Internal-inertial wavesand cross-frontalcircula/ion in the upper ocean

JOAQUÍN TINToRE amI SERGIO ALONSOGeophysical Fluid Dynamics Greup. Dept. de Física

Universitatde les líles Balears07071 Palma,Spain

DONO-PINO WANGMarine Sciences Research Ceriter

State University of New YorkStenyBrook, NY 11794, USA

EMwío GARCIADept. de Física

U niversitat Autónomade Barcelona08193 Bellaterra, Spain

RESUMEN

Se estudiael procesode ajusteentredes masasde aguade propiedadesdiferentesy la circulación secundaríaasociada,por medio de un modelooceánicotridimensionalde ecuacionesprimitivas. En particular,se investigala recirculacióny el movimientovertical cercade un frentede densidadmuymarcado,haciendeénfasisen los efectosde las ondas internasiniciales, conel fin de conocersu influencia en la circulación normal, en doble célula,observadaen los frentesoceánicosde densidad.Se demuestraqueesasondas,generadasen el procesode ajustegeostrófice,jueganun papelesencialen laintensificaciónde la circulaciónnormal al frente y de la mezclaen los frentesde densidad. Los resultadosteóricos obtenidossen aplicablesal frente dedensidad,permanentey mu>’ intenso,observadoen la zonaoriental del Marde Alborán, al sur de Cabe de Gata. Este frente es consecuenciade laconvergenciade aguamediterráneay atlánticaen la zona,ideal paraverificarlos resultadosteóricosde esteestudio.

ABSTRACT

Using a titree dimensionalprimitive equationoceanmedel,we studytiteadjustmentof two ver>’ distinct water massesand tite associatedcirculatienpatterns.In particular,we investigatetite recirculatienand vertical motionsnear a very sitarp density frent witit special empitasis en tite eflects elinterna]-inertialwaves.Titegoau of titis study is te ciarify tite inipertance¿mdtite effectsof internal-inertial wavesen tite observeddeublecelí cress-lrontalcirculatien of oceanicdensity fronts. We slmw diese waves, generatedbygeestrephicadjustment,pía>’ an essentialrole andenitancebethcressfrental

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332 J. Tinturé, 8. Alonso, Dong-Ping Wang o E. Oa,-cia

círculationand-mixing-ineceaniedensityfrenos.Titesetheoreticalresultsareapplicablete tite verv intense and permanentdensity frent presentin titeeastcrnAlteran Sea.Ibis front develepssouthof CapeGataas a resnlt of titeconvergenceof Mediterraneanand Atlantic water asid is titerefozean idealplace te test dic theoreticalresults of titis study.

1. INTRODUCTION

Adjustrnent of a fluid under gravity occurs becauseof the naturaltendency te reach an equlibriuw. Titis problem was inutially studied byRossby (1937) whe was concerned with hew tite pressureasid velecitydistributiosis in the atrnesphereasid tite oceanevelve tewardsgeostrophicequilibrium. Rossby, using tite principie of conservation of potentialvorticity, was ablete fied the final equilibrium statc without knewing Useexact detajis of the transientamI higití>’ nenlinearadjustmentprocess.

Tite preblemwasreviewedby GUI (1982)¿mdení>’ a bricfdescriptionwilLbe given itere. lf a shallew, hemogeneeus,unbeunded,nen-rotatingfluid.initialí>’ at resand with a step-discontinuityin tite surfaceelevationcollapsesat ,‘ mc 0, a graviíy-w’ave freno is generatedte either side. lite final statehasa uevel surfacé, tite initial potential energy being cempleteJycenvertedtokinetic energy. la a rotating fluid, the fluid adjustsrapidí>’ (in a time of titeerder of-tite rotation period)te an equilibrium stateOhar is neO asiate el resoand ohat contains merepotential energy titan doestite rest state.Tite finalsurfaceelevatienis neo Ievel and tite initial discentinuityhasin tite final statea citaracteristicwidth, tite RessbyRadius of deformation.An upes-tasiOfeatureof this caseis that dic kinetic energyof tite final stateis ení>’ ene-titirdof tite petential energy released.Tite etiter two thirds, are cemmon!yassummedte be radiated away by tite gravity w-aves generatedin tite initialstageof tite adjustment.

