## Abstract

The factors controlling the transport of the Antarctic Circumpolar Current (ACC) have recently been a topic of heated debate. At the latitudes of Drake Passage, potential vorticity contours are uninterrupted by coastlines, and large amplitude flows are possible even with weak forcing and dissipation. The relationship between the dynamics of circumpolar currents and inertial recirculations in closed basins is discussed. In previous studies, Sverdrup balance and baroclinic adjustment theories have both been proposed as theories of the ACC transport. These theories predict the circumpolar transport as various simple functions of the surface wind stress. A series of experiments is performed with a simple channel model, with different wind strengths and different idealized basin geometries, to investigate the relationship between wind strength and circumpolar transport. The results show that baroclinic adjustment theories do predict transport in the special case of a periodic channel with no topographic variations, or when the wind forcing is very weak. More generally, the transport is determined by a complex interplay between wind forcing, eddy fluxes, and topographic effects. There is no support for the idea that Sverdrup balance determines the transport through Drake Passage.

## 1. Introduction

There has been much recent debate concerning the dynamical balance of the Antarctic Circumpolar Current (ACC) and in particular the issue of what determines the ACC transport. There are two prevailing viewpoints. Stommel (1957), and most recently Warren et al. (1996), have suggested that the ACC transport is controlled by Sverdrup balance (Sverdrup 1947). The wind stress curl to the north of Drake Passage drives a poleward transport, which Stommel argued passes through Drake Passage and returns equatorward in a western boundary current—the Malvinas Current. The observed ACC transport (134 ± 13 Sv; Nowlin and Klinck 1986) (Sv ≡ 10^{6} m^{3} s^{−1}) is not inconsistent with that predicted by the Sverdrup theory (190 ± 60 Sv; Baker 1982). In contrast, Johnson and Bryden (1989), Straub (1993), and others have suggested that the ACC transport is controlled by an integral angular momentum budget, which can involve a complex interplay between surface wind forcing, buoyancy forcing, eddy fluxes, and friction. Note that Sverdrup balance may still apply locally, but in this interpretation it does not constrain the strength of the ACC [see Hughes (1997), Olbers (1998), and Rintoul et al. (2000) for useful discussions].

The essential difference between the Southern Ocean and other ocean basins is the absence of continental barriers at the latitudes of Drake Passage, and hence the presence of uninterrupted circumpolar potential vorticity contours. As we will argue in section 2 and demonstrate through a series of numerical experiments in sections 3 and 4, this means that a substantial circumpolar transport can exist even in the presence of weak forcing. Thus the ACC may be somewhat analogous to the inertial recirculation subgyres found in the Sargasso Sea and other subtropical basins, which are maintained through an integral balance between forcing and dissipation (Niiler 1966; Marshall 1986).

Several modeling studies have been performed to examine ACC dynamics. These have generally consisted either of simple eddy-permitting models (often quasigeostrophic) with idealized topography and forcing, or realistic-domain GCMs, with parameterized or partially resolved eddies. McWilliams et al. (1978) showed that the resolved eddies can provide interior fluxes of momentum and potential vorticity that allow a balanced mean flow. Wolff et al. (1991) examined the relative role of standing and transient eddies using the same model with different topographic ridges. More recently, Gnanadesikan and Hallberg (2000) and Gent et al. (2001) have shown that water mass transformation can also alter ACC transport and thus they conclude that ACC transport is neither determined by Sverdrup balance nor the mean wind stress. While eddy-permitting general circulation models such as the Parallel Ocean Climate Model, the Parallel Ocean Program model, the Fine Resolution Antarctic Model (FRAM) and the Ocean Circulation and Climate Advanced Model (e.g., Gille 1997; Hughes et al. 1999; Saunders et al. 1999) provide the most realistic representation of the Southern Ocean, such models cannot be run to thermohaline equilibrium and are too expensive to allow a detailed exploration of parameter space.

In this paper we return to a simpler, eddy-permitting model. We consider a purely wind-driven circumpolar current with no buoyancy forcing, although our dynamical framework can be easily extended to include buoyancy forcing if desired. We perform a large number of experiments with different wind forcing and idealized domain geometries to determine the conditions under which the simple theories of ACC transport put forward by previous authors hold, how the different ideas fit together, and where the theories break down. The paper is organized as follows: in section 2 we review the theoretical background; in section 3 we describe the numerical model and the experimental setup; in section 4 we present and analyze the numerical model results; and finally, section 5 contains a brief concluding discussion.

