DYNAMIC OCEANOGRAPHY

Transcription

1 DYNAMIC OCEANOGRAPY JAN ERIK WEER Depame of Geoscieces ecio fo Meeoolog ad Oceaogaph Uivesi of Oslo 9..4

2 CONTENT. ALLOW-WATER TEORY, QUAI-OMOGENEOU OCEAN. Iviscid moio, poeial voici.3. Liea waves i he absece of oaio..8.3 Effec of oaio, geosophic adjusme 3.4 vedup waves, ieial waves ad Poicaè waves 6.5 Kelvi waves a a saigh coas 3.6 Amphidomic ssems Equaoial Kelvi waves Topogaphicall apped waves 3.9 Plaea Rossb waves 35. Topogaphic Rossb waves..4. aoopic isabili 4. aoopic flow ove a ocea idge 45. WIND-DRIVEN CURRENT AND OCEAN CIRCULATION. Equaios fo he mea moio...5. The Ekma elemea cue ssem The Ekma aspo..6.4 Divege Ekma aspo ad foced veical moioekma sucio..6.5 Vaiable veical edd viscosi.65.6 Equaios fo he volume aspo 68.7 The vedup aspo Theoies of ommel ad Muk wese iesificaio AROCLINIC MOTION 3. Two-lae model Coiuousl saified fluid Fee ieal waves wih oaio Ieal espose o wid-focig, upwellig a a saigh coas aocliic isabili...

3 3. ALLOW-WATER TEORY, QUAI-OMOGENEOU OCEAN. Iviscid moio, poeial voici Goveig equaios We sud moio i a iviscid fluid wih desi. The fluid is oaig abou he z-ais wih cosa agula veloci Ωsiϕ, whee ϕ is he laiude of ou sie of obsevaio. Fuhemoe,, ae hoizoal coodiae aes alog he udisubed sea suface, ad he z-ais is dieced upwads. The posiio of he fee suface is give b z η,,, whee η is efeed o as he suface elevaio. The amospheic pessue a he suface is deoed b P,,. The boom opogaph does o va wih ime, ad is give b z, ; see he skech i Fig... Fig.. Defiiio skech. The equaio of moio fo a fluid paicle of ui mass ca ow be wie Dv fk v p gk,.. d whee D/d / Ωsiϕ is he Coiolis paamee. v is he oal deivaive followig a fluid paicle, ad f We ake ha he mass of fluid wihi a geomeicall fied volume ol ca chage as a esul of advecio of fluid paicles. The basic assumpio of cosevaio of mass fo a fluid ca be saed mahemaicall as

4 4 D v... d This is ofe efeed o as he coiui equaio. I his secio we will assume ha he desi of a paicle is coseved D/d. Thus, he coiui equaio educes o v...3 Fuhemoe we will assume ha he desi is he same fo all paicles homogeeous fluid. We oieae ou coodiae ssem such ha he -ais is ageial o a laiudial cicle ad he -ais is poiig ohwads; see Fig... Fig... Oieaio of coodiae aes. I ou efeece ssem f is ol a fucio of. We ma he wie appoimael ha f whee df f f β,..4 d f Ωsiϕ, d β R dϕ Ω R Ωsiϕ cos. ϕ ϕ..5 This is called he bea-plae appoimaio. If f is appoimael cosa i a, - aea, we sa ha he moio occus o a f-plae. The kiemaic ad damic bouda codiios a he suface ca be wie, especivel, as

5 5 D z η, z η,,,..6 d p P, z η,,...7 The kiemaic bouda codiio a he boom ca be epessed as z, v z,...8 We ow iegae he coiui equaio..3 fom z o z η,, : η η η w dz u dz v dz...9 z Uilizig..6 ad..8, we fid ha η η u dz η v dz... The basic assumpio i shallow wae heo is ha he pessue disibuio i he veical diecio is hdosaic. uilizig ha he desi is cosa, ad applig..7, his leads o p g η z P,,... This meas, whe we eu o he veical compoe i.., ha he veical acceleaio Dw/d mus be so small ha i does o oiceabl ale he hdosaic pessue disibuio. We will eu o he validi of his i secio.. The hoizoal compoes of.. ca hus be wie Du fv gη P.. d Dv fu gη P..3 d Thoughou his e we will aleae bewee wiig paial deivaives i full, ad fo ecoomic easos as subscips. We ealize ha he igh-had sides of.. ad..3 ae idepede of z. uilizig ha v D/d ad f f β,.. ca be wie D d β gη P u f..4 Fom..4 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

6 6 z a ime. imilal, fom..3 we fid ha v is idepede of z. We ca accodigl wie u u,,,..5 v v,,. Fuhemoe, i ow follows fom..3 ha w z is idepede of z. ece, b iegaig i he veical: w u v z C,,...6 The fucio C is obaied b applig he bouda codiio..8 a he ocea boom. The veical veloci ca hus be wie w u v z u v...7 ice u ad v ae idepede of z, he iegaios i.. ae easil pefomed. We he obai he followig oliea, coupled se of equaios fo he hoizoal veloci compoes ad he suface elevaio u uu vu fv gη P,..8 v uv vv fu gη P,..9 { u η } { v η } η... 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. Poeial voici We defie he veical compoe of he elaive voici i ou coodiae ssem, e.g. Fig.., b ζ v u... I addiio, eve paicle i his coodiae ssem possesses a plaea voici f, aisig fom solid bod oaio wih agula veloci Ω siϕ. ece, 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..8 ad..9 b / ad /, especivel, ad he add he esulig equaios.

