MAY 2004 1197S P A L L

q 2004 American Meteorological Society

Boundary Currents and Watermass Transformation in Marginal Seas*

MICHAEL A. SPALL

Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

(Manuscript received 17 June 2003, in final form 6 November 2003)

ABSTRACT

The properties of watermass transformation and the thermohaline circulation in marginal seas with topographyand subject to a spatially uniform net surface cooling are discussed. The net heat loss within the marginal seais ultimately balanced by lateral advection from the open ocean in a narrow boundary current that flows cy-clonically around the basin. Heat loss in the interior is offset by lateral eddy fluxes originating in the boundarycurrent. The objectives of this study are to understand better what controls the density of waters formed withinthe marginal sea, the temperature of the outflowing waters, the amount of downwelling, and the mechanismsof heat transport within the marginal sea. The approach combines heat budgets with linear stability theory fora baroclinic flow over a sloping bottom to provide simple theoretical estimates of each of these quantities interms of the basic parameters of the system. The theory compares well to a series of eddy-resolving primitiveequation model calculations. The downwelling is concentrated within the boundary current in both a diffusiveboundary layer near topography and an eddy-driven region on the offshore edge of the boundary current. Formost high-latitude regions, the horizontal gyre is expected to transport more heat than does the overturning gyre.

1. Introduction

Marginal seas are often regions of net buoyancy lossas a result of either cooling or evaporation and, becauseof their semienclosed geometry, are regions of forma-tion of dense intermediate or bottom waters. These wa-ters generally have quite distinct water mass propertiesrelative to the open ocean and make a significant con-tribution to the global thermohaline circulation throughthe transport of heat and salt. They also influence thelarge-scale circulation throughout the water column be-cause they have anomalously low potential vorticity.Understanding what controls the formation rate andproperties of such water masses is an important part ofunderstanding the global oceanic thermohaline circu-lation. In this context, marginal seas can be relativelysmall geographically, such as the Red Sea or the AdriaticSea, or they can be of planetary scale, such as the Med-iterranean Sea or the Nordic seas. The important char-acteristic for the present study is that the marginal seais a semienclosed basin that is subject to a net buoyancyloss.

The areas of deepest convection in marginal seas are

* Woods Hole Oceanographic Institution Contribution Number10942.

Corresponding author address: Dr. Mike Spall, Woods HoleOceanographic Institution, 360 Woods Hole Rd., MS 21, Woods Hole,MA 02543.E-mail: mspall@whoi.edu

generally localized to regions with spatial scales O(100km). The properties of waters formed by such localizeddeep convection have received much recent attention.Idealized numerical and laboratory experiments haveshown that lateral fluxes out of localized deep convec-tion sites by mesoscale eddies can balance surface buoy-ancy fluxes (Visbeck et al. 1996, hereinafter VMJ; Mar-shall and Schott 1999 and the references therein). Bymaking use of stability theory, the properties (density,depth) of deep convection sites in interior regions of theocean subject to localized surface fluxes can be esti-mated with some success (VMJ; Marshall and Schott1999; Chapman 1998). These results demonstrate thatlateral heat fluxes due to mesoscale eddies can play acentral role in controlling the properties of the large-scale thermohaline circulation.

While compelling, these studies make several ratherrestrictive assumptions when compared with the actualproblem of buoyancy loss in a marginal sea. They havegenerally been formulated with a net buoyancy loss thatis localized in the interior of the basin, often with anabrupt transition in its strength. Chapman (1998) al-lowed for a gradual transition in the surface forcingstrength and found that the equilibrium properties hada different scaling, most notably with regard to the Cor-iolis parameter. On the other hand, atmospheric forcingvaries on spatial scales much larger than the oceanicdeformation radius (ice edge effects and polynyas area notable exception) and is often nonzero over much ormost of the marginal seas. The net heat flux out of theocean in these studies has also made it impossible to

1198 VOLUME 34J 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

obtain equilibrium solutions. Last, although it has beenshown that the net vertical motions in interior regionsof buoyancy loss are not significantly different fromzero, it is well known that marginal seas do experiencea net downward flux of water. Theoretical argumentsfavor near-boundary regions for the net downwelling totake place (Spall 2003; Pedlosky 2003; Spall and Pickart2001), an aspect not considered in the previous open-ocean configurations.

For marginal seas connected to the open oceanthrough a narrow strait, the net buoyancy flux at theocean’s surface must be balanced by lateral advectionthrough the strait. This requires both lateral boundariesand a reservoir of buoyant water. Lateral advection alsoappears to be important for less geographically con-strained regions of buoyancy loss such as the Nordicseas (Mauritzen 1996) and the Labrador Sea (Katsmanet al. 2004; Lilly et al. 2003; Straneo 2004, manuscriptsubmitted to J. Phys. Oceanogr.). Understanding theinteraction between the narrow boundary currents thatare required to balance the net buoyancy flux and thevast interior regions over which there are significantsurface fluxes is key to understanding the heat budgetin marginal seas and, hence, the overall process thatcontrols water mass transformation.

In a recent study, Walin et al. (2003, unpublishedmanuscript, hereinafter WBNO) explore the influenceof cooling in a marginal sea on the structure of theresulting boundary currents. Their interest was on thetransformation of a baroclinic inflowing boundary cur-rent into a barotropic outflowing boundary current. Theheat loss was concentrated in the boundary current, andwas strong in the sense that the baroclinicity of theinflowing boundary current was entirely eroded beforeit encircled the marginal sea. The forcing was designedsuch that there was little cooling of the interior of themarginal sea, and so the exchange between the boundarycurrent and the interior was weak. The circulations in-vestigated in the present paper are complementary inthat the outflowing currents are (generally, but not al-ways) baroclinic and the heat loss in the interior of themarginal sea is significant.

Numerous studies have been successful in relatingthe basin-scale buoyancy-forced overturning circulationand meridional heat transport to the basic parameters ofthe system; a recent review is given by Park and Bryan(2000). These studies differ from the current one inseveral important ways. Primarily, they are aimed at thelarge-scale overturning circulation and assume that allof the downwelling at high latitudes is balanced by adiapycnal upwelling at mid- and low latitudes. The pres-ent study makes no assumptions about where the watersupwell, or what dynamics are active in the upwellingregion. For example, the upwelling could take place viadiapycnal mixing of density in the midlatitude deepocean or it could take place within the surface mixedlayer where the isopycnals outcrop at high latitudes. Inaddition, previous studies generally did not properly re-

solve the mesoscale eddies either because of low res-olution in models (e.g., Bryan 1987; Park and Bryan2000; Spall and Pickart 2001) or, in the case of labo-ratory experiments, the deformation radius was the samescale as the basin (e.g., Park and Whitehead 1999). Thetemperature anomaly of the deep waters (required toestimate heat transport and advective speeds) is speci-fied or diagnosed in these previous studies, while it ispredicted as part of the solution in the current study.