## 2. Theoretical background

The dynamics of the Southern Ocean is often regarded as distinct from that of wind-driven gyres in closed ocean basins. However the governing equations are the same. The aim of this section is to briefly review the vorticity dynamics of wind-driven gyres and circumpolar currents within a unified framework and to highlight the dynamical similarities and differences. A more extensive review is given by Rintoul et al. (2000).

### a. Barotropic theory

In a closed barotropic basin the background absolute vorticity (*q*) contours follow latitude circles and are blocked by the lateral boundaries. Free, unforced flow is impossible, but the source of vorticity from the surface winds drives flow across latitude circles and the gyre is closed by a frictional western boundary current. For weak winds (Fig. 1a) the gyre transport is determined by the integrated wind stress curl along latitude circles or “Sverdrup balance” (Sverdrup 1947). However, with realistic levels of wind forcing (Fig. 1b), fluid parcels advect low values of absolute vorticity northward within the western boundary current. If the *q* contours close, then there is the possibility of free recirculating flow, enhancing the overall gyre transport. At leading order vorticity is conserved by fluid parcels, but the strength of the recirculation along closed *q* contours is determined by an integral balance between sources and sinks of vorticity (Niiler 1966; Marshall 1986).

In a barotropic circumpolar channel the background vorticity contours are unblocked by coastlines. Thus in contrast to a closed basin we cannot assume the flow will be weak, even in the limit of weak wind forcing. Instead a more appropriate paradigm is recirculation along the circumpolar *q* contours (Fig. 2). The strength of this recirculation will be determined by an integral balance between vorticity sources and sinks. In a circumpolar channel forced by a surface wind stress, *τ*_{s}, and dissipated by linear friction coefficient, *r,* the circumpolar transport is given by

Here the channel extends from −*L*/2 to *L*/2, *ρ*_{0} is a mean density, and *y* is the meridional coordinate. Taking a weak wind stress of *τ*_{s} ∼ 0.01 N m^{−2}, *ρ*_{0} ∼ 10^{3} kg m^{−3}, *r* ∼ 10^{−7} s^{−1} and a channel width ∼10^{6} m, we obtain a transport of ∼100 Sv, an order of magnitude larger than the Sverdrup transport obtained in a closed basin for the same wind stress. Thus, at leading order the flow is dominated by circumpolar recirculation. As the wind forcing is increased, eddies may play a role in limiting the circumpolar transport.

### b. Baroclinic theory

The barotropic theory is easily extended to a baroclinic ocean; for simplicity we consider an ocean with two isopycnal layers. The dynamics is controlled by the potential vorticity,

where *f* is the Coriolis parameter, *h*_{n} is the isopycnal layer thickness, and *ζ*_{n} the relative vorticity in the *n*th layer. We suppose that the circumpolar channel is blocked by a partial barrier, mimicking Drake Passage in the Southern Ocean. In the lower layer, few potential vorticity contours are able to pass through Drake Passage due to the variations in bottom topography. Consequently, the circumpolar transport in the lower layer is likely to be small. Conversely, in the upper layer, the majority of the potential vorticity contours pass through Drake Passage, allowing a significant circumpolar transport. Hence, we separate the circumpolar transport into two components.^{1}

where *u*_{n} is the zonal velocity, *H* is the total ocean depth, and the integral is evaluated across the open latitude circles of Drake Passage. The barotropic transport is small if *u*_{2} ≈ 0 in the passage. To determine the baroclinic transport we need to determine the velocity shear; by thermal wind balance this is related to the isopycnal slope,

where *g*′ is the reduced gravity.

If we consider an ocean at rest and then switch on an eastward wind stress, the resulting equatorward Ekman transport will gradually increase the tilt of the isopycnals until eddy or diapycnal transports can balance the Ekman transport. There are several closely related approaches to the problem. Here we choose to focus on the mass budget of the upper layer, although similar results are obtained by focusing on the angular momentum or potential vorticity budgets. In the following we neglect diapycnal transfers between the isopycnal layers, which have been studied in some detail by Gnanadesikan and Hallberg (2000), and introduce an additional term into the balances below.

