Localization of Deep Ocean Convection by a Mesoscale · PDF file944 JOURNAL OF PHYSICAL...

944 VOLUME 28J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

q 1998 American Meteorological Society

Localization of Deep Ocean Convection by a Mesoscale Eddy

SONYA LEGG*, JAMES MCWILLIAMS, AND JIANBO GAO

Institute for Geophysics and Planetary Physics, University of California, Los Angeles, Los Angeles, California

(Manuscript received 19 December 1996, in final form 3 September 1997)

ABSTRACT

Observations of open-ocean deep convection indicate that it is a highly localized phenomenon, occurring overareas of tens of kilometers in diameter. The cause of this localization has been ascribed to ‘‘preconditioning’’—the local weakening of the stable density stratification associated with upwardly domed isopycnal surfaces in asurface-intensified cyclonic circulation. However, most numerical and laboratory studies of localized convectionhave prescribed the localization artificially, by confining the surface buoyancy loss to a circular disk. In contrast,in the numerical simulations described here, deep convection forced by horizontally uniform buoyancy loss islocalized within a region of initially weaker stratification than its surroundings. The preconditioned region isassociated with a cold-core cyclonic eddy in geostrophic and cyclostrophic balance.

As in previous studies of disk-shaped cooling, the localized convection region undergoes baroclinic instabilityat late times, causing the breakup of the convection region into multiple eddies. However, since the stratificationstructure determines the extent of vertical buoyancy fluxes, the region of deepest heat loss migrates with theeddy structures. Furthermore, the localized convection generates a secondary circulation in the plane perpen-dicular to the original eddy circulation. This circulation initiates the restratification of the eddy core even beforethe onset of baroclinic instability. Eventually the original preconditioned eddy is destroyed by the barocliniceddy fluxes. The net effect of the eddy on the mean temperature field is to ensure a warmer upper layer andcolder pycnocline than would be achieved with horizontally uniform convection. The authors investigate thedependence of this flow evolution on the size and strength of the preconditioned eddy.

1. Introduction

Open-ocean deep convection, despite its importantrole in renewing deep and intermediate water masses,is a highly localized phenomenon occurring in only afew subpolar and marginal seas: for example, the north-western Mediterranean, the central Labrador Sea, theGreenland Sea, and the Weddell Sea. Within theseregions, active convection is often confined to areas ofstill smaller scale: 50–100 km in the Gulf of Lions(MEDOC Group 1970; Schott and Leaman 1991), 100km in the Maud Rise region of the Weddell Sea, and50-km convecting ‘‘chimneys’’ in the central LabradorSea (Gascard and Clarke 1983). Convection may belocalized by several different mechanisms. The limitedextent of intense buoyancy loss to the atmosphere pro-vides some localization; however, deep convection oftenoccurs over a smaller horizontal area than the flux fields.Alternatively, the ocean may be locally preconditionedto favor deeper convection. One such ‘‘precondition-

* Current affiliation: Woods Hole Oceanographic Institution,Woods Hole, Massachusetts.

Corresponding author address: Dr. Sonya Legg, Woods HoleOceanographic Institution, Woods Hole, MA 02543.

ing’’ mechanism is to trap the fluid within a recirculatingregion over local topography, for example, above theRhone Fan (Hogg 1973; Madec et al. 1996) and aboveMaud Rise (Alverson and Owens 1996), increasing thetime during which fluid is exposed to the atmosphericforcing. The focus of this study is locally deep con-vection within regions of weak stratification relative tothe surroundings.

In several regions weaker stratification occurs withinthe center of a gyre-scale cyclonic circulation on a hor-izontal scale of about 50–200 km, where wintertimedeep convection occurs regularly, for example, in theGulf of Lions (Swallow and Caston 1973), the LabradorSea (Clarke and Gascard 1983), the Greenland Sea (Vis-beck et al. 1995), and in the intermediate convectionregion of the Rhodes gyre in the Levantine Basin (Las-caratos 1993). Within the gyre-scale preconditioningthere may also be smaller-scale stratification anomalies,as observed by Gascard and Clarke (1983), associatedwith mesoscale eddies. Here we will examine the lo-calization of deep convection by an idealized exampleof such a stratification anomaly: a single cold core eddyin gradient wind balance. The center of the eddy isassociated with weaker stratification, and it is thereforepreconditioned for deeper convection.

Numerous studies of localized convection over thelast few years, using both laboratory experiments (Max-

MAY 1998 945L E G G E T A L .

FIG. 1. A schematic diagram showing the simulation volume: anarea 50 km 3 50 km across and 2 km deep. Initially the whole domainis stably stratified, with a region of weaker stratification: a cold coreeddy associated with a cyclonic shear. The eddy anomaly has a Gauss-ian form in the horizontal, decaying on a length scale lr, and decaysexponentially in the vertical over a scale lz. The anomalous stratifi-cation in the center of the eddy is (z) where (z) , 0; the back-2 2N Ne e

ground stratification is (z). A uniform surface buoyancy flux B0 is2N 0

applied over the whole domain.

TABLE 1. Details for all the simulations discussed in this paper. All simulations include a surface buoyancy loss of 1027 m2 s23, a backgroundstratification which decays exponentially as in Eq. (1), and a Coriolis parameter f 5 1024 s21. The eddy anomaly also decays in the verticalover a depth of 1 km. The calculation domain size in all cases except run D is 50 km 3 50 km 3 2 km. Run A has cooling ceased after5 days, while run B is performed without cooling. Run C has cooling applied over a disc of radius lr. Run D has homogeneous coolingapplied to a volume with horizontally homogeneous initial conditions. The symbols in the first column 1 are used in later figures and textto identify different simulations.

Identifyingsymbol

Eddy radius lr

(km)Radius ratio

lr /Lp

Eddy strengtha

Diffusivity k(m2 s21)

Resolutionnx 3 ny 3 nz

Grid scaleDx 3 Dy 3 Dz

(m)

1*●#vCJ3ABCD

810121510101010101010

0

0.91.11.41.71.11.11.11.11.11.11.10

1.01.01.01.00.80.50.21.01.01.00.00.0

0.70.70.70.70.70.70.70.320.70.70.70.32

256 3 256 3 17256 3 256 3 17256 3 256 3 17256 3 256 3 17256 3 256 3 17256 3 256 3 17256 3 256 3 17512 3 512 3 33256 3 256 3 17256 3 256 3 17256 3 256 3 17128 3 128 3 33

195.3 3 195.3 3 125.0195.3 3 195.3 3 125.0195.3 3 195.3 3 125.0195.3 3 195.3 3 125.0195.3 3 195.3 3 125.0195.3 3 195.3 3 125.0195.3 3 195.3 3 125.0

97.7 3 97.7 3 62.5195.3 3 195.3 3 125.0195.3 3 195.3 3 125.0195.3 3 195.3 3 125.0

97.7 3 97.7 3 62.5

worthy and Narimousa 1994; Ivey et al. 1995; Brickman1995; Whitehead et al. 1996) and numerical simulations(Madec et al. 1991; Jones and Marshall 1993; Visbecket al. 1996; Legg et al. 1996) have demonstrated thatlocalized convection, by inducing horizontal densitygradients, can lead to a fundamentally different redis-tribution of density from horizontally homogeneousconvection. Baroclinic instability plays an importantrole in providing lateral fluxes of buoyancy, which mayoffset the local vertical fluxes and alter the evolution ofthe convectively mixed upper layer. All of these studies,however, possess a common simplification: The local-ization was imposed artificially by restricting the surfacebuoyancy forcing to a disk-shaped area and was not a

result of a preconditioned density field. Given that bar-oclinic instability and the subsequent three-dimensionalredistribution of the density field can lead to dramaticchanges in the eventual water mass properties (Visbecket al. 1996), the influence of this simplification requiresevaluation. In particular, localized forcing generates asharp, permanent front between the convecting regionand the exterior flow. When convection is localized byoceanic initial conditions, does the absence of such afront significantly alter the details of the flow? Doesbaroclinic instability still occur when buoyancy forcingis more widespread? How might the boundary betweenthe convecting region and exterior evolve with time?We will answer these questions in this paper.

In our approach the study volume is initialized witha single cyclonic eddy, and uniform cooling is appliedat the top surface over the whole area (see Fig. 1). Anylocally deeper convection will then result from the in-homogeneity in the initial conditions. We examine howhorizontal and vertical variations in stratification influ-ence the localization of convection. We focus in partic-ular on the roles of secondary circulation and baroclinicinstability in providing a three-dimensional redistribu-tion of density, including restratification of the eddycore, in competition with the largely vertical fluxes pro-vided by plume-scale convection. Comparison with pre-vious results for the disk-forcing scenario are made, andthe competition between baroclinic instability and theoutbreak of widespread deep convection is examined.