The objective of this work is to understand better whatcontrols the properties of water mass transformation andthermohaline circulation in marginal seas subject tobuoyancy loss. The basic circulation characteristics arefirst described using an eddy-resolving general circu-lation model. Theoretical arguments are then used toestimate the parameter dependencies of the density ofwater formed in the marginal sea, the density of thewater flowing out of the marginal sea, the amount ofdownwelling within the marginal sea, and the relativeimportance of horizontal versus vertical gyres in the heattransport within the marginal sea. The theory compareswell to results from a series of numerical model cal-culations.

2. Thermohaline circulation in an idealizedmarginal sea

The thermohaline circulation in an idealized semien-closed marginal sea will first be explored using the Mas-sachusetts Institute of Technology (MIT) primitiveequation numerical model (Marshall et al. 1997). Themodel solves the momentum and density equations us-ing level (depth) coordinates in the vertical and a stag-gered C grid in the horizontal. The model retains a freesurface. A linear equation of state is used so that densityis proportional to temperature as r 5 r0 2 a(T 2 Tb),where r0 is a reference density for seawater, Tb 5 48Cis the initial temperature at the bottom, and the thermalexpansion coefficient a 5 0.2 kg m23 8C21. The modelis initialized at rest with a uniform stratification suchthat T0(z) 5 Tb 1 Tz(zb 2 z), with the maximum bottomdepth zb 5 1000 m and vertical gradient Tz 5 1.5 31023 8C m21 (N 2 5 3 3 1026 s21), although the resultshave not been found to be very sensitive to this initialstratification. The initial temperature at the uppermostmodel level (50 m) is 5.358C. The model is configuredwith uniform horizontal grid spacing of 5 km and 10levels in the vertical, each with thickness 100 m. Thisvertical resolution does not resolve the bottom Ekmanlayer but does resolve the first baroclinic mode and, withthe partial cell treatment of bottom topography, the to-pography slope. However, for small Ekman numbers,the growth rate of the most unstable waves, which isthe primary physical process of interest here, are notsignificantly affected by the bottom Ekman layer (Stipa2003, manuscript submitted to Geophys. Astrophys. Flu-id Dyn., hereinafter SGAFD). The Coriolis parameteris 1024 s21 and constant so that the internal deformation

MAY 2004 1199S P A L L

FIG. 1. Basin configuration and bottom topography (contour inter-val 200 m). The lightly shaded region in the lower right is where themodel temperature is restored toward a uniform stratification withtime scale 50 days.

TABLE 1. Model run parameters, units: radius (km); f 0 (s21); Q (W m22); L (km); hr dimensionless; depth (m); Tin2T0 (8C); downwelling(106 m3 s21), and symbols used in Fig. 12.

Run Radius f0 Q L hr Depth Tin2T0 Downwelling Symbol

12345

375250185250250

1 3 1024

1 3 1024

1 3 1024

1 3 1024

1 3 1024

2020202020

6060606060

0.01330.01330.01330.00670.0033

10001000100010001000

1.621.150.930.970.83

0.590.280.190.500.59

squareasterisksquaretriangletriangle

6789

10

250250250250250

1 3 1024

1 3 1024

1 3 1024

1 3 1024

0.25 3 1024

2010406020

6060606060

0.00.01330.01330.01330.0133

10001000100010001000

0.550.811.692.050.92

0.760.230.420.470.90

starcirclecirclecirclediamond

1112131415

250250250250250

0.5 3 1024

2 3 1024

3 3 1024

1 3 1024

1 3 1024

2020202020

60606030

100

0.01330.01330.01330.02660.008

10001000100010001000

0.941.391.791.061.12

0.490.160.110.600.18

diamonddiamonddiamondplus signplus sign

16171819

250250375250

1 3 1024

1 3 1024

1 3 1024

1 3 1024

20204020

60606030

0.00670.02670.01330.0067

500200010001000

2.050.671.990.82

0.140.570.800.65

triangle (down)triangle (down)crosstriangle (left)

radius based on this open-ocean stratification is ap-proximately 17 km and well resolved by the horizontalgrid.

The model domain for the central calculation consistsof a circular basin 750 km in diameter connected to an‘‘open ocean’’ through a channel 200 km wide (Fig. 1).The bottom slopes downward from 200-m depth on theouter boundary to 1000-m depth at a distance 60 km

offshore of the boundary, giving a bottom slope of hr

5 1.33 3 1022 (see Table 1). Parameter sensitivity stud-ies that are carried out in the following section willmostly use a smaller basin of 500-km diameter (forcomputational efficiency); however, the basic circula-tion and governing physics are similar to the larger ba-sin. The temperature within the lightly shaded regionof the open ocean is restored toward the initial uniformstratification T0(z) with a time scale of 50 days. In thisway, the properties of the water flowing into the mar-ginal sea are essentially fixed, independent of what pro-cesses take place in the marginal sea. The exchange ratethrough the strait is not specified, but rather emerges aspart of the solution. The model is forced by an annualcycle of cooling of strength 120 W m22 applied overthe entire marginal sea for 2 months followed by noforcing for 10 months (annual mean heat loss is 20 Wm22). This annual cycle is repeated for an interval of25 years. The sensitivity of the water mass transfor-mation and circulation to changes in the basin size anddepth, bottom slope and width, strength of forcing, andCoriolis parameter will be explored in the followingsection.

Subgrid-scale processes are parameterized using La-placian viscosity and diffusion in both horizontal andvertical directions. Lateral boundary conditions are noslip and there is no heat flux through the solid bound-aries. The horizontal diffusivity is 30 m2 s21 and thehorizontal viscosity is 50 m2 s21. The vertical diffusioncoefficient is 1025 m2 s21, while the vertical viscositycoefficient is 2 3 1024 m2 s21. The bulk characteristicsof the watermass transformation in the marginal sea arenot overly sensitive to these choices, provided that themodel dissipation is not so strong that it suppresses theinstability of the boundary current. The subgrid-scalemixing can, however, influence the details of the down-

1200 VOLUME 34J 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

FIG. 2. Mean upper-level temperature and horizontal velocity (every fourth grid point) for the central calculation(contour interval 5 0.18C). Locations of the time series in Fig. 3 are indicated by the white circles; the boundarybetween the sloping topography and the flat interior is indicated by the white line.

welling boundary layer near the sidewalls, particularlywhen the bottom is flat (Spall 2003). The weaker sen-sitivity found here may be a result of the bottom to-pography. The width of the boundary current over asloping bottom is controlled by the width of the topog-raphy, whereas it scales as the internal deformation ra-dius times the square root of the Prandtl number for aflat bottom (Spall 2003). At high latitudes, where cool-ing is active, the topographic slope is generally muchwider than the deformation radius, and so frictionalboundary layers are likely to be much weaker than inSpall (2003). A similar lack of sensitivity to subgrid-scale mixing was also found in the calculations with asloping bottom by WBNO.