To leading order, the flow is along lines of constant Bernoulli potential, *B* = **u**^{2}/2 + *p*/*ρ*_{0}. In the absence of buoyancy forcing, the total mass flux across a Bernoulli streamline must vanish at statistical equilibrium:

where the overbar represents a time-average over several eddy life cycles, and **n** and **l** are unit vectors perpendicular and parallel to the Bernoulli streamlines. In the upper layer, (6) can be rewritten as

(see appendix). Thus, as sketched in Fig. 3, equatorward Ekman transport can be balanced either by a compensating frictional transport, a Reynolds transport, or an eddy bolus transport. Here *Q̂* = (*f* + *ζ*)/*h*, and **F** represents frictional terms. Note that there is no direct contribution from standing eddies, as the integration is carried out around a time-mean streamline (Marshall et al. 1993). However standing eddies do contribute indirectly in that they alter the background flow; this modifies both the frictional term in (7) and the location and magnitude of the transient eddy fluxes (MacCready and Rhines 2001). Finally, in order to determine the transport, we now need to know how the eddy fluxes are related to the mean interface slope.

#### 1) Weak wind stress (Straub 1993)

Straub (1993) suggests that once the meridional potential vorticity gradient reverses in the lower layer, baroclinic instability will prevent the vertical shear from developing further. Using thermal wind balance (5) and assuming the current is zonal, Straub deduces that the baroclinic transport is given by

independent of the wind stress. Here *L*^{2}_{D} = *g*′*h*_{1}*h*_{2}/*f*^{2}*H* is the baroclinic deformation radius and *L*_{y} is the meridional extent of the channel.

#### 2) Stronger wind stress (Johnson and Bryden 1989)

For stronger wind stress forcing, the onset of baroclinic instability is unlikely to be sufficient to compensate for the Ekman transport. Instead, the layer interface will continue steepening until the eddy transports are of the required magnitude to balance the Ekman transport. Neglecting Reynolds stresses and friction, (7) reduces to a balance between the wind-driven Ekman transport and eddy bolus transport in the upper layer, equivalent to the frequently discussed momentum balance between surface wind stress forcing and the interfacial eddy form stress (e.g., Treguier and McWilliams 1990; Ivchenko et al. 1996). Invoking the eddy closure of Green (1970) and Stone (1972) led Johnson and Bryden (1989) to deduce that

Invoking the alternative eddy closure recommended by Visbeck et al. (1997) leads to the alternative prediction that the baroclinic transport should scale with the cube root of the wind stress (Rintoul et al. 2000).

## 3. Numerical model

To test the competing theories reviewed in section 3, we now present a series of experiments with varying wind strengths using a balanced “geostrophic vorticity” model. The geostrophic vorticity equations are described by Schär and Davies (1988) and Allen et al. (1990); a detailed model description is given in Tansley and Marshall (2000).

Our experiments use two constant density layers within a zonally reentrant channel. The upper-layer thickness is initially 1200 m, and the total ocean depth is 4 km (except above topographic features).

The momentum equations are written

Here **u**_{n} is the full (geostrophic plus ageostrophic) velocity, **u**_{gn} = **k** × ∇*p*_{n}/*ρ*_{0}*f* is the geostrophic velocity, *ζ*_{gn} = ∂_{x}*υ*_{gn} − ∂_{y}*u*_{gn} is the geostrophic relative vorticity, and *B*_{gn} = (*u*^{2}_{gn} + *υ*^{2}_{gn})/2 + *p*_{n}/*ρ*_{0} is the geostrophic Bernoulli potential. The model is driven by a surface wind stress applied to the upper layer:

Biharmonic momentum dissipation is applied to both layers and a linear bottom drag is applied to layer 2 only. Details of model parameters are given in Table 1.

The momentum equations are used to substitute for the full velocity, **u**, in the continuity equations:

where *h*_{n} is the thickness of each model layer. The pressures in each layer are related through hydrostatic balance:

where *g*′ = *g*(*ρ*_{2} − *ρ*_{1})/*ρ*_{0} is reduced gravity. A rigid lid is applied so that

where *H* is the ocean depth. Finite variations in bottom topography are allowed, provided that the Rossby number remains small in the resultant currents. Combining (10)–(16) gives a three-dimensional elliptic equation for pressure tendency, which is solved using a multigrid inverter (see Tansley and Marshall 2000).

