Asymptotical Analysis of Internal Gravity Wave. Dynamics in Stratified Medium

Transcription

1 Alie Mahemaical ciences Vol 8 4 no HIKARI L wwwm-hikaricom h://oiorg/988/ams43637 Asmoical Analsis of Inernal Gravi ave namics in raifie Meium V V Bulaov an Yu V Vlaimirov Insiue for Problems in Mechanics RA Pr Vernaskogo - Moscow 956 Russia fa: Corigh 4 V V Bulaov an Yu V Vlaimirov This is an oen access aricle isribue uner he Creaive Commons Aribuion License which ermis unresrice use isribuion an reroucion in an meium rovie he original work is roerl cie Absrac In aer funamenal roblems of inernal gravi waves namics are consiere ave namics of sraifie meium ocean amoshere is highl eenen on ensi isribuion an boom oograh The eac analical soluion is obaine onl if he waer isribuion ensi an boom shae escribe b some moel funcions The soluion of his roblem is eresse in erms of he Green s funcion an he asmoic reresenaions of he soluions are consiere The uniform asmoic forms of he inernal gravi waves in horionall inhomogeneous an non-saionar sraifie ocean are obaine A moifie saio-emoral ra meho is roose which belongs o he class of geomerical oics mehos KBJ meho The soluion is roose in erms of wave moes roagaing ineenenl a he aiabaic aroimaion an escribe as a non-inegral egree series of a small arameer characeriing he sraifie meium roeries Analical an numerical algorihms of inernal gravi wave calculaions for he real ocean arameers are resene Kewors: KBJ aroimaions wave roagaion sraifie meium inernal gravi waves

2 8 V V Bulaov an Yu V Vlaimirov Inroucion Inusrial aciviies on he coninenal shelf connece wih oil gas an oher minerals eracion became one of he imoran reasons o begin researches of namic inernal gravi waves [-3] his an laforms bus wih rilling an consrucion a he eh use long ubes joining hem wih he sea boom Builers of unerwaer consrucions in euaorial isrics eerience he influence of huge unerwaer inernal waves an srong surface flows which can have he form of see waerfalls ome ime ago when he henomenon of inernal waves an heir srengh were no known i haene ha he builers los heir euimen uch eensive losses mae hem hink ha securi of unerwaer euimen an he influence of inernal gravi waves shoul be conrolle The inernal waves characerisics are use for areciaion of heir influence on he environmen an unerwaer laforms of oil an gas eosis a he shelf Arcic basin China an Yellow eas ec [4-] ecial ineres o research of inernal gravi waves is connece wih also inensive eloiaion of Arcic an is naural wealh Inernal gravi waves in Arcic are oorl suie as he move uner ice an racicall invisible from above bu accessible informaion abou unerwaer objecs movemen show heir eisence omeimes here are eclusions when inernal gravi waves reach ice an ulif an lower i wih efinie erioici which can be fie wih he hel of raiolocaion souning Influence of all kins of waves can be he reason of he ice cover sli in he Arcic Inernal waves make for he movemen of icebergs an ifferen kins of olluion o he research of inernal gravi waves namics is an imoran funamenal scienific an racical roblem aime a ensuring securi while [89] The inernal gravi waves are he oscillaions of a sraifie meium in he gravi force fiel The sraifie meium is such a meium where he ensi increases wih he eh uose ha a volume elemen of he meium is no a he euilibrium for eamle i coul be islace uwar hen i will be heavier han he surrouning meium an herefore Archimeean forces will make i move back o he euilibrium The essenial arameer of an oscillaing ssem is he freuenc I is eermine b he correlaion of wo facors: reurning forces which reurn he erurbe ssem owars is euilibrium an he inerial forces For he inernal gravi waves he reurning forces are roorional o he verical graien of he flui s ensi an he inerial ones are roorional o he ensi iself For he characerisic freuenc of he gravi waves oscillaions we have g he following eression: ρ This freuenc is usuall ρ calle b he Brun-Vaisala freuenc or he buoanc freuenc Here ρ is he ensi consiere as a funcion of he eh g is he acceleraion in he