2. Model formulation and initial conditions

To choose realistic initial conditions for our simula-tions, we consider principally observations from theLabrador Sea where Gascard and Clarke (1983) madeone of the few direct observations of a mesoscale chim-ney of active deep convection, delineated by a thermal


FIG. 2. Initial conditions in the reference simulation, with lr 5 10km, e 5 1.0, shown in the (x, z) plane through the eddy center: (a)temperature, contour spacing 0.04 K; (b) velocity in the y direction,contour spacing 2.5 cm s21, with positive values shown by solidcontours, and negative values by dotted contours.

front and embedded within a larger gyre-scale regionof known wintertime convection. Neither topographicpreconditioning nor local ice cover are important here,so these mesoscale chimney phenomena are most likelythe result of locally weaker stratification.

Gascard and Clarke observed a chimney of diameter;50 km, which we surmise is the horizontal scale ofthe initial stratification anomaly. The background strat-ification in the Labrador Sea (Lazier 1980) consists ofa nearly homogeneous interior overlaid by a stronglystratified seasonal thermocline in the early fall. We ap-proximate this stratification profile with a backgroundstratification , which decays exponentially from its2N 0

maximum near the surface, of the form

z2 2 2N 5 N 1 N exp . (1)0 1 2 1 2lq

Here N 2 5 2(g/r0)]r/]z, where g is the gravitationalacceleration, r0 is a reference density, r is the density,z is the vertical coordinate, and are constants,2 2N N1 2

and lq is the vertical decay scale; z 5 0 at the top surfaceand z 5 2Lz at the bottom of the integration volumewhere Lz is the vertical dimension. In all of our simu-lations N1 5 2.1 3 1024 s21 and N2 5 8.5 3 1024 s21.The effect of the cold core eddy is to locally diminishthis strong surface stratification.

The initial eddy circulation is statically stable and ingradient wind balance with a cyclonic shear. The anom-aly decays at large radius and at depth, with an expo-nentially decaying shape in the vertical and axisym-metric Gaussian shape in the horizontal (see the appen-dix). Parameters that define the eddy (Fig. 1) are lr, thehorizontal length scale; lz, the vertical decay scale; and

, the magnitude of the eddy stratification anomaly.2N e

Here 5 2(g/r0)]r9/]z, where r9 is the anomalous2N e

density of the eddy: , 0 since the eddy is more2N e

weakly stratified than the exterior but is restricted inmagnitude to ensure static stability, | / | , 1. For2 2N Ne 0

simplicity we choose lz 5 lq; Lz also influences thevertical decay of the eddy since the vertical shear andtemperature anomaly are zero at the bottom. If the eddyis to be initially stable, its horizontal extent is limitedrelative to the deformation radius Lr 5 NH/ f (Pedlosky1985), where H is a vertical length scale and f is theCoriolis parameter. Since Lr depends on stratification,scenarios with stronger background stratification wouldallow stable eddies of larger radius.

With B0, the surface buoyancy forcing, as a furtherimportant parameter, the controlling nondimensional pa-rameters are lr/Lr, the ratio of localization scale to de-formation radius; the convective Rossby number Ro 5[B0/( f 3H 2)]1/3, the ratio between rotational and convec-tive timescales; N/ f, the ratio of Brunt–Vaisala fre-quency to Coriolis frequency; and e 5 | / |, the ratio2 2N Ne 0

between stratification amplitudes. All but the final oneare in common with the disk-forcing scenario. In oursimulations N is variable over the domain, so we define

lr/Lr and N/ f in terms of the background stratificationin the upper part of the flow and H as the vertical decayscale of the stratification. All simulations have the sameB0 and f, so the Rossby number of the convection variesonly with the depth of the convection: for depths greaterthan 320 m, Roc , 1, and plume-scale motion may berotationally controlled (Jones and Marshall 1993). HereN/ f is similarly constant for all simulations and greaterthan unity everywhere in the stably stratified part of theflow. Values of the variable parameters for the simu-lations are given in Table 1.

Given these initial conditions, we examine the re-sponse of the flow to surface cooling by integratingforward the Boussinesq, nonhydrostatic, incompressiblefluid equations, using a model described in Julien et al.(1996). The model is pseudospectral in the horizontaland employs second-order finite differences in the ver-tical. Time stepping is achieved using a third-order Ad-

MAY 1998 947L E G G E T A L .

ams–Bashforth scheme for nonlinear terms and aCrank–Nicholson method for linear terms. We use alinear equation of state and for simplicity ignore effectsof salinity: that is, r 5 r0(1 2 aT) where a is a constantexpansion coefficient and T is the temperature anomaly.We ignore variations in the Coriolis parameter and as-sume the rotation axis is aligned with gravity.

Important computational parameters are k, the modeldiffusivity; Pr 5 n/k, the ratio of diffusivity to viscosity;the resolution, nx 3 ny 3 nz; and the size of the in-tegration volume, Lx 3 Ly 3 Lz. In all our calculationsPr 5 1. We run the model at a sufficiently fine gridspacing to resolve the small-scale convective plumesand over a sufficiently large volume to include the bar-oclinic instability following convection. The verticalgrid spacing allows the surface boundary layer to beresolved by two–three grid points. The grid spacing ob-viously limits the maximum volume we can examine:all our simulations are for a volume 50 km 3 50 km3 2 km. The initial eddy is therefore constrained to besomewhat smaller than observed. However, the earlyfall stratification in the Labrador Sea (Lazier 1980) islarger than that applied in our simulations (Nmax ; 403 1024 s21 rather than Nmax 5 8.7 3 1024 s21 as used)so that the ratio lr/Lr is comparable.

Since the convective motion is fully three-dimen-sional, we use resolution that is close to isotropic andtherefore employ isotropic viscosity and diffusivity withvalues, given in Table 1, chosen to maintain smoothsolutions, given the finite resolution, and strong surfaceforcing. Hence, horizontal diffusivities and viscositiesare considerably smaller than typical of many oceangeneral circulation models, while vertical values maybe of the same order. For comparison, Jones and Mar-shall (1993) use a diffusivity of 0.2 m2 s21 in the verticaland 5 m2 s21 in the horizontal at similar resolutions.Since oceanic values are many orders of magnitudessmaller, it is essential to determine whether the artifi-cially high values control the evolution of the simulatedflow. Compared to Julien et al. (1996) we are examininga much larger aspect ratio domain (25 3 25 3 1 ascompared with 4 3 4 3 1) and hence the limitationsof present day computers force us to use coarser reso-lution and hence higher viscosity and diffusivity. Vis-cosity and diffusivity therefore play a potentially greaterrole, and we examine their influence by comparing low-er resolution runs with those at higher resolutions andlower viscosities and diffusivities. As in the earlierstudy, we choose not to use a more complex subgrid-scale formulation since the dependence of the flow onthe variable coefficients may be more difficult to quan-tify.

The model employs periodic sidewall boundary con-ditions and stress-free momentum boundary conditionsand no normal flow at the top and bottom. A fixedstabilizing flux (dT/dz equal to the eddy value at thisdepth, approximately 20.33 the destabilizing surfaceflux) is applied at the bottom boundary, to maintain the

deep stratification with k ± 0, while at the top boundarya destabilizing flux is initiated at the onset of the in-tegration. The balanced eddy satisfies the momentumboundary condition at the bottom boundary since it hasno T anomaly and, therefore, no shear at this depth.However, at the top surface, since the temperature anom-aly is a maximum here, the stresses are initially not zeroas required by the boundary conditions. There is there-fore a brief initial spinup period as the eddy adjusts tothe boundary condition, generating a momentum bound-ary layer with a near-surface inflow into the center ofthe eddy, as predicted by linear Ekman theory. We allowthe flow to spin up prior to applying the cooling; alltime series exclude this spinup period, and show onlythe time elapsed since the onset of cooling.

3. Solutions and analyses

As described in Table 1, we have carried out severalsimulations with varying lr, e, and k in order to examinethe parameter dependencies of the features of the flowevolution. Rotation rate and background stratificationare kept constant, with f 5 1024 s21 and N0 5 8.7 f atthe top surface, fixing the background deformation ra-dius Lr(max) 5 N0(max)lq/ f 5 8.7 km. Therefore, inall cases lr/Lr(max) , 2. Our reference simulation, withwhich most of our discussion is concerned unless statedotherwise, has as initial conditions an eddy of maximumpossible amplitude (e 5 1), radius 10 km, and depthscale 1 km (Fig. 2). A buoyancy flux B0 5 1027 m2 s23

is applied over the domain, where

gaHB 5 , (2)0 r C0 w

H is the surface heat flux, and Cw is the specific heatcapacity of seawater. For temperatures around 28C andsalinity of 35 psu, typical of the Labrador Sea, a 57.81 3 1025 K21, and Cw 5 3.99 3 103 J K21 kg21 atsurface pressure. Hence with g 5 9.81 m s22 and r0 51028 kg m23 this corresponds to a surface heat flux of535 W m22. The integration has a resolution of 512 3512 3 33.

In Fig. 3 we show temperature, vertical velocity, andvertical vorticity fields at various stages of evolution ofthis reference simulation. Horizontal sections just belowthe top surface are shown for temperature and verticalvelocity and at 410 m below the top surface for thevertical component of vorticity. Vertical sectionsthrough the center of the eddy are also shown for thetemperature field.

