Impacts of Localized Mixing and Topography on the Stationary Abyssal Circulation

1660 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 36

CAROLINE A. KATSMAN Royal Netherlands Meteorological Institute (KNMI), De Bilt, Netherlands

(Manuscript received 1 February 2005, in final form 15 December 2005)

ABSTRACT

Stommel and coworkers calculated the stationary, geostrophic circulation in the abyssal ocean driven by prescribed sources (representing convective downwelling sites) and sinks (slow, widespread upwelling through the thermocline). The applied basin geometries were highly idealized with nearly uniform upwelling and gradual bottom slopes. In this paper, the classical Stommel–Arons theory for the abyssal circulation is extended by introducing pronounced bathymetry in the form of a midocean ridge and strongly enhanced upwelling in the vicinity of this ridge, modeled after direct observations of diapycnal mixing rates in the deep ocean. Locally enhanced upwelling over a midocean ridge drives a ␤-plume circulation that is modified by topographic stretching. The dynamics of this abyssal circulation pattern are explained by analyzing the combined impacts of the upwelling pattern and the bathymetry on the stationary circulation, building on their well-known separate impacts. On the western flank of the ridge, the effects of topographic stretching and upwelling oppose, and the direction of the local flow depends on their relative size. In this paper, a simple theoretical estimate is derived that can predict the direction of the flow along the ridge based on the geometry of the basin and the upwelling region. Its applicability is demonstrated for both the idealized model configurations applied in this study and for more realistic model simulations.

1. Introduction welling source, which implied the existence of a strong western boundary current to close the circulation. Inspired by the surprising outcome of a laboratory Several aspects of the model setup applied by SA60a study (Stommel et al. 1958), Stommel and coworkers were highly idealized, as was acknowledged by the au- developed a theory for the geostrophic, stationary cir- thors (Stommel and Arons 1960b). A first criticism re- culation in the abyssal ocean (Stommel et al. 1958; gards the restriction of the study to basins with a flat Stommel 1958; Stommel and Arons 1960a,b). In these bottom or a gradually sloping bottom only. Prominent studies, the abyssal ocean below the thermocline is rep- bathymetric features like the Mid-Atlantic Ridge are resented by a single homogeneous layer in which a cir- expected to have a strong impact on the abyssal circu- culation is driven by prescribing isolated sources (rep- lation, both by steering the flow and by (partially) resentative of convective downwelling sites in the polar blocking exchanges between ocean basins (e.g., Gille et oceans) and sinks (slow, widespread upwelling through al. 2004). Second, SA60a assumed the upwelling was the thermocline). Stommel and Arons (1960a, herein- nearly uniform. However, recent observations have after referred to as SA60a) considered planetary circu- shown that diapycnal mixing rates in the deep ocean are lation patterns for a variety of upwelling patterns and several orders of magnitude larger near rough bathym- basin geometries. For an idealized North Atlantic con- etry (e.g., Ledwell et al. 2000; Mauritzen et al. 2002; figuration (a downwelling source at the Pole, uniform Heywood et al. 2002) than in midocean (Kunze and upwelling elsewhere, and a flat bottom), the interior Sanford 1996), for example as a result of breaking in- circulation appeared poleward, toward the down- ternal tides (St. Laurent and Garrett 2002) or lee waves generated by currents interacting with topography (Naveira Garabato et al. 2004). This pronounced non- Corresponding author address: Caroline A. Katsman, KNMI, Oceanographic Research Department, P.O. Box 201, 3730 AE De uniformity of the diapycnal mixing and the upwelling Bilt, Netherlands. that is associated with it is expected to have a profound E-mail: [email protected] impact on the ocean circulation (e.g., Wunsch and Fer-