7 7 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... usig ha is idepede of ime.. ca be wie D η η u v...3 d ee, η is he heigh of a veical fluid colum. We defie he poeial voici Q b f ζ Q...4 η elimiaig he hoizoal divegece bewee.. ad..3, we fid fo Q ha DQ...5 d This equaio epesses he fac ha a give maeial veical fluid colum alwas moves i such a wa ha is poeial voici is coseved. 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 v Γ abs δ k v δσ cos.,..6 σ whee σ is he aea iside Γ. Fuhemoe, i he suface iegal: k v abs f ζ...7 Whe he suface aea σ i..6 appoaches zeo, we have abs f ζ δσ cos...8 I addiio, he mass of a veical fluid colum wih base δσ mus be coseved, ad hece η δσ cos...9 This is valid fo all imes, sice a veical fluid colum will emai veical; see..5. I ou case he fluid is homogeeous ad icompessible, i.e. is he same

8 8 fo all paicles. Thus, b elimiaig δσ bewee..8 ad..9, we fid as befoe ha o, equivalel, DQ / d. f ζ Q cos.,..3 η I he ocea we usuall have ha ζ << f ad η <<. Fo saioa flow, assumig ha >> η ad f >> ζ,..5 ields appoimael ha v f /...3 O a f-plae, his equaio educes o v...3 Accodigl, he flow i his case follows he lies of cosa 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/; see..3. Take ha he moio is appoimael geosophic. Fo cosa suface pessue, we he obai fom..8 ad..9 ha g v k η...33 f Fo saioa flow he heoem of cosevaio of poeial voici educes o v Q. Combied wih..33, his leads o k η Q...34 This meas ha he sufaces of cosa η ad cosa Q coicide. Accodigl, Q ad η ae uiquel elaed, o η ηq...35 Fo he special case whee we have opogaphic seeig o a f-plae whe Q vaies ol wih, we fid fom..35 ha η η...36 Accodigl, fo saioa, geosophic moio wih a fee suface, he iso-lies fo he suface elevaio ae paallel o he deph coous.. Liea waves i he absece of oaio We assume small disubaces fom he sae of equilibium i he ocea, wodimesioal moio /, v, ad cosa deph. Fuhemoe, we ake he

9 9 amospheic pessue o be cosa alog he suface. The se of equaios ca he be lieaized, educig o u gη, η u. Elimiaig he hoizoal veloci, we fid.. η gη... This equaio is called he wave equaio. iseig a wave soluio of he pe η ep ik ± c, epeseig a comple Fouie compoe, we fid fom.. ha c g /, i.e. all wave compoes move wih he same phase speed egadless of wavelegh. uch waves ae called o-dispesive. We eed o limi ouselves o coside oe sigle Fouie compoe. Fom.. we ealize immediael ha a geeal soluio ca be wie η F c F c...3 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 }...4 Fom.. ad..4 we fid fo he acceleaio g u gη { F' c F' c },..5 whee F ' ξ df / dξ. ece, he hoizoal veloci is give b g u { F c F c }...6 c Fom..4 we ca displa he evoluio of a iiiall bell-shaped suface elevaio F wih pical widh L; see he skech i Fig..3.

10 Fig..3. 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 g /. I a deep ocea 4 m, he phase speed is c m s, while i a shallow ocea m we have c 3 m s. If he maimum iiial elevaio i his eample is h, i.e. F h, we fid fom..6 ha he veloci i he ocea diecl below peak of he igh-had pulse ca be wie gh u,..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: h, >, F..8 h, <. I his case, he veloci ad ampliude developme becomes as skeched i Fig..4.

11 Fig..4. 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. I is eas o show ha he soluio fo he sep poblem saisfies he mass ad eeg cosevaio laws: Choose a geomeicall fied aea D D, whee D c. Fom Fig..4 i is obvious ha he mass of fluid wihi his aea is cosa. We choose z h/ as he level whee he poeial eeg is zeo. The iiial poeial eeg ca hus be wie E p gdh MghG,..9 whee M Dh is he fluid mass above he zeo level, ad h G h/ is he veical disace o he cee of mass of his fluid. The iiial kieic eeg is zeo, sice we sa he poblem fom es. A a lae ime D/c, he poeial eeg becomes while he kieic eeg ca be wie h 4 4 E p Mg gdh,..

12 Thus, fom we fid ha 4 E k D u gdh... E E E, i.e. he oal eeg is coseved. This is of couse also ue fo he bell-shaped elevaio i he fis eample. k p p The hdosaic appoimaio Fiall, we addess 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: whee p g η z P p',.. p' is he o-hdosaic deviaio. The veical compoe of.. becomes o lowes ode: w,..3 p z while he hoizoal compoe ca be wie u gη p...4 The hdosaic assumpio implies ha p << u...5 If he pical legh scales i he - ad z-diecios ae L ad, especivel, we obai fom he coiui equaio ha L u ~ w,..6 whee ~ meas ode of magiude. Fom..3 we he fid p' ~ L Uilizig his esul, he codiio..5 educes o u...7 / L <<...8 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

13 3 ha he ocea deph. Fo a wave, L is associaed wih he wavelegh; fo a sigle pulse, L coespods o he chaaceisic pulse widh..3 Effec of oaio, geosophic adjusme We ow coside he effec of he eah s oaio upo wave moio. Liea heo sill applies, ad we ake he deph ad he suface pessue o be cosa. Fuhemoe, we assume ha f is cosa. Equaios he educes o u fv gη,.3. v fu gη,.3. η u v..3.3 We compue he veical voici ad he hoizoal divegece, especivel, fom.3. ad.3.. uilizig..3, we he obai f v u η,.3.4 ad η f v u g η η..3.5 The voici equaio ca be iegaed i ime, i.e. f f v u η v u η,.3.6 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 Iseig fo he voici i.3.5, we fid ha η,.3.8 c η η f η f η whee c g, 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,, η,,.3.9 whee ~ η ad η fulfil, especivel ~ ~ ~ η ~ c η η f η,.3.

