GEF DYNAMIC OCEANOGRAPHY: Waves and wave-induced mass transport in the ocean

Transcription

1 GEF46 - DYNAMIC OCEANOGRAPHY: Waves ad wave-iduced mass aspo i he ocea JAN ERIK H. WEBER Depame of Geoscieces ecio fo Meeoolog ad Oceaogaph Uivesi of Oslo j.e.webe@geo.uio.o Auum 3

2 CONTENT I. GOVERNING EQUATION FOR THE OCEAN p. 4. Momeum ad mass cosevaio. Equaios fo he Lagagia volume aspo.3 hallow wae damics.4 Cosevaio of poeial voici.5 The som suge equaios II. ADJUTMENT UNDER GRAVITY IN A HOMOGENEOU NON- ROTATING OCEAN p. 4. Liea waves i a ocea of fiie deph. Wave goups ad goup veloci.3 The moio of a pulse i a shallow chael.4 Validi of he hdosaic appoimaio.5 Eeg aspo i suface waves.6 The okes edge wave.7 Wave kiemaics.8 Applicaio o a slowl-vaig medium Ra heo Dopple shif III. HALLOW-WATER WAVE IN A ROTATING NON-TRATIFIED OCEAN...p The Klei-Godo equaio 3. Geosophic adjusme 3.3 vedup ad Poicae waves 3.4 Eeg flu i vedup waves

3 3 3.5 Coasal Kelvi waves 3.6 Amphidomic ssems 3.7 Equaoial Kelvi waves 3.8 Topogaphicall apped waves 3.9 Topogaphic Rossb waves IV. HALLOW-WATER WAVE IN A TRATIFIED ROTATING OCEAN...p Two-lae model 4. Baoopic espose 4.3 Baocliic espose 4.4 Coiuousl saified fluid 4.5 Fee ieal waves i a oaig ocea 4.6 Cosa Bu-Väisälä fequec 4.7 Ieal espose o wid focig; upwellig a a saigh coas V. WAVE-INDUCED MA TRANPORT p The okes dif 5. Applicaio o dif i o-oaig suface waves ad vedup waves 5.3 Relaio bewee he mea wave momeum ad he eeg desi 5.4 The mea Euleia volume flu i shallow-wae waves 5.5 Applicaio o aspo i coasal Kelvi waves Radiaio sess Mea Euleia flues REFERENCE p. 98

4 4 I. GOVERNING EQUATION FOR THE OCEAN. Momeum ad mass cosevaio We sud moio i a ocea wih desi. The ocea is oaig abou he z-ais wih cosa agula veloci Ω siϕ whee ϕ is he laiude ad Ω is he agula veloci of he eah assumed cosa hee). Fuhemoe ) ae hoizoal coodiae aes alog he udisubed sea suface ad he z-ais is dieced upwads. The especive ui vecos ae i j k ). The posiio of he fee suface is give b z η ) whee η is efeed o as he suface elevaio ad is ime. The amospheic pessue a he suface is deoed b ). The boom opogaph does o va wih ime ad is give b z H ) ; see he skech i Fig... P Fig. Defiiio skech. The veloci i he fluid is v u v w) ad he pessue is p. The momeum equaio i a fame of efeece fied o he eah ca he be wie Dv v + v v f k v p gk + F v)..) d

5 5 whee i / + j / + k / z is he gadie opeao. Fuhemoe g is he acceleaio due o gavi ad f Ωsiϕ is he Coiolis paamee. I..) we have egleced he hoizoal compoe of he Coiolis foce he idal foce ad he effec of he ceifugal foce due o he eah s oaio) o he appae gavi. If we le he -ais poi ohwads f is ol a fucio of. We ma he wie appoimael ha f df f + f + β..) d whee f Ωsiϕ d β R dϕ Ω R ) Ωsiϕ cos. ϕ ϕ This is called he bea-plae appoimaio. We have deoed he ficio foce o a fluid paicle b Fv )..3) i..). I ca ake vaious foms depedig o he flow codiios. Fo lamia flow of a icompessible Newoia fluid i becomes F υ + + v v z υ..4) whee υ is he molecula viscosi ad / + / + / z is he Laplace opeao. I cases whe a lage scale mea moio occus i a ubule eviome we ma ake F v A..5) ) ) z) whee A A + A + A z. Hee A ) A ) A z) ae he ubule edd viscosi coefficies i he - - ad z-diecios especivel o fo sho; edd

6 6 viscosiies). The edd viscosiies A ) A ) ad A z) ae geeall diffee bu he ae all much lage ha he molecula viscosi. Usuall we have ) ) z) A ~ A > A >> υ...6) The edd viscosiies ca va i ime ad space bu we hee assume ha he ae cosas. I some cases whee i is impoa o ioduce ficioal dampig wihou complicaed mahemaics we ma ake F v..7) whee is a cosa ficio coefficie. This las vesio is called Raleigh ficio ad is fomall simila o ficioal dampig i a poous medium Dac ficio). Fiall i applicaios whee oe sudies he veicall iegaed fluid popeies he hoizoal ficio foce compoes ae ofe epessed i ems of he hoizoal ficioal shea sesses τ ) τ ) as ) ) ) τ ) τ F F..8) z z The cosevaio of mass fo a fluid paicle ca be epessed mahemaicall as D v + v d..9) As log as we do o coside soud waves we ca eglec he small vaiaio of desi followig a fluid paicle. The cosevaio of mass he educes o v...) This elaio he coiui equaio) acuall epesses he cosevaio of volume. I is of couse eac fo a fluid of cosa desi homogeeous icompessible fluid). Howeve we shall use..) houghou his e fo all oceaic applicaios. ice he fee suface is a maeial suface he kiemaic bouda codiio ca be wie as

7 7 o equivalel D z η) d The kiemaic bouda codiio a he boom becomes z η )..) Dη w z η...) d D z + H ) d z H )..3) o w v H z H...4). Equaios fo he Lagagia volume aspo B iegaig he coiui equaio v i he veical ad applig he bouda codiios..) ad..4) we fid eacl η η H udz η H vdz..) whee a subscip deoes paial diffeeiaio. Thoughou his e we will aleae bewee wiig paial deivaives i full ad fo ecoomic easos) as subscips. The iegals i..) ae volume aspos pe ui legh i he - ad -diecio especivel. ice we hee iegae bewee maeial sufaces he boom ad he fee suface) heses flues ae he Lagagia volume flues: U V L L η H η H udz vdz...) This meas ha..) capues he oal flu of fluid paicles hough veical plaes. Hece..) becomes η U V...3) L L

8 8 I he momeum equaios..) we appl he Boussiesq appoimaio i.e. we assume ha he desi chages ae ol impoa i coecio wih he acio of gavi. This meas ha we ca ake whee is a cosa efeece desi i he hoizoal compoes of..). Iegaig he acceleaio em i..) usig he bouda codiios we fid eacl η H η H u v + v u) dz U + v v) dz V L L + + H η η H u dz + uvdz + H η η H vu dz v dz....4) Assumig ha p P ) z η we obai fom he hoizoal pessue ems i..) η H η H pdz pdz η H η H pdz Pη P H pdz Pη P H B B..5) whee we have defied he boom pessue p H ). We he ma wie fo he hoizoal flues P B U V L L + fv fu L L η H η H P PB pdz + η + H P PB pdz + η + H + + η η η ) F dz u dz H H H η η η ) F dz uvdz H H H vudz v dz...6) I lae applicaios we shall simplif hese eac equaios eac ude he Boussiesq appoimaio) ad fid hem ve useful..3 hallow wae damics

