Potential vorticity conserving flows and vortex-wave interaction : the role of vertical velocity and isopycnal diffusion on plankton heterogeneity

POTENTIAL VORTICITY CONSERVING FLOWS AND VORTEX-WAVE INTERACTION: THE ROLE OF VERTICAL VELOCITY AND ISOPYCNAL DIFFUSION ON PLANKTON HETEROGENEITY Ba celona, Feb e o de 2012 Ma iona Cla e Ilus aci´on de la con apo ada ealizada po Ma c Gasse Rubina . Anexo I D/Dª...............................................................SECRETARIO/A DEL DEPARTAMENTO DE FISICA DE LA UNIVERSIDAD DE LAS PALMAS DE GRAN CANARIA, CERTIFICA, Que el Consejo de Doc o es del Depa amen o en su sesión de echa............................. omó el acue do de da el consen imien o pa a su ami ación, a la esis doc o al i ulada “Po en ial o ici y conse ing lows and o ex-wa e in e ac ion: he ole o e ical eloci y and isopycnal di usion on plank on he e ogenei y” p esen ada po la doc o anda Dª Ma iona Cla e Co és y di igida po el Doc o Ál a o Viúdez Lomba y la Doc o a Y e e H. Spi z. Y pa a que así cons e, y a e ec os de lo p e is o en el A º 73.2 del Reglamen o de Es udios de Doc o ado de es a Uni e sidad, i mo la p esen e en Las Palmas de G an Cana ia, a…......de.........................................de dos mil doce. Po en ial o ici y conse ing lows and o ex-wa e in e ac ion: he ole o e ical eloci y and isopycnal di usion on plank on he e ogenei y (Flujos que conse an la o icidad po encial e in e acción onda- ó ice: el e ec o de la elocidad e ical y la di usión isopicna en la he e ogeneidad planc ónica) Tesis doc o al p esen ada po Ma iona Cla e Co és di igida po el D . Ál a o Viúdez Lomba y codi igida po la D a. Y e e H. Spi z pa a ob ene el g ado de Doc o a po la Uni e sidad de Las Palmas de G an Cana ia, Depa amen o de Física, P og ama en Oceanog a ía (Bienio 2006-2008). En Ba celona, a eb e o de 2012 La Doc o anda El Di ec o La Codi ec o a Ins i u de Ciències del Ma Pe a la me a `a ia, ´unica en su especie. Pe eni un bomb´o semp e a pun , pe iu e el dia a dia amb i onia, pe la il·lusi´o dels seus ulls. THESIS ABSTRACT This hesis in es iga es physical-ecological and o ex-wa e in e ac ions h ough po en- ial o ici y (PV) conside ing a s a i ied and oligo ophic ocean. To his end, a NPZ (Nu ien s-Phy oplank on-Zooplank on) model is coupled o a physical one ha conse es PV explici ly on isopycnals. The physical-ecological coupled model is ini ialized using s a- iona y NPZ solu ions nume ically s able wi h he luid a es . These solu ions a e im- plemen ed homogeneous bo h on ho izon al and isopycnals le els o quan i y he e ec o ho izon al and e ical ad ec ion caused by mesoscale and submesoscale o ex s uc u es, and isopycnal mixing. A he in e io o he o ex sepa a ix, plank on and PV dis ibu ions ansla e in phase a o ex p opaga ion speed. Wi hin cyclones, isopycnal doming enhances plank on biomass a he o ex cen e in di e en ophic condi ions. Fu he mo e, isopyc- nal mixing associa ed o small-scale mo ions maximizes he phy oplank on (P) biomass in cyclones h ough a esonan esponse be ween Pand di usi e imescales. This Pinc ease is signi ican in meso ophic condi ions and occu s whe e he e ical displacemen o isopy- cnals is maximum, and hence whe e e ical g adien s o PV a e la ge. A he sepa a ix ou e , ho izon al and e ical ad ec ion a e o he same o de o magni ude han he ecolog- ical o cing and enhance P h ough di e en mechanisms. Fi s ly, e ical eloci y wupli s nu ien s and P o be e li le els. P esponds wi h some ime lag o his pe u ba ion and he associa ed inc ease in biomass occu s a om he upwelling loca ion due o he ac ion o ho izon al ad ec ion. As a esul , Pco ela es wi h w, and hus wi h ho izon al g adien s o PV, only a ini ial imes. In he pa icula case o ansla ing cyclones, his mechanism explains he de elopmen o a P ail a hei wake. And secondly, he ho izon al ad ec ion o a su ace ecosys em pa ch by subsu ace o ices dec eases Psel -shading a he pa ch on in bene i o Pg ow h. Finally, in e ac ions be ween o ex s uc u es and pu e ine ial and g a i y la ge ampli ude wa es a e in es iga ed. The ad ec ion o PV by wa es causes o ices o be uns eady and modi ies he uppe and lowe bounds o he wa e equency band. The ad ec ion o wa es by o ices Dopple shi s he local wa e equency. When ine ial wa es a e in ol ed, a nea -ine ial igh -handed helical wa e is de eloped due o a non-linea in e ac ion. As a esul , o al winc eases one o de o magni ude and co ela es wi h ho izon al g adien s o PV. These esul s aim o shed u he ligh on he ecological impac o long-li ed cohe en o ices in he open ocean. LIST OF SYMBOLS Physics ρmass densi y ρ′densi y anomaly ρ0cons an densi y α0cons an speci ic olume pp essu e p′p essu e anomaly zs a i ica ion cons an d(x, ) dep h in he e e ence densi y con igu a ion o isopycnal loca ed a (x, ) D(x, ) e ical displacemen o isopycnals in he spa ial desc ip ion D(s, d) e ical displacemen o isopycnals in he isopycnal desc ip ion Co iolis equency ee ec i e Co iolis equency Nbackg ound buoyancy equency N o al buoyancy equency cP and l a io ωllocal wa e equency ωppa icle wa e equency u= (u, , w) eloci y ec o uggeos ophic eloci y wqQG e ical eloci y Qg hgeos ophic Q- ec o Uddipole speed ω≡ωh+ζk ela i e o ici y ωggeos ophic o ici y ω′ageos ophic o ici y PV anomaly Π o al PV ϕ= (ϕ, ψ, φ) ec o po en ial FF oude numbe RRossby numbe κdimensional di usion coe icien Knon-dimensional di usion coe icien Zmin minimum dimensional e ical coo dina e z Lspa ial con e sion ac o T empo al con e sion ac o Ecology Pphy oplank on Zzooplank on Nnu ien s NT o al ni ogen GPphy oplank on p oduc ion a e GZzooplank on p oduc ion a e K0hal -sa u a ion o Pup ake Lpho osyn he ic a e Ia ailable adia ion I0su ace a ailable adia ion Awligh a enua ion by sea wa e Apligh a enua ion by P Ψ0ini ial slope P-Icu e V0Pmaximum up ake a e MPphy oplank on mo ali y MZzooplank on mo ali y Ξ0Pspeci ic mo ali y a e Rzooplank on g azing a e R0zooplank on maximum Λ0I le cons an Γ0 ac ion o Zg azing eges ed Ψ0Zexc e ion/mo ali y a e Θ0Zspeci ic exc e ion/mo ali y a e Chap e 1 In oduc ion Ma ine ecosys ems a e highly sensi i e o ocean physics. Pelagic o ganisms a e embedded in he luid and hus hey a e explici ly a ec ed by i s dynamics. Since mos o hese indi iduals a e non-mo ile, hey d i in he as ocean. These wande e s owe hei name o he G eek equi alen , plank on. Two mic oscopic g oups a e enclosed wi hin plank on, au o ophs and he e o ophs. The o me s a e called phy oplank on, whe e phy o deno es i s plan -like cha ac e due o he pe o mance o pho osyn hesis. The la e s a e he animal-like plank on, he zooplank on, since hey eed om phy oplank on o he smalles zooplank on. Phy oplank on is o g ea impo ance because is he basis o he oceanic ood web. I s g ow h is a ec ed by luid mo ion because g ow h limi ing ac o s, ligh and nu ien s, a e e ically seg ega ed in he ocean. On he one hand, he well-li laye is akin o he ip o an icebe g, i is he uppe ac ion o an ocean which is on a e age wen y imes deepe . On he o he hand, nu ien s slowly and incessan ly sedimen in o he da k deep ocean. As a esul , e ical eloci ies in he ocean play an implici ole in ecosys ems by e ilizing he well-li zone. Many wo ks ha e been conduc ed o cha ac e ize he ecological pa e ns associa ed o spe- ci ic physical s uc u es. In his ega d, Hau y e al. (1978) i s d ew an ecological coun e - pa o he S ommel diag am, which plo s physical ocean a iables in a spa io- empo al log- a i hmic ame, o ep esen zooplank on biomass a iabili y. Nowadays, physical-biological in e ac ions ha e been ex ensi ely epo ed om plane a y o Kolmogo o scales (S eele, 1978;Mann and Lazie ,1991;Denman and Ga ge ,1995). The combina ion o hese wide ange o spa ial scales esul s in complex and de e minis ic plank on pa e ns, which a e e- lec ed in he sea su ace chlo ophyll dis ibu ions as measu ed om space sa elli es. Highly 1 2 CHAPTER 1 p oduc i e egions a e localized a wes e n bounda ies, sub-pola la i udes, and he Equa- o (Yode e al.,1993), whe e coas al upwelling, wa e mass subduc ion, and T ade Winds o cing, espec i ely, o e come he ligh -nu ien e ical seg ega ion. These a eas a e called eu ophic, because communi ies a e no limi ed by nu ien s, as opposed o oligo ophic zones. The la ges pa o he open ocean is oligo ophic. I s sea su ace chlo ophyll concen a ion (CC) has a geome ic mean abou wo o de s o magni ude smalle han ha o eu ophic wa e s, and a a iabili y domina ed by submon hly mesoscale a iance (Doney e al.,2003). Since CC images ca y in o ma ion abou he ocean su ace u bulen low (Nie es e al., 2007), plank on a iabili y in he oligo ophic open ocean may be likely caused by mesoscale phenomena, and o ical s uc u es may be pa icula ly in ol ed. The dynamics and h ee- dimensional s uc u e o ecosys ems in he open ocean emain la gely unknown because synop ic and high spa ial esolu ion sampling he e is mo e cos ly and challenging han in he coas al ocean. In his con ex , he aim o his hesis is o p o ide u he insigh on how ubiqui ous mesoscale eddies al e he basis o he ood web in oligo ophic en i on- men s. To his end, we analyze i s which ac o s shape plank on pa e ns a mesoscales in o de o choose an app op ia e physical desc ip o (sec ion 1.1). We hen e iew he mech- anisms h ough which o ices gene a e ecological he e ogeni y (sec ion 1.2) and iden i y hose which a e ela ed o ou pu pose (sec ion 1.3). Finally, he speci ic objec i es o his hesis a e ou lined by chap e s (sec ion 1.4). 1.1 Plank on pa chiness a mesoscales: Ecological oo - p in o po en ial o ici y Fi s synop ic maps o sea su ace chlo ophyll we e ob ained in he ea ly 80’s (Go don e al., 1980;Gowe e al.,1980). A i s glance, hese ype o maps show high co ela ion wi h sea su ace empe a u e a mesoscales, which has led o conside plank on as a passi e ace o a i s app oxima ion. Howe e , disc epancies be ween physics and ecology a ise when hey a e quan i ied (see e iew Ma in,2003). Two main ac s accoun o hese di e ences, nu ien pumping a smalle scales and plank on ime lag esponse o pe u ba ions. Submesoscale s uc u es ha e gained inc easing a en ion he las decade because hey enhance e ical eloci y wone o de o magni ude wi h espec o mesoscale s uc u es and a e esponsible o injec ion o alloch honous nu ien s o he pho ic zone (see e iew Klein and Lapey e, 1.1. PLANKTON PATCHINESS AT MESOSCALES: ECOLOGICAL FOOTPRINT OF POTENTIAL VORTICITY 3 2009), whe e ligh i adiance is g ea e han one pe cen o ha a i ing a he sea su ace. Since physical and ecological imescales a e simila a he submesoscale, plank on couples o his nu ien upwelling and wcolloca es wi h p ima y p oduc ion (L´e y e al.,2001) and some pho osyn he ic indexes (Falkowski,1983;Cullen and Lewis,1988). Howe e , exac co ela ion wi h wdepends on he ecological pa ame e imescale. Those pa ame e s wi h imescales g ea e han ha o ho izon al ad ec ion, such as phy oplank on biomass, show a spa ial lag wi h w(Lima e al.,2002). The impo ance o inhe en ecological imescales in plank on pa e ns was nicely illus a ed by Ab aham (1998) using a wo-dimensional u bu- lence nume ical model. He obse ed an inc easing pa chiness om physics o phy oplank on, and o zooplank on. The e o e mesoscale plank on dis ibu ions esul om an in e play be- ween ho izon al ad ec ion, e ical ad ec ion, and in insic plank on imescales. Whe he hese dis ibu ions a e caused by ecological o physical phenomena depends on hei ela i e imescale (Mahade an and Campbell,2002). In o de o ela e bo h imescales, we seek o desc ibe he ully h ee-dimensional low na u e wi h a single imescale. A e y use ul physical quan i y ha ela es ho izon al and e ical mo ion is po en ial o ici y (PV). The concep o PV was in oduced by Bel ami in 1871, applied o an adi- aba ic in iscid mul i-laye ed ocean by Rossby in 1936, and ex ended o ba oclinic lows by E el in 1942 (check Vi´udez,2001, o ci es and ela ion be ween PV de ini ions). Acco ding o he la e , speci ic PV is de ined as Π≡ω+ k ρ·∇Tθ,(1.1) whe e he sum o o ici y ω= (ξ, η, ζ) ela i e o a e e ence ame o a ing wi h Ea h’s angula eloci y and he e ical componen o he plane a y o ici y , which is he Co iolis pa ame e , is he absolu e o ici y. Addi ionally, ρis he mass densi y, Tθ he po en ial em- pe a u e, hough i could be any scala luid p ope y ma e ially conse ed (Pedlosky,1987, chap e 2), and ∇ he h ee-dimensional g adien ope a o . The geophysical signi icance o PV lies on i s ma e ial in a iance, ha is, dΠ d = 0 .(1.2) One way o de i e (1.2) comes om he conse a ion o ci cula ion o ic ionless mo ion conside ing ma e ial conse a ion o Tθand mass conse a ion (Pedlosky,1987, chap e 2). 4 CHAPTER 1 Thus (1.2) can be in e p e ed as he conse a ion o ci cula ion along ma e ial ci cui s and angula momen um o a gi en olume, which le us o ela e ho izon al and e ical mo ions. When he luid is ba o opic his ela ion is s aigh o wa d since (1.1) becomes Π≡ζ+ h,(1.3) whe e his he e ical sepa a ion be ween neighbo ing ma e ial isosu aces (eq. 3.4.9. Ped- losky,1987). Thus changes in himply adjus men s in ζin o de o conse e PV, which explains he well-known balle ina and ice ska e e ec s. Ins ead, when he luid is ba oclinic he il ing o ma e ial isosu aces in oduces non i ial changes in ω. Fu he mo e, i some balance condi ion is es ablished in he momen um equa ions (along wi h p ope bounda y condi ions), hen o a gi en dis ibu ion o PV a a ixed ime he eloci y ec o , p essu e, and densi y h ee-dimensional ields associa ed o he balanced low ( oid o ine ia–g a i y wa es IGWs) can be eco e ed in a p ocess known as PV in e sion (Hoskins e al.,1985; McIn y e and No on,2000;Vi´udez,2008a). PV is ele an o ecological dynamics o se e al easons. Fi s ly, i desc ibes he balanced eloci y ec o ield and he e ical displacemen o isopycnals D. Secondly, i ela es e ical upwelling wi h ho izon al ad ec ion, which is c ucial o plank on pa chiness (Ma in e al.,2002). The impo ance o PV in ecology was i s poin ed ou by Woods (1987) and S ass and Woods (1987). They obse ed chlo ophyll inc ease in egions o la ge PV isopycnal g adien s, which we e hough o be a eas o high w. Thei hypo hesis ag ees wi h expe imen al (Pall`as-Sanz and Vi´udez,2005) and nume ical (Vi´udez and D i schel,2003, 2004b) wo ks ha demons a e ha zones wi h la ge ho izon al g adien s o ζand PV a e ela ed o high alues o w. La ge PV ho izon al g adien s in ol e la ge ho izon al g adien s in ρand he e o e in D. Consequen ly, when hese Dg adien s a e ad ec ed, la ge local a es o Dand woccu . Howe e , e ical ad ec ion does no accoun o he whole plank on big pic u e a o emen ioned. 1.2 How do mesoscale o ices al e ecosys ems? Many in-si u obse a ions e eal eddies impac on ma ine biology. A he basis o he ood web, o ices quali a i ely a ec communi y s uc u e (Thompson e al.,2007;Huang e al.,2010), physiological p ocesses (Bibby e al.,2008), and ecosys ems anspo (Ba en 1.2. HOW DO MESOSCALE VORTICES ALTER ECOSYSTEMS? 5 and C aw o d,2005). Addi ionally, hey quan i a i ely al e biogeochemical balances by enhancing new p ima y p oduc i i y (NPP) (Mo ´an e al.,2001). Fo ins ance, hey a e in ol ed in he No h A lan ic ca bon balance, hough hei speci ic con ibu ion is unde scien i ic deba e anging om 50% o less han 10% o NPP (see e iew Oschlies,2008). As a esul , o ices also pe u b highe ophic le els (Mackas e al.,2005;A wood e al.,2010). The e a e di e en mechanisms by which ocean eddies in oduce he abo e men ioned ecological a iabili y. Nex , we analyze hei ecological spa ial signa u e in o de o ela e i wi h PV when his is possible. •Ho izon al ad ec ion. Mesoscale wo-dimensional u bulence induces a conse a i e ans e om la ge o small scales (Ab aham,1998). I we conside ha phy oplank on beha es as a passi e ace , hen i s pa e n is de e mined by o ici y and s ain. Inside o ices, o ici y domina es o e s ain and pa icles a e apped acing nea ly o bi al ajec o ies. This explains long- e m anspo o chlo ophyll ich wa e s o - sho e he Alge ian (A none and LaViole e,1986), Alaskan (Ba en and C aw o d, 2005), and no hwes A ican (Peleg ´ı e al.,2005) coas s among many o he s. Ou side o ices, s ain o e comes o ici y, esul ing in chao ic mo ion (P o enzale,1999). As a esul , eddies s i plank on pa ches in o spi als and ilamen s (Lehahn e al.,2007) and e en wa e-like s uc u es (Menkes e al.,2002) a hei edges. •Eddy pumping. This e m was coined by Falkowski e al. (1991) o deno e NPP en- hancemen due o upwelling o he nu icline in o he pho ic laye caused by he passage o a ansla ing su ace cyclone o subsu ace an icyclone (McGillicuddy e al.,1999). On he one side, isopycnal upli is g ea es a he eddy cen e , which explains high al- ues o sea su ace chlo ophyll a eddy co es (A ´ıs egui e al.,1997;Ba on e al.,1998; Mizoba a e al.,2002;Siegel e al.,2008;Tew-Kai and F ancis,2009;Siegel e al.,2011). On o he side, PV is co ela ed wi h he isopycnal e ical displacemen (Vi´udez and D i schel,2003). Consequen ly, we expec chlo ophyll en ichmen a maxima absolu e PV alues. •Submesoscale. Fully-de eloped o ices a e a ely sphe ical in he iso opic quasi- geos ophic (QG) space. The QG space is he e ically s e ched space o dimensions (x, y, cz), whe e c≡N/ is he P and l a io, N he backg ound buoyancy equency, and he ine ial equency. Ins ead, a a ie y o ellip ical geome ies a e shaped 6 CHAPTER 1 du ing hei li e- ime, which lead o o ex o a ion, and ul ima ely o a quad upo- la pa e n dis ibu ion o e ical eloci y w(Vi´udez and D i schel,2003). Ve ical eloci y maxima a e eached a o ex edges enhancing chlo ophyll concen a ions a he eddy pe iphe y (Mizoba a e al.,2002;Ladd e al.,2005;Siegel e al.,2011). Thus chlo ophyll would be be e ela ed o PV ho izon al g adien s han o PV. Howe e , o un eil his wpa e n, esolu ions smalle han 10 km, which a e close o submesoscale sampling, a e o en equi ed (L´e y e al.,2001). •Eddy-eddy in e ac ions. Vo ices in e ac in a numbe o ways h ough bounding, me ging, o con o ming complex s uc u es (Vo opaye and A anasye ,1994). Since hese in e ac ions modi y he low, hey also pe u b ecological dis ibu ions. Fo ins ance, he ubiqui ous p esence o mush oom-like shapes o sea su ace chlo ophyll in on al coas al zones (Su e al.,1996;S aple on e al.,2002) a e imp in s he in e ac ion be ween wo o ices o opposi e PV, ha is, he o ex dipole. Ano he example is he me ging o an icyclones in he Ku oshio Cu en , which esul s in an inc ease o wwhich igge s phy oplank on enhancemen a o ex edges (Yoshimo i and Kishi,1994). •Isopycnal mixing. As a o eme ioned, Dmaxima occu a he monopole cen e , which c ea es isopycnal g adien o ligh i adiance. In oligo ophic egimes phy oplank on g ow h is mainly limi ed by ligh and nu ien s. Since isopycnal doming modi ies he amoun o ligh a which isopycnal con ined phy oplank on is exposed, i a ec s i s g ow h, and indi ec ly nu ien consump ion. Thus isopycnal di usion may inc ease he phy oplank on esponse o ligh a cyclone co es and an icyclone edges h ough a down-g adien eplenishmen lux o nu ien s. •Vo ex-wa e in e ac ion. On he one hand, in e ac ions be ween o ices and IGWs a e ubiqui ous in he ocean. Some o hese in e ac ions imply ine ial o nea -ine ial wa e equency shi (Pe kins,1976) and apping o wa epacke s wi hin an icyclones (Kunze,1986). On he o he hand, phy oplank on dis ibu ions a e pe u bed by bo h eddies, and high equency oscilla ions (F anks,1995a;G ana a e al.,1995;G´omez e al.,2001;Sang `a e al.,2001). Thus, we expec o ex-wa e in e ac ion o a ec phy oplank on dynamics. 