Initially the flow is dominated by small-scale con-vective plumes, visible in the vertical velocity, concen-trated in the region of weakest stratification in the centerof the eddy. This vertical mixing in the center of theeddy leads to weak, near-homogeneous stratification inthat region, visible in the vertical temperature section.At later times, vertical mixing, indicated by the verticalvelocity field, has spread over the whole domain. How-


FIG. 3. Instantaneous fields in the high-resolution simulation 3: (a) the vertical velocity in the horizontal plane 62.5 m belowthe top surface, with upward velocities (maximum value 7.5 cm s21) shown in yellow and red, downward velocities (minimumvalue 210 cm s21) in green and blue, and negligible velocities in black; (b) the vertical component of vorticity in the horizontalplane 406.25 m below the top surface, with cyclonic vorticity (maximum value 5 12 f ) shown in orange and red, anticyclonicvorticity (minimum value 5 28 f ) in green.

MAY 1998 949L E G G E T A L .

FIG. 3 (Continued) and blue, and negligible vorticity in black; (c) the temperature in the horizontal section 31.25 m below thetop surface; and (d) the temperature in the vertical section through the center of the eddy, with warmer temperatures shown in redand black, and cooler temperatures in green and blue, over a range DT 5 0.5 K, in the horizontal section, and DT 5 0.55 K inthe vertical section. All fields are shown at the times (i) 0.8 days, (ii) 3.1 days, and (iii) 5.9 days after the onset of cooling.


FIG. 4. Time series for the high-resolution reference simulation(3). (a) The root-mean-square perturbation vertical velocityrms(w9) 5 [ u]1/2 at the center of the eddy, averaged overu 2(w 2 w )a strip 0.5 km wide at a depth 62 m below the top surface. (b) Thedepth h(t) at which dT/dz 5 0.0 at a radius of 5 km from the eddycenter. (c) The root-mean-square perturbation azimuthal velocityrms( ) 5 [ u]1/2 at a radius lr 5 10 km from the eddyu 2u9 (u 2 u )u u u

center, averaged over a strip 8 km wide and 1 km deep. (d) Thetemperature change averaged over the top 360 m and a strip 0.5 kmwide centered at a radius (i) 400 m and (ii) 4 km from the eddycenter. (e) The root-mean-square streamfunction in the (r, z) plane:rms(f ) 5 [ . (f ) The maximum magnitude of the lat-

r,z2 1/2(f (r, z, t)) ]

eral eddy heat flux (u9rT 9) u. (g) The mean azimuthal velocity atuuu

a radius lr 5 10 km from the eddy center averaged over a strip 8 kmwide and 1 km deep. Also marked are the convection onset time tc, therestratification time tr, and the baroclinic instability onset time tbc.

ever, whereas small-scale plumes dominate over the ex-terior region, the eddy region has vertical motions on alarger spatial scale, shown in both temperature and ver-tical velocity fields. The eddy interior now has slopingisopycnals and has restratified. The flow is no longeraxisymmetric, with smaller-scale meanders developingaround the main eddy. At a much later stage, the originaleddy has disintegrated into several smaller eddies. Thecoldest regions of these eddies are associated withstrong subsurface cyclonic vorticity, while vortex fila-ments also have counterparts in the temperature field.Regions of intensified horizontal temperature gradientform a discontinuous front separating the interior fromthe exterior, where vertical motion is concentrated. Theeddies migrate away from the center where they orig-inated, destroying the original preconditioning eddy.

The time series shown in Fig. 4 further illustrate thevarious stages of flow evolution. Soon after the appli-cation of cooling, plume convection begins in the eddycenter, indicated by the sudden rise in the perturbationvertical velocity at the convection onset time tc. How-ever, shortly afterward weak restratification occurs

throughout the eddy interior, indicated by the drop inthe depth where dT/dz 5 0 at the restratification timetr. At the baroclinic instability onset time tbc the per-turbation azimuthal velocity begins an exponential in-crease, as meanders develop on the original eddy bound-ary. Finite amplitude instability leads to a warming ofthe upper center of the eddy as fluid is advected in fromoutside, while outside the eddy cooling continues. Thecombined action of localized convection and baroclinicinstability generates a complex secondary circulation inthe r, z plane, where r is the radial distance from theeddy center. The amplitude of this circulation continuesto increase throughout the simulation, as shown by theroot-mean-square value of the circulation streamfunc-tion f. At the breakup of the original eddy describedabove, the magnitude of lateral eddy heat flux reachesa peak. Despite the breakup of the original eddy intoseveral smaller eddies, the mean azimuthal velocity

, (where u is the azimuthal coordinate) averaged overuuu

the upper half of the volume, remains approximatelyconstant, a consequence of the conservation of totalangular momentum. Several important features of theconvecting region have therefore been identified: plumeconvection, restratification, baroclinic instability, anddestruction of the initial eddy. The timing of these pro-cesses depends on the location, with plume convectionin the outer regions existing simultaneously with bar-oclinic instability of the eddy core. We will examine theprocesses responsible for the restratification, the causesof this sequence of events, and the consequences ofbaroclinic instability for the density redistribution below.

a. Upright convection

Initially, the whole domain is stably stratified, but thedestabilizing surface heat flux, communicated by dif-fusion, leads to the development of a negative stratifi-cation in the upper layer. When this reaches a sufficientmagnitude to overcome the effects of diffusion, viscos-ity, and rotation, convective mixing will occur, that is,when the local Rayleigh number (Ra) is greater than thecritical value Rac (Chandrasekar 1961) at the relevantTaylor number (Ta), where

3 2 4gaDTd f dRa 5 , Ta 5 . (3)

2kn n

Here DT is the temperature difference across the neg-atively stratified layer, and d is the depth of the nega-tively stratified layer.

We estimate the time of convective onset by exam-ining the root mean square of the perturbation verticalvelocity field as a function of radial and vertical dis-tance: rms[w9](r, z) 5 ([w(u, r, z) 2 w(r, z)]2 u

)1/2, whererepresents the azimuthal average with respect to the

u

origin at the center of the eddy and w(r, z) is the azi-muthally averaged vertical velocity. The activity ofsmall-scale plumes at the onset of convection leads toa rapid rise in magnitude of rms[w9].

MAY 1998 951L E G G E T A L .

FIG. 5. The convection onset time prediction compared with theactual onset time for all simulations listed in Table 1. The predictionis made by evaluating the diffusive modification of the stratificationand identifying the time at which the Rayleigh number exceeds thecritical Rayleigh number: Ra . Rac.

We evaluate the applicability of linear stability theoryin determining convection onset time by numericallyintegrating the one-dimensional diffusion equation topredict the temperature T in isolation from advectiveand three-dimensional effects,

2]T ] T5 k , (4)

2]t ]z

and diagnosing d, DT, Ra, and, hence the time whichthe critical Rayleigh number is reached, t(Rac), fromthis evolving temperature profile. We use as initial con-ditions the temperature profile in the center of the eddyand apply the same top and bottom boundary conditionsas in the full three-dimensional simulation. The resultsobtained in this way for the predicted convection onsettime t(Rac) are shown compared with the actual con-vection onset time deduced from the Wrms fields in thecorresponding simulations in Fig. 5. The good agree-ment between the two supports the hypothesis that con-vection begins when diffusion has sufficiently alteredthe stratification. However, since the Raleigh number isstrongly dependent on the model diffusivity and vis-cosity and oceanic values of these parameters are severalorders of magnitude smaller than in the model, con-vection in the ocean will begin almost immediatelywhen significant cooling occurs.

After onset of convection, the convective plumes pro-gressively erode the stable stratification and deepen theconvectively homogenized layer. We estimate the depthof this near-homogeneous layer by examining the pointat which a transition occurs from slightly negative strat-ification to positive, stable stratification below. Alter-natively we could use the point at which stratificationis some fraction of its original stable value. In practice,for this strongly rotating case (with convective Rossbynumber Roc ø 0.7) the convectively mixed layer is

slightly negatively stratified, as shown by Julien et al.(1996) for the Rayleigh–Benard problem, and the twochoices of transition point give equivalent results. Weare interested in the ensemble effects of the plumes and,therefore, average the temperature field in the azimuthaldirection. We can then examine the variation of con-vectively homogenized layer depth with r. This estimateof the convectively homogenized layer depth is shownin Fig. 6.