This configuration and forcing differ from most pre-vious studies of localized cooling in several importantways. First, the marginal sea is connected to an openocean, as in Spall (2003) and WBNO, so that the heatloss in the marginal sea can be balanced by an advectiveflux through the strait and equilibrium solutions can beobtained. Unlike Spall (2003), however, the cooling is(more realistically) applied over the entire marginal sea.

This does not introduce an artificially small length scaleinto the problem that arises when the cooling is confinedto a localized region of the ocean. Any spatial variabilitythat arises in the solutions is a result of lateral advectionin the ocean. The second difference from previous stud-ies, except that of WBNO, is that the bottom topographyis sloped near the boundaries. This allows for the de-velopment of strong boundary currents and, as will beshown, plays a fundamental role in the properties of thewater mass transformation in the marginal sea.

The model has been integrated until a statistical equi-librium is achieved. The mean temperature and hori-zontal velocity at 50-m depth averaged over the final 5years of a 25-yr simulation are shown in Fig. 2. Thewhite line marks the region where the topography be-comes flat in the interior. The coldest waters are foundin the interior of the marginal sea, where there aredomed isopycnals and a weak anticyclonic circulation.There is also a strong, warm boundary current flowingcyclonically around the basin. The maximum mean ve-locity in the boundary current is approximately 25 cms21. The temperature of the boundary current decreases

MAY 2004 1201S P A L L

FIG. 3. Time series of temperature (8C) at the center of the basin (solid line) and in theboundary current (dashed line); see Fig. 2 for locations.

monotonically as it flows around the basin such that theoutflowing water is colder than the inflowing water. Itis this decrease in temperature within the boundary cur-rent that balances the net heat loss due to atmosphericforcing within the marginal sea.

Time series of the temperature at 50-m depth at themidpoint of the marginal sea and within the boundarycurrent (x 5 725 km, y 5 525 km; locations are indi-cated by the white circles in Fig. 2) are shown in Fig.3. The boundary current settles down to a fairly regularseasonal cycle very quickly, within a few years. Thetemperature drops rapidly when the surface cooling isapplied and, because of lateral advection from the openocean, rebounds to approximately 4.88C after coolingceases. The temperature in the interior takes much lon-ger, approximately 15 years, to arrive at an equilibriumseasonal cycle. At equilibrium, the seasonal variationin sea surface temperature is O(0.58C). The warming insummer is provided by eddies transporting heat fromthe warm boundary current into the interior. Becausethis restratification mechanism is not effective early inthe calculation, the water initially cools year after year.Once the temperature contrast between the interior andthe boundary current is sufficiently strong, eddiesformed from the boundary current transport warm waterinto the interior to balance heat loss at the surface. Thisbalance is similar to what has been found for localizedcooling over a flat bottom in a stratified ocean by VMJ,Chapman (1998), and Marshall and Schott (1999). Inthose cases, however, the instabilities arise from a geo-strophic rim current that forms around the region oflocalized cooling, not from a boundary current that en-circles the basin.

The mean vertical structure in the marginal sea isindicated by zonal sections at y 5 525 km shown inFig. 4. The interior is filled with cold, low-potential-vorticity water. There is increased potential vorticitynear the surface as a result of weak stratification, and

the gently sloping isopycnals are balanced by a weakcyclonic shear O(1 cm s21). The circulation in the in-terior is anticyclonic, however, in opposite sense to theboundary current circulation. The boundary region ischaracterized by a warm, strong, surface-intensified cy-clonic boundary current with horizontal velocities inexcess of 20 cm s21. The stratification is also strong inthe upper 200–300 m of the boundary current (the small-scale temperature and potential vorticity anomalies verynear the bottom are an artifact of the contouring routineand the definition of bottom topography in the model;they are not active in the flow). The southward-flowingwaters along the western boundary are colder, and haveslower maximum velocity and slightly deeper verticalstructure, than the northward-flowing waters on the east-ern boundary.

The restratification of the interior of the marginal seais indicated by the average seasonal temperature profile(average month 1, month 2, etc.) within the central 50km of the basin calculated over the final 5 years ofintegration in Fig. 5. At the beginning of the year theinterior is stratified with warm water in the upper fewhundred meters. This stratification is eroded through the2-month cooling period so that at the beginning ofmonth 3 the central region of the marginal sea is verycold and unstratified. Within 2 months after cooling hasceased the upper ocean begins to restratify, as indicatedby the arrival of warmer waters by the middle of month4. The later arrival of warm, stratified waters throughoutthe model summer provides the majority of the restra-tification in the interior.

The initial, weak restratification results from watersin the interior of the marginal sea that were not fullyconvected during the winter. Figure 6 shows the in-stantaneous temperature at 50-m depth at the end of thecooling period in year 22. The surface waters are coldthroughout the interior, but there are weak lateral var-iations. The initial restratification in Fig. 5 is provided

1202 VOLUME 34J 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

FIG. 4. Mean zonal sections through the midlatitude of the basin: (a) temperature (8C), (b) meridional velocity (m s21), and(c) potential vorticity.

FIG. 5. Average seasonal profile of temperature (8C) within the central 50 km of the basin.

by the large-scale remnants of warmer waters in theinterior remaining from the previous year. The scale ofthese features is much larger than the internal defor-mation radius and they do not result from baroclinicinstability of the deep convection site during this winter.This is consistent with the observational findings of Lil-ly et al. (1999) that deep convection is patchy on spatialscales O(100 km) in winter in the Labrador Sea. Smaller,

deformation-scale features are also beginning to pene-trate into the interior that do result from instabilitiesalong the offshore edge of the boundary current.

This lateral variation in stratification is also indicatedin Fig. 7 by the mixed layer depth at the end of winter.There is deep mixing throughout most of the interior ofthe marginal sea, but the slightly warmer areas over theflat interior remain weakly stratified at the end of winter.

MAY 2004 1203S P A L L

FIG. 6. Upper-level temperature (8C) at the end of the cooling period in year 22; contour interval 5 0.1.

It is these waters that result in the rapid, weak restra-tification in Fig. 5. Such remnants of the previous sum-mer stratification are generally found at the end of win-ter, although the amount varies from year to year.

The later, stronger restratification results from eddiesformed from the boundary current during spring andsummer months when the waters are warmer and morestratified. Figure 8 shows the 50-m temperature at theend of month 4. Lateral advection from the open oceanhas warmed the boundary current over most of the basin.Filaments and eddies originating from the offshore edgeof the boundary current extend into the interior. Theyare not characterized by pinched-off rings resulting fromvery large meanders, such as is often found along theGulf Stream, for example. There is a turbulent cascadesuch that the length scale of the perturbations growsfrom the deformation scale at their source near the off-shore edge of the boundary current in late winter toO(100 km) in the interior at the end of summer. It isthis cascade of scales that results in the large regionsof warmer, weakly stratified waters at the end of winterseen in Figs. 6 and 7. The mixed layer depth is shallowover most of the basin at this time (not shown).