Ihe linear adjustrnent of an initially motionless fluid with a verticaldensioyinterfacehasbeenstudiedamongothersby Csanady(1971, 1978),Gilu(u976), Ou (1984), Van [-leijst (1985)and Middieten (1987). A» oliesestudieswere mainí>’ cencernedwith tite final equilibrium sitape el the front. Titetransientshavebeenless invesoigated:Kae u aL (¡978) studiedolie circulatienassociatedwith a surfacebuevant influx and found that at thc headof titegr-avity cus-renO,rite frent is fermed it>’ intensesinking of fresit surfacewaoer.Wang (1984) examinedtite dow adjusrmentbetweentwo initial]>’ separatedwater massesand showedthat intensemixing at tite horizontal edgeof titegravity current slowed down tite freno speed.

¡u tite ocean,regionsof iritensified horizontal gradienis(frents) are eftenicuad betweeritwe different water bedies. Statienaryeceanfrentsrcpresenttite near equilibrium soate reached between twe different - waters. Aremarkableexampleof titis type of frents is tite Almeria-Orán freno in theeasterííAlboran Sea(Tintoré «iaL, 1988). Ibis ¡bat developssouthof CapeGata. is alníost statienar>’and representsthe nearequiuibriuni statereached

13. In¡ernal-ineríial waves and rross-Jronlal c.ir.ula¡ion... 333

between tite lighter Atlantic water and tite denser Mediterraneanwater.Associatedwitit tite densitygradient,awell definedsoutheastwardbareclinicjet with velocities around 100 cm/s has beenalso described.

Density frents in tite oceanare well known regionsof increasedbiomass.Titesehigit levels of biomasshavebeen usualí>’ related te tite existenceof aweak, almest permanentcroes-frontalnon linear circulatien (James,1978;Van l-Ieijst, 1985; SimpsonandJames,1986). Brink (1987)indocatedtitat titeexistenceof across-frontalcirculatienas tite «deublecellocirculationpatternproposedby Moerese¡ aL (1976) itas been a controversia1subjet over titeyears. Numericalstudies(Kao e¡ aL, 1978, James,1978, 1984) itave sitowntitis typeof celí patternswitit surfacecenvergence,upwelling anddownwelling.However,tite cross-frontalvelecitiesobtainedseemfar toe smallte acceunt(alone) for tite ebserveditigiter biological activity.

Tite fore-mentionednumericalstudiesaddressedtite questoenof tite nearsteady state circulation but did net censidertite effects of tite oscillator>’motion also presentin an>’ adjustmentpreblem. In fact, inertial escillatiensappearassociatedwitit tite long wavesgeneratedduring tite initial pitaseoftite adjustment(GuI, 1976). Since titeseoscillationsdo not prepagateaway(zero group velocity) tite>’ are ver>’ difficult te damp,and titeir role in titeadjustmentof different fluids sheuld be considered.

Tite aim of titis studyis te investigatetite adjustmentot twe very distinctwater massesand tite associatedcirculation patterns.Mere specifically, titeobjective is te investigatetite recirculation of tite buoyant fluid and titeverticalmotions neara very sitarp densityfrent witit specialempitasisen titeeffectsof gravity-inertial waves.

In a complex systemsucit as tite atmosphereor tite ecean,we need teconsidera simpleexperimentte try te understandtite complicateddynamicsassociatedwitit tite adjustmentte equilibrium. Weitave titereforecensideredan ínitial state in witicit two ver>’ different watersare initialí>’ separatedby avertical barrier. A main differencebetweentitis studyandWang(1984) is thatwitile ite investigatedtite mixing processesat tite front during tite initial stageof tite adjustment,we are interestedin tite near steady-statecross-frontalageostréphiccirculation and en tite effects of inertial-gravity waves. Alsosince titis study was suggestedby tite cross-frontal circulation patternsobserved in the Almería-Orán front (Tintoré et aL, 1988); tite densitydifferenceis 1.5 1V (g/cm3)andtite ligitt waterdeesnet reacit tite bettomoftite basin.Tite recirculation in tite upperlayers(witich is what we are realí>’interestedin) is titereforeless influenced by tite bettominduced circulation.Te investigatetite effect of bottom friction. and its influenceen dawpingthefluctuatingpart of tite cresfrontalcirculatienwe itave studiedtite adjustmentin two different basins. In tite sitallow case(lOO m deep) tite ligitt watereccupiestite upper80 m witile in tite deepene(500 m) tite ligitt water eccupiesthe upper 300 m. Titis last case can be reugití>’ assimilated witit titeadjustmenttaking place in the FasteraAlberan Sea.