At the northern and southern boundaries of the channel, we apply a no-normal-flow lateral boundary condition:

We also apply a no-slip condition to the geostrophic velocity,

together with a higher-order boundary condition for the biharmonic dissipation:

Conservation properties of the model equations are discussed in Tansley and Marshall (2000).

## 4. Results

We perform our experiments in idealized domains, both for pedagogical purposes and to simplify the diagnostics. We start with the simplest case of a flat-bottom channel (Fig. 4a). We then introduce, in turn, ridge a 2 km high (Fig. 4b), partial land barriers to represent Drake Passage (Fig. 4c), and additional topographic features downstream of the model Drake Passage (Fig. 4d).

### a. Flat-bottom channel

#### 1) Standard parameters

As found in previous studies, in the absence of variable bottom topography, the total eastward transport through the channel is very large. There is a large barotropic component that can be predicted from the balance between bottom friction and wind stress, given by a two-layer extension of (1):

Figure 5a shows a time series of the transport in each layer over a 30-yr integration, for a wind stress with *τ*_{0} = 0.1 N m^{−2}. The barotropic component is established rapidly. As found by Wolff et al. (1991), this is followed by a linear increase in the baroclinic transport, corresponding to a gradual increase in the tilt of the thermocline. Eventually the system becomes baroclinically unstable, and the transport fluctuates about an equilibrium value. Figure 5b shows the total kinetic energy in each layer. This shows a rapid increase in kinetic energy when the current becomes unstable, followed by fluctuations about the mean value.

Figure 6 shows the mean barotropic and baroclinic transport values after spinup for a range of wind stress values. Even for very low values of wind stress, there is a considerable baroclinic transport as predicted by Straub (1993). The current only becomes unstable when there is a finite vertical shear, and only then can eddies act to keep the system near an equilibrium state. However, due to the weak Ekman transport, the increase in transport is slow—for *τ*_{0} = 0.01 N m^{−2}, it takes 110 years for the transport to reach quasi equilibrium. Straub's formula for the baroclinic transport (8) predicts 146 Sv for our model parameters. This compares favorably with the baroclinic transport of 151 Sv we find for this case.

As the wind stress increases, so does the transport. Figure 7 is a log–log plot of wind stress against baroclinic transport. For weak wind stress, the dependence on wind stress is small, as predicted by Straub. However, for wind stress values greater than *τ*_{0} = 0.075 N m^{−2} the baroclinic transport is roughly proportional to the square root of the wind stress, as predicted by Johnson and Bryden (1989).

As expected from the results of Panetta (1993) and Sinha and Richards (1999), the flow is organized into zonal jets. Figure 8 shows the time-mean eastward velocity in the upper layer for two different values of wind stress. For a weak wind stress of *τ* = 0.01 N m^{−2} there are several jets across the channel. As the wind stress increases, the number of jets is reduced. For a wind stress given by *τ* = 0.15 N m^{−2} we only find one clear jet. As discussed by Sinha and Richards (1999), the jet spacing is consistent with associated changes in the Rhines scale (Rhines 1975).

As discussed in section 3, we now diagnose the various terms in the mass budget of the upper layer. These are responsible for maintaining the mean isopycnal slope and, thus, the mean baroclinic transport. For the geostrophic vorticity equations, (7) becomes

where *Q̂*_{g} = (*f* + *ζ*_{g})/*h*.

The mass budget across streamlines (21) for this simple case is shown in Fig. 9, with a weak wind stress of *τ*_{0} = 0.01 N m^{−2} and a stronger wind stress of *τ*_{0} = 0.15 N m^{−2}. The horizontal axis shows the mean or “equivalent” latitude of the streamline along which we are integrating. Wind forcing drives a northward Ekman transport, while the eddy bolus transport is southward in both cases. For the case with weak wind forcing, the Reynolds transport is northward in the jets and is of a similar magnitude to the Ekman and bolus transports; the frictional transport is significant and generally acts in the opposite sense. For the stronger wind forcing, the Ekman transport and bolus transport are dominant, and almost cancel, apart from in the center of the jet where there is a significant northward Reynolds transport. Thus, this case with stronger forcing is more in accord with the ideas of Johnson and Bryden, with a dominant balance between the Ekman transport and eddy bolus transport.