14 4 c..3. η η f η f η Equaio.3. fo he asie, fee soluio is called he Klei-Godo equaio ad occus i ma baches i phsics. ee, i descibes log suface waves ha ae modified b he eah s oaio vedup waves. These waves will be discussed i he e secio. Noice ha he iiial codiios fo he fee soluio ae ~ η,, η, η, ad ~ η. As a eample of a saioa soluio of.3.8, we eu o he poblem i he las secio whee he suface elevaio iiiall was a sep fucio: h, >, η.3. h, <, o, fo simplici,, >, η h sg, sg.3.3, <. We assume ha he moio is idepede of he -coodiae. Fom.3. we he obai whee η a η a h sg,.3.4 a c / f. I is eas o show ha he soluio of.3.4 is give b η h{ ep / a}sg..3.5 We have skeched his soluio i Fig..5. Fig..5 Geosophic adjusme of a fee suface.

15 5 The disace a c /f g / /f is called he Rossb adius of defomaio, o he Rossb adius fo sho. A pical value fo f a mid laiudes is 4 s. Fo a deep ocea 4 m, we fid ha a km, while fo a shallow ocea m, a 3 km. Fom.3. ad.3. we fid he veloci disibuio fo his eample, i.e. f v gη,.3.6 ad u..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. Fom.3.5 ad.3.6 he coespodig geosophic veloci i he -diecio ca be wie gh v 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. compaiso wih..7, we see ha ou maimum veloci i is he same as he veloci below a movig pulse wih heigh h/, o as he veloci i he o-oaig sep-poblem. Le us compue he kieic ad he poeial eeg wihi a geomeicall fied aea D D fo he saioa soluio , which is valid whe. The kieic eeg becomes D η D / a E k v dz d gh a e,.3.9 D 8 whee we have used he fac ha >> η. Fo he poeial eeg we fid D η h / D / a D / a E p g z' dz' d gh D gh a 3 4e e,.3. 8 D 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 ha

16 6 E k E p < E..3. 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 Accodigl, he pical imescale T fo he moio mus saisf π T <<..3.4 f A mid laiudes we picall have π/f 7 hous. If he chaaceisic hoizoal scale of he moio is L ad he phase speed is c ~ g /, we fid fom.3.4 ha he effec of eah s oaio ca be egleced if L << a,.3.5 whee a c/f is he Rossb adius of he poblem. I he ope ocea L will be associaed wih he wavelegh, while i a fjod o caal, L will be he widh. Opposiel, whe L a,.3.6 he effec of he eah s oaio o he fluid moio ca o be egleced..4 vedup waves, ieial waves 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. oweve, 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

17 7 η Aep i k l ω..4. This wave compoe is a soluio of.3. if ω f c k l..4. ee k ad l ae eal wave umbes i he - ad -diecio, especivel, ad ω is he wave fequec. Equaio.4. 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,.4.3 whee λ π/k is he wavelegh ad a c /f is he Rossb adius. We oe ha he waves become dispesive due o he eah s oaio, i.e. he phase speed depeds o he wavelegh hee: iceases wih iceasig wave legh. The goup veloci becomes c g dω c d k λ 4π a /..4.4 We oice ha he goup veloci deceases wih iceasig wavelegh. Fom.4.3 ad.4.4 we ealize ha cc g c, i.e. he poduc of he phase ad goup velociies ae cosa. Fom.4., wih l, we ca skech he dispesio diagam fo posiive wave umbes; see Fig..6. Fig..6. The dispesio diagam fo vedup waves.

18 8 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..4.5 If we sill assume ha /, we obai fom.4.5 ad.3. ha f v η, u v. f.4.6 Wih η i.4. depedig ol o ad, we fid, whe we le he eal pa epese he phsical soluio: η η cos k ω, η ω u cos k ω, k η f v si k ω, k z w ηω si k ω..4.7 ee he veical veloci w has bee obaied fom..7. ice ω f fo vedup waves, we mus have ha u v. Fuhemoe, fom.4.7 we fid ha η ω / k η f / k u v..4.8 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.4.7 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.

19 9 Ieial waves Whe discussig he dispesio elaio.4., we ealized ha fo ve log waveleghs he vedup wave was educed o a ieial wave. Fo he ieial wave pheomeo, he effec of gavi is uimpoa. ece, o sud such waves we ma ake ha he suface is hoizoal a all imes, i.e. η. Fuhemoe, fo ifiiel log waves we mus pu /, / i he goveig equaios The he educe o Accodigl, u fv, v fu..4.9 d f, u v,.4. d whee we have chaged o odia deivaives sice u ad v ow ae ol fucios of ime. Wih iiial codiios u u, v, he soluio becomes We oice fom.4. ha u u cos f, v u si f. u v.4. u, i.e. he magiude of he veloci is u evewhee i he ocea, ad ha he veloci veco us clockwise i he ohe hemisphee whee f >. The ajeco of a sigle fluid paicle p, p ca be foud fom he elaios u d p /d ad v d p /d, whee u ad v ae give b.4.. Accodigl o p p u f u f si f, cos f,.4. u p p..4.3 f This meas ha a fluid paicle moves i a closed cicle aoud he poi,, which is he mea posiio of he paicle i quesio. The adius of he cicle is u /f, which is called he ieial adius. If we use he pical values u cm s ad f 4 s, we fid ha km. The ieial peiod T π/f, which is he ime

20 eeded fo a paicle o make oe closed loop, is abou 7 hous i his eample. The paicle moves i a clockwise diecio i he ohe hemisphee. ice he veical compoe of he eah s oaio a a locaio wih laiude ϕ is Ωsiϕ, we ca ioduce he pedulum da T Ω defied b π T Ω..4.4 Ωsiϕ We oice ha he ieial peiod is half a pedulum da. I us ou ha ieial oscillaios ae quie commo i he ocea. Mos ocea cue speca show a local ampliude o eeg maimum a he ieial fequec. Poicaé waves We coside waves i a uifom caal alog he -ais wih deph ad widh. uch waves mus saisf he Klei-Godo equaio.3.. u ow he ocea is laeall limied. A he caal walls, he omal veloci mus vaish, i.e. v fo,. ispecig.4.7, we ealize ha o sigle vedup wave ca saisf hese codiios. oweve, 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, ice he wave umbe l π / i he -diecio ow is discee due o he bouda codiios, he dispesio elaio.4. becomes / π ω ± f c k,,,3, We oice fom.4.5 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 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 Lelod ad Msak 978, p. 7.