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rari 2004). The abyssal circulation does not behave like considered in section 3. The separate impacts of non- a single homogeneous layer, as is obvious from the dif- uniform upwelling and bathymetry are shortly dis- fering pathways of North Atlantic Deep Water and cussed in sections 3a and 3b, respectively, before ana- Antarctic Bottom Water, for example. lyzing their combined effects in section 3c. In particular, The impacts of localized upwelling and of the pres- the circulation on the western flank of the ridge appears ence of bathymetry on the abyssal circulation have of interest because of the competing impacts of the been considered separately in several previous studies. bathymetry and the nonuniform upwelling (section 3d). Localized upwelling is known to give rise to strong hori- For a variety of ridge geometries and upwelling pat- zontal recirculations (␤ plumes: Stommel 1982; Ped- terns, the flow along the ridge is analyzed based on a losky 1996; Spall 2000), governed by the Sverdrup re- simple theoretical estimate that predicts its direction. In lation in the upwelling region and by potential vorticity the discussion (section 4), this theoretical estimate is conservation elsewhere. On the other hand, the abyssal applied to more realistic model simulations. flow forced by uniform upwelling in a basin with pronounced bathymetry is guided by the potential vorticity 2. Model setup and approach contours (e.g., Straub and Rhines 1990; Kawase 1993; Abyssal circulation patterns are calculated by solving Stephens and Marshall 2000), which predominantly re- the full system of stationary, barotropic shallow-water flect the depth contours. equations in Eq. (1) numerically, using an iterative Recently, several detailed numerical studies have method [see Dijkstra and de Ruijter (2001) for an ex- combined the two effects by applying spatially varying tensive description of the model and the method]: mixing (and hence upwelling) in realistic model configurations (e.g., Hasumi and Suginohara 1999; Huang u Ѩu ␷ Ѩu u␷ g Ѩ␩ ϩ Ϫ ␪ Ϫ ␷ ϭϪ Ϫ ␪ Ѩ␾ Ѩ␪ tan f ␪ Ѩ␾ Rbu, and Jin 2002; Simmons et al. 2004; Saenko and Merry- r0 cos r0 r0 r0 cos field 2005). Although the nonuniformity of the mixing ͑ ͒ is shown to have a large impact on the resulting circu- 1a lation, the underlying dynamical causes have received u Ѩ␷ ␷ Ѩ␷ u2 g Ѩ␩ ϩ ϩ ␪ ϩ ϭϪ Ϫ ␷ little emphasis. Marchal and Nycander (2004) pre- ␪ Ѩ␾ Ѩ␪ tan fu Ѩ␪ Rb , r0 cos r0 r0 r0 sented a fairly realistic simulation of the circulation of ͑ ͒ Antarctic Bottom Water in the Brazil Basin forced by 1b upwelling over the Mid-Atlantic Ridge. Despite the and variation in upwelling that they applied, the flow pat- Ѩ͑hu͒ Ѩ͑h␷ cos␪͒ tern appeared to be governed by the topography. ϩ ϭϪ ␪ ͑ ͒ Ѩ␾ Ѩ␪ wr0 cos , 1c This study focuses explicitly on the combined impacts of highly nonuniform upwelling and pronounced where ␾ and ␪ are longitude and latitude, respectively; ␷ bathymetry on the dynamics of the planetary-scale u and are the horizontal velocity components, r0 is the abyssal circulation for parameter settings for which nei- radius of the earth, and f ϭ 2⍀ sin␪ is the Coriolis ther of the two effects clearly dominates. To this end, a parameter. The thickness of the water column h is de- barotropic shallow-water model is applied in a North fined as Atlantic configuration very similar to that used by h͑␾, ␪͒ ϭ H Ϫ h ͑␾, ␪͒ ϩ ␩͑␾, ␪͒, SA60a. Although a comparison is made with observa- b tions and more realistic model simulations, emphasis is where H is the maximum depth of the water column, hb on revealing the dynamics of the circulation rather than is the prescribed bathymetry, and ␩ is the deviation of on simulating it in detail. For several prescribed up- the surface, representing the bottom of the ther- welling fields and bathymetry configurations, stationary mocline. The circulation is forced by the upwelling w circulation patterns are calculated numerically. Al- introduced in Eq. (1c), which has a prescribed ampli- ␾ ␪ though baroclinic effects are of possible importance tude w0 and normalized spatial pattern Q( , ). Dissi- (Pedlosky 1992; Edwards and Pedlosky 1995; Spall pation occurs through bottom friction, with drag coef- ϭ ϫ Ϫ6 Ϫ1 2000, 2001), the current study is limited to the barotro- ficient Rb 1.0 10 s , resulting in a Stommel ␦ ϭ pic circulation. boundary layer width of s 50 km. In section 2 of this paper, the applied model setup is a. Basin geometry and forcing discussed. For a case with uniform upwelling and a flat bottom, the numerical results are validated against the The model domain represents the North Atlantic ⌽ ϭ linear theory presented by SA60a (section 2b). More Ocean and is bounded by meridians E 15°W and ⌽ ϭ complicated upwelling patterns and bathymetry are W 60°W (Fig. 1a). In the meridional direction, it

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FIG. 1. (a) Bathymetry hb (dark gray contours denote the 300-, 1000-, and 2000-m isobath) and upwelling w (light gray shading). A circle marks the location of the downwelling source. (b) Cross section ␾ ␾ ␪ ϭ for w( ) (light gray, left axis) and hb( ) (dark gray, right axis) at 30°N.