A prediction for the convectively homogenized layerdepth can be made using one-dimensional nonpenetra-tive mixing arguments (Turner 1973). In contrast to thesituation prior to convection, it is assumed that onceadvective mixing begins, it dominates diffusion, whichis now neglected. The layer influenced by the surfaceforcing is assumed to be completely mixed, with a neu-tral stratification, and integral heat balance can then beused to determine the depth of that layer. Then,

dh B05 , (5)2dt N hh

where h is the total depth of the homogeneous layer, B0

is the applied buoyancy flux, and is the initial strat-2N h

ification at the depth z 5 2h. In our case N 2 is notinitially uniform with depth. By integrating this equa-tion, we obtain

h12t 5 N h9 dh9. (6)E h9B0 0

Note that if N 2 is constant, this yields the well-knownformula (Turner 1973) h 5 (2Bt)/N. In our simula-Ïtions N 2 is nonuniform, but this integral can still becalculated exactly. Far from the eddy center we have

2 21 N h1 2 2t 5 1 N [l (1 2 exp(2h /l ))2 q q5B 20

2 l h exp(2h /l )] , (7)q q 6as shown previously for a similar stratification by Al-verson and Owens (1996). Unlike their study, the rel-atively long decay-scale lq in our calculations meansthat the mixed-layer deepening rate is always decreasingfor h , Lz.

Since N 2 varies as a function of distance from thecenter of the eddy, as well as with depth, faster con-vective-layer deepening is predicted in the eddy centerthan in the exterior flow. The diagnosed convective-layer depth is compared in Fig. 6 with the one-dimen-sional nonpenetrative prediction at several different ra-dial distances from the eddy center. Initially the con-vective-layer depth closely matches the prediction, al-though smaller in magnitude due to both diffusive ef-fects and the somewhat arbitrary choice of definitionfor the mixed layer depth. However, later the depth ofthe negatively stratified layer levels off and even de-


FIG. 6. The depth of the negatively stratified layer as a function of time at several different radii (r), for the simulation *, (solid line) ascompared with the one-dimensional nonpenetrative mixed-layer depth prediction (dashed line).

creases within the eddy interior, while continuing todeepen in the exterior region. Hence a restratificationprocess, which we will examine in more detail below,occurs at this time. Since restratification is confined tothe eddy region, at late times the homogeneous layer isdeepest in the exterior.

A deeper homogeneous layer outside the eddy mightsuggest that the eddy is providing the reverse of a pre-conditioning mechanism, and actually hindering mixing.However, we stress that the absence of a homogeneouslayer in the eddy center does not imply an end to deepmixing and cooling. As the analysis of heat fluxes willshow, modification of the stratification and vertical mix-ing continue to take place in the eddy interior, but inan environment of slightly stable average stratification.

Given initial correspondence between prediction anddiagnosed convectively homogenized layer depth, one-dimensional mixing can be used to determine scenarioswhere a significant difference in depth of the convectionbetween interior and exterior and hence localized deepconvection may occur. Figure 7 shows the predicted

convective-layer depth both inside and outside the eddyfor several different background stratifications: (a) a uni-form background stratification, (b) a weak, exponen-tially decaying background stratification, and (c) astrong exponentially decaying background stratification,as in the reference calculation. Maximum localization(i.e., the largest possible ratio between mixed layerdepths inside and outside) is achieved when the exteriorregion has slow mixed layer deepening and the interiorregion has continuing rapid mixed layer deepening, asin (c), with surface intensification of the backgroundstratification.

b. Restratification and slantwise convection

As shown above, after restratification the negativelystratified layer decreases in depth within the eddy de-spite the continued forcing. We estimate the restratifi-cation time as the moment at which the homogeneouslayer depth ceases to increase. Figure 8 shows restra-tification time as a function of r for several different

MAY 1998 953L E G G E T A L .

FIG. 7. Mixed layer depths predicted by one-dimensional nonpe-netrative mixing for (a) strong, uniform background stratification; (b)weak, exponential background stratification; and (c) strong, expo-nential background stratification. The solid line shows the mixed layerdepth in the center of the eddy; the dotted line shows the mixed layerdepth far outside the eddy. In all three cases, the eddy anomaly hasa depth scale of 1 km, a horizontal scale of 10 km, and the maximumstrength possible.

FIG. 8. The restratification time as a function of distance from theeddy center for several different simulations, identified by the sym-bols defined in Table 1. The restratification time is determined as thetime at which the depth of the negatively stratified layer ceases toincrease, that is, when dh/dt # 0. (a) The high-resolution case (thereference simulation), (b) four simulations for initial eddies of thesame strength but different radius, and (c) three simulations for initialeddies of the same radius but different eddy strength.

simulations. The restratification begins close to, but notat, the center of the eddy; rapidly spreads throughoutthe eddy interior; and then propagates slowly outwardin the exterior region. The curves are limited by the


FIG. 9. Azimuthally averaged potential vorticity, 5 , for the high-resolution simulation (3) at (a) 0.8u uQ (z 1 f ) ·=aT

days and (b) 2.4 days after onset of cooling. The contour spacing is 1 3 10212 s21 m21; dotted lines show regions of negativepotential vorticity.

FIG. 10. Angular momentum surfaces (solid lines) and isotherms (dotted lines) for the high-resolution simulation (3) at (a)t 5 0 and (b) 1.6 days showing the alignment of isotherms with the sloping angular-momentum surfaces in the upper part ofthe volume after the onset of convection.

duration of the integration, but for most simulations(except for the weakest eddy strength e 5 0.2 wherethere is no restratification) restratification appears to begradually extending to larger and larger radii, suggestingthat restratification would eventually take place every-where.

One restratification mechanism is the lateral heat fluxby the near-surface secondary circulation, which advectsof warmer water into the upper part of the eddy, therebystabilizing the stratification in this region. The natureof this secondary circulation and the heating pattern thatresults are discussed in more detail in later sections.

Another possible cause of interior restratification issymmetric instability, a combined gravitational/centrif-ugal instability, in which fluid is mixed axisymmetri-cally (Ooyama 1966; Hoskins 1974). In the absence ofviscosity and diffusion, the axisymmetric motion mixesfluid along angular momentum surfaces, which, if tilted

as when thermal wind shear is present, implies a slant-wise rather than vertical mixing. Slantwise mixing willtend to produce sloping isotherms and hence positivevertical temperature gradients, providing a means of re-generating a weak stratification. A necessary conditionfor symmetric instability, the presence of negative po-tential vorticity Q, is satisfied by the flow prior to re-stratification (Fig. 9a), while after restratification an areaof marginally positive potential vorticity with weak gra-dients (Fig. 9b) indicates mixing has taken place. How-ever, this mixing of potential vorticity would take placein either upright convection or slantwise convection dueto symmetric instability.

For evidence of symmetric instability we thereforeexamine the angular momentum (defined by 5 fr2/2

uM

1 r ) and isothermal surfaces for alignment and de-uuu

viation from the vertical after restratification. Shown inFig. 10 are overlaid contour plots of angular momentum

MAY 1998 955L E G G E T A L .

FIG. 11. The normalized scalar product of the gradient vectors for temperature and angular momentum for theu u u u(=T ·=M )/(|=T | |=M |)high-resolution simulation (3) shown (a) prior to convection, t 5 0, and (b) after restratification, t 5 1.6 days. Contour spacing is 0.05,and dotted contours indicate where the scalar product .0.9. Note that results are only shown below a depth of z 5 125 m since above thisdepth the gradients pass through zero, reducing the normalization factor (the denominator of the above expression) and increasing the errorof the calculation.

and temperature before and after restratification. Theangular momentum surfaces, which are tilted in thesheared zone of the eddy, change shape very little duringthe course of the integration. Hence mixing along an-gular momentum surfaces in response to the creation ofnegative potential vorticity leads to sloping isothermsin the sheared region in particular. Note that the hori-zontal and vertical axes are not shown on the same scalein Fig. 10 and the slope is, in fact, significantly differentfrom vertical. This slope ensures a small, positive tem-perature gradient, enough to explain the restratificationidentified by examining the sign of that temperaturegradient. Since the angular momentum surfaces are tilt-ed initially, the timescale for this slantwise plume con-vection is determined only by the time necessary togenerate and respond to the negative potential vorticity,that is, the convection onset time. This is in contrast todifferentially forced convection, where the angular-mo-mentum surface tilt is caused by convection and sub-sequent geostrophic adjustment processes and thereforetakes a finite time to occur (Haine and Marshall 1998).

We confirm that the isotherms and angular momentumsurfaces are aligned after restratification by examiningthe normalized scalar product of the gradient vectorsfor temperature and angular momentum,

u u=T ·=M

, (8)u u|=T ||=M |

shown in Fig. 11. The region where angular momentumand temperature gradients are correlated by greater than0.9 increases considerably following the onset of con-vection and corresponds to the restratified area and issignificantly greater than the alignment between tem-perature surfaces and the vertical. The exterior region,

where the angular momentum surfaces are vertical, con-tinues to undergo upright convection while the interiorhas restratified. At much later times, when baroclinicinstability dominates the flow (see the next section), theisotherms become considerably flatter than the angularmomentum surfaces; baroclinic instability, since it isnot axisymmetric, does not conserve angular momen-tum.