21 Eeg cosideaios uilizig he soluio.4.7, we ca compue he mechaical eeg associaed wih vedup waves. The mea poeial eeg pe ui aea of a fluid colum ca be wie π / ω η E p g zdz d gη π / ω The mea kieic eeg pe ui aea becomes π / ω f / ω u v w dz} d g η Ek { π / ω 4 / ω,.4.8 f whee we have uilized ha k <<. We see ha i a oaig ocea f, he mea poeial ad he mea kieic eeg i he wave moio ae o loge equal compae wih he esuls.. ad.. fo he o-oaig case, whee we have a equal paiio bewee he wo. The domiaig pa of he mea eeg is ow kieic. The oal mea mechaical eeg becomes E Ek E p gη c / c..4.9 Eeg flu ad goup veloci The fomula fo he goup veloci, c g dω / dk, used i.4.4, aises fom puel kiemaic cosideaios. supeimposig wo pogessive wave ais wih he same ampliude ad wih wave umbes k ad k k, ad fequecies ω ad ω ω, especivel, oe fids ha he evelope coaiig he wave cess ad wave oughs, defied as he wave goup, will popagae wih he speed ω / k. oweve, he goup veloci has also a ve impoa damical iepeaio. Coside a vedup wave ha popagaes alog -ais. This wave iduces a e aspo of eeg i he -diecio. The eeg aspo hough a veical coss-secio ca be foud b compuig he mea wok doe b he flucuaig pessue a ha secio. The mea wok F, pe ui ime ad ui aea, which is pefomed o he fluid o he igh of a veical coss-secio, ca be wie π / ω η F π ω pudz d,.4. / whee p is he flucuaig pa of he pessue i he fluid. Iseig fom.. ad.4.7, i follows ha

22 F gηc..4. Fo a egula, ifiiel log wave ai he mea eeg cao accumulae i space. ece hee mus be a aspo of mea oal eeg E hough he cosideed coss-secio. Deoe his aspo veloci b c e. Eeg balace he equies ha F c E..4. Equaio.4. ields, b iseig fom.4.9 ad.4., ha e c g e c c / c,.4.3 whee he las equali follows fom.4.4. Accodigl, he mea eeg i he wave moio popagaes wih he goup veloci. The wok doe b he flucuaig pessue pe ui mass ad ui ime is ofe efeed o as he eeg flu, ad he oal eeg pe ui mass as he eeg desi. These quaiies va i space ad ime. oh coceps follow auall fom he eeg equaio fo he fluid. Wih o vaiaio i he -diecio, he lieaized equaios educe o u fv gη, v fu, η. u.4.4 muliplig he wo fis equaios b u ad v, especivel, ad he addig, we obai u v guη guη..4.5 Obviousl, he Coiolis foce does o pefom a wok sice i acs pepedicula o he displaceme o veloci. iseig fom he coiui equaio ha u η io he las em,.4.5 becomes / g u v η guη..4.6 We wie his equaio ed e f,.4.7 whee he eeg desi e d ad he eeg flu e f ae defied, especivel, as g e η d u v,.4.8

23 3 The mea values fo a veical fluid colum become e f guη..4.9 e e d f π / ω π / ω π / ω η π / ω η e dz d d e dz d f η ω g, k c gη ω / k..4.3 compaig wih.4.9 ad.4. we ealize ha e d E / ad e f F /, ad hece e c e. Eve hough we have ol show his o be valid fo vedup f g d waves, i is a quie geeal esul, ad valid fo all kids of wave moio..5 Kelvi waves a a saigh coas We coside a ocea ha is limied b a saigh coas. The coas is siuaed a ; see Fig..7. Fig..7. 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.4.4 becomes u gη,.5. fu gη,.5. η u..5.3

24 4 We ake ha he Coiolis paamee is cosa, ad elimiae u fom he poblem. Equaios.5. ad.5. ield η fη,.5.4 while.5. ad.5.3 ield η ac η..5.5 ee c / g ad a c /f. We assume a soluio of he fom iseig io.5.5, we fid whee η G F,..5.6 F c F ag',.5.7 G G ' dg / d. The lef-had side of.5.7 is ol a fucio of ad, ad he igh-had side is ol a fucio of. Thus, fo.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. ece ag' γ G F γ c F G ep γ / a, F F γ c..5.8 iseig fom.5.8 io.5.4, we fid ha γ ±..5.9 Accodigl, fom.5.8, we have soluios of he fom ad η ep / a F c,.5. η ep / a F c..5. If we have a saigh coas a ad a ulimied ocea fo >, as depiced i Fig..7, he soluio.5. mus be discaded. This is because η mus be fiie 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..5. This pe of wave is called a sigle Kelvi wave double Kelvi waves will be eaed i secio.8. I is apped a he coas wihi a egio deemied b he Rossb

25 5 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.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..8 fo a sigle Fouie compoe i he ohe hemisphee. Fig..8. 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.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 egleced so fa.

26 6 The effec of boom ficio will also modif he Kelvi wave soluio.5. i vaious ohe was. I paicula, i us ou 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..9. Fig..9. Coasal Kelvi waves iflueced b ficio. Ohe ficioal effecs o Kelvi waves ae educed phase speed c < c, ad a ampliude decease as he wave pogesses, which is simila o how ficio affecs vedup waves..6 Amphidomic ssems Wave ssems, whee he lies of cosa phase, o he co-idal lies, fom a sa-shaped pae, ae called amphidomies. The ae wave iefeece pheomea, ad i he ocea he usuall oigiae due o iefeece bewee Kelvi waves. Le us sud wave moio i a ocea wih widh ; see he skech i Fig... Fig... Ocea wih paallel boudaies ifiiel log caal.

