MAY 2004 SPALL 1197
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 ®nal form 6 November 2003)
ABSTRACT The properties of watermass transformation and the thermohaline circulation in marginal seas with topography and subject to a spatially uniform net surface cooling are discussed. The net heat loss within the marginal sea is ultimately balanced by lateral advection from the open ocean in a narrow boundary current that ¯ows cy- clonically around the basin. Heat loss in the interior is offset by lateral eddy ¯uxes originating in the boundary current. The objectives of this study are to understand better what controls the density of waters formed within the marginal sea, the temperature of the out¯owing waters, the amount of downwelling, and the mechanisms of heat transport within the marginal sea. The approach combines heat budgets with linear stability theory for a baroclinic ¯ow over a sloping bottom to provide simple theoretical estimates of each of these quantities in terms of the basic parameters of the system. The theory compares well to a series of eddy-resolving primitive equation model calculations. The downwelling is concentrated within the boundary current in both a diffusive boundary layer near topography and an eddy-driven region on the offshore edge of the boundary current. For most high-latitude regions, the horizontal gyre is expected to transport more heat than does the overturning gyre.
1. Introduction generally localized to regions with spatial scales O(100 km). The properties of waters formed by such localized Marginal seas are often regions of net buoyancy loss deep convection have received much recent attention. as a result of either cooling or evaporation and, because Idealized numerical and laboratory experiments have of their semienclosed geometry, are regions of forma- shown that lateral ¯uxes out of localized deep convec- tion of dense intermediate or bottom waters. These wa- tion sites by mesoscale eddies can balance surface buoy- ters generally have quite distinct water mass properties ancy ¯uxes (Visbeck et al. 1996, hereinafter VMJ; Mar- relative to the open ocean and make a signi®cant con- shall and Schott 1999 and the references therein). By tribution to the global thermohaline circulation through making use of stability theory, the properties (density, the transport of heat and salt. They also in¯uence the depth) of deep convection sites in interior regions of the large-scale circulation throughout the water column be- ocean subject to localized surface ¯uxes can be esti- cause they have anomalously low potential vorticity. mated with some success (VMJ; Marshall and Schott Understanding what controls the formation rate and 1999; Chapman 1998). These results demonstrate that properties of such water masses is an important part of lateral heat ¯uxes due to mesoscale eddies can play a understanding the global oceanic thermohaline circu- central role in controlling the properties of the large- lation. In this context, marginal seas can be relatively scale thermohaline circulation. small geographically, such as the Red Sea or the Adriatic While compelling, these studies make several rather Sea, or they can be of planetary scale, such as the Med- restrictive assumptions when compared with the actual iterranean Sea or the Nordic seas. The important char- problem of buoyancy loss in a marginal sea. They have acteristic for the present study is that the marginal sea generally been formulated with a net buoyancy loss that is a semienclosed basin that is subject to a net buoyancy is localized in the interior of the basin, often with an loss. abrupt transition in its strength. Chapman (1998) al- The areas of deepest convection in marginal seas are lowed for a gradual transition in the surface forcing strength and found that the equilibrium properties had * Woods Hole Oceanographic Institution Contribution Number a different scaling, most notably with regard to the Cor- 10942. iolis parameter. On the other hand, atmospheric forcing varies on spatial scales much larger than the oceanic deformation radius (ice edge effects and polynyas are Corresponding author address: Dr. Mike Spall, Woods Hole Oceanographic Institution, 360 Woods Hole Rd., MS 21, Woods Hole, a notable exception) and is often nonzero over much or MA 02543. most of the marginal seas. The net heat ¯ux out of the E-mail: [email protected] ocean in these studies has also made it impossible to
᭧2004 American Meteorological Society 1198 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 obtain equilibrium solutions. Last, although it has been solve the mesoscale eddies either because of low res- shown that the net vertical motions in interior regions olution in models (e.g., Bryan 1987; Park and Bryan of buoyancy loss are not signi®cantly different from 2000; Spall and Pickart 2001) or, in the case of labo- zero, it is well known that marginal seas do experience ratory experiments, the deformation radius was the same a net downward ¯ux of water. Theoretical arguments scale as the basin (e.g., Park and Whitehead 1999). The favor near-boundary regions for the net downwelling to temperature anomaly of the deep waters (required to take place (Spall 2003; Pedlosky 2003; Spall and Pickart estimate heat transport and advective speeds) is speci- 2001), an aspect not considered in the previous open- ®ed or diagnosed in these previous studies, while it is ocean con®gurations. predicted as part of the solution in the current study. For marginal seas connected to the open ocean The objective of this work is to understand better what through a narrow strait, the net buoyancy ¯ux at the controls the properties of water mass transformation and ocean's surface must be balanced by lateral advection thermohaline circulation in marginal seas subject to through the strait. This requires both lateral boundaries buoyancy loss. The basic circulation characteristics are and a reservoir of buoyant water. Lateral advection also ®rst described using an eddy-resolving general circu- appears to be important for less geographically con- lation model. Theoretical arguments are then used to strained regions of buoyancy loss such as the Nordic estimate the parameter dependencies of the density of seas (Mauritzen 1996) and the Labrador Sea (Katsman water formed in the marginal sea, the density of the et al. 2004; Lilly et al. 2003; Straneo 2004, manuscript water ¯owing out of the marginal sea, the amount of submitted to J. Phys. Oceanogr.). Understanding the downwelling within the marginal sea, and the relative interaction between the narrow boundary currents that importance of horizontal versus vertical gyres in the heat are required to balance the net buoyancy ¯ux and the transport within the marginal sea. The theory compares vast interior regions over which there are signi®cant well to results from a series of numerical model cal- surface ¯uxes is key to understanding the heat budget culations. in marginal seas and, hence, the overall process that controls water mass transformation. 