CHAPTER 5. Shallow water theory

Transcription

1 Damcal Oceaograph CAPTER 5 Shallow waer heor I he shallow waer he characersc eph s much smaller ha he horoal scale o moos (5) I s cosere ha he low oes o epe o eph Ths s eacl rue or baroropc ( cos ) o-rcoal ( ) moos Turbule vscos rouces vercal curre shear (see r curres p 43) owever vercall egrae volume raspors are aece b he surace a boom sresses ol see (468) a (469) I hs chaper we coser mal he small-amplue wave moos We assume ha waer level evao s much smaller ha eph (5) a characersc veloc s much smaller ha characersc phase spee o he wave (53) T where T s characersc wave pero 5 ercall egrae cou equao We wll rouce he ollowg reemes as compare o he sea Sverrup regme (p 44): ) he o-saoar moo vercal veloc a he usurbe sea surace s emacall equal o he me ervave o he waer level ( case o large sea level evaos we have o use he ull ervave) w (54) ) he bas o varable eph a ree-slp boom bouar coo (mssg veloc proeco ormal o he boom) s v u v w (55) ecure oes b Jür Ele Page 5

2 Damcal Oceaograph or w u v (56) Dog he vercal egrao o he cou equao (4) we oba u v w w w u v (57) Due o he small waer level evaos (5) we ma ee he volume raspor b egrag he veloces rom he boom o he usurbe surace u v (58) B ereag he volume raspors we use he rules or ee egrals wh varable bous u u u v v v (59) (5) Combg (57) wh (59) a (5) we oba he vercall egrae cou equao (5) Tme ervave o waer level epes o he vergece o volume raspor Whe he volume raspor s rece owars he coas he waer level wll rse Iee he coas s place o he rgh rom he waer a he a 5 ercall egrae momeum equaos B mog he horoal momeum equaos () () we assume: u u ) veloces are small (53) a o-lear aveco erms u v ma be eglece ) horoal urbule vscos s eglece ecure oes b Jür Ele Page 5

3 Damcal Oceaograph 3) es s cosa 4) ue o hrosac approach a cosa es pressure epes o he sea level a vercal coorae p p g (5) g rom where he horoal pressure graes epe o he sea level graes ol p g p g (53) I he above assumpos we egrae he -compoe momeum equao u u v g a smlar equao or he -compoe Followg (58) we oba he equaos or he volume raspor compoes [m s - ] g u u (54) v g v (55) where vscous sresses appear a he surace u : v (56) a o he boom u v : b b (57) W sress compoes veloc (38) are parameere b a quarac uco o he w Mos problemac em he shallow waer moels s he boom rco parameerao sce he ear-boom curre shears are o eplcl resolve I he Sommel crculao problem (p 44) he lear boom rco ormulao ecure oes b Jür Ele Page 53

4 Damcal Oceaograph r r (58) b b coul be use sce he suao was ahow eale I ca be show ha a quas-sea low regme he mague o he boom sress s proporoal o he mague o he volume raspor owever he vecors are elece epeg o he reco o w a geosrophc low a o he rao o he Ema laers o he whole waer colum hcess I o-saoar wave-omae lows quarac boom rco c b b c (59) gves usuall beer resuls ere he quarac rag coece s ae usuall bu ma also ae o accou he boom roughess c 5 3 Three-mesoal ocea crculao moels base o he ull se o hro-hermoamc equaos ()-(3) use vercall egrae momeum equaos (54) a (55) orer o calculae baroropc pressure graes base o he waer level cou equao (5) Boom rco eecs o o-lear aveco erms a baroclc pressure graes ue o es varaos are calculae b he hree-mesoal par o he moel a her eecs ca be cosere as a correco o he w sress 53 Shallow waer equaos umercal soluo a) equaos ercall egrae momeum equaos a cou equao orm ogeher he shallow waer equaos g b (5) g b (5) (5) where hree uows ca be solve usg he hree equaos (5) (5) I ao o he equaos bouar coos are eee I he case o ree slp he volume raspor ma be rece alog he coasle bu ormal o he coas compoe has o be ero I he ope bouar (rver mouh coeco o he oher sea area) ormal compoe Q w o he volume raspor s prescrbe ecure oes b Jür Ele Page 54