⌰ ϭ ⌰ ϭ extends from S 0° to ⌵ 60°N. All calculations properties and from direct observations (e.g., Wunsch are performed at a resolution of 1⁄3° ϫ 1° (135 ϫ 60 grid and Ferrari 2004). points) in anticipation of large gradients in longitudinal direction in response to the applied basin geometry and b. The Stommel–Arons case forcing (see below). For a layer of constant depth H and uniform up- The maximum depth of the water column is set to welling of strength w0, the analytical solution for the H ϭ 3000 m. A midocean ridge, representing the Mid- frictionless, geostrophic circulation in the interior is Atlantic Ridge, is present near ␾ ϭ 30°W and reaches (SA60a) up to 2400 m (Fig. 1). 2w r The average upwelling velocity over the domain is ͑␾ ␪͒ ϭ 0 0 ␪͑⌽ Ϫ ␾͒ ͑ ͒ uSA , cos E and 2a ϭ ϫ Ϫ5 Ϫ1 H taken to be w0 1.4 10 cm s , corresponding to the canonical value estimated by Munk (1966). Direct w0r0 ␷ ͑␪͒ ϭ tan␪. ͑2b͒ observations of turbulent dissipation rates above rough SA H topography (e.g., St. Laurent et al. 2001) show large This solution was used to validate the numerical solu- variations with depth (maximum dissipation in abyssal tion for the same basin configuration (Fig. 2a). In the canyons) and in horizontal direction (increased dissipa- interior, where the upwelling drives a basinwide cy- tion rates higher up on the ridge flank). The vertical clonic gyre, the two solutions are, indeed, indistinguish- variations cannot be represented in the barotropic able (Fig. 2b). SA60a did not provide a solution for the setup applied here. A positive upwelling velocity is as- boundary currents, but inferred the existence of the sumed throughout the basin, even though enhanced southward boundary current that is found along the mixing rates near the bottom correspond to local down- western boundary. Along the northern edge of the do- welling. The maximum upwelling over the ridge w is max main the numerical solution displays a westward defined to be more than two orders of magnitude larger boundary current, which was also anticipated by than the background value wbg applied in flat areas ϭ ϭ SA60a. (Fig. 1b; wbg 0.02 w0, wmax 4.5 w0). The total upwelling over the domain [which amounts to about 5 Sv (Sv ϵ 106 m3 sϪ1)] is compensated by a downwelling 3. Stationary abyssal circulation source at the grid point in the northwestern corner of The upwelling pattern and basin geometry defined in the basin (circle in Fig. 1a). Assuming a balance be- Fig. 1 results in the abyssal circulation pattern displayed tween vertical diffusion and advection (k␷ ϳ wH; Munk in Fig. 3. As before, the upwelling drives a cyclonic 1966), the above values for the upwelling velocity cor- flow. However, there is hardly any circulation on the respond to variations in the vertical diffusivity between eastern side of the ridge, in contrast to the basinwide ␬ ϭ ϫ 1 2 Ϫ1 ␬ ϭ ϫ Ϫ2 2 Ϫ1 max 1.9 10 cm s and bg 8.4 10 cm s , Stommel–Arons solution (Fig. 2a). The flow speed in ␬ ϭ 2 Ϫ1 Ϫ1 with an average value 0 4.2 cm s . This range is in the west is similar for the two cases (about 0.3 mm s ) line with recent estimates inferred from large-scale but in Fig. 3 it is directed eastward rather than north-

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eastward. A strong cyclonic recirculation is found To explain this abyssal circulation pattern, the im- above the midocean ridge with velocities reaching 1.4 pacts of the upwelling and bathymetry will be consid- cm sϪ1. Along the northern boundary, a westward cur- ered separately as well as combined in the subsequent rent is seen again (thick vectors). Note however that it sections. For this analysis, it is instructive to consider is wider than for the Stommel–Arons case. The western individual terms in the linearized vorticity balance, boundary current is almost identical for the two solu- which is easily derived from Eq. (1) by cross-differ- tions. entiating Eqs. (1a) and (1b) and substituting Eq. (1c):

f u Ѩh ␷ Ѩh wf R Ѩ␷ Ѩ͑u cos␪͒ ␤␷ Ϫ ϩ ϭ Ϫ b Ϫ ͑ ͒ ͩ ␪ Ѩ␾ Ѩ␪ͪ ␪ ͫѨ␾ Ѩ␪ ͬ. 3 h r0 cos r0 h r0 cos