2. Thermohaline circulation in an idealized In a recent study, Walin et al. (2003, unpublished marginal sea manuscript, hereinafter WBNO) explore the in¯uence of cooling in a marginal sea on the structure of the The thermohaline circulation in an idealized semien- resulting boundary currents. Their interest was on the closed marginal sea will ®rst be explored using the Mas- transformation of a baroclinic in¯owing boundary cur- sachusetts Institute of Technology (MIT) primitive rent into a barotropic out¯owing boundary current. The equation numerical model (Marshall et al. 1997). The heat loss was concentrated in the boundary current, and model solves the momentum and density equations us- was strong in the sense that the baroclinicity of the ing level (depth) coordinates in the vertical and a stag- in¯owing boundary current was entirely eroded before gered C grid in the horizontal. The model retains a free it encircled the marginal sea. The forcing was designed surface. A linear equation of state is used so that density such that there was little cooling of the interior of the is proportional to temperature as ϭ 0 Ϫ ␣(T Ϫ Tb), marginal sea, and so the exchange between the boundary where 0 is a reference density for seawater, Tb ϭ 4ЊC current and the interior was weak. The circulations in- is the initial temperature at the bottom, and the thermal vestigated in the present paper are complementary in expansion coef®cient ␣ ϭ 0.2 kg mϪ3 ЊCϪ1. The model that the out¯owing currents are (generally, but not al- is initialized at rest with a uniform strati®cation such ways) baroclinic and the heat loss in the interior of the that T0(z) ϭ Tb ϩ Tz(zb Ϫ z), with the maximum bottom marginal sea is signi®cant. depth zb ϭ 1000 m and vertical gradient Tz ϭ 1.5 ϫ Numerous studies have been successful in relating 10Ϫ3 ЊCmϪ1(N2ϭ3ϫ10Ϫ6 sϪ1), although the results the basin-scale buoyancy-forced overturning circulation have not been found to be very sensitive to this initial and meridional heat transport to the basic parameters of strati®cation. The initial temperature at the uppermost the system; a recent review is given by Park and Bryan model level (50 m) is 5.35ЊC. The model is con®gured (2000). These studies differ from the current one in with uniform horizontal grid spacing of 5 km and 10 several important ways. Primarily, they are aimed at the levels in the vertical, each with thickness 100 m. This large-scale overturning circulation and assume that all vertical resolution does not resolve the bottom Ekman of the downwelling at high latitudes is balanced by a layer but does resolve the ®rst baroclinic mode and, with diapycnal upwelling at mid- and low latitudes. The pres- the partial cell treatment of bottom topography, the to- ent study makes no assumptions about where the waters pography slope. However, for small Ekman numbers, upwell, or what dynamics are active in the upwelling the growth rate of the most unstable waves, which is region. For example, the upwelling could take place via the primary physical process of interest here, are not diapycnal mixing of density in the midlatitude deep signi®cantly affected by the bottom Ekman layer (Stipa ocean or it could take place within the surface mixed 2003, manuscript submitted to Geophys. Astrophys. Flu- layer where the isopycnals outcrop at high latitudes. In id Dyn., hereinafter SGAFD). The Coriolis parameter addition, previous studies generally did not properly re- is 10Ϫ4 sϪ1 and constant so that the internal deformation MAY 2004 SPALL 1199
offshore of the boundary, giving a bottom slope of hr ϭ 1.33 ϫ 10Ϫ2 (see Table 1). Parameter sensitivity stud- ies that are carried out in the following section will mostly use a smaller basin of 500-km diameter (for computational ef®ciency); however, the basic circula- tion and governing physics are similar to the larger ba- sin. The temperature within the lightly shaded region of the open ocean is restored toward the initial uniform
strati®cation T0(z) with a time scale of 50 days. In this way, the properties of the water ¯owing into the mar- ginal sea are essentially ®xed, independent of what pro- cesses take place in the marginal sea. The exchange rate through the strait is not speci®ed, but rather emerges as part of the solution. The model is forced by an annual cycle of cooling of strength 120 W mϪ2 applied over the entire marginal sea for 2 months followed by no forcing for 10 months (annual mean heat loss is 20 W mϪ2). This annual cycle is repeated for an interval of 25 years. The sensitivity of the water mass transfor- mation and circulation to changes in the basin size and depth, bottom slope and width, strength of forcing, and Coriolis parameter will be explored in the following section. FIG. 1. Basin con®guration and bottom topography (contour inter- Subgrid-scale processes are parameterized using La- val 200 m). The lightly shaded region in the lower right is where the placian viscosity and diffusion in both horizontal and model temperature is restored toward a uniform strati®cation with time scale 50 days. vertical directions. Lateral boundary conditions are no slip and there is no heat ¯ux through the solid bound- aries. The horizontal diffusivity is 30 m2 sϪ1 and the radius based on this open-ocean strati®cation is ap- horizontal viscosity is 50 m2 sϪ1. The vertical diffusion proximately 17 km and well resolved by the horizontal coef®cient is 10Ϫ5 m2 sϪ1, while the vertical viscosity grid. coef®cient is 2 ϫ 10Ϫ4 m2 sϪ1. The bulk characteristics The model domain for the central calculation consists of the watermass transformation in the marginal sea are of a circular basin 750 km in diameter connected to an not overly sensitive to these choices, provided that the ``open ocean'' through a channel 200 km wide (Fig. 1). model dissipation is not so strong that it suppresses the The bottom slopes downward from 200-m depth on the instability of the boundary current. The subgrid-scale outer boundary to 1000-m depth at a distance 60 km mixing can, however, in¯uence the details of the down-
Ϫ1 Ϫ2 TABLE 1. Model run parameters, units: radius (km); f 0 (s ); Q (W m ); L (km); hr dimensionless; depth (m); TinϪT0 (ЊC); downwelling (106 m3 sϪ1), and symbols used in Fig. 12.
Run Radius f0 QL hrDepth TinϪT0 Downwelling Symbol 1 375 1 ϫ 10Ϫ4 20 60 0.0133 1000 1.62 0.59 square 2 250 1 ϫ 10Ϫ4 20 60 0.0133 1000 1.15 0.28 asterisk 3 185 1 ϫ 10Ϫ4 20 60 0.0133 1000 0.93 0.19 square 4 250 1 ϫ 10Ϫ4 20 60 0.0067 1000 0.97 0.50 triangle 5 250 1 ϫ 10Ϫ4 20 60 0.0033 1000 0.83 0.59 triangle 6 250 1 ϫ 10Ϫ4 20 60 0.0 1000 0.55 0.76 star 7 250 1 ϫ 10Ϫ4 10 60 0.0133 1000 0.81 0.23 circle 8 250 1 ϫ 10Ϫ4 40 60 0.0133 1000 1.69 0.42 circle 9 250 1 ϫ 10Ϫ4 60 60 0.0133 1000 2.05 0.47 circle 10 250 0.25 ϫ 10Ϫ4 20 60 0.0133 1000 0.92 0.90 diamond 11 250 0.5 ϫ 10Ϫ4 20 60 0.0133 1000 0.94 0.49 diamond 12 250 2 ϫ 10Ϫ4 20 60 0.0133 1000 1.39 0.16 diamond 13 250 3 ϫ 10Ϫ4 20 60 0.0133 1000 1.79 0.11 diamond 14 250 1 ϫ 10Ϫ4 20 30 0.0266 1000 1.06 0.60 plus sign 15 250 1 ϫ 10Ϫ4 20 100 0.008 1000 1.12 0.18 plus sign 16 250 1 ϫ 10Ϫ4 20 60 0.0067 500 2.05 0.14 triangle (down) 17 250 1 ϫ 10Ϫ4 20 60 0.0267 2000 0.67 0.57 triangle (down) 18 375 1 ϫ 10Ϫ4 40 60 0.0133 1000 1.99 0.80 cross 19 250 1 ϫ 10Ϫ4 20 30 0.0067 1000 0.82 0.65 triangle (left) 1200 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34