3 Asmoical analsis of inernal gravi wave namics 9 gravi force fiel he sign - originaes from he increase of he ensi wih he ρ eh an herefore < [-3] The eac soluions of he essenial euaions escribing he inernal gravi waves are onl obaine for secial cases Tha is he reason wh he aroimae asmoical mehos are ssemaicall use for he invesigaion of he inernal gravi wave fiels in sraifie ocean The inernal gravi waves are usuall reresene in he following inegral form: J e λf F λ >> where f an F are analic funcions of γ [ ] he comle variable ; γ is a conour of inegraion on he comle lane The universal wa o consruc he asmoic forms of such eressions is he meho of ealon inegrals [3-7] This aer is evoe o he ssemaical escriion of a generaliaion of he geomerical oics meho KBJ meho ie we iscuss he saioemoral ra meho of ealon funcions This meho allows one o solve he roblem of asmoic moeling of he inharmonic wave acke s namics for he inernal gravi waves in sraifie meia wih slowl varing arameers The main reasons o use he ra mehos are he following: he ra reresenaions are well correlae wih he inuiion an wih he emirical maerial for he roagaion of he inernal gravi waves in naural sraifie meia ocean amoshere These mehos are universal an ver ofen one can use onl hem for he aroimae comuaions of he wave fiels in slowl changing nonhomogeneous sraifie meia [367] As is well known an essenial influence on he roagaion of inernal gravi waves in sraifie naural meia ocean amoshere is cause b he horional inhomogenei an non-saionari of hese meia To he mos ical horional inhomogenei of a real ocean one can refer he moificaion of he relief of he boom an inhomogenei of he ensi fiel an he variabili of he mean flows One can obain an eac analic soluion of his roblem for insance b using he meho of searaion of variables onl if he isribuion of ensi an he shae of he boom are escribe b raher simle moel funcions If he shae of he boom an he sraificaion are arbirar hen one can consruc onl asmoic reresenaions of he soluion in he near an far ones; however o escribe he fiel of inernal waves beween hese ones one nees an accurae numerical soluion of he roblem [-3] Using asmoic mehos one can consier a wie class of ineresing hsical roblems incluing roblems concerning he roagaion of nonharmonic wave ackes of inernal gravi waves in iverse nonhomogeneous sraifie meia uner he assumion ha he moificaion of he arameers of a vericall sraifie meium are slow in he horional irecion From he general oin of view roblems of his kin can be suie in he framework of a combinaion of he aiabaic an semi-classical aroimaions or b using close aroach for eamle ra eansions In aricular he asmoic soluions of

4 V V Bulaov an Yu V Vlaimirov iverse namical roblems can be escribe b using he Maslov canonical oeraor which eermines he asmoic behavior of he soluions incluing he case of neighborhoos of singular ses comose of focal oins causics ec [347] The secific form of he wave acke can be finall eresse b using some secial funcions sa in erms of oscillaing eonenials Air funcion Fresnel inegral Pearce-e inegrals ec The above aroaches are uie general an in rincile enable one o solve a broa secrum of roblems from he mahemaical oin of view; however he roblem of heir racical alicaions an in aricular of he visualiaion of he corresoning asmoic formulas base on he Maslov canonical oeraor is sill far from comleion an in some secific roblems one can use oher schemes o fin he asmoic behavior whose comuer realiaion using sofware of Mahemaica e is raher simle In his aer using he aroaches eveloe in [5] we consruc asmoic soluions of he roblem which is formulae as follows Problem formulaion: horionall inhomogeneous vericall sraifie meium This aer covers he conceual issues of he saio-emoral meho he geomerical oics meho he KBJ aroimaion aking ino consieraion he secifics of inernal gravi waves If we eamine he inernal gravi waves for he case when he unisurbe ensi fiel ρ eens no onl on he eh bu on he horional coorinaes an hen in general erms if he unisurbe ensi is a funcion of horional coorinaes such a isribuion of ensi inuces a fiel of horional flows These flows however are eremel slow an in he firs aroimaion can be neglece o i is commonl suose ha he fiel ρ is efine a riori hus i is assume ha here eis cerain eernal sources or he eamine ssem is non-conservaive I is also evien ha if he inernal gravi waves are roagaing above an non-uniform boom here is no such a roblem because he "inernal wave non-uniform boom" ssem is conservaive an here is no eernal energ flush [-3] e we shall wrie ou he linearie ssem of euaions of hronamics [-35] ~ ~ U U ρ ρ ~ ρ gρ ~ ~ ~ U U ρ ~ ρ ~ ρ ~ ρ U U