9 9 If he hoizoal legh scale of he moio is ve much lage ha he veical legh scale which eve ca be lage ha he ocea deph) he mai balace i he veical momeum equaio..) is hdosaic i.e. p z g..3.) This is he basis fo wha we deoe as shallow-wae damics. I meas whe we eu o he veical compoe i..) ha he veical acceleaio Dw/d ad he ficio foce mus be so small ha he do o oiceabl ale he hdosaic pessue disibuio. A moe quaiaive discussio of his poblem is foud i ec..4. I his case we ca wie he pessue η p g z' ) dz' + P )..3.) Fo a homogeeous ocea he desi is cosa ). The z p g z ) ) η P If we disegad he effec of ficio fo his case he hoizoal compoes of..) ca hus be wie Du d fv gη.3.4) P Dv + fu gη P.3.5) d We ealize ha he igh-had sides of.3.4) ad.3.5) ae idepede of z. B uilizig ha v D/d ad f f + β.3.4) ca be wie D u f β.3.6) d gη P Fom.3.6) i follows ha D u f / ) / d is idepede of z. Thus his is β also ue fo u f β / ) ad heeb also fo u if u ad v wee idepede of

10 z a ime. imilal fom.3.5) we fid ha v is idepede of z. We ca accodigl wie u u ) v v )..3.7) Fuhemoe i ow follows fom..) ha w z is idepede of z. Hece b iegaig i he veical: w u + v ) z C )..3.8) + The fucio C is obaied b applig he bouda codiio..4) a he ocea boom. The veical veloci ca hus be wie w u + v ) z + H ) uh vh..3.9) ice u ad v hee ae idepede of z.3.4) ad.3.5) educe o u + uu + vu fv gη P.3.) v + uv + vv + fu gη P.3.) Fom..3) we easil obai u H + η) ) + v H + η) ) + η..3.) To solve his se of equaios we equie hee iiial codiios e.g. he disibuio of u v ad η i space a ime. If he fluid is limied b laeal boudaies walls) we mus i addiio esue ha he soluios saisf he equiemes of o flow hough impemeable walls. We epea ha he validi of.3.)-.3.) es o i): hdosaic balace i he veical diecio shallow-wae assumpio) ii): cosa desi ad iii): o ficio..4 Cosevaio of poeial voici

11 We eu o he iviscid homogeeous shallow-wae ocea. Fo his case we ma deive a ve poweful heoem goveig he poeial voici. Fis we defie he veical compoe of he elaive voici i ou coodiae ssem b ζ v u..4.) I addiio eve paicle i his coodiae ssem possesses a plaea voici f aisig fom solid bod oaio wih agula veloci Ω siϕ. Hece he absolue veical voici fo a paicle becomes f + ζ. We shall deive a equaio fo he absolue voici. I is obaied b diffeeiaig he equaios.3.) ad.3.) b / ad / especivel ad he add he esulig equaios. Mahemaicall his meas o opeae he cul o he veco equaio o elimiae he gadie ems. ice f is idepede of ime we fid ha D d f + ζ ) f + ζ ) u + v )..4.) B usig ha H is idepede of ime.3.) ca be wie D d H + η ) H + η) u + v )..4.3) Hee H + η is he heigh of a veical fluid colum. We defie he poeial voici Q b f + ζ Q..4.4) H + η B elimiaig he hoizoal divegece bewee.4.) ad.4.3) we fid fo Q ha DQ d..4.5) This equaio epesses he fac ha a give maeial veical fluid colum alwas moves i such a wa ha is poeial voici is coseved.

12 Aleaivel we ca appl Kelvi s ciculaio heoem fo a iviscid fluid o deive his impoa esul. Kelvi s heoem saes ha he ciculaio of he absolue veloci aoud a closed maeial cuve alwas cosisig of he same fluid paicles) is coseved. Fo a maeial cuve Γ i he hoizoal plae Kelvi s ad okes heoems ield Γ v abs δ k v ) δσ cos..4.6) whee σ is he aea iside Γ. Fuhemoe i he suface iegal: σ k v abs) Whe he suface aea σ i.4.6) appoaches zeo we have abs f + ζ..4.7) f + ζ ) δσ cos..4.8) I addiio he mass of a veical fluid colum wih base δσ mus be coseved ad hece H + η) δσ cos..4.9) This is valid fo all imes sice a veical fluid colum will emai veical; see.3.7). I ou case he fluid is homogeeous ad icompessible i.e. is he same fo all paicles. Thus b elimiaig δσ bewee.4.8) ad.4.9) we fid as befoe ha o equivalel DQ / d. f + ζ Q cos..4.) H + η I he ocea we usuall have ha ζ << f ad η << H. Fo saioa flow assumig ha H >> η ad f H >> H ζ.4.5) ields appoimael ha O a f-plae his equaio educes o v f / H )..4.)

13 3 v H..4.) Accodigl he flow i his case follows he lies of cosa H i.e. he boom coous). This pheomeo is called opogaphic seeig. O a bea-plae he flow will follow he coous of he fucio f / H ; see.4.)..5 The som suge equaios Fom epeiece we kow ha whe i comes o compuig he chage of sea level due o amospheic wid ad pessue fields we ca appl he hdosaic appoimaio.3.) ad eglec he desi vaiaio i he veical. Fo such moio efeed o as som suge he wae appeas o be quasi-homogeeous ad we ca use a cosa efeece desi evewhee. Fuhemoe he hoizoal velociies ae fail small which ca jusif he eglec of he oliea covecive acceleaio ems o he igh-had side of..6). This lieaizaio is also cosise wih he assumpio ha η << H. The volume flues i his liea poblem ae he Euleia flues give b H U udz V vdz..5.) E Uilizig a ficio foce of he pe..8) we he fid fo he som suge poblem: E H U V E η U + V ). E fv + fu E E E ghη HP E ghη HP / + τ ) / + τ / τ ) ) B τ B / ).5.) Hee τ τ ) ) suface z ad ) ae he wid sesses alog he mea posiio of he ocea τ τ ) ae he ficioal sesses a he boom z H ). Fo ) B ) B opeaioal use he suface pessue gadies ae obaied fom weahe

14 4 aalses/pogoses ad he wid sesses ae usuall elaed o he wid speed u v ) a m heigh hough τ a c D v v..5.3) Hee a is he desi of ai ad cd is a dag coefficie which is picall i he 3 3 age 3 highe values fo soge wids). The boom ficio is moe difficul o model. omeimes a liea ficio i he flues is applied i.e. τ KV.5.4) B whee K is a cosa boom ficio coefficie. Moe fequel ficio laws ha ae quadaic i he mea veloci ae used a he boom. I is impoa o ealize ha.5.) is a lieaized se of equaios fo he Euleia volume flues.5.). Ulike he oliea Lagagia flues..) he do o coai a mea wave momeum. Hece he som suge equaios ol ield he suface elevaio ad mea cues iduced b wid sess ad amospheic pessue gadies alog he sea suface. I Chape V we eu o he iiguig poblem of mea cues iduced b suface waves i he ocea. E II. ADJUTMENT UNDER GRAVITY IN A HOMOGENEOU NON- ROTATING OCEAN. Liea waves i a ocea of fiie deph Fo a homogeeous fluid a es he suface is hoizoal. If we iiiall esablish a suface elevaio which deviaes fom he hoizoal he subseque moio will be i he fom of suface gavi waves. ice he desi of he ocea is abou oe housad imes lage ha he desi of he amosphee we ca eglec he effec of he ai o he oceaic wave moio. I his chape we coside suface gavi waves

15 5 wih sho peiods much shoe ha he ieial peiod π / f ~6 hs a mid laiude). I is obvious ha he eah s oaio will have ve lile effec o he obial moio i such waves so we ca eglec i. Fo he mome we also eglec he effec of ficio o he wave moio. This is moivaed b he fac ha wid-geeaed waves i he ope ocea swell) ma popagae fo hudeds of kilomees wihou beig seveel damped. Fom..) he momeum equaio ow educes o Dv p gk..) d whee is he cosa desi. Fo his case we have fom Kelvi s heoem fo he veloci ciculaio alog a maeial closed cuve γ : d d γ v d...) If he veloci ciculaio iiiall is zeo which we hee assume i will emai zeo fo all imes i.e. The he veloci ca be deived fom a poeial φ i.e. o v d...3) γ v φ..4) u φ v φ w φ...5) z Accodigl fom he coiui equaio v we obai φ...6) I geeal we have ha Howeve we will see lae o ha he effec of ficio as well as he Coiolis foce will be impoa fo deemiig he oliea mea cue he dif) iduced b suface waves.

