dynamics of the north brazil current retroflection

7/29/2019 Dynamics of the North Brazil Current Retroflection

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. C12, PAGES 28,559-28,583, DECEMBER 15, 2000

Dynamics of the North Brazil Current retroflectionregion from the Western Tropical Atlantic ExperimentobservationsIlson C. A. da Silveira and Wendell S. Brown 2Ocean ProcessAnalysis Laboratory, Institute for the Study of Earth, Oceans, and Space,University of New Hampshire, Durham

Glenn R. FlierlDepartment of Earth, Atmospheric, and Planetary Sciences,Massachusetts nstitute ofTechnology, CambridgeAbstract. Hydrographicand velocityobservations f the North Brazil Current(NBC) retrofiectionegionduring he 1990-1991Western ropicalAtlanticEx-periment WESTRAX) are examinedwith the intent of extractingdynamicalinformation about the NBC eddy shedding. A comparison s performed betweenthe depth structure of empiricalorthogonal unctionsand the dynamicalnormalmodes of the NBC retrofiectionregion on a beta plane centeredon 5N. Thebarotropicand first two baroclinicmodesaccount or about 75% of the verticalstructure of the NBC flow. Thus, a three-layer model of the NBC region is suitablefor dynamical studies of the NBC retrofiection. In terms of flow structure theupper layer represents he retrofiectionof the surface ayersof the NBC to feedthe North Equatorial Countercurrent. The middle layer represents he separatingsubthermocline waters of the NBC which feed the North Equatorial Undercurrent.The third layer representsa weak meandering low that may be thought of as theDeep Western Boundary Current signature n the three-layer ocean. In terms of PV,a well-defined ront separates he NBC waters from the North Equatorial Currentin the upper layer. Both upper and middle layers present closed PV contoursassociatedwith the eddy in the processof pinching off from the retrofiecting NBC.The NBC separation region, although equatorial, complies easonablywell with thebasicquasigeostrophicQG) assumptions. hereforeQG methodsare applied oinvestigate the NBC meander growth. By isolating the effect of PV anomalies ineach of the layers on each of the layers, baroclinic growth is verified to occur duringthe NBC eddy shedding.1. Introduction

The Johns et al. [1990] analysisof moored cur-rent meter measurements and composite satellite im-ages provided convincing evidence that large anticy-clonic eddies pinch off from the North Brazil Current(NBC) retrofiectionat about 7N. The fact that the

Nowat Departamentoe Oceanografiasica,nstitutoOceanogr&fico, Universidade de Sho Paulo, Sho Paulo, SP,Brazil2Now at School of Marine Scienceand Technology, Jni-versityof Massachusetts/Dartmouth, ew Bedford

Copyright 2000 by the American Geophysical Union.Paper number 2000JC900129.0148-0227/00/2000JC900129509.00

NBC retrofiects around two anticyclones to feed theNorth Equatorial Countercurrent NECC) at surfacelevels nd the North EquatorialUndercurrent NEUC)at subthermocline levels was already known from hy-drographic ata analysis Cochrane t al., 1979;Bruce,1984;Bruceand Kerling, 1984]. However, achof theseeddies centered t about 4N and 7N) was hought obe a semipermanent feature.Following ohns t al.'s [1990] indings, therobserva-tional efforts documented the NBC eddy-shedding phe-nomenon.Didden and Schott 1993]analyzedsea, ur-face height anomaly maps obtained from Geosat altime-try and tracked the NBC retrofiection eddies to dis-tances as far as 1000 km from the NBC retrofiection.Their estimates of average translation speeds were of15cms 1. Richardsont al.'s 1994] nalysis f surfacedrifters and SOFAR floats at 900 m during 1989-1992

28,559


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28,560 SILVEIRA ET AL.: NORTH BRAZIL CURRENT RETROFLECTION DYNAMICSalso revealed five retroflection eddies. They found thatthe eddies had mean widths of 250 km at the surfaceand 140 km at 900 m. Average northwest translationspeeds ere9 cm s-..Modeling efforts have also examined he NBC eddy-shedding henomenon. ratantoniet al. [1995] uccess-fully simulated the generationof the NBC retroflectioneddieswith the U.S. Navy nonlinear, primitive equa-tion, layeredoceanmodel orcedby climatologicalwindsand a prescribed hermohalinemeridionaloverturningcirculation MOC). Theseauthorsverified hat at least2-3 rings per year separated from the modeled NBC.Because the model failed to shed eddies in the absenceof an imposedMOC, Fratantoni t al. [1995] uggestedthat the increasedboundary current transport due tothe MOC is an important factor n the NBC ring for-mation.

Ma [1996]usedanalyticaland numericalmethods nan equivalent-barotropic lamework to further investi-gate the formation and translation of the NBC eddies.He found that the formation of the eddies was due tothe short Rossbywaves,nonlinearities, nd the westernboundary. The translation mechanismsseemed o becaused y interaction f the eddieswith the boundaryand the fi effect.Thus the NBC eddy-sheddingrocess as beendoc-umented, ut there s little if any understandingf theNBC retrofiectionobegrowth nd heeddydetachmentprocess tself. There is also virtually no informationavailableon the NBC dynamicalstructure from obser-vations. The purposeof this paper is to use observedfields o extract dynamical nformation about the NBCretroflectionand its eddies. In particular, we addressthe following questions:

1. What is the dynamical modal structure of theNBC?

2. What information about the NBC flow evolutioncanbe nferredrom hepotential orticitymaps?3. What are the dynamicalprocesseseading o theNBC retrofiection xtension nd eddydetachment?We chose the 1990-1991 Western Tropical Atlantic

Experiment WESTRAX) synopticdata set to addressthe issues raised above. This data set consists of fiveship surveys with simultaneoushydrographicand Pe-gasuscurrent profiler measurementsFigure 1). Thereader s referred o Brownet al. [1992] or a descriptionof the WESTRAX program. Bub [1993]and Bub andBrown 1996]presented detaileddescriptive nalysisof the first four hydrographic/observedelocityWES-TRAX fields. These authors showed that the WES-TRAX cruises aptured he NBC eddy shedding t dif-ferentstages.The WESTRAX 1 (WX1, January31 toFebruary 9, 1990)and3 (WX3, January 6-30,1991)cruises measured a retracted NBC front and eddies al-readypinched ff. The WESTRAX 2 (WX2, September

17 to October 4, 1990) cruiseresultsrevealedan NBCretrofiection n the midst of sheddingan eddy. TheWESTRAX 4 (WX4, June 20 to July 3, 1991) cruisehappenedat the time of the onsetof the surface ayerretrofiectionand showed lessclear picture with a muchweakerNBC apparentlyshedding n anticyclone. n thepresentarticle we will focuson the analysisof the WX2and WX3 cruises, s representingwo differentstagesof the NBC eddy retrofiectioncycle.This paper is organized a.s ollows. In section 2 wedecomposehe NBC into dynamicalmodesof a hydro-static, Boussinesq cean on the fi plane. In section3 we discuss he NBC dynamical fields in terms of athree-layermodel. In section4 we stretch he quasi-geostrophic heory to apply it to the NBC retrofiectionregion and search or possible nstability mechanismspresent in the NBC eddy shedding. In section 5 wesummarizeour findings and presentour conclusions.2. Modal Structure for the NBCRetroflection Region

In this sectionwe seek the NBC bulk dynamicalstructure. The NBC observations re decomposedntodynamicalorthonormalmodesof a primitiveequationmodel under the hydrostaticapproximation. This as-sumption is justified based on the typical dimensionsof an NBC eddy: a radius of about L-200 km and avertical extent of 7/=1 kin.

The WESTRAXPegasus/CTD tations10 ..................: - : , ' :, ............: ... , - ......... .......... .. .........

9 ,

6 ''....".-I 3 5z 4

3

1 0 .....53 52 51 50 49 48 47 46 45 44 43

W LongitudeFigure 1. The WesternTropicalAtlantic Experiment(WESTRAX) Pegasus/conductivity-temperature-depth(CTD)stations uring he oursurveys. he thickline representshe actual shapeof the 200-m sobath,and the light gray shadedarea s its rectilinear pprox-imation.


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SILVEIRAET AL.' NORTH BRAZILCURRENTRETROFLECT1ON YNAMICS 28,5612.1. Model Formulation and the NBCDynamical Modes

LeBlond ndMysak 1978] howhat the linearized,inviscid, ydrostatic, oussinesqquationsanbe sep-arated into vertical and horizontal structures:p - p(, y, t)r(), (a)

The DEOF method s analogouso the time domainEOFcalculationf Davis 1976] ndKlinck 1984]. erewe seek vertical structure functions analogous o thedynamical odes. ll deep andv vertical rofilesreusedas the input data to obtaina solution f the type[, ] - [u(), ()]r() (5)

The Westrax N in cph[Jx,y, ), Vi(x,y, )]Fi(z), (lb) o' ' ' 500 r() ()w(,,t)() .where7i,bl,Vi,ndVVaretheith_modepressure,ona1000Ivelocity,eridionalelocity,nderticalelocitym-1500Ilitudes, espectively,nd Fi is the ith-mode erticalmodeorigenfunction).hessumptionfbasic001tratifications ustifiedby considering Brunt-Vgis/ilgPrfileN2(z)--g/PoOa/OzestimatedfrOmtRAX four-cruiseverageFigure2, top) with verticalresolutionf 0 .ndividualruiseveragesndar-000ial averageswhich onsiderednlyWESTRAX cruisesof differentseasons) ere very similar o the global]V2(z)veragendotnfluencingheesultsfheomputationo be describedext.The erticalodesatisfy 4000 '5 10 15N in cph

xr() - 0, (2)dz N 2 z) dzwith the required iat bottomand top rigid id a, sump-tions: () : 0 - 0,-, (a)dzwhere f = fo + 3y, with a central atitude being5N and hus o = 1.27x 10 s s , / = 2.27x 10 (m s) x), and he depth s H -4000 m.The ith-mode eigenvalue i relates o the ith-modeinternal deformation radius Rdi by

Ai - (foRdi)2 (4)The orthonormalized ertical eigenmodesor the firstsix modes re shown n Figure2 (bottom). The numer-ical values or the correspondingeformation adii andequivalent epthsare presentedn Table 1.