5 Asmoical analsis of inernal gravi wave namics ~ ~ ~ Here U U is he veloci vecor of inernal gravi waves an ρ are he ressure an ensi erurbaions g is he acceleraion of gravi - ais is irece ownwars Using he Boussines aroimaion which means he ensi ρ in he firs hree euaions of he ssem is assume a consan value he ssem b aling he cross-iffereniaing will be given as 4 ~ ~ g ~ ρ ~ ρ ~ ρ Δ Δ U U ρ ~ ~ ΔU ~ ~ ΔU Δ / / As he bounar coniions we ake he "rigi-li" coniion a H 3 ~ ~ ~ Consier he harmonic waves U U e i U U Inrouce he non-imensional variable accoring o he formulas: where L is he ical scale of he L L h horional variaions ρ h is he ical scale of he verical variaions ρ for eamle he hermocline wih [35] In non-imension coorinaes he euaion ssem will be wrien as ine is omie hereafer g ρ ρ ρ ε Δ ε εu εu ρ ε ΔU ε ΔU 4 h g where ε << g L h Asmoic soluions The asmoic soluion 4 shall be foun in he form usual for he geomeric oics meho [378-] V iε m Vm e 5 m iε V U U

6 V V Bulaov an Yu V Vlaimirov Funcions an V m m are subjec o efiniion From here on we shall resric ourselves o fining onl he ominan member of he eansion 5 for he verical veloci comonen a ha from he las wo euaions 4 we have i / U 6 U i / ubsiue 5 ino he firs euaion of he ssem 4 an se eual he members of he orer O 7 H g ρ where is he Vaisala-Brun freuenc eening of he ρ horional coorinaes The Vaisala-Brun freuenc buoanc freuenc is a main arameer han eermines inernal gravi wave roeries in real ocean [- 3] The bounar roblem 7 has a calculaion seu of eigenfuncions n an eigenvalues Kn n which are assume o be known [5] From here on he ine n will be omie while assuming ha furher calculaions belong o an iniviuall aken moe For he funcion we have he eikonal euaion [48-] K 8 Iniial coniions for he eikonal for he horional case are efine on he line L : : [8-] For solving he eikonal euaion we L consruc he ras ha is he euaion 8 wih characerisics K 9 K K K where / / is he lengh elemen of he ra The iniial coniions an shall be efine from he ssem

7 Asmoical analsis of inernal gravi wave namics 3 K The euaions 9 an iniial coniions efine he ra Afer he ras are foun he eikonal is efine b inegraion along he ra [569] K ow we are going o fin he eigenfuncion oe ha from 7 we can eermine onl he verical eenenc of he funcion In oher wors he funcion is efine o he accurac of mulilicaion b he arbirar funcion A e shall fin given as ˆ A where ˆ is he soluion of he roblem 7 normalie as follows H ˆ Afer subsiuing 5 ino 4 se eual he members of he orer ε O K g Δ ρ ρ 3 H e we use he orhogonali coniion of he righ-han member of he euaion 3 wih resec o funcion Muliling 3 b an inegraing over from o H we obain H H H g ln ρ 4 Conver he secon erm ino 4 using he inegraion b ars an ero bounar coniions for H H g ln ρ 5 Conver he firs erm ino 4 while accouning for H H A 6 In orer o ransform he hir member ino 4 we al he graien oeraor o he euaion 7 seing Y

8 4 V V Bulaov an Yu V Vlaimirov Y Y K K 7 Muliling 7 b an inegraing over from o H while accouning for we ge ln K A H 8 Finall we wrie over he euaion 4 using 5 6 an 8 ln 3 Δ K A A 9 The ransfer euaion 9 will be solve in characerisics of he eikonal euaion 9 Using he formula for Δ along he ras: JK J Δ where J is he geomeric ra ivergence we reuce he ransfer euaion 9 o he following conservaion law along he ras [38-] ln K J A The conservaion law can be wrien as well in he form suiable for fining he funcion A [58-] a K A a K A where J a is he uni ra ube wih oe ha he wave energ flash is roorional o a K A hus from i follows ha in his case here survives he value eual o he wave energ flash ivie b he wave vecor moulus [ 57] on-harmonic inernal gravi wave ackes To rocee o suing he roblem of non-harmonic wave ackes evoluion in a smoohl non-uniform horionall sraifie meium we resuose he choice of Ana Ana is he German a soluion e [34] which efine he roagaion of Air an Fresnel inernal waves wih cerain heurisic argumens