FIG. 2. Mean upper-level temperature and horizontal velocity (every fourth grid point) for the central calculation (contour interval ϭ 0.1ЊC). Locations of the time series in Fig. 3 are indicated by the white circles; the boundary between the sloping topography and the ¯at interior is indicated by the white line. welling boundary layer near the sidewalls, particularly This does not introduce an arti®cially small length scale when the bottom is ¯at (Spall 2003). The weaker sen- into the problem that arises when the cooling is con®ned sitivity found here may be a result of the bottom to- to a localized region of the ocean. Any spatial variability pography. The width of the boundary current over a that arises in the solutions is a result of lateral advection sloping bottom is controlled by the width of the topog- in the ocean. The second difference from previous stud- raphy, whereas it scales as the internal deformation ra- ies, except that of WBNO, is that the bottom topography dius times the square root of the Prandtl number for a is sloped near the boundaries. This allows for the de- ¯at bottom (Spall 2003). At high latitudes, where cool- velopment of strong boundary currents and, as will be ing is active, the topographic slope is generally much shown, plays a fundamental role in the properties of the wider than the deformation radius, and so frictional water mass transformation in the marginal sea. boundary layers are likely to be much weaker than in The model has been integrated until a statistical equi- Spall (2003). A similar lack of sensitivity to subgrid- librium is achieved. The mean temperature and hori- scale mixing was also found in the calculations with a zontal velocity at 50-m depth averaged over the ®nal 5 sloping bottom by WBNO. years of a 25-yr simulation are shown in Fig. 2. The This con®guration and forcing differ from most pre- white line marks the region where the topography be- vious studies of localized cooling in several important comes ¯at in the interior. The coldest waters are found ways. First, the marginal sea is connected to an open in the interior of the marginal sea, where there are ocean, as in Spall (2003) and WBNO, so that the heat domed isopycnals and a weak anticyclonic circulation. loss in the marginal sea can be balanced by an advective There is also a strong, warm boundary current ¯owing ¯ux through the strait and equilibrium solutions can be cyclonically around the basin. The maximum mean ve- obtained. Unlike Spall (2003), however, the cooling is locity in the boundary current is approximately 25 cm (more realistically) applied over the entire marginal sea. sϪ1. The temperature of the boundary current decreases MAY 2004 SPALL 1201
FIG. 3. Time series of temperature (ЊC) at the center of the basin (solid line) and in the boundary current (dashed line); see Fig. 2 for locations. monotonically as it ¯ows around the basin such that the the gently sloping isopycnals are balanced by a weak out¯owing water is colder than the in¯owing water. It cyclonic shear O(1 cm sϪ1). The circulation in the in- is this decrease in temperature within the boundary cur- terior is anticyclonic, however, in opposite sense to the rent that balances the net heat loss due to atmospheric boundary current circulation. The boundary region is forcing within the marginal sea. characterized by a warm, strong, surface-intensi®ed cy- Time series of the temperature at 50-m depth at the clonic boundary current with horizontal velocities in midpoint of the marginal sea and within the boundary excess of 20 cm sϪ1. The strati®cation is also strong in current (x ϭ 725 km, y ϭ 525 km; locations are indi- the upper 200±300 m of the boundary current (the small- cated by the white circles in Fig. 2) are shown in Fig. scale temperature and potential vorticity anomalies very 3. The boundary current settles down to a fairly regular near the bottom are an artifact of the contouring routine seasonal cycle very quickly, within a few years. The and the de®nition of bottom topography in the model; temperature drops rapidly when the surface cooling is they are not active in the ¯ow). The southward-¯owing applied and, because of lateral advection from the open waters along the western boundary are colder, and have ocean, rebounds to approximately 4.8ЊC after cooling slower maximum velocity and slightly deeper vertical ceases. The temperature in the interior takes much lon- structure, than the northward-¯owing waters on the east- ger, approximately 15 years, to arrive at an equilibrium ern boundary. seasonal cycle. At equilibrium, the seasonal variation The restrati®cation of the interior of the marginal sea in sea surface temperature is O(0.5ЊC). The warming in is indicated by the average seasonal temperature pro®le summer is provided by eddies transporting heat from (average month 1, month 2, etc.) within the central 50 the warm boundary current into the interior. Because km of the basin calculated over the ®nal 5 years of this restrati®cation mechanism is not effective early in integration in Fig. 5. At the beginning of the year the the calculation, the water initially cools year after year. interior is strati®ed with warm water in the upper few Once the temperature contrast between the interior and hundred meters. This strati®cation is eroded through the the boundary current is suf®ciently strong, eddies 2-month cooling period so that at the beginning of formed from the boundary current transport warm water month 3 the central region of the marginal sea is very into the interior to balance heat loss at the surface. This cold and unstrati®ed. Within 2 months after cooling has balance is similar to what has been found for localized ceased the upper ocean begins to restratify, as indicated cooling over a ¯at bottom in a strati®ed ocean by VMJ, by the arrival of warmer waters by the middle of month Chapman (1998), and Marshall and Schott (1999). In 4. The later arrival of warm, strati®ed waters throughout those cases, however, the instabilities arise from a geo- the model summer provides the majority of the restra- strophic rim current that forms around the region of ti®cation in the interior. localized cooling, not from a boundary current that en- The initial, weak restrati®cation results from waters circles the basin. in the interior of the marginal sea that were not fully The mean vertical structure in the marginal sea is convected during the winter. Figure 6 shows the in- indicated by zonal sections at y ϭ 525 km shown in stantaneous temperature at 50-m depth at the end of the Fig. 4. The interior is ®lled with cold, low-potential- cooling period in year 22. The surface waters are cold vorticity water. There is increased potential vorticity throughout the interior, but there are weak lateral var- near the surface as a result of weak strati®cation, and iations. The initial restrati®cation in Fig. 5 is provided 1202 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34
FIG. 4. Mean zonal sections through the midlatitude of the basin: (a) temperature (ЊC), (b) meridional velocity (m sϪ1), and (c) potential vorticity. by the large-scale remnants of warmer waters in the deformation-scale features are also beginning to pene- interior remaining from the previous year. The scale of trate into the interior that do result from instabilities these features is much larger than the internal defor- along the offshore edge of the boundary current. mation radius and they do not result from baroclinic This lateral variation in strati®cation is also indicated instability of the deep convection site during this winter. in Fig. 7 by the mixed layer depth at the end of winter. This is consistent with the observational ®ndings of Lil- There is deep mixing throughout most of the interior of ly et al. (1999) that deep convection is patchy on spatial the marginal sea, but the slightly warmer areas over the scales O(100 km) in winter in the Labrador Sea. Smaller, ¯at interior remain weakly strati®ed at the end of winter.
FIG. 5. Average seasonal pro®le of temperature (ЊC) within the central 50 km of the basin. MAY 2004 SPALL 1203
FIG. 6. Upper-level temperature (ЊC) at the end of the cooling period in year 22; contour interval ϭ 0.1.
