Journalof MarineResearch, 56, 107–139, 1998

The inuence ofthe ambient  ow onthe spreading ofconvected water masses

bySonya Legg 1,2 andJohn Marshall 3

ABSTRACT Weinvestigatethe in uence of a cyclonicvortical  owonthelateral spreading of newlymixed uidgenerated through localized deep convection. Localized open ocean deep convection often occurswithin such a cyclonicgyre circulation, since the associated upwardly domed isopycnals and weakerstratiŽ cation locally precondition the ocean for deeper convection. In theabsence of ambient ow,localizedconvection has been shown to result in stronglateral  uxesof buoyancygenerated by baroclinicinstability, sufficient to offset the local surface buoyancy loss and limit the density anomalyof theconvectively generated water mass. Herewe examine the consequences of acyclonicambient  owonthisbaroclinic instability and lateralmixing. T oisolatethe in uence of thecirculation on this later stage of localized convection, weparameterize the convective mixing by the introduction of baroclinic point vortices (‘ ‘hetons’’) in atwo-layerquasi-geostrophic model, and prescribe the initial  owbyapatchof constantpotential vorticity.Linear stability analysis of the combined system of pre-existing cyclonic vortex and convectivelygenerated baroclinic vortex indicates scenarios in which the pre-existing cyclonic circulationcan modify the baroclinic instability. Numerical experiments with the two-layer QG modelshow that the effectiveness of the lateral heat  uxescan be stronglydiminished by the action ofthepre-existing circulation, thereby increasing the density anomaly of theconvected water mass.

1.Introduction Inseveral recent studies of localized open ocean deep convection (Legg and Marshall, 1993;Send and Marshall, 1995; Visbeck et al., 1996; Ivey et al., 1995;Coates et al., 1995; Brickman,1995; Legg et al., 1996)it has become clear that baroclinic instability of the localizedconvection site leads to signiŽ cant lateral  uxesof buoyancy as  uidis transportedaway from andinto the convectingsite by Žniteamplitude eddies. These lateral uxesmay be largeenough to completely offset the surface forcing. A statisticallysteady statemay result, in which the density anomaly of theoverturned  uidceases to increase, andthe deepening of themixed layer is halted. If thelateral  uxesachieved by baroclinic instabilitywere inany way rendered less effective, different water-mass propertieswould result.

1.Institute for Geophysics and Planetary Physics, University of California,Los Angeles, California, 90024; U.S.A. 2.Present address: Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, 02543,U.S.A. 3.Department ofEarth, Atmospheric and Planetary Sciences, MIT,Cambridge,Massachusetts, 02139,U.S.A. 107 108 Journalof MarineResearch [56, 1

Previousstudies of localizedconvection have considered a convectingregion within a volumeof eitherinŽ nite extent (Legg and Marshall, 1993; Legg et al., 1996)or conŽned by thenumerical domain (Jones and Marshall, 1993) or laboratory apparatus (Coates et al., 1995;Brickman, 1995). Most of these studies assume a quiescentinitial  ow(although Ivey et al. (1995)considered a preconditioning owdriven by arotatingdisc). Observa- tions,however, reveal that convection often occurs within a localgyre circulation (Killworth,1983). This may be associatedwith (i) localtopographic features such as the Rhonefan in the northwest Mediterranean (Swallow and Caston, 1973; Hogg, 1973) or MaudRise in theW eddellSea (Gordon, 1978; Alverson and Owens, 1996), (ii) winter-time intensiŽcation of the wind-driven circulation (Pinardi and Navarra, 1993) or (iii)localized coolingsuch as that generated by winds blowing off theice-sheet in the Labrador Sea (Clarkeand Gascard, 1983; Seung, 1987) or a combinationof all three processes. A cyclonicshear is associated with doming of isopycnalsup tothesurface and provides one ofthe‘ ‘preconditioning’’ factorsnecessary for deepconvection events. Since the eddies generatedby baroclinic instability carry  uidout of the convecting region and into the ambientcirculation, the spreadingphase of deep-waterformation should not be considered inisolation from thebackground  ow.Recentobservations in the Labrador Sea (Peter Rhines,personal communication, 1997) show signiŽ cant barotropic eddy activity occur- ringat thetime of deepconvection, further motivating an investigation of thein uence of barotropic owontheconvection and spreading. Severalrecent studies have begun to examine the preconditioning of the ocean for locallydeeper convection by the ambient circulation, including the  owtrapped over topography(Alverson and Owens, 1996; Madec et al., 1996),isolated geostrophic eddies (Legg et al., 1998),and wind-driven barotropic gyres (Madec et al., 1996).All of these studiesshow that in additionto inuencing the localization of convection,the ambient  ow mayalso in uence the spreading of the convected  uid.Alversonand Owens (1996)show thatbaroclinic instability may be replaced by mean  owadvection in the presence of a uniform ow,while Madec et al. (1996)suggest that a barotropicgyre circulation may suppressbaroclinic instability of theconvecting area. In this study we examinein detailthe inuences of ambient gyre  owsassociated with the preconditioning and wind-driven circulationon the development of baroclinic instability. W einvestigatein particular the possibilitythat this pre-existing circulation can affect the rate at which convected water subsequentlyspreads away from theformation region, and the efficiency with which baroclinicinstability draws buoyancyinto the convecting patch. Thein uence of barotropic  owon baroclinic instability has been examined in the contextof atmospheric baroclinic life cycles, where a ‘‘barotropicgovernor’ ’ (James, 1987)can suppress the eddy activity. In contrast to thesezonal atmospheric  ows,in the oceanicconvection context both baroclinic and barotropic  owsare localized within eddiesor gyres, where the spatial scale of thelocalization can in uence the stability. Weusea highlyidealized model to capturebaroclinic instability and its modiŽ cation by theambient circulation: a pointvortex ‘ heton’model of the convecting process, embedded 1998] Leggand Marshall: In uence of ambient ow 109 withina contour-dynamicalrepresentation of thepre-existing circulation. The point-vortex modelwas introducedby Leggand Marshall (1993) (hereafter LM), andits efficiency and applicabilityin the study of the properties of thespreading phase of convection demon- stratedby Legg et al. (1996)(hereafter LJV). Theadvantages of this Lagrangian formulationinclude its low cost, and the absence of boundariesor viscousprocesses. W e regardthe results of hetonmodel experiments as providing a usefulguide ranging across parameterspace which can be investigated further by other (more expensive)numerical modelscontaining more complete physics. Throughthe combination of linear stability analyses (a varianton the results of Flierl (1988)and Nakamura (1993)) and numerical experiments using the heton model, we Žnd thatthe addition of an ambient circulation comprising a vortexwith a strongbarotropic componentcan suppress the baroclinicinstability of theconvected region. The suppression oftheinstability is aconsequenceof thechange in the absolute vorticity of theconvective region,and along with it, stronger angular momentum constraints on the instability process.Associated with the reductionin instability, the lateral spreading of theoverturned uidis diminished, as are the lateral  uxesof buoyancydrawn into the convecting region from theperiphery. W econcludethat the properties of themixed water will be alteredby thepresence of abarotropicswirl, with a smallervolume of denserwater being generated. InSection 2 we briey reviewthe properties of theheton model of the spreading phase ofoceanconvection from LMandLJV ,andoutline the potential vorticity representation of thepreconditioned  ow.After adiscussionof thepredictions for stabilityof theconvected regiondeduced from lineartheory in Section3, numerical experiments using the Lagran- gianquasi-geostrophic two-layer point vortex/ contourdynamics model are described in Section4, andthe implications for deep-waterformation discussed in Section5.