334 J. Tin¡oré, & Alonso, Dong-Ping Wang o E. García

2. mE MODEL

Tite numericalmodel used itasbeendescribedby Wang(¡982, 1984) andenly a brief descriptien is given here fer cempleteness.It is a titreedimensionalprimitiveequatienmedel but a two dimensionalversion of titemodel is usedin titis studyfor easierinterpretationof tite results.Tite medelis titen two-dimensienal in a vertical plane, representsa cress-sectienperpendicular te tite front and allews tite developmentof tite currentcemponentin tite along-frent direction. We liave also assuwedtitat titemetion is itydrostaticand water is incomprenssible.Titerefere,in cartesianceordinates,tite governingequationsare:

do + 00± dUf ia~~ a 1 au\a( Ou \ (1)

d¡ dx Oz P dx dx k A&~—j+—~A ½)dv dv dv udp al a~fl al Ov\__ ±u—+ w—+fu=--——+— IAh— 1+—lA— ¡ (2)d¡ Ox dz POy dx Ox) 8~k

Op (3)+ pg mc O

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Ou 8w• (4)

dp dp Op _ Of dp\ 8/ dp’~dx +~k K-á--, (5)

8t Ox Oz 8x1 Oz

where t is time, x, y and z tite spacecoerdinates,u and y tite x-and y-compenentsof tite currentvelecity (efíshere,alengshere),p is density, f titeCerielis parameter.Horizontal edd>’ coefficientsare constant,

A» mc Kb = L0~ cm2/s (6)

Beundar>’cenditiensare:

At tite free surfacetites-e is no surfacestress:

A~ (Ou/dz,dv¡Oz) mc 0, - (7a)

K,dp/dzmc O (7b)

AA tite eceanbottom tites-e is bettemfrictien:

A1 (du/dz, dv/dz)mc — A (u, y) (8)

where A is a linear dragceefficient (.1 cm/s).

13. Internal-inertial wavesandcross-fton¡alcirculation... 335

At tite coast,a free slip conditien is assumed(tite normal flew vanishes);

umc0 (9a)

Ah (Ou/Ox, 8v/Ox) = 0 (9 it)

1<,, Op/Ox= O (9c)

At tite openecean,ambientconditionsare specified as

(lOa)

p = p0 (z) (1Gb)

witere ~ is tite free surfaceelevationand p0 is tite ambientdensityprefileTite vertical eddy ceefficientsare tite Munk-Andersentype (cm2/s):

= AV” (1 + IORi)—1/2, (II a)

= K1,o (1 + 3.33Ri)—

3’2, (lIb)

where Ri is tite local Richardsonnumber,

Op

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PO(12)

E( Ou\2±(OvVOz7 \ Oz7 j2

atid

AV~~ = K~I’ mc 50 cm2/s

Initial conditions:we imposea very steepinitial densitydistribution. TitetotaJJenghtof our basinis 120kmasid tite two watermassesareseparatedbya vertical wall, located at x mc x,, mc 72 kw. Tite density difference is¡Ap mc — mc 1.5 u, 0.5 JO—~ g/cm3).

p — P(, x (14)

mc P~, x> x,,

Tite numericaltechniqueused is describedin Wang (1982, 1984) and isstandard for titree dimensional multi-level models (Simmens, 1980). Inessence,equationsu-5 arewritten in finite differenceform in a staggeredgrid,figure 1. Tite sciteme is leapfrog it time, centred in space(except for tite

336 J. lYnu,ré, 5. A tenso, hong-¡-‘¡ng Wang y E. García

Z =0

Z=H

1- igure ¡~—1 ocation el [he basic variables on the fíníte díffercnceU grid.

densio>’ equationwitere an upwind scitemc is used)anduses a mode-splittecitnique in tite vertical direction asid a semi-implicit sciteme in titeiterizental directiente acitieve computationalefliciency.