27 7 ice he ocea ow is limied i he -diecio, boh Kelvi wave soluios.5. ad.5. ca be ealized. ecause 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..6. I geeal he F-fucios i.6. ca be wie as sums of Fouie compoes. I suffices hee o coside wo Fouie compoes wih equal ampliudes: ep / asi k ω ep / asi k ω η A,.6. whee ω c k. Alog he -ais, i.e. fo,.6. educes o η Asik cosω..6.3 This cosiues a sadig oscillaio wih peiod T π/ω. Zeo elevaio η occu whe π,,,, k 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.6. wih espec o ime ields ha he co-phase lies ae give b he equaio ep / acos k ω ep / acos k ω..6.5 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.6.5 he ields ka aω..6.6 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

28 8 souhe hemisphee. We oice fom.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.6. o epess he elevaio as η A k cosω siω..6.7 a Fom.6.6 we fid ha aω / ka alog a co-phase lie. elimiaig he ime depedece bewee his epessio ad.6.7, we fid fo he magiude of he suface elevaio alog a co-idal lie: k / / η A a..6.8 The lies fo a give diffeece bewee high ad low ide ae called co-age lies. These cuves ae give b.6.8, whe η is pu equal o a cosa, i.e. k / a cos..6.9 We hus see ha he co-age lies close o he amphidomic pois ae ellipses. I Fig.. 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. ece 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.

29 9 Fig... Amphidomic ssem i a ifiiel log caal..7 Equaoial Kelvi waves Close o equao we have ha f. The Coiolis paamee i his egio ca he be appoimaed b whee he -ais is dieced ohwads; see Fig... f β,.7. Fig... kech of he co-odiae aes ea he equao. We will 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

30 3 pepedicula o equao. Wih cosa deph, he equaios ae uchaged, bu ow f β i.5.. assumig a soluio of he fom η G F, as befoe,.5.7 becomes Accodigl: iseig io.5.4, we fid F c F cg' γ..7. β G F F γ c, γ β G ep. c.7.3 γ ±..7.4 Fom.7.3 we ealize ha o have fiie soluio whe ±, we mus choose γ i.7.4. The soluio hus becomes η ep / a e F c, g u ep / a F c,.7.5 e c whee he equaoial Rossb adius is defied b a e c β..7.6 / / We oe ha he soluio.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 g /. 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 4 m, we fid fom.7.6 ha he equaoial Rossb adius becomes abou 45 km. 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 will also be efleced i he fom of plaea Rossb waves. This pe of wave will be discussed i secio.9.

31 3.8 Topogaphicall apped waves 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 due o 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. Fig..3. oom opogaph fo escapme waves. The coiui equaio.. ca ow be wie as u v..8.

32 3 Accodigl, we ca defie a seam fucio ψ saisfig u ψ,.8. v ψ. Whe lieaizig, we obai fom he heoem of cosevaio of poeial voici..5 ha ζ u f v f..8.3 We hee assume ha f is cosa. Fuhemoe, we ake ha. iseig ψ fom.8., we ca wie.8.3 as ψ ψ fψ,.8.4 whee d / d. We assume a wave soluio of he fom ψ F ep i k ω..8.5 iseig io.8.4, his ields F k kf F..8.6 ω This equaio has o-cosa coefficies ad is heefoe poblemaic o solve fo a geeal fom of. 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 secio 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..4. The escapme waves eleva fo his opogaph ae ofe called double Kelvi waves. Fig..4. The boom cofiguaio fo double Kelvi waves.

33 33 Fo apped waves, he soluios of.8.6 i aeas ad ae, especivel F A ep k,.8.7 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. v v..8.8, This meas ha ψ ad heeb also ψ mus be coiuous fo, i.e. A A A i.8.7. Fuhemoe, he pessue i he fluid mus be coiuous fo. The pessue is obaied fom he lieaized -compoe of.., i.e. p u fv u fv..8.9 Wiig p P ep i k ω, ad applig.8. ad.8.5, we fid ha ωf f F P..8. k iseig fom.8.7, wih A A, io.8., coiui of he pessue a ields he dispesio elaio ω f..8. We oe ha we alwas have ha ω < f, ad ha he wave popagaes wih shallow wae o he igh i he ohe hemisphee, i.e. ω > whe f >. These wo popeies ae geeall valid fo escapme waves, eve hough we have ol show i fo double Kelvi waves wih a igid lid o op. I he case whee he escapme epeses he asiio bewee a coieal shelf of fiie widh ad he deep ocea, his pe of waves ae ofe called coieal shelf waves. This kid of boom opogaph is foud ouside he coas of Wese Nowa. ee, umeical esuls show he eisece of coieal shelf waves i he aea close o he shelf-beak, e.g. Maise, Gjevik ad Røed 979. The opogaphic appig of log waves ea he shelf-beak ad he cues associaed wih hese waves, ieac wih he wid-geeaed suface waves, which ed o make he sea sae hee paiculal ough. This is a well-kow fac amog fisheme ad ohe sea avelles ha feque his egio.