2.A pointvortex model of the convectionregion Thelocalized convective mixing due to buoyancyforcing at the ocean surface leads to thegeneration of adensityanomaly in theconvected region, which under the in uence of rotationand gravity relaxes to a stateof geostrophic adjustment, with a baroclinicrim currentassociated with the regionof dense uid.The baroclinicstructure of thisgeostrophi- callyadjusted state can be representedmost simply in atwo-layerquasi-geostrophic model witha cyclonic owinthe upper layer and an anticyclonic  owinthe lower layer. The ongoingforcing serves to transform the density of the uidfrom thatof theupper layer to thatof the lower layer, thereby raising the interface between the layers in the forcing region.Prognostic equations in the quasi-geostrophic representation are given in terms of potentialvorticity, which contain information on the velocity Ž eldsin addition to the densityanomaly. W emakea furtherdiscretization by representing the potential vorticity Želdin terms of an ensemble of point vortices. Since the potential vorticity anomaly associatedwith a densityanomaly is baroclinic(the surface density anomaly is replaced by apotentialvorticity anomaly placed beneath an isothermal surface as described by Bretherton(1966) and elucidated further in LM),the point vortex representation involves 110 Journalof MarineResearch [56, 1

Figure1. A schematicdiagram showing the vertical structure of the two-layer quasi-geostrophic 1/2 model.The layer depths, h,areequal, and the deformation radius l 5 (gD r h/(2r 0)) /f, where D r 5 r 2 2 r 1. baroclinicpairs of vortices, with cyclonic vortices in the upper layer and anticyclonic vorticesin thelower layer .Theterm ‘ heton’has been coined to describesuch a pointvortex pair(Hogg and Stommel, 1985). This discretization results in a modelof thedevelopment ofthedensity anomaly and attendant geostrophic dynamics of considerablesimplicity and economy,of which further details are given in Legg and Marshall (1993). a.The heton model: summary of previousresults Inthe heton model a two-layerquasi-geostrophic model is used to represent the geostrophicallyadjusted convectively-mixed water column. The model is formulated in termsof potential vorticity with a baroclinicpotential vorticity anomaly qbc beingrelated to abuoyancyanomaly b through

f qbc 5 2 b; q 5 q bc, q 5 2 q bc (1) N 2H 1 2 where

D r b 5 2 g (2) r 0 where g isthe gravitational acceleration, f isthe Coriolis parameter, r 0 isthe reference density, D r isthe density anomaly, N isthe Brunt-V a¨isa¨la¨frequency,and H is the layer depth. q1 and q2 arethe potential vorticity anomalies in the upper and lower layers 2 1 2 respectively,with units of s . N 5 g/r 0(r 2 2 r 1)/H where r 1 and r 2 arethe densities of theupper and lower layer respectively (See Fig.1). A surfacebuyancy forcing B0 generates abuoyancyanomaly at arate

db B0 5 2 (3) dt H 1998] Leggand Marshall: In uence of ambient ow 111 so that

dq bc 5 R f 2 (4) dt q

2 2 where Rq 5 B0/(H N f )isanondimensionalmeasure of theforcing strength. Rq represents thepotential vorticity anomaly generated in time1/ f ascompared to theplanetary vorticity f. As such Rq issimilar to a Rossbynumber (relative vorticity compared to planetary vorticity).Oceanic values are typically very small ranging from 10 2 5 for weakforcing and strong N, to 102 1 for verystrong forcing. Thispotential vorticity anomaly is discretized into point vortices, to enable efficient computation,and provide a controllableelement of randomness and clumping in the buoyancyforcing, mimicking the spatial uctuationsof theoceanic forcing. The number of pointvortex pairs (hetons), N ,withinthe forcing disc of area A isrelatedto qbc by N 5 qbcA/s where s isthestrength of the individual point vortices. Hence the rate of increaseof hetonsis given by

2 d N AfB0 r Rq f 5 2 2 5 p (5) dt sN H 1 l 2 Sp 2 where l 5 NH/(Î 2f )isa modiŽed deformationradius; Sp 5 s/(l f )isa nondimensional vortexstrength; and r isthe radius of the circular disc over which cooling is applied.

Physicalresults of the problem do not depend on Sp providedit issufficiently small to allow thevortices to actas anensemble. Inthe numerical experiments of LMandLJV thedynamics of theconvecting region in theabsence of anyambient  owwere investigated.In the reference experiment of LMa coolingis applied over the circular disc of aradiusseveral times the deformation radius, generatinga baroclinicpotential vorticity anomaly at arategiven by (4),discretized by the additionof hetons(Fig. 2a). A shearedrim currentis generated around the cooling region (Fig.2b), which eventually becomes baroclinically unstable, allowing tilted clusters of hetonsto break out of the cooling region (Fig. 2c). There is symmetry between the developmentof theinstability in theupper and lower layers, so that the tilted heton clusters haveapproximately equal numbers of hetonsin upperand lower layers, and can therefore self-propagateaway from theconvected region indeŽ nitely (Fig. 2d). A statisticallysteady stateis established after only a few days,in which the rate of generation of hetons is balancedby the rate at which they are  uxedout of the patch by baroclinic instability (Fig.3). The implied lateral heat  uxachieved by the migrating hetons is sufficient to completelybalance the surface cooling. Since the number of hetons within the cooling regionremains roughly constant at latertimes, the density anomaly of theoverturned  uid doesnot increase further .LJV foundsimple scaling relationships for theequilibrium numberof hetons in thepatchand corresponding equilibrium buoyancy anomaly, as well as thetimeat which the breakupof thepatch occurred. These relationships closely match their 112 Journalof MarineResearch [56, 1

Figure2. Four pictures charting the development of anevolvingcluster of hetons in the reference experiment.Each picture shows the trajectories of the hetons over a periodof 6 3 (1/f ). The trajectoriesof theupper-layer vortices are shown by dotted lines, while those of thelower-layer vorticesare shown by solidlines. A crossmarks the position of anupperlayer vortex at theend of itstrajectory, while a lower-layervortex is markedby an asterisk.The horizontal scale is presented in units of l andthe hetons are introduced over a discof diameter 5 l .Thepictures are for the followingperiods; (a) 6 to12 3 (1/f ),(b)12 to18 3 (1/f ),(c)18 to24 3 (1/f ),(d)24 to30 3 (1/f ). predictionsmade using a simpletheory of thestatistically steady state in whichthe  uxof dense uidthrough the boundary of the forcing region balances the generation of dense uidby thesurface buoyancy anomaly, and the root mean square  uxvelocityacross this boundaryis proportional to therim current.(A correspondingset of relationshipsare found by Visbeck et al. (1996)for simulationsand laboratory experiments of convection into a continuouslystratiŽ ed  uid.) 1998] Leggand Marshall: In uence of ambient ow 113

Figure3. The evolution in time of thenumbers of hetons: Ntot,thetotal number of hetons; Nin, the numberof hetonswithin the forcing region; Nout,thenumber of hetons outside the forcing region; forthe reference experiment.