3. RESULTS

3.1. Non rotatíngcase

Titis case, appropriate fer ad~oístuwi processestaking place in shertperieds(cg. river plumes,- --) is p usenLcd te cmpitasizetite role of retation in

k— AXH

13. In¡ernal-iner¡ial Wavc’s aoci <ross-frontal circulation... 337

anyadjustmentproblem.Tite lock betweentite two watermasseswasreleasedato = O andwe itave follewed tite evolutionof tite adjustmentuntil tite frontreacitedtite epen beundar>’.

31.1. Basinloúmdeep

Tite flow ma>’ be citaracterizedby a velocity scaleU mc (g’it)1¡2, witere itis tite deptit of tite inflewing fluid and gis tite reducedgravity. In titis case,U = 107 cmls witile tite surfacefront prepagatesataspeedof 50 cm/s, figure2. Titereforetite frenopropagatesat italf tite gravity wavc speed,a resultthatwas aIread>’ feund by Wang (1984) who alse discussedit in terms of titeenergytransferduring tite adjustment.It is importantte observetitat tite ligitofluid is catchingup tite leadingedgeof tite frent, therefereinducing a strongcenvergenceand downwardmotien. Porexample,att mc 15 it, tite speedoftite frontal bere was 85 cm/s (a, mc 27.25) witile tite speed of the 28.0isopycnalwas significantly lewer, 35 cm/s. Tite maxíwum vertical velecityobtainedwas0.2 cm/s.

3.1.2. Basin500 ni deep

Since tite deptit of tite basin is new 500 m and tite deptit of tite bueyantfluid is 300 m, we expect tite surfacecenvergencete be strengerand as aresult larger vertical velocities can be reacited.The numerical simulationsitows tite prepagatienof tite front is faster and after lO it, tite front itasalreadyreacitedtite limitsofourdomaingiving aspeedof l6Ocm¡s,figure 3.Por comparisen,tite citaracteristi velocity in titis case is 207 cm/s. Titevertical velecitiesinducedare also extremel>’itigit, around 1 cm/s.

Comparisenbetweentitese twe casessitews tite essentialeffect of titedepthof tite buoyant fluid (it is actualiytite fractionaldepthh/ 1-1 which is animportantparameter),a result witich wasalreadyfound in a tankexperimentby Benjamin (1986).

3.2. Rotating case

Tite appearancein tite solution of ver>’ intense inertial escillations is adirecocensequenceof tite drastic initial conditions.Tite adjustmentbetweentwe fluids has been generally studied in sitallew basins and inertialosciJiatienswere titen almost aiwaysdampedafter a few inertia] periods. Ittitis sectienwe sitow tite significant influence of titese escillatiensen titerecírculationof tite buoyant fluid.

Tite lock bctweentite Owo water masseswas releasedat t mc O h and weitave followed tite evolutien of tite adjustmentduring lOO it, after whicit titepositien of tite Freno is almost invaríant. Physically, during tite initial stage,

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Figure 3—Cross-fronta[ circulatien asid density distribution in thenon rotating case.deepbasin during Ohe adjustmentprocess(until lO h).

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¡£ZVSS ERONT DISTAN~ <Ka)

ADJUSTT4fl~T AFPER lh (nosi—rotating fluid, 1 .5 sqt, lcm=200 oT~/s, .25

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Figure 4— Horizontal cross—frent velocioy dríring rite adjustrnenr process (usitil t 500it).

tite evelutien is similar te tite non retatiug case:a gravity current is fermedas a resuir of tite horizontal pressuregradicur induced by tite buoyancydiffercnce. Wioit rotation hewever,tite ferward metienis gradualí>’deflectedtewardstite rigitt, tite propagationof tite frent slewsdewn and a soationar>’state in quasigeestrepiticbalanceis attained(Kae et al., 1978).

