Abstract

A reduced-gravity model is developed to represent the flow of Antarctic Bottom Water (AABW) over realistic bathymetry in an Atlantic domain. The dynamics are based on the steady, planetary–geostrophic, shallow-water equations, including a linear bottom friction and a uniform diapycnal upwelling through the top of the model layer.

The model solutions are broadly consistent with observations of the distribution and transport of AABW. The flows occur predominantly along potential vorticity contours, which are in turn broadly oriented along bathymetric contours. The characteristic weak flow across potential vorticity contours of the Stommel–Arons model is present as a small addition to this stronger forced mode along potential vorticity contours. As a consequence, mass balance is maintained not by hypothesized western boundary currents as in the Stommel–Arons model, but by the interplay between topographic slope currents and interior recirculations. In particular, a transposition is found in the flow of AABW from the western side of the Brazil Basin south of the equator to the western flank of the Mid-Atlantic Ridge north of the equator. This is also consistent with an analytical result derived by extending the Parsons mechanism to an abyssal layer overlying arbitrary bathymetry. The authors suggest that the results provide a more convincing zero-order picture than the Stommel–Arons model for the circulation of AABW and perhaps for abyssal water masses in general.

1. Introduction

Antarctic Bottom Water (AABW) is an abyssal water mass, easily distinguishable in hydrography by its cold (θ < 1.8°C)1 and relatively fresh (S < 34.8 psu) signature, which is also high in silica and low in oxygen. In the Atlantic Ocean, AABW is comprised of Weddell Sea Deep Water, which is formed in the Southern Ocean against the Antarctic continent, and recirculating lower Circumpolar Deep Water, which enters the Atlantic via Drake Passage. In Fig. 1 we present a schematic of the circulation pathways and approximate transports deduced from observations. The bathymetry of the abyssal basins and their interconnecting passages strongly influences the pathways of AABW as it finds its way northward into the western basins of the South Atlantic and then spreads through fracture zones into the eastern basins. For reference, the bathymetric features referred to in the text are represented in Fig. 2.

Fig. 1.

A schematic of the major pathways and transports of AABW. Approximate transports in Sverdrups are shown in circles. The pathways connecting obervations of transport are inferred from the general concensus of obervations. The figure may in reality only represent the actual circulation in the very broadest sense. Shading indicates depths shallower than 4000 m (bathymetry from Row et al. 1995).

Fig. 1.

A schematic of the major pathways and transports of AABW. Approximate transports in Sverdrups are shown in circles. The pathways connecting obervations of transport are inferred from the general concensus of obervations. The figure may in reality only represent the actual circulation in the very broadest sense. Shading indicates depths shallower than 4000 m (bathymetry from Row et al. 1995).

Fig. 2.

Locations of basins and topographic features referred to in the text. Shading indicates depths shallower than 4000 m (bathymetry from Row et al. 1995).

Fig. 2.

Locations of basins and topographic features referred to in the text. Shading indicates depths shallower than 4000 m (bathymetry from Row et al. 1995).

a. Previous theory

The classical view of the abyssal circulation first developed by Stommel and Arons (Stommel 1958; Stommel and Arons 1960a,b) has served as a paradigm for our understanding of the thermohaline circulation. In the Stommel–Arons theory, uniform upwelling in the interior of a flat-bottomed ocean basin balances a localized source of dense water and drives a barotropic poleward interior flow by vortex stretching; boundary currents are hypothesized to exist on the western side of ocean basins to maintain continuity in the circulation. The upwelling in turn balances the downward diffusion of heat from the surface ocean and maintains the permanent thermocline.

However, one of the most interesting aspects of the circulation of AABW, which is not predicted by the Stommel–Arons theory, is the transposition of its flow from the western side of the Brazil Basin south of the equator to the eastern side of the basin north of the equator. Here the current continues northward against the western flank of the Mid-Atlantic Ridge, as represented in Fig. 1. The existence of such an eastern boundary current contradicts the Stommel–Arons theory, and a number of mechanisms have been proposed to account for this. One such mechanism is that of Kawase (1987), who studied the spinup of a source-driven abyssal circulation numerically in a flat-bottomed basin straddling the equator, including a damping term on the interface height of the abyssal layer to represent cross-isopycnal buoyancy mixing. In the limit of weak damping, the deep western boundary current (DWBC) generated by the source in the northwest corner separates along the equator, forming northward and southward flowing currents along the eastern boundary. The eastern boundary currents then radiate long Rossby waves westward from the eastern boundary to set up a Stommel–Arons interior flow. In the limit of strong damping the propagation of long Rossby waves into the interior is prevented and this acts to “freeze” the solution in the state containing eastern boundary currents. Tziperman (1987) proposed a similar mechanism to explain the existence of the eastern boundary Mediterranean Undercurrent.

Another explanation for the equatorial transposition of AABW follows from the work of Nof (1990), who by analogy with the Parsons (1969) model of Gulf Stream separation, demonstrated that a northward-flowing surface geostrophic current above a motionless lower layer and against a western vertical wall can only exist in the Northern Hemisphere if it has no outcrop on the open ocean side. Nof considered a shallow water layer of thickness h in the vicinity of the equator, which is geostrophic in the cross-stream direction; that is,

 
formula

where x and y are eastward and northward distances respectively, f = 2Ω sinθ is the Coriolis parameter, υ is the northward velocity, and g′ is the reduced gravity. Equation (1) was integrated and rearranged by Nof (1990) to yield

 
formula

where T = ∫ υh dx is the northward transport, and he and hw are the depths of the current at its offshore edge and the wall respectively. Here hw is always negative in the Northern Hemisphere for he = 0 [this case was originally considered by Anderson and Moore (1979)]. For a finite value of he a jet can penetrate into the Northern Hemisphere, whereby hw decreases until hw = 0 when the jet must then separate from the wall. Nof and Olson (1993) extended the work of Nof (1990) by including bathymetry in the form of a parabolic channel straddling the equator in a meridional direction. In this case bathymetry relaxes the outcropping constraint, and inertial effects allow their abyssal current to cross isobaths and flip sides in the channel as it crosses the equator, strongly resemblant of the AABW flow.

In a related study to explain the transposition of AABW, Speer and McCartney (1992) introduced variable layer thickness into the Stommel–Arons model (thereby distorting the potential vorticity contours from zonal contours of the Coriolis parameter) and showed that, for a prescribed eastern boundary thickness, separation of the abyssal layer also occured from the western boundary. The separation mechanism is analagous to Parsons (1969) and Nof (1990), although the northward flow in this case is driven by upwelling.