#### 2) Influence of model dissipation

If we increase the bottom friction by a factor of 5, we find a decrease in the depth-integrated circumpolar transport by almost a factor of 5. This is consistent with (20) since the depth-integrated transport is dominated by *T*_{barotropic}. Note that, if the transport of the model circumpolar current were simply determined by Sverdrup balance, then in our experiments we would find only a weak dependence between circumpolar transport and dissipation. However, the flat-bottom case is a special limit due to the absence of zonal pressure gradients at all depths.

The baroclinic transport is also sensitive to the dissipation. In general, we might expect an increase in model dissipation to damp eddy activity and thus reduce the efficiency of the eddies in limiting the north–south interface slope. Similar results are obtained in eddy-resolving simulations of inertial recirculation gyres (e.g., see Barnier et al. 1991). For a wind stress of *τ*_{0} = 0.15 N m^{−2}, an increase in the biharmonic diffusion coefficient by a factor of 4 indeed results in an increase in the baroclinic transport by about 16%. However, if we instead increase the bottom friction by a factor of 5, we find a decrease in the baroclinic transport by about a third. The latter can be explained by the reduced barotropic transport in the presence of increased bottom friction. Lateral shear in the barotropic flow is known to inhibit baroclinic instability (James 1987). Increased bottom friction therefore increases the instability of the current, thus decreasing the circumpolar transport.

### b. 2-km ridge

#### 1) Standard parameters

As found by Wolff et al. (1991) and Krupitsky and Cane (1997), inclusion of a 2-km ridge in the channel (as shown in Fig. 4b) blocks the potential vorticity contours in the lower layer. This effectively eliminates the barotropic component of the circumpolar transport, and we are left with a circumpolar current of a far more realistic strength.

Figure 10 shows the transport for different wind stresses with the 2-km ridge. Even for *τ*_{0} = 0.2 N m^{−2}, the barotropic component of the circumpolar transport is less than 5 Sv, but there are nevertheless barotropic recirculation gyres of considerable magnitude. For the weakest wind stress, *τ*_{0} = 0.01 N m^{−2}, the baroclinic circumpolar transport of 138 Sv is again close to the prediction of 146 Sv given by Straub's formula (8). The slight reduction in baroclinic transport from the flat-bottom case is probably due to the reduced lateral shear in the barotropic flow, which, as mentioned above, is likely to increase the instability of the current (James 1987). For stronger wind stresses, the dependence of transport on wind stress is weaker than that predicted by the standard baroclinic adjustment theories. Figure 11 shows a log–log plot of wind stress against baroclinic transport. Discounting the weakest wind stress experiment, the remaining data fall onto a reasonably straight line of approximate gradient 1/13, compared with the values of 1/2 predicted by Johnson and Bryden (1989) and 1/3 predicted by Rintoul et al. (2000).

The time-averaged Bernoulli potential for *τ*_{0} = 0.01 N m^{−2} (Fig. 12) shows that the flow in the upper layer is still mainly zonal, although there is a slight deflection over the ridge, and downstream of the ridge the current is focused into three tighter jets. As the wind stress increases, the current is deflected further by the ridge until all the flow is deflected toward the northern boundary, and there is one large standing wave downstream of the ridge (Fig. 13). In both cases, the potential vorticity contours closely follow the Bernoulli streamlines (not shown), supporting the interpretation of the ACC as a “free mode.”

To understand the reduced sensitivity of the circumpolar transport to the surface wind forcing in the presence of a topographic barrier, we now consider the integral mass budget across mean Bernoulli streamlines, as detailed in (21). With weak wind stress forcing, the mass budget is similar to that obtained in the flat-bottom case (Fig. 14a). The Reynolds transport is large, and is generally of opposite sign to the eddy bolus transport, while the frictional is transport is also significant. However, with stronger wind forcing, the balance is rather different from the flat-bottom case (Fig. 14b). Here the Reynolds transport is consistently southward, and the eddy bolus transport is therefore slightly reduced compared to the flat-bottom case. While we have eliminated the direct effect of the standing eddies by integrating along a streamline, standing eddies established downstream of the ridge still contribute implicitly to the overall mass budget. Downstream of the ridge the streamlines are focused into a sharp meandering jet, within which the bolus and Reynolds transports are locally enhanced (not shown) as described by MacCready and Rhines (2001). The stronger the wind forcing, the greater the amplitude of the standing eddies, leading to increased local enhancement of the bolus and Reynolds transports. Equilibrium can thus be achieved with a smaller increase in circumpolar transport than in the flat-bottom case, resulting in reduced sensitivity of the circumpolar transport to surface wind forcing. The standing eddies also locally enhance the frictional transport, although the frictional transport remains relatively small in the overall mass budget.