3. Heat balances and residual circulation

The balance of terms in the heat equation is usefulto elucidate the means by which heat is redistributedwithin the marginal sea. The temperature equationsolved by the MIT primitive equation model is writtenas

2]T ](wT ) ] T25 2= · (vT ) 2 1 A ¹ T 1 K , (1)T T 2]t ]z ]z

where v 5 ui 1 yj is the horizontal velocity vector. Theadvection terms [first and second on the right-hand sideof (1)] are further decomposed into mean and eddy com-ponents:

](wT ) ](w9T9)= · (vT ) 1 5 = · (v9T9) 1

]z ]z

](wT)1 = · (vT) 1 , (2)

]z

where the overbars denote a time average and primesare deviations from the time mean.

The balances averaged in the azimuthal direction at

1204 VOLUME 34J 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

FIG. 7. Mixed layer depth (m) at the end of the cooling period in year 22; contour interval 5 100 m.

150-m depth, calculated over the final 5 years of inte-gration, are shown in Fig. 9 as a function of radius. Theshaded area indicates the region of sloping topography.Mean advection acts to warm the region over the slopingbottom as a result of the cyclonic rim current carryingwarm, open ocean water into the marginal sea. Thiswarming is balanced primarily by convection due to heatloss to the atmosphere, and lateral eddy fluxes. Hori-zontal diffusion acts to spread the boundary current lat-erally by cooling the warmest waters near the outerboundary and warming the seaward region of the bound-ary current, but it has very little direct effect on the heatbalance outside the boundary current. Eddy fluxes arenegative over the whole slope and a maximum at theoffshore edge of the boundary current. Synoptic picturesshow that this is where the eddies are being formed(e.g., Figs. 6 and 8). The atmospheric cooling at thisdepth is largest in the center of the boundary currentbecause convection does not penetrate as deeply here,where advection is strong, as it does in the interior,where advection is weak.

The balance in the interior is between cooling by heatloss to the atmosphere (and resulting convection) andheating due to eddy fluxes. The eddy flux divergence

is essentially uniform throughout the interior. Thewarming due to the decay of the eddies takes place overthe entire interior of the basin, far from where they weregenerated at the basin perimeter. In addition, the eddyflux divergence changes sign from the generation regionin the boundary current into the interior, while the bar-oclinic shear is always cyclonic. This indicates that theeddy flux divergence is not simply related to the localmean flow, as is often assumed in eddy flux parame-terization schemes. This redistribution of heat by theeddies is an inherently nonlocal process and thus cannotbe properly parameterized by an eddy flux parameter-ization that depends only on the local mean flow char-acteristics. This is particularly apparent in the interiorwhere the mean flow is driven by the eddies, not theother way around as it is in the boundary current.

The vertical transport # w ds, where s is the along-boundary coordinate, averaged over the last 5 years ofintegration, is shown in Fig. 10. Mean vertical motionseverywhere within the interior are small. The down-welling is concentrated in two areas within the boundarycurrent. The dominant downwelling is found along theboundary and sloping bottom. This is similar to thedownwelling boundary layer discussed by Spall (2003),

MAY 2004 1205S P A L L

FIG. 8. Upper-level temperature (8C) at the end of month 4 in year 22; contour interval 5 0.1.

FIG. 9. Azimuthally averaged terms in the temperature equation (1027 8C s21).

Pedlosky (2003), and Barcilon and Pedlosky (1967) forcases in which the wall is vertical. They used linear,analytic models to show that horizontal mixing of den-sity near the boundary is balanced by vertical advection.In the present model that is connected to the open ocean,

the advection is due to both horizontal and vertical com-ponents and mixing is due to both explicitly resolvededdy fluxes and parameterized horizontal diffusion. Thevertical velocity that advects warm water downward tobalance the diffusive cooling results in stretching in the

1206 VOLUME 34J 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

FIG. 10. Azimuthally integrated vertical velocity as a function of radius (contour interval 5 10m2 s21; solid contours downward).

vorticity equation. This stretching is balanced by lateraldiffusion of vorticity into the solid boundary, which iswhy the near-boundary region is the preferred site fordownwelling. Although such a simple analytic solutionis not possible for the present case of a sloping bottom,the location and scale of this boundary layer, and itscontinuity with the downwelling along the vertical sec-tion of the wall, suggest that its dynamics are analogousto that found in the flat-bottom cases. The downwellingcan also be inferred from the cooling of the temperaturealong the boundary, as discussed by Spall and Pickart(2001).

There is also a second site of downwelling locatedon the offshore edge of the boundary current. Thisdownwelling is located where the eddies that carry heatinto the basin interior are formed and the eddy fluxdivergence is strongly cooling the boundary current.This balance between lateral eddy flux divergence andvertical advection of the mean stratification is expectedfrom quasigeostrophic wave–mean flow interaction the-ory (Pedlosky 1979, 371–377). The vertical advectioncarries warm water downward and so tends to warm thewater column and balance the eddy flux divergence.

The net downwelling in the interior of the basin isnegligible at all depths, so there is no mean baroclinicoverturning circulation that connects the near-boundaryregion to the interior of the basin. The exchange of heatand momentum is carried entirely by the eddies thatoriginate in the boundary current and die in the basininterior. This important role of eddy fluxes from theboundary current into the interior is consistent with theobservational studies of Lilly et al. (1999, 2003) andSJPO, the former of which clearly show the formationof eddies from the Irminger Current along west Green-land that transport boundary current water into the in-

terior of the Labrador Sea, resulting in restratificationof the wintertime deep convection sites.