Further studies investigating the effect of bathymetry on the abyssal circulation include Rhines (1989), who considered planetary-scale zonal flows over topography and demonstrated blocking and resonance structures. Straub and Rhines (1990) investigated isolated regions of closed geostrophic contours due to bathymetric hills and bowls submerged in the abyssal layer and found internal jets linking the closed contour region to the western boundary. Kawase and Straub (1991) examined the spinup of a similar scenario and demonstrated that a vigorous cyclonic circulation is generated over regions of closed geostrophic contours, regardless of the sign of the topography, by a vortex stretching effect due to the convergence of the ageostrophic frictional flow, which balances upwelling over the closed contour region. Johnson (1998) finds support for this mechanism in relation to abyssal flows over ocean trenches; alternatively Dewar (1998) argues for a balance between bottom friction and potential vorticity mixing over the bathymetry determining the strength of the anticyclonic circulation.

b. A generalized Parsons mechanism

In the presence of bathymetry, such as in Nof and Olson (1993), eastern and western boundary currents can occur quite naturally as flows that are geostrophic in the cross-stream direction, and the strong damping limit of Kawase (1987) may thus be avoided. We can generalize the result of Nof and Olson described above to an isopycnal layer that has two outcrops (as abyssal water masses do) against arbitrary bathymetry. We represent the AABW as a shallow homogeneous layer overlain by a motionless upper layer of slightly lesser density, as represented in Fig. 3. With the inclusion of bathymetry, the transport integral of Nof (1990) becomes

 
formula

where H is the resting ocean depth, A is the cross-sectional area of the current, and

 
formula

is the thickness-weighted mean slope on which the fluid sits. Since g′ and A are both positive definite, Eq. (3) demonstrates that, as the current crosses the equator, the average topographic slope beneath it must change sign to maintain a transport in the same direction. We believe that the switching of AABW as it crosses the equator may be understood purely as a consequence of the above integral constraint; the AABW flow actually requires a reversal in the slope to continue northward. This result is independent of the mechanism that is, of course, necessary to modify the sign of the potential vorticity of the current and enable it to cross the equator. In general dissipation in some form is necessary to achieve modification of the potential vorticity [although Nof and Olson (1993) were able to obtain a cross-equatorial flow using inertia alone since their current had zero potential vorticity]. We choose a purely frictional process in order to achieve a cross-equatorial flow, although Edwards and Pedlosky (1998a) and Edwards and Pedlosky (1998b) have recently pointed out the importance of both inertia and dissipation in the vicinity of the equator.

Fig. 3.

Schematic of model configuration showing a layer of thickness h and density ρ + Δρ, overlying topography at depth H and beneath a motionless upper layer of density ρ.

Fig. 3.

Schematic of model configuration showing a layer of thickness h and density ρ + Δρ, overlying topography at depth H and beneath a motionless upper layer of density ρ.

c. Structure of this paper

In this paper we investigate the dynamically simple limit of a reduced-gravity single active layer beneath a motionless upper layer in a numerical study to determine the steady pathways of AABW in an Atlantic domain that incorporates realistic bathymetry. The paper is divided as follows: In section 2 we formulate the numerical model; in sections 3–6, having divided our Atlantic domain into four main subbasins, we discuss the model results for each basin and compare them with observations. In section 7 model sensitivity to various parameters is addressed, and section 8 consists of a concluding discussion.

2. Numerical model

a. Dynamical formulation

We represent the AABW as a shallow homogeneous layer overlain by a motionless upper layer of slightly lesser density. Within this dynamical framework we choose a planetary–geostrophic formulation consistent with the smallness of the oceanic interior Rossby number and include a bottom friction coefficient in the momentum equations to represent the frictional decay of our AABW layer by its interaction with the ocean floor beneath. Our formulation, excepting the bathymetry, and subsequent method of solution is very similar to that of Speer et al. (1993).

Pedlosky (1987) demonstrates the equivalence of a bottom Ekman layer to that of Rayleigh friction with a friction coefficient, R = (κzf/2)1/2/h = /2h, where κz is the vertical eddy viscosity coefficient, h is the layer depth, and δ is Ekman-layer thickness. Since R = f = 0 at the equator and we require friction to allow a cross-equatorial flow, we use the multiplier /h in the momentum equations (hereafter we refer to the constant, r, as the friction coefficient and /h as the friction term). In order to avoid the friction term tending to infinity at the equator we choose δ to be constant. Thus, our friction term incorporates an inverse dependence on layer thickness, as per Pedlosky (1987), and in neglecting f we allow ourselves to retain a nonzero friction coeffcient at the equator where the friction serves as a crude parameterization of the effects of inertia, thus enabling us to retain our simple dynamical form. We include a uniform diapycnal upwelling, represented by the coefficient w in the continuity equation. In contrast to Speer and McCartney (1992), however, we neglect entrainment fluxes. For simplicity we also neglect the effect of eddies since their role in generating topographic recirculation gyres, such as in the Brazil Basin, is not well understood but we will make a posteriori statements about the possible role of eddies at a later stage in the paper.

In spherical coordinates, where θ and ϕ denote latitude and longitude respectively, Re is the radius of the earth, f is the Coriolis parameter, u and υ are velocities in the ϕ and θ directions, h is the (positive definite) layer thickness, H is the ocean depth; and, denoting partial derivatives in the j direction by subscipts j, our momentum and continuity equations take the following form:

 
formula

In the absence of upwelling and in the limit r2f2 (as per Kawase and Straub 1991), the velocity may be eliminated from the above equations to yield a single equation for the interface displacement, η = hH, of the following form:

 
formula

The terms on the left-hand side denote the combined planetary–topographic long Rossby wave propagation. The terms on the right-hand side act as a diffusion operator on η, which arises from the presence of the friction coefficient /h. The magnitude of the diffusion operator is gδr/f2 = L2dr, where Ld is the internal deformation radius based on the reduced gravity g′ at the interface between the AABW layer and the layer above, and the Ekman-layer depth δ.

b. Numerical formulation

We will be seeking to determine the steady distribution of AABW in our domain. This is obtained by integrating Eq. (6) numerically to a steady state, computing the velocities diagnostically from the momentum equations [Eqs. (4), (5)] at each time step.