34 34 The effec of gavi I geeal, we mus allow he sea suface o move veicall. Le us coside a wave soluio of he fom u, v, η ep i k ω..8. Fo such waves, he liea vesios of..8 ad..9, wih cosa suface pessue, ield g u f ω ig v f ω kωη fη, kfη ωη..8.3 We wie he suface elevaio as η G ep i k ω..8.4 Iseig io.., we fid ω f f k G k G..8.5 g ω Fo ω << f, i.e. quasi-geosophic moio, we eve o he gavi-modified escapme wave. Fo f ad aα, his equaio ields he so-called edge waves. These ae log gavi waves apped a he coas. The appig i his case is o due o he eah s oaio, bu is caused b he fac ha he local phase speed g 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, i.e. pepedicula o he co-phase lies, 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. The edge waves ca popagae i he posiive as well as i he egaive -diecio. The lowes possible wave fequec is give b ω gk aα..8.6 Thee ae also a umbe of highe, discee fequecies fo his poblem; see Lelod ad Msak 978, p.. If we ake he eah s oaio io accou f, he

35 35 fequecies fo he edge waves i he posiive ad egaive -diecios will be slighl diffee..9 Plaea Rossb waves Like opogaphic waves, plaea Rossb waves ae a esul of he cosevaio of poeial voici. oweve, i his case he elaive voici of a veical fluid colum chages peiodicall i ime whe he plaea voici chages as he colum moves back ad foh i he oh-souh diecio. We ealize he, ha his moio occus o a bea-plae. The moio is esseiall hoizoal. As fo escapme waves, we ca elimiae gavi compleel fom he poblem b assumig ha he suface is hoizoal fo all imes, i.e. η. The hdosaic assumpio he ields fo he pessue p gz p,,..9. The lieaized equaios hus become u fv p, v fu p, u v..9. Fis, we ake ha is cosa, ad ha f f β. Fom he coiui equaio, we ca defie a seam fucio ψ such ha elimiaig he pessue fom.9. we fid whee u ψ, v ψ..9.3 ψ βψ,.9.4 / / is he hoizoal Laplacia opeao. This equaio also follows diecl fom he lieaized poeial voici cosevaio equaio.8.3, whe is cosa ad f β. We assume wave soluios of he fom Iseig.9.5 io.9.4 ields fo he fequec ψ ep i k l ω..9.5 β k ω..9.6 k l

36 36 We shall discuss his dispesio elaio i moe deail, bu fis we epea some elemes of geeal wave kiemaics. We ioduce he wave umbe veco κ defied b κ k i k i k,.9.7 ad a adius veco, whee 3i i i 3i3 A plae wave of he pe.9.5 ca ow be wie ωκ ψ Aep i κ ω Aep{ iκ }..9.9 κ Accodigl, he vecoial phase speed c ca be defied b c ω κ, κ k k k κ The compoes of he vecoial goup veloci c g ae give b I veco oaio his becomes c c c c κ ω, g g g 3 g ω / k, ω / k, ω / k. i 3 i i.9. κ k k k3 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.

37 37 The gadie κ ω is alwas pepedicula o he cosa fequec suface. Fom.9. 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..9., 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. We ow eu o he discussio of he dispesio elaio.9.6 fo Rossb waves. ice ω ω k, κ, he ssem is aisoopic. ece, he goup veloci ad he vecoial phase speed ae o paallel. Equaios.9.6,.9. ad.9. ield fo he compoes of he phase speed ad goup veloci: ωk β k c, 4 κ κ.9.3 ωl β kl c, 4 κ κ ad ω β k l cg, 4 k κ.9.4 ω β kl c. g 4 l κ We oe ha he -compoe of he phase speed is alwas dieced weswads he - ais is assumed o be paallel o he equao. oweve, he eeg popagaio ca have a easwad o weswad compoe, depedig o he aio l/k. Fo waves popagaig alog laiudial cicles l, he eeg alwas popagaes easwad. The magiudes of he phase speed ad he goup veloci ae easil obaied fom.9.3 ad.9.4: β k c κ, β κ.9.5 c g. κ We oe ha c c. Fo a pical Rossb wave popagaig alog equao we ake g λ π/k km. Wih β m s, he phase speed ad peiod become c.5 m s ad T π/ω 3 das, especivel. We ealize ha such Rossb waves ae log-peiodic pheomea.

38 38 ice ω is egaive we have chose k >, we ca defie a posiive quai γ b: β γ..9.6 ω Equaio.9.6 ca he be wie as γ l γ k..9.7 Fo a pescibed cosa fequec ω, he wave umbe compoes k ad l ae obaied as pois o a cicle i he wave umbe plae. This cicle has a adius γ ad is ceed a γ, ; see Fig..6. The figue also depics he phase speed ad he goup veloci obaied fom.9. ad.9.. Fig..6. kech of he elaio bewee he phase speed ad he goup veloci fo Rossb waves. We ow coside he eflecio of Rossb waves fom saigh coass i he oh-souh diecio. ee, we ake ha he eas coas is siuaed a L, whee he -ais is paallel o he equao; see Fig..7. Fig..7. Reflecio of Rossb waves fom a saigh coas.

39 39 As fa as wave popagaio is coceed, i is he goup veloci ha epeses sigal veloci. Accodigl, fo a eceive, o a coas, o epeiece a phsical impulse, i mus have a compoe of he goup veloci dieced agais i. Accodigl, he icomig wave has a goup veloci wih a easwad -compoe. This wave is labelled wih subscip, i.e. ψ A ep i k l ω..9.8 The efleced wave, labelled wih subscip, mus have a weswad-dieced goup veloci compoe. The efleced wave is wie as ψ A ep i k l ω..9.9 A he eas coas, he omal compoe of he veloci mus vaish, i.e. o ψ ψ, L,.9. il A ep iklep{ i l ω } il A ep iklep{ i l ω}..9. If his is o be valid fo all ad all, we mus have ha l l, ω ω,.9. A ep ep. ikl A ikl ece, he fequec ad he wave umbe compoe i he -diecio ae coseved duig he eflecio pocess. This meas ha he efleced wave umbe veco mus descibe he same cicle as he icide oe, wih he same l; see Fig..8. Fig..8. kech of he elaioship bewee wave chaaceisics of icide ad efleced Rossb waves.