It is these waters that result in the rapid, weak restra- 3. Heat balances and residual circulation ti®cation in Fig. 5. Such remnants of the previous sum- The balance of terms in the heat equation is useful mer strati®cation are generally found at the end of win- to elucidate the means by which heat is redistributed ter, although the amount varies from year to year. within the marginal sea. The temperature equation The later, stronger restrati®cation results from eddies solved by the MIT primitive equation model is written formed from the boundary current during spring and as summer months when the waters are warmer and more 2Tץ (wT)ץ Tץ strati®ed. Figure 8 shows the 50-m temperature at the (vT)ϪϩAٌ2TϩK,(1)´ϭϪ١ z2ץzTTץ tץ end of month 4. Lateral advection from the open ocean has warmed the boundary current over most of the basin. Filaments and eddies originating from the offshore edge where v ϭ ui ϩ j is the horizontal velocity vector. The of the boundary current extend into the interior. They advection terms [®rst and second on the right-hand side are not characterized by pinched-off rings resulting from of (1)] are further decomposed into mean and eddy com- very large meanders, such as is often found along the ponents: (wЈTЈ)ץ (wT)ץ Gulf Stream, for example. There is a turbulent cascade vЈTЈ)ϩ)´vT)ϩϭ١)´ ١ such that the length scale of the perturbations grows zץ zץ -from the deformation scale at their source near the off (wT)ץ shore edge of the boundary current in late winter to (vT)ϩ , (2)´ ١ O(100 km) in the interior at the end of summer. It is ϩ zץ this cascade of scales that results in the large regions of warmer, weakly strati®ed waters at the end of winter where the overbars denote a time average and primes seen in Figs. 6 and 7. The mixed layer depth is shallow are deviations from the time mean. over most of the basin at this time (not shown). The balances averaged in the azimuthal direction at 1204 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34
FIG. 7. Mixed layer depth (m) at the end of the cooling period in year 22; contour interval ϭ 100 m.
150-m depth, calculated over the ®nal 5 years of inte- is essentially uniform throughout the interior. The gration, are shown in Fig. 9 as a function of radius. The warming due to the decay of the eddies takes place over shaded area indicates the region of sloping topography. the entire interior of the basin, far from where they were Mean advection acts to warm the region over the sloping generated at the basin perimeter. In addition, the eddy bottom as a result of the cyclonic rim current carrying ¯ux divergence changes sign from the generation region warm, open ocean water into the marginal sea. This in the boundary current into the interior, while the bar- warming is balanced primarily by convection due to heat oclinic shear is always cyclonic. This indicates that the loss to the atmosphere, and lateral eddy ¯uxes. Hori- eddy ¯ux divergence is not simply related to the local zontal diffusion acts to spread the boundary current lat- mean ¯ow, as is often assumed in eddy ¯ux parame- erally by cooling the warmest waters near the outer terization schemes. This redistribution of heat by the boundary and warming the seaward region of the bound- eddies is an inherently nonlocal process and thus cannot ary current, but it has very little direct effect on the heat be properly parameterized by an eddy ¯ux parameter- balance outside the boundary current. Eddy ¯uxes are ization that depends only on the local mean ¯ow char- negative over the whole slope and a maximum at the acteristics. This is particularly apparent in the interior offshore edge of the boundary current. Synoptic pictures where the mean ¯ow is driven by the eddies, not the show that this is where the eddies are being formed other way around as it is in the boundary current. (e.g., Figs. 6 and 8). The atmospheric cooling at this The vertical transport # w ds, where s is the along- depth is largest in the center of the boundary current boundary coordinate, averaged over the last 5 years of because convection does not penetrate as deeply here, integration, is shown in Fig. 10. Mean vertical motions where advection is strong, as it does in the interior, everywhere within the interior are small. The down- where advection is weak. welling is concentrated in two areas within the boundary The balance in the interior is between cooling by heat current. The dominant downwelling is found along the loss to the atmosphere (and resulting convection) and boundary and sloping bottom. This is similar to the heating due to eddy ¯uxes. The eddy ¯ux divergence downwelling boundary layer discussed by Spall (2003), MAY 2004 SPALL 1205
FIG. 8. Upper-level temperature (ЊC) at the end of month 4 in year 22; contour interval ϭ 0.1.
Pedlosky (2003), and Barcilon and Pedlosky (1967) for the advection is due to both horizontal and vertical com- cases in which the wall is vertical. They used linear, ponents and mixing is due to both explicitly resolved analytic models to show that horizontal mixing of den- eddy ¯uxes and parameterized horizontal diffusion. The sity near the boundary is balanced by vertical advection. vertical velocity that advects warm water downward to In the present model that is connected to the open ocean, balance the diffusive cooling results in stretching in the
FIG. 9. Azimuthally averaged terms in the temperature equation (10 Ϫ7 ЊCsϪ1). 1206 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34
FIG. 10. Azimuthally integrated vertical velocity as a function of radius (contour interval ϭ 10 m2 sϪ1; solid contours downward).
vorticity equation. This stretching is balanced by lateral terior of the Labrador Sea, resulting in restrati®cation diffusion of vorticity into the solid boundary, which is of the wintertime deep convection sites. why the near-boundary region is the preferred site for downwelling. Although such a simple analytic solution 4. Theory and integral constraints is not possible for the present case of a sloping bottom, the location and scale of this boundary layer, and its In this section, basic integral constraints on the mass continuity with the downwelling along the vertical sec- and heat balances within the marginal sea are used to tion of the wall, suggest that its dynamics are analogous derive estimates of the water mass properties and char- to that found in the ¯at-bottom cases. The downwelling acteristics of the circulation within the marginal sea as can also be inferred from the cooling of the temperature a function of the basic parameters of the system. The along the boundary, as discussed by Spall and Pickart approach is very idealized and is intended to provide (2001). general insight into how the bulk properties of the ther- There is also a second site of downwelling located mohaline circulation are determined. The continuous on the offshore edge of the boundary current. This strati®cation of the marginal sea is represented as three downwelling is located where the eddies that carry heat distinct water masses. It is assumed that the waters ¯ow- into the basin interior are formed and the eddy ¯ux ing into the marginal sea from the open ocean have divergence is strongly cooling the boundary current. temperature Tin, the waters formed within the interior This balance between lateral eddy ¯ux divergence and of the marginal sea have temperature T0, and the waters vertical advection of the mean strati®cation is expected ¯owing out of the marginal sea have temperature Tout. from quasigeostrophic wave±mean ¯ow