The model resolution is ½° in a domain extending from 50°N to 30°S, 80°W to 20°E. The model bathymetry is taken from Row et al. (1995) and is generated by computing the local depth maximum in ¼° regions of the domain, smoothing the result with a moving average filter of order 6 and taking every second point. The local maximum is computed to avoid constricting the circulation of our model AABW layer and the smoothing helps to ensure that the horizontal pressure gradient is able to be resolved by the model. Gaps in the Mid-Atlantic Ridge, such as the Vema Fracture Zone and the Romanche and Chain Fracture Zones, are narrower features than we are able to resolve with our model. Even if we could resolve them, their dynamics undoubtedly involve inertial effects, which we neglect and locally enhanced mixing that both warms and modifies the transport of AABW exiting the fracture zones. Thus, in the regions of the above fracture zones we impose the bathymetry manually. We regard the fracture zones purely as a means of connecting the circulation between basins and they have been tuned to yield sensible transports. The sensitivity of the results to this procedure is discussed at a later stage in the paper.

Models such as Straub and Rhines (1990) and Kawase and Straub (1991), where topography is submerged in the active layer, do not capture the “subtle interaction between baroclinicity and bottom slope” (Hallberg and Rhines 1996) that occurs due to isopycnals outcropping against bathymetry. Since the lateral distribution of AABW is defined by outcropping isopycnals, which are generally difficult to model numerically, we use a flux-corrected transport (FCT) numerical formulation according to Zalesak (1979) on a C grid (Arakawa and Lamb 1977). The high-order component of the numerical discretization we choose for the FCT is fourth order and centered in space, and third order Adams–Bashforth in time.

Horizontal pressure gradients are computed according to Scheme 2 in Bleck and Smith (1990), which ensures the reliable computation of velocities where isopycnals overlie variable topography. This method relies on an averaging of the pressure gradient computed within the isopynic layer at surrounding grid boxes, subject to weighting coefficients computed as a function of the number of massless grid boxes involved in each pressure gradient computation.

c. Boundary conditions

The southern boundary of our model is situated purposefully at the latitude of the flow of AABW from the Argentine Basin into the Brazil Basin via the Rio Grande Rise since here the northward transport is horizontally confined and has been determined as 6.6 (±0.4) Sv (Sv ≡ 106 m3 s−1) by Speer and Zenk (1993) (using a level of no motion at a depth close to 2°C). We impose a layer thickness condition to represent this transport, choosing the 1.6°C surface as our layer interface since, throughout the passage of AABW around the Atlantic, most of the transport is contained below this isotherm (McCartney and Curry 1993). We neglect any flow of AABW into the eastern South Atlantic that may occur via the Walvis Ridge, as indicated by cold bottom temperatures in Fuglister (1960) since Warren and Speer (1991) have inferred this to be very small. The southern boundary thickness condition is presented as a zonal section in Fig. 4; outflows through the southern boundary of the model do not occur.

Fig. 4.

Zonal section along 30°S to show the model layer thickness condition at the southern boundary (adapted from Speer and Zenk 1993). Bathymetry is represented by light shading and the AABW layer by dark shading. The latitude and longitude of each end of the section is marked at the base of the figure. The depth of the section in meters is shown at the left of the figure.

Fig. 4.

Zonal section along 30°S to show the model layer thickness condition at the southern boundary (adapted from Speer and Zenk 1993). Bathymetry is represented by light shading and the AABW layer by dark shading. The latitude and longitude of each end of the section is marked at the base of the figure. The depth of the section in meters is shown at the left of the figure.

At the northern boundary of the model it is necessary that the interior flow of AABW determines the boundary condition. We impose a condition of zero meridional gradient in the interface height η; this corresponds to u = 0 in the geostrophic limit, thus requiring flow to be normal to the boundary. In practice, the flow out of the northern boundary of the domain is very weak and has no discernible effect on the interior flow.

d. Parameter values

The reduced gravity g′ for the model is chosen to be constant throughout the domain and is computed as g′ = g[σ4(AABW) − σ4(LNADW)]/ρ0. We take the gravitational acceleration, g = 9.8 m2 s−1, σ4(AABW) = 46.02 kg m−3, and σ4 (LNADW) = 45.85 kg m−3 [which are approximately average values for the density anomalies of AABW below the 1.6°C isotherm and the lower North Atlantic Deep Water (LNADW) above], and ρ0 = 1000 kg m−3, which yields g′ = 1.7 × 10−3 m2 s−1. The Ekman-layer thickness δ is fixed at 25 m.

For the solutions presented in the next four sections, 3–6, the diapycnal upwelling velocity is chosen as w = 2 × 10−7 m s−1, suggested as an area-averaged value by McCartney and Curry (1993), and the friction coefficient as r = 1.15 × 10−6 s−1, which corresponds to a spindown timescale of approximately 10 days for a current that is the thickness of the Ekman layer; this implies that a current moving at 0.1 m s−1 would travel less than 100 km in this time. The observed longevity of abyssal slope currents would appear to suggest that our friction coefficient is too large: however, MacCready (1994) has recently demonstrated that the spindown of abyssal slope currents is lengthened considerably due to the large potential energy to kinetic energy ratio residing in isopycnals upturned against the topography. MacCready interprets the spindown timescale of abyssal currents as the time required for the cross-slope Ekman transport to drain away the isopycnal displacement characterizing a current: the transport of such currents is able to remain fairly constant during this process as the potential energy store replenishes kinetic energy, which is in turn drained by work done in the bottom Ekman layer. We view the frictional coefficient not only as a representation of dissipation within the bottom Ekman layer, but also as a device to allow flow across the equator and through the fracture zones; in reality, these processes will involve inertial effects. We will discuss model sensitivity to the values of upwelling and the prescription of the friction term in section 7.

Steady solutions for the distribution of AABW are determined by initializing the model with the southern boundary condition and allowing AABW to gradually fill up the model abyss as it propagates around the domain in a manner resembling a gravity current, at the speed of a long internal topographic wave (∼1–2 m s−1). We do not consider the spinup process itself since we are unable to resolve the internal deformation radius (typically around 4 km in our model). The model results presented have all been integrated for approximately 500–600 years and are very close to a steady state. For comparison, the time needed to flush our AABW layer out of the model (the residence time) is around 60 yr.

