ioua.

Tbe'r esults o f'! he:Cinite elemen t analYSisare'~ompared

'. f ' ,

Fo r the finite eleme nt"analysis,the,200 0kmby 200 0

. -", " . ,' \ ", ' -I " - -,__'

squ a re·a reaisdiac retize d '9iu to 8x8=64 square,;elemen ts.o:C

"size25,P'k!Jiby 250~. whi l e using'fin 1.teelementmet hod

/.to r

.th~s case; the va.l.ue~ '-o~

^teeveloei.ty

O!lta'iD~d .:c~om 'the

., .exa~t ~;)lU~i;'Q _ar~

prescri bedonl y.at the'boun d aryof a2000

km ,

by 2~0

kin

squa re.a re a.·to,gene~ate'" ~oluiio'n i.na ide,thearea .

10,bot h"~~ses•.an,excell~Dt;agreement is obta iDbd_betw een

.

^, -; " ,;- -" --',,".'-'~-;,"'--,'- -'. .

.:~he.e:XlI.ct,so'l u tion~d_the._re~~t so.t t.1n~1te el~~m t ~alys'is~

The'vel~eityfi e l ds o:c,pack'ice are.·pr esente d-i .li.F i gu re',A-':2a

28.

The external f~rces^which;~akethep'~ck~i~e'^JD'J've mainly cause~.by thegeOS~rOPhlCWind",.:geost:r~p;h~o,cup::~n t'and .0C:;!'lRO"tll t . TheC6ri~li8force-'and ti;l:e drag.fo r c e_al~~:_1n nu_

e~ce. t~~·inov,em.ent5p.ac~ :-~ ce ~~d -th~~e ,f.~rce~,,~re_ ~epen,d~~'~

up o n t.ne- ree wetcc r ty ..Since ,thesys tem'of diff.erentialequa-

.-. i ion~ , ~~verni'ng

^the

·mo~~on.

or". p a.ek

l'~-e are .,~inea~,

^-:'we;

caJ:iS~1",~'e

fort"be,ice,V:eloclti~esdue -t~'each':f~r~.ing"fqnct1o-~'.sep a r aioelY

" ,":' "",' ., ' .- ' . ,'-' . ; ,";,, ";.,-. - ,--' ,;.

an d ad d.tb~..~C,~.V~\OcitY.c o.mpone n t sO~~d"l~d,',fromdi tf ere ni,:

typesof rc e c e e to obt ain"the,co mbined vel~City of.pa ck'i c e .

~ . .

Th eArc~ieBas~nis dlscretize d into 9x13'", 117 square elemeD~sctsize25 0.kD:iby 250·kmas shown'i n 'Fi~re4,

3.'

For th~'·~undarY._cODdti.ons,the veiocit)' of.ice

,ncirmai.

t~.^{bOU.Dda ry:,}

~~dt~e.'t~D~e n,ti!ll shear streas~sa.ttbe'!>.<'undaryare px:-escri- bect,to.be

~eq~.~i to ,~ero ., " .~bla meana,

^{thatthe iceIs}

ali~!"ed,

^.

to'slip at theboundary wi"tho u t anyfr ictio na lreaLetancefro m

,th-~

^{sho re. The}

"v'al~ea

^ot

' ·par~eter~'. ~sed .i~, ~b.e C?J.IlPut~t~~ns

·a..re, s ame as thoee in,thepr~vious.,sect~onex~ePt'fo:r. thevall i e s o~'^{sbear an'd'}bUlJr.'~:iscos1t:1es^{whic h}a~e^{taken'eqUal to} 4XI01'O kg a-I

·tor s~er.

_\

seit.son . an~'.~~10;I 'kg :e~;

^for

w~nter ·s'eas~n .

' " ,

" 4.3 PACK ICE DRIFT IN

ARCTic

OCEAN

- ^-

'"' ) '

.

.

The'win dstressac'tLng on thepa ck,le e can.be,co mp ut e d':' f~C?mthe,a~mospheric,press u re da t aus i n g eq uations (2.3:) and

".'(2.'6 )'. ,".Four 's e t s,of atmos p beric pr es s u r e data:fci~ the

Arctic

"

Ba~o

we; e"ob t a l oe d

frd~,:

^{tb:eCold,Reg i ons'}

.Rese~~<:h ·' an~ /Engi'neer~"

In g'La borator y.:Hano ver•.!'i.B., (Bi bl er.19~~ ).:j~a.c~~,~·t ,.o·t· .,d'ata,1

is

180 dayaverag~~at~sPhe:ricpressure~tp~int s ~n

a '~50

^kDi-

~y

²⁵⁰

.~'~rld- Sy'st_etn '~D,. tbe

^..

~rI '- ~·~ g~S~'Ii·. , ·Th~S ,_ .

^".

