Mean flow velocities and mass transport for Equatorially-trapped water waves with an underlying current

a Xi :2409.08714 1 [ma h.AP] 13 Sep 2024 Mean low eloci ies and mass anspo o Equa o ially- apped wa e wa es wi h an unde lying cu en Da id Hen y and Sil ia Sas e-Gómez Abs ac In his pape we p esen an analysis o he mean low eloci ies, and ela ed mass anspo , which a e induced by ce ain Equa o ially- apped wa e wa es. In pa - icula , we examine a ecen ly-de i ed exac and explici solu ion o he geophysical go e ning equa ions in he β−plane app oxima ion a he Equa o which inco po a es a cons an unde lying cu en . 1 In oduc ion The ques ion o de e mining he luid d i induced by he p opaga ion o su ace wa e wa es is a ascina ing issue and, despi e pionee ing wo k on his subjec being ins iga ed by S okes as a back as he mid-1800’s, i is s ill a highly cu ious and pe plexing ma e a e en he mos undamen al le el. Fo ins ance, in he se ing o pe iodic su ace g a i y wa e wa es S okes demons a ed by way o app oxima ions [39] ha luid pa icles expe ience a (mean) o wa d d i o he o de o ǫ2, whe e ǫ ela es o he wa e s eepness. This d i is in a mean sense, whe eby an a e age is aken o e he wa e pe iod, and i is an inhe en ly nonlinea phenomenon wi h ega d o he o de o wa e ampli ude. The sub le ies o hese d i p ope ies may be illus a ed by conside ing he classical assump ion ha o pe iodic i o a ional wa e mo ion i was assumed, a he linea le el, ha luid pa icles ollow closed ajec o ies [30], whe eas acco ding o he S okes d i phenomenon a he o de o expansion ǫ2i is implied ha a leas some pa icle pa hs a e non-closed. I is no ewo hy ha , wi h ega d o pa icle ajec o ies o pe iodic i o a ional wa e wa es, i was ecen ly p o en by a ious me hods ha all pa icle pa hs h oughou he luid domain a e indeed non-closed o lows induced by a wide- ange o g a i y (and capilla y-g a i y) wa es, bo h in he app oxima e linea egime and o exac solu ions o he ully nonlinea go e ning equa ions [6, 7, 8, 12, 15, 21, 22, 27, 33]. In ecen decades, ollowing he wo k o Longue -Higgins, he s udy o mean d i eloci ies induced by su ace wa e mo ion was placed on a i me heo e ical oo ing o a b oad ange o luid mo ions [1, 3, 31, 32]. I was obse ed ha key ea u es o he mean luid d i eloci y, o so-called S okes’ d i eloci y, could be cha ac e ised in e ms o he mean Eule ian low eloci y and he mean Lag angian low eloci y, whe eby: Lag ange = Eule + S okes. In spi e o ecen p og ess, de e mining he mean luid low eloci ies emains a highly complex and in ica e issue om bo h a heo e ical, and expe imen al [37, 41], iewpoin . 1 In his pape we p esen an analysis o he mean low eloci ies, and ela ed mass anspo , induced by ce ain Equa o ially- apped wa e wa es. In pa icula , we exam- ine a ecen ly-de i ed [23] exac solu ion o he geophysical go e ning equa ions in he β−plane app oxima ion [14, 16, 17] a he Equa o . The o m o his solu ion is explici in e ms o Lag angian a iables, and a bene i inhe en in employing he Lag angian ame- wo k is ha luid kinema ics may o en be desc ibed explici ly and wi h ( ela i e) ease, [2, 4, 5, 9, 10, 11, 13, 18, 19, 20, 