3. Circulation in western South Atlantic (Brazil Basin)

For the purposes of the discussion that follows in this, and the following three sections, we divide up the model domain into four areas: north and south of the equator and east and west of the Mid-Atlantic Ridge. We first present a summary of observations and then compare and contrast the model results in each area in turn.

a. Observations

The ∼7 Sv flow (Speer and Zenk 1993) of AABW from the Argentine Basin into the Brazil Basin via the Rio Grande Rise at 30°S continues northward as a DWBC against the continental slope of South America and fills the abyssal Brazil Basin to a depth of over 1500 m. Recent transport estimates from McCartney and Curry (1993), computed using 1983 CTD data from the R/V Knorr and a level of no motion at the depth 1.9°C, decrease northward, as shown in Table 1. Interestingly, a recent float study by Hogg and Owens (1999) shows little evidence for a DWBC of AABW north of 20°S. This is clear evidence of the uncertainty in our observations of abyssal flows and suggests the need for many more floats over a long timescale in order to obtain a good picture of the mean circulation.

Table 1.

Transport of AABW as a function of latitude, as computed by McCartney and Curry (1993).

Transport of AABW as a function of latitude, as computed by McCartney and Curry (1993).
Transport of AABW as a function of latitude, as computed by McCartney and Curry (1993).

The decline in transport between 30°S and the equator suggests upwelling across isotherms. The decline is uneven (0.3 Sv between 30° and 23°S and 1.2 Sv between 23° and 11°S), which may be due to either variability in the upwelling rate or uncertainty in the transport estimates. The latter can occur due to differences in the selection of a level of no motion, bottom triangle approximation, and temporal variability and directly affect estimates of the area-averaged upwelling. The difference in transport between 23° and 11°S suggests a diapycnal upwelling rate, w = 3.4 × 10−7 m s−1, which is almost identical to an estimate by Warren and Speer (1991) south of 11°S; McCartney and Curry (1993) suggests that this value is suspiciously high for an area-averaged value however. For instance, considering the difference in transports between 30° and 11°S, an estimate of w ∼ 2 × 10−7 m s−1 is obtained. In addition, considering a level of no motion corresponding to 1.8°C, McCartney and Curry (1993) show that the upwelling rate between 23° and 11°S decreases significantly to w = 2.2 × 10−7 m s−1.

In the interior of the Brazil Basin, against the strong northward flowing DWBC, deMadron and Weatherly (1994) and Speer and Zenk (1993) find a weak southward transport of AABW and then a reversal back to weak northward transport adjacent to the Mid-Atlantic Ridge. Superimposed on this flow are abyssal recirculations, which Spall (1994) suggests are driven by eddy fluxes of potential vorticity from instability in the DWBC of AABW flowing north. DeMadron and Weatherly’s circulation schematic for AABW is reproduced in Fig. 5 and is broadly similar to that of Speer and Zenk (1993) except that it includes a strong cyclonic recirculation gyre to the north; deMadron and Weatherly (1994) explained the existence of the northern recirculation gyre as due to the deflection of denser AABW (the Weddell Sea Deep Water component) eastward by shoaling of the bathymetry at the equator.

Fig. 5.

Circulation schematic for Brazil Basin reproduced from deMadron and Weatherly (1994).

Fig. 5.

Circulation schematic for Brazil Basin reproduced from deMadron and Weatherly (1994).

b. Model

In Fig. 6 we show transport vectors in the western South Atlantic from the model. The circulation exhibits both a strong DWBC and a northern cyclonic gyre that is closed by a strong eastward jet along the equator (which is shown separately in Fig. 7), in keeping with deMadron and Weatherly (1994). The interior velocities in our model are very weak: since there are no eddy fluxes in our study, this supports the explanation of Spall (1994) that these interior flows are eddy-driven.

Fig. 6.

Model transport vectors for circulation in the western South Atlantic (Brazil Basin). The friction coefficient r = 1.15 × 10−6 s−1 and the upwelling velocity w = 2.0 × 10−7 m s−1. Arrows corresponding to a very strong eastward jet along the equator (approximately four times larger than the maximum interior flow) are limited in magnitude to enable visualization of the interior flow. The magnitude of the transport vector indicated for scale purposes at the top of figure encompasses all vectors of size greater than or equal to this magnitude. Model bathymetry is shaded at intervals of 500 m from 4000 m and deeper; the darkest shading represents depths greater than 6000 m.

Fig. 6.

Model transport vectors for circulation in the western South Atlantic (Brazil Basin). The friction coefficient r = 1.15 × 10−6 s−1 and the upwelling velocity w = 2.0 × 10−7 m s−1. Arrows corresponding to a very strong eastward jet along the equator (approximately four times larger than the maximum interior flow) are limited in magnitude to enable visualization of the interior flow. The magnitude of the transport vector indicated for scale purposes at the top of figure encompasses all vectors of size greater than or equal to this magnitude. Model bathymetry is shaded at intervals of 500 m from 4000 m and deeper; the darkest shading represents depths greater than 6000 m.

Fig. 7.

Model transport vectors for circulation in western South Atlantic (Brazil Basin) in the vicinity of the equator. The friction coefficient r = 1.15 × 10−6 s−1 and the upwelling velocity w = 2.0 × 10−7 m s−1. Model bathymetry is shaded at intervals of 500 m from 4000 m and deeper; the darkest shading represents depths greater than 6000 m.

Fig. 7.

Model transport vectors for circulation in western South Atlantic (Brazil Basin) in the vicinity of the equator. The friction coefficient r = 1.15 × 10−6 s−1 and the upwelling velocity w = 2.0 × 10−7 m s−1. Model bathymetry is shaded at intervals of 500 m from 4000 m and deeper; the darkest shading represents depths greater than 6000 m.

The corresponding potential vorticity, q = f/h, for the circulation is shown in Fig. 8.2 The transport vectors follow potential vorticity (q) contours closely and in the DWBC also follow the bathymetric contours. Along the equator, however, the flow is across q contours, which vanish at singularities at the eastern and western extents of the basin. Strong cyclonic flows around closed q contours are reminiscent of the Kawase and Straub (1991) mechanism for closed contour spinup. Here, however, the mechanism is different since the recirculation cannot be disconnected from the DWBC and the eastward deflection of part of it across q contours as a frictionally driven jet. The flow is able to conserve q immediately away from the equator because in the potential vorticity equation,

 
formula

where ζ is the relative vorticity, and the frictional drain of q on the right-hand side of the equation has very small contributions from both the friction term and the upwelling.

Fig. 8.

Model potential vorticity, q = f/h, contours for the western South Atlantic (Brazil Basin).