2.2. NBC Statistical ModesNext we aim to identify the dynamicalmodes hataccount or the bulk velocitystructure n the NBC re-gion.To addresshis ssue, ecompute epth mpiricalorthogonalunctionsDEOF) for stations f the WX2cruisewith depthexceeding500m. TheseNBC statis-tical modeswill then be comparedwith the dynamical

modes.

The FirstSix Dynamical odesVerticaltructure0

i /I .1!/t ' , .... Zeroth3000- /'xl ; First' /" ! --- $ocondI - I\ ! Third3500 - / I FurthI li ...... Filth4000 ' ' I; , .4 2 0 2 4 6

PressureEigenmode mplitudes

lOOO

1500

2000.2500

Figure 2. The 10-maverage runt-Vgisglgrequencyprofile or the fourWESTRAX surveystop), and thevertical modal structure for the barotropic and first fivebaroclinicmodes bottom).


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28,562 SILVEIRA ET AL.: NORTH BRAZIL CURRENT RETROFLECTION DYNAMICSTable 1. The First Six Deformation Radii and theCorrespondingEquivalent Depths for the WESTRAXAverage Brunt-Viiisiilii ProfileaMode Rd, km he, (nu'2'r2)m

0b 15,584.6 40001 197.7 0.642 126.2 0.263 74.1 O.O94 54.6 O.O55 44.4 0.03

aThe central latitude and ocean depth considered re 5Nand 4000 m, respectively.bThe values or Rdo = /fo and heo = H are usedinstead of the Rdo = heo = oo obtained from the eigenvaluecalculation.where/} ) x,y, ) andV} ) x, , ) are heorthonor-malizedigenfunctions,ndFi(Z)(z)are he dimen-sional vertical structure functions.The first and secondDEOF modes,explaining78%and 11%, respectively,f the variance f the verticalstructuren WX2, are shownn Figure3 (solid ines).TheFi(Z)(z)modesfor - 1,2)arenondimensional-

ized by their norms to be of the sameorder of magnitudeas the dynamical modes displayed n Figure 2.A visual comparisonbetween the dynamical modes(Figure 2) and the statisticalDEOF modes Figure3)suggests hat DEOF modes 1 and 2 are dominated bydynamical modes l and 2, respectively. For a quan-titative check on this qualitative statement, we opti-really fit each of the first six dynamical modes ndivid-ually, a combination of the first three dynamical modesand a combination of the first six dynamical modes tothe DEOF modes. We also calculate the normalizedroot mean square differences,as defined by Pinardi andRobinson1987],between he DEOF modes nd the re-spective fits using

rms((Fjz)-'i51ziJFi))(Fj(E))2) 'whereai j is the weightobtainedn the optimal ittingof the ith dynamical mode onto the jth DEOF mode.From Table 2 the first (second)dynamicalmode a,c-counts or about 45% (35%) of the verticalstructureofDEOF mode 1 (2). The individual mode fits also n-dicate that the three gravest dynamical modes are themost important in accounting for DEOF mode 1 verti-

-500

- 1000

c: -1500

-2000 -

-2500 -

3-dyn.mode fit on DEOF mode 1i i

rms= 0.21

3-dyn.mode it on DEOF mode20 , ,

-500

-1000

-2000

-2500

rms= 0.28

' ' -3000 -3000 -2 0 2 4 6 -2 0 2 4 6Normalized DEOF mode 1 Normalized DEOF mode 2Figure 3. The vertical tructuref the WX2 depth mpiricalrthogonalunctionDEOF)modesleft)1 and right) . Thedashedine epresentsheoptimalit of he irst hree ynamicalmodeson the DEOF modes. The rms difference aluesare indicated n the lower right corner ofeach panel.


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SILVEIRA ET AL.' NORTH BRAZIL CURRENT RETROFLECTION DYNAIVllCS 28,563

Table 2. Root Mean Square Differences for the Opti-mal Fits of Dynamical Modes Onto the DEOF Modes] and2

Fitted Mode(s)rms differences

DEOF Mode 1 DEOF Mode 2

The vertical divergenceOw/Oz is estimatedby consid-ering a vertical length scale of = 1000 m, and thevertical velocity w evaluated byDr/w- Dr' (9)

0 0.93 0.99I 0.55 O.832 O.93 O.663 0.99 0.964 0.99 0.99$ 0.99 0.99

0-2 0.21 0.280-5 0.03 0.06

cal structure. Regarding the fit using the sum of thesethree dynamical modes, we can see that the rms dif-ferences re generally ess han 25% (seealsoFigure 3,dashed ines). The weightedsum of the first six modesnearly reproduces the vertical structure of both DEOFmodes.

2.3. Three-Mode Approximation to the NBCRetroflection RegionWe now approximate and present the structure ofthe WX2 flow structure with the first three dynami-cal modes, based on the results of section 2.2. We mapthe NBC velocity fields from the sparseWESTRAX ob-servationsFigure 1) using he objectiveanalysis OA)scheme f Brethertonet al. [1976].The horizontal velocity field can be decomposed ntonondivergent and irrotational parts, that is,

f- (u,v) - kxV0-where 0 is the stream function, X is the velocity po-tential, and V is the two-dimensional gradient opera-tot. The objective analysis procedure gives an estimateof the stream function, assuming he divergent flow isnegligible althoughone can show hat it is equivalentto a scheme or estimating both under the assumptionthat the divergent and rotational flows are not corre-lated and the covarianceof X is small compared o thecovariance f 0).The relative sizes of the velocity potential and thestream function can be estimated by noting that

where r/is the depth deviation of a given isopycnalsur-face rr relative to its hydrostatic depth. An upper boundvalue for r/takes place at the pycnocline for nonuniformvertical stratification, with r/ following w values. Bub[1993]estimatedstandarddeviationsof 40 m for therr=27.0kg m 3 isopycnalwith a hydrostatic epthofabout 350 m). If r/ is normally distributed,95% of itsvalues should lie in 2xstandard deviations or 80 m.Considering an advective timescale, an upper bound

scale for w is

w - O r/ - 5X104 ms This estimate implies that the divergent part of the ve-locity is only about 10% of the nondivergent ne. As weshall see ater, interpolation errors are larger than thedivergent velocity field itself. Therefore it is reasonableto approximate 6) by

xOther authors also chose to represent the NBC ve-locity structure approximated by its horizontally non-

divergentpart. For example,Bub [1993]describedheWESTRAX velocity fields n terms of stream functions,disregarding the divergent part based on interpolationerror and measurement oise arguments. Ma [1996]modeled the NBC by its nondivergent velocity field tosimulate successfullyhe NBC ring shedding.The relationships between ith-mode velocity ampli-tude functions nd follow rom (lb) and (10):

v,] or' % 'with i = 0,1,2 for the present case.

The b/, and Vi values a.t one particular location aredeterminedby projecting he ith dynamicalmode ontothe u and v profiles.Mathematically, his is

X V'X wOv (7)The vorticity can be estimated by consideringa. near-surfacemaximum tangential velocity of a.n NBC eddyin the processf pinching ff, U - O(1 m s-l), and alength scaleof L=200 km. This yields

0VO --)-X 0 S--1. (s)To complete the information required for mappingthe WX2 stream function directly from the Pegasus e-locity data, we must also specify he form of the corre-lation function. This is estimatedby bin-averaging llthe correlation airs n the data, following he method-ologyof Carter and Robinson 1987]),and fitting themto an assumed sotropic Gaussianshape given by


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28,$64 SILVEIRA ET AL.: NORTH BRAZIL CURRENT RETROFLECTION DYNAMICS

C(r) - (1 )er2/c2 (13)where = v/x2+ y2 is the radialdistance, 2 s therandom sampling error variance, and lc is the correla-tion length. As shown n Figure 4 (top), best fits areobtained for lc 400 km. All the OA gridded fields aremapped with this correlation length scale. The inter-polarion error field, which dependssolelyon the stationlocationsnd he [lc,2] arameteret, s presentednFigure 4 (bottom) and is representative f the WES-TRAX fields.

We force the interpolated velocity maps to satisfy thezero-normal-flow condition at the western boundary byemploying the method of images. As shown n Figure l,the almost rectilinear 200-m isobath is approximated bya straight line and regarded to be the line of symme-try. Then the velocity observationsare symmetricallyprojectedon the land side (i.e., their mirror images)ofthe WESTRAX domain before their input to the OAscheme.The total stream function field at a given level z = z.is obtained by directly OA mapping the velocitiesmea-sured at z.. The three-mode approximation requiresthat the amplitude functions L/i and i for i - 0, 1, 2be calculatedusing (12). The modal stream functionamplitudes are shown in Figure 5.Figure 6 shows he maps or ) (left panels)and thecorrespondinghree-mode pproximation right panels)at three different evels:50 m (in the mixed ayer), 350m (embeddedn the thermocline),and 1000 m (at thelower boundaryof the NBC eddy activity). The three-mode approximation is very similar to the total streamfunction field. It even captures the northwestward tiltdescribed riginallyby Bub [1993] n the retrofiectioneddy and also observed n the total stream function field(Figure 6, left panels).