interaction the- ory (Pedlosky 1979, 371±377). The vertical advection a. Temperature of the interior water carries warm water downward and so tends to warm the water column and balance the eddy ¯ux divergence. The ®rst quantity to be estimated is the temperature The net downwelling in the interior of the basin is (or density) of the waters formed within the marginal negligible at all depths, so there is no mean baroclinic sea relative to the temperature of the open ocean. The overturning circulation that connects the near-boundary heat budget for the interior (¯at bottom portion) of the region to the interior of the basin. The exchange of heat marginal sea may be written as and momentum is carried entirely by the eddies that RQ2 originate in the boundary current and die in the basin 2RH uЈTЈϭ , (3) in C interior. This important role of eddy ¯uxes from the 0 p boundary current into the interior is consistent with the where R is the radius of the interior of the marginal sea, observational studies of Lilly et al. (1999, 2003) and Hin is the vertical scale of the boundary current, Q is SJPO, the former of which clearly show the formation the mean surface heat ¯ux, 0 is a reference density of of eddies from the Irminger Current along west Green- seawater, and Cp is the speci®c heat of seawater. The land that transport boundary current water into the in- eddy temperature ¯ux is represented by uЈTЈ, where uЈ MAY 2004 SPALL 1207
FIG. 11. (a) Growth rate of the most unstable wave as a function of wavenumber and slope parameter ␦ ϭ topographic slope/isopycnal slope. (b) Correlation uЈTЈ (solid line) and e2␦ (dashed line), normalized to 1 for a ¯at bottom (␦ ϭ 0). Growth rate and solid line in (b) are from Blumsack and Gierasch (1972). and TЈ are the deviations of the horizontal velocity and function of wavenumber (nondimensionalized by the temperature from their time means, and the overbar in- internal deformation radius) and bottom slope parameter dicates a temporal average. This balance assumes that ␦, which is the ratio of the bottom slope to the isopycnal the net annual heat exchange with the atmosphere is slope, may be written as balanced by lateral eddy ¯uxes originating at the perim- 2 1/2 eter of the basin interior, as seen in Fig. 9. A similar Ϫ tanh()1␦ ϭ (1 Ϫ ␦) ϪϪ,(5) balance has been used to derive the properties of lo- i Ά·tanh() 4[]tanh() calized deep convection by VMJ, Marshall and Schott (1999), and others. It is assumed that there is no mean and is shown in Fig. 11a. The ¯at-bottom case is given advection through the ¯at interior of the basin. This is by ␦ ϭ 0. The present situation of a warm boundary reasonable for the ¯at bottom f-plane calculations con- current ¯owing cyclonically around a deep basin has ␦ sidered here. Calculations in which the Coriolis param- Ͻ 0 (see Fig. 4a). The growth rate decreases with in- eter varies with latitude give very similar results, so this creasingly negative ␦. The wavenumber of the maxi- analysis is relevant to the more general case as well. mum unstable wave also increases, indicating that the The source of the eddies is the boundary current that most unstable waves decrease in scale. As the bottom ¯ows over the sloping bottom. Here the general ap- slope increases, it becomes increasingly more dif®cult proach of VMJ and Stone (1972) is followed such that for parcels to move across the topography. Bottom the eddy heat ¯ux is assumed to be proportional to the slopes greater than about Ϫ1.5 times the isopycnal slope essentially stabilize the ¯ow. Currents with the topog- mean velocity Vin and temperature anomaly of the boundary current (T Ϫ T )as raphy parallel to or steeper than the isopycnals (␦ Ͼ 1) in 0 are also stabilized. This is generally the case for deep uЈTЈϭcVin(Tin Ϫ T0), (4) western boundary currents. The boundary current struc- where c is a nondimensional correlation coef®cient. ture found in the central calculation (Fig. 4) has ␦ ഠ VMJ and Spall and Chapman (1998) empirically found Ϫ1, typical of boundary currents in the Labrador Sea. c ഠ 0.025 over a wide range of ¯at-bottom cases based The linear theory of Blumsack and Gierasch (1972) on laboratory and numerical model experiments. The also provides an estimate of the eddy heat ¯ux as a function of the bottom slope parameter, growth rate, and temperature anomaly Tin Ϫ T0 is the temperature change across the boundary current and so represents the den- mean ¯ow, which may be written sity of the water formed in the interior of the marginal tanh() sea relative to the open-ocean temperature. uЈTЈϰ iVin(Tin ϪT0). (6) Ϫ tanh() The present case differs from those considered by Stone (1972), VMJ, Visbeck et al. (1997), and Spall and The linear theory does not provide the constant of pro- Chapman (1998) because the baroclinic ¯ow resides portionality but does give its dependence on ␦ (through over topography. Blumsack and Gierasch (1972) present the relationship to i and plotted in Fig. 11a). The a simple quasigeostrophic stability analysis for a bar- ratio of uЈTЈ to Vin(Tin Ϫ T0), given by (6), is shown oclinic ¯ow over a sloping bottom (see also the recent in Fig. 11b as a function of the bottom slope parameter study SGAFD). Their solution for growth rate i as a ␦ for the most unstable wave. The value has been ar- 1208 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 bitrarily set to 1 for a ¯at bottom. The eddy heat ¯ux decreased to zero. This width in his scaling is replaced is greatly reduced as the bottom slope increases relative by the width of the topography L in the present case, to the isopycnal slope. For typical values of ␦ ϭ O(Ϫ1), but leads to the same result. The scaling of VMJ is the eddy heat ¯ux is less than 20% of what would be reproduced if the width of the topography is replaced expected for the same strength of baroclinic ¯ow over by the internal deformation radius. The width of the a ¯at bottom. The growth rate i also decreases with topographic slope is, in general, much wider than the increasingly negative ␦, as seen in Fig. 11a, but it does deformation radius at high latitudes, and so the present not decrease as rapidly as the eddy ¯ux. This indicates scaling is more appropriate for eddy ¯uxes driven by that the decrease in eddy ¯ux with increasing bottom instability of the boundary current. This is true even if slope is due both to a decrease in the growth rate of the the scale of the eddies is on the order of the deformation unstable waves and a decrease in the correlation between radius (as is expected from linear instability theory). It the perturbation velocity and the perturbation temper- is the width of the slope that determines the baroclinic ature. The eddy heat ¯ux for ␦ Ͻ 0 can be closely shear of the mean boundary current, which in turn in- approximated by the exponential e2␦, as shown by the ¯uences the eddy heat ¯ux. dashed line. This simple exponential estimate of the A series of numerical model calculations have been eddy heat ¯ux will be used to close the heat budget (4). carried out in which the basin radius, surface heat ¯ux, Much work has been done testing balances similar to Coriolis parameter, width and amplitude of the topo- (4) for a ¯at bottom, which is just a special case with graphic slope, and depth of the basin have been varied, ␦ ϭ 0. In these studies, the relationship between the as summarized in Table 1. For computational ef®ciency, eddy heat ¯ux and the mean ¯ow parameters is re- the diameter of the marginal sea has been decreased to markably constant such that c ഠ 0.025 over