1.3. RELEVANT PHYSICAL-ECOLOGICAL INTERACTIONS IN COHERENT VORTICES 7 Se e al o he o ical mechanisms ha e been p oposed o explain plank on pa e ns. Fo ins ance, wind o cing inc eases p ima y p oduc ion ei he a he o ex edges h ough eddy-wind in e ac ion (Ma in e al.,2001;Mahade an e al.,2008) o an icyclone co es by deepening o he mixed laye (Thompson e al.,2007). The con ibu ion o hese mechanisms as well as diapycnal mixing o phy oplank on pa chiness is beyond he scope o his wo k. 1.3 Rele an physical-ecological in e ac ions in cohe - en o ices Vo ices li e cycle in ol e h ee di e en s ages wi h di e en dynamics: o ma ion, ma u i y, and decay (Sang `a e al.,2005). High u bulen a eas, such as coas al on s, a e ho -spo s o o ex o ma ion and decay. In con as , he s a i ied open ocean is mainly domina ed by long-li ed ma u e o ices. Fully-de eloped cohe en o ices ansla e ac oss he ocean conse ing hei PV. The ou e PV isosu ace de ines he o ex sepa a ix, which ma ks o ex limi s and ac s as an impe meable ba ie . Wi hin he sepa a ix he luid o a es app oxima ely as a solid body. Ou side he sepa a ix, he ho izon al eloci y dec eases exponen ially wi h adius. Thus o ices can ap wa e s in hei in e io and anspo ecosys ems long dis ances. I hose ecosys ems a e only a ec ed by ho izon al ad ec ion, he anspo ing dis ance will depend on he sinking a e o nu ien s o he apho ic laye . This leads o long- e m chlo ophyll deple ion wi hin o ices unless some nu ien upwelling occu s. In his ega d, eddy pumping assumes en ainmen o new nu ien s in o he o - ex (Ma in and Ponda en,2003), which implies mixing ac oss o ex bounda ies and hus es ic s he mechanism o de eloping cyclones and decaying an icyclones (F anks e al., 1986b). This mechanism causes he ini ial seed o ecological he e ogenei y hough does no explain how i is main ained du ing o ex ansla ion. In con as , he e ical eloci y o mesoscale and submesoscale monopoles o complex o ical s uc u es may con ibu e o ecosys em subsis ence in one yea li ing o ices. Addi ionally, nu ien injec ion by isopyc- nal di usion is likely o occu in he open ocean, whe e he low is mainly wo-dimensional. Though he con ibu ion o isopycnal mixing in upwelling new nu ien s is minimal compa ed o he men ioned mechanisms (Siegel e al.,1999;Ledwell e al.,2008), i may play some ole in upwelling egene a ed nu ien s. Finally, he way o ex-wa e in e ac ion a ec s ecology emains la gely unknown. 2.1. INTRODUCTION 15 2.1 In oduc ion The dynamics o oceanic plank onic ecosys ems is o en in es iga ed using nu ien -phy o- plank on-zooplank on (NPZ)- ype nume ical models (W oblewski,1977;F anks e al.,1986a; F anks,2002;Newbe ge e al.,2003). These models a e disc e e e sions o he con inuous pa ial di e en ial equa ions o he ield a iables NPZ. In he absence o low and ho izon al g adien s his sys em o equa ions has a numbe o one-dimensional (1D) s eady analy ical solu ions which a e e y use ul bo h o cha ac e ize di e en plank onic egimes and o se e as ini ial condi ions in coupled physical-ecosys em nume ical modeling. I may happen, howe e , ha some o hese 1D s eady solu ions be only con inuous, bu no con inuously di e en iable unc ions o he e ical coo dina e zalong he wa e column. This po en ial lack o di e en iabili y makes hese NPZ solu ions inapp op ia e as ini ial condi ions in h ee-dimensional (3D) coupled physical-ecosys em models. These 3D coupled models, i o mula ed as i is usual in he spa ial desc ip ion, equi e exis ence o e ical de i a i es o he e ical ad ec i e e ms, p esen in he ma e ial a e o change o he ecosys em quan i ies, make sense. In his pape we p esen a pa icula example o a mo ionless, 1D s eady NPZ solu ions which, being con inuous bu no con inuously di e en iable a se e al dep hs, do howe e admi nume ical equi alen s ha a e con inuously di e en iable in he nume ical sense (he e meaning con e gence o he e ical de i a i e wi h espec o inc easing e ical esolu ion). In he nex sec ion he NPZ sys em o equa ions is b ie ly in oduced. Though mo e sophis- ica ed ecological models exis a simple one is used he e because we seek o keep he numbe o ee pa ame e s as small as possible, while e aining he essen ial beha io o he eco- logical ields, pa icula ly he de elopmen o NPZ anomalies due o he e ical ad ec ion o nu ien s in o he eupho ic zone. The NPZ- ype models a e ecological bulk algo i hms subjec ed o signi ican e o s in he ma hema ical pa ame iza ion o he di e en ecological p ocesses (see Ande son,2005;Flynn,2005;Mi a e al.,2007). The mo ionless 1D con inuous s eady ecological solu ions and hei nume ical con inu- ously di e en iable equi alen s a e ob ained in sec ion 2.3. Con e gence o his di e en iable solu ion is eached howe e a e ical esolu ions o a ew cen ime e s. As an applica ion example, in sec ion 2.4 he di e en iable s eady NPZ solu ions a e used as ini ial condi ions in a 3D coupled physical-ecosys em model (wi h poo e ical esolu ion) o add ess he ole 16 CHAPTER 2 o he e ical eloci y in a case o oceanic ba oclinic ins abili y. Concluding ema ks a e gi en in sec ion 2.5. 2.2 NPZ Model The dependen a iables o he NPZ model a e he dissol ed ino ganic ni ogen (N), he phy oplank on (P), and he zooplank on (Z) biomass. These a iables a e exp essed in uni s o concen a ion o ni ogen (he e always in mmol N m−3), and sa is y he sys em o equa ions (W oblewski,1977;Newbe ge e al.,2003) dP d =N K0+NL P |{z } GP −R0(1 −e−Λ0P)Z |{z } R −Ξ0P |{z} MP ,(2.1) dZ d = (1 −Γ0)R |{z } GZ −Φ0Z |{z} MZ ,(2.2) dN d =−GP+ Γ0R+MZ+MP.(2.3) Abo e, he ma e ial a e o change dχ/d ≡∂χ/∂ +u·∇χis he sum o he local and ad ec i e a es o change o χ,u= (u, , w) is he 3D eloci y, and ∇is he 3D g adien ope a o . The Pinc eases due o i s p oduc ion a e (GP), which depends on bo h he ino - ganic dissol ed ni ogen up ake and he pho osyn hesis, and dec eases due o he he bi o e g azing (R) and he phy oplank on mo ali y a e (MP). Cons an K0is he hal -sa u a ion concen a ion o phy oplank on up ake o nu ien s. The Zinc eases due o he inges ion o phy oplank on (GZ) and dec eases due o he zooplank on speci ic exc e ion and mo ali y a e (MZ). The Pp oduc ion (GP) ollows he Michaelis-Men en kine ics and depends on he ol- lowing pho osyn he ic a e, L(x, )≡V0Ψ0I(x, ) pV2 0+ Ψ2 0I2(x, )(2.4) (Newbe ge e al.,2003), which depends on he pho osyn he ically a ailable adia ion I(x, )≡I0exp ½Awz−ApZ0 z P(x, y, z′, )dz′¾,(2.5) 2.3. STEADY SOLUTIONS 17 whe e z≤0. Abo e, V0is he phy oplank on maximum up ake a e, Ψ0is he ini ial slope o he P-Icu e, Awis he ex inc ion coe icien o seawa e in he absence o phy oplank- on ( ollowing he Lambe -Bee law and ligh a ia ion wi h day ime is no conside ed), I0exp{Apκ(x, )}is he ligh a enua ion by phy oplank on sel -shading, whe e cons an I0 is he su ace pho osyn he ically a ailable adia ion, and Apis he ex inc ion coe icien pe uni concen a ion o phy oplank on. Adding (2.1)+(2.2)+(2.3) he o al ni ogen NT≡N+P+Zis ma e ially conse ed, dNT d = 0 .(2.6) I is con enien he e o e o de ine he sys em o independen equa ions as (2.1)-(2.2)-(??), ins ead ha he o iginal se (2.1)-(2.2)-(2.3). In he nex sec ion wo pa icula solu ions o hese equa ions a e ob ained in he case o s eady and ho izon ally homogeneous ecosys em dis ibu ions in he absence o low. 2.3 S eady Solu ions 2.3.1 Analy ical S eady Solu ions In o de o ob ain analy ical s eady solu ions o ho izon ally homogeneous dis ibu ions we se dP/d = dZ/d = 0 in (2.1) and (2.2), so ha spa ial unc ions depend only on z. The wa e column is di ided in o h ee laye s. In he uppe laye P6= 0 and Z6= 0, in he mid laye P6= 0 and Z= 0, and in he lowe laye P=Z= 0. In he uppe laye (z∈(z1,0]) he s eady P, ob ained di ec ly om (2.2) (e.g., Busenbe g e al.,1990, eq. 8; Newbe ge e al.,2003, eq.18) is independen o z P(z) = P1=−1 Λ0 ln ·1−Φ0 R0(1 −Γ0)¸, z ∈(z1,0] .(2.7) Wi h he commonly used pa ame e s o upwelling condi ions gi en in Table 2.1,P1= 8.468 mmol N m−3. The alues and a sensi i i y analysis o hese pa ame e s a e gi en in (Newbe ge e al.,2003). The maximum dep h z1a which his solu ion is easible is de- e mined below. F om (2.1) and (??) he zooplank on Z(z) = Z1(z) in he uppe laye 18 CHAPTER 2 desc ip ion alue uni s Awligh a enua ion 0.067 m−1 Apligh a enua ion by P0.0095 m2mmol N−1 Ψ0ini ial slope o P-Icu e 0.025 m2(W d)−1 V0Pmaximum up ake a e 1.5 d−1 I0su ace a ailable adia ion 158 W m−2 K0hal -sa u a ion o Pup ake 1 mmol N m−3 Ξ0Pspeci ic mo ali y a e 0.1 d−1 R0Zmaximum g azing a e 0.52 d−1 Λ0I le cons an 0.06 m3mmol N−1 Γ0 ac ion o Zg azing eges ed 0.3 Φ0Zexc e ion/mo ali y a e 0.145 d−1 Table 2.1: Lis o cons an s (Newbe ge e al.,2003). Pand Zs and o phy oplank on and zooplank on, espec i ely. z∈(z1,0] is ob ained sol ing he quad a ic equa ion A0Z2 1+BZ1+C= 0 ,(2.8) whe e he coe icien s A0≡Φ0 (1 −Γ0)P1 ,(2.9) B(z)≡ −L(z)−A0(K0+NT−P1) + Ξ0,(2.10) C(z)≡[L(z)−Ξ0] (NT−P1)−Ξ0K0.(2.11) Thus, Z1(z) = −B(z)−pB2(z)−4A0C(z) 2A0 .(2.12) The nega i e oo solu ion abo e ensu es ha Z(z)< NT. To u he simpli y he p oblem we conside NTas cons an , independen o z.NTmus be such ha he disc iminan B2(z)−4A0C(z)≥0. The maximum dep h z1o he uppe laye is de ined as he shallowe dep h a which Z(z1) = 0. Using (2.1) o (??) his condi ion implies C(z1) = 0, and he e o e L1≡L(z1) = Ξ0µ1 + K0 NT−P1¶.(2.13) 2.3. STEADY SOLUTIONS 19 In e ing L(z1) using (2.4) he dep h z1de ining he lowe bounda y o he uppe laye is z1=1 2(ApP1+Aw)ln ·V2 0L2 1 (V2 0−L2 1)Ψ2 0I2 0¸.(2.14) In he mid laye , z∈(z2, z1], Z(z) = Z2(z) = 0 and P(z) = P2(z)6= 0. In his laye , due o (2.3) o (2.1), P2sa is ies he ela ion P2(z) = NT−Ξ0K0 L(z)−Ξ0 ,(2.15) whe e L(z) is gi en by (2.4) and (2.5). The solu ion P2(z) is ound he e sol ing (??) nu- me ically. This solu ion is easible as long as P2(z)≤NTwhich, om (??), implies ha L(z)>Ξ0. Consequen ly, he maximum dep h z2o his mid laye is de ined as ha a which L(z2) = Ξ0. In e sion o his equa ion implies ha z2sa is ies he ela ion z2−Ap AwZz1 z2 P2(z′)dz′= =1 Aw ln ÃV0Ξ0 Ψ0I0pV2 0−Ξ2 0!−Ap Aw P1z1.(2.16) Fo i s la e use i is con enien o de ine he maximum dep h z3o his mid laye as ha ob ained by neglec ing he phy oplank on sel -shading only in his mid laye , z3≡1 Aw ln ÃV0Ξ0 Ψ0I0pV2 0−Ξ2 0!−Ap Aw P1z1.(2.17) Clea ly, z3< z2. Finally, in he lowe laye , z∈[zmin, z2], he e is nei he Pno Z(P3= Z3= 0), so ha he dissol ed ino ganic ni ogen N(z) = N3=NT. The choice o NTis pa icula ly impo an in his NPZ solu ion since i mus be such ha he esul ing dis ance be ween z1and z2be la ge enough o be p ope ly disc e isized using a ini e g id size. Based on he beha iou o z1and z3as unc ions o NT(Fig. 2.1), NT mus be close o P1= 8.468 mmol N m−3. We selec NT= 8.8 mmol N m−3. Wi h his choice he alues o he ansi ion dep hs a e z1=−15.26 m, z2=−25.13 m, and z3=−36.52 m. The maximum Z, ob ained om (??) a z= 0, becomes Z0=Z(0) = 0.25 mmol N m−3. The solu ions NPZ a he h ee laye s a e shown in Fig. 2.2. 20 CHAPTER 2 6 8 10 12 14 16 −50 −40 −30 −20 −10 0 z* 1 z* 3 N* 0 z1 z2 z3 P1N0 z (m) Figu e 2.1: Dep hs z∗ 1(??) and z∗ 3(2.6) as unc ions o he a iable o al ni ogen N∗ 0. Uni s a e mmol N m−3. No e ha z1=z∗ 1(NT) and z3=z∗ 3(NT). In he uppe laye he amoun o Zde- c eases wi h dep h ( om Z0 o 0) which is compensa ed (since bo h Pand NTa e con- s an ) by an equally small inc ease o N. In he mid laye , Pdec eases om P1 o 0, and is compensa ed (since Z= 0) by an equally la ge inc ease o N. In his s eady solu ion, and in he up- pe laye , he la ge z-dependen GP(z) is mainly balanced by he la ge cons an MP (Fig. 2.3). The emanen , smalle pa o GP(z) is balanced by he small z-dependen g azing R(z). In he Zbalance, he small g ow h GZ(z) o zooplank on is balanced by MZ(z). In he mid laye R= 0, so ha GP(z) and MP(z) exac ly balance. These s eady NPZ solu ions a e con inuous unc ions o z, bu hey ha e been ob ained wi hou any equi emen on di e en iabili y condi ions. I is clea , a leas isually om Fig. 2.2, ha hese unc ions a e no e ically di e en iable a z=z1o z=z2. As a simple p oo conside di e en iabili y o P(z) a z2. The e ical de i a i e o (??) is ∂P2 ∂z =Ξ0K0 L(z)−Ξ0 ∂L ∂z =Ξ0K0 L(z)−Ξ0 V3 0Ψ0 p(V2 0+ Ψ2 0I2)3 ∂I ∂z =Ξ0K0 L(z)−Ξ0 V3 0Ψ0[Aw+ApP2(z)] p(V2 0+ Ψ2 0I2)3I(z).(2.18) As z→z2, we ha e P2(z)→0 and L(z)→Ξ0, and he e o e ∂P2/∂z → ∞ in he mid laye , as obse ed in Fig. 2.2a. This limi does no ma ch wi h he e ical de i a i e o P as z→z2in he lowe laye , whe e ∂P3/∂z = 0. This lack o di e en iabili y ( he unc ions a e con inuous bu no con inuously di e en iable) implies ha hese s eady solu ions a e ques ionable as ini ial condi ions in many coupled physical-ecosys em models. Mos o hese models a e o mula ed in he spa ial (Eule ian) desc ip ion and equi e ha solu ions NPZ be con inuously di e en iable o he e ical g adien s p esen in he ad ec i e de i a i e on he igh hand side o (2.1)-(2.2) make sense. In he nex sec ion NPZ solu ions, simila o he ones desc ibed abo e bu con inuously di e en iable, a e nume ically ob ained. 2.3. STEADY SOLUTIONS 21 −2 0 2 4 6 8 10 P(z) −60 −40 −20 0 z (m) z1 z2 (a) N0 P1 uppe laye mid laye lowe laye 0.0 0.1 0.2 0.3 Z(z) −60 −40 −20 0 Z0 (b) −2 0 2 4 6 8 10 N(z) −60 −40 −20 0(c) N0 Figu e 2.2: S eady e ical p o iles o (a) P(z), (b) Z(z), and (c) N(z) in he h ee laye s. The cons an concen a ions P1,NT, and Z0, as well as he ansi ion dep hs z1 and z2a e indica ed. Uni s a e mmol N m−3. −0.2 0.0 0.2 0.4 0.6 0.8 1.0 −40 −30 −20 −10 0 z (m) GP RGZ (a) z1 z2 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 −40 −30 −20 −10 0 PM ZM (b) Figu e 2.3: Ve ical p o iles o (a) phy o- plank on p oduc ion GP(z), g azing R(z), and zooplank on g ow h GZ(z); (b) phy o- plank on mo ali y MP(z) and zooplank on mo ali y MZ(z). Uni s a e mmol N m−3d−1. 22 CHAPTER 2 2.3.2 Nume ical S eady Solu ions In o de o ob ain con inuously di e en iable s eady P(z) and Z(z) solu ions, he p ognos ic equa ions (2.1)-(2.2) a e nume ically in eg a ed in ime, using as ini ial condi ions smoo h p o iles P(0)(z) and Z(0)(z), un il a s eady s a e is eached. The ini ial p o iles P(0)(z) and Z(0)(z) a e iden ical o he solu ions ob ained in he p e ious sec ion (Fig. 2.2a,b) excep ha in he non homogeneous laye s, ha is, he mid laye in he case o Pand he uppe laye in he case o Z, he z-dependen p o iles a e eplaced by ansi ion cosine unc ions. Since ∂cos(z)/∂z =−sin(z) = 0 a z={0, π},P(0) and Z(0) ha e con inuous (ze o) de i a i es a z1and z3, and a 0 and z1, espec i ely. Speci ically, we de ine he ini ial p o iles P(0)(z) =          P1, z ∈(z1,0] 1 2P1h1 + cos (z−z1)π z1−z3i, z ∈[z3, z1] 0, z < z3, (2.19) and Z(0)(z) =    1 2Z1³1 + cos zπ z1´, z ∈[z1,0] 0, z < z1. (2.20) These ini ial p o iles a e shown in Fig. 2.4. Nex , P(0)(z) and Z(0)(z) a e e ically disc e isized and in eg a ed o wa d using (2.1)-(2.2) wi h u= 0. Nine di e en nume ical esolu ions, anging om cons an g id-size δz = 4 m (i= 1) o δz = 1.5625 cm (i= 9), a e implemen ed (Table 2.2). 0 2 4 6 8 P(z) −40 −35 −30 −25 −20 −15 −10 z (m) (a) P(0) P(1) z3 z2 z1 −0.1 0.0 0.1 0.2 0.3 Z(z) −20 −15 −10 −5 0(b) Z(0) Z(1) z1 Figu e 2.4: (a) P(z), and (b) Z(z). The ini ial p o iles a e P(0) and Z(0). The s eady solu ions o he di e en nume ical esolu ions a e P(i)and Z(i)(i={1,...,9}). Close up iews o P(i)a z∼z2, and o Z(i)a z∼z1a e shown in Figs. 2.5a and 2.6a, espec i ely. Uni s a e mmol N m−3. 