4. Theory and integral constraints

In this section, basic integral constraints on the massand heat balances within the marginal sea are used toderive estimates of the water mass properties and char-acteristics of the circulation within the marginal sea asa function of the basic parameters of the system. Theapproach is very idealized and is intended to providegeneral insight into how the bulk properties of the ther-mohaline circulation are determined. The continuousstratification of the marginal sea is represented as threedistinct water masses. It is assumed that the waters flow-ing into the marginal sea from the open ocean havetemperature Tin, the waters formed within the interiorof the marginal sea have temperature T0, and the watersflowing out of the marginal sea have temperature Tout.

a. Temperature of the interior water

The first quantity to be estimated is the temperature(or density) of the waters formed within the marginalsea relative to the temperature of the open ocean. Theheat budget for the interior (flat bottom portion) of themarginal sea may be written as

2pR Q2pRH u9T9 5 , (3)in

r C0 p

where R is the radius of the interior of the marginal sea,Hin is the vertical scale of the boundary current, Q isthe mean surface heat flux, r0 is a reference density ofseawater, and Cp is the specific heat of seawater. Theeddy temperature flux is represented by , where u9u9T9

MAY 2004 1207S P A L L

FIG. 11. (a) Growth rate of the most unstable wave as a function of wavenumber and slope parameter d 5 topographicslope/isopycnal slope. (b) Correlation (solid line) and e2d (dashed line), normalized to 1 for a flat bottom (d 5u9T90). Growth rate and solid line in (b) are from Blumsack and Gierasch (1972).

and T9 are the deviations of the horizontal velocity andtemperature from their time means, and the overbar in-dicates a temporal average. This balance assumes thatthe net annual heat exchange with the atmosphere isbalanced by lateral eddy fluxes originating at the perim-eter of the basin interior, as seen in Fig. 9. A similarbalance has been used to derive the properties of lo-calized deep convection by VMJ, Marshall and Schott(1999), and others. It is assumed that there is no meanadvection through the flat interior of the basin. This isreasonable for the flat bottom f -plane calculations con-sidered here. Calculations in which the Coriolis param-eter varies with latitude give very similar results, so thisanalysis is relevant to the more general case as well.

The source of the eddies is the boundary current thatflows over the sloping bottom. Here the general ap-proach of VMJ and Stone (1972) is followed such thatthe eddy heat flux is assumed to be proportional to themean velocity Vin and temperature anomaly of theboundary current (Tin 2 T0) as

u9T9 5 cV (T 2 T ), (4)in in 0

where c is a nondimensional correlation coefficient.VMJ and Spall and Chapman (1998) empirically foundc ø 0.025 over a wide range of flat-bottom cases basedon laboratory and numerical model experiments. Thetemperature anomaly Tin 2 T0 is the temperature changeacross the boundary current and so represents the den-sity of the water formed in the interior of the marginalsea relative to the open-ocean temperature.

The present case differs from those considered byStone (1972), VMJ, Visbeck et al. (1997), and Spall andChapman (1998) because the baroclinic flow residesover topography. Blumsack and Gierasch (1972) presenta simple quasigeostrophic stability analysis for a bar-oclinic flow over a sloping bottom (see also the recentstudy SGAFD). Their solution for growth rate hi as a

function of wavenumber m (nondimensionalized by theinternal deformation radius) and bottom slope parameterd, which is the ratio of the bottom slope to the isopycnalslope, may be written as

1/22m 2 tanh(m) 1 d

h 5 (1 2 d) 2 2 m , (5)i 5 6[ ]tanh(m) 4 tanh(m)

and is shown in Fig. 11a. The flat-bottom case is givenby d 5 0. The present situation of a warm boundarycurrent flowing cyclonically around a deep basin has d, 0 (see Fig. 4a). The growth rate decreases with in-creasingly negative d. The wavenumber of the maxi-mum unstable wave also increases, indicating that themost unstable waves decrease in scale. As the bottomslope increases, it becomes increasingly more difficultfor parcels to move across the topography. Bottomslopes greater than about 21.5 times the isopycnal slopeessentially stabilize the flow. Currents with the topog-raphy parallel to or steeper than the isopycnals (d . 1)are also stabilized. This is generally the case for deepwestern boundary currents. The boundary current struc-ture found in the central calculation (Fig. 4) has d ø21, typical of boundary currents in the Labrador Sea.

The linear theory of Blumsack and Gierasch (1972)also provides an estimate of the eddy heat flux as afunction of the bottom slope parameter, growth rate, andmean flow, which may be written

tanh(m)u9T9 } h V (T 2 T ). (6)i in in 0

m 2 tanh(m)

The linear theory does not provide the constant of pro-portionality but does give its dependence on d (throughthe relationship to hi and m plotted in Fig. 11a). Theratio of to Vin(Tin 2 T0), given by (6), is shownu9T9in Fig. 11b as a function of the bottom slope parameterd for the most unstable wave. The value has been ar-

1208 VOLUME 34J 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

bitrarily set to 1 for a flat bottom. The eddy heat fluxis greatly reduced as the bottom slope increases relativeto the isopycnal slope. For typical values of d 5 O(21),the eddy heat flux is less than 20% of what would beexpected for the same strength of baroclinic flow overa flat bottom. The growth rate hi also decreases withincreasingly negative d, as seen in Fig. 11a, but it doesnot decrease as rapidly as the eddy flux. This indicatesthat the decrease in eddy flux with increasing bottomslope is due both to a decrease in the growth rate of theunstable waves and a decrease in the correlation betweenthe perturbation velocity and the perturbation temper-ature. The eddy heat flux for d , 0 can be closelyapproximated by the exponential e2d, as shown by thedashed line. This simple exponential estimate of theeddy heat flux will be used to close the heat budget (4).

Much work has been done testing balances similar to(4) for a flat bottom, which is just a special case withd 5 0. In these studies, the relationship between theeddy heat flux and the mean flow parameters is re-markably constant such that c ø 0.025 over a wide rangeof applications (VMJ; Visbeck et al. 1997; Marshall andSchott 1999). This empirical finding is also consistentwith a simple quasigeostrophic theory for how eddiesself-propagate, which gives the same scaling but directlyestimates an upper bound of c ø 0.045 (Spall and Chap-man 1998). Combining this understanding of the flat-bottom behavior with the theory for the sloping bottom,c 5 0.025 e2d, and thus the eddy heat flux is parame-terized as

2du9T9 5 0.025e V (T 2 T ). (7)in in 0

The velocity of the boundary current Vin is related tothe temperature anomaly of the boundary currentthrough the thermal wind relation:

ag(T 2 T )Hin 0 inV 5 , (8)inr f L0 0

where L is the width of the sloping topography. Thevertical scale Hin is the depth of the inflowing water,taken to be the average depth of the water over thesloping bottom.

An estimate of the equilibrium temperature of thedense water in the basin interior is found by combining(3), (7), and (8) to give

1/21 Rf LQ0(T 2 T ) 5 . (9)in 0 1 2H 2agC cin p

This result differs significantly from the analogous scal-ing for isolated deep convection derived by VMJ. Itspower dependence on the surface buoyancy forcing andthe radius of the basin are different, but more importantis that this scaling is also dependent on the rotation ratef 0, the width of the topography L, and the slope of thebottom (through c). This scaling has the same parameterdependence as the scaling of Chapman (1998), who al-lowed for a finite-width region over which the cooling

decreased to zero. This width in his scaling is replacedby the width of the topography L in the present case,but leads to the same result. The scaling of VMJ isreproduced if the width of the topography is replacedby the internal deformation radius. The width of thetopographic slope is, in general, much wider than thedeformation radius at high latitudes, and so the presentscaling is more appropriate for eddy fluxes driven byinstability of the boundary current. This is true even ifthe scale of the eddies is on the order of the deformationradius (as is expected from linear instability theory). Itis the width of the slope that determines the baroclinicshear of the mean boundary current, which in turn in-fluences the eddy heat flux.