Fig. 8.

Model potential vorticity, q = f/h, contours for the western South Atlantic (Brazil Basin).

In Fig. 9 we show a zonal section along 23°S from the model simulation. This is to be compared with the observed hydrography in Fig. 10 (recall that the 1.6°C isotherm represents the top of our AABW layer). The depth of the isopycnal in midbasin is both flatter and ∼200 m greater in the model than in the data. The uptilt of isopycnals at the western boundary representative of the DWBC is clearly evident in both figures, although it is the colder components of AABW that flow more strongly northward in the data.

Fig. 9.

Zonal section along 23°S from the model simulation shown in Fig. 6. Bathymetry is represented by light shading and the AABW layer by dark shading. The latitude and longitude of each end of the section is marked at the base of the figure. The depth of the section in meters is shown at the left of the figure.

Fig. 9.

Zonal section along 23°S from the model simulation shown in Fig. 6. Bathymetry is represented by light shading and the AABW layer by dark shading. The latitude and longitude of each end of the section is marked at the base of the figure. The depth of the section in meters is shown at the left of the figure.

Fig. 10.

Zonal section along 23°S reproduced from McCartney and Curry (1993).

Fig. 10.

Zonal section along 23°S reproduced from McCartney and Curry (1993).

4. Circulation in western North Atlantic

a. Observations

AABW exits the Brazil Basin into the western North Atlantic via a zonal equatorial channel in the northwest corner of the Brazil Basin. AABW encounters a sill at the entrance to the equatorial channel: the combined effect of this and the gradual shoaling of the Ceara Abyssal Plain as AABW exits the channel prevents the densest AABW (colder than 1.0°C) from flowing northward (Whitehead and Worthington 1982). McCartney and Curry (1993) estimate that 4.3 Sv of AABW flows across the equator from the Brazil Basin. A more recent and smaller estimate at the equator of 2.0 Sv has been made by Hall et al. (1997); the equator is a particularly difficult place to calculate the transport because conventional geostrophic estimates cannot be made.

North of the equator AABW no longer flows as a DWBC, and between ∼8°N and 16°N the northward flow of AABW is concentrated against the western slope of the Mid-Atlantic Ridge (Fuglister 1960). The western boundary flow at 10°N is, in fact, reported by Johns et al. (1993) and McCartney (1993) to be southward; Johns et al. (1993) compute a transport of about 3 Sv for waters colder than 1.8°C. Additionally Molinari et al. (1992) report a southeastward flow of waters colder than 1.8°C of 2–3 Sv concentrated against the boundary at 14°N. Thus, the circulation of AABW north of and in the vicinity of the equator has been interpreted as a recirculation by Friedrichs and Hall (1993), and in Fig. 11 we reproduce their circulation schematic, which is broadly consistent with these observations.

Fig. 11.

Circulation schematic for AABW flow in tropical North Atlantic reproduced from Friedrichs and Hall (1993). Estimated transports in Sverdrups are indicated in circles.

Fig. 11.

Circulation schematic for AABW flow in tropical North Atlantic reproduced from Friedrichs and Hall (1993). Estimated transports in Sverdrups are indicated in circles.

At 11°N almost half of the AABW transport flows into the eastern basin via the Vema Fracture Zone, and McCartney et al. (1991) estimate the transport of AABW colder than 2.0°C here as 2.1–2.3 Sv. This supports the larger estimate of McCartney and Curry (1993) for the cross-equatorial transport of AABW.

North of 24°N there is no confined current of AABW against the Mid-Atlantic Ridge (Fuglister 1960), indicating that the water from it spreads out onto the sea floor in some manner. Traces of AABW beneath the Gulf Stream on 50°W, which show up clearly in the silica and temperature fields (Clarke et al. 1980), and north of the Bahama Banks and Hispaniola (Tucholke et al. 1973) have been interpreted by Warren (1981) as fairly definite evidence for a general counterclockwise circulation of AABW in the western North Atlantic. AABW has also been found in the vicinity of the Blake–Bahama outer ridge near 30°N (Amos et al. 1971).

b. Model

In Fig. 12 we show transport vectors from the model for the western North Atlantic. Potential vorticity (q = f/h) contours are shown in Fig. 13. In this region, in contrast to the Brazil Basin, there is much structure in the transport vectors, also evidenced in the q field which is likely to reflect nonlinear control of the flow in the manner of Rhines (1989).

Fig. 12.

Model transport vectors for circulation in the western North Atlantic. The friction coefficient r = 1.15 × 10−6 s−1 and the upwelling velocity w = 2.0 × 10−7 m s−1. The magnitude of the transport vector indicated for scale purposes at the top of figure encompasses all vectors of size greater than or equal to this magnitude. Model bathymetry is shaded at intervals of 500 m from 4000 m and deeper; the darkest shading represents depths greater than 6000 m.

Fig. 12.

Model transport vectors for circulation in the western North Atlantic. The friction coefficient r = 1.15 × 10−6 s−1 and the upwelling velocity w = 2.0 × 10−7 m s−1. The magnitude of the transport vector indicated for scale purposes at the top of figure encompasses all vectors of size greater than or equal to this magnitude. Model bathymetry is shaded at intervals of 500 m from 4000 m and deeper; the darkest shading represents depths greater than 6000 m.

Fig. 13.

Model potential vorticity, q = f/h, contours for the western North Atlantic.

Fig. 13.

Model potential vorticity, q = f/h, contours for the western North Atlantic.

The northward transport at 1°N is 4.6 Sv, which conforms well with the estimate for a cross-equatorial transport of 4.3 Sv by McCartney and Curry (1993). By about 3°N almost the entire northward transport of AABW lies against the Mid-Atlantic Ridge, as shown in Fig. 14. Thus, friction is sufficient to break the along-isobath confinement of the outcrop lines of the current under geostrophic balance, and the current switches to bathymetry of opposite slope as it crosses the equator, as predicted by the transport integral [Eq. (3)]. Without friction in the model, the current is unable to cross the equator at all. In reality we expect inertia to be important in this region too (as in Nof and Olson 1993).

Fig. 14.

Zonal section along 3°N from model simulation shown in Fig. 12. Bathymetry is represented by light shading and the AABW layer by dark shading. The latitude and longitude of each end of the section is marked at the base of the figure. The depth of the section in meters is shown at the left of the figure.

Fig. 14.