͑i͒ ͑ii͒ ͑iii͒ ͑iv͒

The left-hand side represents vorticity changes due to effects. Hence, the flow is forced to turn westward in a (i) the advection of planetary vorticity and (ii) topo- narrow jet (marked with a “2” in 4a), whose strength is graphic stretching, respectively. The sources and sinks equal to the northward transport just south of the edge of vorticity, namely, (iii) the input provided by the up- of the forcing region (Spall 2000). For the case dis- welling and (iv) the dissipation by bottom friction, are played here, the forcing region extends to the equator. found on the right-hand side. There, the meridional velocity vanishes [Eq. (4a)]. When the southern edge of the forcing region is located a. Localized upwelling over a flat bottom at any other latitude, the upwelling will induce a me- First consider the abyssal circulation pattern forced ridional flow there as well, and a second zonal jet arises by nonuniform upwelling over a flat bottom (Fig. 4a). that carries the same transport as this meridional flow. In the upwelling region (gray shading) the flow is north- On a single-hemispheric domain, this southern jet op- (east)ward and increases with latitude. As in Fig. 3, poses the northern jet because ␷␤ is unidirectional (Ped- there is an eastward flow west of the forcing region, losky 1996; Spall 2000). On a double-hemispheric do- hardly any flow east of the forcing region, and a wide main, the meridional flow changes sign at the equator northern boundary current. This circulation pattern is [Eq. (4a)] and, as a consequence, both jets are directed an example of a ␤-plume circulation, which was first westward (not shown). described by Stommel (1982). This and later studies For all configurations of the upwelling region, mass considered ␤ plumes at midlatitudes, either on the ␤ conservation requires a flow toward the forcing region plane (Stommel 1982; Spall 2000) or on a spherical do- (marked with a “3” in Fig. 4a) to compensate for the main (Pedlosky 1996). Its dynamics are well known and losses due to the upwelling and the (net effect of the) are only briefly repeated here based on an analysis of zonal jet(s) (Pedlosky 1996; Spall 2000). The linearized, the vorticity balance [Eq. (3): Fig. 4b]. frictionless solution for this eastward flow is derived In the upwelling region (marked with a “1” in Fig. from Eq. (1c) by substituting Eq. (4a) and yields 4a), the local input of vorticity [(iii), solid line in Fig. 4b] ⌽ r E Ѩ is balanced by the advection of planetary vorticity in- ͑␾ ␪͒ ϭ 0 ͓ ͑␾ ␪͒ 2␪͔ ␾ u␤ , ␪ ͵ Ѩ␪ w , sin d duced by a meridional flow that arises in response to H sin ␾ ⌽ the upwelling [(i), dashed line: Stommel (1982)]. The E Ѩ ͑␾ ␪͒ r0 w , linearized, frictionless solution for the meridional ve- ϭ ͵ ͫ2w͑␾,␪͒ cos␪ ϩ sin␪ ͬ d␾ H Ѩ␪ locity in the interior is simply the Stommel–Arons so- ␾ ␾ ␪ lution [Eq. (2b)] with w0 replaced by w( , ): ͑4b͒ ͑␾ ␪͒ w , r0 (Pedlosky 1996). ␷␤͑␾, ␪͒ ϭ tan␪. ͑4a͒ H b. Uniform upwelling over a ridge Bottom friction plays a minor role [(iv), dotted line in Fig. 4b]. The impacts of bathymetry are illustrated in Fig. 5a, Around 50°–55°N, the upwelling reduces to its weak which shows the circulation forced by uniform up-

background value, so that the forcing term on the right- welling of strength w0 over a midocean ridge. Away hand side of Eq. (3) nearly vanishes, and the (potential) from the ridge, the Stommel–Arons case (Fig. 2a) is vorticity of the flow is conserved except for frictional recovered: the flow is (north)eastward and decreases

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FIG. 3. Stationary abyssal circulation forced by nonuniform upwelling over a midocean ridge (vectors defined as in Fig. 2a). The upwelling region is indicated by light gray shading; dark gray contours outline the 300-, 1000-, and 2000-m isobaths.