40 4 We oice ha he goup veloci saisfies he law of eflecio i a heo; ha is he agle wih espec o he omal of he eflecig plae is he same fo he icide ad fo he efleced wave. We also oe ha a icide sho wave is efleced fom a eas coas as a log wave, while eflecio fom a wes coas iechage subscips ad i Fig..8, esuls i he asfomaio fom a log wave o a sho wave. Fo his easo he eas coas is said o be a souce of log waves, while he wes coas is a souce of sho waves. ice sho waves ae subjec o soge dissipaio ha log waves dissipaio is popoioal o he squae of he veloci gadies, his ma lead o a iceased asiio of mea momeum fom waves o cues a he wes coass of he oceas. This has bee used Pedlosk, 965 as basis fo eplaiig wh he ocea cues ae moe iese i such egios wese iesificaio. We shall eu o ha quesio i secio. To see how he effec of a fee suface modifies he Rossb wave ha we have sudied up o ow, we coside he lieaized vesio of he voici equaio..: ζ f u v β v..9.3 Elimiaig he hoizoal divegece u v i his equaio b usig he lieaized fom of.., we fid f v u η β v..9.4 We ow simplif he poblem b assumig ha he ime scale fo he Rossb wave is so lage seveal das ha he pessue gadie foce ad he Coiolis foce alwas have ime o adjus o a sae of geosophic balace. This meas ha he veloci field i.9.4 ca be appoimaed b v gη / f, u gη / f,.9.5 whee f is a mea cosa value of f. This elaio is ofe called he quasigeosophic appoimaio ad equies ha.9.4 ad iseig fom.9.5, we obai ω << f. eplacig f wih f i η βη a η,.9.6 whee / a g / f. assumig soluios of he fom

41 4 η ep i k l ω,.9.7 he dispesio elaio is obaied fom.9.6: β k ω..9.8 k l a If he wavelegh is much smalle ha he Rossb adius k, l >> a, he effec of he suface elevaio ca be egleced, ad we ae back o ou igid lid case.9.6. oweve, if he wavelegh is lage k, l << a, we fid fom.9. ha β k a k l β k κ c κ a, β kla k l β kl κ I his case, he goup veloci compoes become c κ a c g β a,.9.3 β kl κ κ 4 c g a ice κa ow is a small paamee, we ealize ha he eeg mail popagaes weswad i his case. If he wave umbe veco is paallel o he equao l we have o-dispesive waves, i.e. c cg a, ad c g β c.. Topogaphic Rossb waves Le us assume ha somewhee he elaive voici is zeo. Fom he heoem of cosevaio of poeial voici..5 wih η, we fid ha a displaceme ohwads, whee f is iceasig, geeaes egaive ai-ccloic elaive voici. oweve, we ealize ha he same effec ca be achieved b a ohwad displaceme if f is cosa ad he deph deceases ohwad. This gives ise o he so-called opogaphic Rossb waves. Of couse, he eisece of such waves does o equie ha he boom does slope i oe paicula diecio. Topogaphic Rossb waves ae ol a special case of escapme waves whe he boom has a ve weak epoeial slope evewhee i he ocea. leig ep α, equaio.8.6 educes o assumig αfk F α F k F... ω

42 4 F Aep iκ,.. iseio io.. ields he comple dispesio elaio α fk i ακ k κ...3 ω I geeal we ma allow fo a ve weak chage of wave ampliude i he diecio omal o he coas, i.e. we ake κ i.. o be comple: κ l iγ...4 iseio io..3, he imagia pa leads o γ α/ whe l. Fom he eal pa of..3 we he obai α fk ω...5 k l α / 4 We oe ha he alog-shoe phase speed c ω / k is posiive. This meas ha he wave popagaio i he alog-shoe diecio is such ha we have shallow wae o he igh i he ohe hemisphee. Fo a boom ha slopes gel compaed o he wavelegh k >> α, we see fom he..5 ha hese waves ae simila o sho plaea waves popagaig i a ocea of cosa deph. The epessios fo he fequec ω ae ideical, e.g..9.6, if β α f...6 This similai is ofe used i laboao epeimes i ode o simulae plaea effecs. Whe l, he equaios ae saisfied fo γ, i.e. cosa ampliude waves. uch waves popagae alog he boom coous wih shallow wae o he igh, ad mimic plaea Rossb waves alog laiudial cicles i a ocea of cosa deph.. aoopic isabili Rossb waves i he ocea ca be geeaed b amospheic focig, as will be show i secio. The ca also develop due o eeg asfe fom usable ocea cues. Assume ha we have a saioa mea flow alog a laiudial cicle v U i,.. ad ha his iiial sae is i geosophic balace, i.e. P f U. We disub peubae he iiial sae such ha he velociies ad he pessue ca be wie

43 43 u U u~,,, v v~,,, p P ~ p,,... ee we have assumed ha he moio is wo-dimesioal, i.e. idepede of z. If he peubaios deoed b ~ ae sufficiel small, he equaios goveig he moio ca be lieaized. Wih w, i.e. cosa deph ad hoizoal suface i he model depiced i Fig.., equaios.. ad..3 educe o u~ ~ ~ ~ ~ U u vu f v p, ~ v ~ ~ ~ U v f u p, u~ v~...3 ioducig he seam fucio ψ fo he peubaio, defied b u ~ ψ, v~ ψ, ad elimiaig he pessue fom he equaio above, we fid ψ U ψ β U ψ...4 Fo U,..4 educes o he equaio fo Rossb waves.9.4. We assume ha he soluios ca be wie as ψ ϕ ep ik c...5 Fuhemoe, he ocea is esiced b wo saigh coass, defiig a chael i he - diecio; see Fig..9. Fig..9. oizoal shea flow i a ocea wih paallel boudaies.