Althougheach baroclinic point vortex pair is associated with a localizeddensity anomaly,as anensemble the hetons generate a densityanomaly which decays over a Žnite distance,determined by the deformation radius, similar to that generated in a primitive equationmodel. (See LJVfor comparisonof density Ž eldsat similarstages of evolutionfor aprimitiveequation calculation and a hetonmodel calculation of localizedconvection.) LJVestimatethe  uxvelocityfrom linearstability arguments (as inStone,1972), with theratio between  uxvelocityand rim currentvelocity proportional to the growth rate of theinstability (nondimensionalized by the potential vorticity anomaly). If theselinear stabilityarguments to estimate  uxvelocity are applicable during the Ž niteamplitude developmentof theinstability, we wouldexpect a decreasein growthrate to (a) increase theequilibrium density of the convected region (b) increasethe time taken to reach equilibrium. Inthe experiments to be described, the parameters will match those of LM, witha 2 forcingparameter of Rq 5 0.08,(corresponding to a surfacecooling of H 5 800 W/m intoa two-layer uidof depth 2 kmandstratiŽ cation N 2 5 5.0 3 102 8 s2 2 ),apatchradius of r 5 5l (where l 5 1.58km) and a nondimensionalvortex strength of Sp 5 1.9.It should benoted that the value of forcing given is stronger than usually found in convection 114 Journalof MarineResearch [56, 1 regions,while N 2, r/l , and l areall smaller than typical values. The evidence of scaling behaviorgiven in LJV allowsus tohave conŽ dence that results will apply even for more typicalvalues, and the present choice is a convenientcomputational one, allowing more rapidcalculations. (A widerange of r/l and Rq valuesare accessible— see LJV— but larger r/l requiresmore individual vortices, while smaller Rq takeslonger to reach equilibrium, bothincreasing the time of computation). The breakup time scale in this reference experimentis Tb 5 15(1/f ),towhich the development time scales in the presence of ambient owwillbe compared. To represent the larger scale preconditioned  ow,we usea patchof piece-wiseconstant potential vorticity to induce a large-scaleswirl andemploy a mixtureof contourdynamics (see Appendix B) andpoint vortex techniques. b.A potentialvorticity representation of theambient  ow Wefocuson two different aspects of the pre-existing circulation in the convecting region.The Ž rst isthesurface-intensiŽ ed cyclonic circulation in geostrophic wind balance withthe domed ‘ ‘preconditioned’’ densityproŽ le, of whichnumerous observations exist. Lazier(1973) and Clarke and Gascard (1983) in the Labrador Sea, recorded a surface 2 3 densityanomaly of s T8 , 0.6 kg m overan area200 km across,associated with a doming ofisopycnalsand supporting a cycloniccirculation. Similarly, Swallow and Caston (1973) 2 3 observesurface density anomalies of magnitude s T8 , 0.1 2 0.2 kg m inthe Mediterra- nean,(Fig. 4), over an area75 km across in the east-west direction, and 30 km acrossin the north-southdirection. Associated with the densityanomaly are cyclonic geostrophic shears of , 5 cm s2 1 overa radiusof 20 km,conŽ ned mainly to the upper 200 m, in the Mediterranean,and , 15 cm s2 1 inthe Labrador Sea. Inspection of thevertical proŽ le of thedensity structure reveals that the density anomaly extends to about 500 m inthe Mediterranean,and 1000– 1500 m inthe Labrador Sea. The vertical structure of the velocityproŽ le is less clear, but a deepcyclonic shear of about 0.05 cm/ sat1500 m, relativeto 2000m isdeduced in theMediterranean (Swallow and Caston, 1973). Thesecond aspect of the pre-existing  owwhich we willconsider is the barotropic componentof the circulation, poorly estimated from observations,but available from numericalsimulations forced by climatological surface wind stresses. Numerical simula- tionssuch as that of Zavatarelliand Mellor (1995) reveal a barotropiccyclonic gyre in the Gulfof Lions of maximum intensity during the winter, with velocities of a few cms 2 1. Modelingof the circulation associated with convection over Maud Rise (Alverson and Owens,1996) indicates that a strongbarotropic (anticyclonic) circulation can develop as a ‘‘Taylorcap’ ’ ow. Werepresentthese two aspects of theambient  ow—the surface density anomaly, and thebarotropic vortical  ow—using piece-wise constant potential vorticity distributions. Theanomaly is chosen to be of a similarspatial dimension to the convective chimney, sinceit is likely that the preconditioned  owcoincides with the area where overturning takesplace. W ewillassume that the ambient circulation takes the form ofa circular cyclonicvortex, of radius b,withthe baroclinic convectively generated vortex of radius a 1998] Leggand Marshall: In uence of ambient ow 115

Figure4. Observations of (a) the preconditioned potential density section through the region of highestsurface density in theGulf of Lions,Mediterranean Sea (from Swallow and Caston, 1973) and(b) thepreconditioned potential density anomaly at 100 db in the Labrador Sea (from Gascard andClarke, 1983), showing the existence of aregionof weakerinitial stratiŽ cation associated with thedoming of isopycnalsto the surface. 116 Journalof MarineResearch [56, 1

Figure5. The potential vorticity structure of the baroclinic convectively generated vortex of radius a containedwithin a preconditioned owofradius b,withpotential vorticity anomaly qb. embeddedwithin it (Fig.5). The two different representations of theambient  owthatwe willexamine are (see Fig. 6): (case i) anomalously high potential vorticity extending throughboth layers, equivalent to a barotropiccylonic  owwith no density anomaly

(q1 5 q2 . 0)(where we areconsidering the preconditioned vortex potential vorticity anomalyonly), and (case ii) high potential vorticity conŽ ned to the upper layer only,

(q1 . 0; q2 5 0),representing a positivesurface density anomaly, invoking the ideas of Bretherton(1966). Case (ii) is associated with a cyclonicshear with height, corresponding toa reductionin the stratiŽ cation in the center of the vortex, consistent with the preconditioningcontext. The ambient  owin the convecting region in nature probably containsa mixtureof thesetwo features. The results for ananticyclonicbarotropic vortex wouldbe completely analogous to those of the cyclonic barotropic vortex. Prior tothe generationof abaroclinicpotential vorticity anomaly by surface buoyancy forcing, both forms ofpreconditionedcirculation are stable to baroclinicinstability.