In conclusion, the rapid restratification of the interiorof the eddy is consistent with slantwise plume convec-tion in the presence of tilted angular momentum sur-faces. The restratification effect of this symmetric in-stability is reinforced, as we shall note later, by theeffects of the secondary circulation in the near-surfacelayers.

c. Baroclinic instability

After restratification of the interior, smaller eddiesdevelop along the edge of the original eddy throughbaroclinic instability. The zone of greatest eddy activitymigrates outward, gradually restratifying the domain.We estimate the growth rate of the instability generatingthese eddies by examining the growth of perturbationsin the velocity field, that is, by examining

uu 2 1/2rms[u9] 5 [(u (u, r, z) 2 u (r, z)) ] , (9)u u u

where uu is the velocity in the azimuthal direction andsimilarly for ur. The onset of convection gives rise toan initial jump in the values of these perturbation ve-locities, associated with the plume velocities. Followingthis initial rise, these quantities grow exponentially, asshown by the straight-line sections of the lognormalplot, indicating increasing deviation from axisymmetric


FIG. 12. The rms perturbation azimuthal velocity, rms( ) 5u9u[ u]1/2, as a function of time for (a) the high-resolutionu 2(u 2 u )u u

reference simulation, (b) initial eddies of the same strength but dif-ferent radius, and (c) initial eddies of the same radius but differentstrength. In all cases rms( ) has been averaged over a horizontalu9udistance of 4 km centered at the edge of the eddy where the shearis strongest and over the upper half of the fluid where most of thebaroclinic activity is concentrated. The dashed lines show the bestfit to an exponential function from which the growth rate s is cal-culated. The baroclinic instability onset time is determined as thetime when rms( ) first matches this exponential function.u9u

FIG. 13. The baroclinic-instablility onset time, as deduced from theperturbation velocities in Fig. 12. compared with the convection onsettime. The solid line shows where they are equal.

motion (Fig. 12). We associate this growth in the hor-izontal velocity perturbations with baroclinic instability,rather than increasing plume-scale activity, since theperturbation vertical velocity does not show such anexponential increase. The onset time for the baroclinicinstability can be estimated by identifying the point atwhich the exponential growth begins in the lognormalplot. Baroclinic instability begins shortly after convec-tion (Fig. 13), indicating that convection generates theconditions necessary for baroclinic instability.

By including random perturbations to excite insta-bility we have verified numerically that, in the absenceof cooling, no instability is generated for these initialconditions, unless the background stratification N2 isreduced or the rotation rate is increased (thereby re-ducing the deformation radius Lr). Hence we concludethat the necessary condition for instability to occur isa reduction in overall stratification sufficient to increasethe ratio lr/Lr to a sufficiently large value. [For a qua-sigeostrophic two-layer vortex, lr/Lr $ 2 is a necessaryrequirement for instability (Pedlosky 1985).] Note thecontrast to the disk-forcing scenario (e.g., Coates et al.1995) where lr/Lr is determined by the initial configu-ration of stratification and cooling area, but a baroclinicshear must be developed through the cooling. In thelocalized eddy configuration, shear is present in the ini-tial conditions, but the lr/Lr ratio must be decreased byerosion of the stratification.

We estimate the instability growth rate s over thetime period in which the exponential growth occurs bya least squares fit to the lognormal plot. The deducedgrowth rates are compared in Fig. 14 with the Eadyformula (Eady 1949) for the growth rate of a baroclinicdisturbance

f f ]u /]zs ; ; , (10)

NÏRi

where

MAY 1998 957L E G G E T A L .

FIG. 14. The baroclinic-instability growth rate, as deduced fromthe perturbation velocities in Fig. 12, compared with the Eady for-mula, v } f / Ri. Also shown (solid line) is the best linear fit. AÏlinear fit with intercept of 0 would also satisfy the contraints of thedata.

2]u

2Ri 5 N .@1 2]z

Since the vertical shear changes little during the courseof the integration, we use the ]uu/]z values prior toconvection but after the boundary layer flow has beenspun up, averaged over the eddy sheared zone, and theaveraged stratification in the upper part of the flow. Wenote a surprisingly good agreement in both the scalingand the constant of proportionality (0.3 for the Eadyproblem), given that the Eady formula strictly appliesonly for quasigeostrophic flows with spatially uniformshear. Eddies of different radii but the same strengthhave approximately the same growth rate since all havesimilar average N (all have N 2 5 0 at the top and centerof the eddy and N 2 5 outside the eddy) and small2N 0

differences in ]u/]z. As the strength of the eddy de-creases, the growth rate decreases due to the increasein the average N 2 and decrease in average ]u/]z. Thedifference between the two cases, * and 3, which havethe same parameters except for resolution and viscosi-ties, is due solely to the differences in the strength ofthe integrated velocity field, which is affected by dif-ferences in the viscous boundary-layer flow after theinitial spinup has occurred.

By comparison, disk-forcing scenarios, where theshear is generated through the geostrophic adjustmentof the locally cooled region, have ]u/]z proportional toN (velocity magnitudes increase due to the forcing, butso too does the depth scale h) so that s ; f (Coates etal. 1995). The different mechanism for generation ofthe shear means a different growth rate, although in bothscenarios one which is well predicted by the Eady for-mula.

The length scales of the smaller eddies generated bythe instability are difficult to estimate since the presenceof the initial cyclonic flow distorts their shape. To within

the large measurement error we cannot distinguish be-tween an instability eddy radius independent of the ini-tial eddy size and one that scales as , as suggested1/3lr

by Visbeck et al. (1996) for the disk-forcing scenario.The average instability eddy diameter is about 7.5 kmfor all low-resolution cases with f 5 1024 s21. This maybe compared with an approximate upper limit on thedeformation radius of Lr(max) 5 N0(max)lq/ f 5 8.7km.

d. Secondary circulation

In addition to baroclinic eddies and small-scaleplumes, localized convection may generate a large-scalesecondary circulation in the r, z plane, which we in-vestigate by examining the streamfunction f in the r,z plane, defined by

]f 1 ]u uu 5 ; w 5 2 (rf), (11)r ]z r ]r

where ur is the radial velocity and w is the verticalvelocity. Figure 15 shows instantaneous images of f(a) prior to convection when it is driven solely by theadjustment of the geostrophic eddy to the surfacestress-free boundary condition and (b) during plumeconvection. Note that azimuthally averaged velocitiesare small (of the order of a few millimeters per sec-ond) compared to both the plume-scale velocities andthe geostrophic azimuthal velocity, in agreement withresults of previous studies (e.g., Send and Marshall1995). However, this small secondary circulation, as-sociated with flow at the surface toward the center ofthe eddy across isopycnal surfaces, leads to a signif-icant net flow of heat toward the center of the eddy(see Fig. 20 and the later discussion of heat fluxes).After the onset of baroclinic instability (Fig. 15c) thesecondary circulation at the edge of the eddy is nowin the reverse direction to the exterior, with the con-vergence of these two cells in the upper layer at theeddy edge. With further development of baroclinicinstability (Fig. 15d) this pattern migrates outwardand becomes even more complicated with multiplecells of circulation. To examine the factors that maylead to the generation of a secondary circulation, weconsider the azimuthal vorticity tendency equation:

2]z ]u ] u ]T ]u u u5 f 1 2 ga 2 [u ·=u ]r1 2[ ]]t ]z ]z r ]r ]z

]21 [u ·=w] 1 n¹ z , (12)u]r

where zu is the azimuthal vorticity, f is the Coriolisparameter, uu is the azimuthal velocity, and u is the fullvelocity vector. Averaging this equation in the azimuthaldirection and making use of the nondivergence relation= ·u 5 0, we have


u u uu 2 2]z ]u ] u ]T ] 1 ] ] ] 1 ] ] ]u u u uu u u 2 2 25 f 1 2 ga 1 r 2 (wu ) 2 r (u ) 1 (w ) 1 n¹ z ,r r u21 2 5 6[ ] [ ] [ ] [ ]]t ]z ]z r ]r ]r r ]r ]z ]z r ]r ]r ]z

(A) (B) (C)(13)

where term A represents the deviation of the flow fromaxisymmetric cyclostrophic balance, term B representsthe combined effects of advection and stretching of vor-ticity, and term C is the diffusion of vorticity. The vor-ticity and streamfunction are related through

1u 2z 5 ¹ 2 f. (14)u r,z 21 2r

The terms may be further separated into those due tothe mean circulation and those due to eddies:

u uu u 2 2]z ]u ] (u ) ]T ] 1 ] ] 1 ] ] ] ]u u uu u u 2 25 f 1 2 ga 2 r (u9 ) 1 r 2 (w9u9) 1 (w9 )r r21 2[ ] [ ] [ ] [ ]]t ]z ]z r ]r ]z r ]r ]r r ]r ]z ]r ]z

(1) (2) (3) (4)

u2 2] 1 ] ] 1 ] ] ] ] ] (u9 ) uuu 2 u u u 2 22 r(u ) 1 r 2 (w u ) 1 (w ) 1 1 n¹ z , (15)r r u2 1 2[ ] [ ] [ ]]z r ]r ]r r ]r ]z ]r ]z ]z r

(9)(5) (6) (7) (8)

where primes represent the perturbation from the azi-muthal average.