term (i) and the topographic stretching term (ii) cancel; the upwelling term (iii) is much smaller (Fig. 6). Fric- tion (iv) causes the flow to deviate from the f/h contours, by decreasing the negative vorticity of the flow on the flanks of the ridge and its positive vorticity on top of the ridge. Its effects are strongest in the southern part of the domain. Along the eastern flank of the ridge, both the upwelling and the topographic stretching induce a northward flow. On its western flank, the two effects oppose and the direction of the flow depends on their relative

FIG. 2. (a) Stationary abyssal circulation forced by uniform up- size. In section 3d, this competition is analyzed in detail. welling over a flat bottom (Stommel–Arons case). Thin (thick) For the case presented in Fig. 5, topographic stretching Ϫ Ϫ vectors denote speeds |u| Յ 0.15 cm s 1 (|u| Ͼ 0.15 cm s 1), vec- clearly dominates so that the resulting flow is south- ϫ tors are shown at 2° 2° for clarity. For plotting purposes, the ward. maximum (minimum) speed is set to 1.5 cm sϪ1 (0.01 cm sϪ1). (b) Validation of the solution in (a) (symbols) with Eq. (2) (solid lines). Shown are u(␪) and ␷(␪) for ␾ ϭ 45°W (marked by *) and c. Localized upwelling over a ridge ␾ ϭ 30°W(छ). Based on the analysis presented in the previous two sections, the dynamics of the abyssal circulation pattern toward the eastern boundary. This is confirmed by the displayed in Fig. 3 can now be explained. Away from vorticity balance, which reveals that the upwelling is the midocean ridge, where the upwelling is very weak, balanced by the planetary vorticity term in the interior the circulation behaves like a ␤ plume emerging from (not shown). the region of enhanced upwelling located over the ridge Figure 5b shows the circulation in the vicinity of the (section 3a). West of the midocean ridge, the flow is ridge; Fig. 6 shows the relevant terms in the vorticity nearly zonal and the velocity is constant, as for the balance over the same region. The flow near the ridge circulation forced by nonuniform upwelling over a flat is dominated by topographic stretching, which steers bottom. Near the ridge, topographic stretching comes the flow closely along f/h contours (Fig. 5b): on the into play (section 3b). This is clear from Fig. 7, which western flank of the ridge the circulation turns south- shows a section of the vorticity balance at 30°N (the ward as the water column shoals; the opposite occurs on spatial pattern of the various terms is very similar to the eastern flank. To first order, the planetary vorticity those for the circulation forced by uniform upwelling

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FIG. 4. (a) As in Fig. 3 but forced by nonuniform upwelling over a flat bottom. (b) Individual terms in the vorticity balance [Eq. (3)] at 30°N for the solution in (a) [(i) planetary vorticity (dashed line), (iii) upwelling (solid), and (iv) dissipation (dotted); Eq. (3) FIG. 5. (a) As in Fig. 3 but forced by uniform upwelling over a states (i) ϭ (iii)Ϫ(iv)]. All terms are normalized with the maxi- midocean ridge. (b) Detail from (a) for ␾ ∈ (40°W, 20°W); vectors mum of the upwelling over the section. are shown at 1° ϫ 2°. Gray lines denote f/h contours (contour interval is 5.0 ϫ 10Ϫ9 msϪ1); depth contours are omitted for clarity. displayed in Fig. 6). The planetary vorticity term [(i) in Eq. (3), dashed line] and the topographic stretching term [(ii), dash–dotted line] again dominate over the northward flow induced by the upwelling. As a conse- upwelling term [(iii), solid line]. As before, the flow quence, the resulting flow can either be southward or mainly follows f/h contours except where friction [(iv), northward, depending on the geometry of the ridge and dotted line] plays a role. Stretching at the bottom ap- of the upwelling region. From Eq. (3) one can estimate pears to have a net effect on the poleward flow across that the topographic effect is compensated by that of potential vorticity contours over the basin. In compari- the upwelling when son to the flat-bottom case (Fig. 4), the poleward flow f U ⌬H fW is reduced when a ridge is present. The reduction is as ϳ ͑ ͒ ␪ ⌬␾ , 5a large as 20% at midlatitudes. Hr r0 cos H Hr where U and W are typical scales for the local zonal d. Circulation on the western flank of the ridge velocity and upwelling, ⌬H is the height of the ridge, ⌬␾ On the western flank of the ridge, topographic H is the horizontal scale of the ridge flank, and Hr is stretching induces a southward flow that opposes the the mean depth of the water column on the flank (see