23, 25, 26, 28, 38, 40]. A signi ican complica ing ac- o o he analysis unde aken in his pape , pa icula ly wi h ega d o de e mining he mean Eule ian low eloci y and subsequen ly he S okes d i eloci y, is he p esence o a cons an unde lying cu en e m in he solu ion gi en in [23]. This is in spi e o he unde lying cu en assuming a ela i ely simple mani es a ion in he Lag angian o mula- ion o he solu ion, along he lines o he unde lying cu en e ms which we e in oduced by Mollo-Ch is ensen [36] in o Ge s ne - ype solu ions in an a emp o model billows and a ious o he complica ing e ec s in bo h a mosphe ic and oceanog aphic si ua ions. We also no e ha i is well es ablished ha cu en s play a i al ole in Equa o ial dynamics [9, 11, 14, 16, 29], and in e es ingly a ans e se Equa o ial cu en can be inco po a ed in o a Ge s ne -like exac solu ion in he Equa o ial −plane o mula ion, c . [24]. The pape is concluded wi h a b ie discussion o some mass- anspo p ope ies o hese Equa o ially apped wa es. 2 The Equa o ially apped wa e solu ion 2.1 Go e ning equa ions We conside geophysical wa es in he Equa o ial egion, whe e we assume ha he ea h is a pe ec sphe e o adius R= 6378 km, and wo k in a e e ence ame o a ing wi h he ea h whose o igin is ixed a he ea h’s su ace, wi h he {x, y, z}-coo dina e ame chosen so ha he x-axis is poin ing ho izon ally due eas ( he zonal di ec ion), he y-axis is due no h (me idional di ec ion), and he z-axis is poin ing e ically upwa ds and pe pendicula o he ea h’s su ace. The go e ning equa ions o geophysical ocean wa es a e gi en by u +uux+ uy+wuz+ 2Ωwcos Φ −2Ω sin Φ = −1 ρPx(2.1a) +u x+ y+w z+ 2Ωusin Φ = −1 ρPy(2.1b) w +uwx+ wy+wwz−2Ωucos Φ = −1 ρPz−g, (2.1c) oge he wi h he mass conse a ion equa ion ρ +uρx+ ρy+wρz= 0 (2.2a) and he equa ion o incomp essibili y ux+ y+wz= 0.(2.2b) 2 He e Φ ep esen s he la i ude, (u, , w)is he luid eloci y, Ω = 73.10−6 ad/s is he (cons an ) o a ional speed o ea h [16], g= 9.8m/s−2is he g a i a ional cons an , ρ is he wa e densi y, and Pis he p essu e. We a e in e es ed in Equa o ial wa es, ha is, geophysical ocean wa es in a egion which is wi hin 2ola i ude o he Equa o . Since he la i ude is small, we may use he app oxima ions sin Φ ≈Φ, and cos Φ ≈1, and hus linea ising he Co iolis o ce leads o he β-plane app oxima ion o equa ions (2.1) gi en by u +uux+ uy+wuz+ 2Ωw−βy =−1 ρPx +u x+ y+w z+βyu =−1 ρPy w +uwx+ wy+wwz−2Ωu=−1 ρPz−g, (2.2c) whe e β= 2Ω/R = 2.28·10−11 m−1s−1. The ele an bounda y condi ions a e he kinema ic bounda y condi ions w=η +uηx+ ηyon z=η(x, y, ),(2.2d) P=Pa m on z=η(x, y, ),(2.2e) whe e Pa m is he (cons an ) a mosphe ic p essu e, and η(x, y, )is he ee su ace. The bounda y condi ion (2.2d) s a es ha all he pa icles in he su ace will s ay in he su ace o all ime , and he bounda y condi ion (2.2e) decouples he wa e low om he mo ion o he ai abo e. Finally, we assume he wa e o be in ini ely deep, wi h he low con e ging apidly wi h dep h o a uni o m zonal cu en , ha is, (u, , w)→(−c0,0,0) as z→ −∞.(2.2 ) The se o equa ions (2.2) comp ises he go e ning equa ions o he β−plane app oxima ion o geophysical ocean wa es wi h a cons an unde lying cu en . 