16 6 v v v + v v )...7) ice he las em hee he voici) is zeo fom..4) we ealize ha..) ca be wie p + φ + φ) + gz...8) Whe iegaig his equaio i space he iegaio cosa ca be se equal o zeo. Hece p φ φ) This is he Eule equaio fo he pessue. g z...9) If he ocea bed is fla which we assume hee ad siuaed a z H we mus have a he ocea boom w φ z H...) z This cosiues he kiemaic bouda codiio a he ocea boom. I his chape we coside waves wih small ampliudes. As a fis appoimaio we eglec ems i he goveig equaios ha ae popoioal o he squae of he wave ampliude i.e. we lieaize ou equaios. I his appoimaio he kiemaic bouda codiio a he suface becomes η φ w z..) z We coside a wave soluio i he fom of a comple Fouie compoe η Aep i k ω))...) Fom..6)..) ad..) we he obai iωacosh k z + H ) φ epi k ω)...3) k sih kh )

17 7 Hece he eal pas of he velociies i he ocea ca be wie ωacosh k z + H )) u cos k ω) sih kh ) ωasih k z + H )) w si k ω). sih kh )..4) Fo he eal pa of he pessue we fid fom he lieaized vesio of..9) ha p ω Acosh k z + H )) cos k ω) gz...5) k sih kh ) Fo suface waves i he ocea we ca eglec he effec of he ai above he wae. This meas ha we ca ake p a he suface. Hece fom he damic bouda codiio p η) he lieaized vesio of..9) ields ω Acosh k η + H )) η Acos k ω) cos k ω)...6) gk sih kh ) Uilizig ha η << H we obai fo he fequec ω gk ah kh )...7) Fo waves popagaig i he posiive -diecio we fid fo he phase speed ha / ω gλ ahπh / λ) c...8) k π I is eadil see ha c iceases moooicall wih iceasig wavelegh. uch waves ae called dispesive waves posiive dispesio). Hece fo a esemble of waves wih vaious waveleghs geeaed a a ceai locaio he loge waves will move fase ad disappea fom he geeaio aea. This is like ocea swell escapig fom he som cee. The eeme cases of..8) ae a): Deep-wae waves kh >> ). The / c ω gλ...9) k π

18 8 b): Waves i shallow wae kh << ). The / c gh )...) To fis ode i wave ampliude we fid ha idividual fluid paicles i suface wave moio moves i closed pahs. If he Lagagia coodiaes of a sigle paicle is z ) we ca wie L L whee u ad w ae give b..4). Defiig we fid fom..) ha zl u w..) L Acosh k z + H )) Asih k z + H )) R R..) sih kh ) sih kh ) L ) z ) L z +...3) R R We ealize ha he paicle pah is ellipic wih cee i ). The majo half ais z is R ad he mio half ais is R. The boh decease wih deph. Fo ifiiel deep wae R R ad he paicles move i cicles. We shall see i Chape V ha whe we coside oliea wave moio he paicle pah is o closed. Each paicle has a fowad spiallig moio which gives ise o a mea fowad dif of paicles. This meas ha waves do iduce a cue i he medium hough which he popagae.. Wave goups ad goup veloci Up o ow we have cosideed oe sigle wave compoe. If we have wo wave compoes he same ampliude bu wih slighl diffee wave umbes ad fequecies he ca be wie i comple fom as

19 9 η+ Aepi{ k + k) ω + ω) } η Aepi{ k k) ω ω) }..) whee k / k << ω / ω <<. Each of he wo compoes above is a soluio o ou wave poblem. ice we wok wih liea heo also he sumη of he wo compoes becomes a soluio. This supeposiio ca be wie + +η η + + η Aepi k ω) ω Acos k epi k ω). k [ epi k ω) + ep i k ω)) ]..) We deoe he eal pa of..) bη epeseig he phsical soluio. We he fid ω ω η Acos k cos k...3) k k This shows ha η is a ampliude-modulaed wave ai cosisig of seies of wave goups as show i Fig.. whee we have ploed η / A as a fucio of fo k / k.. Fig.. kech of wave goups. The idividual waves i he goup will popagae wih he odia phase speed c ω / k while he goup iself will popagae wih he goup veloci ω / k. c g

20 I he limi whe k he goup veloci becomes he deivaive of he fequec wih espec o he wave umbe i.e. c g dω...4) dk ice ω kc ad k π / λ we oe ha..4) ca be wie as c g dc c λ...5) d λ o if he phase speed iceases wih iceasig wavelegh omal dispesio) he c g < c. If he phase speed is idepede of he wavelegh o-dispesive waves) we have ha c g c. I is a simple eecise o show fom..7) ad..4) ha he geeal elaio bewee he goup veloci ad he phase veloci fo suface waves becomes c g kh +...6) c sihkh ).3 The moio of a pulse i a shallow chael I he pevious aalsis we have used he cocep of Fouie compoes o descibe he wave fom. Howeve fo shallow-wae waves which ae o-dispesive we ca easil deive soluios fo abia suface displacemes. We assume small disubaces fom he sae of equilibium i he ocea wo-dimesioal moio / v ) ad cosa deph. Fo lieaized shallow-wae waves i he -diecio..) educes o u gη η Hu..3.) Elimiaig he hoizoal veloci we fid η ghη..3.)

21 This equaio is called he wave equaio ad appeas i ma places i phsics. Isead of assumig a sigle Fouie compoe as soluio of his equaio we ealize immediael ha a geeal soluio ca be wie η F + c ) + F c ).3.3) whee c gh / ). If a ime he suface elevaio was such ha η F) ad η i is eas o see ha he soluio becomes η { F + c) + F c) }..3.4) Fom.3.) ad.3.3) we fid fo he acceleaio u g gη { F' + c) + F' c) }.3.5) whee F ' ξ ) df / dξ. Hece he hoizoal veloci is give b g u { F + c) F c) }..3.6) c Fom.3.4) we ca displa he evoluio of a iiiall bell-shaped suface elevaio F) wih pical widh L; see he skech i Fig...