2.3. STEADY SOLUTIONS 23 label g id poin s esolu ion (m) i ni= 25 ∗i+ 1 |Zmin|/(ni−1) 1 26 4 2 51 2 3 101 1 4 201 0.5 5 401 0.25 6 801 0.125 7 1601 0.0625 8 3201 0.03125 9 6401 0.015625 Table 2.2: Lis o nume ical esolu ions. He e Zmin =−100 m. Time in eg a ion is ca ied ou using an explici leap- og scheme, oge he wi h a Robe - Asselin ime il e o a oid he compu a ional mode (see e.g., Du an,1998, p. 62). The ime in eg a ion was 104days, a he end o which he maximum o cing e m in he local a e o change o Po Zwas o he o de o 10−7mmol N m−3d−1. Du ing he in eg a ion ime Pand Zmono onically con e ged o he s eady solu ions P(i)and Z(i)(i= 1,...,9) shown in Fig. 2.4. A e a i s look hese solu ions seem o be e y simila o he non con inuously di e en iable solu ions ob ained in he p e ious sec ion. Howe e , a close iew a ound he laye bounda y dep hs (zooms on z=z2and z=z3in Figs. 2.5a and 2.6a, espec i ely) e eals ha o la ge esolu ions (i≥7) he solu ions P(i)and Z(i)become con inuously di e en iable unc ions o z. F om he nume ical poin o iew his means ha he e ical g adien s ∂P(i)/∂z and ∂Z(i)/∂z, he e compu ed using a simple second o de cen e ed scheme, ha e con e ged o ini e alues and no longe depend on he nume ical esolu ion (Figs. 2.5b and 2.6b). −0.2 0.0 0.2 0.4 0.6 0.8 1.0 P(z) (mmol N m −3) −25.4 −25.3 −25.2 −25.1 −25.0 (a) z (m) z2 3 4 5 6 7, 8, 9 −10123456 dP/dz (mmol N m −4) −28 −27 −26 −25 −24 −23 −22 (b) 123 4 56 7, 8, 9 Figu e 2.5: A close up iew a z∼z2o (a) P(i)(z), and (b) ∂P(i)/∂z. 30 CHAPTER 2 −1 −0.5 0 0.5 1 x (km) −1 −0.5 0 0.5 1 y (km) −5.0 −4.2 −3.3 −2.5 −1.7 −0.8 0.0 0.8 1.7 2.5 3.3 4.2 5.0 (a) P’ (mmol N m −3) −1 −0.5 0 0.5 1 x (km) −1 −0.5 0 0.5 1 −11.3 −9.4 −7.5 −5.7 −3.8 −1.9 0.0 1.9 3.8 5.7 7.5 9.4 11.3 (c) Z’ (mmol N m −3) (x10 2) −1 −0.5 0 0.5 1 y (km) −50 −25 0 z (m) (b) −1 −0.5 0 0.5 1 y (km) −50 −25 0 (d) Figu e 2.10: (a) P′(x, y) a zb≃ −23.4 m (P′∈[−3.3,3.1] mmol N m−3). PV con ou s =±0.05 a z= 0 a e included o e e ence. (b) P′(y, z) on e ical sec ion x= 0 (P′∈ [−3.3,4.8] mmol N m−3). (c) Z′(x, y) a za=−12.5 m (Z′∈[−9.0,8.3] ×10−2mmol N m−3). (d) Z′(y, z) on e ical sec ion x= 0 (Z′∈[−8.9,11.2] ×10−2mmol N m−3). Time = 8 Tip. he o al Pand Zchanges ( he hs o (2.1) and (2.2), espec i ely) o ced by wand a e independen o he e ec o me e ad ec ion. Ascending luid pa icles expe ience an inc ease o hei Pcon en while descending pa icles expe ience a dec ease o P. No e pa icula ly he la ge Pdec ease a he no he n side o he domain (Fig. 2.11a,b), whe e w < 0 (Fig. 2.8c,d). Posi i e Pbudge s occu a dep hs a bi shallowe han nega i e Pbudge s, which explains why dP/d is mos ly nega i e a he dep h shown in Fig. 2.11a. The ma e ial a e o change o Z(Fig. 2.11c,d) and wa e also clea ly co ela ed. Howe e dZ/d usually displays a minimum and a maximum along he wa e column, which is consis en wi h he wo maxima in σ{dZ/d }(Fig. 2.9d). The analysis o dP/d and dZ/d in o hei local and ad ec i e changes (Fig. 2.12) shows ha , as in e ed om hei s anda d de ia ions (Fig. 2.9c,d), he e is a la ge cancela ion be ween he local change and he ho izon al ad ec ion o Pand Z. The e ical ad ec ion is smalle . Consis en ly also wi h he P′and Z′dis ibu ions (Fig. 2.10) he local change and ho izon al ad ec ion o Zp esen pa e ns mo e elonga ed han hose o P. This is a 2.4. COUPLED PHYSICAL-ECOSYSTEM NUMERICAL SIMULATIONS 31 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 y (km) −5.1 −4.3 −3.4 −2.6 −1.7 −0.9 0.0 0.9 1.7 2.6 3.4 4.3 5.1 (a) z=−18.7 m (mmol N m −3 d −1) (x10) −1 −0.5 0 0.5 1 x (km) −1 −0.5 0 0.5 1 y (km) (b) z=−23.4 m −1 −0.5 0 0.5 1 y (km) −50 −25 0 z (m) (c) Figu e 2.11: Pp oduc ion anomaly G′ P(x, y) on ho izon al planes (a) z=−18.75 m (iz= 53, G′ P∈[−0.12,0.33] mmol N m−3d−1). (b) z=zb≃ −23.4 m (iz= 50, G′ P∈ [−0.51,0.035] mmol N m−3d−1). (c) G′ P(y, z) on e ical sec ion x= 0 (G′ P∈ [−0.51,0.34] mmol N m−3d−1). Time = 8Tip. consequence o he be e ma e ial conse a- ion o Zin compa ison con P. La ge local a es occu in he on al a eas, whe e bo h ho izon al eloci y and ho izon al g adien s o Pand Za e la ge. The e ical ad ec ion o Pand Zha e howe e simila pa e ns. This is so because ∂P/∂z ≃∂Ps/∂z > 0 a z=zband ∂Z/∂z ≃∂Zs/∂z > 0 a z=za, so ha he e ical ad ec ion pa - e ns (Fig. 2.12c, ) esemble he wpa e n (Fig. 2.8c). The ime e olu ion o σ{dP/d },σ{dZ/ d }, and σ{w}(Fig. 2.13) show ha he eco- sys em ime esponse o wmaxima is abou 5Tip. The second wmaximum a ≃37 Tip is ela ed o he low enhancemen due o he usion o wo an icyclones. The e ical es- olu ion used in his simula ion (δz = 65 cm) is no good enough o ully esol e he la ge e ical g adien s o Pand Za ansi ion dep hs. Based on Figs. 2.5b and 2.6b, e - ical g adien s a e unde es ima ed by a 50%. La ge e ical esolu ions would co ec ly e- sol e he e ical ad ec ion o Pand Zwhich would cause an impo an inc ease in P′and Z′dis ibu ions. Howe e , as ano he con- sequence o an inc eased e ical esolu ion, hese la ge anomalies would be es ic ed o hinne ocean laye s, so ha only quan i a- i e changes a e expec ed in he ecosys em a iables. 32 CHAPTER 2 These nume ical esul s co espond o a non di usi e NPZ ecosys em model coupled o an adiaba ic in iscid physical model. These esul s will no apply when e ical mixing is added o he NPZ model (see Edwa ds e al.,2000) since in such a case he la ge e ical NPZ g adien s ound he e would u n Psand Zsin o uns eady solu ions. We no e ha he me e exis ence o e ical eddy di usion in a nume ical model al eady equi es e ical di e en iabili y. Ve ical di usion is no included he e because he NPZ model is kep as simple as possible in o de o analyze he e ical eloci y o cing o NPZ anomalies. Including e ical di usion will add new ee pa ame e s ( he e ical di usi i y coe icien s) o he al eady la ge lis o NPZ pa ame e s on Table 2.1. Fu he mo e, he absence o NPZ di usion This is also consis en wi h he in iscid na u e o he PV-conse ing dynamical model (only a e y small amoun o nume ical di usi i y is included o a oid g id-size noise). −1 −0.5 0 0.5 1 x (km) −1 −0.5 0 0.5 1 y (km) −8.2 −6.8 −5.5 −4.1 −2.7 −1.4 0.0 1.4 2.7 4.1 5.5 6.8 8.2 (a) dP d (mmol N m −3 d −1) (x10) −1 −0.5 0 0.5 1 x (km) −1 −0.5 0 0.5 1 −2.0 −1.7 −1.3 −1.0 −0.7 −0.3 0.0 0.3 0.7 1.0 1.3 1.7 2.0 (c) dZ d (mmol N m −3 d −1) (x10 4) −1 −0.5 0 0.5 1 y (km) −50 −25 0 z (m) (b) −1 −0.5 0 0.5 1 y (km) −50 −25 0 (d) Figu e 2.12: (a) dP/d a zb≃ −23.4 m (iz= 50,dP/d ∈[−0.82,0.081]). (b) dP/d a x= 0 (dP/d ∈[−0.82,0.42]). (c) dZ/d a z=−12.5 m (iz= 57,dZ/d ∈[−19.8,5.3] ×10−5). (d) dZ/d a x= 0 (dZ/d ∈[−19.6,10.5] ×10−5). Time = 8 Tip. 2.4. COUPLED PHYSICAL-ECOSYSTEM NUMERICAL SIMULATIONS 33 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 y (km) −9.5 −7.9 −6.3 −4.8 −3.2 −1.6 0.0 1.6 3.2 4.8 6.3 7.9 9.5 (a) (mmol N m −3 d −1) ∂P ∂ −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −5.2 −4.3 −3.4 −2.6 −1.7 −0.9 0.0 0.9 1.7 2.6 3.4 4.3 5.2 (d) (mmol N m −3 d −1) (x10) ∂Z ∂ −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 y (km) (b) uh·∇hP −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 (e) uh·∇hZ −1 −0.5 0 0.5 1 x (km) −1 −0.5 0 0.5 1 y (km) −4.2 −3.5 −2.8 −2.1 −1.4 −0.7 0.0 0.7 1.4 2.1 2.8 3.5 4.2 (c) (mmol N m −3 d −1) w∂P ∂z −1 −0.5 0 0.5 1 x (km) −1 −0.5 0 0.5 1 −14.0 −11.6 −9.3 −7.0 −4.7 −2.3 0.0 2.3 4.7 7.0 9.3 11.6 14.0 ( ) (mmol N m −3 d −1) (x10 2) w∂Z ∂z Figu e 2.13: (a) ∂P/∂ a z≃zb=−23.4 m (∂P/∂ ∈[−8.3,7.4]). (b) uh·∇ hP(∈ [−9.4,8.3]). (c) w∂P/∂z (∈[−1.8,4.2]). (d) ∂Z/∂ a za=−12.5 m (∂Z/∂ ∈[−0.51,0.30]). (e) uh·∇ hZ(∈[−0.32,0.53]). ( ) w∂Z/∂z (∈[−0.042,0.14]). Time = 8 Tip. 34 CHAPTER 2 dP/d dZ/d w 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 z (a) (T ) ip (m) ime 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 (b) (T ) ip ime 0 10 20 30 40 50 −50 −40 −30 −20 −10 0 (c) (T ) ip ime Figu e 2.14: (a) σ{dP/d }(z, ) (max = 0.11, ∆ = 0.01). (b) σ{dZ/d }(z, ) (max = 5×10−6, ∆ = 5.4×10−5). Uni s a e mmol N m−3d−1. (c) σ{w}(z, ) (max = 1.2×10−3, ∆ = 10−4). 2.5 Concluding Rema ks We ha e i s shown ha 1D s eady and con inuously di e en iable (in a nume ical sense) solu ions o he NPZ equa ions a e possible. These solu ions a e po en ially use ul as ini ial s eady ecosys em condi ions o in es iga e he ole o ho izon al and e ical ad ec ion in 3D coupled physical-ecosys em nume ical models. An example o ecological de elopmen due o e ical eloci y enhancemen du ing a ba oclinic ins abili y p ocess has been p esen ed. This example shows ha once phy oplank on and zooplank on anomalies de elop locally o ced by balanced e ical eloci y hey a e ho izon ally ad ec ed away om he upwelling o downwelling egions so ha spa ial dis ibu ions o e ical eloci y and ecological ields become e en ually unco ela ed ( o expe imen al e idence o his p ocess see e.g. Ruiz e al., 2001). Thus he biological dis ibu ions a e mo e ela ed o PV g adien s han o PV i sel . This ac , and he submesoscale e ical o igin o NPZ anomalies, is consis en wi h (L´e y e al.,2001), who used a p imi i e equa ions model wi h e ical di usion. Howe e , he expe imen al wo k o (L´e y e al.,2005) ques ions he con ibu ion o submesoscale o al ad ec ion in he phy oplank on a iabili y o e la ge ime scales. The physical-ecological model used he e has se e al limi a ions. On he one hand, hese NPZ solu ions equi e e y good e ical esolu ions, wi h a g id size o ew cen ime e s, o be p ope ly disc e isized. This imposes a se e e handicap o he a ailable andom ac- cess memo y o cu en compu e s unning 3D coupled physical-ecosys em models. Though om a s ic nume ical pe spec i e his ac is a se ious modeling limi a ion, om a wide 2.5. CONCLUDING REMARKS 35 pe spec i e o he handicaps, o ins ance, e o s in he ma hema ical pa ame iza ion o he di e en NPZ p ocesses, a e likely o be o la ge ele ance (see Ande son,2005;Flynn, 2005;Mi a e al.,2007). I is ne e heless impo an o know he deg ee a which ecosys- em modeling solu ions ai h ully ep oduce he unde laying ecosys em dynamics and ha , e en wi h poo e ical esolu ion, i is possible o ob ain good quali a i e esul s om hese models. On he o he hand, ou esul s show ha Pand Zapp oxima ely beha e as passi e ace s while o ganisms a e in ac ac i e ace s. This is so because he simple ini ial s eady NPZ p o iles le li le in e ac ion be ween Pand Z. In u u e wo k we will add ess hese in e ac ions using bo h mo e ealis ic ini ial NPZ p o iles and a mo e complex biological model. To conclude, hese esul s a e a i s app oxima ion owa ds a be e unde s anding o biological p ocesses o ced by e ical eloci y a he submesoscale. Many ques ions s ill emain o be answe ed in his con ex . How much does he submesoscale e ical eloci y con ibu e o p ima y and seconda y p oduc i i y in compa ison o he mesoscale? Does he e ical ad ec ion induce di e en biological pa e ns in eu ophic and oligo ophic egimes? Is he submesoscale impo an in he seasonal biological a iance? Chap e 3 Phy oplank on enhancemen by oceanic dipoles Chap e submi ed as M.Cla e , A.Viúdez, and Y. H. Spi z, 2012: Phy oplank on enhancemen by oceanic dipoles. Al cim d’un p omon o i que domina les ones de la ma , quan l’as e ei cap a ponen declina me’n pujo a medi a . Amb la cla o d’aqueixa ll`an ia encesa con emplo mon no- es; con emplo el ma i el cel, i llu g andesa m’aixa a com un pes. Vo a la ma , Jacin Ve dague 39 ABSTRACT Phy oplank on dis ibu ions associa ed o oceanic dipoles a e in es iga ed using a high- esolu ion, non-hyd os a ic, h ee-dimensional nume ical model coupled o a NPZ (Nu ien - Phy oplank on-Zooplank on) oligo ophic model. Two scena ios a e conside ed, a subme- soscale su ace and a mesoscale subsu ace dipoles, in o de o cha ac e ize bo h he e ec o o ices in ansla ion and he dis an ac ion o po en ial o ici y (PV) on phy oplank on dynamics. We obse e ha he dipole sepa a ix ac s as an impe meable ba ie di iding wo di e en ecological niches. On he one hand, plank on apped wi hin o ices eaches a nea s eady s a e, which depends on he communi y en ained a he o ma ion s age o o ices. On he o he hand, phy oplank on ad ec ion by he dipole in oduces ecological he e ogenei y ou side he sepa a ix. A balance be ween e ical ad ec ion (VA) and biologi- cal o cing gene a es a subsu ace ail o phy oplank on a he wake o he ansla ing dipole. The leng h o his ail is nea ly cons an and may be used o es ima e he phy oplank on mo ali y a e o a known dipole speed. When nega i e g adien s o PV exis in he uppe ocean, ho izon al ad ec ion accoun s indi ec ly o a la ge phy oplank on inc ease, hough much localized, han ha caused by VA. I a subsu ace dipole is conside ed, an ecosys em pa ch is s i ed such ha a ilamen uns along i s axis. The ilamen ex en inc eases wi h dep h and as a esul phy oplank on sel -shading dec eases a he ilamen on in bene i o ligh i adiance, and hus o phy oplank on p oduc ion. Finally, he ho izon al ex ension o he ilamen is app oxima ed o an analy ical exp ession de i ed o a dipole o known geome y and in ensi y. 46 CHAPTER 3 since he dynamical ABmodel is adiaba ic so ha changes in wa e densi y due o sola adia ion a e igno ed. Phy oplank on is emo ed by zooplank on g azing R(x, ), modeled wi h an I le esponse (Pa sons e al.,1967), and a linea e m (MP) ha ep esen s dea h, exuda ion, exc e ion o o he p ocesses. A po ion (1 −Γ0) o he Pg azed is con e ed o Zp oduc ion (GZ), which is balanced by a densi y dependen dea h a e (MZ). A quad a ic closu e e m o Zhas a double pu pose. On he one hand, i elimina es he oscilla o y beha io o he NPZ model wi h u= 0 ha appea s when a linea closu e e m is ins ead conside ed (Edwa ds and Yool,2000). On he o he hand, i in oduces he e ec o cannibalism (Pi ch o d and B indley,1998;Ohman e al.,2002), and p eda ion by highe ophic le els (Ohman and Hi che,2001). Finally, de i al ma e (Γ0R+MP+MZ) is emine alized and nu ien s a e again a ailable o Pup ake. The e o e, he o al ni ogen NT≡P+Z+Nis ma e ially conse ed, dNT d = 0 .(3.11) The abo e equa ion le s us o de ine he o iginal NPZ equa ions (3.5)–(3.7) depending on only wo a iables, Pand Z, which sa is y (3.5), (3.6), and (3.10). The ecological model is coupled o he physical one such ha he la e p o ides he 3D eloci y ield o ad ec he a iables o he o me . Addi ionally, di usion p ocesses a e neglec ed so ha PV is ma e ially conse ed. 3.2.3 Nume ical pa ame e s We wo k in he quasigeos ophic (QG) space, whe e e ical dimension is s e ched by c. Thus he nume ical domain is iso opic wi h e ical ex en LZ= 2π(which de ines he uni o space) and ho izon al ex en s LX=LY=cLZ. T iply-pe iodici y is also imposed bu only in he dependen a iable ϕ(x, ). Fo example, hough D(x, ) is iply-pe iodic he o al densi y ield ρ(x, ) is no . The numbe o g id poin s is (nX, nY, nZ) = (128,128,128), and he numbe o isopycnals nL= 128. Equa ions (3.4), (3.5), and (3.6) a e in eg a ed o wa d in ime using an explici leap- og scheme combined wi h a Robe -Asselin ime il e o a oid he compu a ional mode. Time-s ep is δ = 7×10−4d in all he cases conside ed. Finally, a biha monic hype di usion ope a o µ∇4 q o Ah,P, and Zis added o hei espec i e diagnos ic equa ions in o de 3.2. PHYSICAL-ECOLOGICAL COUPLED MODEL AND IMPLEMENTATION 47 o dump he ampli ude o g id-size scale noise due o spa ial disc e iza ion on a ixed g id. Abo e, ∇q≡c∇h+k∂zis he g adien ope a o in he QG space, and he hype iscosi y coe icien µis de ined by speci ying he e- olding ime o he la ges wa e numbe in spec al space pe ine ial pe iod, which co esponds o abou 0.7 d o a mean la i ude o 45o. In he AB-model e = 100, while in he NPZ model e = 10. 3.2.4 Ini ial condi ions The AB-model is ini ialized using he so called PV ini ializa ion app oach (Vi´udez and D i schel,2003), which is unique o he PV conse ing algo i hm used in his nume ical model. I consis s in a g adual inc ease o PV in e e y luid pa icle un il a p esc ibed alue is eached. This ini ializa ion echnique la gely a oids he gene a ion o ine ia–g a i y wa es due o he ini ial imbalance be ween densi y and eloci y ields, which o he wise could con amina e he balanced e ical eloci y. In he dipole cases he e conside ed, an ini ializa ion ime pe iod o abou 3.5 d is su icien o a oid he appea ance o he imbalance. The NPZ model is ini ialized wi h s eady s a e solu ions in o de o isola e he p ocesses ha con ol plank on dynamics when he physical sys em is pe u bed by an oceanic dipole. S a iona y s able ecological solu ions ˆ N(z),ˆ P(z), and ˆ Z(z) a e ob ained nume ically by ime in eg a ion o he NPZ model (3.5)–(3.7) in he s a e o es (u= 0) wi h he ollowing ini ial {NT, P, Z}p o iles NT(z) =    T1cos h³z z1−1´2π 5i, z ∈[z1,0], 1, z ∈[zmin, z1), (3.12) (P, Z)(z) =    (P1, Z1) sin h³z z2−1´π 2i, z ∈[z2,0], 0, z ∈[zmin, z2), (3.13) whe e T1= 1 mmol N m−3,P1= 0.22 mmol N m−3, and Z1= 0.1 mmol N m−3. In con as , he alues o zmin,z1, and z2di e whe he a su ace o a subsu ace dipole is conside ed. Independen ly o he {zmin, z1, z2}choice, ime in eg a ion o (??)-(??) con e ges o a s eady s a e (Fig. 3.1). The esul ing s a iona y p o iles {ˆ P, ˆ Z, ˆ N}, whe e ˆ Nis eco e ed by in e - sion o (3.10), a e used o ini ialize he ecological model in he AB-NPZ coupled cases. Bo h models a e coupled once he PV ield is ully ini ialized a = isince, in he 48 CHAPTER 3 PV ini ializa ion app oach, PV is no ma e ially conse ed du ing he ini ializa ion pe iod ( < i). No e ha a ini ial ime, 0= 0, isopycnals a e la and he e o e d(x, 0) = z since D(x, 0) = 0. Howe e , a he end o he ini ializa ion pe iod doming o isopycnals implies D(x, i)6= 0. As a esul , wo di e en ecological ini ial condi ions can be u he dis inguished depending on whe he NTis conside ed homogenous on isopycnals (cons an d) o on ho izon al laye s (cons an z). In his wo k, we ha e conside ed he o me ini ializa ion when zP∼ =−61 m (sec ion 3.3.1) and he la e when zP∼ =−90 m (sec ion 3.3.2). 