9 Asmoical analsis of inernal gravi wave namics 5 Air wave Le s inrouce he slow variables ε ε ε no slowness is suose over he ine is omie hereafer where ε λ / L << is he small arameer ha characeries he sofness of ambien horional changes λ is he ical wave lengh L is he scale of a horional non-uniformi Then he ssem in slow variables will be wrien as follows: 4 g ρ ρ ρ ε Δ εu εu ρ ε ΔU ε ΔU e we eamine he suerimosiion of harmonic waves in slow variables m i iε m e [ m ] 3 m ε ih resec o funcions m i is assume ha he are o-numbere on an min / is reache a for all an ubsiuing 3 ino we can easil have i rove ha he funcion m has a a ole of he m-h orer Therefore as he moel inegral R m for iniviual erms in 3 will serve he following formulas: m 3 i R i m e π where he inegraion conour is going 3 aroun he oin from overhea which enables he funcions R m o eoneniall eca a >> [-] The funcions R m have he following feaure: Rm R m a ha R Ai R Ai R Ai u u ec I is evien consiering cerain roeries of Air inegrals ha he funcions R m relae wih each oher as: R R R 3 R R [] Fresnel wave As he moel inegrals R m ha escribe he roagaion of Fresnel waves aking ino accoun he soluion srucure for he islacemen in he horionall uniform case we use he following formulas: * R Re e i i ReΦ Φ Is eas o see ha funcion * Φ has he following feaure * Φ i e i i

10 6 V V Bulaov an Yu V Vlaimirov e e * Φ i i i i i i i i From which we can obain for insance: i i * * 3 * 3 Φ Φ Φ or in erms of funcions m R : R i R 3 R i R i R [] Base on he above аn as well on he firs member srucure of he Air an Fresnel uniform wave asmoics for a horionall uniform meium he soluion of he ssem in can be foun for insance in he form for an iniviuall aken moe n n U furher omiing he ine n [5] K ε ε ε R R R a a 4 K 3 ε ε ε R R R a a U U U U where he argumen a a a ε is assume o be of he orer of uni Eansion Eression 4 agrees wih a common aroach of he geomeric oics meho saial ime ra meho KBJ meho [-3367] oe also ha from such a soluion srucure i follows ha he soluion for a horionall non-uniform an non-saionar meium shall een on boh he "fas" verical coorinae an "slow" ime an horional coorinaes variables e we generall are going o fin a soluion in "slow" variables a ha he soluions srucural elemens which een on he "fas" variables aear in he form of inegrals of some slowl varing funcions along he sace-ime ras This soluion choice allows us o efine he uniform asmoics for inernal gravi wave fiels roagaing wihin sraifie laer wih slowl varing arameers which hols rue eiher near or far awa from he wave frons of a single wave moe If we nee onl o efine he behavior of a fiel near he wave fron hen we can use one of he geomeric oics mehos he "rogressing wave" meho an a weakl isersive aroimaion in he form of aroriae local asmoics an fin he reresenaion for he hase funcions argumen in he form: ε ε a where he funcion efines he wave fron osiion an is eermine from he eikonal euaion soluion: c [35] A funcion c is he maimum grou veloci of a resecive wave moe ie he firs member of he isersion curve eansion in ero [5] The funcion he secon member of he eane isersion curve escribes he sace-ime imulse wih evoluion of Air or Fresnel non-harmonic waves an hen i will be efine from some arbirar laws of conservaion along he eikonal euaion characerisics wih heir acual form o be eermine b he roblem hsical coniions [-3 4]

11 Asmoical analsis of inernal gravi wave namics 7 Eikonal euaion: ras an iniial coniions formulaion e we focus on he eikonal euaion an formulaion of iniial seings for i In his secion we shall eamine generall wo basic eikonal euaions an relaive characerisic ssems for solving hese euaions The firs e eikonal euaion is obaine b consrucing he asmoics wihin a limie sace omain ie near he wave fron of a single moe ha is o sa we use a "weak isersion" aroimaion [5] In his case he isersion curve having is roeries efine b he wave fiel near he fron is aroimae b aroriae Talor eansion limie b he hir member of he asmoic series Then as i will be illusrae in [5] o efine he momen of he wave fron arrival we have he following eikonal euaion c 5 where c is he maimum grou veloci of roagaing iniviual inernal gravi waves moe The soluion of he verical secrum roblem wih aroriae bounar coniions A A c is assume o be known [-3] Then for solving he euaion of his eikonal 5 he ssem of characerisic euaions will be wrien as [8-] c c c 6 c c c Hence we ge for eamle he relaionshi as follows: c c c 7 which inicaes ha for his case he eikonal is he linear funcion of ime As he iniial seings for solving he ssem 6 we ma use he aa on some line The roblem uner invesigaion is wo-imensional because i is assume ha he meium arameers are slowl varing in ime an horionall an no slowness in he verical coorinae is reuire Therefore le he lane have a aramericall efine line: l λ λ on which we can lo he eikonal iniial isribuion: λ The hsical meaning of seing he iniial coniions in such a form is raher evien: he horional inernal wave falls on he bounar beween horionall uniform an non-uniform meiums