Figure 16 shows the significant terms in this equation(terms 1, 2, and 3) and the total, almost entirely deter-mined by these three terms, (a) during plume convec-tion, (b) after onset of baroclinic instability, and (c) laterin the development of baroclinic instability. Duringplume convection, the generation of negative azimuthalvorticity is dominated by the small deviation from ax-isymmetric cyclostrophic balance in the boundary layer,with an Ekman-like inflow, which tends to force theflow back to a balanced state. Mixing along angularmomentum surfaces generates a relative warming in theeddy interior and a tendency for positive azimuthal vor-ticity through the imbalance term, which is approxi-mately cancelled by the second derivative of , re-

uw9u9

sulting from slantwise transports. The initial inflow cir-culation in the surface layers carries fluid across iso-thermal surfaces, hence leading to a warming in theinterior of the eddy and reinforcing the restratificationdue to symmetric instability.

After the onset of baroclinic instability the generationof positive azimuthal vorticity by the imbalance termdominates in the eddy interior, generating a reverse sec-ondary circulation cell. Horizontal eddy transportscaused by baroclinic instability, indicated by the secondderivative of the horizontal eddy velocity variance

, export the azimuthal vorticity from the eddy centeru2u9r

to the exterior. Eventually, since individual eddies move

at different speeds the scattered distribution of the ed-dies leads to the appearance of multicellular patterns inthe azimuthally averaged signal, as seen in term 3.

For comparison, the secondary circulation generatedin a disk-forcing simulation, similar to that of Visbecket al. (1996) (run C in Table 1), is shown in Fig. 17before and after the onset of convection and during thedevelopment of baroclinic instability. The magnitude ofthe secondary circulation is similar to the eddy local-ization simulation of identical forcing magnitude, back-ground stratification, and horizontal length scale but isinitially concentrated at the front between the convect-ing region and exterior. There is surface inflow towardthe convecting region and downward flow at the edgeof the convection disk. As time proceeds, this secondarycirculation pattern expands inward with net inflow even-tually occurring over most of the upper layer. As theflow becomes dominated by multiple eddies, the sec-ondary circulation shows a multicellular pattern as inthe eddy localization scenario. A more quantitative com-parison is difficult since the difference in initial strati-fication in the two cases leads to different timescalesfor evolution and different depths of the convected layer.

e. Gradient wind balance

We have already considered how deviations from anaxisymmetric cyclostrophically balanced circulationforce the azimuthal vorticity. Is the circulation as a

MAY 1998 959L E G G E T A L .

FIG. 15. The streamfunction of the mean circulation in the r, z plane for the high-resolution reference simulation (3) shown (a) prior toconvection t 5 0.3 days, (b) during plume convection t 5 1.6 days, (c) after onset of baroclinic instability t 5 3.1 days, and (d) duringfurther development of baroclinic instability t 5 4.5 days. The contour spacing 5 0.2 m2 s21. Solid lines indicate positive values of f(anticlockwise motion), and dotted lines indicate negative f (clockwise motion).

whole unbalanced, or simply no longer axisymmetric?We anticipate that for motion on the geostrophic scales,dominated by the effects of rotation and stratification,deviations from balance will be small. For full 3D bal-ance, without assuming axisymmetry, we would have

] ]2ga¹ T 2 ( f z ) 2 (2J (y , 2u)) 5 0, (16)h z]z ]z

where 5 ]2/ 1 ]2/ is the horizontal Laplacian,2 2 2¹ ] ]h x y

and J is the horizontal Jacobian; J (y , 2u) incorporatesthe cyclostrophic terms. This form of the balance re-lation is obtained from the divergence of the horizontalmomentum equation assuming geostrophic and cyclos-trophic balance, followed by the vertical derivative,eliminating pressure through the hydrostatic relation(Norton et al. 1986). To determine the degree of balancewe therefore calculate these three terms and their re-

sidual. Since small-scale convective plumes are obvi-ously far from balanced, we examine the larger-scalemotions only by filtering velocity and temperature fieldswith a low-pass filter designed to eliminate variationson the plume scale but retain motions on the scale ofthe instability eddies. The flow is largely balanced inthe interior (Figs. 18 and 19), with a ratio between therms values of the residual and component terms of about30%. Near the top surface, the flow is not balanced, dueto the reduction ]( fzz)/]z there, as required by the mo-mentum boundary conditions. There is a strong corre-lation between the regions of imbalance in this upperlayer and the regions of strong convergence and diver-gence of the horizontal velocity field. Hence a balancedmodel may reproduce many of the aspects of the re-distribution the density field within the interior of thedomain, provided the vertical plume-scale mixing pro-


FIG. 16. The dominant terms in the azimuthal vorticity equation for the high-resolution reference simulation (3) (a) during plume convectiont 5 1.6 days, (b) after onset of baroclinic instability t 5 3.2 days, and (c) later in the development of baroclinic instability t 5 4.5 days.Contour spacing is 2 3 1029 s22. Positive vorticity tendency is shown in solid contours, negative vorticity tendency in dotted contours.

cesses are parameterized. However, balance does notapply in the momentum boundary layer in the presenceof density anomalies, and this region would have to betreated with care.

f. Heat fluxes

Ultimately the features of the flow described above—plume convection, baroclinic instability, and secondarycirculation—are of most interest for the way in whichthey redistribute the density field. The important pro-

cesses at each stage in the evolution of the localizedconvection region can be identified by examining thecomponents of the temperature tendency equation

]T2 25 2u ·=T 1 k¹ T 5 2= · (uT ) 1 k¹ T, (17)

]t

where we have used the fact that the flow is incom-pressible and = ·u 5 0. If we azimuthally average allterms and separate both the flow and the temperaturefield into azimuthally averaged and perturbation com-

MAY 1998 961L E G G E T A L .

FIG. 17. The streamfunction of the mean circulation in the r, z plane for a simulation of convection localized by a disk of forcing (radiuslr 5 10 km) and horizontally uniform initial conditions (a) prior to convection t 5 0.05 days, (b) during plume convection t 5 6.6 days,(c) at the onset of baroclinic instability t 5 11.9 days, and (d) during later development of baroclinic instability t 5 16.3 days. The contourspacing is identical to that in Fig. 15. The parameters used are identical to those of run * but with horizontally uniform stratification.

ponents, we can rewrite the temperature tendency equa-tion as

u]T ] 1 ] ]u u u

u u5 2 (w T ) 2 r(u T ) 2 (w9 T9)r]t ]z r ]r ]z

(2)(1)

1 ] u u22 r(u9T9) 1 k¹ T , (18)rr ]r

(3) (4)

where w is the vertical velocity and ur the radial velocity.Here we can associate each term on the right-hand sidewith a different process: the first two terms (1) representthe temperature redistribution by the secondary circu-lation, (2) represents vertical fluxes by both convectiveplumes or baroclinic eddies, (3) represents the lateralheat fluxes by baroclinic eddies, and the final term (4)

represents the effects of diffusion. Of particular signif-icance for restratification and the renewal of deep andintermediate water is the distribution of these fluxes inthe vertical. In horizontally homogeneous convection,all the density redistribution is effected by plume-scalevertical fluxes .

uw9T9

Figure 20 shows contour plots of heat flux diver-gence after azimuthal averaging for several differencestages in the flow evolution: during plume convection,after onset of baroclinic instability, and later in thedevelopment of baroclinic instability. The diffusivecomponent is not shown, but is included in the total.Initially strong cooling in the center of the eddy byvertical eddy fluxes indicates plume-scale convection,while the secondary circulation provides a lateral heatflux in the upper layers. The dipolar pattern of heatflux is generated by a single secondary circulationcell, with inflow in the upper layers and outflow be-


FIG. 18. The components of the gradient-wind balance relation for the run *, evaluated at t 5 10 days at a depth of z 5 62.5 m belowthe top surface. (a) The residual of the gradient-wind balance relation ga T 2 ]( fvz)/]z 2 ](2J (y , 2u))/]z and (b) ga T, (c) ]( fvz)/]z,2 2¹ ¹h h

and (d) ](2J (y , 2u))/]z. At this time the flow is dominated by baroclinic eddies. Contour spacing is 1.5 3 1029 s22 m21. Velocity andtemperature fields have been filtered to remove plume-scale signals.

low, transporting fluid horizontally across isopycnals.Since the secondary circulation is strong from theonset of convection and the horizontal temperaturegradients are large at this time, the heat fluxes by thesecondary circulation are comparable to those of theplume-scale quantities.

After the onset of baroclinic instability, the secondarycirculation generates a surface thermal front (at the lo-cation marked by an arrow) through cooling on the in-terior side by surface flow moving from the cold eddyinterior outward and warming on the exterior side byinward flow in the surface layers. Baroclinic eddies,

MAY 1998 963L E G G E T A L .