3.Predictions from linear stability analysis Detailsof the linear stability analysis of a circularbaroclinic vortex representing the circulationgenerated through convection, embedded in a barotropicvortex representing thepreconditioned circulation, are given in AppendixA. Here onlythe salient points are outlined,followed by a Žniteamplitude investigation using contour dynamics. Weexaminethe stability of twopiece-wise constant concentric vortices in eachlayer, of radius a (thebaroclinic vortex of potentialvorticity qbc) and radius b (thebarotropic vortex ofvorticity anomaly qbt incase i, or the vortex of potential vorticity anomaly q0 in the upperlayer, with no signalin thelower layer in caseii) (Fig. 5). This piece-wise constant representationforces the change in surface density anomaly to be very abrupt, and is 1998] Leggand Marshall: In uence of ambient ow 117

Figure6. The total potential vorticity anomaly as a functionof radius for two different precondi- tionedvortex scenarios. Case i: The preconditioned circulation is independent of z, so that in additionto the baroclinic component of the potential vorticity anomaly, qbc of radius a,abarotropic component, qbt of radius b,isadded to both layers. Case ii: A preconditionedpotential vorticity q0 of radius b isaddedto the upper layer only. 118 Journalof MarineResearch [56, 1 thereforerather more extreme than found in nature. However, an advantage of this representationis therelative simplicity of thelinear stability analysis. In the atmospheric contextNakamura performed a linearstability analysis of a two-layerQG zonal ow separatedinto three distinct regions, analogous to our inner vortex, outer vortex and exteriorregion. However, he examined jet-like baroclinic and barotropic  owonly for coincidentbaroclinic and barotropic jets. By contrastwe examinecases where baroclinic andbarotropic jets need not be coincident.Like Flierl (1988), we examineaxisymmetric eddy owinwhich the eddy size is anadditional important parameter, rather than zonal owasinNakamura (1993). Thelinear stability analysis indicates that in the case of a pre-existingcirculation independentof height (case i), instability of the convected region can be completely suppressedif the baroclinic convectively generated potential vorticity anomaly is of a lessermagnitude than the pre-existing barotropic anomaly; as shown by Flierl (1988), bc bt * q * . * q * isanecessarycondition for instability(see Fig. 6). (This also corresponds to thenecessary condition for instabilityfrom theCharney-Stern theorem (Charney and Stern,(1962).) When the barotropic potential vorticity is of lowermagnitude, instability occurswith a reducedgrowth rate (Fig. 17a). Extensionof Flierl’s analysisto barotropicand baroclinic circulations of different radii (see AppendixA) showsthat the suppression of theinstability can only occur when the pre-existingvortex and convectively generated vortex are of similar size. If theboundaries ofthe two vortices are separated by more than a few deformationradii, they no longer inuence one another. Incase ii— the preconditioned vortex with cyclonic vorticity in the upper layer only—instability is always possible, no matter how large the preconditioned potential vorticityanomaly. Furthermore, for thefastest growing mode, the growth rate is increased (Fig.17b) when a 5 b. Purelybarotropic instability could occur if thepotentialvorticity jumps were ofopposite sign at r 5 a and r 5 b,butof thesame sign in each of the two layers. However, this does notfall within the regime we haveselected, where the potentialvorticity of vortexradius a isentirely baroclinic. Motivatedby theresults of thelinear stability analysis, we examinethe Ž nite-amplitude developmentof thebaroclinic instability of acircularpiece-wise constant vortex, in each scenario.A similarstudy was performedby Helfrichand Send (1988), and we extendtheir workto include concentric vortices of different sizes. As inthe stability analysis, we examinethe development of two piece-wise constant concentric vortices in each layer, withpotential vorticities corresponding to case (i) orcase (ii) as above. In case i, in agreementwith the predictions of the linear stability analysis, we Žndthat when b 5 a, instabilityis prevented for values qbt/qbc . 1(Fig.7d). At lowervalues of qbt/qbc, the instabilityis suppressed particularly in theupper layer (which has potential vorticity of the samesign as thebarotropic potential vorticity), while the perturbation grows in the lower layer(Fig. 7b,c). However, the mechanism for self-propulsionof potential vorticity 1998] Leggand Marshall: In uence of ambient ow 119

Figure7. Results of numerical integrations using contour dynamics, showing the evolution of a bt baroclinicvortex of radius r 5 a containedwithin a vortexof radius r 5 b. Case i: qb 5 q , independentof z.Inallcases vortex a is of radius a 5 5l andhaspotential vorticity anomaly qa 5 bc bc q intheupper layer, and qa 5 2 q inthe lower layer. The radii of thebarotropic and baroclinic circulationsare equal: a 5 b,andhence the vortex boundaries are coincident at alltimes, since theyare material surfaces. qbt takesvalues (a) 0, (b) 0.2 qbc, (c) 0.5qbc, (d) 1.0qbc.Thecontour positionsat a timeof about 8 3 (1/qbc)areshown for all cases, with the solid and dotted lines representingthe upper and lower layer boundaries respectively. The tendency for the meanders to bt bc growin the lower layer only as q /q increasesis demonstrated, while when qbt/qbc 5 1, the instabilityis completelysuppressed. anomaliesoutward is no longerpresent. Any clusters which break off from themainvortex containpredominantly lower-layer vorticity, so thatthey remain close to, and circle around themain vortex, in accordwith the results of Helfrichand Send (1988). If b ismuchlarger than a,we Žndthat, as predicted, instability takes place even at large values of qbt/qbc (Fig.8). However, as the tilted vortex dipoles move outward, they eventually come close enoughto the boundaryof the barotropicvortex to interact with it, and any further outward movementis prevented. For lowervalues of qbt/qbc thetranslating baroclinic dipoles cause fragmentsof thebarotropic vortex to be ejected in theirpath. Inthe Ž niteamplitude integrations corresponding to case(ii), where the preconditioned vortexof radius b haspotential vorticity anomaly q0 intheupper layer, and zero potential 120 Journalof MarineResearch [56, 1

Figure8. As for Figure 7, but with larger outer vortex radii b; b 5 8l . (a) qbt 5 0.2qbc, t 5 8 3 (1/qbc) (b) qbt 5 0.2qbc, t 5 11 3 (1/qbc) (c) qbt 5 1.0qbc, t 5 8 3 (1/qbc), (d) qbt 5 1.0qbc, t 5 14 3 (1/qbc).Initiallythe development of theinstability is unhindered, and the lobes of the baroclinic vortexare symmetrical between upper and lower layers, but at later times, greater development of thelower layer lobes, (b), or inability of theinstability to develop further, (d), is evidenceof the inuence of thebarotropic outer vortex. vorticityanomaly in the lower layer, the baroclinic vortex of radius a andbaroclinic potentialvorticity qbc isalways unstable, as predicted by the linear stability theory. However,the meanders develop predominantly in the lower layer, which has potential vorticityof oppositesign to the barotropic component of the ow,resultingin aspreading ofuidin the lower layer only. During the instability of apurelybaroclinic vortex (Fig. 7a) dipolareddy structures are created possessing a symmetrybetween meander development inupper and lower layers; the self propagation of thesedipoles  uxes uidvery efficiently outof thepatch. However, when the meanders occur in thelower layer only, as incase(ii), thismechanism for propulsionof uidout of thepatch is removed (Fig. 9). Similarcalculations (not shown) with anticyclonic barotropic vorticity conŽ rm that analogousbehavior occurs, but with meanders occurring predominantly in theupper layer (where qbc isofoppositesign to qbt). 1998] Leggand Marshall: In uence of ambient ow 121

Basedon these results, one would expect baroclinic instability to be prevented from occurringin aconvectingregion embedded in abarotropiccirculation at sufficiently high values of qbt/qbc when b 5 a.Lateralspreading of the lower layer anomaly would be greaterthan that of the upper layer for intermediatevalues of qbt/qbc,for cyclonic qbt. (Spreadingof the cyclonic upper layer would be greater for anticyclonic qbt.) When the preconditionedvortex is signiŽ cantly larger than the baroclinic convectively generated vortex,we donotexpectthe instability to be suppressedinitially. When the meandersof the convectivevortex boundary have propagated out to the boundary of the preconditioned vortexthe instability may then be suppressed(Fig. 8). A pre-existingcirculation consisting ofcyclonicvorticity in the upperlayer only, while it doesnot suppress the initial instability andmay even enhance it, alters the Ž niteamplitude development.