22 Fig.. Evoluio of a bell-shaped suface elevaio. We oe ha he iiial elevaio splis io wo ideical pulses movig igh ad lef wih veloci c gh) /. I a deep ocea H 4 m) he phase speed is c m s while i a shallow ocea H m) we have c 3 m s. If he maimum iiial elevaio i his eample is h i.e. F) h we fid fom.3.6) ha he veloci i he ocea diecl below peak of he igh-had pulse ca be wie gh u.3.7) c whe >> L/c ha is afe he wo pulses have spli. If we ake h m as a pical value he deep ocea eample ields u.5 cm s while fo he shallow ocea we fid u 7 cm s. As a secod eample we coside a iiial sep fucio:

23 3 h > F ).3.8) h <. I his case he veloci ad ampliude developme becomes as skeched i Fig..3. Fig..3 Evoluio of a suface sep fucio. I is obvious ha we i a eample like his wih a sep i he suface a ) mus be caeful whe usig liea heo which equies small gadies. I a moe ealisic eample whee diffeeces i heigh occus he iiial elevaio will have a fial a quie small) gadie aoud. Qualiaivel howeve he soluio becomes as discussed above..4 Validi of he hdosaic appoimaio Le us coside he validi of he hdosaic appoimaio i he case of waves i a o-oaig ocea. We ewie he pessue as a hdosaic pa plus a deviaio:

24 4 whee p g η z) + P + '.4.) p p' is he o-hdosaic deviaio. The veical compoe of..) becomes o lowes ode: w.4.) p z while he hoizoal compoe ca be wie u gη p..4.3) The hdosaic assumpio implies ha p << u..4.4) If he pical legh scales i he - ad z-diecios ae L ad H especivel we obai fom he coiui equaio ha L u ~ w.4.5) H whee ~ meas ode of magiude. Fom.4.) we he fid p' ~ H L u..4.6) Uilizig his esul he codiio.4.4) educes o H / L <<..4.7) Thus we ealize ha he assumpio of a hdosaic pessue disibuio i he veical equies ha he hoizoal scale L of he disubace mus be much lage ha he ocea deph. Fo a wave L is associaed wih he wavelegh; fo a sigle pulse L coespods o he chaaceisic pulse widh..5 Eeg aspo i suface waves

25 5 As meioed i ecio. a local wid eve i he ope deep ocea geeaes wid waves wih ma diffee waveleghs. ice such waves ae dispesive he loges waves will avel fases. Fo eample fo a wavelegh of 3 m we fid ha he phase speed is eal m/s. These waves ma popagae fase ha he low pessue ssem ha geeaed he ad hece escape fom he som egio. uch waves ae called swell ad ma popagae fo hudeds of kilomees hough he ocea ill he fiall each he coas gaduall asfomig o shallow-wae waves. Fiall he beak i he suf zoe o he beach ad loose hei mechaical eeg. I his wa we udesad ha waves ae caies of eeg. The ge hei eeg fom he wid popagae he eeg ove lage disaces ad loose i b doig wok o he beaches i he fom of beach eosio pocesses ec. If hee is a es mechaical eeg i is asfeed o hea i he beakig pocess. The oal mechaical eeg E pe ui aea i suface waves is he sum of he mea kieic eeg whee E Ek ad he mea poeial eeg E p. Pe defiiio T η T u w dz d u w dz d T T + ) + ).5.) H H k T π / ω is he wave peiod. Fo peiodic wave moio we assume ha he poeial eeg is zeo a he mea suface level. Hece E p T T η g zdz d..5.) Iseig fom..) ad..4) we obai afe some algeba ha E k 4 E p ga..5.3) Hece he mechaical eeg is equall paiioed bewee kieic ad poeial eeg. The oal eeg pe ui aea ofe efeed o as he eeg desi becomes

26 6 E E E ga k + p..5.4) The mea hoizoal eeg flu F e is he wok pe ui ime doe b he damic flucuaig) pessue i displacig paicles hoizoall. B defiiio F e T T η H pudz d T T H pudz d..5.5) Applig he hoizoal veloci i..4) ad he damic pessue i..5) leavig ou he saic pa gz ) we fid 3 ω A F e + 8k sih kh sihkh ) kh )..5.6) Uilizig he dispesio elaio..7) ad he goup veloci give b..6) we ca wie he mea eeg flu.5.6) as F e cge..5.7) I ou ealie eame of he goup veloci i was defied fom a puel kiemaic poi of view. We udesad fom.5.7) ha he goup veloci has a much deepe sigificace: I is he veloci ha he mea eeg i he wave moio avels wih. Accodigl o eceive a sigal ha popagaes ove a disace L i he fom of a wave we mus wai a ime L / c befoe he eceive picks up he sigal. g.6 The okes edge wave okes 846) discoveed a suface wave ha could eis i a ocea whee he boom was slopig lieal; see he skech i Fig..4 whee he slope agle is β.

27 7 Fig..4 kech of he okes edge wave. I he absece of viscosi ad oaio he soluio ca be deived fom he Laplace equaio..6). Fo a wave i he -diecio we ca wie: φ i k ω) F z) e..6.) The Laplace s equaio educes o F F + z k F..6.) We coside epoeiall apped waves i he diecio omal o he coas ad assume ha he soluio decas epoeiall wih deph i.e. F Ce a+ bz a b >..6.3) Hece fom.6.) The kiemaic bouda codiio a he slopig boom is: o Fom.6.6) we obai ha a + b k..6.4) w v h z h.6.5) φ a β ) φ z a β..6.6) z b a a β. Iseig io.6.4):

28 8 Hece we ca wie he veloci poeial a k cos β b k si β..6.7) k cos β + kz si β + i k ω) ) φ C ep..6.8) Fom he lieaized kiemaic bouda a he suface..) we fid fo he suface elevaio ha k cos β + i k ω) ) η Aep..6.9) whee A ick si β / ω. The damic bouda codiio a he suface is p z η). Fom he lieaized vesio of..9) we obai φ + gη z..6.) B iseig io his equaio we fid he dispesio elaio ω gk si β..6.) This esul is valid fo < β < π /. We oe ha his apped wave called he okes edge wave ca avel alog he coas i boh diecios due o he wo possible sigs i.6.). Whe he beach slope is small β << ) we ca aalse his poblem b usig shallow wae heo. We he ealize ha he appig ca be eplaied b he fac ha he local phase speed gh iceases wih iceasig disace fom he coas. If we epese he wave b a a which is dieced alog he local diecio of eeg popagaio e.g. ecio.8 he a will alwas be gaduall efaced owads he coas. A he coas he wave is efleced ad he efacio pocess sas all ove agai. The oal wave ssem hus cosiss of a supeposiio bewee a icide ad a efleced wave i a aea ea he coas. The widh of his aea depeds o he agle of icidece wih he coas fo he a i quesio. Ouside his egio he wave ampliude deceases epoeiall.

29 9 Whe we aalse his poblem moe hooughl we fid ha he okes edge wave is he fis mode i a specum of shelf modes ha coais boh discee ad coiuous pas; see LeBlod ad Msak 978) p.. If we ake he eah s oaio io accou f ) he fequecies fo he edge waves i he posiive ad egaive -diecios will be slighl diffee..7 Wave kiemaics We ca geealize he esul i his chape o wave popagaio i hee dimesios. Le ψ deoe he veloci poeial o he seam fucio of a plae wave. B ioducig a wave umbe veco κ defied b κ k.7.) i + ki + k3i3 ad a adius veco whee.7.) i + i + 3i3 we ca wie a plae wave as ωκ ψ Aep i κ ω)) Aep{ iκ }..7.3) κ The vecoial phase speed c is ow defied b c ω κ κ κ + k + k k ) Fuhemoe we ca wie he compoes of he vecoial goup veloci c g as c c c ) g ) g 3) g ω / k ω / k ω / k ) I veco oaio his becomes c κ ω g κ i + i + i3..7.6) k k k 3

30 3 If he fequec ω ol is a fucio of he magiude of he wave umbe veco i.e. ω ωκ ) we efe o he ssem as isoopic. If we cao wie he dispesio elaio i his wa he ssem is aisoopic. We ow coside he suface i wave umbe space give bω ω k k k ) C whee C is a cosa; see Fig..5 3 whee we displa a wo-dimesioal eample. Fig..5 Cosa- fequec suface i wave umbe space. The gadie κ ω is alwas pepedicula o he cosa fequec suface. Fom.7.6) we oe ha his meas ha he goup veloci is alwas dieced alog he suface omal as depiced i Fig..5. ice he phase veloci is dieced alog he wave umbe veco e.g..7.4) we ealize ha if he phase speed ad goup veloci should become paallel he he cosa fequec suface mus be a sphee i wave umbe space. Mahemaicall his meas ha ω ωκ ) i.e. we have a isoopic ssem..8 Applicaio o a slowl-vaig medium