Zonal section along 3°N from model simulation shown in Fig. 12. Bathymetry is represented by light shading and the AABW layer by dark shading. The latitude and longitude of each end of the section is marked at the base of the figure. The depth of the section in meters is shown at the left of the figure.

The DWBC against the Mid-Atlantic Ridge continues northward, where approximately 1.8 Sv is diverted eastward through the Vema Fracture Zone at 11°N. At 16°N the current bifurcates westward and northward. The northward branch continues as a weakened DWBC, where beyond ∼25°N it peels off westward to feed a counterclockwise interior flow. Both the westward arm of the bifurcation and the interior flow feed a strong recirculation associated with a deep basin along 20°N in the west and a southeastward DWBC from 17° to 8°N, consistent with the recirculation of Friedrichs and Hall (1993) and also Johns et al. (1993) and McCartney (1993).

It is striking, in fact, that the broad characteristics of the flow field are consistent with each of the observations above. We also note the lack of any coherent signal of poleward flow in the interior of the basin as predicted by the Stommel–Arons model.

5. Circulation in eastern North Atlantic

a. Observations

The ∼2.1 Sv transport through the Vema Fracture Zone flows eastward. McCartney et al. (1991) finds that the flow subsequently bifurcates into a northward (1.3–3.0 Sv), and a weaker eastward flow (1.8–3.9 Sv) along the southern boundary of the Gambia Abyssal Plain. Thus there is an increase in transport of the AABW by at least 50% and a necessary warming of the layer results; this is in large part due to the presence of a sill [first hypothesized by Heezen et al. (1964)]. McCartney et al. (1991) find no communication of AABW between the Cape Verde and Sierra Leone Basins (and therefore the northeastern and southeastern Atlantic Ocean) via the Kane Gap.

The circulation schematic from McCartney et al. (1991) for the AABW flow in the eastern North Atlantic is reproduced in Fig. 15. In this schematic the southern boundary flow loops counterclockwise and westward to join the northward branch. Friedrichs and Hall (1993) present a different picture for the AABW circulation in the vicinity of the Gambia Abyssal Plain, as shown in Fig. 11. They find weak southward flow in the vicinity of McCartney et al.’s (1991) northward branch, which forms a closed cyclonic recirculation around the perimeter of the Gambia Abyssal Plain. The northward flow postulated by McCartney et al. (1991) is motivated by a northward bulge in contours of bottom potential temperature, although they note the uncertainty in their transport estimate arising from the rough topography above the Mid-Atlantic Ridge. The confusion is enhanced since a meridional density section near 35°W in McCartney et al. (1991) clearly exhibits isopycnals in the AABW density range tilted upward against the Mid-Atlantic Ridge, which would appear to indicate southward flow unless there is a strong influence from overlying waters.

Fig. 15.

Circulation schematic for AABW flow in the eastern North Atlantic reproduced from McCartney et al. (1991).

Fig. 15.

Circulation schematic for AABW flow in the eastern North Atlantic reproduced from McCartney et al. (1991).

The AABW flow continues northward and is little documented until the Canary Basin, where at 35°N and beyond Saunders (1987) finds that the northward flow becomes concentrated in the east.

b. Model

In Fig. 16 we show transport vectors from the model in the eastern North Atlantic. We recall that the bathymetry of the Vema Fracture Zone has been widened since the model resolution is insufficient to resolve this feature. We have tuned the bathymetry to yield a transport consistent with the 2.1-Sv estimate of McCartney et al. (1991); in this simulation it is approximately 1.8 Sv.

Fig. 16.

Model transport vectors for circulation in eastern North Atlantic. The friction coefficient r = 1.15 × 10−6 s−1 and the upwelling velocity w = 2.0 × 10−7 m s−1. The magnitude of the transport vector indicated for scale purposes at the top of figure encompasses all vectors of size greater than or equal to this magnitude. Model bathymetry is shaded at intervals of 500 from 4000 m and deeper; the darkest shading represents depths greater than 6000 m.

Fig. 16.

Model transport vectors for circulation in eastern North Atlantic. The friction coefficient r = 1.15 × 10−6 s−1 and the upwelling velocity w = 2.0 × 10−7 m s−1. The magnitude of the transport vector indicated for scale purposes at the top of figure encompasses all vectors of size greater than or equal to this magnitude. Model bathymetry is shaded at intervals of 500 from 4000 m and deeper; the darkest shading represents depths greater than 6000 m.

The circulation field exhibits the strong eastward flow of AABW along the Gambia Abyssal Plain and the lack of flow through the Kane Gap, as noted by McCartney et al. (1991). There is no bifurcation northward against the eastern flank of the Mid-Atlantic Ridge however, and, in fact, the flow against the Mid-Atlantic Ridge is both southward and weakened by northward bifurcation, consistent with the cyclonic recirculation of Friedrichs and Hall (1993). We will discuss this further in a later section when we see that for different parameters in our model it is possible to observe the northward branch of McCartney et al. (1991).

There are two northward bifurcations around the perimeter of a shallower plateau at around 18°N, which merge to form a broad northward flow at 25°N. Some of this flow turns southward against the Mid-Atlantic Ridge and the majority continues into the Canary Basin where there are a number of recirculations and the flow structure is quite complicated. Beyond 35°N the northward flow becomes more concentrated in the east of the Canary Basin, as noted by Saunders (1987).

Potential vorticity, q = f/h, contours for the model circulation are shown in Fig. 17. South of approximately 28°N, q contours, transport vectors, and bathymetry are closely aligned. North of 28°N, however, q gradients are much weaker and there are recirculating elements to the flow, particularly north of 35°N where we suspect that there is some degree of free mode resonance operating (Rhines 1989).

Fig. 17.

Model potential vorticity, q = f/h, contours for the eastern North Atlantic.

Fig. 17.

Model potential vorticity, q = f/h, contours for the eastern North Atlantic.

6. Circulation in eastern South Atlantic

a. Observations

AABW enters the Guinea Basin via the Romanche Fracture Zone and the Chain Fracture Zone. The combined Romanche and Chain Fracture Zone transport has recently been estimated at 1.22 (±0.25) Sv by Mercier and Speer (1998), divided fairly equally between fracture zones, from two, two-year-long moored-current-meter arrays. These are the first long-term measurements of transport at these points.

