MAY 10

CENTRE FOR NEWFOUNDLAND STUDIES

TOTAL OF 10 PAGES ONLY MAY BE XEROXED

(Without Author's Permission)

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---A NUMERICAL METHOD FOR THE

---·_.

COASTAL-T~~D

^WAVE

P~OSL~M

' \ '

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BY

\.\,___.'

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@ D~uid Michoel Hollon(

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-A Thesia Submitted to the- School or Graduate

...

Studies in partial (ulftllment of·tb._~

·,,

. _ _)

requirements ror the de.gree or Master or 'science

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·\l Department or, fbyaica

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Memorial yniveraity or Newroundland

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February 1087

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· ~erm~ssion .}las

beEm

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t ... oAhe ·National Library of

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_author (co~yright oWner) h a s r e s -e ·r v e d

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n~ither the thesi~- nor extensive ^I extracts from it may be printed or ot~erwise reproduced~ without hia/her written.per~i~siq~

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de vendre des exemplaires du film •. · · . ·

' L' auteur (titulaire! du droit' d'auteur) se r6serve ~~s

autr.es droits de· publ,J.catiom- ni 'la . these ni de longs

·extraits de cell·e-ci-· ne doivent . etr~ ' impri~6s· 6u autrement reproduits sans,son autorisation ecri"te. .

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A numerical me~od is presented to solve the ·coastal~trapp~d wav~. (CTW) ·

I .. -

j:_~ . _ . /,. ; j,ro~lem. Th~ met~Od

iS thO exj,lkitly, shirt:"

1

inverse power algo,ritbm .for the

r .

^{~->-~~·~ ___ / genera~ze}~d eig~nvalue problem. Problems -~C??·line~r in t~e eigenval~e, su.ch as

r. . . . - . . . .

~,~:

^· · · · . ' / . the

dispersi.V~

baroclil1it' CTW ··problem, are addressed

~Y

expressing them as

\j-- · i , ' I ' '

;~)·"-.,, '. I

•o • o • I '

:!·. · :. / li~ear:-prohlems of expanded dimen~io.n. The novelty' or the tech~iqu·e is it~ abil-

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ap~lieations

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m~thod ar~~ented.

First, the

elJ~t

pf stratification on CTW's~is investig~ted by ^·det-ermining their dispersion 4-~rac­

teris~ics. ^Th\se a~e

^•

-~om~uted

^• for 'both real propagating and ^{• • •}

co~plex ev~n~cent

^{' ' 0}

. '

.

^{'· .} . , .

w&Venumber

~nd;--the-inllue~l-a-me...tongs!J6re

^{eurrent on}

CTW~s

is stjldi . .'

-~ ~h·e· result~

are compared to

observabo~s

^qtaae

alo~·tne.

eastern Au'str'alian coast . l

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· Acknowlediemen.t!l

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It is wjth

q~p\ appr~ciation.

^{that I}

~cknowledge

my tesearch supervistr,

Dr,:

^l

LT,. ·Webster, for his invaluable assistance,· his availability, an~ his insight into· .

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^{. '}

.

~

cor:ec:ting many

~f.

the

o~staeles enc~untered

alOng. ihe. WaY.

! -~~h

to. thank the .

- ~any :

staff.

~r

the. Phys1cs Department

1 and

Co~put1ng Se~!if~

^Jt.t Memorial

' Unive~it)'or N~~t;ndland

^{whO have}

lls~isted

^me

~ith

^this

proj1~t.-

I woUld alsc;>. ··

:like to

than~'

Dr.

H.J.

Freeland for providing

· experim~ntal

data. \ ·

. . . ·. . ' . : ._ . ' . : . . J-· · . - . . . . · · ·. ' ' ' . :;

This'

work was _{. .}

· support~d

^{by a}

graduate

_.

· scholarshi~

_. ^{from. the}

Natur~l

_. ^Sci· _.

.

enc~

and Engine-ering.

.

_. Research

Council~~ Can~da.