31 3 If he medium hough which he waves popagae is o compleel spaiall uifom o cosa i ime he wave ai will va as i popagaes. If he legh ad ime scales ove which he medium vaies ae lage compaed o he wavelegh o wave peiod he local popeies of he wave will va slowl houghou he field. If ζ epeses he displaceme of a fluid eleme he wave ai ca be specified b ζ Aep iθ ) whee A is he local ampliude which is a slowl vaig fucio of posiio ad ime ad θ ) is he phase fucio. The wave umbe κ ad he adia fequec ω which boh ma be slowl vaig fucios of space ad ime ca ow be defied as Fom his i follows ha κ θ ω θ..8.) κ..8.) Hece he disibuio of he local wave umbe i space is ioaioal. Fuhemoe fom.8.) κ + ω..8.3) This ca be cosideed as a kiemaical cosevaio equaio fo he desi of waves. I a adom field of lieal supeposed waves.8.3) holds fo each Fouie compoe. Fo a sead wave field ω. If he waves popagae i he - diecio ad he dispesio elaio have he fom ω ω k H )) we have fo his case ha dω ω dk ω dh + d k d H d..8.4) Fo eample fo shallow wae waves o a gel slopig beach we have fom..7) ha ω gh )) / k. B iseig io.8.4) ad iegaig we eadil fid fo his case ha

32 3 / H ) k whee k H ae he wave umbe ad he deph a k.8.5) H ). We oe fom.8.5) ha whe he wave popagaes io shallowe wae like a suami appoachig he shoe he wave umbe iceases. Accodigl he wavelegh becomes smalle. Togehe wih iceasig wave ampliude his is seepes he wave which ulimael leads o beakig i he suf zoe. Ra heo The wave eeg popagaes i he diecio of he goup veloci veco. We ca defie he eeg pah o a as he cuve i o-dimesioal space whee he age a each poi is alog he goup veloci i.e. d c g..8.6) Fo eample i he hoizoal plae d di + dj ad hece he equaio fo he a becomes d c..8.7) d c ) g ) g If he goup veloci compoes ae idepede of ad he a F) becomes a saigh lie. Howeve if we fo eample coside shallow wae waves i a ocea wih a slowl vaig deph he goup veloci compoes will va slowl wih he hoizoal coodiaes. The he a will be cuved as meioed i coecio wih edge waves i ecio.6. Dopple shif

33 33 I his aalsis he fequec ω is he fequec fo waves popagaig i a medium a es. If ow he fluid moves wih a veloci U which ca be a slowl vaig fucio of space ad ime ω is he fequec ha will be foud b a obseve movig wih he udisubed fluid veloci. I is called he iisic fequec ad ca be obaied fom he dispesio elaio. Howeve he fequec measued b a obseve a es o he appae fequec will be ω + κ U..8.8) Whe he wave ad he medium move i he same diecio he las em is posiive ad he fequec appeas o icease highe oe) fo a fied obseve while i deceases lowe oe) whe he move i opposie diecios. This pheomeo is kow as Dopple shif. III. HALLOW-WATER IN WAVE IN A ROTATING NON-TRATIFIED OCEAN 3. The Klei-Godo equaio We ow coside he effec of he eah s oaio upo wave moio i shallow wae. Liea heo sill applies ad we ake he deph ad he suface pessue o be cosa. Fuhemoe we assume ha f is cosa. Equaios.3.)-.3.) he educe o u fv gη 3..) v + fu gη 3..) η H u + v ). 3..3) + We compue he veical voici ad he hoizoal divegece especivel fom 3..) ad 3..). B uilizig 3..3) we he obai

34 34 f v u ) η 3..4) H ad η H + f v u ) g η + η ). 3..5) The voici equaio ca be iegaed i ime i.e. v f f u η v u η 3..6) H H whee sub-zeoes deoe iiial values. We assume ha he poblem is saed fom es which meas ha hee ae o velociies o veloci gadies a. Thus f v u η η). 3..7) H Iseig fo he voici i 3..5) we fid ha η 3..8) c η + η ) + f η f η whee c gh ad η is a kow fucio of ad he suface elevaio a ). The soluio o 3..8) ca be wie as a sum of a asie fee) pa ad a saioa foced) pa η ~ η ) + ˆ η ) 3..9) whee ~ η ad ηˆ fulfils especivel ~ ~ ~ η + ) + ~ c η η f η 3..) ˆ η ˆ ) ˆ + η + f η f η c. 3..) Equaio 3..) fo he asie fee soluio is called he Klei-Godo equaio ad occus i ma baches i phsics. Hee i descibes log suface waves ha ae modified b he eah s oaio vedup o Poicaé waves). These waves will be discussed i he e secio. Noice ha he iiial codiios fo he fee soluio ae ~ η ) η ) ˆ η ) 3..)

35 35 ad ~ η. 3..3) 3. Geosophic adjusme As a eample of a saioa soluio of 3..8) we eu o he poblem i ecio.3 whee he suface elevaio iiiall was a sep fucio: o fo simplici h > η ) 3..) h < > η ) h sg ) sg ) 3..) <. We assume ha he moio is idepede of he -coodiae. Fom 3..) we he obai Hee we have defied fo f > ): ˆ η ˆ a η a h sg ). 3..3) a c / f 3..4) which is called he baoopic Rossb adius of defomaio o simpl he baoopic Rossb adius. I ses a impoa legh scale fo he ifluece of oaio i a quasihomogeeous ocea. The soluio of 3..3) is easil foud o be We have skeched his soluio i Fig. 3. ˆ η h ep / a) ) sg ). 3..5)

36 36 Fig. 3. Geosophic adjusme of a fee suface. A pical value fo f a mid laiudes is 4 s. Fo a deep ocea H 4 m) we fid fom 3..4) ha a km while fo a shallow ocea H m) a 3 km. Fom 3..) ad 3..) we fid he veloci disibuio fo his eample i.e. f vˆ gηˆ 3..6) u ˆ. 3..7) We oe fom 3..6) ha we have a balace bewee he Coiolis foce ad he pessue-gadie foce geosophic balace) i he -diecio. Uilizig 3..5) he coespodig geosophic veloci i he -diecio ca be wie gh vˆ ep / a). 3..8) c This is a je -like saioa flow i he posiive -diecio. Alhough he geosophic adjusme occus wihi he Rossb adius we oice fom 3..8) ha he maimum veloci i his case is idepede of he eah s oaio. B compaiso wih.3.7) we see ha ou maimum veloci i is he same as he

37 37 veloci below a movig pulse wih heigh h/ o as he veloci i he o-oaig sep-poblem i ecio.3. Le us compue he kieic ad he poeial eeg wihi a geomeicall fied aea D D fo he saioa soluios 3..5)-3..8) which is valid whe. The kieic eeg becomes D D a E ˆ η k vˆ dz d gh a e / ) 8 D H 3..9) whee we have used he fac ha H >> η. Fo he poeial eeg we fid E p / z' dz' d gh D ˆ η + h g D + 8 D gh a 3 + 4e D / a e D / a ) 3..) whee we have ake z h / as he level of zeo poeial eeg ad ioduced z ' z + h /. Iiiall he oal mechaical eeg wihi he cosideed aea equals he poeial eeg o E E p gh D. 3..) Le us choose D >> a. We he oice fom 3..9)-3..) ha E k E p < E ) Thus whe he oal mechaical eeg iside he cosideed aea is less ha i was a. The easo is ha eeg i he fom of fee vedup waves soluios of he Klei-Godo equaio) has leaked ou of he aea duig he adjusme owads a geosophicall balaced sead sae. We will coside hese waves i moe deail i he e secio. Fiall we discuss i a quaiaive wa whe i is possible o eglec he effec of eah s oaio o he moio. Fo his o be possible we mus have ha >> fk v. 3..3) v