FIG. 19. As for Fig. 18 but at a depth 312.5 m below the top surface.

which feed off this front, result in increased magnitudelateral eddy fluxes, which partly balance the mean flowfluxes. The vertical eddy fluxes extend over the wholedomain through the surface layer with their deepest ex-tent at the edge of the frontal zone and shallower extentin the interior. Due to the dominance of the mean flowwarming near the surface just outside the front, there is

a net surface warming in this region that, as the frontmigrates outward, propagates through the volume, re-stratifying the surface layer.

Later in the development of baroclinic instability, thesecondary circulation is more complicated with multi-cellular patterns of motion due to variability in locationof the instability eddies generating this circulation. This


FIG. 20. The components of the heat equation for the high-resolution reference simulation (3) (a) during plume convection t 5 1.6 days,(b) after onset of baroclinic instability t 5 3.2 days, and (c) later in the development of baroclinic instability t 5 4.5 days. Contour spacingis 16 3 1028 K s21 and regions of cooling are shown with solid contours, while regions of warming are shown with dotted contours. Theapproximate position of the front referred to in the text is marked by an arrow.

complexity is reflected in the patterns of heat flux. Re-stratification is therefore not simply a uniform inflowof stratified water from the exterior near the surface withan outflow of dense water below.

In addition to the changes in temperature induced by

the advective fluxes, diffusion also plays a role, confinedprincipally to cooling in the uppermost, diffusive bound-ary layer, and gradual weakening of the stratification inthe interior, particularly in lower resolution, more dif-fusive simulations.

MAY 1998 965L E G G E T A L .

FIG. 21. The maximum amplitude lateral eddy heat flux as a func-tion of time for (a) the high-resolution simulation, (b) simulationswith identical eddy strength and different radius, and (c) simulationswith same radius but different strength.

g. Eddy breakup

As baroclinic instability continues, eventually the am-plitude of the small-scale eddies becomes so large thatthe original eddy is completely torn apart. In the disk-forcing scenario, the large amplitude instability leads toa quasi-equilibrium between loss of heat from the forc-ing region and the lateral transport of heat by the eddiesinto this region. Does such a quasi equilibrium exist inthe eddy-localization scenario? As shown, in this sce-nario the site of greatest vertical heat flux is not fixedby the forcing, but migrates outward as the baroclinicinstability progresses. Hence, we cannot separate thedomain into interior and exterior regions as in the disk-forcing scenario. However, examination of the time se-ries of the eddy lateral heat fluxes (Fig. 21) shows that,after a period in which the heat flux increases expo-nentially, it then decreases. The peak heat flux may cor-respond to the maximum possible extraction of energyfrom the large-scale flow. An important difference be-tween this scenario and the disk-forcing case is that thebaroclinic current is not continuously maintained. Asthe instability eddies homogenize the flow, the lateralheat transports may become negligible at large times(although our limited duration simulations do not allowas to verify whether this is the case). From Fig. 21 theinitial growth rate is identical for simulations with initialeddies of different radius but identical strength, as de-duced earlier from the eddy velocities. Initial eddies ofdecreasing strength give rise to decreasing growth rates.The time at which the maximum value is reached ap-pears to increase with increasing radius (as shown forthe disk-forcing scenario by Visbeck et al. 1996) andwith decreasing strength, but cannot be determined withsufficient accuracy to identify a scaling exponent. Thehigh-resolution case appears to reach the maximumsomewhat faster than its low-resolution counterpart; thiscan be qualitatively explained by the much earlier onsetof convection and hence baroclinic instability.

h. Net effects of localized convection

To understand the role of localized convection in thelarge-scale ocean circulation, the net effect of the het-erogeneous initial conditions on the final water massproperties must be quantified. We have shown that thereis a complicated three-dimensional redistribution ofdensity as a result of the eddy in the initial state: doesthis transition from one-dimensional to three-dimen-sional mixing occur if strong surface forcing is only oflimited duration (as is often the case in the ocean)? Byrepeating one of our simulations (with eddy radius 510 km, f 5 104 s21 and maximum eddy amplitude) atintermediate resolution (nx 5 ny 5 256; nz 5 17) withthe cooling switched off at t 5 43/ f (run A in Table 1),(Fig. 22) and without cooling at all to compare the ef-fects of diffusion alone (run B in Table 1), we confirmthat this short duration of cooling is sufficient to initiate

the three-dimensional evolution of the density field bythe mean circulation and baroclinic instability (indicatedby the maximum amplitude of the streamfunction of thesecondary circulation). For similar simulations with ces-sation of cooling before significant baroclinic instability,the secondary circulation decayed rapidly after coolingwas switched off.


FIG. 22. The maximum amplitude of the streamfunction f (r, z) for(a) no cooling (dotted line), (b) continued cooling (solid line, with*), and (c) cooling switched off at t 5 5 days (dashed line).

FIG. 23. The temperature change for the high-resolution simulation(3) as a function of time averaged over a strip 0.5 km wide and 0.4 kmdeep at the center of the eddy (solid line), 4 km from the center of theeddy (dotted line), and 8 km from the center of the eddy (dashed line).

FIG. 24. The temperature field, averaged over the top 400 m, as afunction of radius for the high-resolution simulation (3) at the beginning(solid line) and end (t 5 6.1 days) (dotted line) of the integration.

The net effect of the heterogeneity on the redistri-bution of density can be illustrated by time series of thechange in temperature (Fig. 23). Important spatial vari-ations in the temperature change are evident: coolingpersists until the end of the simulation far from the eddycenter, but diminishes when finite amplitude baroclinicinstability develops close to the eddy center. In the verycenter of the eddy there is a net warming toward theend of the integration, when the original eddy breaksup. Hence, even though early convection penetratesmost deeply in the eddy region, the secondary circu-lation and baroclinic instability reduce the net coolingthere and the greatest cooling occurs outside the eddy.As a result the horizontal temperature gradients (Fig.24) and vertical shear of the cyclonic flow (Fig. 25) arereduced.

What impact do these spatial variations in densityredistribution have on the volume-averaged quantities?The horizontally averaged temperature profile for thereference run within the central region (of radius 18.75km) to which the baroclinic instability is confined overthe length of our integration is shown in Fig. 26, alongwith that obtained by horizontally averaging the initialconditions over the same area, and then applying theforcing (run D in Table 1). Important qualitative dif-ferences are visible: in the absence of an eddy anomalyin the initial flow, the upper part of the flow is approx-imately neutrally stratified down to the depth to whichconvection has penetrated. Below that depth the onlydensity changes compared to the initial conditions arethe result of diffusion at the bottom boundary. In con-trast, the averaged temperature field in the eddy case isstably stratified everywhere. The difference between thetemperature fields in the two cases (Fig. 27) shows thatthe upper layer is warmer in the case with the eddyinitial conditions than in the homogeneous case, whilea region below is cooler. This implies that the net effect

of the eddy is to enhance the penetration of the con-vection not through vertical boundary-layer entrain-ment, as in homogeneous convection, but through the3D processes of secondary circulation and baroclinicinstability. The magnitude of these differences in theupper layer is equivalent to 0.5 days of cooling (10%of the total cooling duration). This obviously has im-plications for parameterization of convection in GCMs,where the vertical resolution is usually much higher thanthe horizontal resolution: conventional convective par-ameterizations assume that fields within the grid-cell arehomogeneous and can therefore be represented by theirhorizontal average.

4. Discussion and conclusions

Convection may be localized by the presence of geo-strophic eddies, the long hypothesized ‘‘preconditioning’’of the ocean. Many of the features in our simulations find

MAY 1998 967L E G G E T A L .

FIG. 25. The mean azimuthal velocity in the high-resolution simulation (3) averaged over (a) z 5 2250 m → 0 m,uuu

(b) z 5 2500 m → 2250 m below the top surface, as a function of radius, at the beginning (solid line) and end (t 5 6.1days) (dashed line) of the integration.

FIG. 27. The difference between horizontally averaged temperaturefields as a function of time and depth for high-resolution simulationswith and without eddy heterogeneity in the initial temperature field. Thecontour spacing is 0.000 66 K. Solid lines indicate a warmer temperaturewith the eddy present, while dotted lines indicate cooler temperatures.

FIG. 26. The horizontally averaged temperature field as a functionof time and depth with (dotted contours) and without (solid contours)the eddy heterogeneity in the initial temperature field for the high-resolution reference simulation (3). Contour spacing is 0.0198 K.

counterparts in the observations of oceanic deep convec-tion. Thermal fronts associated with mesoscale eddy ac-tivity have been noted by Gascard and Clarke (1983) inthe Labrador Sea and Gascard (1978) in the Mediterranean.Schott and Leaman (1991) also observed a front betweenthe deeply mixed fluid and the exterior stratified fluid inthe surface salinity field. This front showed considerableevolution with time, developing large meanders, suggest-ing the presence of baroclinic instability eddies. The strongvertical velocities associated with the baroclinic instabilityeddies in our simulations are also found in the observationsof Gascard and Clarke (1983), while Schott and Leaman(1991) found narrow downwelling elements in the Med-iterranean, which could be associated with either plumesor baroclinic eddies. Eddy activity adjacent to the deepconvection region is suggested by the increased variancein the temperature field and the presence of subsurfacehomogeneous water parcels (Schott et al. 1996). The lat-eral heat fluxes found in our simulations are also indicatedin estimates of the heat content of the convecting region

from acoustic tomography measurements in the GreenlandSea (Pawlowicz et al. 1995) and Mediterranean Sea (Sendet al. 1995), while rapid restratification of the convectingregion is found in both the Mediterranean (Leaman andSchott 1991; Send et al. 1995) and Greenland Sea (Paw-lowicz et al. 1995) with a surface layer of less dense watermoving into the convected region.