. :::~~:::::t::n:.:::::~:.,::m:~4~S~:::: ,::::::~p:::r" . .. ;'~ .,:

Wint~r ^i"

^{SUlIIDe r2''and}

,w~nt~r" 2 a~·d _U~i~ co~espondlng

^{val u es -.}

" :;.:i:::;:·~t; s~:;\~.~tI~o;1ti_ :i.~dU~.tO . , .'

"The.i,o r c e s, '~ai.tsed.^bythe.geo~fi:.oPhic;c~t"r,e.~ts_·'a~rt·tbe'^'.

' ~eea~

"uit'

'C~D' b~~ ~-~lCUla~ed t.r~m , 'tch'e~~~~'fal;l:hel~bt,~-,:d~ta _~sil1g

^.

e~tiilt'~q'~~

^{:(2 :4 )}

:·;a~.d, .. _{(_,~ .:5~} ...." _~~ o:ce~n het~ht '

^d;,-tB._L

at P.Ol,~t;: :

OD thesame 2501mrby'250km"grid1i\rst e m wasals o 'ob~ alned,'

\

fr~~

^the

'ec;ld R~gion's ' ~ese~rCh an~ Englneerln~ La~or~tjry

(Hl~1~r;:U~76

^{)." Thi'''s}

'~ata'

^was'gener at ";d'fr om,th e

l(m~ "i~rm' i::oa~hman'- ~

^{'( 19.74)}

. ocea~

^{heightval uea;'wbich' -iepresEmted'1n}

Tabie~: '5 . ·,' .,.TwO ~eta

^of

p'~ck ~.ce

^ve locit,tescausedby

:a"~~~~:.

" .

.. ' . .

curze n t'a n d;t1~t.we re compu t:e d.u~ingdl/ferE:nt·v~tu~s ~~'vis-

'~ ~81tY

^for summ'"er'and winter.: Figur es4. 5"and

4 ~6 , sbow.tbe

v:e'ltici't;,t._.1_,el ds

'~~~sed' _.

^{by'oceancur;eQ.t·}. ^'.^{a o'd}"

'~i l

^t

' i~: s~r',-'

'

. .

^1.

I • • •

In'o r der toob t ain·the'v e l o city ,f ield due to th e

.'ne d

fQr~'eS

^0'[·W.1nd,.

cJ~re:nts

^{'an d:}

C:~'ean ·.~it~,.·.we ~d·ci·

'a'll',i he

vei'~~~ty" fi~ld.s ·'~o'r :~h 'o~ tb~'-'f'~ur ' ^{se a:'l0Da} .'~ons.i~~~ed

^{bes-e .:}

for,.SUmmer 1,

-Wi~ter l,S~er

^,2

-and.'1I'1'Dt~i.2,

^..

re.s'p~etivei.~·; C

^...;

'<The-reB'ult~,o'r ^f.lnlt-eel~iie~t'arlal;~is a~~'C~mpaFed,:::'.

-.'wi t h that·of,Fourier:Trans!onnmethod(Hibler,19 76. )' eompar iso n is sbon

In ·Figurie , ~·.u ·. ·

, ~,

Transt"or:ai-method.thello1uti~nre pe a t s itselfev~ry^{4000km"i n'}

,c.a n,;a1so no t e'from:Figur e,4.11-th a t'"the.1ni i uenc e ofdif f ere nt .be~D^assumed,for solu~ loll.^{by the"}t,,~^{methods: .III}the FOu rie r

."' .are dueto,"the tact thatdifferentbounda ry'CODdltl?D;Sha v e

..no':"" sl1pbound ary.'co ndition s areused. It~canb~noted that~

.

^'

",".-

.... ,''t .'

.-

..

'

..

^{;: :}

, ~' .

,' ,

,.(.

.,".; l~_.

- ,

^.'

t~~

^-

eU'~6~ ^of b~undar.Y ·

^con"dit10n.

e~:"endl:! ~.~ ,

^a'dl!tance

~f 750~

^•

. ,.

The Douodary.

e~~di~~O~,$ , t~~t':. ;;~n' be

>mp os e d

~i~ '

the

~~,I?i

^te

~lem~nt', s~~u..~~,~~r:f7, ,~f~~o'-i:;'p~s,',:' eJtbe~, ~

^t.be'

,VE!\~ei

^ty

o~.the,'~~~~ss'i n tb.enormaj,an~the,.~.aDge,~ti~l:?ir ec ttoo:,'Of

~h~ bo~i;~; })can ,

^be:'

preseri~~;~.~T~~'S!~--~~D,~ai; . ~O~dition~

ma ynotbe-vattd'when :he pac k,ice'edge is moving.awa yf r o m~ the sho re,be c a us e't~e,pack' ice'can no t of 1 er r e s l l:j:taoe eio .