44 44 A ad D he omal veloci mus be zeo, i.e. ψ. ice his is valid fo all alog he boudaies, we mus accodigl have ha ψ a, D. Iseio fom..5 io..4 ields U β ϕ k ϕ,..6 U c wih bouda codiios ϕ,, D...7 If β, equaio..6 is educed o he well-kow Raleigh equaio fo wodimesioal peubaios of shea flow i a homogeeous, iviscid fluid. We assume ha he wave umbe k i..5 is eal, while he phase speed c ad he ampliude fucio ϕ ae geeall comple, i.e. c c ici,..8 ϕ ϕ iϕi. Fom..5 we see ha if c i >, ψ will gow epoeiall i ime. The iiial sae is he said o be usable, ad we have a eeg asfe fom he mea flow o he disubace, which i his case is a Rossb wave. ice equaio..6 fo he peubaio ampliude ϕ has o-cosa coefficies ad a sigulai fo U c he em coaiig he highes deivaive vaishes a locaios i he fluid whee U c, i is i geeal quie complicaed o solve. We will o aemp o do so hee. Isead of a deailed ivesigaio of he sabili codiios, we will be coe wih he deivaio of a ecessa codiio fo he occuece of isabili. Fis, we make he deomiao i..6 eal, i.e. U β U c ici U c c ϕ k ϕ...9 i Now we mulipl his equaio wih he comple cojugae ampliude fucio ϕ ϕ iϕ i, ad oe ha ϕϕ ϕ ϕ ϕ. iegaig he esulig equaio fom o D, ad applig he bouda codiios ϕ ϕ a, D, we fid ha D U β U c U c c i U β U c D ϕ ϕ ϕ k ϕ d ici d... i ci Fo his equaio o be saisfied, boh eal ad imagia pas mus be zeo. I suffices hee o coside ol he imagia pa, i.e.

45 45 D i c U β U c ϕ c i d... If c i >, i.e. he basic flow is usable, he iegal i.. mus be zeo. oweve, he iegad is posiive, possibl ecep fo he faco U β i he umeao. ece, fo he iegal o become zeo, fluid, which meas ha hee mus be a leas oe place whee U β mus chage sig i he U β. This epesses a ecessa bu o sufficie codiio fo isabili of he basic flow. A equivale saeme is ha he absolue voici of he basic flow, ζ f U, abs mus have a eeme somewhee i he ocea whee ζ / d. This kid of d abs isabili is elaed o he eeg asfe fom he kieic eeg of he mea flow via he veloci shea o he peubaios, ad is called baoopic isabili. Pehaps eve moe commo i he ocea is baocliic isabili, whee eeg is asfeed fom he poeial eeg of he basic sae o he peubaios via he desi saificaio. We will eu o discuss baocliic isabili i secio 3. Fiall, we meio ha o a f-plae β, he ecessa codiio fo isabili is ha he basic flow pofile mus have a poi of ifleio U somewhee i he fluid. Also, he umeical value of he voici U mus have a maimum whee U he Raleigh-Fjøof cieio. The las codiio ca be foud b applig ha he eal pa of.. is zeo, ogehe wih.. ad c i.. aoopic flow ove a ocea idge We coside a ocea cue ha cosses a sub-sea idge. Fo simplici, we assume ha he idge is ifiiel log i he -diecio, ad ha he deph fa awa fom he idge is cosa ; see Fig..,

46 46 Fig... Model skech. The suface fa upseam is hoizoal ad he cosa upseam flow U is geosophic, i.e. Ps fu... O a f-plae, which we assume hee, his meas ha he suface pessue P s of he basic flow mus va lieal wih. We sud he effec of he idge b applig he poeial voici heoem..5. I a sead sae we have v Q... A possible soluio of his equaio is Q cos., i he, -plae. We assume ha he effec of he idge is o fel fa upseam, i.e. Q f / fo all values of whe. ice Q is a coseved quai, we he mus have f ζ Q η f..3 We ioduce he shape of he idge q, i.e. q...4 The pical widh of he idge is L; see Fig... Iseig..4 io..3, ad assumig ha /, we obai v f η q...5

47 47 I he lieaue oe ofe fids discussios of his poblem whee he suface is modeled as a igid lid, i.e. η. I his case we oice fom..5 ha aiccloic voici is geeaed ove he eie idge. iegaio we obai Fo v f L qξ dξ...6 L, we fid ha v. Fo L, we obai fa v v,..7 whee A is he coss-secio of he idge, give b A The coespodig seamlies ae skeched i Fig... L q ξ dξ q ξ dξ...8 L Fig... Deflecio of flow ove a idge whe he suface is appoimaed b a igid lid. The deflecio agle θ of he seamlies o he lee side of he idge is v fa θ aca aca...9 U U The soluio..6 ma o be ealisic, sice i hough geosophic balace i he -diecio implies ha p / mus be cosa fa dowseam. This meas ha he peubaio pessue becomes ifiiel lage whe. oweve, he soluio obaied hee ma be of iees close o he idge. I mos geophsical applicaios he suface ca move feel i he veical diecio. This meas ha a fluid colum ca be seched, ad o jus be equal o, o

48 48 shoe ha he upseam value, as i ou pevious eample. Fom he lieaized vesios of , we obai fo he suface elevaio ha is iduced b he idge: U f η v v... gf g iseio io..5, we fid f F v v... q a ee F is he Foude umbe, ad a g / / f is he Rossb / U /g adius. I he ocea F <<, while we picall have ha a 3 km. I is he easoable o assume ha L << a. ece, o he lef-had side of.., he fis em domiaes. Accodigl, a paicula soluio, whe F <<, ca be wie f C... v p qd C The soluio of he homogeeous equaio is: v h C3 ep / a C4 ep / a...3 The coefficies C, C, C 3, ad C 4 ae deemied fom he bouda codiios. We assume ha he idge is smmeic abou, ad we equie ha he veloci ad he pessue ae coiuous a. Fuhemoe, we assume ha v,, ad ha he peubaio pessue becomes fiie whe. The las codiio is equivale o equie vaishig v a ifii i he case of o waves behid he idge. The soluio becomes: fa v fa ep / a f { ep / a} q ξ dξ, f, q ξ dξ,...4 ee A is he coss-secioal aea defied b..8. This soluio ields v. Fo < L, we fid ha v faep / a / >, while fo > L we have ha v faep / a / <. Fo a smmeical idge, v becomes ai-smmeic abou he ceelie of he idge. The seamlies i his case ae skeched i Fig...