A series of numerical model calculations have beencarried out in which the basin radius, surface heat flux,Coriolis parameter, width and amplitude of the topo-graphic slope, and depth of the basin have been varied,as summarized in Table 1. For computational efficiency,the diameter of the marginal sea has been decreased to500 km for most of these sensitivity studies. The generalproperties of the circulation in all of these calculations(except for the flat bottom, as discussed further below)are the same as for the central case discussed earlier.For comparison with the theory, the vertical scale heightHin was taken to be one-half the depth of the basin,consistent with the average depth of the inflowing warmwater. The slope parameter d was taken to be 20.8 forall cases in which the bottom depth on the boundarywas 20% of the basin depth (200 m for the standard1000-m-deep basin). This is based on the finding thatthe isopycnals intersect the bottom at about the midpointof the slope and outcrop on the offshore side of thetopography. For the cases in which the bottom depth onthe boundary was 600 and 800 m (runs 4 and 5), as-suming that the isopycnals slope from the midpoint ofthe topography to the surface on the offshore side ofthe front, d 5 20.4 and 20.2, respectively. The resultsare not overly sensitive to how d is defined as long asit is consistent with the mean shear found in the model.

The density of the interior waters formed in the mar-ginal sea, relative to the mean temperature of the in-flowing water, was calculated for each of the runs andis compared to the theoretical estimate (9) in Fig. 12a.The central case is indicated by the asterisk at (Tin 2T0) 5 1.158C. The temperature anomaly of the watersformed in the marginal sea varies from approximately0.58 to over 28C for all of the calculations. There is veryclose agreement between the theory and the model re-sults, with a correlation of 0.95 and a least squares fitslope of 1.23. The density of the waters formed in theinterior of the basin varies by a large amount, even forthe same heat loss over the marginal sea. The controllingfactor is the effectiveness of the instability process intransporting the warm waters from the boundary currentinto the basin interior.

The flat-bottom calculation, indicated by the star, re-

MAY 2004 1209S P A L L

FIG. 12. Comparison between the theory and the series of numerical model calculations summarized in Table 1: (a) temperatureanomaly of waters formed in the interior Tin 2 T0 (8C), (b) temperature anomaly of outflowing waters Tout 2 T0 (8C), (c)downwelling in the basin (106 m3 s21), and (d) comparison between downwelling and interior temperature anomaly. Correlationcoefficients are given in the lower right of each panel and a least squares fit line to the points is given in the upper part of(a)–(c).

quires special treatment and will be discussed after theanalysis of the sloping bottom cases.

b. Temperature of the outflowing water

The waters flowing out of the marginal sea are, ingeneral, cooler than the inflowing waters, as requiredto balance the heat budget in the marginal sea. However,the outflowing waters can be much warmer than thecoldest waters formed in the interior of the basin. Theaverage temperature of the outflowing waters can beestimated by requiring that the net heat flux into themarginal sea through the boundary current balances thenet heat loss to the atmosphere and that the net massflux into the marginal sea be zero. The heat budget maybe written as

2pR QT V H L 2 T V H L 5 , (10)in in in out out out

r C0 p

where the subscripts ‘‘in’’ and ‘‘out’’ refer to the prop-erties of the inflowing and outflowing boundary cur-rents.

The inflowing heat transport can be written, by mak-ing use of (8) and (9), as

RQLT V H L 5 . (11)in in in 2r C c0 p

The requirement that mass be balanced within the mar-ginal sea requires that VinHinL 5 VoutHoutL. This maybe combined with (10) and (11) to obtain an expressionfor the temperature anomaly of the outflowing water,relative to the temperature of the cold waters formed inthe interior of the basin, as

T 2 T 5 (T 2 T )(1 2 2pRc/L)out 0 in 0

5 (T 2 T )(1 2 e). (12)in 0

The factor e 5 2pRc/L may be interpreted as the fraction

1210 VOLUME 34J 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

of the inflowing waters that are fluxed into the interiorby eddies. For the standard configuration, e 5 0.13. Forstrongly sloping topography (c K 1), small basins, orwide topographic slopes, the outflowing water is closeto the temperature of the inflowing water. The densityof the outflowing water increases as the basin size in-creases, the width of the topographic slope decreases,or as the topographic slope itself decreases (c increases).The implication is that the transport in the boundarycurrent, and the exchange between the marginal sea andthe open ocean, is enhanced as e decreases. As a ref-erence, e for the Labrador Sea is approximately 0.1.

The average temperature of the outflowing waters wascalculated from the series of model calculations and iscompared to the theoretical estimate (12), making useof (9), in Fig. 12b. Once again there is general agreementbetween the model and theory, although the comparisonis not as close as it was for Tin 2 T0. This is not sur-prising since the theory for the outflowing temperatureuses the theory for (Tin 2 T0) as part of the solution.In particular, the theory predicts that the outflowing tem-perature should increase with increasing f 0 but the mod-el results in much less variation with f 0. The theoryalso predicts an increase with increasing L, largely dueto an increase in Tin 2 T0, but both the inflowing andoutflowing temperature anomaly show little dependenceon the slope width. Nonetheless, the general tendencyis well predicted by the theory, and the correlation re-mains high: r 5 0.87.

c. Downwelling in the marginal sea

The requirement that the net cooling in the marginalsea be balanced by advection through the strait does notimmediately reveal the means by which the heat balanceis satisfied. The outflowing colder water could be in theupper water column, in which case the heat transportwould be carried primarily by a horizontal gyre. Thedense waters could also flow out of the marginal sea ata depth much greater than the inflowing waters, indi-cating that the heat transport is achieved by an over-turning, or sinking, circulation. The distinction betweenthese two heat transport mechanisms is significant, asthe sea surface temperature can be much colder for thecase in which the heat transport is carried by the hor-izontal gyre rather than the overturning gyre, as dem-onstrated by Spall and Pickart (2001).

The maximum mean downwelling in the marginal seacan be estimated by considering the thickness of theoutflowing water Hout in comparison with that of theinflowing water Hin. The net downwelling transport Wis given simply by this change in thickness times thewidth of the outflowing boundary current L and thevelocity of the outflowing boundary current Vout:

W 5 (H 2 H )LV .out in out (13)

Mass conservation in the marginal sea, and the use

of (12) and thermal wind, allows the thickness of theoutflowing waters to be written as

V H Hin in inH 5 5 . (14)out 1/2V (1 2 e)out

For e 5 O(0.1), the change in thickness between theinflowing boundary current and the outflowing bound-ary current is only O(5%), and so would be difficult toobserve directly, in either the model or observations.