(4)where the hats are used o distinguish he layered modelfrom the continuously stratified model quantities. The/5 referso heamplitudef he th modentohe thlayer.

The vertical orthonormal modes n an N layer systemare found by solving:

z_lgHz-tj -igHi )-jFj 0,where Hi is the rest thicknessof the ith layer and i =(a,+ -cr,)/fi are the densitystepsbetween ayers andi + 1. The assumed values of e0 = N = oc are used toeliminate the extraneous stretching terms.The eigenvaluesare found by requiring that the Nequation system has a nontrivial solution: the determi-nant of the N x N matrix formedby (15) is zero. The/ functionsre heorthonormalizedigenvectors.Theamplitudeunctions3 are ound sing

HAn obvious consequenceof the conditions above isthat an N layer model is able to resolveN layered dy-namical modes. However, layered models must be cal-ibrated to reproduce quantitatively the correct physicsof the continuously tratified NBC [Flierl, 1978]. Inparticular, the fundamental ength scales i.e., the de-formation adii) of the modelmustbe similar o thoseof

the continuously stratified model. This means that therest depths Hi and the layer densitiescrimust be chosencarefully o make the eigenvaluesf (15) approximatelythe sameas thoseof (2). We havesomeguidancen thismatter, as we shall see later.3. NBC Retrofiection Region as aThree-Layer Ocean

When a given current system can be reasonablyap-proximated by a. small number of modes, a more usualapproach is to employ the so-called layered model in-stead of a continuouslystratified configuration. As theresults of section 2 indicate, this seems o be the caseof the NBC systemwhere the first six (three) modesaccount or 90% (75%). In such ayeredmodels, hevertical density structure s simplified o a step-likepro-file with successiveomogeneousayers. Consequently,the Brunt-Vi, s/51/5rofile becomes sum of Dirac deltafunctions and there is no velocity vertical shear withinthe layer.3.1. Model Formulation

FollowingFlierl [1978], we assumea separationofvariables n analogy o the solution or the continuouslystratified model:

3.2. Layered NBC Models From the LiteratureIn section 2 we have shown that the NBC vertical

structure is dominated by the first three dynamicalmodes, while a six-mode NBC reproducesvery closelythe total velocity ield (seeTable 2). Coincidentally rnot, two of the NBC layered models found in the liter-ature are six-layer models.Fratantoniet al.'s [1995]six-layerprimitiveequationmodel successfully eproduced he NBC eddy shedding.In Table 3 we present the H, and or, values used bythose authors. We then apply these values n the eigen-value problem (15) for N = 6 and f0 for 5N. We useH=3000 m to be consistent with our assumption of afiat bottom, 4000-m-deep oceanwithin the WESTRAXregion. The results or the deformation adii (Table 3)agree reasonably well with those obtained for the con-tinuouslystratifiedocean Table 1). Becausehe exter-nal deformation radius does not depend on the strati-fication, it is perfectly matched. The first internal de-formation radius differs from the "true" one by only


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SILVEIRA ET AL.' NORTH BRAZIL CURRENT RETROFLECTION DYNAMICS 28,565

1

0.5

0

Isotropic,Gaussian CorrelationFunction or WX2! ! i !

Ic=411.5m 2=0.0776

-0.5 ......... ' ' '0 200 400 600 800lags in km

1000

The WX2 Error Field10 , ,. ,

9

3

2 ..........

-52 -50 -48 -46 -44W Longitude

Figure 4. The sample correlation unction for the WX2 Pegasus elocity magnitude at 100 m(top). The correlationength c is 411.5km and he random ampling rrorvariance2 s 0.077.The typical normalized root mean squareerrors associatedwith the OA horizontally griddedfields .bottom),using he data.of the stationsdisplayedn Figure 1 and the OA parametersabove. The asterisks epresent he WESTRAX CTD/Pegasus station locations.


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28,566 SILVEIRA ET AL.' NORTH BRAZIL CURRENT RETROFLECTION DYNAMICS

6

WX2 Mode 0 Stream FunctionAmplitude

o -52 -5o -48 -46WX2 Mode I Stream FunctionAmplitude

8

65-J4

1

-52 -50 -48 -46 -44WX2 Mode 2 Stream FunctionAmplitude9876 -0.

z 3,2

0 -52 -50 -48 -46 -44W Longitude

Figure 5. The stream function amplitude functions ifor the first three dynamicalmodes top) i-O, (middle)i=l, and bottom) =2. Unitsare 1x 10 m2 s . R,eadle+05 a.s1 x 10s.10%. According to our analysis n section2 the NBCdynamical structure is dominated by the first baroclinicmode. Thus the approximately correct Rd within theNBC region might have played a role in the agreement

of Fratantoniet al.'s [1995]NBC eddy-shedding odel-ing with observations. The average differencebetweenthe six deformation radii in Table 3 and Table 1 is 20%.Bub and Brown's 1996]descriptivemodelverticallysliced the NBC region according o water mass crite-ria,. Using the WESTRAX data set, they defined a

150-m-thicksurface ayer, split the intermediate ayersinto four layers, and defined a deeper ayer extendingfrom 1500m to the bottom. The Bub and Brown 1996]model values for Hi and ai are shown in Table 4. Thesame eigenvalue calculation done for the Fratantoni etal. [1995]model is carried out for Bub and Brown's[1996]model. The resultsobtained or the first threebaroclinicmodes Table 4) differ from the "true" onesin Table 2 by only about 5%. Thus the vertical struc-ture of Bub and Brown's 1996]model,basedsolelyonwater mass dentification, seemsalso very appropriatefor six-layer dynamical studies of the NBC.3.3. WESTRAX Three-Layer Model

The analysis n section2 indicates hat 75% of theNBC vertical structure is explained by the barotropicand first two baroclinic modes. Therefore we probablydo not need six layers o capture the essence f the NBCdynamics, and a three-layer model should be adequate.We now approximate the WESTRAX observations o athree-layer configuration, map stream function and po-tential vorticity for each layer, and seek nformation ofthe flow evolution via. the interpretation of the quasi-synoptic maps.

3.3.1. Vertical structure parameters and dy-namical modes. To build a three-layer WESTRAXmodel,weestablishcalibrationchemehichorces - A and 2 - A2. The mathematicaletails fthe calibration scheme are presented in Appendix A.The three-layer calibra.tion procedure requires the in-put of A, A2, and the rest depths H_, H2, a. d H3. Wechoosehe Bub and Brown 1996]surfaceayerdepthofH=150 m. The choice of subthermocline layer depthH2 is based on the zero crossingdepth of DEOF mode1. As seen in Figure 3, DEOF mode 1 changessignbetween 800 and 1700 m. We then choose H2:850 m.With a deep layer thickness of H3:3000 m, our fiatbottom ocean depth H = H + H2 + Ha is 4000 m.The calibrationprocedure see Appendix A) yieldstwo possiblesets of density jump values:

setl - 2.32x 10 3. , 2 - 0.76x 10 3set2' - 3.96x 10 3 2 --0.45 x 10 3We then compute "observed" a values by averagingthe mean WESTRAX density profile over the three lay-ers. The same density profile with which we generatedthe N2(z) (Figure2a) and computedhe eigenvaluesand eigenmodes in the continuously stratified model

(Figure 2b) is used. The observed valuesare com-puted from the observeda values, yielding


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SILVEIRA ET AL.' NORTH' BRAZIL CURRENT RETROFLECTION DYNAMICS 28,567

WX2Pegasust50m9 -08 . '' .7

5 '"'L-..-.J4Z 3 '.

.'. :...._. . .-52 -50 -48 -46 -44 -50 -48 -46WX2Pegasust350 n WX23-modet350 n

6

2 -.-'. --- ,':":"?1 ,,,,-.:,..,.,.,--"-'"'.%...-52 -50 -48 -46 -44

WX2Pegasust1000m9 ...... .7 0 .5.4z 3

0

WX2 3-modet50m

.... . :-52 49876

:35..J4z 3

.

0 -52 -50 -48 -46 -44WX2 i/3_modet 1000m

-52 -50 -48 -46 -44 -52 -50 -48 -46 -44W Longitude W Longitude

Figure 6. Comparisonetweenhe (left) Pegasus-derivedtream unction nd the (right) corre-spondinghree-moderuncation .t he (top) 50 m, (middel)350m, and (bottom)1000m levels.Read le+05 as 1 x 105.


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28,568 SILVEIRA ET AL.' NORTH BRAZIL CURRENT RETROFLECTION DYNAMICSTable 3. The Fratantoni et al. [1995]Model' RestThicknesses,Layered Density Values, and the Corre-sponding nternal Deformation Radiii Hi, m ai, kg m a Rdi, km1 80 24.97 179.42 170 26.30 91.33 175 26.83 61.24 250 27.12 45.75 325 27.32 34.36 3000 27.77 -

2.85x 10 a , e2 0.68x 10 aWe choose the set of calibrated e values which betterfits the observedvalues above. In the present case thefirst set is a clear choice.