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Table of-Contents

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Dd. t-. -~ .r .

e 1ca zoo.,, ... -' ... :~· .... ..

.J I' •

Abstract ... .-... : ... : ... : ... .

I~

..

~cknowledgements ... ~ ... · .... ~.: ... :~:. ... : ... _ ... :., ... .

. . ;11 .. • . . . •

. . .

_ ,

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Table

-or

Contents· ... : .... : ... t.: ... .-: ... ; ... .-... .-... : ... .' ... .

.

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^{, - -}

L.· t ·

r ·

^{F' · • · · ·}

. ts o · tgures ... _. ...•... :··········•······. :··~···;···.···~···.. . ^~ . ···

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I "' • • • - ·- ' ' I ' • ' ' o '• • • ' . . '

LISt f-'Fables .; ... _.: ... : .... : .. ::-:: ... : ... _ ... ..

'

..

I r. o' '

. . . I

. Chapter 1 INTRODUCTION ...

:.r ... ... .' ... _ ... ... : .... ... .' ..

' r

,J4">-- :.

Chapter 2 EQUATIONS OF MOTION ... : _. ... , ....... ~:

.

.. , ..... ..

2.1 · ·Momentum E;quations ... , ... _. ... : ... , ... ..

2.2 Mass Coo.servatioo and

Co~1uity E'!"~~i9o

... .' .... :

. ~ ··- . "-.. '

2.3 The Press~re . Eqtultion ^{. . .} ... ~ ... ~ ... ~ ... : ... ,. .... : ... ..

,.

_'

.

^{\ '}

2.4 · Th·e·Eigenvahie·Probl~m ... , ... : ... ~ ... ; ... · ... .

- . . to- . .

.

.

Chapter 3 NUMERICAL M~HODS ... : ... : ... : ... . 3.1 Coordinate Transfo~matioq · ... ..

• .

'

3.2 Finite Difference Forms ...

_.. L ... : ... ... : ... \ . . .. .. .

~: ...

3.3 Matrix Formulation ... · ... ~ ... ) ... .

. ^.

.

^~~

3.4 Aitlrnate"#Formulation ...

~

^{... ; ... .':} . .' ... : .. .' ... : ...

~

...

;~

^{..... .}

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. . .

-

^~

is Converg~nee

Criteria ... .' ... ,:: ... :.:, ....

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^{;· '., , ... (< ... . ' •' 1-·, • • •}

-'•• "'·1" ~ ~~~~~~l•~'~·· .... •.t..,\-..,_...,_,_.tf•:,.~.,.·~l.:l\~~i11}~·-~f~.._\'' rr • ..;~• ~·:. -~~~r. ~tot .. ,~~· :.•:•.}, n.·;·,..,·:;.~.ST.j! ¹\t>•:.:,

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4.a. Fixe~ Frequ~ut,:r: ... t ... -1--... : ... : ... , .... ( 40)

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-~4.4 Rate-or Co~vergence ... : ... :.;...~~: .... : ... : ... ;.; ... ; ... (43}

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. A'. . - - . . . ' . \ . . .

- ·Chapter 5 INCLUSION,OF

AllAROCLI~C

SHEAR CuRRENT : ..

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...

~5)

^{., .}

. . ' . . I . . , . . ' J . ·.

^~

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•, _....

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5 1 E- t' . \ .... ·. . . f ·' - • .(45)' . ' '

. qua IOnS -... i ... : ... :...... · . , : . :-

. . . · · ·: . , . .. . . ; . \ . . I . , . : . . . . .

^{t ... ·-}

5.2· Long-Wave Apprfxima.tion : ... · ... ~.: .. , ... ( ~sr.--:-:-' . . ·.. . \ ' . 1-.' '.. . : ' ': . ·, . . . ' '. . . ,. · ..

5.~ . Pr~ure Formulati\~n ... ~ ... : ...

1

.. ~ .... _. ... : .. · .. :::. .... : .. ··: .. : ... ( 49)

. p . ' ' . . . ' ^~- ' .