5 Damcal Oceaograph a he rg bouar Q a he ope bouar (53) A he ope bouar sea level or sea level grae ma be prescrbe sea o ormal volume raspor (53) b) umercal soluo I a realsc bas wh complcae coasle a boom opograph shallow waer equaos (5)-(53) are solve umercall I a mos commol use e erece meho emporal a spaal ervaves are wre as ereces alog he e me a/or gr seps Eplc me seppg s mos eas o mpleme Coser evoluo equao or a varable a orm where s a spaal operaor rom he moele varables Eplc me scheme s wre as (54) where superscrp eoes he umber o subseque me laer a s he value o scree me sep Ths wa all he spaal ervaves ec o coag he me ervave are ae rom he prevous me laer For he umercal sabl reasos me screao o he shallow waer equaos s que oe oe as ollows g b (55) g b (56) (57) where he cou equao (57) he volume raspors a me laer are ae rom (55) (56) For he spaal screao o (55)-(57) he oma s covere b a gr wh a sep Wh a scree se o values he subscrp rus alog he -as a alog he -as I he Araawa C-gr he pos o eo o are saggere as show Fg 5 Dephs are gve a - pos ecure oes b Jür Ele Page 55

6 Damcal Oceaograph ecure oes b Jür Ele Page 56 Fgure 5 Placeme o pos or volume raspor a sea level o he Araawa C-gr use or umercal soluo o he shallow waer equaos Sea level ervave alog s ee a he -po (58) a ervave alog s ee a he -po (59) We reach he ollowg scheme g b (53) g b (53) (53) where ue o he sh o a -pos spaall average volume raspors have o be ae o accou he erms o Corols orce 4 (533)

7 Damcal Oceaograph 4 (534) For he umercal sabl me a gr sep mus ollow he relao g ma (535) whch phscal coe s: log gravaoal wave s o allowe o propagae urg oe me sep urher ha oe spaal gr sep I he problems o waer level a coasal crculao he umercal moel (53)-(534) que oe gves more ha 9% o accurac as compare o he much more comple hreemesoal moels 54 og grav waves a) oe-mesoal waves a arrow chael Coser rcoless ( b b ) ree moos a log a arrow chael oree alog he -as The cross-chael volume raspor s mssg a Corols orce wll o appear The equaos (5)-(5) oba he orm o oe-mesoal waves g (536) (537) Dereag (536) b a (537) b we oba hperbolc ereal equao g (538) I case o cosa eph search or he wave soluo a orm e (539) Replacg (539) o (538) we oba ha (538) s sase he wave parameers ollow g (54) From he coo (54) we ge he sperso relao or oe-mesoal log grav waves ecure oes b Jür Ele Page 57

8 Damcal Oceaograph g (54) ha relaes requec o he wave umber Rao o requec a wave umber eermes he phase spee c c g (54) og oe-mesoal grav waves are o-spersve e he phase spee oes o epe o he wave umber(s) I we a (varable) w sress o he equaos (536)-(537) he we oba orce waves ere resoa orcg appears he w paer moves wh a phase spee o waves c g Resoa orcg s oe a precoo o creag hghes sorm surges Tag he mea eph o he Gul o Fla 6 m we oba loos S Peersburg whe he ccloe s movg wh a spee m/s Coser e he case o arrow chael o a cera legh A he es o he chael he volume raspor has o vash (543) Search or he soluo o (538) a orm cos s (544) Bouar coos (543) ca be sase a scree se o wave umbers (545) The soluo (544) (545) represes he sag waves he chael These egeoscllaos are calle sesches Oscllaos correspog o he ere scree wave umbers are calle moes (Fg 5) 3 3 Fgure 5 Frs seco a hr moes o egeoscllaos o a arrow chael: volume raspor (le) a waer level (rgh) ecure oes b Jür Ele Page 58