The values or the calibrated 41, 42, and 43 are foundby choosing 3 to be the observed 3 and using he cal-ibrated e values o get the other two a values. The ca.l-ibra, ed density profile is shown n Figure 74. The cor-respondingdynamical modes or the three-layermodelare displayed n Figure 7b.3.3.2. Stream function and potential vorticityfields. We compute the stream function in each ayerby a procedure similar to that described n section 2to map the stream function on the continuouslystrati-fied model as a truncation of the first three dynamicalmodes. The obvious difference is that here the veloc-ity amplitude functions are obtained by projecting thediscrete modes of Figure 7b onto the u and v profiles.We apply the model for WX2 (fall 1990) and WX3(winter 1991) surveys,which are only 3 monthsapartand, as discussedy Bub 1993], robably epresentwodifferent stages n the NBC eddy-sheddingprocess.The stream functions in the three layers for WX2 andWX3 are shown in Figure 8. The first layer shows heretrofiection of the surface layers of the NBC to feedthe NECC. In the second ayer, the separatingsubther-mocline waters of the NBC feed the NEUC. The thirdlayer flow consistsof a meandering,basicallysouthwardmoving current that may be related to the Deep West-ern BoundaryCurrent (DWBC) signaturen the three-layer approximation. Consequently,we can see hat theNBC system and its eddy structure are confined o theupper two layers. The velocities in the middle layerare about a fourth of those in the surface ayer in bothsurveys. In addition, the layered retrofiectionof theNBC is reproduced n this three-layer approximation:the subthermocline waters retroflect at more southernlatitudes 3-5N)han the surface aters 6-8N). sreportedby Bub and Brown 1996], he latitudinaldif-ferences n the NBC retrofiection between upper andsubthermoclineayers s somewhatsmaller during theeddy-sheddingrocessFigure 8, left). It can alsobeobserved hat when the NBC retracts after shedding he

eddy,as n WX3 fields Figure8, right), a cyclonic ir-culation exists in the subthermocline ayer, northwest-ward of the retrofiection bulge. This feature, which isabsent during the eddy-sheddingphase, has been re-ported by Cochrane t al. [1979],Bruce and Kerling[1984],and Bub and Brown [1996], he last referringto it as the "FrenchGuiana,Low." Bub [1993] oundamore coherentand strongercutoff eddy than seen n ourmaps. Combiningwater massanalysisand estimates orthe eddy translation speed,he suggestedhe WX3 eddywas probably the same one observedduring WX2.The potential vorticity fields are computed byI i+ fhi , (17)

where hi is the ith layer thickness. We map (i directlyusing an OA scheme imilar to that used or the streamfunction seeHua et al. [1986] or details). The thick-nesses f the three layersare delimited by a rigid surfacelid, the topographyof the a=26.10 surface, he a=27.45surface, and the idealized flat bottom sequentially. Thehydrostatic depths of the a:26.10 and 27.45 surfacesare 150 and 1000 m, respectively. Thus the rest thick-nesses orrespond o H=150, H2=850, and Ha = 3000m. We acknowledge hat the topographicbeta is largerthan the planetary beta. next to the continental slopeand due to the presenceof a geological eature referredas to the "Amazon Deep Sea Fan." Aside from thosefeatures, the assumed lat bottom is a reasonableap-proximation in the WESTRAX interior domain. Weshould also mention that the "vertical wall" approxi-mation to the continental border is necessary o satisfythe zero-normal-flowcondition at the boundary.It is convenient to express the potential vorticity(PV) fields n vorticity units. Hencewe redefine hePV expression ccording o

- - f0. (18)The constant 0 is subtractedout of (18) since t is notdynamically relevant.The Ij fieldsorWX2 andWX3areshownn Fig-ure 9. A PV front defining the region of confluencebetween he retroflectingNBC and the flow associatedwith the cyclonic urn of the NEC water is evident inTable 4. The Bub and Brown [1996] Model' RestThicknesses, ayered Density Values, and the Corre-sponding nternal DeformationRadiii H, rn cry, g m-3 /d, km1 150 24.13 194.62 440 26.97 129.93 240 27.28 71.14 445 27.48 37.55 225 27.74 32.26 2500 27.87 -


11/25

SILVE[RA ET AL.' NORTH BRAZIL CURRENT RETROFLECTION DYNAMICS 28,569a

-5OO

- 1000

- 1500

-2000

-2500

-3000

-3500

-4000 20

The WESTRAX 3-layer model24.76

0=27.07

27.84

I I I I ,,22 24 26 28 30density1000 g/m

b

-500

-1000

-1500

-2000

-2500

-3000

-35OO

-4000

The Mode 0(dotted),1 solid), (dashed)Amplitudes,

, - - Radii of deformation'

(in km)

- Rd0= 15584 -

Rd1=197.7

Rd2=126.2

i I i-2 0 2 4 6Streamfunction igenmodeNormalizedAmplitudes

Figure 7. (a) The calibratedWESTHAX densityprofileunder the three-layerapproximation,and (b) the correspondinghree discretedynamicalmodes.Mode 0, dotted; 1, solid;2, dashed.the upper-layerPV fields Figure 9, top). It separatesa region of low PV to the south from high PV to thenorth.

In the WX2 upper layer (Figure 9, left top), thechangen M1 overonedeformationadius ndacross

the sheddingddy s 1.74x105 S 1 (f0 is 1.27x105s 1). In termsof strengthhe upper ayerPV frontgradient is about 4.8/.The beta, term seems to dominate the PV fields in

the secondand third layers. Nevertheless, n the WX2


12/25


WX2 Layer 1 Stream Function9 ...65 3 . .- ..:.., ....-52 -50 -48 -46 -44

WX2 Layer2 Stream Function' , , ,

0 258 2

654 I

1o -52 -50 -48 -46 -44

wx2 Layer 3 Stream Function8

5

3

,-52 -50 -48 -46 -44W longitude

WX3 Layer 1 Stream Function

76

54 2z32 0 -52 -50 -48 -46 -44

WX3 Layer 2 Stream Function876

4z ? ,..,21

-52 -50 -48 -46 -44WX3 Layer 3 Stream Function9

87 ?;::.. 0156 '54

3

1o -52 -50 -48 -46 -44W longitude

Figure S. The three-layer tream unction ields or (left) WX2-and (right) WX3. (top) Layer1, (middle)a,yer , a, d (bottom)a,yer . Unitsa,re x 105m2 s l-. R,ea,le+05 a,s x 105.


13/25

SILVEIRA ET AL.: NORTH BRAZIL CURRENT RETROFLECTION DYNAMICS 28,571

WX2 Layer1 fullPV WX3 Layer1 full PV9 9 ,87 76 6

25 25.3 32 21 1 1

-52- L - f - -449 98 87 7

5 - 5

0 0-52 -50 -48 -46 -44 -52 -50 -48 -46 -44WX2 Layer3 full PV WX3 Layer3 full PV9 9

8 87 76 6

5 .-:---. ....... 5

0 0-52 -50 -48 -46 -44 -52 -50 -48 -46 -44W longitude W longitudeFigure 9. The potentialvorticity fields,a,sdefinedby (18), for (left) WX2 a, d (right) WX3.Unitsa,re1x 10 5 s . (top) Layer1, (middle) a,yer , a, d (bottom) a,yer . Read e+05 a,s1x 105.


14/25


PV fields Figure 9, left), both upper and middle ayersexhibit closedPV contours associatedwith the sheddingeddy. In terms of strength he middle ayer PV front isabout 1.3fi.We have said earlier that Bub [1993]and Bub andBrown [1996] nterpreted he WX2 and WX3 veloc-ity fields as representing wo sequentialstages n theNBC eddy-sheddingphenomenon. WX2 captured theNBC in the processof sheddingan eddy. WX3 docu-mented an already pinched-offeddy on the northwestcorner of the WESTRAX domain and a regeneratedNBC retrofiection. These authors used water mass anal-ysis and kinematic arguments o support the above in-terpretation. We can corroborate their statement byexamining the dynamical fields of Figures 8 and 9. Ifwe assume hat potential vorticity is conserved, he PVcontours n Figure 9 representmaterial lines. A steadystate is reached when the PV contours are parallel tothestreamlinesi.e.,hj- Ij(l}j)). Thereforeegionswhere streamlines cross PV contours demark advectiveregions where the PV and flow will be changing withtime. Following this rationale, in Figure 10 we overlaythe } field dashedines) nd omeIcontourssolidlines). Two regionswith larger crossing nglescan beidentified: the northwestern border and the southeast-ern part of the pinching-off eddy. In the former regionthe retrofiection bulge will be pushed northwestwardby the NBC flow. In the latter region the cross-frontalvelocities will close additional PV contours, suggestingthat eddy detachmentwill occur sometime ater. If wenow eexamineheWX3 ) field Figure , top ight),

WX2 Full PV and Stream Function fields

6 '\ ,,,. r .

. \ \\\% \ \ \

.,.:...-=.........:::l.l:,:...lll ........'.l3[''.'":l::.l "'::!l''t'' XXX''4X......, ll.......... :.'4:;l:*... l C': '. :l.ll'.'.RI'J::'.":ll?Fo::':I.I0-53 -52 -5 -50 -40 48 -47 -4 -45 -44

W longitudeFigure 10. The upper-layer WX2 potential vorticity(thick olidines, ontourntervals0.5x10 s 1) andstream unction ields dashed ines,contour nterval s0.2x10 m2 s-l).

we see that the eddy has migrated northwestwardandthe whole PV structure seems to have been stretchedin the samedirection. This picture s dynamicallycon-sistentwith the previousanalysis,despite he fact thatfriction s important close o the westernboundaryandPV conservationmay be only a crude approximation.We acknowledgehat the northwestward ropagation fthe eddy may be enhancedby other processeshan ad-vectionby the NBC flow, as for example, the eddy inter-actionwith its image $ilveiraet al., 1999].Quantifyingthe eddy motion is beyond the scopeof this paper. Thishas beendone by Fratantoniet al. [1995] rom modeloutputsand Bub [1993] rom the WESTRAX data set.