38 38 Accodigl he pical imescale T fo he moio mus saisf T π <<. 3..4) f A mid laiudes we picall have π / f 7 hs. If he chaaceisic hoizoal scale of he moio is L ad he phase speed is c we fid fom 3..4) / gh ) ha he effec of eah s oaio ca be egleced if L << a. 3..5) I he ope ocea L will be associaed wih he wavelegh while i a fjod o caal L will be he widh. Opposiel whe he legh scale is lage ha he Rossb adius i.e. L a 3..6) he effec of he eah s oaio o he fluid moio ca o be egleced. 3.3 vedup ad Poicaé waves We coside log suface waves i a oaig ocea of ulimied hoizoal ee. uch waves ae ofe called vedup waves vedup 97). The ae soluios of he Klei-Godo equaio 3..). Acuall vedup s ame is usuall elaed o ficio-modified log gavi waves bu hee we will use i also fo he ficioless case. I lieaue log waves i a iviscid ocea ae ofe called Poicaé waves. Howeve his em will be eseved fo a paicula combiaio of vedup waves ha ca occu i caals wih paallel walls. vedup waves A suface wave compoe i a hoizoall ulimied ocea ca be wie η Aep i k + l ω)). 3.3.)

39 39 This wave compoe is a soluio of he Klei-Godo equaio 3..) if ω f + c k + l ). 3.3.) Hee k ad l ae eal wave umbes i he - ad -diecio especivel. Equaio 3.3.) is he dispesio elaio fo iviscid vedup waves. Fom his elaio we oe ha he vedup wave mus alwas have a fequec ha is lage ha o equal o) he ieial fequec f. Fo simplici we le he wave popagae alog he -ais i.e. l. The phase speed ow becomes / λ 4 + π ω c c k a 3.3.3) whee λ is he wavelegh ad a is he Rossb adius. We oe ha he waves become dispesive due o he eah s oaio. The goup veloci becomes c g dω c d k λ + 4π a / ) We oice ha he goup veloci deceases wih iceasig wavelegh. Fom 3.3.3) ad 3.3.4) we ealize ha cc g c i.e. he poduc of he phase ad goup velociies is cosa. Fom 3.3.) wih l we ca skech he dispesio diagam fo posiive wave umbes; see Fig. 3..

40 4 Fig. 3. The dispesio diagam fo vedup waves. Fo k << a i.e. λ >> a) we have ha ω f. This meas ha he moio is educed o ieial oscillaios i he hoizoal plae. Fo k >> a gavi domiaes i.e. ω c k ad we have suface gavi waves ha ae o iflueced b he eah s oaio. Coa o gavi waves i a o-oaig ocea he vedup waves discussed hee do possess veical voici. Fo a wave soluio ep iω) ) 3..4) ields f ζ η 3.3.5) H whee he elaive veical voici ζ is defied b.4.). If we sill assume ha / we obai fom 3.3.5) ad 3..) ha f v η H u v. f 3.3.6) Cosideig eal soluios wih η Acos k ω) 3.3.7) we fid fom 3.3.6):

41 4 Aω u cos k ω) kh Af v si k ω) kh z + H w Aω si k ω). H 3.3.8) Hee he veical veloci w has bee obaied fom.3.8). ice ω f fo vedup waves we mus have ha u v. Fuhemoe fom 3.3.8) we fid ha u + Aω / kh )) Af / kh )) v ) This meas ha he hoizoal veloci veco descibes a ellipsis whee he aio of he majo ais o he mio ais is ω / f. Fom 3.3.8) i is eas o see ha he veloci veco us clockwise ad ha oe ccle is compleed i ime π / ω. vedup 97) demosaed ha he idal waves o he ibeia coieal shelf wee of he same pe as he waves discussed hee. I addiio he wee modified b he effec of boom ficio which leads o a dampig of he wave ampliude as he wave pogesses. Fuhemoe ficio acs o educe of he phase speed ad i causes a phase displaceme bewee maimum cue ad maimum suface elevaio. I his coecio i is ieesig o coside he mos eegeic idal cosiue i he Baes ea egio which is M. This idal compoe has a peiod T.4 hs ad he coespodig fequec becomes.4 4 ω s. Accodig o he esuls above i ca ol eis as a fee vedup wave if ω Ωsiϕ ϕc f. This meas ha we have a ciical laiude si ω / Ω) 75 o.8' N fo his compoe. A highe laiudes ha ϕ c he M compoe cao eis as a vedup wave. Howeve we shall discove lae o ha his ϕ c o

42 4 compoe ideed ca eis a highe laiudes bu he i he fom of a coasal Kelvi wave o be discussed i ecio 3.5. Poicaé waves We coside waves i a uifom caal alog he -ais wih deph H ad widh B. uch waves mus saisf he Klei-Godo equaio 3..). Bu ow he ocea is laeall limied. A he caal walls he omal veloci mus vaish i.e. v fo B. B ispecig 3.3.8) we ealize ha o sigle vedup wave ca saisf hese codiios. Howeve if we supeimpose wo vedup waves boh popagaig a oblique agles α ad α sa) wih espec o he -ais we ca cosuc a wave which saisfies he equied bouda codiios. The veloci compoe i he - diecio mus he be of he fom π v v si )ep i k ω)) 3.. B 3.3.) ice he wave umbe l π / B i he -diecio ow is discee due o he bouda codiios he dispesio elaio 3.3.) becomes / π ω ± f + c k ) B We oice fom 3.3.) ha he spaial vaiaio i he coss-chael diecio is igoomeic. uch igoomeic waves i a oaig chael ae called Poicaé waves. The ca popagae i he posiive as well as he egaive -diecio. We shall see ha his is i coas o coasal Kelvi waves which we discuss lae i his secio. I geeal he deivaio of he complee soluio fo Poicaé waves is oo legh o be discussed i his e. Fo a deailed deivaio; see fo eample LeBlod ad Msak 978) p. 7.

43 Eeg flu i vedup waves We have peviousl i ecio.5 calculaed he mea eeg flu i suface waves wihou oaio. I is ieesig o do a simila calculaio fo shallow-wae waves i a oaig ocea. B uilizig he soluios 3.3.7)-3.3.8) we ca compue he mechaical eeg associaed wih vedup waves. The mea poeial eeg pe ui aea of a fluid colum ca be wie E p T η g zdz) d g A T 3.4.) 4 whee T π / ω. The mea kieic eeg pe ui aea becomes E k + f / ω u v w ) dz d g A ) 4 f / H ω T T whee we have uilized ha kh <<. We see ha i a oaig ocea f ) he mea poeial ad he mea kieic eeg i he wave moio ae o loge equal. This is i coas o he o-oaig case e.g ) whee we have a equal paiio bewee he wo. The domiaig pa of he mea eeg is ow kieic. The eeg desi becomes E Ek + E p ga c / c ) Coside a vedup wave ha popagaes alog -ais. This wave iduces a e aspo of eeg i he -diecio. The mea hoizoal eeg flu is he wok pe ui ime b he damic flucuaig) pessue i displacig paicles hoizoall. I shallow wae he damic pessue is gη. The mea eeg flu o secod p ode i wave ampliude ca he be wie F e T T d ) g ηudz H Iseig fom 3.3.7) ad 3.3.8) i follows ha

44 44 c Fe gca E cg E ) c As could be epeced also i vedup waves he mea eeg popagaes wih he goup veloci. This is i fac a quie geeal esul fo wave moio. I his case i is ve simple o deive he coceps of eeg desi ad eeg flu diecl fom he eeg equaio fo he fluid. Wih o vaiaio i he -diecio he lieaized equaios 3..)-3..3) educe o u fv gη v + fu η. Hu 3.4.6) B muliplig he wo fis equaios b u ad v especivel ad he addig we obai u + v ) guη) + guη ) Obviousl he Coiolis foce does o pefom a wok sice i acs pepedicula o he displaceme o he veloci). B iseig ha u η H io he las em / 3.4.7) becomes g u + v + η + ) gηu ) H We wie his equaio e + d e f 3.4.9) whee he eeg desi e d ad he eeg flu e f pe ui volume ae defied especivel as g e d u + v + η 3.4.) H e f gηu. 3.4.)