The downwelling W may now be derived:

1/2H RLQagin 1/2W 5 [1 2 (1 2 e) ]1 2r 2C cf0 p 0

1/25 [1 2 (1 2 e) ]V H L. (15)in in

The downwelling within the marginal sea is simply re-lated to the inflowing transport VinHinL by a factor of 12 (1 2 e)1/2. As eddies flux all of the inflowing bound-ary current water into the interior, e → 1, and all of theinflowing boundary current water downwells within themarginal sea. For small e (steep slopes, small basins,or wide topographic slopes), only a small fraction ofthe inflowing transport downwells within the basin. Thisis made clear by considering the limit of e K 1, inwhich case (15) may be approximated as

eW ø V H L. (16)in in2

The maximum downwelling within the marginal seahas been calculated for each of the model calculationsand is compared to the theory in Fig. 12c (correlation0.92). The downwelling rate varies by almost an orderof magnitude even for those calculations with the samesurface heat flux. The downwelling rate increases withincreasing transport in the boundary current, which re-sults in these calculations mainly from increasing theinflowing velocity or the basin depth. The inflowingvelocity, in turn, is determined by the density of thewater in the interior of the basin, Tin 2 T0, and thethermal wind relation. Because the thermal wind resultsin larger velocities for the same temperature gradient atlower latitudes, the downwelling increases with decreas-ing latitude. The downwelling also increases with in-creasing e, which results primarily from decreasing thebottom slope or increasing the radius of the basin. Theability of the bottom slope to stabilize the boundarycurrent influences not only the density of the watersformed in the interior of the basin, but also the totalamount of downwelling found within the marginal sea.The theory and model demonstrate the somewhat coun-terintuitive result that increasing the bottom slope resultsin more dense waters in the interior of the marginal sea,but less total downwelling within the marginal sea.

This result indicates that the downwelling rate is notsimply related to the amount of cooling or the densityof the water masses formed. This is clear from Fig. 12din which the downwelling rate is plotted against the

MAY 2004 1211S P A L L

density anomaly of the waters formed in the interior ofthe marginal sea. The two properties have a weak, neg-ative correlation.

d. Heat transport

The mechanism of heat transport within the marginalsea is also of interest. For some marginal seas, such asthe Mediterranean Sea, the heat transport is carried pri-marily by an overturning gyre, where light water entersthe basin in the upper ocean and dense water leaves thebasin in the deep ocean. In other basins, such as theLabrador Sea, the horizontal circulation appears to alsobe an important heat transport mechanism, where thewater simply becomes more dense as it encircles thebasin with little net downwelling (Spall and Pickart2001). Understanding the dynamical reasons for thesedifferences is important to understanding the differencesin the thermohaline circulation in a variety of marginalseas.

The ratio of the heat transport by the horizontal gyreto the heat transport by the overturning gyre can beestimated from the basic properties derived thus far. Itis important to note that the downwelling takes placenear the boundary, and so the heat flux carried by theoverturning gyre is proportional to the temperature dif-ference of the inflowing water and the outflowing water,Tin 2 Tout. Making use of the theoretical estimate forW, the ratio of horizontal to overturning heat transportcan be estimated as

V H L(T 2 T ) 1 2in in in out5 ø k 1. (17)

1/2W(T 2 T ) 1 2 (1 2 e) ein out

The final approximation makes use of (1 2 e)1/2 ø 1 2e/2 for e K 1. The relative importance of the horizontaland overturning gyres depends only on the parametere. In the limit of weak eddy fluxes from the boundarycurrent (e K 1), the horizontal gyre transports muchmore heat than the overturning gyre. For large basinsor strong eddy fluxes, e → 1, the final approximationin (17) is not valid, and the heat transport is equallydivided between the horizontal and overturning gyres.These estimates are consistent with the findings in high-resolution numerical models that the horizontal gyretransports more than one-half of the total meridionalheat transport at high latitudes in the North AtlanticOcean (Boning and Bryan 1996).

e. Flat-bottom theory

The flat (or very weakly sloping topography such thate $ 1) case requires special consideration. The heatbudget (3) that was used to calculate the temperature ofwaters formed within the marginal sea assumed that theeddy flux takes place all along the perimeter of themarginal sea. If the eddy fluxes are strong enough orthe marginal sea is large enough, then the heat carriedinto the marginal sea in the boundary current will be

fluxed into the interior by eddies before the boundarycurrent has extended around the marginal sea. This iswhat happens for a 500-km diameter basin with a flatbottom, as indicated by the mean upper-level temper-ature shown in Fig. 13. In this case, the eddy contri-bution will be nonzero only over the portion of theboundary where the boundary current exists.

The distance s that the boundary current will penetrateinto the marginal sea is determined by the distance ittakes for the eddy flux to balance the advective heatflux into the marginal sea:

su9T9 5 L U (T 2 T ) . (18)flat flat in 0 flat

The length scale L, velocity, and temperature anomalyof the boundary current now have the subscript ‘‘flat’’because these will, in general, be different from theprevious results for sloping topography near the bound-ary. Making use of the eddy flux parameterization (4),the distance s is simply

Lflats 5 , (19)c

where Lflat is the width of the boundary current. The flatbottom theory of Spall (2003) shows that the width ofthe boundary current in this case scales approximatelyas the internal deformation radius. For the open-oceanstratification used here, the internal deformation radiusis 17 km. Equation (19) then predicts that the boundarycurrent over a flat bottom, with c 5 0.025, would beeroded in 680 km. This is 43% of the perimeter of the500-km diameter marginal sea and is consistent withthe penetration found in Fig. 13. For e 5 1, the distanceover which the boundary current is eroded exactlymatches the perimeter of the basin. The results derivedhere are applicable when the parameter e $ 1.

The interior heat budget may then be used to estimatethe temperature of the water in the interior as

1/221 pR Qf(T 2 T ) 5 . (20)in 0 flat 1 2H agCin p

The temperature in the interior for the flat bottom hasthe same parameter dependence as the sloping cases forthe heat flux and Coriolis parameter, but now dependslinearly on the radius of the basin. It is independent ofthe eddy flux parameter c and the width of the slope L,which in effect control the strength of the eddy flux. Ingeneral, the temperature anomaly for the flat or weaklysloping bottom cases will be less than it is for basinswith a steep slope near the boundaries because all ofthe boundary current heat transport in the flat bottomcase goes toward balancing the surface heat loss. Thetemperature anomaly for the flat bottom is simply relatedto the temperature anomaly for the sloping bottom bythe parameter e as

1/2(T 2 T ) 5 e (T 2 T ).in 0 flat in 0 (21)

The downwelling within the marginal sea for the flat-

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FIG. 13. Mean upper-level temperature (8C) for the flat-bottom calculation; contour interval 5 0.1.

bottom case is the same as the transport into the marginalsea because the boundary current is entirely eroded byeddy fluxes before it can exit the basin. In this case, thedownwelling is

1/22H agpR QinW 5 V H L 5 . (22)flat in in flat 1 2r f C0 0 p

Expressions (20) and (22) were used for the stars in Fig.12. The theory also predicts that the outflowing densewaters will have the same temperature as the watersformed in the interior of the marginal sea (Tout 2 T0 50), which is close to what is found in the model (Fig.12b). In this case the dense waters flow out of the mar-ginal sea as bottom-trapped currents along both the left-and right-hand sides of the strait, as in Spall (2003).