5.4 Mean Velocity and ~ell~·ty Fields _{. I ,} ... _. : ..._. ... : ... (54) _{. .}

. I • • .. I -

5.5 Appiication · ... : ... : .... : ...

L ... , ... \ ... ~

^(55}

. . • \ I . . ' •

5.6·; Results .: ... · ... J.' ... :.: ... (60)'

- ! . '

- . . . I • 1 ' ..

Chapter 6 CONCLUSI0NS ... J.. .. j ... : ... : ... : .. :~ ... (66)

. ~ I

• J' ' -~ i .

\ • I

Appendix

i

Derivation or

Transrormatio~

Equations ... ; ... ::-. (69,--'.

. 1 • ' •

Apj\~ndix

^B

~erivation

of Fini1Diffe;en:b

For~

... , ... (72).

Ap~endix . ,

^-.

^.c

^.

Ve~ificatiC>n

^{. . of}

kfmerical~~.

^.

.

erne .... : ...

~~.'

... ,! .... ; ... : ... :.: ... : ... ·. (74} ··

I .

Appendix.,P Analytical Kelvin W_ave Solu ion .: ... , ... ." ... : ... (79)

I . . . , ~

i . - \. '

R~ferences ... : ... _ ... : ... : ... :··· ... (82)

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Figure 3.2

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.Figure 5.3

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' Llst

ot

Figures

~·

•

,

Schematic of a shelf profiie showing the· coord'i~ate

syst~m. i

' ·

I

An example of t.he numerical discretization. of the .~er- turbatio~ preQure over the s~ell profile. .

I

Di.agr~mniatk. retation bet'Yee~ formulation of·. e1gen-

• value problem and the alter-nate. submatrix fo~rula­

tion. ·

...

. '•

' '

.../

Disp,rsion diagra~ .ror . t~e re~l . par.~ or/. t.he.

· wayenumber for propagatmg and evan~cent_CTW · . ··modes for. a barotropic ocean._ · · ·

• . • • • .... / 4 ~.

Dispersio~ diagram for · the · re~l part of the

·wavenumber ·for propagating and ev4'Descent CTW modes for a baroclinic ocean;

s =

~·

. ·

Dispersion. piagram for the rea~ part of the wavenumber for. propagating and evanescent CTW modes.for

!i

b~roclinic ocean;

s

^::~ 1.0.

-' . . .

Loci of wavenumber 'solutions ,on the complex plane for ,; ~ 0.3~ ~nd tor;

. s : .

^va_ried between

0.01

an~ ·(\4.

1 ·The mod~ ·nurnb~rs are-indicated.-:F--designat.es a f9r-

ward propagating mode,: B a· backward propaga~.ing

mode, ,ana E · and evan~cenr mode. ,. - ··

.• ,. • ' •. I '. • 'J

Compariso~ :

of acttmt:"Shel( profile

at~~~~

^Newcastle

line with the· fitted exponential shelf.

.~, . .

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^{. -- .--~ -~- - -}

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~

·.

C<?mpar~n

^of

cu~rent

^meter

~~~ .urements .wi~

mocleled·mean alongsho.re 'cu.rrent.

. .

Behavior·

or

alongshore. ~venumbef! ~ fo~ mode one CTW:. for ··varying .~ren&\h and direction of mean al.Qngshore .. c~rrent·. (NOO-diinensional frequency equal to 0.1~ · . • . . ,.

) . •'

5

'18

27

::.";',·.·.· 35.:

38

39

42

5'7

59

61

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Behavior or alongsho_re ·wavenumber " for /mode two

CTW

ror:_ .• eying strebgth . and djreetion or mean . alongsbore current. Non-dimensional .frequency ~qual

' '

'

to.O.l: , · · "' ^{a • •}

~havior

or

alongsh~re

wavenumber k for. mod'e three CTW for varying strength . and direction' or mean , -alongshore current. Non-dimensional frequency. equal

to O.l. .,

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63

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=.: ~

"

··"A_., ·

-'..t.~

'

·,

. ·'

. ^,. ,_';

0~

~ .,

' .

.

^.

:> . . .. --··:.

'• .. -

. '•

.. '

- ·-

0 . lo

.1'

.

.·

. '

, I

.

.

_..

...

~

-~

"

. •'i

\' J •••

:,, ···

~-:-~

~~>;.

. .

.

-

~(;:

..

)•

r::r::, •·.