4.Numerical experiments with hetons inapreconditioned ow Thenumerical experiments to be described are based on the‘ ‘reference’’ experimentof LMoutlinedearlier, with hetons of strength s 5 1.9l 2f beingadded at aconstantrate given by(5) randomlypositioned over a disc10 l indiameter.(See Leggand Marshall (1993) for details.)However, now a barotropiccirculation is added, represented by an initially circularpatch of piece-wise constant potential vorticity. Three different possible pre- existingvortex scenarios are investigated, selected using results from ourlinear stability analysisas a guide.It should be recalled that a preconditionedvortex with a potential vorticitylarger than the magnitude of the baroclinic potential vorticity anomaly of the convectivelygenerated circulation is required, in orderto suppress the baroclinic instabil- ity.Hence, since the maximum value of baroclinic potential vorticity anomaly in the bt referenceexperiment is q < 0.95f0,we selecta preconditionedvortex with q 5 f0 (case i) 0 or q 5 f0 (caseii). (Later discussion will show that for moretypical forcing strengths and lengthscales, smaller, more realistic values of qbt willbe sufficient.)In order to investigate thein uence of thevortex size, we examinecase (i) witha radiusof b 5 6l and b 5 8l . (Theradius of the baroclinic vortex is a 5 5l .)Henceour three different ambient circulationscenarios are as follows:a vortexof radius6 l ,andpotential vorticity anomaly of f0,independentof z,(Fig.10); a vortexof radius8 l andpotential vorticity anomaly of f0, independentof z,(Fig.12); and a vortexof radius6 l witha potentialvorticity anomaly of f0,inthe upper layer only, (Fig. 14). a.The evolution of theheton cluster in thepresence of abarotropicvortex of radius6 l TheŽ rst experiment(Fig. 10) lies in the regime where linear theory suggests baroclinic instabilitywill be absent until the baroclinic potential vorticity anomaly is greater in magnitudethan the barotropic, pre-existing anomaly. The results of theexperiment show that,as expected, the ability of baroclinicinstability to form self-propellingdipoles which uxthe overturned  uidefficiently out of the cooling patch is severely reduced. The 122 Journalof MarineResearch [56, 1

Figure9. Results of numerical integrations using contour dynamics, showing the evolution of a baroclinicvortex of radius r 5 a containedwithin a vortexof radius r 5 b. Case ii: qb 5 q0 in the upperlayer and qb 5 0inthe lower layer. In all cases the inner vortex has radius a 5 5l and bc bc potentialvorticity anomaly qa 5 q inthe upper layer, and qa 5 2 q inthelower layer. The radii ofthepreconditioned and baroclinic circulations are equal: a 5 b,sothatone boundary encloses bc boththe baroclinic and additional potential vorticity anomalies. q0 takesvalues (a) 0.2 q , (b) 0.5qbc, (c) 1.0qbc and (d) 2.0qbc.Asin Figure 7, the dotted line represents the lower layer contour, withthe solid line representing the upper layer, and positions are shown at about8 3 (1/qbc). As q0/qbc increases,the meanders tend to growonly in the lower layer. behaviorof vorticesis markedly different in theupper and lower layers, with lower-layer vorticesbeing  uxedefficiently out of the patch, the numbers of lower-layer vortices withinthe patch remaining approximately steady after t 5 15/f;whilethe number of vorticesin the upper layer continues to increase steadily (Fig. 11). The symmetry between thedevelopment of the instability in the upper and lower layers present in the reference experiment(Fig. 2d) has therefore been removed by the presence of the barotropic anomaly. Thetime scale for thelower-layer vortices to reach a quasi-steadystate is similarto the timescale for bothlayers in thereference experiment (Fig. 3), suggesting that in thislayer, theefficiency of the uxvelocity is unaltered. 1998] Leggand Marshall: In uence of ambient ow 123

Figure10. Four pictures charting the development of anevolvingcluster of hetons contained within abarotropicvortex of radius 6 l andpotential vorticity qbt 5 f0,independentof z.Eachpicture showsthe trajectories of the hetons over a periodof 0.3 3 (1/f ),andthe position of theboundary ofthe barotropic vortex at the end of thatperiod. The trajectories of the upper layer vortices are shownby dottedlines, with a crossmarking the endof thetrajectory, while those of thelower layer vorticesare shown by solid lines, with an asterisk marking their Ž nalposition. The vortex boundaryis shown by a solidline in the upper layer, and a dottedline in the lower layer. The horizontalscale is presentedin units of l andthe hetons are introduced over a discof diameter5 l , witha strength s 5 0.6p l 2f,ata ratecorresponding to a coolingof 800 W atts/m 2, as in the referenceexperiment. The pictures are for the periods ending at the following times; (a) 12 3 (1/f ), (b) 18 3 (1/f ), (c) 24 3 (1/f ), (d) 30 3 (1/f ).

If thepopulationof vorticesand their rate of increase(Fig. 11) are interpreted in termsof theimplied buoyancy  uxes,about half of thesurface buoyancy loss is offset by thelateral uxesaffected by the lower-layer vortices, while the other half is servingto increasethe densityanomaly of the  uidin theoverturning region. A statisticallysteady state exists in thelower layer, but not in the upper layer .Henceunlike the reference experiment, the densityanomaly of theconvecting region continues to increase indeŽ nitely. b.The evolution of theheton cluster in thepresence of abarotropicvortex of radius8 l Inthe second experiment (Fig. 12), the heton cluster initially behaves much the same as inthe reference experiment, although with a solidbody rotation superimposed upon the 124 Journalof MarineResearch [56, 1

Figure11. The evolution in time of Ntot, Nin, Nout in(a) the upper layer, and (b) the lowerlayer, in thepresence of a barotropicpreconditioned circulation of strength f0 and radius 6l . The lower layervalues were only sampled every 6 time-steps,resulting in a lessnoisy signal. velocities.The number of vortices contained within and without the cooling region is initiallysymmetric between the two layers (Fig. 13), and identical to that of thereference experiment.Breakup of thecluster begins at about 15 3 (1/f ),indicatedby a risein the numberof vorticeslying outside the cooling region, as inthe reference experiment (Fig. 3). However,at about t 5 18 3 (1/f ), (Fig.12b), the tilted heton clumps have propagated 1998] Leggand Marshall: In uence of ambient ow 125