The circulation of AABW in the eastern South Atlantic has generally been little documented. However, Warren and Speer (1991) have conducted a detailed study and produced circulation schematics as different depths based on Stommel–Arons dynamics. In Fig. 18 we reproduce their circulation field for depths greater than 4000 m, which shows the characteristic Stommel–Arons interior poleward flow and a DWBC, which closes the mass balance; here the DWBC is poleward in the north and equatorward in the south. Warren and Speer argue from comparison with sections along 11° and 24°S, that this circulation field is consistent with data.

Fig. 18.

Angola Basin circulation beneath 4000 m computed according to Stommel–Arons dynamics and reproduced from Warren and Speer (1991). The interior flow is poleward (southward); the western boundary current flow (hypothesized to maintain continuity) reverses at 15°S.

Fig. 18.

Angola Basin circulation beneath 4000 m computed according to Stommel–Arons dynamics and reproduced from Warren and Speer (1991). The interior flow is poleward (southward); the western boundary current flow (hypothesized to maintain continuity) reverses at 15°S.

b. Model

As a result of the along-equatorial proximity of the Romanche Fracture Zone (along 0°S) and the Chain Fracture Zone (along 1°S), combined with the purely frictional nature of the model dynamics at the equator, we encountered problems representing the Romanche and the Chain Fracture Zones in the model. For this reason we have positioned the throughflow from the Brazil Basin into the eastern South Atlantic entirely via a single fracture zone located at approximately 3°S. We believe that this introduces no more than a cosmetic difference to the resulting transport vectors in the eastern South Atlantic, although clearly the magnitude of the flow into the basin is effectively prescribed. In Fig. 19 we present transport vectors from the model in the eastern South Atlantic. Potential vorticity, q = f/h, contours are shown in Fig. 20.

Fig. 19.

Model transport vectors for circulation in the eastern South Atlantic. The friction coefficient r = 1.15 × 10−6 s−1 and the upwelling velocity w = 2.0 × 10−7 m s−1. The magnitude of the transport vector indicated for scale purposes at the top of figure encompasses all vectors of size greater than or equal to this magnitude. Model bathymetry is shaded at intervals of 500 m from 4000 m and deeper; the darkest shading represents depths greater than 6000 m.

Fig. 19.

Model transport vectors for circulation in the eastern South Atlantic. The friction coefficient r = 1.15 × 10−6 s−1 and the upwelling velocity w = 2.0 × 10−7 m s−1. The magnitude of the transport vector indicated for scale purposes at the top of figure encompasses all vectors of size greater than or equal to this magnitude. Model bathymetry is shaded at intervals of 500 m from 4000 m and deeper; the darkest shading represents depths greater than 6000 m.

Fig. 20.

Model potential vorticity, q = f/h, contours for the eastern South Atlantic.

Fig. 20.

Model potential vorticity, q = f/h, contours for the eastern South Atlantic.

Figure 19 shows an intense boundary current that follows the bathymetry around a shallow plateau situated at approximately 6°S and flows southward. This eastern boundary current broadens and weakens between 10° and 20°S and then intensifies as the bathymetry steepens between 20°S and the southern boundary of the model. There is a northward flow along the eastern side of the Mid-Atlantic Ridge as a cyclonic continuation of the eastern boundary current. The flows in the interior of the relatively flat Angola Basin are very weak.

The circulation field in Fig. 19 is very different from the circulation schematic shown in Fig. 18, which was computed by Warren and Speer (1991). Of particular interest is the flow at 24°S, along which we show a zonal section of potential density reproduced from Warren and Speer (1991) in Fig. 21. This very clearly shows the eastern confinement and uptilt of dense isopycnals (below around σ4 = 45.87), which appears consistent with the strong southward eastern boundary current in our model. Warren and Speer interpret this signal differently as newer water carried southward by the Stommel–Arons interior flow, which is then recirculated as older water to be carried in the northward branch of their hypothesized western boundary current against the Mid-Atlantic Ridge. There is clearly much room for interpretation in the limited observations available; however, the consistency of our model thus far in our basins leads us to believe that our interpretation may well be valid.

Fig. 21.

Zonal potential density section along 24°S reproduced from Warren and Speer (1991).

Fig. 21.

Zonal potential density section along 24°S reproduced from Warren and Speer (1991).

7. Model sensitivity

a. Friction coefficient

We have investigated the sensitivity of our results to both the formulation and the magnitude of the friction term in the model. It is common in formulations of a bottom friction term to neglect the effect of the Coriolis parameter f and the layer thickness h which should both be present if representation of the effect of a bottom Ekman layer is desired (Pedlosky 1987). In the solutions presented in sections 3–6 we have neglected the Coriolis parameter dependence since the friction term including an f dependence has zero magnitude at the equator, which precludes flow across the equator in our model:in reality, inertia has to be important at the equator. We have investigated the dependence of layer spindown on the layer thickness, and have found little detectable difference between model solutions incorporating a friction term as either /h or just as the coefficient r itself (not shown).

For friction coefficients smaller than we have used in the model solutions presented in sections 3–6, the model is unstable, as mass is unable to cross the equator efficiently. Recent work by Borisov and Nof (1998) and Nof and Borisov (1998), in which cross-equatorial flow determines its own frictional coefficient, supports larger values at the equator than in the interior, consistent with our observation of instability at the equator for low values of the frictional coefficient. For a friction coefficient twice as large as that used for the solutions thus far (i.e., with a spindown timescale of around 5 days at the Ekman layer depth as opposed to 10 days) we find some differences in the model solutions. In Fig. 22 we present the western North Atlantic solution for this increased friction coefficient for comparison with Fig. 12. Flows are generally broader as a result of the effect of the diffusive action of friction on the interface height [see Eq. (7)] and weaker as a result of the enhanced spindown, although they are qualitatively very similar in the two cases. In Fig. 23 we present the eastern North Atlantic solution for the increased friction coefficient for comparison with Fig. 16. In this case we note more than a qualitative difference since there is now a northward branch to the circulation at the western edge of the Gambia Abyssal Plain, as found by McCartney et al. (1991), as opposed to the cyclonic circulation we observed originally, consistent with Friedrichs and Hall (1993). It is surprising that we see this difference in both the interpretation of observations and the model solutions, although, in view of the uncertainity in both magnitude and the parameterization of our frictional term, to suggest the possibility of time variability in the path of the real AABW flow would certainly be premature.

Fig. 22.