4. Analysis of the DynamicsNext we continue using a PV-based analysis to at-

tempt to extract dynamical information about the NBCmeanderextensionand eddy detachment rom the WES-TRAX observations.QuasigeostrophyQG) more eas-ily enablesus to employ classical heory to examine theNBC dynamics. However, the QG theory requires thatthe shallowwater form of the PV expression17) be lin-ea.rized, ssuming he flow is essentiallygeostrophic ndthat the changes n layer thicknesses re small. Cer-tainly, since we are dealing with a near-equatorial re-gion, we should show that the QG is reasonably appli-cable to the NBC retrofiection before we proceed withthe dynamical analysis.4.1. QG Assumption in the NBC RetrofiectionRegion

The kernel of the QG approximation is that theRossby umber Ro) is small. Three differentapproxi-mations re required seePedlosky1979]or Gill [1982]for a completederivation).4.1.1. Geostrophic approximation. This ap-proximation requires that the horizontal velocity isapproximated y its geostrophicomponent9accord-ing to - v + O(Ro), (19)where G -/; x v p (20)pofoWe can verify the validity of the first approximationin two different ways. The simplest is to use scalingarguments and compute Ro as the ratio between theCoriolisand inertial/local terms n the horizontalmo-mentum equations. Using the scalesof section 2.2, thisestimateyields Ro = U/foL=0.36.A more thorough examination of the geostrophic p-proximation, one which involves the full range of spa-tial scales, can be made by comparing the observedvelocity shear between two levels to the correspond-ing geostrophic shear derived from the geopotentialanomaly fields. We compute


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SILVEIRA ET AL.' NORTH BRAZIL CURRENT RETROFLECTION DYNAMICS 28,57312 -

wherei s the geopotentia.1nomaly or dynamicheight)at the ith level relative to the surface, that is,c) = 5vdp , (22)

in which 5v is the specificvolume anomaly.The left-handsideof (22) is obtainedby directly OAmapping the Pegasusstream function "differences"be-tween levels I and 2 from the observedvelocity shear.The right-hand side is obtained by OA mapping thegeopotentia.1nomalies normalizedby the Coriolispa.-rameter) derived from the WESTRAX hydrographicdata set. The mean of the latter field is removed beforeinterpolation and the method of images is also appliedto satisfy the =const boundary condition.We chooseone level near the depth of maximum ve-locity and away from the Ekman layer (100 m) andthe other at a. depth of substantially reduced velocities(1200m). The resultsare shownn Figure 11. The 100-to 1200-m Pegasusand geostrophicstream functions areremarkably similar. The major difference between thetwo maps is the strength of the Amazon eddy in thesoutheast side. It turns out that this difference is lo-cated inside of the region of the larger OA interpolationerrors Figure 4b), and furthermore, ur assumption fa constant (the 5N value) s moresuspect onear theequator. The normalized rms differencebetween the ve-locity magnitudes of the two fields of Figure 11 is 0.31(or 31%) if the 0-IN latitudinalband s excluded.Asshownn Figure4b,'one hird or moreof this differencecan be attributed to interpolation error.4.1.2. The "beta plane" approximation. Thisapproximation requires that

: (23)f0We have assumed he secondapproximation n sections2 and 3. However, we can explicitly evaluate it usingthe scaling a,rgumentsof section 2.2. For L=200 km,a length scale correspondingnot only to the averageradius of the NBC eddies m200 kin) but also to thefirst baroclinic adiusof deformationn the region seeTable 2), and a.central atitude of 5N, we get

f0 - 1.27x1_05 s , - 2.27x 10-TM m s)-'Ro = L = 0.35.f0

While not small by midlatitude standards, the beta,term is about a third of f0 and significantly ess han 1.4.1.3. Layer thickness approximation. For a

continuously stratified ocean, this approximation re-quires that

Oz02aa= O(Ro), (24)where - 2(a) representshe hydrostaticdepth of agiven sopycnal , and z - z(x, y, a, t) are the fiuctu-

WX2 100-1200 Pegasus Streamlines

5

2

-52 -50 -48 -46W longitude

-44

WX2 100-1200 Geost. Streamlines,,,

8

7

5

4

3

2

-52 -50 -48 -46 -44W longitude

Figure 11. Oompa,risonbetween the 100- to 1200-m (,) g,u r,m faio difrr, ,d (b)geosrophic stream function differences for the WX2survey. Contours n l xlOs m2 s . Read ]e+05 a,s1 x l0 s.


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28,574 SILVEIRA ET AL.: NORTH BRAZIL CURRENT RETROFLECTiON DYNAMICS

ations from 2. However, it is perhaps simpler to thinkof the latter in terms of a layered QG ocean, yieldingO(o) ,

where 5Hi are the fluctuations from the rest thicknessHi We can check he third approximation y computingthe standard deviations for the upper two layer thick-nessesof the three-layer model. For WX2 the upperlayer standarddeviation std) is 26.2 m, while the mid-dle layer is 68.9 m. Within 95% confidence, ssuminga normal distribution for the layer thicknesses,we es-timate Rio by 2 x stdi/Hi. We then obtain Rio=0.37for the upper layer, and Ro=0.16 for the middle layer.Note that the upper layer estimate is remarkably closeto the Ro estimate for the first approximation.The previousanalysis ndicates hat the QG assump-tion is borderline but still usable in the WESTRAXregion. There is evidence in the literature which corrob-orates this statement. Qualitatively, it was the similar-ity between he climatologicaldynamicalheightmapsofCochranet al. [1979] ndBruceandKerling 1984] ndthose derived from the WESTRAX Pegasusobserva-tionsby Bub and Brown [1996] hat motivated he cal-culationsdescribed n the previousparagraphs. Quan-titatively, by comparingGeosat-derived eostrophic e-locities o results rom simulationsby the World OceanCirculation Experiment Community Modeling Effort(WOCE CME) [Schott ndBSning, 991],DiddenandSchott 19921 stimated hat the NBC retrofiection e-gion low structurewas about 80% geostrophic.Fratan-toni et al. [1995]alsoestimateda Ro=0.2 for the NBCeddies,an indication hat they are nearly as geostrophicas the Gulf Stream rings.4.2. QG Potential Vorticity Fields

LeBlond and Mysak [1978] showed hat the samemethodologyemployed for the hydrostatic, primitiveequationset (equation 1)) separateshe vertical andhorizontal structure in the QG system: the same formof separationof variablesand samevertical mode equa-tions are valid. The additional step is to assumea smallRossby number which idetifies the pressurewith thezeostrophic stream function.We choose o discuss he QG system in terms of thethree-layer model. Given that the QG vertical modesare also oundby solving 15), the stream unction ieldsshown n Figure 8 also represent the geostrophic ields.However, he PV expression17) is linearized,yielding

shown n Figure 9 resideon the stretching erm. To ob-tain Figure 9, we used the layer thicknessesmappedfrom hydrography.Here the stretching erm (last termon the right-handsideof (26)) representshe order Rochanges n the rest thicknessesHi and is mapped fromthe previously gridded stream function fields.The results for WX2 and WX3 shown in Figure 12reveal patterns identical to the PV fields in Figure 9.For the upper layer fields, the rms differences,defined

as

rms- /((lIiare 0.36 for WX2 and 0.39 for WX3. Also, PV gradi-ents are dynamically more important than the actualvalues of PV itself, and we should look for cross-stream(or cross-frontal) V gradients.We then estimate herms differences etween he meridionalgradientsalong44Wof I and to be 0.29 or WX2 and0.22 orWX3. Both calculations are consistent with the esti-mates in section 4.1, in which the ageostrophic ermsare of 0(15'o) 0.35 in the WESTRAX region.The cross-frontal changesover one deformationra-diusaresimilar o the PV maps Figure9) andare arge:1.37f0. However, his violation of the QG assumptionsis not unusual;Hall [1985], or example,modeled om-parable cross-frontal alues or the Gulf Stream, a mid-latitude current where the QG approximation s moretraditionally accepted.The variation with latitude along 44W is dis-played in Figure 13. It can be seen hat the regionof maximum PV gradient is between 4N and 6N.The meridional variation of the three componentsofthe PV field (the relative vorticity, the beta. erm, andthe stretching erm) are alsoplotted in Figure 13. Therelative vorticity is about 20% smaller than the stretch-ing term. However, the cross-frontal change n relativevorticity is 2 times larger than that of the stretchingterm.

We can again compare our results to those of Hall[1985].Unlike the NBC configurationFigure13), theGulf Stream relative vorticity is about only 25% of thestretching term. Such differences should be examinedcarefullybecauseHall's [1985] ransectwas ocated15deformation radii away from the coast, while in theNBC case he transect is close o the westernboundary.An expected difference is that the beta term contribu-tio is virtually negligible in the Gulf Stream case.

- v2 + f02f02

The differences n mapping the quasigeostrophicpo-tentialvorticity QG PV) fields elative o the PV fields

4.3. QG Potential Vorticity-Stream FunctionScatterplots' Next we consideran analysisbasedon the QG poten-tial vorticity and stream function fields as described orthe primitive equation fields n section3.3.2. However,

herewe followRhinesand Schopp1991] nd buildscat-terplots of potential vorticity against stream function.