12 8 V V Bulaov an Yu V Vlaimirov he inerface beween he meiums is efine b his line all wave fiel arameers a his borer line are known Each value of he arameer λ ienicall eermines is value λ an he Caresian orinaes λ λ from which he characerisics is rouce ha is he ssem 6 soluion In he eamine ssem 6 b reason of is efiniion he arameer ha is varing along he characerisics is he ime elece as he iniial value is he ime of ei of raiaion Le here be a oin on he line l from which here originaes a characerisics ra wih arameer λ for which i akes ime o reach an arbirar oin Evienl we ma assume ha hese values are he funcions of he arameers λ an ha is λ λ If he ransformaion Jacobian from ra coorinaes λ o he Caresian iffers from ero hen hese euaions are solvable wih resec o variables λ : λ λ The funcion a λ λ is given as λ an as he iniial value we ake he momen ime of ei for he characerisics ra Hence we have he ssem λ a λ λ Then using he relaion 7 we obain ha The secon e eikonal euaion resuls from consrucing he roorional asmoics of he far fiels of inernal gravi waves The eikonal euaion for eermining he hase of non-harmonic wave ackes ha escribe hese roorional asmoics shall be wrien as K 8 k / olving he verical secral roblem wih resecive bounar coniions A k A is also assume o be known The isersion eenenc enoe as K is efine from he soluion of his verical secral roblem is he secral arameer The euaion 8 is he Hamilon-Jacobi euaion wih Hamilonian oeraor k K [8-] an he characerisic ssem of his euaion is given b K K K K K K K K Consier he characerisic ssem as follows

13 Asmoical analsis of inernal gravi wave namics 9 9 where he freuenc an he ime of ei of raiaion are he ra coorinaes The soluions of his ssem are: The Jacobian of ransiion from he Caresian orinaes o ra coorinaes is wrien as: e we calculae: Using he characerisic ssem 9 we can obain Then we have e we show ha here is he relaionshi Hence he following euali will be realie ivc c 3

14 3 V V Bulaov an Yu V Vlaimirov c ivc In he case uner invesigaion he oeraor ivc accouns for he elici eenence of c c on he variables an he imlici eenence on hese variables since i is evien ha he freuenc is also a funcion of hese variables ow we calculae on he characerisics 9 he values of funcions iffereniaing he euaion of characerisics on variables we have conseuenl From which we obain he following relaionshis / / / / Then we have: / / B his means he relaionshi 3 has been rove I is evien ha in he sace such roving is no necessar whereas he euali ivc ln in he sace is evien b virue of he oeraor efiniion iv wihin his sace The reuiremen of roving he relaionshi 3 comes abou from he fac ha he characerisics ras are no viewe in he four-imension sace bu in he wo-imension Caresian sace [ 3 8-] Take a ifferen seing of he iniial aa for solving he characerisic ssem in 9 such as he iniial aa seing a he momen of ime on he aramericall efine line are he ra coorinaes

15 Asmoical analsis of inernal gravi wave namics 3 Then we have: Ω a On his line he eikonal is he well-known funcion Then he characerisics euaion can be wrien as Ω a 3 a For he eamine case we wrie ou a formula for he Jacobian ransiion from he Caresian orinaes o he ra coorinaes : e we calculae 3 Using he characerisic ssem 9 an iniial aa in he form 3 we obain he following relaionshis: Ω Ω Ω Ω Using hese relaionshis we can calculae Ω Ω Ω Ω Laer iffereniaing he iniial coniions 3 on resecivel an b reeaing he above algorihm we can ge / / Ω Ω Ω Ω / / Ω Ω Hence resuling from 3 we also come o he formula 3 e have o make a few oins wih regar o he Jacobian geomeric meaning of he iscusse roblem in relaion o non-harmonic ackes of inernal gravi waves generae b isurbing sources in moion [7] Consier a characerisic ssem for he horionall uniform saionar scenario an a oin isurbing source in moion wih veloci V The euaions of characerisics ras are efining he sraigh lines

16 3 V V Bulaov an Yu V Vlaimirov V VK K ν K ν K K K V The Jacobian ransiion from he ra coorinaes o Caresian orinaes is given b ν ν V VK K K K VK K VK K / Fig The geomerical meaning of he Jacobian L is he source moion rajecor wihin he Caresian sace; А is he characerisic ra wih a consan value of ra coorinaes ; C is he characerisic ra wih a consan value of ra coorinaes Δ ; B is he characerisic ra wih a consan value of ra coorinaes Δ ; is he characerisic ra wih a consan value of ra coorinaes Δ Δ Then on accoun of he geomeric consrucion shown in Fig i is evien ha in he firs aroimaion he area Σ of he surface elemen ABC euals: Σ Δ Δ Here lines AC an B are he consan value lines ha is he conour lines of ra variables an in he Caresian sace I is evien ha in non-isersive meiums because of no ivergence of characerisics ras he area sie Σ of he surface elemen ABC b virue of is efiniion is eual o ero If aar from he isersion he sraifie meium is