2.2 Exac solu ion In his sec ion we b ie ly desc ibe he exac solu ion o he β-plane go e ning equa ions (2.2) which was p esen ed in [23]. This solu ion p esc ibes a h ee-dimensional eas wa d- p opaga ing s eady geophysical wa e in he p esence o a cons an unde lying cu en o magni ude |c0|. The wa e-like e m is pe iodic in he zonal di ec ion and i has a cons an phasespeed c > 0. Fu he mo e, he wa e is Equa o ially apped, exhibi ing a s ong ex- ponen ial decay away om he Equa o . Equa o ially apped wa es which a e symme ic abou he Equa o and p opaga e eas wa d a e known o exis , and hey a e ega ded as an impo an ac o in a possible explana ion o he El Niño phenomenon (c . [14, 16, 17]). The solu ion o (2.2) we p esen is o mula ed in he Lag angian amewo k, whe eby he e olu ion in ime o indi idual luid pa icles is p esc ibed [2]. In his Lag angian o mu- la ion he Eule ian coo dina es o luid pa icles (x, y, z)a e exp essed as unc ions o he 3 Lag angian labelling a iables (q, , s)∈(R,(−∞, 0),I), and ime , as ollows: x=q−c0 −1 kek[ − (s)] sin [k(q−c )],(2.3a) y=s, (2.3b) z= +1 kek[ − (s)] cos [k(q−c )],(2.3c) whe e 0<0and kis he wa enumbe de ined by k= 2π/L, and whe e Lis he ( ixed) wa eleng h. Fo c0>0 he unde lying cu en is ad e se, while o c0<0 he cu en is ollowing, and we see below ha he sign o he cu en de e mines whe he Iis he eal line Ro a ini e in e al. The unc ion (s)de e mines he decay o he pa icle oscilla ions in he la i udinal di ec ion away om he equa o and i is gi en by (s) = cβ 2γs2,(2.4) whe e γ:= 2Ωc0+g(>0) is a “modi ied g a i y” e m and we make he (physically eason- able) assump ion ha c0>−g 2Ω . Fo no a ional con enience le us choose ξ=k( − (s)) , θ =k(q−c ). Then he Jacobian ma ix o he ans o ma ion (2.3) is gi en by ∂(x, y, z) ∂(q, s, )=  1−eξcos θ0−eξsin θ seξsin θ1− seξcos θ −eξsin θ0 1 + eξcos θ  ,(2.5) which has he ime-independen de e minan 1−e2ξ. Consequen ly he low de ined by (2.3) is olume p ese ing, ensu ing ha (2.2b) holds in he Eule ian se ing [2]. Since he solu ion (2.3) is explici in he Lag angian o mula ion, we may immedia ely disce n some quali a i e p ope ies o he physical luid mo ion. Indeed, a signi ican bene i o wo king in he Lag angain amewo k is ha he luid kinema ics can o en be desc ibed explici ly and wi h ela i e ease. In he case abo e we calcula e he eloci y ield di ec ly om (2.3) o ge u(q, , s; ) = Dx D =ceξcos θ−c0,(2.6a) (q, , s; ) = Dy D = 0,(2.6b) w(q, , s; ) = Dz D =ceξsin θ, (2.6c) whe e D/D is he ma e ial (o con ec i e) de i a i e wi h espec o Eule ian a iables. Fo ixed la i udes, ha is o e e y ixed s, he sys em (2.3) desc ibes he low benea h a su ace wa e p opaga ing eas wa ds a cons an speed cde e mined by he dispe sion ela ion (2.10) 4 below. Addi ionally, o ixed la i udes he ee su ace z=η(x, y, )is ob ained by se ing = 0(s)in (2.3c), whe e 0(s)< 0is he unique solu ion o e2k[ (s)−cβ 2γs2] 2k− (s) + c0β 2γs2−e2k 0 2k+ 0= 0,(2.7) The exis ence o a unique solu ion (s) o (2.7) o |s|>0is equi alen o he condi ion e2k[ 0−cβ 2γs2] 2k+c0β 2γs2<e2k 0 2k,(2.8) c . [23] o de ails. Fo c0≤0, i is easy o see ha condi ion (2.8) holds o all s∈R. Fo c0>0, condi ion (2.8) will hold o es ic ed alues o son a ini e in e al Iwhich depends on he magni ude |c0|o he cu en . Fo ou p esen pu poses we ema k ha , gi en a cu en wi h c0>0, o a unique solu ion o (2.7) o exis i is necessa y ha c0< ce2k 0,(2.9) and acco dingly (2.3) ep esen s a dynamically possible solu ion o (2.2). Since c06=c(by (2.9)) i ollows ha he dispe sion ela ion o he wa e akes he o m c=pΩ2+kγ −Ω k=pΩ2+k(2Ωc0+g)−Ω k>0.