9 Damcal Oceaograph b) wo-mesoal waves Corols orce ma become mpora or wo-mesoal waves Assume cosa eph cos Search or he peroc soluo o (5)-(5) a orm e e (546) e For he spaal pars o (546) we oba he equaos g g (547) (548) (549) Solve or he spaal pars o volume raspors rom (547) (548) g (55) g (55) Subsug (55) (55) o he cou equao (549) we oba g (55) I case o peroc wave soluo has he orm ep l (553) Replacg hs o (55) we oba sperso relao o log grav waves o a roag rame he - plae appromao g l (554) Whe boh he wave umbers l are real e he wave prole s peroc boh he a recos he requec s hgher ha he eral requec Cosequel peroc ( boh recos) grav waves ca o have a loger pero ha he eral pero T T 4h ecure oes b Jür Ele Page 59

10 Damcal Oceaograph ecure oes b Jür Ele Page 6 I case o loger peros l mus ae place or a leas oe o he wave umbers have o be magar I hs reco he wave prole s epoeal I s possble ol ear he coas whereas he wave amplue s mamal a he coas a ecas epoeall seawar Waves wh a epoeal prole are calle Kelv waves Kelv waves appear or eample a Balc Sea sesches a also a ural es For he sesches o small bass (le he Gul o Rga) roaoal eecs o o have remarable eec 55 Waer level equao Whe vesgag he wave properes appare he shallow waer equaos a ere coasle a boom geomer he s useul o rasorm he equaos o oe equao Frcoless ( b b ) small amplue moos are escrbe [see (5)- (5)] as g (555) g (556) (557) e us ereae (557) b me (558) Replacg (558) rom (555) (556) we oba g (559) For elmag we ae oce more he me ervave rom (559)! " # $ % & g (56) e us rasorm he las erm o (56) replacg rom (555) (556) a ag o accou cos

11 Damcal Oceaograph ecure oes b Jür Ele Page 6 J g g (56) where a b b a b a J (56) s he Jacob operaor Tag o accou (56) he equao (56) s rewre! " # $ % & g g (563) ha s equvale o he al equaos (555)-(557) Coser a bouar coo or (563) a wall whch s perpecular o he -as I ha case a From (555) (556) we oba g g or ' (564) Aalogcall he bouar coo s obae or he wall perpecular o he -as ' (565) 56 Geeral wave soluo a chael o cosa eph Wh cosa eph cos he equao (563) s smple! " # $ % & g (566) The hghes orer o ervaves s wo a (566) s urher reuce o he hperbolc wave equao g (567)

12 Damcal Oceaograph Coser a chael alog -as wh a wh Bouar coos a he chael walls a (568) are rasorme accorg o (565) o a orm a (569) Search or he soluos peroc b a Re e (57) where s comple wave amplue across he chael s alog-chael wave umber a s wave requec Replacg (57) o he equao (567) a bouar coos (569) we oba a egevalue problem or g (57) a (57) whch geeral soluo s As l B cosl (573) whereas b (57) l g (574) ere l s he cross-chael wave umber ( -compoe o he wave vecor l) Subsug geeral soluo (573) o he bouar coos (57) we ge a homogeeous ssem o lear equaos wh respec o A B la B (575) A l cosl s l B cosl l s l (576) The above ssem has o-rval soluos ol he he eerma s ero ecure oes b Jür Ele Page 6

13 Damcal Oceaograph g s l (577) B he coo (577) he egevalue problem (57) (57) has hree pes o soluos whereas he case or ( s spurous 57 Pocare waves Coser he waves correspog o he case sl (578) he coo (577) o he egevalue problem (57) (57) Coo (578) s sase cross-chael wave umber l aes scree values l (579) Seeg he eo (574) o l base o he equao (567) a wave parameers (57) we oba g l (58) where requec s ou as ( g (58) or ( g l (58) Dsperso relao o Pocare waves (58) coces wh ha o he wo-mesoal waves he uboue bas (554) ol he cross-chael wave umbers oba ol scree values ue o he bouar coos Frequec o Pocare waves s alwas hgher ha he eral requec a he requec o oe-mesoal cross-chael egeoscllaos g l or he pero s shorer ha he eral pero T T 4 h a he pero o cross-chael egeoscllaos T T g ecure oes b Jür Ele Page 63