3.3 Nume ical simula ions The e ec s o bo h o ex ansla ion and he dis an PV ac ion on ecological dynamics a e in es iga ed conside ing wo oligo ophic scena ios ypical o he open ocean. We i s ocus on how a su ace dipole pe u bs a s a iona y ecosys em a he submesoscale (sec- ion 3.3.1), whe e high- esolu ion emo e sensing obse a ions ha e ecen ly un eiled many o ical s uc u es (Munk e al.,2000). In his case phy oplank on subsu ace maximum ˆ Pmax ≡ˆ P(zP) is ini ially placed unde nea h he dipole, so ha zP≃ −61 m (Fig. 3.1a), by choosing (z1, z2, kw) = (−120 m,−150 m,0.03 m−1). Secondly, he dis an in luence o a subsu ace dipole on a plank on 3D pa ch is in es iga ed a he mesoscale (sec ion 3.3.2), he scale a which subsu ace long-li ed o ices, such as meddies, a e widesp ead in he ocean (Richa dson e al.,2000). Since he shallowes dipole edge is a z∼ =−200 m, we ha e deep- ened ˆ Pmax o zP≃ −90 m (Fig. 3.1b) by se ing (z1, z2, kw) = (−290 m,−187 m,0.023 m−1). 3.3.1 Submesoscale su ace dipole The submesoscale is he e in oduced as c= 50, and only hal o he e ical domain LZis conside ed wi h zmin =−150 m. Thus he ho izon al domain is 15 km. The o ex dipole consis s o a ba oclinic cyclone (+) and an icyclone (−) wi h PV anomalies ± max = 0.5 (∼ =0.35 d−1). The maximum leng h o he ho izon al semi-axes o he ellipsoids o cons an PV in bo h o ices a e aX∼ =1.9 km and aY∼ =2.9 km. The leng h o he e ical semi- axes a± Za e di e en in he ini ial con igu a ion wi h la isopycnals, being a+ Z∼ =62 m and a− Z∼ =52.5 m. Howe e , du ing he ini ializa ion ime he isopycnals s e ch (sh ink) in he an icyclone (cyclone), so ha a i∼ =3.5 d he o ices ha e simila e ical ex en and he dipole desc ibes a s aigh ajec o y along he x-axis (Dubosq and Vi´udez,2007). A 3.3. NUMERICAL SIMULATIONS 49 0.0 0.2 0.4 0.6 0.8 1.0 −140 −120 −100 −80 −60 −40 −20 0 mmol N dm−3 (m)d N ^ P ^ Z ^NT ^ (a) 0.0 0.2 0.4 0.6 0.8 1.0 −300 −250 −200 −150 −100 −50 0 mmol N dm−3 (m)z N ^ P ^ Z ^ NT ^ (b) Figu e 3.1: Ve ical p o iles o o al ni ogen NT(z), and i s associa ed s a iona y and s able p o iles o {P, Z, N}ob ained wi h he pa ame e s gi en in Table 3.1 and wi hou physical o cing. In he su ace dipole case he p o iles co esponding o (a) a e ini ialized homoge- neous on isopycnals (d), while in he subsu ace dipole scena io hose o (b) a e homogeneous on ho izon al le els (z). he end o he ini ializa ion ime = i, he ho izon al speed con ou s co espond o hose o concen ic de o med double o i in e sec ing an ho izon al plane wi h a maximum speed |u|max = 4.76 cm s−1along he dipole axis (Figs. 3.2a,b). We no e ha submesoscale dipoles, as long as hey emain isola ed and do no in e ac wi h o he submesoscale o ices, ha e a quad upola pa e n o w(Fig. 3.2c) simila o mesoscale dipoles (Pall`as-Sanz and Vi´udez, 2007). The wmaximum absolu e alue |w|max = 71.23 cm d−1is placed a z=−35 m (iZ= 50, Fig. 3.2d). As he dipole mo es o wa d, he isopycnals a e displaced upwa ds (downwa ds) a he on ( ea ) o he cyclone. The opposi e changes occu in he an icyclone. The dipole low emains always s a ically and ine ially s able since he Rossby numbe R≡ωh/N, whe e he squa ed o al B un -V¨ais¨al¨a equency N2(x, )≡ −gα0∂ρ/∂z = N2[1−∂D/∂z(x, )], minimum is Rmin =−0.43, and he F oude numbe F≡ζ/ maximum Fmax = 0.19. The NPZ model is coupled o he abo e desc ibed dynamical condi ions a = i. In his case, o al ni ogen is se homogeneous on isopycnal le els by p ojec ing he ini ial p o iles 50 CHAPTER 3 −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 (km)y 0.0 0.8 1.6 2.4 3.2 4.0 4.8 | h| (x10−2 m s−1)u (a) −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 (c) −7.5 −3.25 0 3.25 7.5 −100 −50 0 (km)x (b) −7.5 −3.25 0 3.25 7.5 −100 −50 0 (km)x (d) Figu e 3.2: (a) Ho izon al dis ibu ion o |uh|a z= 0 (iZ= 65). Only e e y o he ec o is plo ed. A s aigh solid line is d awn whe e he e ical dis ibu ion o is shown in (b) a y∼ =−2.6 km (iY= 43, ∈[−4.76,2.76] cm s−1, con ou in e al δ ≃0.4 cm s−1). Ho izon al (c) and e ical (d) dis ibu ions o w(con ou line ange is |w|<71.23 cm d−1 wi h δw ≃1 cm d−1) a z∼ =−33 m (iZ= 50) and a y∼ =−0.82 km (iY= 58), espec i ely. Dis ibu ions co espond o he ini ial ime i∼ =3.5d. PV con ou s =±0.2 (∼ =0.28 d−1) a e included o e e ence. He eina e , solid and dashed con ou s a e used o posi i e alues and nega i e alues, espec i ely. We obse e ha he dipole ansla es sou hwa ds as a solid body since uhhas a su ace maximum along he dipole axis. In con as , ex eme wbounds a e eached a he subsu ace. (Fig. 3.1a) on isopycnals as NT(x, ) = ˆ NT(d(x, )) .(3.14) No e ha his unc ion does no depend explici ly on because ˆ NTis ma e ially conse ed. In acco dance o his NTini ializa ion, {P, Z, N} a iables a e also se cons an on isopycnals a = isa is ying P(x, i) = ˆ P(d(x, i)) ,(3.15) wi h simila ela ions o Z(x, i) and N(x, i). Thus he ini ial dis ibu ions o hese a iables depend on hei e ical g adien and he isopycnal e ical displacemen D. Fo ins ance, Pshows a quad upola pa e n a = i(Fig.3.3a) because Pihas a subsu ace maximum a zP∼ =−61 m, and hus a dipola pa e n is obse ed wi hin each o ex. This also explains 3.3. NUMERICAL SIMULATIONS 51 −7.5 −3.25 0 3.25 7.5 −150 −100 −50 0 (km) (m) x z 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 −7.5 −3.25 0 3.25 7.5 −150 −100 −50 0 (a) P’ D = ip −7.5 −3.25 0 3.25 7.5 −150 −100 −50 0 (km) (m)z −7.5 −3.25 0 3.25 7.5 −150 −100 −50 0 (b) Z’ = ip −7.5 −3.25 0 3.25 7.5 −150 −100 −50 0 (km) (m)z −7.5 −3.25 0 3.25 7.5 −150 −100 −50 0 (c) =24.7 d −7.5 −3.25 0 3.25 7.5 −150 −100 −50 0 (km) (m) y z −7.5 −3.25 0 3.25 7.5 −150 −100 −50 0 (d) =88.4 d Figu e 3.3: (a) Ve ical sec ion c ossing he o ex cen e s, a y∼ =−2.6km, (iY= 43), o P′(x, z) in g ey scale and D(|D|<4.3 m, con ou in e al δD≃0.6 m) a e PV ini ializa ion a = i. (b, c, d) Ve ical sec ions pa allel o he dipole axis o P′(y, z) and Z′(y, z) in con ou lines (|Z′|<1.06, δZ′= 1%) a di e en imes and (b) x∼ =2.1 km (iX= 83), (c) x∼ =2.3 km (iX= 85), and (d) x∼ =3.5 km (iX= 95). Phy oplank on subsu ace maximum zP(s aigh line in (a) ) and con ou s o =±0.2 a e indica ed o e e ence. No e ha P′inc eases abou 2% a he cyclone wake. 52 CHAPTER 3 he quad upola ini ial dis ibu ion o Z, hough o di e en magni ude (no shown). In con as , Nhas a dipola pa e n (no shown) since Niinc eases g adually wi h dep h. The ini ial ields o P, and Za e ad ec ed wi h he 3D eloci y a imes > i, while N is ob ained using (3.5), (3.6), and (3.10). In o de o quan i y he dipole impac on he ecosys em, ecological p ope ies a e adimensionalized such ha hey a e no malized by a alue ou side he dipole whe e isopycnals a e la , ha is, by he s a iona y solu ion a he isopycnal dep h in he e e ence densi y con igu a ion (d). So ha , any ecological a iable χ is cha ac e ized h ough he ollowing enhancemen ac o , χ′ (x, )≡χ (x, ) ˆχ (d(x, )) ,(3.16) while any ecological a e χ is e e ed o he s eady s a e Pp oduc ion (GP) as χ′ (x, )≡χ (x, ) ˆ DP(d(x, )) .(3.17) The P′maximum is loca ed a he cen e o he cyclone, whe e isopycnal displacemen is he la ges , and mo es wi h he ansla ing dipole (Figs. 3.3b–d). I ep esen s abou a 6% inc ease e e ed o i s s eady s a e ˆ P(d). Zooplank on enhancemen Z′is o he same o de as P′and i s dis ibu ion is posi i ely co ela ed wi h ha o P′. Based on he biological o cing balance (Eq. 3.5), we ound ha zooplank on g azing is much smalle han he o he e ms. In ligh o ha , we shall ocus on he phy oplank on esponse only. Two di e en phy oplank on dynamics a e obse ed depending on a c i ical dep h zc≃ −80.8 m, which co esponds o he maximum dep h o he dipole. Abo e zc, he physical o cing domina es o e he ecological o cing and Pbeha es mainly as a passi e ace . In con as , below zc e ical ad ec ion igge s an ecological esponse o simila magni ude o physical e ms and aP′ ail de elops a he dipole wake. Abo e zc, he dipole sepa a ix, which is de ined by he minimum isosu ace, ac s as a dynamical ba ie . On he one hand, he ho izon al ad ec ion o Pis g ea e han i s e ical ad ec ion (Fig.3.4a). On he o he hand, e ical upli o P o ligh e le els igge s aPinc ease, howe e i is much smalle han he ho izon al ad ec ion (no shown). Thus Pis ansla ed a he dipole phase speed. Fu he mo e, Pco ela es wi h D(Fig.3.4b), and ul ima ely wi h  h ough (??). This ac sugges s ha s a iona y s able ecological solu ions a e eached wi hin o ices, which a e close o hei ini ial condi ions a iwhen 3.3. NUMERICAL SIMULATIONS 53 −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 7.5 (km)x (km)y −4.38 −3.12 −1.88 −0.62 1.25 2.50 3.75 −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 7.5 (a) =88.4 d (w P z)’ (%) u h hP( )’ −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 7.5 (km)x 0.94 0.95 0.97 0.98 1.00 1.02 1.03 1.05 1.06 −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 7.5 (b) P’ D Figu e 3.4: Ho izon al sec ion a z≃ −34.6 m (iZ= 50) o (a) (w ∂P/∂z)′in g ey scale and (uh· ∇hP)′in con ou lines (|∆|<0.054, δ= 6.25 ×10−3), and (b) P′in g ey scale and Din con ou lines (D∈(−4.15,3.75) m, δD= 0.5 m). PV con ou s ±=±0.2 a z= 0 ( hickes line) a e included o e e ence. A e δ = 88 d, P′co ela es wi h Dwi hin o ices as in he ini ial con igu a ion a ime i. 20 40 60 80 0.0 0.5 1.0 1.5 2.0 2.5 -0.2 0.0 0.2 0.4 0.6 < P’>-1 (%) (x10-1 d-1)< ϖ > <( h h P)’> <(w P z)’> (%) u<( P/ )’> d d days Figu e 3.5: Time e olu ion o hP′i(solid line), hi(dash do ed line), h(dP/d )′i(do ed line), h(uh·∇hP)′i(dash iple do ed line) and h(w ∂P/∂z)′i(dashed line), a e aged o e he cyclone olume. A e an ini ial adjus men , hP′i eaches a s eady s a e close o he ini ial con igu a ion. 54 CHAPTER 3 NTis ini ialized homogeneous on isopycnals and so ecological a iables. In o de o un eil his s eady s a e, any p ope y χis in eg a ed on he cyclone olume as hχi( ) = 1 nZΩ χ(x, ) dV,(3.18) whe e he spa ial domain Ω comp ises hose g id poin s nwi h  > 0. As expec ed, a e an ini ial adjus men o abou δ ∼ =20 d, hP′i emains nea ly cons an oscilla ing a ound a ime a e aged alue o 1.9% (Fig. 3.5). The oscilla ion in hP′ihas a pe iod T≃5.6 d di e en om ha o ad ec i e e ms. Ac ually, hP′iis in phase wi h hi, which oscilla es due o o ex Rossby wa es (VRWs) (Rod ´ıguez-Ma oyo and Vi´udez,2009). VRWs cause a pe iodic azimu hal oscilla ion o he dipole geome y, ela i e o i s ime a e aged con igu a ion, wi h a pe iod o 8 ine ial pe iods, so ha close o T o a midla i ude, and hus also al e Pdis ibu ions. A dis inc plank on dynamics occu s below he dipole a z < zc. In his case, P′inc eases a he cyclone wake as a combina ion o physical and biological p ocesses, which ha e simila magni ude. Fi s ly, (w∂P/∂z)′is nega i ely co ela ed wi h P′−1 a he dipole ea ( = 7 d, Fig. 3.6a). This sugges s ha P′inc eases (dec eases) due o subduc ion o shallowe dep hs iche (poo e ) in P′a he an icyclone (cyclone) wake. As dipole ansla es, he luid pa cels ha expe ience hese P′changes decele a e ela i e o o ices and each a s a e o es a la e imes. Howe e , P′does no inmedia ely con e ge o i s s a iona y ini ial condi ion (P′= 1) since he ecosys em esponds wi h some ine ia o a pe u ba ion. As a esul , changes in P′ e ain some memo y o hei causing mechanism and hus P′co ela es in his case wi h he ime in eg a ed e ical ad ec ion R(w ∂P/∂z)′d (Fig. 3.6b, = 10.6 d). La e on, a e δ ∼ =15.5 d o he inc ease in P′, biological o cing (dP/d )′ eaches he o de o magni ude o (w∂P/∂z)′(Fig. 3.6c). Since ecological and ad ec i e e ms a e opposi e, we conclude ha P′is induced a he dipole wake due o a balance bo h and hus i becomes co ela ed wi h he ime in eg a ed local a e R(∂P/∂ )′d (Fig. 3.6d, = 24.7 d). The long e m e olu ion o his P′ ail is quan i ied by olume in eg a ion o p ope ies using (3.11). In his case, he spa ial domain Ω is enclosed wi hin isosu ace P′= 1.002 a he dipole wake. We obse e ha hP′i eaches a nea ly s a iona y s able s a e a e an ini ial adjus men o δ ∼ =35 d, howe e h(dP/d )′i ≡ h(∂P/∂ )′i+h(u·∇P)′i+h(w∂P/∂z)′i 6= 0 (Fig. 3.7). In o de o un eil wha we a e missing in he abo emen ioned balance, we ake a La- g angian app oach. To his end, pa icles ha expe ience a smalle P′inc ease han ha 3.3. NUMERICAL SIMULATIONS 55 P’ 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 7.5 (km)y −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 7.5 (a) =7 d −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 7.5 (b) =10.6 d −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 7.5 (km)x (km)y −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 7.5 (c) =15.5 d −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 7.5 (km)x −7.5 −3.25 0 3.25 7.5 −7.5 −3.25 0 3.25 7.5 (d) =24.7 d Figu e 3.6: Ho izon al sec ions o P′in g ey-scale and con ou lines o (a) (w ∂P/∂z)′ wi h maximum absolu e alue |∆|max ∼ =0.016 and con ou in e al δ∼ =3×10−3, (b) R(w ∂P/∂z)′d wi h |∆|max ∼ =0.24 and δ= 0.025, (c) (dP/d )′wi h |∆|max = 0.019 and δ∼ =3×10−3, and (d) R(∂P/∂ )′d wi h |∆|max = 0.32 and δ∼ =0.04. Sec ion laye s a e z≃ −71.5 m (iZ= 34) in (a) and z≃ −85.4 m (iZ= 28) in (b–d). Finally, PV con ou s = 0.2 a e included o e e ence. No e ha he P′inc ease esul s om a balance be ween e ical ad ec ion and biological o cing. 20 40 60 80 1.000 1.002 1.004 1.006 1.008 1.010 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 < P’> <( h h P)’> <(w P z)’> (%) u<( P/ )’> d d days Figu e 3.7: Time e olu ion o hP′i(solid line), h(dP/d )′i (do ed line), h(w ∂P/∂z)′i (dashed line), and h(uh·∇hP)′i (dash do ed line) a e aged wi hin isosu ace P′= 1.002 below he deepes PV edge o he dipole. No e ha hP′i eaches a nea ly s eady s a e, hough ecological e ms do no balance ou . 62 CHAPTER 3 cen e , espec i ely, is δ =Z i d ′ g e( ′, z)(3.23) =γ0 3[R3( , z)−R3 i( i, z)] + γ0z2[R( , z)−Ri( i, z)] +γ0z3ln ¯¯¯¯ [R( , z)−z][Ri( i, z) + z] [R( , z) + z][Ri( i, z)−z]¯¯¯¯, whe e cons an γ0= 3/(2N0R3 0). Since all he e ms abo e a e s ic ly inc easing con inu- ous unc ions o , (z, i+δ ) is easily ob ained by nume ical in e sion. This app oxima ion p edic s a maximum sepa a ix dep h zAclose o he obse ed zS(Fig.3.12b). Abo e zA he p edic ed NT on ansla ion is abou hal he obse ed one, a disc epancy ha may be explained by he absence o non-linea e ms in he QG PV dynamics. Below zA, he on line g adually con e ges o he sepa a ix cu a u e wi h a maximum displacemen a z=−500 m close o wice he o ices adius. Though he di e ences obse ed, equa ion (3.15) may be used o ob ain a i s es ima e o he leng h o he pa ch ilamen a ion a any dep h o a gi en dipole geome y and in ensi y. Finally, he ecological dynamics inside his NTpa ch is in es iga ed. To his end, ecologi- cal p ope ies a e quan i ied h ough he enhancemen ac o s (??) and (??), which a e e e ed o a loca ion ou side he dipole whe e isopycnals a e la . In his case, he ecosys em mo es wi h a spa ial and empo al dependen speed since he pho ic dep h is shallowe han ha o he dipole sepa a ix, ha is, z=−240 m < zS. As in he p e ious sec ion, we ocus only on Pdynamics since g azing is much smalle han he Pg ow h. We obse e ha wo di e en mechanisms cause a Pinc ease o di e en o de o magni ude. On he one hand, a P′maximum inc ease o abou 7% de elops a he wake o he an icyclone, since his o ex ype has isopycnals upli ed in he pho ic zone. As discussed in he p e ious sec ion, his inc ease is gene a ed by e ical ad ec ion (Fig. 3.13a) and diminishes a e wa ds by biological o cing (Fig. 3.13b). No e ha he obse ed Pinc ease is independen o whe he NTis ini ialized homogeneous on ho izon al laye s o isopycnals, and hus o he ecological ini ial condi ions. On he o he hand, posi i e P′inc eases up o a 42% ˆ Pa he ilamen on . Since he pa ch ansla ion speed has nega i e e ical shea , deep Playe s ansla e u he sou hwa ds han shallow ones. As a esul , Psel -shading dec eases a he ilamen on in bene i o Pne g ow h (dP/d )′(Fig. 3.13c). 3.3. NUMERICAL SIMULATIONS 63 −20 −15 −10 −5 0 5 10 15 20 25 −500 −400 −300 −200 −100 0 (km) (m) zS y z (a) −20 −15 −10 −5 0 5 10 15 20 25 −500 −400 −300 −200 −100 0 (km) (m) zA Y z (b) Figu e 3.12: Posi ion o he NT on al edge along dipole axis, x= 0, cen e ed a y1and ela i e o he dipole. Nume ical esul s (a) a e app oxima ed anali ically (b) using he ex e io geos ophic eloci y ˜ g e( , ) o he QG app oxima ion, equa ion (3.12), whe e Y is he dis ance be ween he on edge and he dipole cen e . In bo h cases, he e ically s aigh NT on il s g adually wi h ime a ound he minimum sepa a ix dep h zS. Each line co esponds o consecu i e imes anging om ∈( i,15.9) d wi h δ ∼ =0.7 d. In o de o quan i y he con ibu ion o bo h mechanisms o he Pinc ease, ecologi- cal p ope ies a e in eg a ed on he pa ch olume abo e he pho ic dep h using (3.11). A ini ial imes, hP′i apidly inc eases up o 7% due o e ical ad ec ion (Fig. 3.13a), hough (w∂P/∂z)′nea ly balances ou since whas a symme ic dis ibu ion wi hin he pa ch (Fig. 3.14). A imes > 12 d, he pa ch has been decele a ed ela i e o he dipole and hus P e ical ad ec ion dec eases while pa ch ilamen a ion inc eases. The elonga ion o his ilamen enla ges wi h dep h causing Psel -shading o diminish a i s on in bene i 64 CHAPTER 3 −50 −25 0 25 50 −50 −25 0 25 50 (km)y 0.80 0.88 0.95 1.05 1.12 1.20 −50 −25 0 25 50 −50 −25 0 25 50 (a) P’ =2.8 d −50 −25 0 25 50 −50 −25 0 25 50 (km)y −50 −25 0 25 50 −50 −25 0 25 50 (b) =12.7 d −50 −25 0 25 50 −50 −25 0 25 50 (km)y (km)x 0.89 1.00 1.11 1.21 1.32 1.42 −50 −25 0 25 50 −50 −25 0 25 50 (c) P’ =28.3 d Figu e 3.13: Ho izon al dis ibu ions a z=−125 m (iZ= 112) and a he indica ed imes o P′(g ey scale) and (a) (w ∂P/∂z)′(con ou lines ange is (−0.28,0.48) wi h δ= 0.05), and (b)-(c) (dP/d )′(con ou line ange is ∈(−0.14,0.4) wi h δ= 0.02 in b and δ= 0.1 in c). PV con ou s = 0.2 a e included o e e ence. Two mechanisms a e esponsible o P′ inc ease, and biological o cing a he ilamen on . 