(2.10) We ema k ha i c0=c hen he dispe sion ela ion o he wa e would ake he o m c=pg/k. Hence, in his si ua ion geophysical Co iolis e ec s ha e no bea ing on he dispe sion ela ion, which ins ead ma ches ha o he celeb a ed Ge s ne ’s wa e solu ion [5, 7, 20] o deep-wa e g a i y wa es. This obse a ion leads us o in e ha p ecluding he case c0=c, as is consis en wi h condi ion (2.9), is na u al in he con ex o geophysical wa e wa es (c . [23] o de ails on he dispe sion ela ions). Finally, we no e ha a ixed la i udes s=s∗ he c es and ough le els o he wa e su ace p o ile a e p esc ibed in e ms o he Lag angian pa ame e s by z±(s∗) = 0(s∗)±1 kek[ 0(s∗)− (s∗)]. 3 Mean eloci ies and S okes d i In his sec ion we analyse he e ec ha he cons an unde lying cu en has wi h espec o bo h he mean Lag angian and Eule ian low eloci ies induced by he exac solu ion (2.3). In [13] i was shown ha in he absence o a cu en , ha is o c0= 0, he mean Lag angian eloci y is ze o and he mean Eule ian eloci y lows wes wa ds. Hence, in he absence o he cu en he S okes d i (o mean S okes low eloci y), which which is he di e ence in he mean Lag angian and Eule ian eloci ies [31, 32], is eas wa ds. He e we show ha he si ua ion is a mo e complex in he p esence o an unde lying cu en , in pa icula when de e mining he mean Eule ian eloci y. Th oughou he ollowing conside a ions we ix he la i ude by se ing s=s∗. 5 3.1 Mean Lag angian low eloci y The mean Lag angian low eloci y (also known as he mass- anspo eloci y [31]) a a poin in he luid domain is he mean eloci y o e a wa e pe iod o a ma ked luid pa icle which o igina es a ha poin . Fo he exac solu ion (2.3) we may calcula e he a e age o he ho izon al eloci y uin (2.6a) o e a wa e pe iod T=L/c as ollows: huiL=1 TZT 0 u(q−c , s, )d =ceξ TZT 0 cos [k(q−c )] d −1 TZT 0 c0d =−c0, (3.1) whe e we ha e used he ac ha he i s in eg al on he le -hand side abo e anishes. I is immedia ely appa en ha he mean Lag angian low eloci y is ei he wes wa ds o eas wa ds, depending on whe he he sign o c0is posi i e o nega i e espec i ely. When c0= 0 he mean Lag angian eloci y is ze o, which concu s wi h he esul o [13], and in his ligh he o m o he mean Lag angian low eloci y abo e is no pa icula ly su p ising conside ing he explici manne in which c0appea s in he exp ession o he Lag angian eloci y (2.6a). We no e ha he exp ession o he mean Lag angian eloci y is independen o bo h he la i ude s, and he loca ion in he luid domain whe e he luid pa cel o igina es. 3.2 Mean Eule ian low eloci y When wo king in he Eule ian se ing ma e s a e g ea ly complica ed by he p esence o he unde lying cu en . The mean Eule ian low eloci y a a ixed-poin in he luid domain is he Eule ian