~~~~.: ' ...

·.•· _,, ....

-

· -

}··· _,·. . .

·.:.

.

.

.

. . . .

....

-6.

:.1

-.

· . .

,

~ -'~-·-7rr·\-

..

~~ ·-i~~~:·~~·. i~. ^{·: ..} ~~;;: ·:! ~<·::·.~.:ry.:·: ~1 ^{..• : ..} ,.~::.~ :-::~ ^{· .•} :·::?~-~~: ·;~\~·:: ~:;::_~: ·.~::,.:·~ ~:-:·.: ^r. :·.>? ;:·· ,:~~-s ·<·~~~~:;:·~:~:·:

.. · ·

^~~·: ·:.T_:}·.-~>~:.;1;

-

. .

Table C.l

•

/ l .

..

-

^.

Comparison of_ wavenumber solutions or the numerical model with known analytica'tsolutions and solution or a different numerical mode_l for a mode one CTW . The

~fl'ect

^or.

horiz~ntal

^and

~eitical

grid· spacillg on the wayenumb~r ·solutio·D. fo~ a barotropic ocea~ Cor

~-mo4•. one, two,. _an<Vthree. ^: --. ..

'The efl'~t" or horizo~tal and vertical gri~ spac:i~g en ih.e ;.vavenumber s_olution for a baroclinic oc.ean Cot

:modes one, two, and three. J ^0•

- ^.//'

r

I .. ^\

t ~ ~.

. .

^.

/

-

'

-;.c.

] '1\-

_(..

. "

•

. ^'·

' ·.,

.-i

_\,

,. )

..

^~

·., ..•

\

·.'

- - ---

.

^.

.,.).

!

•

75

'77

,

.~ .

.,

'

. I

.

.

..

• .

~-

•

. .

r • • .

.

.

·. ^:,. .

. ^.... "·

l'

~:

'to ••

__

^{,. .}

'

.

. ......

..

^{.. ~}

. .

.. ^', r: ...

·:· ·. ,·,-

• ... ^{: '·}

. ' ' l • \.·

:.(.. /"' .. -:-V

.• . ' .... ~

.

..

.

-

. ^~

,.,

~.

' ^' ...

: ^-~

. . ,

' . ^~

•• "

' .

.

.

.:. '1

\· ' t ' I o

; :~-... .

.

^~ . . . ': .•

....

J

..

}

.

INTR'ODUC'I:J6N

•

^•

•

- ..

.

. ...

^:

.

^.,

, ·'

•

_. ^,

.

_'

....

_ _ r

~ ....

. A continental. shelf is the main topographic feature found along the co3.St-

'

. ^.

^.

. -

^{~ . . .}

.. ^-

^{. . . . .}

--- ^.

^{. ,·.-.:}

. lines ff the wo~ld. ·

A

s~elf ~tends from t~~ sho~e~ine:-t~ v~iable distance or .. ,:·:

· :.J . · \

, . --J ·. . W: ~ \ . • ^{I a< .,. -.. ,}

0 ,. , . ' • •

'be.tween lo. and 3oo:km offshore. where th~ water d~ptb is or the .order of··-200.

m. ·

• D • • • . . • • • • - • I .

. • . . • _..,. . ·: .t . . • . . : . • .• . • . "' • .

. Bey9nd ~his distance·: exists. th~. CQntinental s·Iope wbere the depth increases much :

·.·.". ..

. ·.~ .. .• ^{~ ~}

. ^.;\

.. ..