14 Damcal Oceaograph Pocare waves ca propagae boh posve a egave recos o -as Phase spee c s eerme rom he sperso relao (58) as c g l ( g (583) See ha Pocare waves are spersve Rem ha o-roaoal grav waves have ospersve phase spee c g Spaal paer o Pocare waves s erve rom (573) where coeces A B As l B cosl mus sas bouar coos (57) pre-assumg scree wave umbers (579) The ravelg wave s cos s cos (584) or l l l cos s cos (585) Cross-chael waer level paer o Pocare waves s asmmerc a epes o he reco o wave propagao Wh he wave propagaes o he rgh a hgher waer levels appear he upper hal o he chael Wh he suao s oppose (Fg 53) Fgure 53 Waer level srbuo a chael ue o Pocare waves propagag o he rgh (above) a o he le (below) ecure oes b Jür Ele Page 64

15 Damcal Oceaograph Superposo o wo Pocare waves o equal amplue ravelg oppose recos els sag Pocare wave cos cos cos (586) l where he waer level srbuo s smmerc a he oal les (les o ero waer level) are e I s useul o scale he wave requec b eral requec a rewre he sperso relao (58) a o-mesoal orm 4 & R $ $ % ) R #!!" (587) where [m] s ) s wavelegh [m] a he baroropc (eeral) Rossb eormao raus g R (588) Pero o Pocare waves s eerme rom (587) as ) +* he pero o Pocare waves asmpocall approaches T T T A large waveleghs R 4 B creasg he chael wh + * he pero approaches o he eral pero T * + T A small waveleghs ) he wave pero s proporoal o he wavelegh ) T (Fg 54) R sg he wavelegh ) alog he -as a he relao (579) l we oba he rao o he phase spees o he Pocare waves (583) a he oe-mesoal waves g a orm * T c R ) g & $ $ % 4 R #!!" (589) ecure oes b Jür Ele Page 65

16 Damcal Oceaograph No-mesoal peros wave legh ) R =5* -4 s - we oba T c a phase spees T g a chael wh R as he ucos o o-mesoal are show Fg 54 Wh = m a g = 33 m/s (phase spee o oe-mesoal waves) a g R 5 m (baroropc Rossb eormao raus) Coserg pcal wh o he Balc Sea = 5 m we ge he o-mesoal wh or he rs moe = a R or he seco moe = R 4 6 Pero T/T / R = 3 / R = / R = Phase spee c / (g) ½ / R = / R = / R = Wave legh ) /R Wave legh ) / R Fgure 54 No-mesoal peros (le) a phase spees (rgh) o Pocare waves as a uco o o-mesoal alog-chael wave legh case o ere o-mesoal chael wh 58 Kelv waves Kelv waves are obae rom he egevalue problem (57) (57) coserg he case ( g (59) rom he coo (577) The phase spee c ( g umber a hereore he Kelv waves are o-spersve oes o epe o he wave From (574) we oba l g (59) whch meas ha he cross-chael wave umber s magar ecure oes b Jür Ele Page 66

17 Damcal Oceaograph l ( g (59) a he cross-chael srucure (573) s epoeal Aep B ep (593) g g For he wave ravelg posve reco o he -as ( ) he bouar coo (57) ca be sase ol he he waer level amplue ecreases wh -as ep cos ˆ (594) g where ˆ Kelv wave ravelg he oppose reco ca es ol he he waer level amplue s mamal a he wall ep cos ˆ (595) g Ths wa reco o cross-chael amplue eca epes o he reco o wave propagao oog he reco o Kelv wave propagao he amplue creases o he rgh he Norher emsphere (Fg 55) c Fgure 55 Waer level evaos he Kelv wave ravelg he reco c ecure oes b Jür Ele Page 67

18 Damcal Oceaograph g The scale or he amplue eca a cross-chael reco s R ha s also ow as he baroropc (eeral) Rossb eormao raus For eample a = m he phase spee s c g =33 m/s a he e-ol amplue eca occurs a sace R =5 m I he shallow waer bo wh = m (or eample ae o Peps) he phase spee o Kelv wave s c = m/s a he e-ol eca sace s R = 8 m Wh he Kelv wave he cross-chael volume raspor compoe s eacl ero (596) everwhere he chael Sce he cross-chael varao o properes s moooc he he bouar coos (568) cao be oherwse sase Due o (596) he shallow waer equaos escrbg Kelv waves are g (597) (598) g (599) As eve rom (599) he alog-chael volume raspor s a geosrophc balace wh he cross-chael sea level grae 59 Amphromc ssems o Kelv waves Superposo o wo Kelv waves o equal amplue propagag oppose recos s [see (594) a (595)] & $ ep $ % & $ ep $ % R R # ep cos! cos ˆ R!" # ep s! s ˆ R!" (5) where R s he eormao raus (588) Epresso (5) represes he sag Kelv wave whch geeral orm s cos ˆ A s ˆ A A s ˆ A (5) ecure oes b Jür Ele Page 68