Figure12. As for Figure 10, but wherethe barotropic vortex now hasaradiusof b 5 8l . sufficientlyclose to thelarger pre-existing vortex boundary to interactwith it. As above,the lower-layervortices continue to be  uxedout of the region, but the cyclonic vortex preventsthe upper layer vortices from escaping.The number of lower-layer vortices within theforcing region reaches an equilibrium, while the number of upper layer vortices continuesto rise. At laterstages several processes combine to balance the surfacebuoyancy loss: half of it isbalanced by the lateral  uxof buoyancy achieved by migration of the lower-layer vortices;one quarter is balanced by the lateral  uxof buoyancy due to the upper-layer vorticesmigrating out of the cooling region, but unable to move further out than the boundaryof thebarotropicvortex at 8 l ;onequarterserves to increasethe density anomaly oftheoverturned region. Hence the density anomaly does not reach a statisticallysteady state. 126 Journalof MarineResearch [56, 1

Figure13. The evolution in time of Ntot, Nin, Nout in(a)the upper layer, and (b) the lowerlayer, in thepresence of abarotropicpreconditioned circulation of strength f0 and radius 8l . c.The evolution of theheton cluster in the presence of aupper-layerpreconditioned vortexof radius6 l Thethird experiment contains a preconditionedvortex in the upper layer only. In this case,at about 12 3 (1/f )(Fig.15b), when the baroclinic circulation is of sufficient magnitude,clusters of lower-layervortices move outward (Fig. 14). This breakup of the lowerlayer is more rapid than in the reference experiment (Fig. 3), suggesting a more efficient uxgenerated by the instability, in accordance with the higher growth rate 1998] Leggand Marshall: In uence of ambient ow 127

Figure14. As for Figure 10, but where the heton cluster in contained within a vortexof radius 6 l havinga potentialvorticity anomaly in the upper layer only, of strength q0 5 f0,correspondingto caseii in thetext. indicatedby linear theory. However, without a cyclonicpartner, these clusters are unable to selfpropellvery far andthey circle the cooling region. On the long term, it appearsthat, as inthe Ž rst heton/contourexperiment, the lateral buoyancy  uxis achieved solely by the lower-layervortices, which have reached a statisticallysteady state. However, the density anomalycontinues to risedue to the trapped upper-layer vortices. d.Discussion Theseheton experiments, (Figs. 10, 12, 14), exhibit much of the behavior of the experimentsperformed with contour dynamics alone, described in Section3. For example, whenthe barotropic vortex is of approximately the same size as the cooling region, instabilityis suppresseduntil there are sufficient hetons to inducea baroclinicPV greater thanthe barotropic PV .Wheninstability occurs, lower-layer hetons preferentially migrate outward,but remain close to the cooling region, so that the efficiency of the instability in transporting uidfar from thecooling region is suppressed. When the barotropic vortex is muchlarger than the baroclinic vortex, instability is not suppresseduntil migration of heton 128 Journalof MarineResearch [56, 1

Figure15. The evolution in time of Ntot, Nin, Nout in(a) the upper layer and (b) the lower layer in thepresence of apreconditionedcirculation of strength f0 inthe upper layer only, of radius6 l . clustersout to theboundary of thebarotropic vortex has occurred. The heton experiments clearlydemonstrate that even when the barotropic component of the  owis not large enoughto preventinstability, the lateral buoyancy  uxesare sufficiently suppressed as to havea signiŽcant in uence on thedensity anomaly of the overturned  uid. Theheton model, due to its inherently discrete formulation, produces some spurious interactionsbetween the vortices and the sharp front of the piece-wise constant vortex. 1998] Leggand Marshall: In uence of ambient ow 129

Thesetend to encourage the ejection of theanticyclonic lower-layer vortices and reduce furtherthe baroclinicity of theconvecting region. In the realocean it islikelythat the fronts ofpotential vorticity are not so well deŽ ned as in this piece-wise constant case; the baroclinicpotential vorticity anomaly will be more evenly distributed throughout the forcingregion in nature. However, despite this shortcoming, these experiments provide motivationfor thefurther investigation of the in uence of the ambient  owon the subsequentdevelopment of the dense  uidchimney in a morecomplete numerical formulation.They clearly suggest that the effect of theambient  owcannotbe ignoredin investigationsof thespreading phases of deep convection. Undercontinued forcing, the baroclinic potential vorticity anomaly is continuously increasing.Obviously if thebarotropic preconditioned anomaly is very small, it willhave verylittle in uence on thedevelopment of theinstability, because the baroclinic anomaly willvery rapidly be greaterin magnitudethan the pre-existing barotropic anomaly. W ecan estimatethe magnitude of thebarotropic potential vorticity anomaly necessary to inuence theinstability by consideringthe equilibrium potential vorticity anomaly achieved in the experimentsof LJVwhenthe lateral  uxdueto baroclinicinstability completely balances thesurface forcing. Then the potential vorticity anomaly of theconvective region qbc 5 1/2 3 3 1/2 1.0(Rqr/l ) f 5 1.2f(B0r/(H N )) (where thenumerical constants were empirically determined).If thebarotropic anomaly is greaterthan this value, equilibrium will not be 3 3 1/2 achieved.In other words a barotropicrim currentof U . 0.6 3 (B0r/(H N )) rf will be 2 8 necessaryto prevent instability. In the ocean, we expect B0 tobeoftheorder of 5 3 10 to 5 3 102 7 m2 s2 3, N , 5–50 3 102 4 s2 1 and r , 104–105 m.Hence barotropic currents between1– 100 cm/ swillbe necessaryto preventinstability, depending on themagnitude ofthe forcing and size of the vortex. Smaller radius cooling regions will have their instabilitysuppressed by smaller magnitude barotropic currents. The corresponding barotropicvorticity anomaly may be as low as 0.003 f,for smallradius, low forcing, and strongambient stratiŽ cation (at the limits of these parameters given above). Clearly the importantnondimensional parameter describing the in uence of theambient  owis

q bt q bt N 3H 3 1/2 bc 5 (6) q (eq) 1.2f 1 B0r 2 theratio of the barotropic potential vorticity to the equilibrium baroclinic potential vorticitywhich would be obtainedin theabsence of ambient  ow.For ourexamples this ratiois 1.6.Future work will investigate the dependence of  owpropertieson this ratio.