3.4. CONCLUDING REMARKS 65 10 20 30 40 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 -2 -1 0 1 2 3 66 67 68 69 70 71 72 73 < P’> < (GP)’> < ( P/ )’>d d -1 <I> (W m-2) <(w P z)’> x 10-2 days Figu e 3.14: Time e olu ion o hP′i(solid line), h(dP/d )′i(do ed line), hG′ Pi(long dashed line), h(w ∂P/∂z)′i(sho dashed line), and hI′i(dash-do ed line) a e aged on he pa ch abo e he pho ic dep h z=−240 m (iZ= 97). The obse ed hP′iinc ease is caused by wo mechanisms. A ini ial imes, e ical ad ec ion accoun s o an enhancemen o abou 6%. La e on, a dec ease in Psel -shading in bene i o sola adia ion inc eases GP, which doubles hP′ia = 45 d. o sola adia ion I. As a consequence, GPcon e ges o a alue a ound 1.23, which o ces he ecosys em o be uns eady (hdP/d i>0) and ul ima ely doubles hP′iwi hin he ollowing = 45 d. 3.4 Concluding ema ks Th ee-dimensional plank on dis ibu ions we e ela ed o po en ial o ici y (PV) using a nume ical physical-biological coupled model ha explici ly conse ed PV on isopycnals. In pa icula we in es iga ed how o ex ansla ion and he dis an PV ac ion pe u bed an oligo ophic sys em in s eady s a e. The o me ac was add essed by conside ing a ma u e dipole embedded in he ecosys em, while he la e by ini ializing a subsu ace dipole and a su ace ecosys em pa ch. We s a ed ha dipole sepa a ix, which was de ined by PV edges, ac ed as an impe me- able ba ie , c ea ing wo di e en ecological niches. Fi s ly, phy oplank on and zooplank on apped wi hin o ices con e ged o an equilib ium s a e, which was sus ained by egene - 66 CHAPTER 3 a ed p oduc ion. This may explain he subsis ence o communi ies wi hin long-li ed o ices mo ing ac oss he open ocean (Mackas e al.,2005;Whi ney e al.,2005). The s eady s a e obse ed depended on bo h he isopycnal e ical displacemen , and hus on PV, and he ecological ini ial con igu a ion. This was in ag eemen wi h p e ious nume ical and expe i- men al wo ks which concluded ha o ex geome y changes (L´e y,2003) and biogeochem- ical p ope ies o sou ce wa e s (Thompson e al.,2007), espec i ely, al e phy oplank on dis ibu ions. One ca ea wo h no ing is ha we assumed a ully- ecycling ecosys em and slow mixing p ocesses and sedimen a ion a e likely o occu as o ices ansla e. The way hese ac s al e he ecological s eadiness wi hin o ices is le o u u e esea ch. Secondly, plank on he e ogenei y a he dipole ou e was gene a ed by e ical and ho i- zon al ad ec ion, which we e o he same o de o magni ude as biological o cing. Plank on enhancemen h ough e ical ad ec ion occu ed a he wake o he dipole. As he dipole ansla ed, luid pa icles we e displaced om he o ices on o hei ea a ound PV edges. The e o e, hey we e upwelled o be e li dep hs when mo ing an iclockwise and clockwise in su ace and subsu ace dipoles, espec i ely. Phy oplank on esponse o his o cing had a ime lag o abou δ = 1.4 d causing a spa ial uncoupling be ween he inc ease in phy oplank on and i s causing mechanism. As a esul , luid pa icles wi h enhanced phy- oplank on biomass accumula ed a he o ices wake and since hey mo ed slowe han he dipole phase speed, a ail o phy oplank on was obse ed behind i . The ex en o his ail was p opo ional o he phy oplank on ma e ial a e o change dP/d . So ha , o a gi en leng h o he phy oplank on ail and dipole speed, dP/d could be in e ed. The o ma ion o a phy oplank on ail behind o ices was i s concep ualized by Olaizola e al. (1993). In his wo k, we ook a s ep u he by concei ing wo upwelling mechanisms ocu ing si- mul aneously in a single o ex. A pe manen upwelling o apped wa e s wi hin he o ex sepa a ix and a ansien upwelling along he o ical pe iphe y (Flie l and McGillicuddy, 2002). Bo h upwellings may explain he chlo ophyll ail de eloped below a shallow chlo o- phyll peak (Ning e al.,2004, Fig. 9d) a he wake o a pai o o ices o opposi e sea le el anomaly (Chang e al.,2010, Fig. 10). We also e idenced ha ho izon al ad ec ion was indi ec ly esponsible o a phy oplank- on inc ease when nega i e e ical g adien s o PV exis ed. Since ho izon al speed |u|h was maximum along he dipole axis and inc eased wi h dep h, a ilamen o phy oplank on unned along he di ec ion o he dipole ajec o y, he ex en o which was maximum a 3.4. CONCLUDING REMARKS 67 dep h. The ho izon al ex ension o his ilamen was quali a i ely in ag eemen wi h he analy ical ela ion be ween he on displacemen and o ex PV de i ed using a simple heo e ical quasigeos ophic dipole model composed o sphe ic o ices o ixed adius and dep h. The e o e, he ilamen elonga ion was app oxima ed o a PV s uc u e o known geome y and in ensi y. As a esul o he nega i e |u|hshea , phy oplank on inc eased a he ilamen on due o a dec ease in sel -shading in bene i o ligh i adiance. Though he ligh change was small, i accoun ed o signi ican enhancemen s o phy oplank on as was epo ed wi hin an an icyclone o e which wa e s wi h highe anspa ency looded (Bai d e al.,2011). Thus, mesoscale subsu ace dipoles a e esponsible o o -sho e anspo o su - ace chlo ophyll ilamen s Se a e al. (2010) and also o phy oplank on enhancemen . The mechanism desc ibed may explain he subsis ence o subsu ace communi ies as ilamen s pene a e in o he open ocean, which o he wise would decay h ough mo ali y and sinking. Finally, sha p phy oplank on on s such as he one desc ibed he e a e o en associa ed o well-de ined densi y on s (Jones e al.,1991), which induce upwelling along he ilamen edges (Moisan and Ho mann,1996). The simul aneous con ibu ion o a change in he ligh egime and localized upwelling o he communi y subsis ence in coas al ilamen s is le o u he in es iga ion. Chap e 4 Plank on esonan esponse o ligh and nu ien s in mesoscale o ices Chap e submi ed as M. Cla e , A. Viúdez, and Y. H. Spi z, 2012: Plank on esonan esponse o ligh and nu ien s in mesoscale o ices. Nom´es iu qui p egun a. P`o ic, Miquel Ma ´ı i Pol 71 ABSTRACT The esponse o a ully- ecycling ecosys em o ligh and nu ien s a iabili y wi hin mesoscale cyclones is in es iga ed. A simple physical-ecological coupled nume ical model is used o s udy he ho izon al and s eady low o ellipsoidal o ices. Ecological a i- ables ha e only e ical spa ial dependence h ough phy oplank on g ow h. This enables us o in oduce mesoscale o cing in e ms o e ical displacemen o isopycnals in a NPZ (Nu ien s-Phy oplank on-Zooplank on) model. Small-scale mo ions is pa ame ized as a Fickian- ype di usion along isopycnals. We obse e ha s a iona y s able ecological solu- ions a e possible wi hin mesoscale o ices wi h non-ze o Pand Zp o iles. The dependence o hese solu ions on small-scale mo ion, o ex in ensi y, and ophic egime is explo ed. Fi s ly, small-scale mo ions inc ease he spa ially in eg a ed Pbiomass a a cha ac e is ic di usion coe icien K∗, a which esonance be ween phy oplank on and di usi e imescales occu s. Two mechanisms a e in ol ed in his Pinc ease, namely nu ien s upli o be e li le els, and Zg azing p essu e dec ease. Secondly, o ex in ensi y exe s a double e ec on he ecosys em. On one side, i enhances he i adiance a ec ing isopycnals, which induces a posi i e linea esponse in P. On he o he side, i inc eases he di usi e luxes. As a esul , K∗is una ec ed by o ex in ensi y, ha is, Penhancemen s caused by isopycnal doming and small-scale mo ions a e app oxima ely addi i e. Finally, he ophic egime de e mines he magni ude o he ecosys em esponse. The inc ease in Pbiomass caused by small-scale mo ion is signi ican only in meso ophic egimes, and i emains independen o he ophic condi ion when he o ex in ensi y a ies. 78 CHAPTER 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 z Figu e 4.1: Ve ical dis ibu ion o D( , z) (dashed line) co esponding o he polyno- mial unc ion (??) is compa ed wi h ha ob ained using he CASL algo i hm (solid line) o a cyclone wi h Dmax = 0.39. PV con ou s wi h maximum = 1.8 and δ = 0.3 ( hickes line) a e included o e e - ence. (b) Ve ical dis ibu ion o D(s, d) co esponding o Dmax = 0.39 (uppe case, λ0= 0.027, solid line), and Dmax = 0.16 (λ0= 0.011, dashed line). Con ou in e al δD= 0.025. places Dmax a z=−π/3≃zD. On he ho izon al, Ddec eases exponen ially wi h he adius, simila ly o he PV dis ibu ion. Howe e , a adial linea dependence o ype π− is only conside ed in he polynomial exp ession o D. This simpli ica ion allows a closed- o m solu ion o he isopycnal adius, which acili a es he physical-ecological coupling (explained in sec ion 4.2.3). The esul ing polynomial exp ession o D( , z) is D( , z) = −λ0z(z+π)2(π− ),(4.11) whe e λ0is a dimensionless scaling ac o . This pa ame e adjus s he magni ude o ou analy ical solu ion o D o he nume ical one ˆ D h ough he ela ion λ0= 27/(4π4)ˆ Dmax. 4.2.2 The NPZ model The ecological model consis s o h ee dependen a iables, namely, phy oplank on (P), zoo- plank on (Z), and dissol ed nu ien s (N). Each a iable ep esen s a unc ional g oup and hey a e ela ed h ough physiological p ocesses. Func ional g oups a e composed ei he by one dominan species o se e al species wi h simila physiological cha ac e is ics. This sim- pli ica ion o he ood web minimizes he e o associa ed o uncons ained ee pa ame e s, while main ains he ecological dynamics o ou in e es . The addi ion o complexi y o he ecological model is ound con enien when ep oducing speci ic da a se s (F ied ichs e al., 2006), a he han when used in a heu is ic way as in his wo k. 4.2. PHYSICAL-ECOLOGICAL COUPLED MODEL 79 We conside i s an oligo ophic en i onmen ypical o he open ocean (Lawson e al., 1996). In his case, he ligh -nu ien s seg ega ion implies an acu e sca ci y o esou ces, and species compe e by op imizing hei nu ien assimila ion. This ecological sys em is ully- ecycling and sa is ies he se o equa ions dP d =N K0+NL P |{z } GP −R0(1 −e−Λ0P)Z |{z } R −Ξ0P |{z} MP ,(4.12) dZ d = (1 −Γ0)R |{z } GZ −Θ0Z2 |{z} MZ ,(4.13) dN d =−GP+ Γ0R+MP+MZ,(4.14) whe e d( )/d is he ma e ial ime de i a i e. Phy oplank on p oduc ion (GP) is ni ogen and ligh limi ed. The nu ien up ake ollows a Michaelis-Men en kine ics (Dugdale,1967) go e ned by K0, an indica o o he Pa ini y o N. The pho osyn he ic a e Lhas also a sa u a ing esponse (Smi h,1936;Jassby and Pla ,1976) acco ding o L( , z, )≡V0Ψ0I( , z, ) pV2 0+ Ψ2 0I2( , z, ),(4.15) whe e he a ailable adia ion I( , z, )≡I0exp ½Awz−ApZ0 z P( , z′, )dz′¾(4.16) is he su ace sola adia ion a enua ed by bo h sea wa e and Psel -shading. Phy oplank- on is emo ed by zooplank on g azing R, which has an I le esponse (Pa sons e al.,1967), and by a linea e m (MP) including dea h and exuda ion. Only a ac ion (1 −Γ0) o he Pg azed is con e ed in o Zp oduc ion GZ, while he emainde is ans e ed o he N compa men . In con as o P,Zhas a quad a ic closu e e m o wo pu poses. Fi s ly, i gi es s abili y o he model (Edwa ds and Yool,2000), and secondly, i in oduces implici ly he e ec o in a ophic cannibalism (Ohman e al.,2002;Pi ch o d and B indley,1998) and p eda ion by highe ophic le els (Ohman and Hi che,2001). Finally, de i al ma e (Γ0R+MP+MZ) is emine alized making nu ien s again a ailable o Pup ake. 80 CHAPTER 4 4.2.3 Physical ecological coupling Ecological s eady-s a e solu ions a e ound he e nume ically by conside ing ma e ial conse a- ion o o al ni ogen NT. The small-scale and sub-g id ecological dynamics a e pa ame ized h ough Fickian di usion. We ocus nex on he spa ial he e ogenei y in oduced a he mesoscale, which depends on h ee main ac o s. Fi s ly, Pg ow h is a ec ed by he e ical ligh egime, which is de e mined by he isopycnal dep h. Secondly, he di usi e lux de- pends on he isopycnals slope. And hi dly, NTis homogeneous on isopycnals bu we assume a nega i e diapycnal g adien o NT. Consequen ly, he pe u ba ion in he ecosys em caused by he mesoscale o cing can be cha ac e ized h ough he spa ial dis ibu ion o isopycnals, which is gi en by D( , z). We ocus nex on exp essing he NPZ equa ions in e ms o D( , z). In o de o simpli y he de i a i es along isopycnals we swi ch om cylind ical ( , z) o isopycnal (s, d) coo di- na es. An exp ession o he isopycnal adius s( , z) is ob ained by combina ion o (4.7) and he line elemen de ini ion ds2≡d 2+ dz2, ha is, s( , z) = Z 0s1 + µ∂D ∂ ′( ′, z)¶2 d ′.(4.17) Conside ing (??), he abo e in eg al has he ollowing exac solu ion s( , z) =   p1 + β2 , γ = 0 , −1 γ³τ+β 2ln ¯¯¯τ−β τ+β¯¯¯´+C(z), γ 6= 0 ,(4.18) whe e β(z)≡λ0z(z+π)2,(4.19) γ(z)≡3λ0(z+π)(z+π/3) ,(4.20) τ( , z)≡p(1 + γ(z)(π− ))2+β(z)2,(4.21) and C(z) is ob ained om he bounda y condi ion s( = 0, z) = 0. The e ical displacemen o isopycnals D(s, d) as a unc ion o sand d, ha is D(s( , z), d( , z)) ≡D( , z), is ound using a nume ical i e a i e p ocedu e. 4.3. INITIAL CONDITIONS 81 Finally, pa ame izing he small-scale isopycnal mixing as a Fickian di usion, he con- cluding NPZ equa ions o sol e a e GP−R−MP+κ∂2P ∂s2= 0 ,(4.22) GZ−MZ+κ∂2Z ∂s2= 0 ,(4.23) −GP+ Γ0R+MP+MZ+κ∂2N ∂s2= 0 ,(4.24) whe e κis he isopycnal mixing coe icien , and he spa ial dependence o he a ailable adia ion is compu ed using z(s, d) = d+D(s, d). Abo e, and he eina e , unless o he wise speci ied, all he dependen a iables a e assumed o be unc ion o he independen a iables (s, d), e. g. , P=P(s, d), e c. No e ha he ma e ial a e o change o he he o al ni ogen NT≡P+Z+Nbalances i s isopycnal di usion dNT d =κ∂2NT ∂s2.(4.25) Equa ions (4.16)–(4.18) a e used in he nex sec ion o ob ain s a iona y ecosys em solu ions wi h he luid a es . 4.3 Ini ial condi ions 4.3.1 Nume ical pa ame e s 4.3.1.1 Spa io- empo al domain The uni o space in he QG domain is de ined by he e ical ex en LZ=π. Thus he adial ex en LR=cLZ. The numbe o g id poin s is (nR, nZ) = (64,128), and he numbe o isopycnals nL= 128. Va iables can be adimensionalized by speci ying wo dimensional pa ame e s. Fi s ly, he maximum dep h Zmin ≡ −150 m de e mines he leng h con e sion ac o acco ding o L≡ |Zmin|/LZ. Secondly, he maximum pho osyn he ic a e Lmax, co e- sponding o ha a he sea su ace Lmax =α0I0V0/pV2 0+ (α0I0)2∼ =3 d−1wi h cons an sea su ace i adiance I0, gi es he ime con e sion ac o as T≡L−1 max. Consequen ly, ecological a iables a e exp essed in e ms o mmol N. Dimensional physical alues can he e o e be 82 CHAPTER 4 eco e ed by mul iplying he adimensional alues by he spa io- empo al con e sion ac o s Land Tele a ed o he app opia e powe s o ma ch physical dimensions. Fo exemple, he dimensional equi alen o a Pp oduc ion GP= 700 mmol N is 700/(L3×T) = 0.02 mmol N m−3d−1. 4.3.1.2 Pa ame e λ0 The in ensi y o cyclones is modeled h ough he magni ude o he e ical displacemen o isopycnals D, and hus h ough he dimensionless scaling ac o λ0. We ha e conside ed sphe ical cyclones om = 0.75 o = 2, which is a he limi o hei s abili y. This ange co esponds o Dmax ∈[0.16,0.39], using he CASL algo i hm a o emen ioned (sec ion 4.2.1), and hus o λ0∈[0.11,0.27]. An a bi a y δλ0= 1.7×10−3is chosen. No e ha λ0does no al e he pa e n o D( , z) bu ha o D(s, d). As Dmax inc eases, he co esponding isopycnal le el deepens (Fig. 4.3). 4.3.1.3 Isopycnal di usion coe icien κ Sugges i e eddy di usi i ies a e ex ac ed om he No h A lan ic T ace Release Expe - imen (Ledwell e al.,1998;Polzin and Fe a i,2004). Scale-dependen measu emen s o la e al dispe sion we e pe o med a 300 m and es ima ed κ o scales anging 1–30 km a abou 2 m2s−1. Subine ial o ical s uc u es had a majo con ibu ion o he men ioned 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 d Figu e 4.2: Ve ical dis ibu ion o D(s, d) co esponding o Dmax = 0.39 (uppe case, λ0= 0.027, solid line), and Dmax = 0.16 (λ0= 0.011, dashed line). Con ou in e al δD= 0.025. 4.3. INITIAL CONDITIONS 83 isopycnal coe icien wi h κ∈(0.5,2.5) m2s−1, ollowed by mesoscale-submesoscale in e ac- ions (κ≃0.1 m2s−1), ul ima ely de eloping in o ine ia–g a i y wa es (κ≃0.01 m2s−1) (Polzin and Fe a i,2004). The es ima es abo e a e based on mesu eamen s o an ine ace . Howe e , he dis- pe sion a e o κo a eac i e ace , such as plank on, depends on i s eac i e imescale (Pasque o,2005). Fu he mo e, spec al slopes be ween ine o ac i e ace s and chlo o- phyll di e ge a a c i ical leng hscale 2–7 km, whe e he low and ecological imescales ha e he same o de o magni ude (see e iew Ma in,2003). Many heo ies may accoun o his di e gence such as he scale o he ace supply (B acco e al.,2009), and hin laye s (F anks,2005). As a esul o he unce aini y abou bounding an isopycnal di ussion coe i- cien o ecological a iables, we cons ain κwi hin he wides plausible ange κ∈[0.01,10] m2s−1. Acco ding o he abo e men ioned scaling, he isopycnal di usion coe icien is adimen- sionalized such ha K≡κT (Lc)2.(4.26) No e he squa ed dependence on cdue o he ac ha he QG space is e ically s eched. I mus be kep in mind also ha he pa ame e cspeci ies QG mesoscale o ices as well as submesoscale o small scale o ices. 4.3.1.4 T ophic egime The nu ien s a ailabili y o he sys em is modi ied h ough he maximum o o al ni ogen T1. We ha e conside ed a ange o T1 om 1L3, which is ep esen a i e o an oligo ophic condi ion, o 8L3, wi h inc emen s δT1= 1L3mmol N (see Fig. 4.4 o associa ed e ical p o iles). 