a wide range 500 km for most of these sensitivity studies. The general of applications (VMJ; Visbeck et al. 1997; Marshall and properties of the circulation in all of these calculations Schott 1999). This empirical ®nding is also consistent (except for the ¯at bottom, as discussed further below) with a simple quasigeostrophic theory for how eddies are the same as for the central case discussed earlier. self-propagate, which gives the same scaling but directly For comparison with the theory, the vertical scale height estimates an upper bound of c ഠ 0.045 (Spall and Chap- Hin was taken to be one-half the depth of the basin, man 1998). Combining this understanding of the ¯at- consistent with the average depth of the in¯owing warm bottom behavior with the theory for the sloping bottom, water. The slope parameter ␦ was taken to be Ϫ0.8 for c ϭ 0.025 e2␦, and thus the eddy heat ¯ux is parame- all cases in which the bottom depth on the boundary terized as was 20% of the basin depth (200 m for the standard 2␦ uЈTЈϭ0.025eVin(Tin ϪT0). (7) 1000-m-deep basin). This is based on the ®nding that the isopycnals intersect the bottom at about the midpoint The velocity of the boundary current V is related to in of the slope and outcrop on the offshore side of the the temperature anomaly of the boundary current through the thermal wind relation: topography. For the cases in which the bottom depth on the boundary was 600 and 800 m (runs 4 and 5), as- ␣g(T Ϫ T )H suming that the isopycnals slope from the midpoint of V ϭ in 0 in , (8) in the topography to the surface on the offshore side of 00fL the front, ␦ ϭϪ0.4 and Ϫ0.2, respectively. The results where L is the width of the sloping topography. The are not overly sensitive to how ␦ is de®ned as long as vertical scale Hin is the depth of the in¯owing water, it is consistent with the mean shear found in the model. taken to be the average depth of the water over the The density of the interior waters formed in the mar- sloping bottom. ginal sea, relative to the mean temperature of the in- An estimate of the equilibrium temperature of the ¯owing water, was calculated for each of the runs and dense water in the basin interior is found by combining is compared to the theoretical estimate (9) in Fig. 12a. (3), (7), and (8) to give The central case is indicated by the asterisk at (Tin Ϫ 1/2 T0) ϭ 1.15ЊC. The temperature anomaly of the waters 1 Rf0LQ (Tin Ϫ T0) ϭ . (9) formed in the marginal sea varies from approximately Hin 2␣gCpc 0.5Њ to over 2ЊC for all of the calculations. There is very This result differs signi®cantly from the analogous scal- close agreement between the theory and the model re- ing for isolated deep convection derived by VMJ. Its sults, with a correlation of 0.95 and a least squares ®t power dependence on the surface buoyancy forcing and slope of 1.23. The density of the waters formed in the the radius of the basin are different, but more important interior of the basin varies by a large amount, even for is that this scaling is also dependent on the rotation rate the same heat loss over the marginal sea. The controlling
f 0, the width of the topography L, and the slope of the factor is the effectiveness of the instability process in bottom (through c). This scaling has the same parameter transporting the warm waters from the boundary current dependence as the scaling of Chapman (1998), who al- into the basin interior. lowed for a ®nite-width region over which the cooling The ¯at-bottom calculation, indicated by the star, re- MAY 2004 SPALL 1209
FIG. 12. Comparison between the theory and the series of numerical model calculations summarized in Table 1: (a) temperature
anomaly of waters formed in the interior Tin Ϫ T0 (ЊC), (b) temperature anomaly of out¯owing waters Tout Ϫ T0 (ЊC), (c) downwelling in the basin (106 m3 sϪ1), and (d) comparison between downwelling and interior temperature anomaly. Correlation coef®cients are given in the lower right of each panel and a least squares ®t line to the points is given in the upper part of (a)±(c). quires special treatment and will be discussed after the where the subscripts ``in'' and ``out'' refer to the prop- analysis of the sloping bottom cases. erties of the in¯owing and out¯owing boundary cur- rents. b. Temperature of the out¯owing water The in¯owing heat transport can be written, by mak- ing use of (8) and (9), as The waters ¯owing out of the marginal sea are, in general, cooler than the in¯owing waters, as required RQL TVHLin in in ϭ . (11) to balance the heat budget in the marginal sea. However, 2 0Ccp the out¯owing waters can be much warmer than the The requirement that mass be balanced within the mar- coldest waters formed in the interior of the basin. The ginal sea requires that VinHinL ϭ VoutHoutL. This may average temperature of the out¯owing waters can be be combined with (10) and (11) to obtain an expression estimated by requiring that the net heat ¯ux into the for the temperature anomaly of the out¯owing water, marginal sea through the boundary current balances the relative to the temperature of the cold waters formed in net heat loss to the atmosphere and that the net mass the interior of the basin, as ¯ux into the marginal sea be zero. The heat budget may be written as Tout Ϫ T0 ϭ (Tin Ϫ T0)(1 Ϫ 2Rc/L)
2 RQ ϭ (Tin Ϫ T0)(1 Ϫ ⑀). (12) TVHLin in in ϪTVHLout out out ϭ , (10) 0Cp The factor ⑀ ϭ 2Rc/L may be interpreted as the fraction 1210 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34 of the in¯owing waters that are ¯uxed into the interior of (12) and thermal wind, allows the thickness of the by eddies. For the standard con®guration, ⑀ ϭ 0.13. For out¯owing waters to be written as strongly sloping topography (c K 1), small basins, or VH H wide topographic slopes, the out¯owing water is close H ϭϭin in in . (14) out 1/2 to the temperature of the in¯owing water. The density Vout (1 Ϫ ⑀) of the out¯owing water increases as the basin size in- For ⑀ ϭ O(0.1), the change in thickness between the creases, the width of the topographic slope decreases, in¯owing boundary current and the out¯owing bound- or as the topographic slope itself decreases (c increases). ary current is only O(5%), and so would be dif®cult to The implication is that the transport in the boundary observe directly, in either the model or observations. current, and the exchange between the marginal sea and The downwelling W may now be derived: the open ocean, is enhanced as ⑀ decreases. As a ref- erence, ⑀ for the Labrador Sea is approximately 0.1. 