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A ncarsreadvstate is reachedafter appreximaoelylOO it of simulatien.Tite position of tite fronr is titen almest soationar>’, altitougit inertialescollationsaresoilí presenoin tite velecity field Titis is particularí>’ evident intite evolution of tite horizontal cross-frenovelecio>’, figure 4, whicit geesthreugh a sequenceel escillaoionsof decayisigamplitude. Ato = lOO it, titeescíllaoiensare soilí very significant, ami en!>’ att mc 450. it, a weak (.2 cm/s)brít permanenocros-frontalcirculatien is esoablished.

It is iniporoant te note that aftes- lOO h, tite steady stare reached (it isacouallya pracoicalcitoicefer cquilibriuw) is still ver>’ mucit influencedby titeelltcts of tite gravity-inertiauwavestitat inducehorizontal velecitiesat lcastan erderof magnirudeitigiter titan tite mean.Figure 5 presentstite evolutienof tite adjustmentever>’ ¡O it and [he sorong influence of titese inertialoscíllatiens en tite cross—freíítal ci rculation is clearí>’ demonstrated.Ar

/3. Iníernal-inertial waves and cross-fronial circula ¡ion... 34!

= 100 it, tite alongfront circulation, figure 6, featuresa surfaceintensifiedjet (40 cm/s) witit a widtit of approximately20 km.

3.2.2. ¡Jeepbasin

Like in tite previouscase,tite positienof tite frontis almostinvariantafterLOO it andtite crossand along-frentcirculationsarealsestrongly affectedbytite inertial oscillations. At t mc 70 ansI 90 it (figure 7a) tite streng surfaceconvergenceinducestite subductionof tite ligitt fluid. Titis downwardmetienis clearí>’ diapycnalin tite upperlayersbut tends te becomealong-isopycnalbelow 200 m. Tite recirculation patternsitews a celí xvith strong upwardmotion en tite side of tite ligtit waters.Tite patternsobservedat t mc 80 andJOO it, figure 7b, shewtite samefeatureswitit reservedsings, tite ciretijatienbeing now eppositete witat was found in figure 7a becauseof tite stronginertial oscillations (period around 19h) titat dominate tite cress-frentalcirculatien.

Tite alongfront circulation induced by such an intensedensity front isfermed by an intense baroclinic jet. At t = 90 it, tite mean along-frontvelocity was around 70 cm/s and tite widtit approximately40 km. TitebaroclinicRossbyRadiusof deformationRd = NO/fis 20km and titereferetite result obtained agreeswell witit tite expectedtiteoretical result. It isimportant te empitasizetite ordersof magnitudeof tite velocities induced.Maximum valuesfor tite cross-frentalvelecity are of tite order of 50 cm/s,witile titey are around75 cm/sTer tite a]ongfrontvelocity. Maximum verticalvelocities of tite order of .3 cm/s are reacitedat areund200 m.

4. DISCUSSION

Tite presenceof a recirculating pattern close te tite «deuble cellocirculation prepesedit>’ Mooerse¡ al. (1976) is wititout an>’ deubt, figure 7.A similar type of recii-culation wasfeund by James(1977, 1984) and Wang(1984) but tite induced velocity fields were at least an order of magnitudesmallertitan tite enesobtaineditere. An essentialpoint is thatwiten titis cres-frontal circularion is real»’ significant(t < >00 it), it is ves-ystrongí>’modifiedby tite gravity-inertial oscillations(tite fluetuatingpart)generatedduring titeearly stageof tite adjustment.In otiter words, tite permanentcross-frontalcirculatien in tite sense of James(1984) is strongly dominated it>’ titefluctuating internal-inertialoscillations.Tite intensevertical metionsobtainedin titis numerical study suppert tite observatiensof regions of surfacecenvergenceor divergence often feund in frontal regiens (Simpson andJames,1986; Tintoré e¡ aL, 1988; Beuchezet aL, 1989).