17/25

SILVEIRA ET AL.' NORTH BRAZIL CURRENT RETROFLECTION DYNAMICS 28,575

WX2 Layer 1 quasi-geostrophic V9 '1.76

4

2 :g'-;?

-52 -50 -48 -46 -44WX2 Layer 2 quasi-geostrophicPV

9 '-" '_-.. ........"'"--


18/25

28,576 S1LVEIR,A ET AL.' NORTH BRAZIL CURRENT RETROFLECTION DYNAMICS

1.5 WX2 Layer 1 Potential Vorticity! ! !

0.5

:E -0.5o._ c- -1o

-1.5

-2

-2.5 ' ' ' '2 3 4 5 6Latitude

1.5 WX2 Layer2 PotentialVorticity! ! !

0.5

-0.5o._ c- -1o

-1.5

-2

-2.5 ' ' ' '2 3 4 5 6 7Latitude

Figure 13. Meridionalvariationof the upper ayer QG potentialvorticity (solid ine) along44W. The relativevorticity (dashedine), the stretchingdashed-dottedines), a,nd he beta,(dotted ine) termsare also epresented.top) Layer and (bottom)layer . Read e-05 as 1 x10 -5 '


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S[LVE1RA ET AL.' NORTH BRAZIL CURRENT RETROFLECTION DYNAMICS 28,577

According to these authors, mean q- field scatter-plots summarize some of the dynamics of the flow sys-tem. For example, tight curves on these plots indicatesteady states of nearly inertial flows. Hence dispersionon q- scatterplots, o severaldegrees,may be asso-ciated with dissipative lows, the propagation of vortic-ity waves hrough the system, or simply the absenceofsteady states.The scatterplots or WX3 /which seems o have thesmoothest etachment nd the leastdispersionare dis-played in Figure 14. The scatter evident in these plotsshows that the WX3 state is not steady, even allowingfor the possibility that the PV-stream function rela-tionship could be multivalued and for the vertical linesshowingvariations in PV associatedwith/ in regionsof weak flow. One interpretation of these results s thatthe NBC system might not have a steady state. Norand Pichevin 1996]haveargued, n the more estrictive1 layermodel ontext,hatsteadytatesreunlikelyin retrofiecting currents, such as the NBC, becauseofmomentum irabalances. A retrofiecting current would"paradoxically" shed eddies as an attempt to reach thebalance in this case.

Another interpretation is to consider he plots in Fig-ure 14 as showing a flow with fluctuations around asteady state due; o, for example, Rossbywavespassingby the region. As a matter of fact, Johnsat al. [1990]found a 50-day oscillatory behavior in their current me-ter data deployed n the NBC retrofiection egion. Theyspeculated hat the NBC eddy sheddingcould eitherbe modulated or triggered by short first-mode baro-clinic Rossby waves of the same period. Richardsonet al.'s [1994]analysisof surfacedrifters and SOFARfloats corroborated the idea that short Rossby wavestrigger the NBC retrofiectionpulsationand eddy shed-ding. Johnset al. [1998]also ndicated hat NBC eddysheddingmay not be a local process nd that precursorsof theseeddies/related o shortequatorialRossbywavesor Yanai waves n this case)may be presentupstreamofthe retrofiection. On this basis the dispersion depictedin the upperand middleWX3 scatterplotsFigure 14)could represent he signatureof thosebaroclinicwaves.However, if there is a steady state, the general slopeof the q- plots can give us additional informationabout the geophysical tability of the flow. For a fiat-bottomed, three-layerocean, the first Arnol'd [1966]theorem conditions or stability of arbitrary curved lowsare reduced to simply

- >0.0In other words, f there is a negative slope n the meanq- curveon at least one of the layers, nstability spossible.Ripa [1991]has shown hat equivalent ondi-tions with additionalestrictionsn he maximumlowvelocity)apply or the shallow-water odel.From Figure 14 it appears hat the slopeof the - relationshipould enegative.ndeed,heslopesf

the best fit lines are negative in all three layers. UsingFisher'sZ transformationSpiegel, 984],we find thatall correlation oecients are significant t the 95% con-fidence level. Thus, within the scope of analyzing thenonsteadyWX3 scatterplots, t seems hat the negative-slopeq- relationships avor geophysicalnstability nthe NBC retrofiection region.4.4. Growth Mechanisms' Baroclinic andBarotropic Interactions

Next we present an analysis that aims to use theWESTRAX "snapshots" o build a dynamical pictureof the NBC eddy-shedding phenomenon. In particu-lar, we search or evidenceof baroclinic and barotropicinteractions that may contribute to explain the NBCmeander extension and growth. In this sense hese in-teractions can be thought of as resulting from pertur-bations in the PV contours in several areas, such thatthe flows associated with one anomaly tend to increasethe perturbation in other regions.In order to pursue this analysis,we can take advan-tage of the linear relationship between stream functionand potentialvorticity n the QG senseequation 26)).We start by exploring he barocliniceffectsand applythe invertibility and superpositionprinciples o deter-mine whether or not there are interactions among thelayerswhich lead to mutual enhancement.The methodto be described next follows the ideas of Davis andEmanuel 1991].We will use he WX2 three-layerieldsin these calculations.

First, we define the ith-layer PV field as a sum of ameanpart (i.e., the beta, erm) and an anomalousart'

- + (28)From the inversion elationship 26), it can be notedthat there is no flow associatedwith the mean PV part.Hencehe th-la.yertreamunction is duesolelyothe PV anomaly ieldsas defined n (28).Second, using the superpositionprinciple, we parti-tion the ith-layer stream function field into four parts:

3(j) + i. (29)

j=l^!Wedefinei(j) as he th-layerinterior"omaintreamfunction nducedby jth-layer PV anomalies nly, andi as he th-layer boundary"omain treamunction.It ispossibleoorthogonalize(j) and using14),yielding - , (30)

where ) is the ruth-modenterior omain treamfunction nduced y PV anomaliesn the jth layeronly,and %5,, s the ruth-modeboundarydomain streamfunction.Third, using he inversion xpressionsn the WES-TRAX egionomain,esolveor ) and%5, y


20/25


Layer 1 Scatter PlotI I I I I I''=''''$"J'-?..,,,.,.., corrcoef=-O.2'. ' ".,.,,. ' .' C '%';: r;'""'::". "....':;; : r'"""-."-::'.-.\ 'X'--- ."e,.",-',....'_......--":.:.: -...

,, -. "..'.;..: :.-'o..- . .... ........-:.:..._,, ,, x .'P,t.. t:=;.;; ; :" ....... .... :;J.',,'. '..* ;* ' *, .,e .t * '*: .... .'.**l ) I N, ., '.. '-.'.:..-.'..-'.,..,.'

-0.5 0 0.5 1 1.5 2 2.5 3Streamunctionle+05m2/s]Layer 2 Scatter Plot

3.5

2 I I I I I I I I, 1.5- - 11- , ,,, corrcoef=-O.83 -.......0.5 -' l *"-, . l.' ' u[- .,.;.....: ...,:-...., ,:.,.... - I .".' .' '. '. -.' :%'%,.. .' ' ' . . .-o.- i '"-.'-:."':. ' '-'.'Y'..:'...('.....'. '-' - ., i %. ". ".. ". ' '-.' '.' , ".: ':_ ,[ = %... . -.. . ..-: .': - e 1.5 -_21 I I I I I I I I-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Streamunctionle+05m2/s]

Layer 3 Scatter Plot2 i i1.5 corrcoef=-0.26

1 ,..?;}.''- $ '...- .' %,.:; , - z ," ..0.5F_..'.... P' ;' ' '. ':.-.,-'-.-.'.-,..-.. 1 . ' . ' .' .' - '. . . . '.' . ..' ' .'. .' . O. ', -. ........ . . . .' : '.'%':: '.%:':' "" '" '" ''"' ''' '" '" "".-0.5 " '"'..'.' :'. ''. '..''. ''. '."'"'4' ' "/.-' /.- i-1.5-2

0.8

I i-0.2 -0.1 0 O.1

Streamunctionle+05m2/s]Figure 14. Scatterl)lots f QG potentialvorticity agaist stream unction or the (top) layer 1,(middle) layer 2, alld (bottom) layer 3 in WX3. The correlation oefficientsre indicatedon theupper right cor]erof eachpanel. Read le+05 as 1 x l0 snumerical iteratio]l'

() - 0 at, heboundaries,31b)

-- 0, (32a)-- , at heboundaries.32b)

In (31a),)' representshePVanomalyieldn theruth mode due to PV anomalies n the jth layer only,being mathematically defined by(33)

The ast teps o ma,p5, ndi(5) sing30).Onthe left (right) panelsof Figure 15, we show the re-sultsor hesumof three nterior treamunctions)(j)


21/25

S1LVEIRA ET AL.: NORTH BRAZIL CURRENT RETROFLECTION DYNAMICS 28,579

WX2 Layer Interior tream unction WX2Layer Interior tream unction9 .... ' ........ 98 87 76 6

5 5?-...,...,....,.'"',?................. ..... . //I !1

-52 -50 -48 -46 -44 -52 -50 -48 -46 -44WX2 Layer1 Boundary treamFunction WX2 Layer2 BoundaryStream Function

8'7

:. ' "'"''-- 5.

-52 -50 -48 -46 -44 -52 -50 -48 -46 -44W longitude W longitudeFigure 15. The WX2 (top) interior domainstream unctionand (bottom) boundarydomainstreamunction. left) Upper ayer ields nd right)middleayer ields.Read e+05 a.s x 10 .