1/2 Hin RLQ␣g The average temperature of the out¯owing waters was W ϭ [1 Ϫ (1 Ϫ ⑀)]1/2 2Ccf calculated from the series of model calculations and is 0p 0 1/2 compared to the theoretical estimate (12), making use ϭ [1 Ϫ (1 Ϫ ⑀)]VHLin in . (15) of (9), in Fig. 12b. Once again there is general agreement between the model and theory, although the comparison The downwelling within the marginal sea is simply re- is not as close as it was for T Ϫ T . This is not sur- lated to the in¯owing transport VinHinL by a factor of 1 in 0 1/2 prising since the theory for the out¯owing temperature Ϫ (1 Ϫ ⑀) . As eddies ¯ux all of the in¯owing bound- ary current water into the interior, ⑀ → 1, and all of the uses the theory for (Tin Ϫ T0) as part of the solution. In particular, the theory predicts that the out¯owing tem- in¯owing boundary current water downwells within the perature should increase with increasing f but the mod- marginal sea. For small ⑀ (steep slopes, small basins, 0 or wide topographic slopes), only a small fraction of el results in much less variation with f 0. The theory also predicts an increase with increasing L, largely due the in¯owing transport downwells within the basin. This is made clear by considering the limit of ⑀ K 1, in to an increase in Tin Ϫ T0, but both the in¯owing and out¯owing temperature anomaly show little dependence which case (15) may be approximated as on the slope width. Nonetheless, the general tendency ⑀ is well predicted by the theory, and the correlation re- W ഠ VHLin in . (16) 2 mains high: r ϭ 0.87. The maximum downwelling within the marginal sea has been calculated for each of the model calculations c. Downwelling in the marginal sea and is compared to the theory in Fig. 12c (correlation 0.92). The downwelling rate varies by almost an order The requirement that the net cooling in the marginal of magnitude even for those calculations with the same sea be balanced by advection through the strait does not surface heat ¯ux. The downwelling rate increases with immediately reveal the means by which the heat balance increasing transport in the boundary current, which re- is satis®ed. The out¯owing colder water could be in the sults in these calculations mainly from increasing the upper water column, in which case the heat transport in¯owing velocity or the basin depth. The in¯owing would be carried primarily by a horizontal gyre. The velocity, in turn, is determined by the density of the dense waters could also ¯ow out of the marginal sea at water in the interior of the basin, T Ϫ T , and the a depth much greater than the in¯owing waters, indi- in 0 thermal wind relation. Because the thermal wind results cating that the heat transport is achieved by an over- in larger velocities for the same temperature gradient at turning, or sinking, circulation. The distinction between lower latitudes, the downwelling increases with decreas- these two heat transport mechanisms is signi®cant, as ing latitude. The downwelling also increases with in- the sea surface temperature can be much colder for the creasing ⑀, which results primarily from decreasing the case in which the heat transport is carried by the hor- bottom slope or increasing the radius of the basin. The izontal gyre rather than the overturning gyre, as dem- ability of the bottom slope to stabilize the boundary onstrated by Spall and Pickart (2001). current in¯uences not only the density of the waters The maximum mean downwelling in the marginal sea formed in the interior of the basin, but also the total can be estimated by considering the thickness of the amount of downwelling found within the marginal sea. out¯owing water H in comparison with that of the out The theory and model demonstrate the somewhat coun- in¯owing water H . The net downwelling transport W in terintuitive result that increasing the bottom slope results is given simply by this change in thickness times the in more dense waters in the interior of the marginal sea, width of the out¯owing boundary current L and the but less total downwelling within the marginal sea. velocity of the out¯owing boundary current V : out This result indicates that the downwelling rate is not simply related to the amount of cooling or the density W ϭ (Hout Ϫ Hin)LVout. (13) of the water masses formed. This is clear from Fig. 12d Mass conservation in the marginal sea, and the use in which the downwelling rate is plotted against the MAY 2004 SPALL 1211 density anomaly of the waters formed in the interior of ¯uxed into the interior by eddies before the boundary the marginal sea. The two properties have a weak, neg- current has extended around the marginal sea. This is ative correlation. what happens for a 500-km diameter basin with a ¯at bottom, as indicated by the mean upper-level temper- d. Heat transport ature shown in Fig. 13. In this case, the eddy contri- bution will be nonzero only over the portion of the The mechanism of heat transport within the marginal boundary where the boundary current exists. sea is also of interest. For some marginal seas, such as The distance s that the boundary current will penetrate the Mediterranean Sea, the heat transport is carried pri- into the marginal sea is determined by the distance it marily by an overturning gyre, where light water enters takes for the eddy ¯ux to balance the advective heat the basin in the upper ocean and dense water leaves the ¯ux into the marginal sea: basin in the deep ocean. In other basins, such as the Ј Јϭ Ϫ Labrador Sea, the horizontal circulation appears to also su T LUflat flat(Tin T0)flat. (18) be an important heat transport mechanism, where the The length scale L, velocity, and temperature anomaly water simply becomes more dense as it encircles the of the boundary current now have the subscript ``¯at'' basin with little net downwelling (Spall and Pickart because these will, in general, be different from the 2001). Understanding the dynamical reasons for these previous results for sloping topography near the bound- differences is important to understanding the differences ary. Making use of the eddy ¯ux parameterization (4), in the thermohaline circulation in a variety of marginal the distance s is simply seas. L The ratio of the heat transport by the horizontal gyre s ϭ flat , (19) to the heat transport by the overturning gyre can be c estimated from the basic properties derived thus far. It where L¯at is the width of the boundary current. The ¯at is important to note that the downwelling takes place bottom theory of Spall (2003) shows that the width of near the boundary, and so the heat ¯ux carried by the the boundary current in this case scales approximately overturning gyre is proportional to the temperature dif- as the internal deformation radius. For the open-ocean ference of the in¯owing water and the out¯owing water, strati®cation used here, the internal deformation radius Ϫ Tin Tout. Making use of the theoretical estimate for is 17 km. Equation (19) then predicts that the boundary W, the ratio of horizontal to overturning heat transport current over a ¯at bottom, with c ϭ 0.025, would be can be estimated as eroded in 680 km. This is 43% of the perimeter of the VHL(T ϪT )12 500-km diameter marginal sea and is consistent with in in in out ϭ ഠ k 1. (17) W(T Ϫ T )1Ϫ(1 Ϫ ⑀)1/2 ⑀ the penetration found in Fig. 13. For ⑀ ϭ 1, the distance in out over which the boundary current is eroded exactly The ®nal approximation makes use of (1 Ϫ ⑀)1/2 ഠ 1 Ϫ matches the perimeter of the basin. The results derived ⑀/2 for ⑀ K 1. The relative importance of the horizontal here are applicable when the parameter ⑀ Ն 1. and overturning gyres depends only on the parameter The interior heat budget may then be used to estimate ⑀. In the limit of weak eddy ¯uxes from the boundary the temperature of the water in the interior as current (⑀ K 1), the horizontal gyre transports much more heat than the overturning gyre. For large basins 1 RQ2 f 1/2 or strong eddy ¯uxes, ⑀ → 1, the ®nal approximation (Tin Ϫ T0)flat ϭ . (20) Hin ␣gCp in (17) is not valid, and the heat transport is equally divided between the horizontal and overturning gyres. The temperature in the interior for the ¯at bottom has These estimates are consistent with the ®ndings in high- the same parameter dependence as the sloping cases for resolution numerical models that the horizontal gyre the heat ¯ux and Coriolis parameter, but now depends transports more than one-half of the total meridional linearly on the radius of the basin. It is independent of heat transport at high latitudes in the North Atlantic the eddy ¯ux parameter c and the width of the slope L, Ocean (BoÈning and Bryan 1996). which in effect control the strength of the eddy ¯ux. In general, the temperature anomaly for the ¯at or weakly sloping bottom cases will be less than it is for basins e. Flat-bottom theory with a steep slope near the boundaries because all of The ¯at (or very weakly sloping topography such that the boundary current heat transport in the ¯at bottom ⑀ Ն 1) case requires special consideration. The heat case goes toward balancing the surface heat loss. The budget (3) that was used to calculate the temperature of temperature anomaly for the ¯at bottom is simply related waters formed within the marginal sea assumed that the to the temperature anomaly for the sloping bottom by eddy ¯ux takes place all along the perimeter of the the parameter ⑀ as marginal sea. If the eddy ¯uxes are strong enough or (T Ϫ T ) ϭ ⑀1/2(T Ϫ T ). (21) the marginal sea is large enough, then the heat carried in 0 flat in 0 into the marginal sea in the boundary current will be The downwelling within the marginal sea for the ¯at- 1212 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 34