~,en~~on.:h,e'.impos~tton

^of

, ~c;un,da~~ CO,~d~~.~on mus1: ·.t~ke

^ca/e

~

of tbecondi tio n wbetbe r tbepack.vc e edgeis r,etr e a t'ingfrom . the'sho reor

p·usbing':-:g·~l'~S t

^{'the'sho re, .on}

' ~h~ 'one"':ha'nd;" Wb~n th'-e _pac~'

ic"e boun'da r-yis moving away 'fromthe

~~'~~~",···~be ·b·~~'n.;.

s'

dar ys'hOUl d befree'to'mo vewi

t'b~ut o:ri'.; ~tr~~s 'impo's~'d ~~ . 1

^t,

'OD'theI

O~h~~,

^{'b; n d /}

:".h~~:.t:h~, '~,~C,k

^{lc e.}

b~UDda;Y·

^1:

PUS'hin~agairi'~t

t'he ,;bo r e "~beJeloc1ty'no rma litoi!teShor~'must be zero -an d the tangl;lDttal~tr~ss^at'the~oundary^{must oat}

'b e'

^{zer o'(p re- \'}

~erablylproRortioDai· .ectbe'Do r ma,l.pz-e eeur- e'~ccordin g:t d

."",'Coulo~bl:' iel d,c.riteri a ) ."

....

.

.

' . ..

.The ab ove'boundarycOOd}tiODsc~nnot:b~'expr-eeeed'U n e a r -' Iy ,/.10.'~e'r.msOf~~knowD V~lOC?i

t:y

~?~p~ne'nts~~~paek'ic e , H~.nc~;^v"a~^.1t~rat iv~f~ooed~~e'~,B. ;fo ~lowe,~to:::Corpora.~etbes~

Don-linear· bounda~y .co n d i tions, 'wi t b t he help O.fboundary

.'el~~en·t'~whose, mll,terl al-~r~p~'rties

^{(L·e .' .}

~:l~C~BitieB'j e~D

^be

:,Cha~ge,d' ~~b~

ⁱ

t~;at1v~

?Ol ut io n

,p~~res~e:

^{.' "·}

Thk~,e,bo,~~da~Y

^.

.

e.I~m~Di~ .~are~~ong, ~~.~ n~,t·

^{'so wide}

e'1E;~l)ts p~,~~.,ed·, 'at /~~e

^{,.;o u.,n.-"}

•dary;llS:sbOWn"± \

Fl,~~r~·

^4.13"..'The.ve l oc i tY,·a t.Dt he

bOUDd~1'Y

n~d:7~ '

are mlide ·e qu a l,t ?',ze.r o. .The.

mat~r~aI '

pr o pe rt les

~f

t'bes~/elements,.~'ar~·fitrst;takep. to'~~ the's~e;)

' , I

~-

.

.'

32·

·~s,t h o s e in? . ·the' .dom~ln., ,

The

..nodalveloct'Ues of- ' 'p a c k l'ee'. .'~re obta.1ned

_.

and the stressesin theboundaryereeenes

: a re'

^caj.cu-,

' . ,. ' :. '.'. -

... .

^{' ,-','}

lated. If a__tensile st'ressexistsin the dlrection normal to

, " ' , ' . - ', .'

. '

- ' ", "

the boundary. .the"el emen tproper ties'

.

ot the element areredu- eadto'alowv8:1ue. r e.a"h~~ld be_^reeuceet~ - zero^but.th'l'~^is not,:d o n e ip'order'tc'

a~Old4iific~1~1es

^in.

num~ric~~ -~al c~~~..:.

^.

t10'~s.

Tbis procedure'

~f

,ob t ai n i n"s

th~' SOlU~lon

an.d'c h e c ki n g the st r e s/?'in the boundaryelemen~to change,1ts material prC!;- perites-

is

repeated anumbe r- of"times·~n.ul·wesee,~he's~me SOi:utl0~'as"in,t~e',previous.~ier.;Uv~',s t e p•.

i~

^{Figure 4.13,}

" the ' bou~~ary' elements ' tiave

^a

';~dth

^;;;f

25kmand--.l'~ngth,of250~: T~~'com~ute,r.p~dgram:wasmodit1'ed

·,tofucor-por-ate,t h e above.it'e~at1V:e,pro'cedur~; The re.sutts .df Conip~t'a.t1on'ofpack'i c e'velociti~Bdue-to-'wiodstres sin

" . " _ : '.,' . " ,

.

.SumDerl,ispresented inFigure'4.14; Ooly·two1t e'rll.tion s w~r,ene~esBaryto obtdo tbisresuit .··,The'v~l~C1ty^{field io.} Figure4.14,can be comparedwitb the velocity fteld,in Figure

4.4 .in'whi~h

^a

.Sl1~, b~u~dary COndit1~n i~

^{imposed .}

Tbe~e

'c a l c u l a ti o n s wereI:iC:t·repeatedtorall'cas~sof'drivingtor c e s