luid eloci y a ha ixed-poin a e aged o e a wa e pe iod. In he case o he eloci y ield (2.6) he mean Eule ian low eloci y may be compu ed by aking he mean o e a wa e pe iod o he ho izon al eloci y (2.6a) a any ixed-dep h benea h he wa e ough. Le ing z=z−(s∗)deno e he e ical posi ion o he wa e ough le el, we ix a dep h z=z0< z−(s∗). This ixed dep h z=z0may be cha ac e ised in e ms o Lag angian a iables, using (2.3c), by he equa ion z0=R+1 keξ(R)cos θ, (3.2) whe e we deno e by =R(q−c ;s∗, z0) he unc ional ela ionship induced by ela ion (3.2) be ween he o he wise independen a iables and q, as ollows om he implici unc ion heo em. We no e ha a consequence o (3.2) is ha Ris pe iodic in he q− a iable, wi h pe iod L. Di e en ia ing (3.2) wi h espec o qyields 0 = Rq+Rqeξ(R(q)) cos θ−eξ(R(q)) sin θ, ha is Rq=eξsin θ 1 + eξcos θ.(3.3) 6 We no e om (3.3) ha Ris maximised o minimised wi h espec o qwhene e sin θ= 0, and he e o e o a ixed-dep h z0 he maximal and minimal alues achie ed by Ra e gi en implici ly by he ela ions z0=R±1 keξ(R), whe e he posi i e (nega i e) sign co esponds o he minimal (maximal) alue o R, e- spec i ely. To compu e he Eule ian mean eloci y huiE(s∗, z0)a la i ude s∗and dep h z0≤z−(s∗)we examine c+huiE(s∗, z0) = 1 TZT 0 [c+u(x−c , y, z0)] d . =1 LZL 0 [c+u(x−c , y, z0)] dx, which upon ans o ming, by way o (2.3), o he labelling a iables (q, s, ), and in oking unc ional pe iodici y wi h espec o he q− a iable, we ge c+huiE(s∗, z0) = 1 LZL 0 [c+u(q−c , s∗, R(q−c ;s∗, z0))] ∂x ∂q dq. By di e en ia ing xin (2.3a) wi h espec o q, using (2.6a), and aking in o accoun (3.3), we ob ain c+huiE(s, z0) = 1 LZL 0hc+ceξ(R(q)) cos θ−c0ih1−eξ(R(q)) cos θ−eξ(R(q))Rqsin θidq =1 LZL 0 c1 + eξ(R(q)) cos θ1−e2ξ(R(q)) 1 + eξ(R(q)) cos θdq −c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) cos θdq =c−c LZL 0 e2ξ(R(q))dq −c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) cos θdq. The e o e he mean Eule ian eloci y is gi en by he ela ion huiE(s∗, z0) = −c LZL 0 e2ξ(R(q))dq −c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) cos (k[q−c ])dq. (3.4) The p esence o a non-ze o unde lying cu en c0adds a signi ican complica ing ac o o exp ession (3.4), and in pa icula he sign (and hence di ec ion) o he mean Eule ian e- loci y is no easily disce nible om he abo e exp ession in gene al. Ne e heless, depending on he size and di ec ion o he cu en c0, we may ob ain es ima es which de e mine he di ec ion o he mean Eule ian eloci y ollowing om he inequali ies ZL 0 1−e2ξ 1 + eξdq ≤ZL 0 1−e2ξ 1 + eξcos θdq ≤ZL 0 1−e2ξ 1−eξdq. (3.5) 7 3.2.1 The case c0>0: Fi s o all le us s udy he case when c0is posi i e, which ep esen s an unde lying ad e se cu en in he Lag angian a iables. The second in eg al e m on he igh -hand side o inequali y (3.4) sa is ies −c0 LZL 0 1−e2ξ 1−eξdq ≤ −c0 LZL 0 1−e2ξ 1 + eξcos θdq ≤ −c0 LZL 0 1−e2ξ 1 + eξdq. (3.6) Since 0< c0< ce2k 0< c om (2.9), equa ion (3.6) yields huiE≤ − c LZL 0 e2ξdq −c0 LZL 0 1−e2ξ 1 + eξdq ≤ −c0 LZL 0 1 + e3ξ 1 + eξdq < 0.