4.3.2 S a iona y ecosys em wi h he luid a es A i s s ep in sol ing he ime e olu ion o he ecosys em is o de ine he app op ia e ini ial condi ions o he NPZ dynamics. S able s eady-s a e solu ions o he coupled model a e ob ained by ime in eg a ion o (4.16)–(4.18), which con e ge o an equilib ium s a e when using an ini ial solu ion close enough o ha s a e. To his end, we i s seek o ecological s eady s a e p o iles wi h he luid a es , ha is, ou side he o ex. In his case, isopycnals 84 CHAPTER 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 d P ^ x 3(mmol N)L 01234 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 d Z ^ x 3(mmol N)L 02468 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 d N ^ x 3(mmol N)L 02468 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 d NT x 3(mmol N)L Figu e 4.3: S eady-s a e p o iles ˆ P,ˆ Z, and ˆ N o a gi en p o ile o NTwi h he luid a es . Ecological a iables a e adimensionalized and exp essed in e ms o mmol N. He eina e , p og essi e line hickness indica es an inc ease in NTmaximum δT1= 1 L3mmol N (solid lines) and δT1= 0.2L3mmol N (dashed lines). 4.4. RESULTS 85 a e la (D= 0), so ha ligh a ailabili y is ho izon ally homogeneous and he ecosys em has only e ical a iabili y. The e ical dependence o {NPZ}is ob ained by ime in eg a ion o (4.16)–(4.18), wi h K= 0 and using he oligo ophic non-dimensional pa ame iza ion (Table 4.1) o Lawson e al. (1996) and he ollowing ini ial p o iles NT(d) =    T1cos h³1−d d1´2π 5i, d ∈[d1,0], 1, d ∈[−π, d1), (4.27) (Pi, Zi)(d) = S1µ1−d dmin ¶,(4.28) whe e T1∈(1,8) L3mmol N, S1= 0.1L3mmol N, d1∼ =−0.91 (isopycnal index, iL= 91), and dmin =Zmin. Time in eg a ion is pe o med using an explici leap- og scheme, oge he wi h a Robe -Asselin ime il e o a oid he compu a ional mode. The ini ial p o iles smoo hly con e ge o s eady-s a e solu ions (Fig. 4.4), which a e he eina e e e ed o as {NT,ˆ P, ˆ Z}. Phy oplank on and zooplank on ha e simila e ical dis ibu ions (Fig. 4.4), hough he la e is sligh ly la ge han he o me , wi h a subsu ace maximum a d∼ =−0.86 (iL= 92). The ob ained {NT,ˆ P, ˆ Z}dis ibu ions ha e a double pu pose. On he one hand, hey a e used as a i s guess in he coupled model wi h gi en dis ibu ion o D(s, d) by p ojec ing hem on isopycnals. On he o he hand, hey de ine he bounda y condi ion a smax, ha is, ou side he o ex whe e isopycnals a e la . 4.4 Resul s In his sec ion we in es iga e he phy oplank on esponse o ligh and nu ien s. To his end, di e en cases a e explo ed conside ing he isopycnal e ical displacemen D, he non- dimensional di usion coe icien K, and he maximum o o al ni ogen T1(see sec ion 4.3.1 o pa ame e s ange). The dis ibu ion o Dmodi ies he phy oplank on ligh egime o a gi en isopycnal, while di usion ends o homogenize ecological a iables along isopycnals. The ac o T1de ines he ophic egime o he sys em. The ecosys em dynamics wi hou di usion is i s analyzed conside ing o ices o a iable isopycnal e ical displacemen embedded in a ious ophic egimes (sec ion 4.4.1). Nex , di usion is conside ed in hese 86 CHAPTER 4 symbol desc ip ion alue uni s Awligh a enua ion by sea wa e 0.03 m−1 Apligh a enua ion by phy oplank on 9.5×10−3m2mmol N−1 I0su ace a ailable adia ion 234.11 W m−2 Ψ0ini ial slope o he P-Icu e 0.025 m2W−1d−1 V0phy oplank on maximum up ake a e 3.6 d−1 K0hal -sa u a ion o phy oplank on up ake 0.5 mmol N m−3 Ξ0phy oplank on speci ic mo ali y a e 0.1 d−1 R0zooplank on maximum g azing a e 0.8 d−1 Λ0I le cons an 0.4 mmol N−1m3 Γ0 ac ion o zooplank on g azing eges ed 0.25 Θ0zooplank on exc e ion/mo ali y a e 0.05 d−1 Table 4.1: Lis o ecological cons an s co esponding o an oligo ophic en i onmen as ex ac ed om Lawson e al. (1996). The Zspeci ic exc e ion/mo ali y a e Φ0is compu ed no malizing Θ0by he spa ially a e aged Z. All cons an s a e made adimensional in he physical-ecological coupled model. di e en ophic condi ions (sec ion 4.4.2) and o ex ypes (sec ion 4.4.3). One gene al esul is ha ecological s a iona y p o iles nume ically s able a e eached in all conside ed cases. 4.4.1 Isopycnal e ical displacemen and ophic egime Isopycnal doming deepens he isolines o N,P,and Z owa ds he o ex cen e (Fig. 4.5) because he i adiance ecei ed by an isopycnal inc eases when i shallows. Thus Pinc eases in de i men o N, and so does he g aze communi y. In addi ion, when T1inc eases om 1L3mmol N o 8L3mmol N, he maximum o Pis doubled (Figs. 4.5a,b), while ha o Zis enhanced by one o de o magni ude (Figs. 4.5c,d). In ac , his is no an o iginal con ibu ion since phy oplank on g ow h is limi ed by ligh and nu ien s. Ou aim is o quan i y his limi a ion when he ligh egime is modula ed by cyclones. To his end, we de ine he anomaly o any a iable χ(s, d, K,Dmax, T1) wi h espec o a e e ence s a e whe e isopycnals a e la (do ed lines in Fig. 4.5) as χ′(s, d, K0,0.39, T1)≡χ(s, d, K0,0.39, T1)−χ(s, d, K0,0, T1),(4.29) whe e K0= 0. The la ges isopycnal e ical displacemen , and hus i adiance change, occu s a he o ex cen e a isopycnal dI=−1 (Fig. 4.3). Acco dingly, in he T1= 8L3 mmol N case, he maximum o P′is loca ed close o dI, bu i u ns ou o be much deepe 4.4. RESULTS 87 (d∼ =−1.9) when T1= 1 L3mmol N (Figs. 4.6a,b). This pa adox is add essed by subse- quen ly spli ing he anomalies o he physiological p ocesses. Anomalies o phy oplank on g ow h GPand g azing Ra e posi i ely coupled as a e P′and Z′in bo h ophic condi- ions. Howe e , hey a e one o de o magni ude la ge in he T1= 8 L3mmol N case han he lowes T1case (Figs. 4.6c,d). The p oduc ion o Pdepends upon he ligh bu also upon nu ien s. While he o me a enua es wi h dep h, he la e inc eases. When T1= 1 L3mmol N, he dis ibu ion o P′ esembles ha o he nu ien up ake. In con as , when T1= 8 L3mmol N he pa e n o P′co ela es be e wi h he pho osyn he ic a e L (Figs. 4.6e, ). Thus we ha e cha ac e ized wo di e en ecological dynamics depending on he ligh egime and he ophic condi ion. We ha e ocused so a on one o ex ype and on wo ophic egimes. In o de o NT= 1 L3mmol N NT= 8 L3mmol N 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 (a) d 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.00 11.41 22.81 34.22 45.63 57.04 68.44 79.85 (b) d x 103 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 2.75 17.91 33.07 48.22 63.38 78.54 93.69 108.85 (c) d x 103 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 16.94 138.92 260.90 382.88 504.85 626.83 748.81 870.79 (d) d x 103 Figu e 4.4: Ve ical dis ibu ions wi h K= 0 and Dmax = 0.39 o (a,b) P(s, d) in shaded con ou s and ˆ P(d) (do ed line, max ∼ =70.5×103mmol N, δ≃105mmol N); and (c,d) N(s, d) and Z(s, d) (solid line, Zmax ∼ =52.79 (592.57) ×103mmol N, δZ ≃104(105) mmol N in c (d)). Le panels co espond o NTmaximum T1= 1 L3mmol N, while igh panels o T1= 8 L3mmol N. 94 CHAPTER 4 4.4.3 Isopycnal e ical displacemen and di usion We analyze nex whe he he cha ac e is ic esonan di usion coe icien depends on he isopycnal e ical displacemen . To his end, we ix he ophic condi ion o an a bi a y alue T1= 1.4L3mmol N wi hin he ange 1 < T1<2L3mmol N, whe e K∗shows he highes sensi i i y (Fig. ??a). The maximum P esponse o di usion occu s when Kis O(0.1). Thus we explo e a ange o Kcomp ised wi hin (0,0.4) conside ing a subse o o ices wi h Dmax ∈(0.16,0.39). Each case is cha ac e ized by he enhancemen ac o wi h espec o he s a e wi hou di usion, ha is, ´ P(K,Dmax,1.4L3mmol N). We obse e ha Dmax a ec s he magni ude o ´ Pmaximum bu no he alue o K(Fig. 4.9a). Fi s ly, as s a ed in he i s sec ion, Pinc eases linea ly wi h Dmax and acco dingly a linea inc ease is also expec ed in ´ P. Secondly, K∗shi s om 0.08 o 0.1 a Dmax = 0.16 and Dmax = 0.4, espec i ely. The esonan beha iou is eached when Pand di usi e imescales a e compa able. So ha , he ac o s de e mining he loca ion o ´ Pmaximum in he (K,Dmax) space should be ela ed o he Pdi usi e a e and he inc ease in he Pg ow h a e caused by di usion. In his ega d, ´ Pmaximum is eached when η≡K∂2P/∂s2|Ω (G′′ P−M′′ P)|Ω∼ =1.08 ,(4.36) which co esponds o he cha ac e is ic non-dimensional di usion coe icien ˜ Ka which ´ Pis maximum o his ophic condi ion. Isopycnal doming inc eases di usion, since i s eepens isopycnals slopes, and also phy oplank on g ow h, because i enhances he i adiance ecei ed by he upli ed phy oplank on. As a esul , K∗is nea ly una ec ed by Dmax. Howe e , phy- oplank on g ow h and di usion do no accoun o ally o ´ P(Fig. 4.9b). We obse ed ha di usion uncouples Pand Z, and hus dec eases he g azing p essu e on P. Consequen ly, when Zg azing is included, he esul ing balance co ela es wi h ´ P(no shown). 4.5 Concluding ema ks We cons uc ed a simple nume ical model o in es iga e he dynamics o a ully- ecycling ecosys em wi hin mesoscale o ices. Pa icula ly, we ocused on su ace cyclones and sub- su ace an icyclones since hei isopycnals a e domed in he uppe laye . The no el y o 4.5. CONCLUDING REMARKS 95 0.0 0.1 0.2 0.3 0.4 0.16 0.19 0.22 0.25 0.27 0.30 0.33 0.36 0.39 99.7 100.0 100.2 100.4 100.6 100.9 101.1 101.3 101.5 101.7 102.0 102.2 P (x 10−2) ′ (a) K Dmax 0.0 0.1 0.2 0.3 0.4 0.16 0.19 0.22 0.25 0.27 0.30 0.33 0.36 0.39 97.3 97.6 97.9 98.2 98.5 98.9 99.2 99.5 99.8 100.1 100.5 100.8 GP’’+ 2P/ s2 (x 10−2) _______________ K′ ∂ ∂ (b) K Dmax Figu e 4.9: Dis ibu ions o (a) ´ P(K,Dmax) and η′=η−1.08 (con ou line, η′∈(−0.4,0.6), δη′∼ =0.035), (b) ´ G′′ P+K∂2P/∂s2(K,Dmax) and ´ R′′ (K,Dmax) (con ou line, ∈(−4,0.2) × 10−2,δ∼ =1.5×10−3). The ophic condi ion is T1= 1.4L3mmol N. his model was ha he mesoscale o cing was in oduced in e ms o e ical displace- men o isopycnals in he NPZ (Nu ien s-Phy oplank on-Zooplank on). Using his physical- ecological coupled model, he impac o he ophic egime, o ex in ensi y, and small-scale mo ions, pa ame ized as a Fickian- ype isopycnal di usion, on he ecosys em was explo ed. In all conside ed cases, s a iona y ecological solu ions nume ically s able, wi h coexis ence o phy oplank on (P) and zooplank on (Z), we e ob ained. The ophic egime, de e mined by he maximum o o al ni ogen, cha ac e ized wo di - e en ecological dynamics. In he oligo ophic egime case, Pand Zanomalies (wi h espec o a e e ence s a e whe e isopycnals a e la as ou side he o ex) had simila magni ude. 96 CHAPTER 4 These anomalies we e localized a he o ex bo om edge, whe e he nu ien s (N) lux was he la ges . When a meso ophic egime was ins ead ini ialized, Zanomalies we e one o de o magni ude la ge han hose o P. Bo h anomalies had hei maximum localized a he isopycnal wi h he la ges e ical displacemen , and hus he one expe iencing he la ges change in i adiance. Small-scale mo ions, pa ame ized as a Fickian- ype di usion along isopycnals, inc eased he spa ially in eg a ed Pbiomass up o 1% o 15% depending on he ophic condi ion. This inc ease occu ed a a cha ac e is ic di usion coe icien (K∗) due o a esonance mechanism be ween di usi e and phy oplank on imescales. Tha is, when he a io be ween di usi e and phy oplank on g ow h a es was close o one. Two main mechanisms we e in ol ed in his Pinc ease. Fi s ly, di usion upli ed N o be e li le els h ough a Nin lux ha coun e balances he Pou lux. Secondly, Zuncoupled o i s p ey inc ease because he g ow h a e o Pwas la ge han ha o Z. As a esul , he Zou lux was la ge han ha o Pand he g azing p essu e diminished in bene i o he la e . The la ges con ibu ion o di usion in maximizing Pwas obse ed in meso ophic egimes. Howe e , la ge K∗ alues we e equi ed and some cau ion should be aken when in e p e ing hem. Conside ing K∗= 0.3 and he uppe bound o he small-scale mo ion pa ame iza ion (κ= 10 mmol N m3), esul s in a P and l a io c∼ =20, which is comp ised wi hin he submesoscale. Since ou basic equa ions we e inapp opia e o model submesoscale iscid dynamics, we cons ain ins ead c o he lowes mesoscale bound (c= 35). In his case, κ∼ =30 m2s−1, which app oaches o he pa ame iza ion o mesoscale mo ions (Ma in e al.,2001;Ledwell e al.,2008). This sugges s ha esonance be ween plank on g ow h and ad ec i e a es occu s wi hin non-axisymme ic o ices, which has been al eady epo ed by Pasque o (2005). Finally, he o ex in ensi y, modeled h ough he maximum isopycnal e ical displace- men , exe ed an an agonis ic e ec on he ecosys em. On he one side, i induced a posi i e linea P esponse since Pand Nwe e upli ed o be e li dep hs. Isopycnal doming ac- coun ed o an inc ease abou 8% o 15% o he Pbiomass co esponding o a scena io wi h la isopycnals depending on he ophic condi ion. On o he side, i also inc eased di u- si e luxes since s eepened he isopycnal slopes. As a esul , K∗was nea ly independen o he isopycnal e ical displacemen . This sugges s ha he Penhancemen ac o may be app oxima ed by addi i e con ibu ions o isopycnal doming and di usion a K∗. Chap e 5 Ve ical eloci y in he in e ac ion be ween ine ia-g a i y wa es and submesoscale ba oclinic s uc u es This chap e has been published as: Cla e , M., and A. Viúdez, 2010: Ve ical eloci y in he in e ac ion be ween ine ia-g a i y wa es and submesoscale ba oclinic s uc u es. J. Geophys. Res., 115, C12060, doi:10.1029/2009JC005921. A la ida o al co quelcom li p enen les ones que se’n an; si no inc es, les ones que a a ´enen dieu-me qu`e old an? Vo a la ma , Jacin Ve dague 99 ABSTRACT The in e ac ion be ween submesoscale ba oclinic o ical s uc u es and la ge ampli ude ine ia–g a i y wa es (IGWs), wi h emphasis on he e ical eloci y, is nume ically in es- iga ed using a high- esolu ion h ee-dimensional non-hyd os a ic model. A ich a ie y o o ex-wa e in e ac ions a e possible depending on he po en ial o ici y (PV) con en and leng h scale o he submesoscale monopoles o dipoles, and on he ampli ude and wa enum- be o he IGWs. On he one hand he la ge ampli ude IGWs cause ho izon al and e ical ad ec ion o he o ices, which conse e hei s abili y hough hei geome y is la gely mod- i ied by he wa e mo ion. On he o he hand he ho izon al o ical mo ion Dopple shi s he local equency o he IGWs. The o ical angula eloci y and o ex densi y s a i i- ca ion lead o a wa e dispe sion ela ion in ol ing he e ec i e Co iolis equency (Co iolis equency plus he o ical angula eloci y) and he o al B un -V¨ais¨al¨a equency. This inhomogeneous change in he local wa e equency causes he IGWs depa om hei ini ial plane geome y. In he pa icula case o ine ial wa es, he non-linea o ex-wa e in e - ac ion gene a es spi al IGWs, ha ing e ical eloci ies one o de o magni ude la ge han he submesoscale o ical low in he absence o wa es. 5.1. INTRODUCTION 101 5.1 In oduc ion Recen nume ical wo ks ha e ep oduced he ully h ee-dimensional na u e o submesoscale lows (e. g., Cape e al.,2008), whe e e ical eloci y can each alues one o de o magni- ude g ea e han hose a he mesoscale (Mahade an,2006). Submesoscale s uc u es ha e been epo ed bo h in he uppe ocean (Rudnick and Luy en,1996;Shay e al.,2003;Cape e al.,2008) and deep ocean (McWilliams,1985;Tes o and Gasca d,2003;S e en and D’Asa o,2004;Kasajima e al.,2006). On he o he hand, ine ia–g a i y wa es (IGWs) a e also ubiqui ous in he ocean (Ga e and Munk,1979;Mi opol’sky,2001;Pedlosky, 2003), and consequen ly in e ac ion be ween submesoscale lows and IGWs is a equen phenomenon. He e we add ess his in e ac ion, ocusing on he e ical eloci y, in he special case whe e o ical and wa e lows ha e simila ampli udes. The submesoscale e e s o lows wi h ho izon al scales Lo o de 1, and Rossby Rand F oude Fnumbe s also o o de 1. I plays an impo an ole in he ocean because i a- cili a es he ene gy ans e om he mesoscale o smalle scales (Molemake e al.,2005), and he e ical lux o momen um, buoyancy, po en ial o ici y (PV), and biogeochemical p ope ies (L´e y e al.,2001;Thomas e al.,2008). In he deep ocean, long-li ed subme- soscale o ices a e also esponsible o bo h deep con ec ion (Gasca d e al.,2002), and ho izon al anspo , as hey a e ad ec ed away om hei o igin by mean cu en s (Tes o and Gasca d,2003). In he pa icula case o nea -ine ial oscilla ions, aniso opy o he wa e ield caused by he mesoscale geos ophic mo ions has been ex ensi ely epo ed (Mooe s,1975a,b;Pe kins, 1976;Welle ,1982;Kunze,1984; an Meu s,1998;Niwa and Hibiya,1999). Se e al o ex- wa e in e ac ions ha e been p oposed o explain his wa e he e ogenei y, like wa e apping o IGWs inside o ices (Kunze,1985), wa e cap u e (B¨uhle and McIn y e,2005), dispe sion o nea -ine ial ene gy by geos ophic eddies (Young and Jelloul,1997; an Meu s,1998), ine ial pumping (Rubens ein and Robe s,1986), o esonance mechanisms (Niwa and Hi- biya,1999;Danioux and Klein,2008b). He e we add ess bo h he o ex-wa e mechanisms ha explain he wa e equency shi by submesoscale o ices and he PV s uc u es ha emain cohe en a e being ad ec ed by la ge ampli ude IGWs. Ou esul s ex end he wo ks men ioned abo e by using a non-hyd os a ic nume ical model, which conside s he ully nonlinea h ee-dimensional momen um equa ions and esol es he e ical eloci y 102 CHAPTER 5 wi h high accu acy, o simula e ba oclinic assymme ic PV lows o leng h scales simila o hose o he p e-exis en wa e ield. The i s o ex-wa e in e ac ion we in oduce implies o ical mo ion a ec ing plane IGWs. This occu s bo h h ough he wa e equency Dopple -shi (Kh·ubh), whe e Khis he ho izon al wa enumbe and ubh he ho izon al o ical (balanced) eloci y, and h ough he o ical angula eloci y and o ex densi y s a i ica ion anomaly, which lead o e ec i e Co iolis and B un -V¨ais¨al¨a equencies in he dispe sion ela ionship o Kunze (1985). The second in e ac ion is he ad ec ion o he o ical low by la ge ampli ude IGWs. In his case, he o ex geome y is la gely de o med gi ing ise o new ci cula ion gene a ed in he p ocess owa ds geos ophic adjus men . Finally, nonlinea o ex-wa es in e ac ions igge a spi al IGW when a pu e ine ial wa e is p esen in a o ex low. In his case ho izon al g adien s o he e ical o ici y ζgene a e g adien s in he e ec i e Co iolis equency, h ough he ζ/2 shi (Mooe s,1975a;Kunze,1985;Rubens ein and Robe s, 1986) o igina ing di e gence o he wa e ield om which e ical eloci y de elops. In his wo k we use a iply pe iodic non-hyd os a ic nume ical model unde he Boussi- nesq and -plane app oxima ions (sec ion 5.2) o examine he in e ac ion be ween subme- soscale ba oclinic o ex s uc u es wi h Rossby numbe R.1 and p e-exis en IGW ields. We pa icula ly ocus on he gene a ion o spi al pa e ns o e ical eloci y. The low has cons an backg ound P and l a io N/ = 10, whe e Nand a e cons an backg ound B un - V¨ais¨al¨a and Co iolis equencies, espec i ely. We conside wo ypes o o ical s uc u es and wa es, namely he monopola o ex (cyclonic and an icyclonic) and he o ex dipole, and wo ypes o wa es, namely pu e ine ial and g a i y plane wa es (sec ion 5.3). Nex we in es iga e he low o a monopole embedded in ini ially plane ine ial and g a i y wa e ields (sec ions 5.4.1 and 5.4.2). The o ex, al hough no longe homogeneous no s eady, emains always s able despi e he subs an ial ad ec ion by he la ge IGWs. In e ac ions be- ween a submesoscale dipole, he simples o ical s uc u e ha ing linea momen um, and la ge ampli ude IGWs o di e en wa enumbe s a e add essed in sec ion 5.5. The ba oclinic dipole emains cohe en despi e he p esence o la ge ampli ude wa e ields. The balanced and unbalanced componen s o he low a e ex ac ed om he o al low and a e sepa a ely analyzed. Finally, conclusions a e gi en in sec ion 5.6. 