Model transport vectors for circulation in the western North Atlantic. The friction coefficient r = 2.3 × 10−6 s−1 and the upwelling velocity w = 2.0 × 10−7 m s−1. The magnitude of the transport vector indicated for scale purposes at the top of figure encompasses all vectors of size greater than or equal to this magnitude. Model bathymetry is shaded at intervals of 500 m from 4000 m and deeper; the darkest shading represents depths greater than 6000 m.

Fig. 22.

Model transport vectors for circulation in the western North Atlantic. The friction coefficient r = 2.3 × 10−6 s−1 and the upwelling velocity w = 2.0 × 10−7 m s−1. The magnitude of the transport vector indicated for scale purposes at the top of figure encompasses all vectors of size greater than or equal to this magnitude. Model bathymetry is shaded at intervals of 500 m from 4000 m and deeper; the darkest shading represents depths greater than 6000 m.

Fig. 23.

Model transport vectors for circulation in the eastern North Atlantic. The friction coefficient r = 2.3 × 10−6 s−1 and the upwelling velocity w = 2.0 × 10−7 m s−1. The magnitude of the transport vector indicated for scale purposes at the top of figure encompasses all vectors of size greater than or equal to this magnitude. Model bathymetry is shaded at intervals of 500 m from 4000 m and deeper; the darkest shading represents depths greater than 6000 m.

Fig. 23.

Model transport vectors for circulation in the eastern North Atlantic. The friction coefficient r = 2.3 × 10−6 s−1 and the upwelling velocity w = 2.0 × 10−7 m s−1. The magnitude of the transport vector indicated for scale purposes at the top of figure encompasses all vectors of size greater than or equal to this magnitude. Model bathymetry is shaded at intervals of 500 m from 4000 m and deeper; the darkest shading represents depths greater than 6000 m.

b. Upwelling coefficient

Upwelling in the model acts to control the spatial extent of the AABW layer. We find that increasing the upwelling coefficient from 2 × 10−7 to 3 × 10−7 m s−1 has little observable effect on the circulation in the western basins; however, it does act to severely limit the extent of the AABW layer in the eastern basins. Similarily decreasing the upwelling coefficient to 1 × 10−7 m s−1 has little observable effect on the western basins but it enhances the transport of the circulation in the eastern basins (hardly affecting its pathways), particularly for the eastern North Atlantic where the excess flow leaves the model through the northern boundary at 50°N.

c. Fracture zones

In our model, the fracture zones do little more than to connect basins together: In reality, there is significant mixing within fracture zones that modifies water masses, unlike our simple model. We require the model fracture zones to be at least two grid boxes wide to obtain a transport effectively. Their transport is approximately tuned to observations by increasing their width (which generally has a large effect) or by varying their depth (which has a smaller, but still significant, effect). The transport in our model is sensitive to the fracture zone geometry, particularly in the case of the Romanche and Chain Fracture Zones near the equator. Here the combination of our poor representation of the fracture zones and our simple formulation for the friction term provides additional difficulties and minor changes to the bathymetry in the fracture zones (∼100 m) can easily change the transport by an order of magnitude. Since the fracture zones effectively provide the eastern boundary for the circulation in the western basins (Straub et al. 1993), it is important to emphasize that the circulation of AABW may well be sensitive to the detailed dynamics within these fracture zones. A proper sensitivity study will require a model with vastly improved horizontal and vertical resolution and with a full treatment of the nonlinear accelerations. A reviewer has also pointed out that the fracture zones not only control transport magnitudes, but they also segregate water masses. As AABW makes its way around the Atlantic, it entrains LNADW to make a range of water masses with intermediate properties. The fracture zones largely block the passage of the densest, pure AABW, but allow these lighter hybrid water masses to pass through. The consequences for the circulation of AABW within the eastern basins could well be significant. The sensitivity of the fracture zone geometry emphasizes the problems of representing throughflows in coarse-resolution ocean models, particularly where there are a number of water masses in the vertical negotiating a fracture zone.

8. Concluding discussion

We have developed a reduced-gravity model consisting of a single active layer beneath a motionless upper layer to represent the flow of AABW over realistic bathymetry in an Atlantic domain. The model dynamics are based on the steady, planetary–geostrophic, shallow-water equations, including a linear bottom friction and a uniform diapycnal upwelling through the top of the model layer. Our model solutions are notably consistent with observations of the pathways and transports of AABW, even in the eastern basins where we might have anticipated that the model representation of the AABW pathways would be somewhat poorer due to mixing within fracture zones.

The modeled flow occurs predominantly along potential vorticity, q, contours (except in the immediate vicinity of the equator), which are in turn generally oriented along bathymetric contours. This demonstrates the importance of bathymetry in shaping the circulation pathways of abyssal currents, which occurs because bathymetry modifies the characteristics along which long Rossby waves propagate information [see Eq. (7)].

The existence of a significant flow component along q contours presents a very different picture for the circulation to that of the Stommel–Arons model. The reason for the difference is that in the Stommel–Arons model the small flow across q contours that results from upwelling constitutes the entire interior flow, whereas here it is masked by the larger forced mode along q. Flows are also driven across q by the bottom friction term, but again this effect is small except at the equator. In addition, rather than having to hypothesize western boundary currents to close the mass balance as in the Stommel–Arons picture, topographic slope currents are obtained in our model as part of the solution as in the work of Straub et al. (1993). The mass balance at each latitude is maintained by the interplay between these currents and interior recirculations.

We suggest that our results provide a more convincing zero-order picture than the Stommel–Arons model for the circulation of AABW, and perhaps for abyssal water masses in general. Work is currently underway to extend this model to more layers and to include the overlying North Atlantic Deep Water and possibly Antarctic Intermediate Water.

Acknowledgments

The authors wish to thank Ric Williams for thought provoking discussions prior to submission and for providing JCS with a timely and productive visit. The authors wish to thank Doron Nof and an anonymous reviewer for helpful comments on the original manuscript. NERC Grants GR3/10157 and GR8/03760 are gratefully acknowledged.

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Footnotes

Corresponding author address: Dr. James C. Stephens, Geophysical Fluid Dynamics Laboratory, NOAA/Princeton University, Post Office Box 308, Forrestal Campus, Princeton, NJ 08542.

1

All temperatures referred to in this paper are potential temperatures.

2

Our potential vorticity, q = f/h, differs from the Ertel potential vorticity, Q = −σ−1fσ/∂z, in that our potential vorticity implicitly includes a contribution from bottom density gradients. This precludes comparison with maps of abyssal potential vorticity recently produced by O’Dwyer and Williams (1997).