Our resulis show the adjustmentiii a sitallow basin is ciearly differentfrem tite adjustmentin a deepene. In titis last case,tite itigher depth allowsfer itigiter vertical velocities witicit cempletel>’ distert tite initial density

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Figure 6.—-Alongfront velocity asid density distribution with rotation and in a shallowbasin att = OOh.

distribution at lower levels. Near tite surface,tite front is alse less sitarp as aresult of tite ver>’ intensevertical metions.Titis suggeststitat witen titis typeof adjustmenteccursin deepbasins(like in tite EasternAlboran Sea,whereMediterraneanandAtíantie watersconvergesoutit of CapeGatawitit deptitsaround1500 m) ver>’ strongverical velecitiescan be reacitedas a result of titenear-inertialmotien. Even in shallow seas(as a result of tidal forcing) titeeffectsof titese inertial oscillatienshaste be consideredsince tite asseciatedvertical motienscan itave a significant influence en the increasedbiomassfound in tidal fronts.

Tite equilibrium stateebtainedin titis numericalsimulation is very closete tite situationdescribedby Tinteré«taL (1988)for tite easternAlteran SeaAlmeria-Oránfront. Por a v¡eakerinitial densitygradient,tite final positionof tite surfacefront wpuld itave beencloser te x« (On, 1984). Titerefore itappearsfrom titis numerical study titat tite formation of titis ver>’ strongdeusity can be explained in terms of tite geestropiticadjustmentof titeModified Atlantic Waterandtite MediterraneanWatertitat collapsesoutit ofCape Gata. For a weaker initial distribution titere are still citancestitat asitarp frent would form becauseof tite centribution of otiter pitysicalmecitanisw(not includedin titis simulation)sucit asconvergence,existenceofa itorizontal deformatienfield (Heskinsasid Bretiterten, u972).

5. SUMMAI{Y AND CONCLUSIONS

Tite dynamics of tite initial adjustmentbetween twe different watermassesis representativeof tite formation of frents it mostsiteuf regiensasidof tite type of precessesecurring in straits.Tite final steadystatedynamícsis

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meregeneraland is prebablyrelevantte alí kinds of densityfronts. Tite ver>’sharpinitialdensity distributionimposedin titis numericalstt’dy gives risc tea steadystatecitaracterizedby a sharpdensityfrent witit a permanentandver>’ slow ageostrepiticcirculatien.However, we itave found titat in tite earlvstagesof tite adjustment(T mc 5/E), titis cressfrontalcirculation witich depietsa «deubleccli» pattern, is strongly niodified it>’ tite internal-inertial wavcsradiatedduring tite initial adjustínent.Tite velecitiesebtainedwere at leasteneorder of magnitudeitigiter titan iii previcussiudies. clearly showing titefundamentalimportanceof theseinertial escillations.Their rouewill itave tebe censideredte fulí>’ understandtite regicus of sus-faceconvergenceordivergenceor tite higl-íer biomassciten feun in frental regicus.

Cemparisenof titesetiteoretical resultswith ebservatiesisin tite AlboraííSeaindicatestitat tite fermationof tite permanentdensityfrent in tite EasternAlboran Sea can be explained in terms of tite geostropitie adjusimeníofModified Atlantic Water asid MediterraneanWater titat cellapsesoutit ofCapeGata.Tite intense vertical wetions ebtainednumericalí>’ alse supporttite ebservationsof very close regleusof suriarecenvergenceansIdivergencewith strongverticalniotions andsignificant biemasslites-case.A quantitativestudyof tite influence of inertial oscillatiens on uppereceancírculationasidmixing is not yet av-aliable.mis study is neededfor a betterunderstadingoftite dynawicsof eceaniefrontal regiens.

Acknowuedgements

Titis werkwaspartialí>’ fundedit>’ tite DGICYT (PB89-0428),tite MASEFC prograw under preject FUROMODEL (O043~-C) asid tlíe CICYTProgramaNacionaldc RecursosMarinos>’ Acuicultura(MAR 89-0550)andMAR9O-l229E.

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209-48.Bonchez, 1., E. Ibáñezasid L. Prieur(¡987): DaiLy and seasonalvariations si tite spaíial

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