5.2. NUMERICAL MODEL AND PARAMETERS 103 5.2 Nume ical Model And Pa ame e s 5.2.1 AB-model The non-hyd os a ic nume ical model (he eina e e e ed o as he AB-model) simula es he isocho ic ( olume-p ese ing) low o a s a i ied o a ing luid unde he Boussinesq and -plane app oxima ions (D i schel and Vi´udez,2003). He e he low is ini ialized wi h 1) a localized o ical low speci ied by he po en ial o ici y PV using he PV ini ializa ion app oach (Vi´udez and D i schel,2003), and 2) a plane IGW backg ound ield (desc ibed in sec ion 5.3.1). The heo e ical basis o he nume ical model is explained in de ail in he e e ences abo e, succinc ly he e in appendix Aand only a b ie de ini ion o he physical quan i ies is gi en nex . The F oude numbe F≡ωh/Nand he Rossby numbe R≡ζ/ , whe e ωhand ζ a e he ho izon al and e ical componen s o he ela i e o ici y ω≡ωh+ζk, and N is he o al B un -V¨ais¨al¨a equency. The e ical displacemen o isopycnals is de ined as D(x, )≡z−d(x, ), whe e d≡(ρ−ρ0)/zis he dep h ha an isopycnal loca ed a xa ime has in he e e ence densi y con igu a ion de ined by ρ0+zz. Abo e ρ(x, ) is he mass densi y, and ρ0>0 and z<0 a e cons an alues ha do no need o be speci ied in he Boussinesq app oxima ion. The squa ed o al B un -V¨ais¨al¨a equency is he e o e N2(x, ) = N2µ1−∂D ∂z (x, )¶.(5.1) S a ic ins abili y occu s when he s a i ica ion numbe Dz≡∂D/∂z > 1, and ine ial ins abili y when R<−1. The AB-model in eg a es he dimensionless ageos ophic ho izon al o ici y Ah= (A,B)≡˜ ωh−c2∇ hD, dAh d =− k×Ah+ (1 −c2)∇ hw+˜ ω·∇uh+c2∇ hu·∇D,(5.2) whe e N2≡ −gz/ρ0, he P and l a io c≡N/ , he ela i e o ici y ω≡∇×u, he eloci y u=uh+wk,∇is he g adien ope a o , subsc ip hdeno es he ho izon al componen , and ˜χ≡χ/ , o any quan i y χ. The ma e ial de i a i e dχ/d ≡∂χ/∂ +u· ∇χ. The hi d p ognos ic equa ion is he explici conse a ion o PV anomaly  h ough 110 CHAPTER 5 V¨ais¨al¨a equency. In he cyclone case (C1) an e ec i e wa e equency ωl∼ =0.13 is p edic ed om (??). This is con i med by nume ical esul s which show ha he wa e equency peak e ol es om ine ial ωl= 0.1 o nea -ine ial equency ωl= 0.12 ±0.01 (Fig. 5.3). So ha he local equency shi ζ/2 (Mooe s,1975a;Kunze,1985;Rubens ein and Robe s,1986) is also caused by ba oclinic PV s uc u es ha emain no longe axisymme ic a e he o ex- wa e in e ac ion. In o de o analyze how he balanced low a ec s he wa e mo ion and ice e sa, we ha e ex ac ed he balanced low om he o al low. The balanced ec o po en ial ϕb= (ϕb, ψb, φb) is he e diagnosed using he Op imal PV Balance (OPVB) app oach (Vi´udez and D i schel,2004a), and he balanced quan i ies a e de i ed he e om. F om a gi en PV ield anomaly (x, y, z), he OPVB app oach diagnoses a low ha ing only hose IGWs ha ha e been spon aneously gene a ed du ing he p ocess o acqui ing i s own PV ( ha is, du ing a ime in e al se equal o he ini ializa ion ime i= 5 Tip). The OPVB low does no con ain mos o he IGWs, which emain, almos en i ely, in he unbalanced ec o po en ial ϕi≡ϕ−ϕb. The unbalanced eloci y and e ical displacemen o isopycnals, a e ob ained di ec ly om ϕi h ough he usual ela ions ui=− ∇×ϕiand Di=−ǫ2∇·ϕi. An al e na i e way o ob ain he in e ac ion be ween he ine ial wa es and he o ical low in his case is using he nea -ine ial oscilla ion (NIO) equa ion o Young and Jelloul (1997), which is alid o small Rossby numbe s. We no e ha hei geos ophic s eam unc ion Ψ Figu e 5.3: Clockwise o a- o y spec og am o u+i . The spec og am comp ises 194 spec a om = 0 o = 14.25 Tip using a win- dow o 5 Tip and a ime lag o δ = 0.5Tbp. The dis ibu ion shows he Fou ie ans o m magni ude o he compo- nen s pˆu2(ωF) + ˆ 2(ωF) o he Fou ie equencies ωF< 0. The ini ial ine ial peak e ol ed o nea -ine ial, ωl= 0.12 ±0.01. 5.4. VORTEX-WAVE INTERACTION 111 is simila o ou e ical po en ial φ. Howe e , we use he e he OPVB because is alid o la gely ageos ophic lows. The wa e equency shi men ioned be o e is no iceable because he e ical wa e phase eloci y a he o ex cen e (σZ= ( + Ω)/m) is la ge han ou side he o ex (σZ= /m). The e o e, phase lines o uia x∼ =y∼ =0 accele a e inside he o ex (Figs. 5.4a,b). As a esul , he o iginally s aigh phase lines o uih a e b oken by he cyclone, and he e ical dis ibu ion o he speed anomaly o he unbalanced ho izon al eloci y is dis o ed (Fig. 5.4c), eaching nega i e alues a he o ex cen e . As we ha e seen, he ini ial ine ial wa e ield is s ongly modi ied by he o ex, bu a he same ime he o ex is also de o med by he wa e eloci y, which causes he PV con ou s o depa om he sphe ical geome y, modi ying he e ical dis ibu ion o D(Fig. 5.5a). Since Dis ela ed o he geos ophic eloci y shea ug hz ≡∂ug h/∂z by he he mal-wind ela ion ug hz =−N2 k×∇hD,(5.11) he |ug h|con ou s (Fig. 5.5b) depa om he ci cula geome y ypical o a o ex in he absence o a wa e ield. This is also con i med when ex ac ing he balanced low om he o al low (Fig. 5.5c). An impo an esul o he o ex-wa e in e ac ion is he gene a ion o win he o m o spi al wa es (Fig. 5.6). Since he mo ion o he isola ed sphe ical o ex on he one hand, and he mo ion o he isola ed ine ial wa es on he o he hand, a e pu ely ho izon al, he de elopmen o win he o ex-wa e sys em is a clea esul o non-linea o ex-wa e in e ac ion. The maximum wampli ude eaches |w|max = 5 ×10−2, ha is 5% o he ho izon al ine ial wa e speed, om = 5 Tip o = 6 Tip (Figs. 5.6a,b). This spi al wpa e n seems o be ela ed o wa e mo ion a he han o balanced mo ion since he QG e ical eloci y wq ob ained by sol ing he QG omega equa ion (Hoskins e al.,1978) c2∇2 hwq+∂2wq ∂z2= 2∇ h·Qg h,(5.12) whe e Qg h≡c2∇ hug h·∇ hDis he geos ophic Q- ec o and ug his he geos ophic eloci y. Though ha ing a spi al pa e n as well, is abou one o de o magni ude smalle han he o al w(no shown). The unbalanced o igin o he o al wis con i med by spli ing i in o wband wiusing he OPVB app oach. The e ical dis ibu ion o w(Fig. 5.6b) ollows he 112 CHAPTER 5 (a) (b) (c) Figu e 5.4: Ve ical dis ibu ions in he x-z plane iY= 65 (y= 0) a = 5 Tip o (a) ui (ui∈[−1.3,1.28], δui= 0.15), (b) i( i∈ [−1.17,1.19], δ i= 0.15), and (c) he speed anomaly o he unbalanced ho izon al eloc- i y U′ i=|ui| − 1 (U′ i∈[−0.82,0.31], δU′ i= 0.09). Domain ex end is x∈[−π, π]c,z∈ [−3/4π, 0]. The PV con ou = 0.2 ( hick line) is included o e e ence. He eina e , solid and dashed lines indica e posi i e and nega i e alues, espec i ely. Figu e 5.5: Ve ical dis ibu ions in he x- zplane a iY= 65 (y= 0) and = 5 Tip o he (a) isopycnal displacemen D(D∈ [−0.06,0.17], δD= 0.017), (b) ho izon al geos ophic speed anomaly Ug′=|Ug h| − 1 (Ug′∈[−0.15,0.82], δUg′= 0.07). and (c) balanced ( b∈[−1.55,1.18], δ b= 0.15). Domain ex en is x∈[−π, π]c,z∈[−π, 0]. The PV con ou = 0.2 is included. (a) (b) (c) 5.4. VORTEX-WAVE INTERACTION 113 pa e n o wiand bo h ha e he same o de o magni ude, which is wo imes la ge han ha o wb(Fig. 5.6c). The mo ion o he spi al IGWs is ha o a igh -handed helix ( he heigh inc easing wi h inc easing phase, Fig. 5.7), o a ing an icyclonically so ha he phases p opaga e upwa ds. The wa e packe p opaga es downwa d and ho izon ally lea ing he o ical egion in a ew ine ial pe iods (no shown). This spi al IGW has a local equency anging om o ea ini ial imes and ex ending o 2 and 3 equencies a e wa ds (Fig. 5.8). Nea -ine ial wis gene a ed by di e gence o he ui ield, which becomes ho izon ally inhomogeneous because ζshi s he equency o pu e ine ial wa es. When sepa a ing unbalanced om balanced lows we obse e ha ζiand ζbha e he same o de o magni ude a e > i. On he one hand, ζiis in phase wi h w(no shown), as p edic ed om (5.6). Howe e pu e ine ial wa es ha e ζi= 0 a = 0. On he o he hand, we obse e ha wco ela es wi h |∇ζb|maxima a e > iand in deepe laye s, whe e ho izon al ad ec ion is minimum (Fig. 5.9a). As a esul , wide elops a o ex edges (no e ha F>1, Fmax = 1.37, occu s once he spi al wa e has been al eady gene a ed). Since he o ex geome y is la gely ho izon ally ad ec ed by an ini ially pu e ine ial wa e, |∇ζb|isosu aces become spi alized wi h dep h (Fig. 5.9b) gene a ing an helical IGW. Thus, while he equency o he o al wis di ec ly ela ed o ζ, i s 3D s uc u e is explained by |∇ζb|, in acco dance wi h he s a ed co ela ion be ween wand he eddy ela i e o ici y (Danioux and Klein,2008a). Finally, supe ine ial wobse ed a la e imes is due o esonance mechanisms, in ag eemen wi h he esul s o Niwa and Hibiya (1999) and Danioux and Klein (2008b), ha occu when PV s uc u es and IGWs ha e simila leng h scales. Analogous esul s we e ob ained wi h an axisymme ic an icyclone (case C2), ha ing min =−0.5 and semi-axes ah/c =aZ= 1, ini ialized wi h an ine ial wa e ield o |uih|= 0.1 and m= 6 (wa eleng h λZ∼ =1). In his case he o ex has Ω <0 and he local equency ωl∼ =0.79 < . Consequen ly, σZa he o ex cen e is smalle han ha a away om i , which is he opposi e e ec o ha desc ibed in he cyclonic case, and wa e phase lines accumula e a eddy edges (Fig. 5.10a). The o al walso shows a igh handed helical s uc u e (no shown), consis en wi h he an icyclonic o a ion wi h ime o he ine ial wa e eloci y, which p opaga es ho izon ally and downwa ds a ini ial imes bu i is apped a he eddy bo om la e on (Figs. 5.10b–d). This spi al IGW has a subine ial equency ωl≃0.08±0.01 (Fig. 5.12) and he e o e close o he p edic ed e. 114 CHAPTER 5 (a) (b) (c) Figu e 5.6: Dis ibu ions o o al e ical eloci y w(w∈[−4.81,4.23]×10−2,δw = 5×10−3) a = 5 Tip (a) in he x-yplane a iZ= 45 (z=−0.98) and (b) in he x-zplane a iY= 65 (y= 0). (c) Ve ical dis ibu ion a he same ime o wb(wb∈[−5.52,3.6] ×10−4, δwb= 5×10−5). Domain ex end is x, y ∈[−π, π]c,z∈[−3/4π, 0]. PV con ou = 0.2 ( hick line) is included. S aigh lines ma k he ho izon al (a) and e ical (b,c) sec ions plo ed. x/c z Figu e 5.7: Isosu aces o o al e ical eloci y w(w=±0.02) a = 5 Tip. The iew is om he sou h. 5.4. VORTEX-WAVE INTERACTION 115 Figu e 5.8: Domain a e - aged spec og am w(ω, k)≡ 1 nPn j=1 ˆw(xj, ω; k) om = 0 o = 15 Tip, whe e ˆw(xj, ω; k) is he Fou ie ans o m o he ime se- ies w(xj, ) wi h ∈[ k− ∆ /2, k+ ∆ /2]. The spa ial a e age comp ises n= 83 ime se ies equally dis ibu ed in he 3D domain. The spec o- g am window is ∆ = 5Tip and he ime lag δ = 0.5Tbp. The e ical dashed lines ma k he equencies , +ζ/2∼ =0.12, 2 , and 3 . 0.0 0.9 1.7 2.6 3.5 4.3 5.2 (x100) (a) (b) z x/c Figu e 5.9: (a) As in igu e 5.6 bu a iZ= 43 (z=−1.03) (w∈[−4.18,3.48] ×10−2, δw = 5 ×10−3). Shaded con ou s a e |∇hζb|(max = 5.1×10−2, ∆ = 8.7×10−3). Domain ex end is x, y ∈[−3/4π, 3/4π]. (b) Isosu ace |∇hζb|= 0.04 a he same ime = 5 Tip. The iew is om he sou h. 116 CHAPTER 5 Though we do no conside in de ail he long e m o ex-wa e in e ac ion we no e ha , s a ing a = 8.25 Tip and du ing he nex 16 Tip, he ine ial wa es may cause he o ex become uns able in he sense ha he o ex losses PV by PV ilamen a ion (no shown). This PV ilamen a ion inc eases he ho izon al PV g adien s emaining in he o ex and as a esul , la e on a = 13.43 Tip, he low becomes ine ially uns able (R<−1). This long e m ins abili y is le o u u e esea ch. (a) (b) (c) (d) Figu e 5.10: Ve ical dis ibu ions a iY= 65 (y= 0) o (a) i(x, z) ( i∈[−0.24,0.2], δ i= 0.025) a = 5 Tip and o o al w(x, z) (w∈[−8.5,8.7] ×10−3,δw = 8.3×10−4) a (b) = 5 Tip, (c) = 5 Tip, and (d) = 15 Tip. PV con ou =−0.2 is included (solid hick line). The domain ex end is x∈[−π, π]c,z∈[−2/3π, 0]. 5.4. VORTEX-WAVE INTERACTION 117 Figu e 5.11: Domain a e - aged spec og am w(ω, k) as in Fig. 5.8. Spa ial a e age comp ises n= 92ho izon- al poin s equally dis ibu ed o e 17 e ical le els. In his case e ical dashed lines ma k equencies and − ζ/2∼ =0.08. 5.4.2 Vo ex And G a i y Wa es In e ac ion In his sec ion a sphe ical (in he QG space) an icyclone, ha ing min =−0.75 and semi- axes ah/c =aZ= 1.5 in he ini ial con igu a ion, is ini ialized (Fig. 5.13a) embedded in a g a i y wa e ield wi h k= 8/c (wa eleng h λX/c = 2π/(ck)∼ =0.78) and Di= 10−2(case C3, Fig. 5.13b). Thus, he wa e spa ial scale is smalle , hough o he same o de , han he o ical low scale. No e ha he o ical D(due o he o he wise balanced o ex) is ze o a z= 0. The maximum ampli ude o Dcaused by he o ical mo ion is |Db|max = 0.20 a he end o he ini ializa ion pe iod (Fig. 5.13c), which is abou 20 imes la ge han Di. Howe e , he ampli ude o he e ical wa e mo ion is wi= 0.06, which is 10 imes la ge han he ypical mesoscale QG e ical eloci y wq( ha is, wqis abou 10−3 imes he ho izon al o ical speed). The mos no iceable esul is he de o ma ion o he ini ially s aigh phase lines o wo he g a i y wa es (Fig. 5.14). This occu s because he oscilla ing luid pa icles a e ho izon ally ad ec ed by he o ex gi ing a new local (absolu e) wa e equency ωlwhich is he Dopple shi ed pa icle (in insic) wa e equency ωpby he o ex mo ion, acco ding o ωl=ωp+Kh·ubh .(5.13) Since |Dz|max = 0.2, we ha e N∼ =N, and hus ωpis app oxima ely homogeneous. Consequen ly, ωlis a ec ed mainly by he Dopple shi Kh·ubh. The an icyclonic o ical mo ion ubh =ubi+ bjhas ub>0 (ub<0) a y > 0 (y < 0), implying a posi i e (nega i e) 118 CHAPTER 5 (a) (b) Figu e 5.12: Dis ibu ions a = 4 Tbp o he e ical displacemen Da (a) iZ= 40 (z=−1.2,D∈[−18,4.2] ×10−2, δD= 1 ×10−2), and (b) iZ= 65 (z= 0,D∈ [−1,1] ×10−2, δD= 2.5×10−3). The PV con ou =−0.2 a z= 0 is included. The domain ex en is δx =δy = 2πc in (a), and δx =δy = 3.62cin (b). equency Dopple shi . Hence ωlinc eases (dec eases) in he no he n (sou he n) egion o he o ex, so ha he ini ially s aigh phase lines acqui e an an icyclonic pa e n. The de o ma ion o wa e phase lines caused by he o ex is no con ined o he o ical egion bu is ans e ed h ough all he wa e column (Fig. 5.14). As a i s app oxima ion we assume ha he o ical e ical mo ion can be neglec ed, so ha ∇ h·ubh = 0, since uih =0 o g a i y wa es, and he non-di e gence condi ion yields ∂w/∂z = 0. The e o e he ho izon al phase eloci y σh≡(−∂w/∂ )/|∇hw|is cons an along he wa e column, ∂σh/∂z = 0. Simila esul s we e ob ained o a cyclone wi h max = 0.75 and semi-axes ah/c =aZ= 1.5 in a g a i y wa e ield iden ical o he case abo e (case C4, no shown). In his case he ini ial con igu a ion has |D|max = 0.18, ha is, 18 imes la ge han Di. Con a y o he an icyclonic case he ini ially s aigh phase lines acqui e a cyclonic pa e n because now ωl dec eases (inc eases) in he no he n (sou he n) side o he o ex due o he Dopple shi equency (??). 5.5 Dipole-Wa e In e ac ion We add ess he e he in e ac ion be ween a o ex dipole, which, unlike he monopola o ex, possesses a ne linea momen um, and la ge ampli ude IGWs. Wi h ha pu pose we i s 5.5. DIPOLE-WAVE INTERACTION 119 (a) (b) Figu e 5.13: Dis ibu ions o w(w∈[−6.8,8.5] ×10−2, δw = 0.13) a = 26.8Tbp (a) in he x-yplane a iZ= 65 (z= 0) and (b) in he x-zplane a iY= 65 (y= 0). PV con ou s =−0.2 a (a) z= 0 and (b) y= 0 a e shown (dashed hick line). Domain ex en is x, y ∈[−π, π]c, and z∈[−π, 0]. desc ibe he low cha ac e is ics o he dipole (sec ion 5.5.1) ini ialized when pu e ine ial wa e (sec ion 5.5.2) o g a i y wa e (sec ion 5.5.3) ields a e included. 