^{. .. ~ ~,,}

. more quickli..to~reach ·its :average abyssal value. of ·4000 m.:-·Becaus~ the shelf. and. .~:·

. . ' . · . . . . . •. \ r. , . \. ~ · ~ . · .. ~ · i ~-^••

:

~10~~· ~~tuat~d

o:pon a fotatini>;

e~:f~tbey a~t '!:"

^a

one-~i,eD.SiO~al

Viave,guid'. 0

\ 0

•• )

. , Cor. the propagatiOn pr coastal-trapped

w~ves

(CTWJs).__JI'he Importance or .\ ;

.~ ~

.

• r . , ,

' ' . . I' .

• 0

CTW's

is .. that' they·

:Co~m th~· d~mina·nt

circulation respohse of

-co~

^tal

waters~~

^:

·:-> ·:

• ofprcingo by the

at~osphere,

⁰

~

^{_ _ ) .1, . \,}

~

^•I

..

_;.___..

• '

CTW phenomeDJl.· have· been investiga.ted in .detail since Hamon 1Ig62l first · '

·~

.

^'

^{. ·}

dete,cte~

_,a tea'

l~v.el signa~~~ ~subi'nerti~l

'frequency

pr~pagating a.lon~

the eas!.

~

,. . coast

· ol

Australia.

Th~first . theoretic~(-

treatment or· Hamori's

observatio~:;

w-as

• , I " . -

. .

by "Robinson. (IQ6.4).

~-

_t:he

<(epth-int~grat~in·eariz:d

·

sh~w wa~er

equa- ^{! ~··• ../ . ·}

. · - - :· . . '· • _,. . .. ,. I

tions for

~-

rotating ocean,

iv~velike ~solutions

We're obtained which prop.ated .

· ~

_-.. I • • 00 \... ..r '

CL }t , ' ,. • ~,., , • ~-. • • ,

. phase with· the coast on the right (feft) Jn the northern. (soOthern) hemisp~ere.

·Later', Bus:hw~ld and. AdaDJs· (lg6s) developJid an analytical dispersion ,relation Jor.

'

-

' .

· thes~ topogr~phic~ Rossby waves· which showed -that they existed as ·discret¹e

• d • t • • • • . . .. - -~"" ^•

inodes·' ~nd that· &a.ch mode.

had

a sub inertial cuf:off.. frequency ab.ove which that

"

\:i

I _,. • .. t:

~ •:':

, ;I

':J . \

.

..

•' •.. • r, ,'' t

. .

^~--

..

^{:· ··.: .}

. ..

^{,. '}

. ·

•• ~ ; • • •• •• ' • • • -.. • • • • '• ' • • 1!. • ' ., '

\· ·. . ·n:tOd~ could QOt ,exist. Brink. [19801· showeq that wavelike s_olutioris .do o~u(' /::~~

-~·, • o o I 0 ~ o' ' - ' ' . . . '"' :?'

{: .. · '

a~ove

^{: th_e.}

~ut-oH

^.

rr~~~~ncy,~·

^but

<i'hatr. -~hese

^.

~od~t

^{we;e .}

evanes~~~'t:

^"

^{The •} ·~ ^{(/ ' .} ~}~

110::::: , ' .

^{0 '}

0

' ' 0 . :. ' . 0 .. ' .

0

. ; :

'\- . I . , : '' ..,· ,' • ' • , ,. • :~ .,.·

,~·l_ :t(~J;J~~;.;A:~:-:\~~\~is.~r~V:; _, · : · . ; ,:· .-.: · . . . .-. ,>.-· . . : ... ;·· ;. ' .: : .~ .. ~.~:·:; r+·:<·~. ~~{k.~;;.~c . . ~:~:A~; / .4·::.:~.~i.~:.%i. .·~~\ . : : · · ::.:·:\\:~ . . · .. ... : · ~:-~. '-~ ·

^J·.:

~ ^{<.: . . ,: · : ·.;,. ; :} ~ .·~~- ^{. ..} ··~~~~:~~:::~

^:,f;

c