17 Asmoical analsis of inernal gravi wave namics 33 also horionall non-uniform hen he characerisics ras are no sraigh lines he surface elemen ABC onl in he firs aroimaion an a uie small Δ an Δ is aroimae b a resecive arallelogram For he case of a horionall uniform meium he characerisics ras are alwas sraigh lines an he elemen ABC is alwas a arallelogram Thus he Jacobian of ra coorinaes ransiing o he Caresian orinaes oulines he geomeric ivergence in he Caresian sace of he characerisics ras for he resecive eikonal euaion [38-] ow we go back o he wo rincial mehos of efining he iniial seings for solving he characerisic ssem of an arbirar eikonal euaion: F / which eermines he hase funcion [358-] The firs meho consiss in ha we efine an arbirar iniial sace-ime omain : λ λ λ where λ are some ra variables in which all iniial aa is efine incluing he hase funcion: λ The characerisics ras are «le ou» of his sace-ime omain which is he beginning of furher namics for he characerisics ras I is evien ha he following relaionshis are holing for his omain λ λ λ λ λ λ λ Using he eikonal euaion for he iniial values F 33 we obain hree euaions o eermine he remaining unknown iniial values wihin his iniial sace-ime omain Us a rule he hsical saemen of a roblem relae o non-harmonic ackes of inernal gravi waves generae in real-worl naural environmens involves he following There are wo meiums: a horionall uniform an a horionall non-uniform which have a cerain inerface borer line in beween he form of which is well-known [5] On his borer line base he roblem soluion for a uniform meium we assume as known all roeries of he eamine wave fiel ha is he hase funcion is consiere as known for all momens of he ime Accouning for ha we have o efine he evoluion of non-harmonic wave ackes in a horionall non-uniform meium To his en i seems more convenien o se he iniial aa for solving a ssem of characerisics in oher forma ha is in he form of arbirar "bounar" coniions as efine on his borer line beween wo meiums ha his means is ha we efine he line a over which he hase funcion is well-known a an momen of ime : Is

18 34 V V Bulaov an Yu V Vlaimirov worh o menion ha he bounar line beween wo meiums is no ime eenen On his line evienl we can b virue of efiniion realie he following ar: Then o eermine he iniial values on his line we have which ogeher wih he relaionshi 33 allows us o efine he iniial aa values for hese wo variables In wha follows we shall al his meho o formulae he iniial aa for solving a characerisic ssem [578-] Asmoical analsis of sraifie non-saionar meium wave namics Uner he real oceanic coniions he Vaisala-Brun freuenc g ln ρ / efines he basic characerisics of inernal gravi wave namic an i shall no een solel on saial variables bu also on he ime The mos characerisic es of ime-o-ime variabili are he hermocline going u or own an changing is wih ec [479] There is a number of ime scales for variaions of hro-hsical fiels in he oceans an seas: a small-scale wih erios of abou minues a meso-scale wih erios of abou a a wen-four hours аs well as snoical an global variaions wih erios of a few monhs o a few ears [5] In wha follows we shall anale he inernal gravi fiel roagaion in non-saionar meiums wih arameer variaion erios of a a an over which allows us o use he geomeric oics aroimaion because he erio of inernal gravi waves is ens of minues an less The ssem of linearie euaions of hronamics when he non-erurbae ensi ρ eens on variables an reuces o a single euaion for eamle for he verical veloci comonen: ln ρ ln ρ ln ρ g Δ Δ Δ If ln ρ / is neglece we obain an euaion in he Boussines aroimaion: ln ρ Δ Δ I aears naural o neglec as well he member wih ln ρ / which woul correson o a conseuen alicaion of he Boussines hohesis [-3] I means he ensi characeriing he fuis s iner mass can be assume consan Then we have: Δ Δ The resuling euaion iffers from a sanar