(?2(j)) nd (2). It is seenhat, in the vicinityofthe pinching-off ddy in the two layers, he flow is al-most entirely due to the interior stream function com-ponent. Then, on the left (right')panelsof Figure 16weshowhe ndividualhi(j) or heupper subthermo-cline) layer. The solid ine representshe (total) PVcontour in the center of the front for each layer. It canbe seen hat 1(2) (left center anel) nd 2(z) (rightupper panel) are enhancing he retrofiectionbulge inthe two layers. In other words, he flow induced n theupper layer by the PV anomalies n the middle layeris increasing he PV anomalies n the upper layer andvice-versaseearrows n Figure16). This layer nterac-tion causesbaroclinic growth.It is also seen that cross-frontalvelocities on the up-per layer, that is, the velocitieson the right-hand cor-ner of the bulge in the PV contour that account or theeddy pinch-offonly occur in the upper layer when in-duced by PV anomalieson the upper layer itself. Thismay indicate either that another nteraction mechanism

is also contributing or tha, nonlinear vortex dynamicscauses he eddy occlusion. As an example, Springer andKawase 1993] ound that barotropic nstability playedan important role in meander growth and eddy for-mation of separating low-latitude boundary currents intheir1-layer odel.In order to verify the possibility of ba,rotropic nter-action, we perform a,calculation analogous o the layer-interactionanalysis.We choosehe 01 field (Figure 12,top left corner)and invert ndividualPV features o de-termine the flow individually induced by them. We findthat the cross-frontal velocities are mainly induced bythe already closededdy. We do not find any compellingevidence of interactions between the high and low PVfeatures that can justify a,barotropic roll-up with coop-erating enhancement, typical of barotropic instability.On the other hand, the cross-frontal velocities in Fig-

ure 16 resemble hose of the theoretical studies by Prattand Stern [1986]and Meacham 1991]. The formerau-thors howed1-layerddy inchingffdueoa. orn-


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28,580 SILVEIRA ET AL.: NORTH BRAZIL CURRENT RETROFLECTION DYNAMICSbination of nonlinear vortex induction mechanisms andadvection by the basic flow on a,n already finite am-plitudedisturbance.Meacham 1991]showedmeandergrowth by ba, oclinic instability and eddy detachmentby nonlinear vortex dynamics. Thus a combination ofba, oclinic growth and nonlinear vortex dynamics couldaccount for the NBC pinching-offeddy depicted by theWX2 da.ta.

While originally describing he NBC eddy shedding,Johnsat al. [1990]speculated hat a local instabilityprocess, probably modulated by the influence of thewave background in the retrofiection area, should ex-plain the phenomenon. The analysis described aboveshows that indeed baroclinic growth occurs in the re-gion. We cannot consider his growth a,san instabilityprocess,a.t least not in the classicalsenseof growing un-stable waves draining energy from a basic steady flow.No steady state is known for the NBC system. But cer-tainly there is mutual enhancementbetween the uppertwo layers of the WX2 fields in Figure 16.The possible ole of catalystsof the NBC retroflectionrings raised by the observationalargumentsof Richard-sontal. [1994] nd Johnstal. [1998] oesnot conflictwith the results presented here. Eddies associatedwithYanai or short equatorial waves can be thought a,sa. re-mote sourceof potential vorticity to the NBC retrofiec-tion region, one that could trigger the baroclinic growthmechanism.

The roleof suchcatalystswasexplored y Ma's [1996]successful simulations with an equivalent-barotropicmodel of the NBC eddy shedding triggered by an in-coming equatorial Rossby wave packet. However, thesimplifiedvertical structureof Ma's I1996]model leftno possibility of baroclinic interactions. Thus the rel-ative importance of the baroclinic growth in the eddyshedding phenomenon, as revealed by the data analysispresented n this section, remains to be assessed.5. Summary and Conclusions

In this work we investigated the dynamical structureof the NBC retrofiection region and the dynamical pro-cesseselated to the NBC eddy shedding rom the West-ern TropicalAtlantic Experiment WESTRAX) synop-tic data set. We searched for the NBC modal struc-ture considering a hydrostatic, fiat-bottomed planeocean model centered at 5N. The dynamical modes andthe correspondingdeformation radii were computed us-ing the average WESTR,AX Brunt-Viisili profile. Thefirst two depth empiricalorthogonal unction DEOF)modes, computed from the Pegasusvelocity profiles,explained about 90% of the variance in vertical struc-ture of the NBC region. Optimal fits of the dynamicalmodes onto the DEOF modes suggested hat DEOFmode 1 wasdominatedby (dynamical)baroclinicmode1, and DEOF mode2 by (dynamical)baroclinicmode2.The sum of the first three dynamical modes was foundto account for about 75% of the vertical structure of

the first two DEOF modes.A six-mode pproximationnearly reproduces he WESTRAX profiles. These re-sultscorroboratedhose rom the Johns t al. [1990]time domain EOF analysis on their moored currentmeter velocities and the dynamical decomposition fthe CME model esultsby McCleanandKlinck 1995].Both works, ike the results rom the presentpaper, in-dicated a dominant first (internal) mode. We foundthat a secondary ole is played by the secondbaroclinicmode.

Basedon these findings, a. hree-layer model was builtfor the WESTRAX region. The density values withineach ayer were carefully chosen o reproduce he defor-mation radii of the continuouslystratified model. Thethree-layer approximation for the WX2 surveydepictedthe NBC in the eddy-sheddingphase, while the NBCfront had retracted in the WX3 survey. The obtainedthree-layer strea.m function fields restricted the NBCeddy activity to the upper two layers. The third anddeepest ayer is the domain of a southward, weak me-andering flow that could be associatedwith the three-mode approximation of the DWBC. The NBC retrofiec-tion in the upper layer occurred at about 7N, feedingthe NECC. The middle layer was characterizedby theNBC separation near 4N to feed the NEUC. The po-tential vorticity fields were presented for both cruises.Over one deformation radius, the variation of the upper-layerPV wasabout timesheCoriolisarameter.This variation might define a PV scale. Midlatitudesystemssuch as the Gulf Stream present similar cross-streamvariations Hall, 1985].We stretched he quasigeostrophicQG) theory andapplied it to the NBC retrofiection region to search forpossible dynamical processes hat could contribute tothe NBC eddy shedding. The estimate of a Rossbynumber, away from the 0-IN latitudinal band andbased on the size of the three terms that form the QGpotential vorticity, indicated that Ro 0.36 within theWESTRAX domain. QG potential vorticity fields werethen mapped, using the previously gridded stream func-

tions to compute the stretching term. Therefore onlyquantities derived from the Pegasusobservedvelocitieswere used to build the QG PV maps.FollowingRhinosand $chopp 1991],we built QG po-tential vorticity-stream function scatterplots as a toolto evaluate the NBC system dynamics. We presentedthe results for the WX3 survey, which seemed o havethe smoothestdetachment and NBC front mostly re-tracted. The dispersiveaspect of the WX3 scatterplotsin all three layerssuggestshat the scenariodepicted snot steady. This result might reflect the Rossbywavebackground nown to exist in the area [Johnset al.,1990; McC'leanand Klinck, 1995]or simplymight cor-roborate the ideas of the nonexistenceof steady statesfor retrofiecting urrents Nof andPichevin, 996].How-

ever, if there is a steady state, a negative slopeor trendon only one of the la.yers' scatterplots might suggest


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S[LVEIRA ET AL.' NORTIt BRAZIL CURRENT RETROFLECTION DYNAMICS 28,581

1/'i)-ANOMALIESNLAYER

4' -, ....... ::


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28,582 S1LVEIRAET AL.' NORTH BRAZIL CURRENT RETROFLECTIONDYNAMICSthat instability processes re likely. Indeed, the slopesof the best fit lines in the scatterplots of all three layerswere negative,and the negativecorrelationcoeicientswere significant t a 95% confidenceevel.We performed, on the WX2 data set, a PV-basedanalysis o search or mechanismshat couldcontributeto the NBC retrofiection lobe extension and eventualdetachment. The analysiswas designed o isolate theeffect of PV anomalies in each of the layers on each ofthe layers. Our results suggested hat baroclinic nter-actionsoccurred,with upper (middle) ayerPV anoma-lies nducing xtension f the middle upper) ayereddy.Johnset al. [1990]speculated hat a local instabilityprocess auses he NBC meander growth, and our re-sults indicated that localized baroclinic growth was in-deedoccurring. This baroclinicgrowth s not the onlyprocess esponsibleor the eddy formationand detach-ment. Ma's [1996]equivalent-barotropic odel of theNBC shed eddieseither when an incomingRossbywavepacketreached he regionor via, wind forcingover thetropical Atlantic. However, our data-basedanalysessuggestedhat a three-layermodel etains he basicdy-namical structure of the NBC region. In order to assessthe relative importanceof the baroclinic nteractions nthe NBC meander growth and eddy detachment, sim-ulations with the Miami Isopycnic Coordinate Model,configuredwith the vertical structure proposed n thiswork,are currentlybeingconductedPatti et al., 19981.

Appendix A: Three-Layer CalibrationSchemeThis calibration rocedure xtends lierl's 1978]deasof calibrating two-layer models. Its goal is to matchthe two internal deformation radii of the three-layermodel to those of the continuouslystratified ocean byobtaining specialdensity ump valuesbetween he lay-ers. Choices for the rest depths Hi, H2, and Ha arerequired,as well as a previous omputation f the eigen-valuesof the continuouslystratified model.The determinantof (15) is calculated, iving

^3 1 1 1 1H H2 2H21 I 192 ele2HH2 te2HH3=0 .