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FIG. 6. Individual terms in the linearized vorticity balance [Eq. (3)] for the circulation displayed in Fig. 5: (i) advection of planetary vorticity; (ii) topographic stretching; (iii) input by upwelling; and (iv) dissipation. All terms are normalized with the maximum of the upwelling term over the basin [contour interval is 0.3; gray shading denotes negative values; Eq. (3) states (i)Ϫ(ii) ϭ (iii)Ϫ(iv)]. Only the vicinity of the ridge is shown [␾ ∈ (40°W, 20°W)].

Fig. 8a for definitions). The scale for the zonal flow U ⌬␾ ϳ Ϫ ⌬ H can be linked to that for the upwelling W by assuming Hw H H ⌬␾ . w [in accordance with Eqs. (2a) and (4b)] Substitution of the two expressions above in Eq. (5a) 2Wr cos␪ ϳ 0 ⌬␾ yields, after some manipulation, an equation for the U w Hw critical ridge height ⌬H: ⌬␾ with w and Hw as measures for the width of the ⌬␾ ⌬␾ ⌬ ϳ w H ͑⌬␾ Ͼ⌬␾ ͒ upwelling region and the depth of the water column H H w H . ͑5b͒ 2⌬␾2 ϩ ⌬␾2 over this region, respectively. Since only the part of the w H upwelling that takes place on the ridge flank and to the For ridge heights larger than ⌬H, topographic stretch- east of it determines the zonal velocity on the flank, the ing effects overcome the local upwelling and the flow is ⌬␾ ϭ⌬␾ ϩ⌬␾ ⌬␾ former is defined as w H bdy, with bdy directed southward (Northern Hemisphere). A plot of ⌬ ⌬␾ ⌬␾ being the distance between the ridge and the eastern H as a function of H and w is shown in Fig. 8b. boundary of the upwelling region (Fig. 8a). Here Hw Symbols mark parameter settings applicable to model can be approximated by configurations discussed in this section (see legend).

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tends all the way to the eastern boundary of the do- ⌬␾ ϭ ⌬␾ ϭ main. Hence, bdy 15° (see Fig. 1) and w 20°. For this case, compensation is estimated to occur for a ridge height ⌬H ϳ 3.6 ϫ 102 m (circle in Fig. 8b). This is much smaller than the height of 2.4 ϫ 103 m applied for the circulation in Fig. 5, in line with the earlier conclusion that the flow is dominated by topographic stretching. Figure 9a shows the meridional velocity along the ridge for a series of stationary circulation patterns forced by uniform upwelling over a ridge that varies in height between 0 and 600 m (increments of 120 m). The flow at the western flank is seen to reverse when the maximum ridge height is close to 360 m, in excellent ⌬ FIG. 7. Individual terms in the vorticity balance [Eq. (3)] at 30°N agreement with the estimate for H. A second series of for the circulation forced by nonuniform upwelling over a mid- circulation patterns using a wider ridge is shown in Fig. ocean ridge displayed in Fig. 3 [(i) planetary vorticity (dashed ⌬␾ ϭ ⌬␾ ϭ 9b ( H 10°, w 25°). According to Eq. (5b), line), (ii) topographic stretching (dash–dotted), (iii) upwelling compensation is expected when ⌬H ϳ 5.6 ϫ 102 m. The (solid), and (iv) dissipation (dotted)]. All terms are normalized with the maximum of the upwelling over the section; Eq. (3) states numerical solution indeed shows a reversal of the flow (i)–(ii) ϭ (iii)Ϫ(iv). Only the vicinity of the ridge is shown [␾ ∈ for a maximum ridge height between 480 and 600 m. (40°W, 20°W)]. Also for cases forced by a narrow upwelling region (section 3c) the critical ridge height ⌬H can be estimated using Eq. (5b), keeping in mind that the correct ⌬ ϳ ϳ ϫ 3 ⌬␾ ⌬␾ The maximum value for H H/3 1.0 10 m choices for both w and H depend on the width of ⌬␾ ϭ⌬␾ ⌬␾ ϭ occurs for H w ( bdy 0). the upwelling region and its location with respect to The standard ridge (section 2a) has a characteristic the western flank of the ridge. For the circulation dis- ⌬␾ ϭ width H 5°. For circulations forced by uniform played in Fig. 3, the upwelling region is narrow but upwelling (section 3b; Fig. 5), the upwelling region ex- still encompasses the ridge flank (gray and black line in