19 Asmoical analsis of inernal gravi wave namics 35 euaion of inernal gravi waves in a saionar sraifie meium jus b he ime aramerical inclusion ino he Vaisala-Brun freuenc The asmoic soluion is foun in he form of a sum of moes wih ever one of hem roagaing ineenenl of each oher he aiabaic aroimaion e are going o eamine a single iniviuall aken moe while omiing is ine e we focus solel on he sace region near he wave fron which means ha we consier he ime as being close o he arrival ime of he wave fron henceforh enoe b τ ie we use a weakl isersive aroimaion Consier he wave roagaion in a laer of sraifie meium H < < wih he Vaisala-Brun freuenc e shall seek he soluion wih bounar coniions H in he form O ε A ε ε τ A ε ε τ ε τ L F ϕ τ F m ϕ Fm ϕ τ τ ε ε B ε ε τ B ε ε τ ε τ L F ϕ τ where / 3 F ϕ Ai is he Air erivaive having is argumen ϕ ε ε ε τ ε ε of he orer of uni The funcion τ efines he ϕ ε wave fron osiion funcion escribes he evoluion of he Air wave wih he small arameer ε secifies "slow variables" ince our focus is onl on slow imes ε being close o he ime of he wave fron arrival τ hen all funcions receing funcions F m are given in he form of Talor series bε τ ε owers Le ε be wrien as: τ ε τ ε τ O ε e consier he far inernal τ gravi waves generae b moving source Afer comlicae analical calculaions we can obain an eression for he erm in he form: 5 / c τ f τ f τ τ Ai 3 / / / 4 / 3 c τ R V c τ ε where c f - eigenvalue an eigenfuncion of inernal gravi verical secral roblem V- source see - eh of source moion R - some funcion ha are eermine b he arameers of he roblem [57] The figures emonsrae he numerical resuls of inernal gravi wave calculaions for ical oceanic arameers [569] The Fig shows a ssem of ras hin line causics bol line generae b source moving in a nonsaionar sraifie ocean I is a general rule ha causic of a famil of ras single ou an area in sace so ha ras of ha famil canno aear in he marke area There is also anoher area an each oin of ha area has wo ras ha ass hrough his oin One of hose ras has alrea asse his oin an anoher is

20 36 V V Bulaov an Yu V Vlaimirov going o ass he oin Formal aroimaion of geomerical oics or KBJ aroimaion canno be alie near he causic ha is because ras merge ogeher in ha area afer he were reflece b causic If we wan o fin wave fiel near he causic hen i is necessar o use secial aroimaion of he soluion an in he aer a moifie ra meho is roose in orer o buil uniform asmoic eansion of inegral forms of he inernal gravi wave fiel Afer he ras are reflece b he causic here aears a hase shif I is clear ha he hase shif can onl haen in he area where mehos of geomerical oics which were use in revious secions can be alie If he ras ouch he causic several imes hen aiional hase shifs will be ae Phase shif which was creae b he causic is raher small in comarison wih he change in hase along he ra bu his shif can consierabl affec inerference aern of he wave fiel The Fig3 emonsraes an evoluion of inernal gravi wave acke in a nonsaionar sraifie ocean wihin he ssem of coorinaes ha is in moion ogeher wih he isurbing source Time inerval for calculaions is eual hours A ha i s evien ha if here were no Vaisala-Brun ime-o-ime freuenc variaions such a wave coorinae woul be saionar umerical resuls show ha inernal gravi waves namic in he real ocean is subsaniall influence b non-saionari of hro-hsical fiels The obaine asmoic soluions are uniform an allow far inernal gravi wave fiels o be escribe boh near an far from urning oins an wave frons Fig Ras an causics in sraifie non-saionar meium

21 Asmoical analsis of inernal gravi wave namics 37 Fig 3 Evoluion of inernal gravi wave acke in non-saionar sraifie meium iscussion Thus we have he following scheme for far inernal gravi waves asmoic calculaing in heerogeneous an non-saionar sraifie meium: for an arbirar ensi isribuion inernal gravi waves verical secral roblems is solve numericall b a shooing meho an he corresoning normalie eigenfuncions an eigenvalues are obaine [5] eikonal euaion is solve numericall along hese characerisics ras wih aroriae iniial coniions [3-5] 3 afer eerminaions of characerisics ras eikonal hase value hase funcions is calculae b numerical inegraion along hese ras [-3] 4 geomeric ivergence ra ubes is eermine for eamle numerical iffereniaion of closel sace ras [3 5-7] 5 inernal gravi waves amliue is calculae from he corresoning conservaion laws energ conservaion along he ras characerisics [ ] In his aer a moifie sace-ime ra meho is roose which belongs o he class of geomerical oics mehos KBJ aroimaion The ke oin of he roose echniue is he ossibili o erive he asmoic reresenaion of he soluion in erms of a non-ineger ower series of he small arameer