5. Summary and conclusions

Basin-scale heat budgets and linear stability theoryhave been used to provide theoretical estimates of theproperties of the watermass transformation and ther-mohaline circulation in marginal seas subject to buoy-ancy loss at the surface. Specific quantities of interest

include the density of waters formed in a marginal sea,the density of the outflowing waters, the strength of theboundary currents that encircle the marginal sea, themagnitude of downwelling within the marginal sea, andthe mechanisms of heat transport. The theory compareswell to results from a high-resolution primitive equationmodel over a wide range of parameter space.

Although the general approach taken here is similarto what has been used to characterize localized open-ocean convection (e.g., Visbeck et al. 1996), the inclu-sion of boundaries and an open ocean allows for equi-librium solutions and a full three-dimensional thermo-haline circulation. The heat loss in the marginal sea isultimately balanced by lateral advection in boundarycurrents from the open ocean. The strength of the bound-ary current is determined as part of the solution. Bound-aries also support the downwelling component of thethermohaline circulation. The downwelling is locatedwithin a narrow boundary layer over the sloping bottomand along the offshore edge of the boundary current,and is not directly related to the density of waters formedin the interior of the marginal sea. The downwellingwithin the boundary layer is analogous to the boundary

MAY 2004 1213S P A L L

layers over a flat bottom discussed previously by Bar-cilon and Pedlosky (1967), Spall (2003), and Pedlosky(2003). The downwelling on the offshore side of thefront is required to balance the eddy heat flux divergencethat supplies the interior of the basin with warm water.

The key process that determines the properties of thewatermass transformation and thermohaline circulationin the marginal sea is the interaction between the cy-clonic rim current along the basin perimeter and theinterior of the marginal sea. This exchange is controlledby baroclinic instabilities of the boundary current that,for realistic configurations, are on scales somewhatsmaller than the internal deformation radius. The ratioof the isopycnal slope to the bottom slope is the fun-damental parameter that quantifies this exchange. Foreven moderately steep topography, the eddy fluxes re-sulting from surface-intensified fronts are much weakerthan would be expected for the analogous flat-bottomtheory. Most marginal seas are in such a topographicallycontrolled state. Bottom-trapped currents, such as deepwestern boundary currents, can be completely stabilizedby the topography. The theory presented here providesa simple modification of the flat-bottom parameteriza-tion of eddy fluxes due to baroclinic instability thataccounts for such topographic effects. However, it isalso evident that the seasonal restratification of the mar-ginal sea interior is due to nonlocal sources of eddiesand is thus very difficult to represent using any localparameterization scheme.

Acknowledgments. This work was supported by theOffice of Naval Research under Grant N00014-03-1-0338 and by the National Science Foundation underGrant OCE-0240978.

REFERENCES

Barcilon, V., and J. Pedlosky, 1967: A unified linear theory of ho-mogeneous and stratified rotating fluids. J. Fluid Mech., 29, 609–621.

Blumsack, S. L., and P. J. Gierasch, 1972: Mars: The effects of to-pography on baroclinic instability. J. Atmos. Sci., 29, 1081–1089.

Boning, C. W., and F. O. Bryan, 1996: Large-scale transport processes

in high-resolution circulation models. The Warmwatersphere ofthe North Atlantic Ocean, W. Krauss, Ed., Gebruder Borntraeger,91–128.

Bryan, F. O., 1987: Parameter sensitivity of primitive equation oceangeneral circulation models. J. Phys. Oceanogr., 17, 970–985.

Chapman, D. C., 1998: Setting the scales of the ocean response toisolated convection. J. Phys. Oceanogr., 28, 606–620.

Katsman, C. A., M. A. Spall, and R. S. Pickart, 2004: Boundarycurrent eddies and their role in the restratification of the LabradorSea. J. Phys. Oceanogr., in press.

Lilly, J. M., P. B. Rhines, M. Visbeck, R. E. Davis, J. R. N. Lazier,F. Schott, and D. Farmer, 1999: Observing deep convection inthe Labrador Sea during winter 1994–95. J. Phys. Oceanogr.,29, 2065–2098.

——, ——, F. Schott, K. Lavender, J. Lazier, U. Send, and E. d’Asaro,2003: Observations of the Labrador Sea eddy field. Progress inOceanography, Vol. 59, Pergamon, 75–176.

Marshall, J., and F. Schott, 1999: Open-ocean convection: Obser-vations, theory, and models. Rev. Geophys., 37, 1–64.

——, C. Hill, L. Perelman, and A. Adcroft, 1997: Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling. J. Geophys.Res., 102, 5733–5752.

Mauritzen, C., 1996: Production of dense overflow waters feedingthe North Atlantic across the Greenland–Scotland Ridge. Part 1:Evidence for a revised circulation scheme. Deep-Sea Res., 43,769–806.

Park, Y.-G., and J. A. Whitehead, 1999: Rotating convection drivenby differential bottom heating. J. Phys. Oceanogr., 29, 1208–1220.

——, and K. Bryan, 2000: Comparison of thermally driven circu-lations from a depth-coordinate model and an isopycnal-layermodel. Part I: Scaling-law sensitivity to vertical diffusivity. J.Phys. Oceanogr., 30, 590–605.

Pedlosky, J., 1979: Geophysical Fluid Dynamics. Springer-Verlag,624 pp.

——, 2003: Thermally driven circulations in small ocean basins. J.Phys. Oceanogr., 33, 2333–2340.

Spall, M. A., 2003: On the thermohaline circulation in flat bottommarginal seas. J. Mar. Res., 61, 1–25.

——, and D. C. Chapman, 1998: On the efficiency of baroclinic eddyheat transport across narrow fronts. J. Phys. Oceanogr., 28,2275–2287.

——, and R. S. Pickart, 2001: Where does dense water sink? Asubpolar gyre example. J. Phys. Oceanogr., 31, 810–826.

Stone, P., 1972: A simplified radiative–dynamical model for the staticstability of rotating atmospheres. J. Atmos. Sci., 29, 405–418.

Visbeck, M., J. Marshall, and H. Jones, 1996: Dynamics of isolatedconvective regions in the ocean. J. Phys. Oceanogr., 26, 1721–1734.

——, ——, T. Haine, and M. Spall, 1997: Specification of eddytransfer coefficients in coarse-resolution ocean circulation mod-els. J. Phys. Oceanogr., 27, 381–402.