45 45 The mea values fo a veical fluid colum become o uepecedl: T T T T H H gc A ed dz) d E c e f dz) d gca Fe 3.4.) whee F e cge. 3.5 Coasal Kelvi waves We coside a ocea ha is limied b a saigh coas. The coas is siuaed a ; see Fig Fig. 3.3 Defiiio skech. Fuhemoe we assume ha he veloci compoe i he -diecio is zeo evewhee i.e. v. Wih cosa deph ad cosa suface pessue 3..)- 3..3) become u gη 3.5.) fu gη 3.5.) η Hu ) We ake ha he Coiolis paamee is cosa ad elimiae u fom he poblem. Equaios 3.5.) ad 3.5.) ield

46 46 η fη 3.5.4) while 3.5.) ad 3.5.3) ield η ac η 3.5.5) whee c is he shallow wae speed ad a is he Rossb adius. We assume a soluio of he fom B iseig io 3.5.5) we fid η G ) F ) ) c F F ag' 3.5.7) G whee G ' dg / d. The lef-had side of 3.5.7) is ol a fucio of ad ad he igh-had side is ol a fucio of. Thus fo 3.5.7) o be valid fo abia values of ad boh sides mus equal o he same cosa which we deoe b γ γ fo a o-ivial soluio). Hece ag' γ G F γ c F G ep γ / a) F F + γ c ) ) B iseig fom 3.5.8) io 3.5.4) we fid ha Accodigl fom 3.5.8) we have soluios of he fom ad γ ± ) η ep / a ) F + c ) 3.5.) η ep / a ) F c ). 3.5.) If we have a saigh coas a ad a ulimied ocea fo > as depiced i Fig. 3.3 he soluio 3.5.) mus be discaded. This is because η mus be fiie

47 47 evewhee i he ocea eve whe. The soluio fo he suface elevaio ad he veloci disibuio i his case he become η u ep / a) F c ) g fa ep / a) F c ). 3.5.) This pe of wave is called a sigle Kelvi wave double Kelvi waves will be eaed i secio 3.8). I is apped a he coas wihi a egio deemied b he Rossb adius. I is heefoe also efeed o as a coasal Kelvi wave. The Kelvi wave popagaes i he posiive -diecio wih veloci c like a gavi wave wihou oaio. The diffeece fom he o-oaig case howeve is ha ow we do o have he possibili of a wave i he egaive -diecio. This is because he Kelvi wave soluio equies geosophic balace i he diecio omal o he coas; see 3.5.). This is impossible fo a wave i he egaive -diecio i he ohe hemisphee. I geeal if we look i he diecio of wave popagaio alog he wave umbe veco) a Kelvi wave i he ohe hemisphee alwas moves wih he coas o he igh while i he souhe hemisphee f < ) i moves wih he coas o he lef; see he skech i Fig. 3.4 fo a sigle Fouie compoe i he ohe hemisphee.

48 48 Fig. 3.4 Popagaio of Kelvi waves alog a saigh coas whe f >. ice he wave ampliude is apped wihi a egio limied b he Rossb adius he wave eeg is also apped i his egio. The eeg popagaio veloci he goup veloci) is hee c g dω / dk d c k) / dk c ad he eeg is popagaig wih he coas o he igh i he ohe hemisphee. We oe ha fo Kelvi waves he fequec has o a lowe limi fo vedup waves ω f). The oceaic ide ma i ceai places maifes iself as coasal Kelvi waves of he pe sudied hee. We will discuss his fuhe i coecio wih amphidomic pois pois whee he idal heigh is alwas zeo). Fom 3.5.) we oice ha he suface elevaio ad veloci ae i phase i.e. maimum high ide coicides wih maimum cue. I us ou fom measuemes ha he maimum idal cue a a give locaio occus befoe maimum idal heigh. This is due o he effec of ficio a he ocea boom which we have

49 49 egleced so fa. I ode o iclude he effec of ficio i he simples possible wa we model he ficio foce as i..7). The lieaized -compoe ow becomes u gη u ) ice v 3.5.) ad 3.5.3) emai as befoe. B elimiaig u bewee 3.5.3) ad 3.5.3) we obai η ghη + η ) We ow assume a soluio i ems of he comple Fouie compoe η G )ep i κ ω)) ) Hee ω is eal while he wave umbe κ i he -diecio is comple: κ k + iα ) We ake ha k > is he eal wave umbe while α is he spaial dampig coefficie i he -diecio. We shall assume houghou his aalsis ha α << k i.e. he wave dampig is small ove a disace compaable o he wavelegh. Iseig 3.5.5) io 3.5.4) we obai he comple dispesio elaio ω + i ω ghκ ) Uilizig ha α / k << he eal pa of 3.5.7) ields o lowes ode ha ω ± gh ) / k as befoe. We coside waves ha popagae i he posiive - diecio i.e. ω gh ) / k ck. Fom he imagia pa of 3.5.7) we he obai α ) c The value of depeds amog ohe higs o he boom oughess. A pical value deived fom he idal lieaue could be 5 ~.5 s. The geosophic balace codiio i 3.5.) ow ields

50 5 dg + il G 3.5.9) d a whee l α /ka) is a small wave umbe i he -diecio iduced b he combied acio of ficio ad oaio. This ields a coasall apped soluio: G Aep / a)ep il). 3.5.) If we le he eal pa epese he phsical soluio we he obai fo his case η Aep α / a)cos k + l ω) c A α u ep α / a) cos k + l ω) + si k + l ω). H k 3.5.) We oe fom his soluio ha a a give locaio ) sa he cue maimum is ahead i ime of he suface elevaio maimum as kow fom obsevaios. We also oe ha he lies descibig a cosa phase he co-idal lies) ae o loge dieced pepedicula o he coas bu ae slaig backwads elaive o he diecio of wave popagaio Maise ad Webe 98). This siuaio is skeched i Fig. 3.5: Fig. 3.5 Coasal Kelvi waves iflueced b ficio. Hee c c is he phase speed i he -diecio. 3.6 Amphidomic ssems Wave ssems whee he lies of cosa phase o he co-idal lies fom a sashaped pae ae called amphidomies. The ae wave iefeece pheomea ad

51 5 i he ocea he usuall oigiae due o iefeece bewee Kelvi waves. Le us sud wave moio i a ocea wih widh B; see he skech i Fig Fig. 3.6 Ocea wih paallel boudaies ifiiel log chael). ice he ocea ow is limied i he -diecio boh Kelvi wave soluios 3.5.) ad 3.5.) ca be ealized. Because we ae wokig wih liea heo he sum of wo soluios is also a soluio i.e. η ep / a) F + c) + ep / a) F c). 3.6.) I geeal he F-fucios i 3.6.) ca be wie as sums of Fouie compoes. I suffices hee o coside wo Fouie compoes wih equal ampliudes: ep / a)si k + ω) + ep / a)si k ω) ) η A 3.6.) whee ω c k. Alog he -ais i.e. fo 3.6.) educes o This cosiues a sadig oscillaio wih peiod occus whe η Asik cosω ) T π / ω. Zeo elevaio η ) π... k 3.6.4)