Consequently,

t

0 -- 0 ,

1 1etH2 e2H2

1+ H3 X +2ee2H2H3

(A1)

(A2a)

+ (A2b)e2H3 '

32 ee2H1H2ele2H1/:t31 ] . A2c)l e2/--/2/3EquationsA2b) and (A2c) canbe easilymanipulatedto obtain a quadratic expression or et'

11[ H(H+H2)l2 Hl2H2(H2 -H3)

g(] -2)H1Hh-- 0. (A3)

The expression or e is simply given by

1 _ %H3 (k+5,2) H,+H22- H2h-H3 g I - -- (A4)Finally, l andexareobtainedyusing,t - At and)x - Axand therebymatchinghe deformationadiiof the layeredmodel to the valuesof the correspondingcontinuouslystratified ocean.It is evident rom (A3) and (A4) that there s somefreedom eft in the system and the calibration outputstwo sets of values for el and e2. We suggest hat deci-sion between he two pairs of values be made throughcomparisons ith e valuescalculated rom averaging hedensity profile used on the Rd computation.

Acknowledgments. We deeply hank F. L. Bub (UNH)for sharing his knowledge,his unpublished results and manu-scripts of the WESTRAX region with us. We are verygrateful to D. M. Fratantoni (WHO[) and W. E. Johns(RSMAS/UM) for sharing heir unpublishedmodeloutputswith us and discussing our research on several occasions.We also thank M. Wirebush for his careful review of ourmanuscript. F. O. Smith's help with English editing andthe manuscript camera-ready preparation was greatly ap-preciated.ReferencesArnol'd, V.[., On the a priori estimate in the theory of hy-drodynamical instability, Izv. Vyssh. Uchebn.Zaved Mat.,5/(5), 3-5, 1966. (Am. Math. Soc. Transl., Ser. 2, Engl.Transl., 79, 267-269, 1969.)Bretherton, F.P., [I.E. Davis and C.B. Fandry, A techniquefor objective analysis and design of oceanographic xper-iments applied to MODE-73, Deep Sea Res., 2S, 559-582,1976.Brown, W.S., W.E. Johns, K.D. Leaman, J.P. McCreary,R.L. Molinari, P.L. Richardson, and C. Rooth, A WesternTropical Atlantic Experiment (WESTRAX), Oceanogra-phy, 5(1), 73-77, 1992.Bruce, J.G., Comparison of eddies off the north Brazilianand Somali coasts, J. Phys. Oceanogr., 14, 825-832, 1984.Bruce, J.G., and J.L. Kerling, Near equatorial eddies inNorth Atlantic, Geophys. es. Lett., 11(8), 779-782,1984.


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SILVEIRA ET AL.: NOFlTH BRAZIL CURRENT RETROFLECTION DYNAMICS 28,583Bub, F.L., The structure of water mass, salt, and tempera-ture transports within intermediate depths of the westerntropical Atlantic Ocean, Ph.D. dissertation, 222 pp., Univ.of N.H., Durham, 1993.Bub, F.L., and W.S. Brown, tritermediate layer water massesin the western tropical Atlantic Ocean, J. Geophls.Res.,101, 11,903-11,922, 1996.Carter, E.F., and A.R. Robinson, Analysis models for the

estimation of oceanic fields, J. A tmos. Oceanic Techrol.,4(1), 49-74, 1987.Cochrane, J.D., F.J. Kelly, and C.R. Olling, Subthermoclinecountercurrents n the western equatorial Atlantic Ocean,J. Phls. Ocearogr., 9, 724-738, 1979.Davis, C.A., and K.A. Emanuel, Potential vorticity diagno-sisof cyclogenesis, or. WeatherRev., 119(8), 1930-1953,1991.Davis, R.E., Predictability of sea surface temperature andsea level pressure anomalies over the North Pacific Ocean,J. Phls. Ocearogr., 6, 249-266, 1976.Didden, N., and F. Schott, Seasonal variations in the west-ern tropical Atlantic: Surface circulation from Geosat al-timetry and WOCE model results, J. Geophls. Res., 97,3529-3541, 1992.Didden, N., and F. Schott, Eddies in the North Brazil Cur-rent retroflection region observed by Geosat altimetry, J.Geophls.Res., 98, 20,121-20,131, 1993.Flierl, G.R., Models of vertical structure and the calibra-tion of two-layer models, Dyr. Atmos. Oceans, 2, 341-381,1978.Fratantoni, D.M., W.E. Johns, and T.L. Townsend, Rings ofthe North Brazil Currents: Their structure and behavior

inferred from observations and a numerical simulation, J.Geoph!/s.Res., 100, 10,633-10,654, 1995.Gill, A.E., Atmosphere-Ocear !/rmmics,652 pp., Academic,San Diego, Calif., 1982.Hall, M.M., Horizontal and vertical structure of velocity, po-tential vorticity and energy in Gulf Stream, Rep. WH01-85-16, 165 pp., Woods Hole Oceanogr. nst. / Mass. nst.of Technol. Joint Program in Oceanogr., Cambridge, 1985.Hua, B.L., J.C. McWilliams, and W.B. Owens, An objec-tive analysis of the POLYMODE local dynamics exper-iment, II, Stream function and potential vorticity fieldsduring the intensive period, J. Phys. Ocearogr., 16, 506-522, 1986.Johns, W.E., T.N. Lee, F. Schott, R. Zantopp, and R.H.Evans, The North Brazil Current retrofiection: Seasonalstructure and eddy variability, J. Phys. Ocearogr., 95,22,103-22,120, 1990.Johns, W.E., T.N. Lee, R.C. Beardsley, J. Candela, R. Lime-burner, and B.M. Castro lilho, Annual cycle and variabil-ity of the North Brazil Current, J. Ph!/s. Ocearogr., 28,103-128, 1998.Klinck, J.M., EOF analysis of Central Drake Passage Cur-rents from DRAKE 79, J. Phys. Ocearogr., 15, 288-298,1984.LeBlond, P.H., and L.A. Mysak, Waves in the Ocear, 602pp., Elsevier Sci., New York, 1978.Ma, H., The dynamics of the North Brazil Current retrofiec-tion eddies, J. Mar. Res., 54, 35-53, 1996.

McClean, J.L., and J.M. Klinck, Description and vorticityanalysis of the 50-day oscillations in the western tropicalAtlantic region of the CME model, J. Phys. Ocearogr.,25, 2498-2517, 1995.Meacham, S.P., Meander evolution on piecewise-uniform,quasi-geostrophic ets, J. Phys. Ocearogr., 21, 1139-1170,1991.Nof, D., and T. Pichevin, The retrofiection paradox, 3. Phys.

Ocearogr. 26, 2344-2358, 1996.Patti Jr., E., I.C.A. da Silveira, E.J. Campos, and M.R.F.Guimaraes, Simulations of the North Brazil Current retro-fiection and eddy-shedding using the Miami isopycnic co-ordinatemodel (abstract), Eos Tr'ars.AGU, 79(1), OceanSci. Meet. Suppl., OS47, 1998.Pedlosky, J., Geophlszca! luid Dlrmmics, 770 pp., Springer-Verlag, New York, 1979.Pinardi, N., and A.Fl. Flobinson, Dynamics of deep thermo-cline jets in the POLYMODE region, J. Phls. Ocearogr.,17, 1163-1188, 1987.Pratt, L.J., and M. E. Stern, Dynamics of potential vorticityfronts and eddy detachment, J. Phys. Ocearogr., 16, 1101-1120, 1986.Rhines, P.B., and Fl. Schopp, The wind-driven circulation:Quasi geostrophic simulations and theory for nonsymmet-ric winds, J. Ph!/s. Ocearogr., 21, 1438-1469, 1991.Richardson, P.L., G.E. Hufford, R. Limeburner, and W. S.Brown, North Brazil Current retrofiection eddies, J. Geo-ph!/s. Res., 99, 5081-5093, 1994.Ripa, P., General stability conditions for a multi-layer model,J. Fluid Mech., 222, 119-137, 1991.Schott, F., and C.W. B6ning, The WOCE model in thewestern equatorial Atlantic: Upper layer circulation, J.Geoph!/s.Res., 96, 6993-7004, 1991.Silveira, ].C.A. da, G.R. Flierl, and W.S. Brown, Dynam-ics of separating western boundary currents, J. Phys.Ocearogr., 29, 119-144, 1999.Spiegel, M.R., Schaum's Outline Series-Theor!/ arid Prob-lems of Statistics, 454 pp., McGraw-Hill, New York, 1984.

Springer, S.R., and M. Kawase, Nonlinear and dissipa-rive dynamics in the connection region between westernboundary currents and equatorial currents, J. Geoph!/s.Res., 98, 12,511-12,525, 1993.

W.S. Brown, School of Marine Science and Technology,University of Massachusetts/Dartmouth,706 S. RodneyFrench Boulevard, New Bedford, MA 02744-1221.I.C.A. da Silveira, epartamentoe Oceanografiasica,lnstituto Oceanografico, Universidade de So Paulo, SoPaulo, SP 05508-900,Brazil. ([email protected])G.R. Flierl, Department of Earth, Atmospheric, and Plan-etary Sciences,Massachusetts nstitute of Technology, Cam-bridge, MA 02139.

(ReceivedMarch 30, 1999; revised June 30, 2000;acceptedJuly 12, 2000.)

dynamics of the north brazil current retroflection

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