5.5.1 The dipole A submesoscale ba oclinic dipole is he e ini ialized, in he absence o wa es, as wo ellipsoidal PV dis ibu ions wi h max = 0.75 and min =−0.75, ho izon al semi-axes a± X= 0.6cand a± Y= 0.4c, and e ical semi-axes a+ Z= 0.4 and a− Z= 0.27 o he cyclone (+) and he an icyclone (−), espec i ely (case C5, Fig. 5.15a). The ini ial asymme y in he p esc ibed a± Zis due o he ac ha hese o ices a e de ined in he ini ial ( e e ence) con igu a ion which has la isopycnals. Du ing he ini ializa ion ime he isopycnals s e ch (sh ink) in he an icyclone (cyclone), so ha a he end o he ini ializa ion pe iod ( i= 5 Tip) he 126 CHAPTER 5 (a) (b) Figu e 5.20: Ho izon al dis ibu ions o wa iZ= 65 (z= 0) and a = 2.08 Tip wi h di e en wa enumbe s (a) (k, l) = (8/c, 0) (w∈[−8.2,7.3] ×10−2,δw = 18 ×10−2), and (b) (k, l) = (0,8/c) (w∈[−7.5,7.5]×10−2,δw = 18×10−2). Domain ex en is δx =δy = 4.32c. 5.6 Concluding Rema ks In his wo k we ha e nume ically in es iga ed he in e ac ion be ween idealized ba oclinic o ical s uc u es and p e-exis en plane ine ia–g a i y wa es wi h simila ho izon al e- loci y o isopycnal e ical displacemen ampli udes a he submesoscale. The e is a la ge numbe o di e en possible in e ac ions depending on he ini ial pa ame e s o he o ical s uc u es and IGWs, and we ha e no a emp ed o exhaus he e y la ge pa ame e space. Two main mechanisms a e usually in ol ed in his o ex-wa e in e ac ion. The i s mech- anism is he ad ec ion o PV by he wa es, which makes he o ical s uc u e uns eady and o ces i o be pe manen ly in a s a e o geos ophic adjus men , a he same ime ha i modi ies he uppe and lowe limi s o he IGW equency wa e band. The second mech- anism is he ad ec ion o wa es by he o ices, which changes he local wa e equency h ough he Dopple -shi equency ela ion. These mechanisms ope a e on submesoscale o ical s uc u es wi h Rossby numbe s close o, bu smalle han 1, which emain always s able despi e he la ge ampli ude wa es. A ema kable esul is he enhancemen o he o al e ical eloci y by an o de o magni ude when ine ial wa es a e p esen in o ical lows. This is a clea example o a 5.6. CONCLUDING REMARKS 127 non-linea o ex-wa e in e ac ion, which esul s in he gene a ion o igh -handed helical wa es.The e o e, he wa e equency anges a ini ial imes om he Co iolis quency o an e ec i e equency eand a ewa ds eaches also sup aine ial equencies due o esonance mechanisms. Finally, we ha e conside ed only in e ac ions be ween wo kinds o submesoscale o i- cal s uc u es (monopola and dipole o ices) and wo kinds o plane wa es (ine ial and g a i y wa es) and many o he in e ac ions emain s ill unexplo ed. Some examples a e he in e ac ion be ween localized wa e packe s o IGWs and submesoscale o ical s uc u es, and he long e m o ex ins abili y o hese o ical lows in p esence o an ine ia–g a i y wa e ield. We also lea e o u he esea ch he ca aly ic beha iou o o ical s uc u es igge ing IGWs. Chap e 6 Discussion This hesis aims o cha ac e ize plank on pa e ns associa ed o uns able je s and long-li ed o ices a mesoscales and submesoscales. Speci ically, he e ec s o ho izon al ad ec ion, e ical ad ec ion, and ecological isopycnal mixing a e nume ically in es iga ed. Addi ion- ally, he low esul ing om a pa icula kind o o ex-wa e in e ac ion is also analyzed due o i s likely ole a in oducing plank on he e ogenei y. We summa ize he e he main esul s aised in he con ex o he hesis pu pose. 6.1 Ecological ini ializa ion Fi s o all we add essed he non- i ial p oblem o ecological ini ializa ion using a h ee- a iable NPZ (Nu ien s-Phy oplank on-Zooplank on) ecological model (chap e 2). We sough e ical p o iles in s able equilib ium wi h he luid a es because he pu pose o his wo k was o quan i y he ecological esponse o physical dis u bances. To his end, analy ical s eady solu ions wi h a cons an p o ile o o al ni ogen NTwe e ound. We obse ed ha hese p o iles a e non con inuously di e en iable, which implies an e o sou ce when com- pu ing e ical g adien s in he ad ec ion e m o he Eule ian physical-ecological coupled equa ions. To o e come his e o , we ound nume ical solu ions h ough ime in eg a ion o a con inuously di e en iable e ical p o ile, which con e ged o a s able s a iona y s a e. In con as o hei analy ical coun e pa s, hese nume ical solu ions a e con inuously di e - en iable bu equi e e ical esolu ions o a ew cm, which a e compu a ionally un easible. Howe e , he nume ical e o in oduced when using la ge esolu ions is smalle han hose associa ed o unce ain ies in ecological pa ame iza ion. Thus, nume ically s able s eady- s a e e ical p o iles a e sui able o ini ialize one dimensional NPZ models, bu how a e hey implemen ed in o h ee-dimensions? Is NThomogeneous on ho izon al o isopycnal le els? Is 129 130 CHAPTER 6 NTpa chy o con inuously dis ibu ed in he domain? We obse ed ha hese ini ial condi- ions led o di e en plank on dis ibu ions (chap e 3). Thus ca e mus be aken in choosing p ope ini ial condi ions pe inen o he objec i es o he speci ic physical-ecological coupled easea ch. 6.2 Ho izon al and e ical ad ec ion The ole o ho izon al and e ical ad ec ion on plank on dynamics was in es iga ed h ough di e en h ee-dimensional ini ializa ions o NT. When NTwas assumed homogeneous on ho izon al le els we we e able o un eil he plank on he e ogenei y caused by e ical ad ec- ion. Ins ead, when NTwas homogeneous on isopycnals o ully-de eloped o ices, an ini ial seed o ho izon al he e ogenei y was c ea ed. Thus we we e able o compa e he e ec s o e ical and ho izon al ad ec ion on his ini ial he e ogenei y. We conside ed i s he o me ini ializa ion o quan i y he ecological impac o submesoscale e ical eloci ies associa ed o a ba oclinic uns able je (chap e 2). We obse ed ha phy oplank on anomalies de elop by e ical ad ec ion. Howe e , hese anomalies a e unco ela ed om e ical eloci y because phy oplank on esponds o he upwelling slowe han he imescale o ho izon al ad ec ion. Based on ha , we nex conside ed a submesoscale su ace o ex dipole whe e NTwas ini ialized cons an on isopycnals (chap e 3). This scena io could be concei ed as a consequence o an eddy pumping e en and led us o in es iga e i s long e m e olu ion. We obse ed wo di e en plank on dynamics spa ially di ided by he o ex sepa a ix, and hus by ou e PV isosu aces. Wi hin o ices plank on dis ibu ion is domina ed by ho izon al p ocesses. The gene a ion o plank on anomalies due o e ical ad ec ion was in- signi ican compa ed o p eexis ing plank on anomalies, which we e apped inside o ices and hus ansla ed a dipole phase speed. In con as , ou side o ices plank on he e o- genei y was in oduced by e ical ad ec ion, hough plank on biomass was dispe sed om upwelling egions, analogously o he je case. As a esul , a ail o phy oplank on de el- oped a he cyclone wake. When on al luid pa icles mo ed an iclockwise o he o ex ea hey we e upli ed. Since phy oplank on esponse o he pe u ba ion had some ime lag, biomass inc eased a he cyclone wake. Once he e, i decayed a cons an mo ali y a e. This mo ali y a e was app oxima ed o he pa ame ic alue conside ing a cons an ail ex en and dipole speed. Fi s ly, he dominance o ho izon al o e e ical ad ec ion on plank on dynamics when ho izon al g adien s o plank on exis was in ag eemen wi h L´e y 6.3. ECOLOGICAL ISOPYCNAL MIXING 131 (2003). Secondly, spa ial unco ela ion be ween e ical eloci ies and he plank on inc ease caused by hem had been al eady s a ed a mesoscales and submesoscales (L´e y e al.,2001; Lima e al.,2002). In his wo k, we ook a s ep u he by ep esen ing an scena io whe e bo h p ocesses occu ed, and sugges ed ha po en ial o ici y con ou s may ac as a spa ial di ide be ween hem. The abo e men ioned esul s ga e us insigh on he dynamics o a plank on pa ch pe - u bed by a subsu ace mesoscale dipole (chap e 3). We obse ed ha he dis an ac ion o PV de o med his pa ch in such a way ha a ilamen unned along he dipole axis, whe e ho izon al speed was maximum. Since his speed inc eased exponen ially wi h dep h, so did he ilamen elonga ion. An analy ical app oxima ion o his ilamen ex en was de i ed using a quasi-geos ophic model wi h o ices o gi en adius and cons an PV. As a esul o he nega i e e ical shea , a gi en phy oplank on laye was ad anced om he laye abo e, phy oplank on sel -shading dec eased a i s on , and plank on anomalies de eloped. Ho - izon al ad ec ion has been o en conside ed incapable o in oducing he e ogenei ies unless hey al eady exis (see e iew Ma in,2003), as in he p e ious dipole case. We de eloped u he his concep , showing an indi ec way h ough which e ical shea o ho izon al speed c ea es phy oplank on he e ogenei y. 6.3 Ecological isopycnal mixing We ha e obse ed ha ecosys ems ge apped a he in e io enclosed by he o ex sep- a a ix. This ini ial seed o he e ogenei y is o en in oduced by isopycnal doming when o ex o ma ion, such as occu s in eddy pumping. Since i s long- e m e olu ion emains la gely unknown we in es iga ed whe he i eaches a s eady-s a e conside ing h ee ac o s: isopycnal e ical displacemen D, ophic condi ion, and isopycnal di usion (chap e 4). In o de o explo e a wide ange o he abo e men ioned ac o s, an isopycnic physical- ecological coupled model, less complex han he AB-NPZ model, was i s cons uc ed. We conside ed a sphe ical mesoscale o ex in he QG space, whe e he low is in g adien balance, and hus, is s eady and ho izon al. As a esul , ecological dynamics is only e ically dependen , which le us o in oduce he mesoscale o ex o cing in o he NPZ model in e ms o D. To his end, an analy ical exp ession o Dwas ound by app oxima ing a polynomial unc ion o a known Ddis ibu ion. Finally, his physical-ecological coupled model was ini ialized using s a iona y e ical p o iles nume ically s able wi h he luid a 132 CHAPTER 6 es (sec ion 6.1). We obse ed ha hese p o iles always con e ged o a s eady-s a e wi h coexis ence o phy oplank on and zooplank on. When isopycnal mixing was no conside ed, phy oplank on biomass inc eased linea ly wi h D, due o an enhancemen o ligh i adiance, and loga i h- mically wi h o al ni ogen, sugges ing a sa u a ing esponse wi h nu ien s. Though hese esul s a e es ic ed o QG sphe ical o ices, phy oplank on biomass eached also a nea ly s eady-s a e in he submesoscale dipole case (Chap e 3). Thus hese esul s may gi e us some insigh in mo e complex cases ha conside h ee-dimensional luid mo ion. Ano he ca ea wo h no ing is ha we assumed a ully- ecycling sys em, while in ac ni ogen is los ac oss o ex bounda ies due o phy oplank on sedimen a ion. Sinking a es o phy o- plank on ange om 0.5 m d−1 o 10 m d−1in labo a o y expe imen s (Smayda,1970). This explains why o ices become nu ien deple ed a e a ew mon hs. When isopycnal mixing was conside ed, phy oplank on biomass was maximized a a cha ac e is ic isopycnal di usion coe icien K∗due o a esonance mechanism be ween di - usi e and plank on imescales. Two main e ec s we e in ol ed. Isopycnal doming induced phy oplank on g ow h, and hus nu ien consump ion. This c ea ed isopycnal g adien s o phy oplank on and zooplank on opposi e o hose o nu ien s, which esul ed in an ou - wa d di usi e lux o he o me s and an inwa d o he la e . As a esul , phy oplank on biomass inc eased di ec ly h ough an upwa d lux o nu ien s, and indi ec ly h ough a g azing dec ease, which was a consequence o a di usion o zooplank on as e han ha o phy oplank on. Phy oplank on esonan esponse o nu ien s and ligh has al eady been s a ed when including ho izon al (Pasque o,2005;McKi e and Neu eld,2011) and e ical (Huisman e al.,1999;Ghosal and Mand e,2003) di usions, indi idually. A s ep u he was aken he e by explo ing a ange o nu ien condi ions and o ex ypes conside ing bo h di u- sions. We obse ed ha he ophic egime de e mined he magni ude o he phy oplank on esponse, being meaning ul in meso ophic condi ions, in which he la ges K∗was ob ained. Con e sely, K∗was nea ly una ec ed by isopycnal doming. In his case, he phy oplank on g ow h caused by isopycnal upli o be e li le els was balanced by an enhancemen o phy oplank on ou wa d lux. We ha e s a e he e o e ha isopycnal mixing upwells nu ien s o be e li le els, bu can i balance he long- e m nu ien deple ion wi hin o ices? To answe his ques ion, we 6.4. THE ROLE OF PV IN VORTEX-WAVE INTERACTIONS 133 can compa e he imescales o he e ical componen o he isopycnal di usion coe icien wi h a cons an plank on sinking a e. The obse ed nondimensional esonan isopycnal di usion coe icien anged om 0.06 o 0.3, o oligo ophic o meso ophic condi ions, e- spec i ely. Conside ing he isopycnal le el wi h he g ea es slope, his in e al co esponds o a e ical di usion coe icien Kz∈[0.35,1.8] ×10−2. Es ima es o phy oplank on sinking a es a e abou wP=−0.65 m d−1(Spi z e al.,2003). This alue has been calib a ed o a coas al ecosys em and is chosen as an uppe bound o sinking a es in oligo ophic en i- onmen s, whe e phy oplank on sedimen a es slowe han in coas al sys ems. App opia ely adimensionalizing wPin o WPwe ob ain a |Kz/WP|- a io comp ised wi hin [0.78,4]. Thus isopycnal mixing could accoun o long- e m plank on subsis ence in meso ophic condi- ions. The alida ion o his g oss app oxima ion wi h he simple physical-ecological model cons uc ed is le o u u e esea ch. 6.4 The ole o PV in o ex-wa e in e ac ions In gene al e ms, we ha e s a ed ha plank on dis ibu ions a e ela ed wi h PV when ho izon al p ocesses a e dominan , which occu s i plank on he e ogenei ies al eady exis . Wi hin o ices plank on and PV dis ibu ions a e in phase since bo h p opaga e a o ex ansla ion speed. Ou side o ices, he dis an ac ion o PV de o ms plank on pa ches in a non-linea way. In con as , plank on is co ela ed o ho izon al g adien s o PV when e ical p ocesses a e ele an . Howe e , his co ela ion is only obse ed a ini ial imes since ecological anomalies a e as ly anspo ed by ho izon al ad ec ion a om he upwelling loca ion. Addi ionally, isopycnal mixing accoun ed o an inc ease in phy oplank on biomass close o he o ex cen e . Speci ically, his occu s whe e isopycnal e ical displacemen eached i s maximum, and hus whe e e ical g adien s o PV a e he la ges . In o de o ge an insigh on how o ex-wa e in e ac ion may al e plank on dynamics we in es iga ed he ela ion be ween PV and he low esul ing om a pa icula ype o in e ac ion (chap e 5). We conside ed sphe ical o ex monopoles, in he QG space, and o ex dipoles embed- ded in an ini ial pu e ine ial and g a i y wa e ield. When ine ial wa es we e in ol ed, a nea -ine ial igh -handed helical wa e was de eloped h ough a non-linea o ex-wa e in- e ac ion. Fi s ly, o ices shi s he ine ial equency o an e ec i e equency e= +ζ/2 (Mooe s,1975a;Kunze,1985;Rubens ein and Robe s,1986), whe e ζis he e ical com- ponen o he ela i e o ici y. Since he ini ial con igu a ion o ζwas spa ially dependen , 134 CHAPTER 6 ho izon al g adien s o ewe e gene a ed. Secondly, wa es o ced he o ical s uc u e o be in pe manen geos ophic adjus men h ough PV ho izon al ad ec ion. As a esul , PV con ou s emained no longe axysimme ic and unbalanced e ical eloci ies de elop, which co ela ed wi h ho izon al g adien s o PV. When as g a i y wa es we e in ol ed, he wa e ad ec ion by o ices caused a Dopple shi o he local wa e equency ωl. Thus he la ges ωlshi s we e obse ed whe e ho izon al speeds we e maximum, and hence whe e ho izon al g adien s o PV we e la ge. Ine ia–g a i y wa es (IGWs) ha e imescales om minu es o hou s, o en smalle han he phy oplank on g ow h imescale, which anges om hal a day o a couple o days. Howe e , a combina ion o ho izon al and e ical ad ec ion may allow plank on coupling o IGWs (F anks,1995b). In ac , some in-si u obse a ions mani es he ole o IGWs c ea ing he e ogenei ies (F anks,1995a;G ana a e al.,1995). The impac o he obse ed nea -ine ial spi al wa e on plank on dynamics is le o u u e esea ch. O e all, he p esen wo k con ibu es o cha ac e ize he h ee-dimensional plank on s uc u e associa ed o mesoscale and submesoscale o ices h ough PV. I is hough as a p ocess in es iga ion a he han an a emp o simula e any pa icula ecosys em. Ou esul s a e he consequence o some ini ial nume ical assump ions, which ha e been designed o widen he comp ehension plank on dynamics in he s a i ied and oligo ophic open ocean. Chap e 7 Conclusions 1. Ecological s eady-s a e e ical p o iles nume ically s able we e ound sui able o ini- ialize physical-ecological coupled models in he Eule ian desc ip ion. 2. Ini ializa ion o he abo e men ioned p o iles, homogeneous on ho izon al o isopycnal le els in he whole domain o in spa ial pa ches, led o di e en plank on dis ibu ions. Thus, h ee-dimensional ini ial condi ions should be ca e ully chosen based on he p oposed esea ch objec i es. 3. Vo ex sepa a ix di ided wo di e en plank on dynamics. Inside he sepa a ix, plank on dis ibu ion was domina ed by ho izon al ad ec ion and was in phase wi h po en ial o ici y (PV) since bo h ansla ed a o ex phase speed. In con as , ou side he sepa a ix, plank on e ical and ho izon al ad ec ions we e o he same o de o magni ude han he ecological o cing. Ve ical ad ec ion gene a ed plank on anomalies, which we e immedia ely anspo ed a away om he upwelling loca ion by ho izon al ad ec ion. As a esul , plank on was ini ially co ela ed wi h e ical eloci y, and hence wi h ho izon al g adien s o PV. 4. A phy oplank on ail was de eloped a he wake o a ansla ing su ace cyclone due o ho izon al ad ec ion, e ical ad ec ion, and phy oplank on in insic imescale. The ho izon al leng h o his ail was nea ly s a iona y and depended linea ly on he o ex p opaga ion speed and phy oplank on mo ali y a e. 5. A ba oclinic subsu ace o ex dipole de o med su ace ecosys em pa ches such ha a ilamen an along i s axis. The elonga ion o his ilamen inc eased wi h dep h because he e ical shea o he ho izon al speed was nega i e. As a consequence, phy oplank- on sel -shading dec eased a he ilamen on and phy oplank on biomass inc eased. 135