52 5 A he locaios give b π / k) he suface elevaio is zeo a all imes. These odal pois ae efeed o as amphidomic pois. We coside he shape of he co-phase lies ad choose a paicula phase e.g. a wave ces o ough). A a give ime he spaial disibuio of his phase is give b η ; i.e. a local eeme fo he suface elevaio. Paial diffeeiaio 3.6.) wih espec o ime ields ha he co-phase lies ae give b he equaio ep / a)cos k + ω ) ep / a)cos k ω) ) We oice igh awa ha he co-phase lies mus iesec a he amphidomic pois π / k fo all imes. As a eample we coside he amphidomic poi a he oigi. I a sufficiel small disace fom oigi ad ae so small ha we ca make he appoimaios ep ± / a) ± / a cosk si k k. Equaio 3.6.5) he ields ka aω) ) This meas ha he co-phase lies ae saigh lies i a egio sufficiel close o he amphidomic pois. ice a ω is a moooicall iceasig fucio of ime i he ieval o π / ω) we see ha a co-phase lie evolves aoud he amphidomic poi i a coue-clockwise diecio i his eample f > ). I us ou ha as a mai ule he co-phase lies of he amphidomic ssems i he wold oceas oae coue-clockwise i he ohe hemisphee ad clockwise i he souhe hemisphee. We oice fom 3.6.6) ha if we have high ide alog a lie i he egio > > a some ime we will have high ide alog he same lie i he egio < < a ime + π / ω o half a peiod lae. We ow coside he umeical value of η alog a co-phase lie. Close o a amphidomic poi hee he oigi) we ca use 3.6.) o epess he elevaio as

53 53 η A k cosω + siω) ) a Fom 3.6.6) we fid ha aω / ka) alog a co-phase lie. B elimiaig he ime depedece bewee his epessio ad 3.6.7) we fid fo he magiude of he suface elevaio alog a co-idal lie: k / ) / η A + a ) The lies fo a give diffeece bewee high ad low ide ae called co-age lies. These cuves ae give b 3.6.8) whe η is pu equal o a cosa i.e. k + / a cos ) We hus see ha he co-age lies close o he amphidomic pois ae ellipses. I Fig. 3.7 we have depiced co-phase lies solid cuves) ad co-age lies boke cuves) esulig fom he supeposiio of wo opposiel avellig Kelvi waves boh wih peiods of hous ad ampliudes of.5 m. The wavelegh is 8 km he widh of he chael is 4 km he deph is 4 m ad he Coiolis paamee is 4 s. The Rossb adius becomes 98 km i his eample. Hece he igh-had side of he chael is domiaed b he upwad-popagaig Kelvi wave he oe wih mius sig i he phase) ad he lef-had side is domiaed b he dowwad popagaig Kelvi wave.

54 54 Fig. 3.7 Amphidomic ssem i a ifiiel log chael. 3.7 Equaoial Kelvi waves Close o equao we have ha f. Fom..) he Coiolis paamee i his egio ca he be appoimaed b whee he -ais is dieced ohwads; see Fig f β. 3.7.)

55 55 Fig. 3.8 kech of he co-odiae aes ea he equao. We shall fid ha i is possible o have equaoiall apped gavi waves aalogous o he appig a a saigh coaslie. Assume ha he veloci compoe i he - diecio is zeo evewhee i.e. we assume geosophic balace i he diecio pepedicula o equao. Wih cosa deph he equaios 3.5.)-3.5.3) ae uchaged bu ow f β i 3.5.). B assumig a soluio of he fom η G ) F ) as befoe 3.5.7) becomes F c F cg' β G γ. 3.7.) Accodigl: F F + γ c) γ β G ep. c 3.7.3) B iseig io 3.5.4) we fid γ ± )

56 56 Fom 3.7.3) we ealize ha o have fiie soluio whe ± we mus choose γ i 3.7.4). The soluio hus becomes η ep / a e ) F c) g u ep / a ) F c ) e c 3.7.5) whee he equaoial Rossb adius ae is defied b a e c β ) / / ) We oe ha he soluio 3.7.5) efeed o as a equaoial Kelvi wave is valid a boh sides of equao ad ha i popagaes i he posiive -diecio i.e. easwads wih phase speed c gh) /. The eeg also popagaes easwads wih he same veloci sice we have o dispesio. A he equao β is appoimael m s. Fo a deep ocea wih H 4 m equaoial Rossb adius becomes abou 45 km. we fid fom 3.7.6) ha he Equaoial Kelvi waves ae geeaed b idal foces ad b wid sess ad pessue disibuios associaed wih som eves wih hoizoal scales of housads of kilomees. Whe such waves mee he ease boudaies i he ocea he wes coas of he coies) pa of he eeg i he wave moio will spli io a ohwad popagaig coasal Kelvi waves i he ohe hemisphee ad a souhwad popagaig coasal Kelvi wave i he souhe hemisphee. ome of he eeg ma also be efleced i he fom of log plaea Rossb waves i such waves he eeg popagaes weswad if he wavelegh is much lage ha he Rossb adius). 3.8 Topogaphicall apped waves

57 57 We have see ha gavi waves ca be apped a he coas o a he equao due o he effec of he eah s oaio. Tappig of wave eeg i a oaig ocea ca also occu i places whee we have chages i he boom opogaph. I his case howeve he wave moio is fudameall diffee fom ha associaed wih Kelvi waves. While he veloci field iduced b Kelvi waves is alwas zeo i a diecio pepedicula o he coas o equao i is i fac he displaceme of paicles pepedicula o he boom coous ha geeaes waves i a egio wih slopig boom. We call hese waves escapme waves ad he aise as a cosequece of he cosevaio of poeial voici. Rigid lid The escapme waves ae esseiall voici waves. The moio i hese waves is a esul of he cosevaio of poeial voici. Moe pecisel he elaive voici fo a veical fluid colum chages peiodicall i ime whe he colum is seched o squeezed i a moio back ad foh acoss he boom coous. To sud such waves i hei pues fom we will assume ha he suface elevaio is zeo a all imes i.e. we appl he igid lid appoimaio. I his wa he effec of gavi is elimiaed fom he poblem. Le us assume ha he boom opogaph is as skeched i Fig. 3.9.

58 58 Fig. 3.9 Boom opogaph fo escapme waves. The lieaized coiui equaio.3.) ca ow be wie as Hu ) Hv). 3.8.) + Accodigl we ca defie a seam fucio ψ saisfig Hu ψ Hv ψ. 3.8.) Whe lieaizig we obai fom he heoem of cosevaio of poeial voici.4.5) ha ζ + u H f H + v f H ) We hee assume ha f is cosa. Fuhemoe we ake ha H H ). B iseig ψ fom 3.8.) we ca wie 3.8.3) as H ψ ψ + fψ ) 3.8.4) H whee H dh / d. We assume a wave soluio of he fom ψ F )ep i k ω)) ) B iseig io 3.8.4) his ields

59 59 F k kf H F ) H H ω H This equaio has o-cosa coefficies ad is heefoe poblemaic o solve fo a geeal fom of H). We shall o make a aemps o do so hee. Isead we deive soluios fo wo eeme pes of boom opogaph. Oe of hese cases whee he boom ehibis a weak epoeial chage i he -diecio will be deal wih i sec 3.9 i coecio wih opogaphic Rossb waves. The ohe eeme case whee he slope eds owads a sep fucio will be aalsed hee; see he skech i Fig. 3.. The escapme waves eleva fo his opogaph ae ofe called double Kelvi waves. Fig. 3. The boom cofiguaio fo double Kelvi waves. Fo apped waves he soluios of 3.8.6) i aeas ) ad ) ae especivel F A ep k) F A ep k) ) We oe ha hese waves ae apped wihi a disace of oe wavelegh o each side of he sep. A he sep iself ) he volume flu i he -diecio mus be coiuous i.e.