⌬ ⌬␾ ⌬␾ FIG. 8. (a) Schematic defining the scales used in Eqs. (5a)–(5b). (b) Critical ridge height H as a function of H and w [Eq. (5b); contour interval is 100 m]. A circle and a square mark the parameter values for the circulation forced by uniform upwelling and nonuniform upwelling over the ridge (Figs. 5a and 3), respectively. Asterisks indicate parameter settings for the various configurations discussed in section 3d.

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FIG. 9. Sections for ␷(␾ ) along 30°N for the circulation forced by uniform upwelling; heights of the midocean ridge range from 0 to 600 m (increments of 120 m). A thick black line outlines the ridge, which has a ⌬␾ ϭ ⌬␾ ϭ ⌬␾ ϭ characteristic width (a) H 5° and (b) H 10°. For both cases, bdy 15°. A dotted line marks the reversal of the flow.

Fig. 10a). The characteristic width of the forcing region hand side of Eq. (5a)] is overestimated, and, as a con- ⌬␾ ϭ ⌬␾ ϭ ⌬␾ ϭ ⌬ is now w 13° ( bdy 8°), and H 5° as before. sequence, so is H in this case. Equation (5b) predicts a reversal of the flow for a ridge As a last example, a narrow upwelling region over a height ⌬H ϳ 5.4 ϫ 102 m (square in Fig. 8b). Figure 10a wide ridge is considered (Fig. 10c). When the upwelling shows the meridional velocity along the ridge for a se- and the topography are aligned (solid lines), the flank ries of stationary circulation patterns with a ridge of the ridge extends farther westward than the up- height ranging from 0 to 720 m. Reversal of the flow welling region. Hence, the relevant scale for the ridge is ⌬␾ ϭ occurs for a ridge height between 480 and 600 m, in line H 8° rather than its full width of 10°. The scale for ⌬ ⌬␾ ϭ ⌬␾ ϭ with the estimate for H. the upwelling region is w 16° ( bdy 8°,asfor Figure 10b shows the meridional velocity along the the case displayed in Fig. 10a). The critical ridge height ridge for heights of 720, 840, and 960 m (solid lines) that follows from Eq. (5b) using these values is ⌬H ϳ for a case in which the upwelling region was shifted 6.7 ϫ 102 m. The numerical calculations already display westward by 5° (solid gray line). Although the width southward flows for smaller ridge heights on the west- of the forcing region is the same as in Fig. 10a, the ern part of the flank, again because the local upwelling distance between the ridge and the eastern boundary of there is weaker than estimated and hence cannot com- ⌬␾ ϭ the upwelling region is now reduced to bdy 3° so pensate the topographic stretching. Higher up on the ⌬␾ ϭ ⌬␾ ϭ that w 8° ( H 5° as before). As a consequence, ridge flank, reversal does occur for ridge heights be- compensation is now expected for a larger ⌬H ϳ 7.8 ϫ tween 600 and 720 m, in line with the estimate for ⌬H. 102 m. The numerical calculations again confirm this When the upwelling region is shifted westward by 5° estimate. When the upwelling region is shifted eastward (dashed lines in Fig. 10c), it does cover the western ⌬␾ ϭ by 5° (dashed gray line in Fig. 10b), the western flank of flank of the ridge so that H 10° as for the case the ridge is no longer fully enclosed by the upwelling shown in Fig. 9b. The width of the upwelling region is ⌬␾ ⌬␾ ϭ⌬␾ ϩ ϭ region. Hence, the scale H needs to be reduced ac- approximated by w H 3° 13°, and hence ⌬␾ ϭ ⌬ ϳ ϫ 2 cordingly ( H 3°). At the same time, this eastward H 8.9 10 m. In close agreement with this esti- ⌬␾ ϭ ⌬␾ ϭ shift leads to a larger bdy 15° so that w 18°. mate, the meridional velocity along the flank (dashed The critical ridge height now becomes only ⌬H ϳ 2.5 ϫ lines, off set by ϩ0.3 cm sϪ1) reverses for a ridge height 102 m. However, the numerical simulations (dashed of about 840 m. lines, velocities are off set by ϩ0.2 cm sϪ1) already show In summary, the simple estimate for the critical ridge a reversal of the flow on the western flank for a ridge height derived in this section appears very accurate in height close to 120 m. Because the upwelling has a predicting the reversal of the flow on the western flank hyperbolic rather than a linear shape, as sketched in of the ridge, except when the location of the ridge flank Fig. 8a, the upwelling near the western edge of the coincides with the far edge of the upwelling region. This forcing region is underestimated. Therefore, the input holds for a substantial range of parameter settings (as- of vorticity by the upwelling on the ridge flank [right- terisks in Fig. 8b).