FIG. 13. Mean upper-level temperature (ЊC) for the ¯at-bottom calculation; contour interval ϭ 0.1. bottom case is the same as the transport into the marginal include the density of waters formed in a marginal sea, sea because the boundary current is entirely eroded by the density of the out¯owing waters, the strength of the eddy ¯uxes before it can exit the basin. In this case, the boundary currents that encircle the marginal sea, the downwelling is magnitude of downwelling within the marginal sea, and the mechanisms of heat transport. The theory compares 2 1/2 Hin ␣gRQ well to results from a high-resolution primitive equation Wflat ϭ VHLin in flat ϭ . (22) 00fCp model over a wide range of parameter space. Expressions (20) and (22) were used for the stars in Fig. Although the general approach taken here is similar 12. The theory also predicts that the out¯owing dense to what has been used to characterize localized open- waters will have the same temperature as the waters ocean convection (e.g., Visbeck et al. 1996), the inclu- sion of boundaries and an open ocean allows for equi- formed in the interior of the marginal sea (Tout Ϫ T0 ϭ 0), which is close to what is found in the model (Fig. librium solutions and a full three-dimensional thermo- 12b). In this case the dense waters ¯ow out of the mar- haline circulation. The heat loss in the marginal sea is ginal sea as bottom-trapped currents along both the left- ultimately balanced by lateral advection in boundary and right-hand sides of the strait, as in Spall (2003). currents 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 the 5. Summary and conclusions thermohaline circulation. The downwelling is located Basin-scale heat budgets and linear stability theory within a narrow boundary layer over the sloping bottom have been used to provide theoretical estimates of the and along the offshore edge of the boundary current, properties of the watermass transformation and ther- and is not directly related to the density of waters formed mohaline circulation in marginal seas subject to buoy- in the interior of the marginal sea. The downwelling ancy loss at the surface. Speci®c quantities of interest within the boundary layer is analogous to the boundary MAY 2004 SPALL 1213 layers over a ¯at bottom discussed previously by Bar- in high-resolution circulation models. The Warmwatersphere of cilon and Pedlosky (1967), Spall (2003), and Pedlosky the North Atlantic Ocean, W. Krauss, Ed., GebruÈder Borntraeger, 91±128. (2003). The downwelling on the offshore side of the Bryan, F. O., 1987: Parameter sensitivity of primitive equation ocean front is required to balance the eddy heat ¯ux divergence general circulation models. J. Phys. Oceanogr., 17, 970±985. that supplies the interior of the basin with warm water. Chapman, D. C., 1998: Setting the scales of the ocean response to The key process that determines the properties of the isolated convection. J. Phys. Oceanogr., 28, 606±620. Katsman, C. A., M. A. Spall, and R. S. Pickart, 2004: Boundary watermass transformation and thermohaline circulation current eddies and their role in the restrati®cation of the Labrador in the marginal sea is the interaction between the cy- Sea. J. Phys. Oceanogr., in press. clonic rim current along the basin perimeter and the Lilly, J. M., P. B. Rhines, M. Visbeck, R. E. Davis, J. R. N. Lazier, interior of the marginal sea. This exchange is controlled F. Schott, and D. Farmer, 1999: Observing deep convection in by baroclinic instabilities of the boundary current that, the Labrador Sea during winter 1994±95. J. Phys. Oceanogr., 29, 2065±2098. for realistic con®gurations, are on scales somewhat ÐÐ, ÐÐ, F. Schott, K. Lavender, J. Lazier, U. Send, and E. d'Asaro, smaller than the internal deformation radius. The ratio 2003: Observations of the Labrador Sea eddy ®eld. Progress in of the isopycnal slope to the bottom slope is the fun- Oceanography, Vol. 59, Pergamon, 75±176. damental parameter that quanti®es this exchange. For Marshall, J., and F. Schott, 1999: Open-ocean convection: Obser- vations, theory, and models. Rev. Geophys., 37, 1±64. even moderately steep topography, the eddy ¯uxes re- ÐÐ, C. Hill, L. Perelman, and A. Adcroft, 1997: Hydrostatic, quasi- sulting from surface-intensi®ed fronts are much weaker hydrostatic, and nonhydrostatic ocean modeling. J. Geophys. than would be expected for the analogous ¯at-bottom Res., 102, 5733±5752. theory. Most marginal seas are in such a topographically Mauritzen, C., 1996: Production of dense over¯ow waters feeding controlled state. Bottom-trapped currents, such as deep the North Atlantic across the Greenland±Scotland Ridge. Part 1: Evidence for a revised circulation scheme. Deep-Sea Res., 43, western boundary currents, can be completely stabilized 769±806. by the topography. The theory presented here provides Park, Y.-G., and J. A. Whitehead, 1999: Rotating convection driven a simple modi®cation of the ¯at-bottom parameteriza- by differential bottom heating. J. Phys. Oceanogr., 29, 1208± tion of eddy ¯uxes due to baroclinic instability that 1220. ÐÐ, and K. Bryan, 2000: Comparison of thermally driven circu- accounts for such topographic effects. However, it is lations from a depth-coordinate model and an isopycnal-layer also evident that the seasonal restrati®cation of the mar- model. Part I: Scaling-law sensitivity to vertical diffusivity. J. ginal sea interior is due to nonlocal sources of eddies Phys. Oceanogr., 30, 590±605. and is thus very dif®cult to represent using any local Pedlosky, J., 1979: Geophysical Fluid Dynamics. Springer-Verlag, 624 pp. parameterization scheme. ÐÐ, 2003: Thermally driven circulations in small ocean basins. J. Phys. Oceanogr., 33, 2333±2340. Acknowledgments. This work was supported by the Spall, M. A., 2003: On the thermohaline circulation in ¯at bottom Of®ce of Naval Research under Grant N00014-03-1- marginal seas. J. Mar. Res., 61, 1±25. 0338 and by the National Science Foundation under ÐÐ, and D. C. Chapman, 1998: On the ef®ciency of baroclinic eddy heat transport across narrow fronts. J. Phys. Oceanogr., 28, Grant OCE-0240978. 2275±2287. ÐÐ, and R. S. Pickart, 2001: Where does dense water sink? A subpolar gyre example. J. Phys. Oceanogr., 31, 810±826. REFERENCES Stone, P., 1972: A simpli®ed radiative±dynamical model for the static stability of rotating atmospheres. J. Atmos. 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