22 38 V V Bulaov an Yu V Vlaimirov ε Λ / L where Λ is he characerisic wave lengh an L is he characerisic scale of he horional heerogenei The elici form of he asmoic soluion was eermine base on he rinciles of locali an asmoic behavior of he soluion in he case of a saionar an horionall homogeneous meium The wave acke amliues are eermine from he energ conservaion laws along he characerisic curves A ical assumion mae in suies on he inernal wave evoluion in sraifie meia is ha he wave ackes are locall harmonic A moificaion of he geomerical oics meho base on an eansion of he soluion in moel funcions allows one o escribe he wave fiel srucure boh far from an a he vicini of he wave fron The universal characer of he asmoic meho roose for moeling inernal gravi fiels makes i ossible o effecivel calculae wave fiels an in aiion ualiaivel anale he obaine soluions This meho offers broa ooruniies for he analsis of wave fiels on a large scale which is imoran for eveloing correc mahemaical moels of wave namics an for assessing in siu measuremens of wave fiels in he ocean The aricular role of he roose asmoic mehos is eermine b he fac ha he arameers of naural sraifie meia are usuall known aroimael an aems a heir aeuae numerical soluion using he iniial euaions of hronamics an such arameers ma resul in a noable loss of accurac for he resuls obaine In aiion o heir funamenal significance he obaine asmoic moels are also imoran for alie invesigaions since he roose meho of geomerical oics allows soluion of a wie secrum of roblems relae o moeling wave fiels In such a siuaion he escriion an analsis of wave namics ma be realie hrough eveloing asmoic moels an using analical mehos for heir soluion base on he roose KBJ moifie meho The resuls of his work reresen a significan ineres for hsics mahemaics an engineers Besies ha ineres analical asmoic an numerical soluions which were obaine in his aer can resen significan imorance for engineering alicaions since resene meho which were o calculae he inernal gravi waves fiel make i ossible o calculae ifferen wave fiels in he raher big class of anoher roblems Acknowlegmens The resuls resene in he aer have been obaine b research erforme uner rojecs suore b he Russian Founaion for Basic Research o Program of he Russian Acaem of ciences Funamenal Problems of Oceanolog: Phsics Geolog Biolog Ecolog

23 Asmoical analsis of inernal gravi wave namics 39 References [] VV Bulaov YuV Vlaimirov ave namics of sraifie meiums Moscow auka Publishers [] J Pelosk aves in he ocean an amoshere: inroucion o wave namics Berlin- Heielberg ringer [3] BR uherlan Inernal gravi waves Cambrige Cambrige Universi Press [4] C Garre E Kune Inernal ie generaion in he ee ocean Rev Flui Mech [5] J Grue JKveen A scaling law of inernal run-u uraion Ocean namics [6] ZJong BYGou L Lua ZM hi YXiao Y Qu Comarisons of inernal soliar wave an surface wave acions on marine srucures an heir resonses Alie Ocean Research 33-9 [7] MK Hsu AK Liu C Liu A su of inernal waves in he China eas an Yellow ea using AR Coninenal helf Research [8] J Grue A Jensen Orbial veloci an breaking in see ranom gravi waves J Geohs Res 7 C7-C3 [9] J Grue RH Gabrielsen Es Marine ransor in he High orh Oslo ovus Forlag e orske Vienskas-Akaemi orges ekniske Vienskasakaemi [] J oong BC Lee CC Kao ave measuremen using GP veloci signal ensors [] Yang-Yih Chen GYChen Chia-Hao Lin Chiu-Long Chou Progressive waves in real fluis over a rigi ermeable boom Coas Eng J [3] VM Babich V Bulrev Asmoic mehos in shor-wavelengh iffracion heor Ofor Alha cience 7 [4] VIArnol Caasrohe heor Berlin Heielberg ringer 99

24 4 V V Bulaov an Yu V Vlaimirov [5] VVBulaov YuV Vlaimirov The uniform asmoic form of he inernal gravi wave fiel generae b a source moving above a smoohl varing boom J Eng Mah [6] J Asen J Vannese Ineria-gravi-wave generaion: a geomeric-oics aroach II/IUTAM roceeings 9 [7] Yu obrokhoov A Lohnikov CA Vargas Asmoics of waves on he shallow waer generae b saiall-localie sources an rae b unerwaer riges Rus J Mah Phs 3-4 [8] RB hie Asmoic analsis of ifferenial euaions Lonon: Imerial College Press 5 [9] FC Hoensea Quasi-sea sae analsis of ifferenial ifference inegral an graien ssems Couran Lecure oes Amer Mah oc [] JAaners F Verhuls JMurock Averaging mehos in nonlinear namical ssems n e ew York ringer-verlag 7 [] M Abramowi IA egun Hanbook of mahemaical funcions Rerin of he 97 e ew York over Publicaions Inc 99 [] G ason A reaise on he heor of Bessel funcions Rerin of he n e Cambrige Cambrige Universi Press 995 Receive: ovember 3