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4. Summary and discussion

In this paper, solutions for the stationary abyssal circulation forced by prescribed localized upwelling are considered for basin geometries that incorporate pronounced bathymetry. For this, a planetary-scale barotropic model on the Northern Hemisphere is used. It is shown that the abyssal circulation driven by enhanced upwelling over a midocean ridge (Fig. 3) differs sub- stantially from the classical Stommel–Arons circulation forced by uniform upwelling over a flat bottom (Fig. 2a: Stommel and Arons 1960a). Away from the ridge, a ␤ plume arises (e.g., Stommel 1982); near the midocean ridge, topographic stretching dominates the flow pattern. An interesting aspect of the circulation is the flow on the western flank of the ridge. There, the upwelling induces a northward flow through vortex stretching, while topographic stretching in the shoaling water column directs the circulation southward. Based on the vorticity equation, one can estimate the critical ridge height for which both effects compensate and the direction of the flow reverses. For a series of model configurations covering a substantial range of parameter values, the accuracy of this estimate was demonstrated in this paper. The question arises of whether the simple estimate presented in this paper can predict the direction of the abyssal flow along the western flank of a midocean ridge in a more realistic setting. As an example, the circulation in the Brazil Basin is considered, as direct observations of the turbulent dissipation rate (Polzin et al. 1997; Ledwell et al. 2000) motivated numerical studies specifically focusing on the circulation forced by nonuniform mixing for this area (St. Laurent et al. 2001; Huang and Jin 2002) or on a global scale (Simmons et al. 2004). On the Southern Hemisphere, upwelling forces a southward flow through vortex stretching, while the topographic effects of a shoaling water column induce a northward flow on the western flank of a ridge (both effects induce a southward flow on its eastern flank). The derived estimate for the critical ridge height ⌬H [Eq. (5b)] can be compared to the change in FIG. 10. Sections for ␷(␾ ) along 30°N for the circulation forced by nonuniform upwelling, for various model configurations and depth over the area of study to predict the flow direc- ridge heights. Black (gray) lines outline the shape of the ridge tion: southward (northward) for ridge heights smaller (upwelling pattern). The displayed range of ridge heights is given than (larger than) ⌬H. It should be noted that the es- in parentheses; increments of 120 m are used. Dotted lines mark timate for ⌬H was derived using a barotropic model. In the reversal of the flow for the various cases. (a) Standard ridge such a model, the bathymetry is “felt” throughout the and localized upwelling (0–720 m); (b) standard ridge and localized upwelling shifted westward by 5° (solid lines, 720–960 m) and water column, while in reality the stratification of the eastward by 5° (dashed, 0–360 m, velocities offset by ϩ0.2 water column will restrict its influence (Pedlosky 1987). cm sϪ1); (c) wide ridge and localized upwelling centered over the Therefore, the barotropic model probably overesti- ridge (solid, 480–840 m) and shifted westward by 5° (dashed, 720– mates the impacts of topography on the vorticity bal- ϩ Ϫ1 1080 m, velocities offset by 0.3 cm s ). ance and hence is expected to underestimate the critical value for ⌬H.

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The model simulation presented by Huang and Jin resent the main features of the abyssal flow as simu- (2002) show a northward flow of several millimeters per lated by St. Laurent et al. (2001). second along the western flank of the Mid-Atlantic Ridge in the South Atlantic Ocean (see their Fig. 9c), Acknowledgments. The numerical continuation code indicative of a circulation dominated by topographic used in this paper was kindly provided by Henk Dijk- stretching. The global-scale simulation discussed by stra. Discussions of the results with Sybren Drijfhout Simmons et al. (2004) displays a similar flow pattern and Gerrit Burgers are greatly appreciated. (their Fig. 7). Both circulation patterns are forced by enhanced mixing over the eastern half of the Brazil REFERENCES

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