19 Damcal Oceaograph where he al phase epee o s A arca A (5) Epresso (5) meas ha he amplue o oscllaos a ever po s gve b A A A (53) a he co-rage les are eerme b A cos I he case whe he amplue (53) s permael ero A a some solae pos he waves orm amphromc ssems a he pos o ero amplue are oal pos o he amphromc ssem Noe ha al phase (5) cao be eerme he oal pos Ouse he oal pos he phase o oscllaos s ere epeg o he al phase (5) Co-al les are eerme b he equao ˆ or A a ˆ A (54) Co-al les coverge he oal pos The les are usuall mare b agle -36 or me - Amphromc ssems ma appear a ere waves I he oceaograph he are mal relae o he Kelv waves arsg rom he al orces a rom he ecao o sesches b he chagg ws I vew o (5) a (5) sag Kelv waves have specc amplue srbuo he chael A ep cosh cos R A ep sh s R R R where oal pos are sg he epaso o Talor seres sh R a cosh R he equao or co-al les R R (54) s rasorme as a ˆ a R (55) Sce a a he co-al les become sra les ear he oal po R a ˆ (56) ecure oes b Jür Ele Page 69

20 Damcal Oceaograph where age o he le R a ˆ creases wh me Thereore he Kelv waves he oscllao phase roaes he Norher emsphere couerclocwse arou he oal pos (Fg 56a) Amphromc ssems o Kelv waves appear also he close a semeclose bass (Fg 56b) a) b) Fgure 56 Amphromc ssems: a) Kelv waves a el log chael b) M es he Norh Sea 5 Topographc Rossb waves Coser a chael wh slopg boom (57) where he boom slope s small We ca ae he equao o waer level (563) ecep or he ervaves where we oba The waer level equao aes he orm & $ % g # g g! (58) " ecure oes b Jür Ele Page 7

21 Damcal Oceaograph Search or he wave soluo as earler (57) Re e (59) The cross-chael srucure mus sas & g # g $ g g! (5) % " or aer smplcaos g (5) Bouar coos are he same as earler (57) (5) Geeral soluo o he equao (5) has he orm e A s l B cosl (53) where l g (54) 4 Bouar coos (5) el he egevalue problem g s l (55) ha s smlar o (577) he case o chael o a cosa eph We wll see ha Pocare a Kelv waves rema earl uchage a small boom slopes ol a slowl chagg mulpler e appears he wave amplue From he coo s l (56) o he problem (55) we oba scree cross-chael moes ecure oes b Jür Ele Page 7

22 Damcal Oceaograph l (57) From (54) we oba cubc equao or he requec g g 4 (58) g I case o low-requec waves g (59) usg also coo or small boom slope 4 g sperso relao or he (baroropc) opographc Rossb waves g we oba he (5) Frequec o opographc wave s mamal (pero s mmal) a he scales o Rossb eormao raus or g R (5) Topographc Rossb waves ca propagae oe reco ol: he shallower waer mus rema o he rgh rom he reco o wave propagao Topographc waves are que slow a he phase spees are usuall less ha m/s For he small boom slopes he requec o opographc waves s alwas less ha he eral requec All he baroropc waves cosere above p 57-5 are summare Fg 57 aope rom J Pelos ( Geophscal Flu Damcs 979) ecure oes b Jür Ele Page 7

23 Damcal Oceaograph Fgure 57 A scheme o sperso relaos o he baroropc waves appearg he el log chael: o-moal Kelv waves propagag boh recos crosschael moes o Pocare waves propagag boh recos cross-chael moes o opographc Rossb waves propagag wh shallower waer o he rgh (a gure b J Pelos) ecure oes b Jür Ele Page 73