(3.7) The e o e he mean Eule ian low eloci y is wes wa ds o all admissible alues o c0 o which (2.9) holds. To ge an idea o he ange o he mean Eule ian low we no e ha huiE≥ − c LZL 0 e2ξdq −c0 LZL 0 1−e2ξ 1−eξdq ≥ − c LZL 0 1−e3ξ 1−eξdq. (3.8) Hence, since ξ≤kR < k 0<0, we see ha o all la i udes sand dep hs z0< z−(s) he mean Eule ian low eloci y is in he ange huiE(s, z0)∈−c1−e3k 0 1−ek 0,0.(3.9) Tha he mean Eule ian low is wes wa d o an ad e se cu en is no su p ising, since in he absence o he cu en he mean Eule ian low is wes wa d (c . [13]) and he p esence o he ad e se cu en e m in (3.4) me ely se es o exace ba e his e ec . 3.2.2 The case c0≤0: The case when c0is nonposi i e, c0≤0, ep esen s an unde lying ollowing cu en . In his case he in luence ha he cu en has on he mean Eule ian low in (3.4) is complex and di icul o disce n, and i is no gene ally possible o analy ically de e mine i s e ec di ec ly om exp ession (3.4). None heless, we can deduce some b oad cha ac e is ics o he low by wo king as ollows. The mean Eule ian eloci y (3.4) is wes wa ds, ha is huiE(s∗, z0)<0, i −c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) cos θdq ≤ −c0 LZL 0 1−e2ξ(R(q)) 1−eξ(R(q)) dq ≤ −c0max q∈[0,L] 1−e2ξ(R(q)) 1−eξ(R(q)) < c min q∈[0,L]e2ξ(R(q)) ≤c LZL 0 e2ξ(R(q))dq. These se ies o inequali ies hold, and acco dingly huiE(s∗, z0)<0, i c0>−cmin q∈[0,L] e2k(R(q;z0)− (s∗))1−ek(R(q;z0)− (s∗)) 1−e2k(R(q;z0)− (s∗)) .(3.10) 8 We no e ha in he absence o an unde lying cu en , ha is when c0= 0, condi ion (3.10) always holds and so he esul ing mean Eule ian eloci y is always in he wes e ly di ec ion, an obse a ion which acco ds wi h [13]. The mean Eule ian low (3.4) is eas wa ds, huiE(s∗, z0)>0, i −c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) cos θdq ≥ −c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) dq ≥ −c0min q∈[0,L] 1−e2ξ(R(q)) 1 + eξ(R(q)) > c max q∈[0,L]e2ξ(R(q)) ≥c LZL 0 e2ξ(R(q))dq. These inequali ies hold, and hence huiE(s∗, z0)>0, i c0<−cmax q∈[0,L] e2k(R(q;z0)− (s∗))1 + ek(R(q;z0)− (s∗)) 1−e2k(R(q;z0)− (s∗)) .(3.11) 3.3 S okes d i The S okes d i (o mean S okes) eloci y US(z0)is de ined (c . [1, 13, 31, 32, 37, 39]) by he ela ion huiL(z0) = huiE(z0) + US(z0). We de i e an exp ession o he S okes d i by compu ing US=huiL− huiE=c LZL 0 e2ξ(R(q))dq +c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) cos (k[q−c ])dq −c0. Fo an ad e se cu en , c0≥0, i ollows om (2.9) ha US=1 LZL 0ce2ξ(R(q)) −c0dq +c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) cos (k[q−c ])dq > 0. The e o e o c0≥0 he S okes d i is eas wa ds h oughou he luid domain. In he case a ollowing cu en , c0<0, he exp ession o S okes d i is al oge he mo e complica ed and in ac able. Ne e heless we ema k ha , o c0<0, i he magni ude o he cu en is such ha (3.11) holds hen he S okes d i mus be wes wa ds. 4 Mass lux We conclude wi h a b ie discussion o mass- anspo p ope ies o he low (2.6), whe e we ecall ha huiL, being he mean eloci y o a ma ked pa icle, is some imes called he mass- anspo eloci y. Fo a non-ze o unde lying cu en , c06= 0, we in ui i ely expec he o al mass lux below he ee-su ace wa e pas a poin x=x0 ixed in Eule ian coo dina es o be in ini e. To see his di ec ly we compu e he in eg al m(x0−c , s) = Zη(x0−c ,s) −∞ u(x0−c , s, z)dz. (4.1) 9