Our simulations raise several questions that we hopemay be addressed by future observational programs. Inparticular, greater temporal and spatial coverage of baro-clinic instability eddy structures is required to understandtheir evolution from small-scale meanders to finite am-plitude structures of larger spatial scale. The correlationbetween these eddies and both the thermal front and therestratifying secondary circulation need to be verified.Fine-resolution temporal evolution of the mixed layer tem-perature structure is needed to examine whether slantwiseconvection may be occurring. Since we have shown thatthe presence of a single preconditioned eddy may signif-


icantly alter the average response of the ocean to a buoy-ancy loss, it is important for observational programs toascertain the mesoscale eddy activity prior to convectionas well as the fraction of the large-scale gyre circulationoccupied by deep convection.

The previous paradigm for studying the effects oflocalization on deep convection in both laboratory andnumerical simulation has been to impose a disk-shapedsurface buoyancy loss on a horizontally homogeneneousfluid. In contrast, we have generated localized convec-tion by applying a horizontally uniform forcing to avolume containing a stratification inhomogeneity. Inboth scenarios, horizontal inhomogeneities in the den-sity field and associated baroclinic currents affect theevolution. However, in the disk-forcing scenario theseinhomogeneities only develop over time in response tothe forcing, while in our simulations they are presentfrom the beginning. Hence, slantwise convection andlateral heat fluxes may occur from the onset of con-vection. In contrast to the disk-forcing scenario, theeddy initial conditions determine the growth rate of thebaroclinic instability, while the convective erosion ofstratification provides the necessary condition for in-stability to occur: a large ratio of baroclinic eddy radiusto deformation radius. In the disk-forcing case the lo-cation of deep convection is fixed, while in the eddyscenario it is free to evolve in response to changes inthe stratification. Hence, the site of deepest vertical heatflux migrates outward with the baroclinic eddies. Thethermal front is generated by the secondary circulation,rather than being imposed by the forcing.

The principal effect of high diffusivity and viscosity isto delay the onset of convection, and hence of baroclinicinstability. Large diffusive erosion of the stably stratifiedthermocline also occurs when diffusion is strong. How-ever, the qualitative features—restratification, baroclinicinstability, secondary circulation—and some quantitativescaling results—baroclinic instability growth rate—are un-affected by diffusion values, and we expect these to berelevant to the low k, n regimes of the ocean.

Observations of deep convection are often associatedwith considerable temporal variability in the surfaceforcing, whereas we have here assumed a constant, uni-form forcing. We have also considered only a singleisolated eddy, whereas several such geostrophic circu-lations may exist within the large-scale convection gyre.Future research must therefore consider the effects ofincreasing complexity in the spatial and temporal dis-tribution of forcing and ocean circulation.

Here we have ignored the processes by which the pre-conditioned eddy may be formed. The mechanisms arelikely to vary according to location: baroclinic instabilityof large-scale currents is a possibility. Regions of out-cropping isopycnals and weakened stratification, albeitwith a bottom-intensified anticyclonic flow rather than sur-face-intensified cyclonic flow, can be generated by inter-action between flow and topography (Alverson and Owens1996). Gascard and Clarke (1983) suggest topographic

Rossby waves as a preconditioning mechanism. Tidal mix-ing over topographic features would also weaken the localstratification. Cyclonic wind stress may also, through Ek-man flow and upwelling, generate a domed, outcroppingisopycnal structure on a larger spatial scale. Variations incooling on scales below the ocean deformation radiuscould also lead to small-scale-adjusted dense water cores(larger-scale localized cooling would lead to baroclinicinstability as discussed in Visbeck et al. 1996). Also ig-nored in our calculations are the later interactions betweenthe baroclinic eddies and the large-scale flow, which even-tually mix the convected fluid into the general circulation.These processes have a greater temporal and spatial spanthan our present simulations.

We have shown, through a series of numerical simu-lations, that deep convection may be localized within theweakly stratified interior of a cold-core mesoscale eddy.The convective layer depth initially agrees approximatelywith predictions using one-dimensional nonpenetrativemixing arguments. However, surface inflow of heat andslantwise convection rapidly regenerate a weak stable strat-ification in the eddy interior. Baroclinic instability beginsto develop soon after the onset of convection, with agrowth rate approximately predicted by applying linearEady theory to the baroclinic properties of the initial eddy.Lateral heat fluxes result, spreading outward as the bar-oclinic instability eddies migrate away from the originalpreconditioned eddy. A secondary circulation associatedwith the baroclinic eddies advects lighter fluid over thedenser convected fluid, gradually restratifying the domain.A thermal front develops on the outer edge of these eddies,associated with strong downwelling in a narrow region.The strong vertical heat fluxes associated with this regionmigrate outward with the instability eddies, controlling theregion of deepest cooling. The net effect of this three-dimensional redistribution of buoyancy initiated by theinhomogeneity in the original flow is an averaged densityprofile that is stably stratified everywhere with a warmersurface layer and cooler middepth than in horizontallyhomogeneous convection. Convection parameterizationschemes for coarse-resolution models unable to resolvethese inhomogeneities therefore need to be modified toinclude these effects.

Acknowledgments. We would like to acknowledgeuseful discussions in our continuing collaboration withKeith Julien and Joseph Werne. Martin Visbeck and twoanonymous reviewers provided constructive commentson an earlier version of the manuscript. SL is supportedby a NOAA Climate and Global Change postdoctoralfellowship, administered by the Universities Corpora-tion for Atmospheric Research. Support for this studywas provided by ONR Grant N00014-95-1-0316. Sim-ulations were carried out on local Silicon Graphics ma-chines and the Pittsburgh Supercomputing Center Cray-90, on Metacenter Allocation MCA93S010P.

MAY 1998 969L E G G E T A L .

APPENDIX

Eddy Initial Conditions

Our axisymmetric initial eddy is in cyclostrophic andhydrostatic balance, satisfying the following equations:

21 ]P9 uu5 1 f u (A1)ur ]r r0

1 ]P9gaT9 5 , (A2)

r ]z0

where uu is the azimuthal velocity of the eddy, T9 is thetemperature anomaly, P9 is the pressure anomaly as-sociated with the eddy, g is the gravitational accelera-tion, a is the coefficient of expansion, r is the radialdistance from the center of the eddy, and z is the verticaldistance. Consistent with the rest of this study, z 5 0at the top surface, and z 5 2Lz at the bottom, whereLz is the total depth of the volume.

We choose a form of uu that decays in an exponentialform away from the top surface, has identically zeroshear at the bottom boundary, and decays away fromthe center of the eddy:

22r r (z) L 1 z Lz zu 5 A exp 2 exp 2 exp 2 ,u 21 2 1 2 1 2[ ]l l l l lr r z z z

(A3)

where A is the magnitude of the eddy velocity, lz is thevertical decay scale of the eddy, and lr is the horizontaldecay scale of the eddy. The temperature anomaly inbalance with this flow is

2A r 1 z LzT9 5 2 f l exp 2 exp 2 exp 2r 27 1 2 1 2 1 2[ ]ga l l l lr z z z

2A 2r2 exp 2

21 2l lr r

2 2z z 2 Lz3 exp 2 exp5 1 2 1 2[ ]l l lz z z

2(L 1 z) 2L z 2Lz z z1 exp 2 2exp .2 1 2 1 2 68[ ]l l lz z z

(A4)

The maximum possible value of the eddy amplitude isdetermined from the stratification, given the constraint ofstatic stability; that is, for this maximum amplitude eddy

]T dT ]T905 1 5 0 (A5)]z dz ]z

at the top surface z 5 0, where T0 is the horizontallyhomogeneous background temperature field, and dT0/dz5 1/(ga)[ 1 exp(z/lq)], where lq is the vertical decay2 2N N1 2

scale of the background stratification. Inserting the analyticform of T9 we find the maximum magnitude of A

2 21 f l f l 2ag dTr r 02A 5 2 1 1 l , (A6)max r21 2!2 2g 4g g dz

where

2L L Lz z zg 5 2 1 exp 2 2 2 1 exp 2 . (A7)1 2 1 2 1 2l l lz z z

Other eddies have amplitude A 5 eAmax where 0 , e, 1. The ratio of eddy to background stratification atthe top surface | / | is approximately equal to e with2 2N Ne 0

a correction due to the cyclostrophic term.

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