5.Conclusion Recentnumerical and laboratory studies of localized open ocean convection have identiŽed baroclinicinstability as providing a lateral uxofbuoyancy,which in somecases completelyoffsets the surface buoyancy loss, and solimits the depth of themixed layer and densityanomaly of the overturned  uid.Here we haveidentiŽ ed and assessed the 130 Journalof MarineResearch [56, 1 importanceof aprocesswhich is likely to in uence the efficiency of thelateral  uxes,by preventingthe onset, and suppressing the development of the baroclinic instability: the barotropicvortical component of the ambient circulation. Such a barotropiccirculation mayexist as a componentof thepreconditioning necessary to localize deep convection. Thepreconditioned  owappears to be self-perpetuating:the net result of convectionis a reinforcedcyclonic vortex, because the anticyclonic component in the lower layer of the convectivelygenerated  owis preferentially removed by thebaroclinic instability. Analo- gously,if the preconditioned circulation is comprised predominantly of an anticyclonic component,we wouldexpect the upper-layer  uxesto bemostsigniŽ cant, removing the cycloniccomponent of the convectively generated circulation, and hence reinforcing the anticyclonicpreconditioned circulation. If thepreconditioned  owhas a signiŽcantly largerdimension than that of the convectively generated circulation, then that  owmay trapthe  uidwithin its conŽ nes, and force its continued exposure to thesurface forcing in the region. Inthe oceanthe interaction between baroclinic instability and barotropic circulation may beconsiderablymore complex than studied here. The eddies generated through baroclinic instabilitymay interact so asto generate larger-scale barotropic circulation, through merger andalignment, and this circulation may then in uence instability generated by subsequent convectiveactivity. In this sense, an oceanic analogue to the atmospheric ‘ ‘barotropic governor’’ mayexist, with barotropic circulation generated through baroclinic instability ultimatelyin uencing the efficiency of thatinstability. Toconclude, the presence of thepreconditioned  owhasa signiŽcant in uence on the resultingwater masses and large-scale  ow,andan accurateparameterization scheme for theeffects of open-ocean deep convection on the large-scale ocean circulation must accountfor thesefeatures correctly.

Acknowledgments. SL ispresently supported by aNOAA climateand global change post-doctoral fellowship,administered through the Universities Corporation for Atmospheric Research. Much of thiswork was carried out while SL wasthe recipient of a NERC studentship,while a studentat ImperialCollege, and an exchange visitor at MIT.JMwassupported by theNSF .Wewouldlike to thankGlenn Flierl for his help in the understanding of thelinear stability problem and Wim Verkley forproviding the original version of the contour dynamics code. Peter Rhines pointed out the similarityto the barotropic governor in atmospheric baroclinic lifecycles, and together with an anonymousreviewer made useful suggestions for improvements to the text.

APPENDIX A Linearstability analysis of acirculartwo-layer vortex Wesummarizethe results of stabilityanalyses of abaroclinicpiece-wise constant vortex embeddedwithin a concentricvortex containing a strongbarotropic component. W ebriey showhow the stability analysis of Flierl(1988) can be appliedto allthe cases relevant to thiswork, and make an extension of his analysis to include a barotropicvortex which is 1998] Leggand Marshall: In uence of ambient ow 131 largerthan the baroclinic vortex. The investigation of stabilityis conŽ ned to the scenario shownin Figure 6: abaroclinicvortex A of radius a,representingthe potential vorticity anomalygenerated through convective overturning, is contained within a vortex B of radius b whichrepresents any preconditioned circulation. V ortex A haspotential vorticity bc bc qa 5 q inthe upper layer, and qa 5 2 q inthe lower layer. V ortex B haspotential bt vorticity qb ofwhich two scenarios will be considered:(i) qb isindependentof z (qb 5 q ); 0 (ii) qb ispositive in the upper layer, and zero in thelower layer ( qb 5 q ,0respectively). Usingthe formulation of Flierl (1988), the matrix equation for perturbationsof the vortexboundary of azimuthalmode number m canbe obtained:

m (V 2 V I 2 M Q)n 0 5 0 (7) where Mm istheGreens function matrix appropriate for themode number m, and I is the identitymatrix, V isthematrix of basic state boundary velocities, Q isthematrix of basic statevortex potential vorticities, n 0 isthevector of boundaryperturbation amplitudes, and V isthefrequencyof theseperturbations (See Flierl(1988) for thedetailed derivation). The vortexwill be unstableto theperturbation if thedisturbance grows exponentially; i.e. V is imaginaryand V /i . 0.For nontrivialeigenvectors we obtainthe eigenvalue equation

determinant( V 2 V I 2 MmQ) 5 0. (8)

For ourtwo-layer, two-vortex conŽ guration, the matrices take the following form

bt n a bt n b n 0 5 bc (9) n a bc 1 n b 2 Vbt (a)/a 0 Vbc (a)/a 0

0 Vbt (b)/b 0 Vbc (b)/b V 5 (10) Vbc (a)/a 0 Vbt (a)/a 0 1 0 Vbc (b)/b 0 Vbt (b)/b2 bt bc q a 0 q a 0 bt bc 0 q b 0 q b Q 5 bc bt (11) q a 0 q a 0 bc bt 1 0 q b 0 q b 2 132 Journalof MarineResearch [56, 1

1 1 0 0 2m b m2 1 2m 1 a2 1 1 0 0 b m1 1 2m Mm 5 2m (12) 1 a2 a a b a b 0 0 I K I K m 1 l 2 m 1 l 2 1 a2 m 1 l 2 m 1 l 2 a a b b b 0 0 I K I K 1 b2 m 1 l 2 m 1 l 2 m 1 l 2 m 1 l 2 where Im(x) and Km(x) are1 the mthorder modiŽ ed Besselsfunctions, and l istheeffective 2 deformationradius, l 5 NH/(Î 2f ).Thesubscripts bt and bc refer tothe barotropic and barocliniccomponents respectively. Furthermore, the velocities can be expressed,in terms ofthepotential vorticity anomalies thus:

bt bt Vbt (a) q a q b 5 1 (13) a 2 2

2 bt bt Vbt (b) a q a q b 5 1 (14) b 1 b2 2 2

Vbc (a) a a b a b 5 q bcI K 1 q bcI K (15) a a 1 1 l 2 1 1 l 2 1 a2 b 1 1 l 2 1 1 l 2

Vbc (b) a a b b b 5 q bc I K 1 q bcI K . (16) b a 1 b2 1 1 l 2 1 1 l 2 b 1 1 l 2 1 1 l 2 Wethereforehave an eigenvalue problem dependent on thefollowing variables: a, b, l , bt bc bt bc m, q a , q a , q b , q b .However,we willlimit our parameter space by investigating cases bt bc bc bt bc bc where q a 5 0,andeither q b 5 0 or q b 5 q b .For simplicitywe cannow deŽ ne q 5 q a , bt bt bt bc 0 and either q 5 q b , (case i) or q b 5 q b 5 q /2(caseii). Then the eigenvalue problem will bt bc 0 dependon four independent variables b/a, a/l , m and q /q or q /qbc. If a 5 b, and qbt 5 0we obtainthe sufficient condition for instabilitydeduced by Pedlosky(1985) and Saunders (1972):

a a 1 I K , (17) 1 1 l 2 1 1 l 2 2m 1998] Leggand Marshall: In uence of ambient ow 133

For m $ 2thisapproximates to

a m , . (18) l

Howeverthe vortex is not unstable to mode number 1, since then V ; 0. Hence a baroclinicvortex is unstable to perturbations of mode numbers m $ 2givenby (18), if a/l . 2. Thegrowth rate of theinstability, when the above condition is satisŽed is

1 V 5 q bc I K 2 (I K 2 I K ) (19) Î 31 1 1 2m2 1 1 m m 4 whereall Bessels functions have as their argument a/l . When a 5 b and qbt Þ 0thesufficient condition for instabilitybecomes:

1 4(I K 2 I K ) 2 I K qbt 1 1 m m 2m 1 1 , Î 1 1 22 (20) q bc 1 2 I K 2m m m when qbt/qbc . 0.Thelimiting value of qbt/qbc for instabilityis shown plotted in Figure16a, asa functionof mode number m,for thespeciŽ c caseof a/l 5 5.0(as inour heton simulations).If thebarotropic component qbt isgreaterthan this limiting value, baroclinic instabilitywill be prevented. For themost unstable mode-number m,(20)reduces to qbt/qbc , 1,which is the necessary condition for instabilitygiven by the Charney-Stern theorem(Charney and Stern, 1962). For smallervalues of qbt/qbc,thegrowth rate of the instabilityis reduced(Fig. 17):

1 1 1 2 V 5 4(q bc )2 I K 2 (I K 2 I K ) 2 (q bt )2 2 I K (21) 2 Î 3 1 1 1 2m2 1 1 m m 1 2m m m2 4 Incontrast, if the vortex consists of a baroclinicpart qbc plusan additional positive component q0 inthe upper layer only, the sufficient condition for instabilitybecomes:

1 1 1 4(I K 2 I K ) 2 I K 1 2 2 I K (I K 2 I K ) 2 I K q 0 1 1 m m 2m 1 1 2m m m 1 1 m m 2m 1 1 , 1 2 1 2Î 1 2 (22) q bc 1 1 2 1 2 I K 2 4(I K 2 I K ) 2 I K 2 11 2m m m2 1 1 m m 1 2m 1 122 134 Journalof MarineResearch [56, 1

bt bc bc Figure16. The limiting values of (a) q /q and (b) q0/q abovewhich instability does not occur,as afunctionof mode number m, for a/l 5 5.0,when the radiiof theadditional vortex component is equalto that of the baroclinic vortex ( b 5 a).Thefastest growing mode ( m < 4)is suppressed bt bc bc when q /q 5 1.0,but only at inŽ nitely high q0/q . 1998] Leggand Marshall: In uence of ambient ow 135

Figure17. The growth rate, nondimensionalized by thegrowth rate for a purelybaroclinic vortex, as bt bc bc afunctionof (a) q /q and (b) q0/q ,forthe fastest growing mode. The radii of theadditional vortexcomponent is equal to thatof thebaroclinic vortex ( b 5 a), m 5 4 and a/l 5 5. 136 Journalof MarineResearch [56, 1 withthe limiting value for instabilityshown plotted against mode number m inFigure16b for a/l 5 5.0.For themost unstable mode number, there is nolimitingvalue of q0 at which instabilityis suppressed.Note that the Charney-Stern necessary condition for instabilityis alwayssatisŽ ed in this case. Althoughthe instability is never suppressed, the growth rate is modiŽ ed:

2 2 2 1 q0 1 q0 1 V 5 4 q bc 1 I K 2 (I K 2 I K ) 2 2 I K (23) 2 Î 3 1 2 2 1 1 1 2m2 1 1 m m 4 1 2m m m2 4 However,for themost unstable mode number, this corresponds to an increase in growth rate,rather than a decrease. bt bc Finally,we examinethe case where a Þ b.If, asin case (i), q a 5 0 and q b 5 0, the eigenvalueequation (8) becomes

q 2 q 1 2 v I K 2 0 2 2mb m2 1 1 1 2m q 1 2 1 1 b 0 1 2 2 v I1K1 2 m 2mb m1 1 b l det 1 2 1 2 5 0 (24) q b I K 2 I K 0 2 v 2 qb I K 1 1 m m 2 m m 1 l 2 2 1 b 1 b 1 b b I K I K 0 q 2 I K 2 v b m m 1 l 2 b 1 1 1 l 2 1 2 m 1 l 2 m 1 l 22 where q 15 qbt/qbc, b 5 b/a, v 5 V /qbc.AllBessels functions have a/l astheir2 argument unlessotherwise stated. Numerical solution of thisfourth order equation for v shows that, asin the case where b 5 a,asufficientlylarge barotropic component qbt,willsuppress the instability,provided that b 2 a isnot much greater than a deformationradius l . Figure 18a showsthe numerically obtained values of v asafunctionof qbt/qbc and b/a for a/l 5 5.0 and m 5 4(thefastest growing mode if qbt 5 0).When the barotropic vortex is large comparedto the baroclinic vortex, the inner vortex feels only what amounts to a solidbody rotation,which does not reduce the baroclinic instability. bt bt bc For case(ii), where qb 5 0inthelower layer, we have q a 5 0, q b 5 q0/2, and q b 5 q0/2. Substitutingthese values into the eigenvalue equation, we againŽ ndthat suppression of theinstability is possiblefor b < a,althoughnot over as largea partof thedomain as in casei, and not for themost unstable mode number .Figure18b shows the numerically bc obtained v asafunctionof q0/q and b/a for m 5 4 and a/l 5 5.0. Weconcludethat the instability of a circularbaroclinic vortex in atwo-layerformulation ofequallayer depths may be suppressed by abarotropiccomponent of potentialvorticity greaterin magnitude than the baroclinic component. This result holds if thebarotropic 1998] Leggand Marshall: In uence of ambient ow 137

Figure18. The growth rate of the instability v /qbc formode number m 5 4 and a/l 5 5.0as a functionof bt bc bc (a) q /q or (b) q0/q and b/a.Theshaded region indicates no exponential growth ( v 5 0). vortexis largerthan the baroclinic vortex provided the radius of thelarger vortex is not muchmore than a deformationradius greater than the radius of the inner vortex. By contrastthe additionof acycloniccomponent in theupper layer alone does not suppressthe instabilityof thefastest growing mode. 138 Journalof MarineResearch [56, 1

APPENDIX B

Contourdynamics Weusecontour dynamical techniques as developed by Zabusky et al. (1979),to study theevolution of thepiece-wise constant vortex and its interaction with the point vortices. Theboundary of thevortex is represented by a Žnitenumber of discretenodes and it is sufficientto determine the self-advection of these boundary nodes to study the  ow evolution,thereby reducing the problem from oneof three dimensions to one of two dimensions.As theintegration proceeds, the boundary of thevortex becomes increasingly contorted.In order to resolve these contortions it is necessary to increase the number of nodesand to rearrange them so that they are evenly distributed, an operation which is carriedout at eachtime-step. However, this increase in node number causes the timefor the integrationto run to increase geometrically as it proceeds. Hence in order to reduce the integrationtime to manageable proportions, a ‘‘surgery’’ algorithm(Dritschel, 1989) is adopted,eliminating thin Ž lamentsof vorticityand thereby signiŽ cantly reducing the total lengthof the vortex boundary. W ehaveemployed a particularlysevere form ofthis algorithm,so that Ž lamentsare removed immediately. Since an important motivation behindthe heton model is the efficiency with which solutions may be computedon small platforms,we donot wish to combine it with a form ofcontourdynamics which would eliminatethis advantage. This reduces the accuracy of the integration, but these experi- mentsstill provide a usefulguide to possiblefeatures of the large-scale behavior of boththe oceanand its representation in morecomplex numerical models.

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Received: 9December,1996; revised: 23September,1997.