#### 2) Influence of model dissipation

As in section 4a, we take a wind stress of *τ*_{0} = 0.15 N m^{−2} and examine the influence of changing the dissipation coefficients on the baroclinic transport.

As for the flat-bottom case, when we increase the biharmonic diffusion coefficient by a factor of 4, we find an increase in the baroclinic transport, but this time only by 7%. However, in contrast to the flat-bottom case, if we increase the bottom friction by a factor of 5, we find an increase (by about 15%) in baroclinic transport. This last result is the opposite to that found in the simple buoyancy-forced model of Gnanadesikan and Hallberg (2000). Gent et al. (2001) also find a reduction in baroclinic transport with an increase in dissipation coefficient since increasing the dissipation coefficient in their eddy bolus transport parameterization leads to an increased slumping of the isopycnals. In our experiments with a 2-km ridge, increasing the dissipation also damps the standing waves so that the current becomes more zonal. The mass budget across mean Bernoulli streamlines shows a change in the Reynolds transport so that for high bottom friction it is northward over most of the channel. Thus, the dependence on model dissipation is complex. This sensitivity to model parameters highlights the delicate balance between the surface forcing, topography, eddies, and frictional processes, each of which affects the model ACC transport.

### c. 2-km ridge with land barriers

Including idealized land barriers to represent Drake Passage leads to a decrease in the baroclinic transport (Fig. 15). However, even with a very small wind stress, a reasonable circumpolar current is generated (e.g., Fig. 16 shows the 42 Sv current for a wind stress of *τ*_{0} = 0.01 N m^{−2}). Note that if we scale the predicted transport for a baroclinically adjusted current from (8) by a factor of 0.22, to represent the reduced extent of the meridionally connected part of the domain, we get a transport of 32 Sv. Note also that Sverdrup balance, as advocated by Warren et al. (1996), predicts a transport of only 6 Sv. The circumpolar current is confined to the open latitude circles. As the wind stress increases, the interior Sverdrup flow increases. Figure 17 shows the time-mean streamlines for a wind stress given by *τ*_{0} = 0.15 N m^{−2}. Part of the flow through “Drake Passage” is deflected north along the model coastline, as seen in data and the FRAM model (Best et al. 1999). A standing wave develops near the line of zero wind stress curl, which can be identified as a “Moore wave” (Moore 1963). For large wind stress values, the transport through Drake Passage only increases slightly with increasing wind stress, as for the case above without the land barriers. Figure 18 shows the mass budget for experiments with weak and stronger wind forcing for this model configuration. The mass budgets are shown only across circumpolar streamlines. With weak wind forcing the balance is very similar to that found across the inertial jets in the flat-bottom case: the Reynolds transport is almost as large as the Ekman transport and is northward (Fig. 18a). With stronger wind forcing, the bolus transport is greatly reduced compared to the flat-bottom and ridge cases (Fig. 18b). Where the streamlines follow the land boundary as the current turns north to the east of Drake Passage, the Reynolds transport and frictional transport make significant contributions and further decrease the sensitivity of the circumpolar transport to surface wind forcing.

### d. Downstream topography

The path of the model circumpolar current is strongly modified by topography downstream from Drake Passage. To investigate the importance of downstream topography in influencing the circumpolar transport, we now introduce an idealized topographic seamount (representing the Kerguelen Plateau) and a downstream ridge (representing the Pacific–Antarctic Ridge) into the model (as shown in Fig. 4d). Figure 19 shows the time-average streamlines with a wind stress given by *τ*_{0} = 0.15 N m^{−2}. The downstream topography focuses the flow into tighter jets. The bump has a clear upstream effect, with jets appearing north and south of the bump, as described by Webb (1993). The flow is also deflected over the second 1-km ridge. However, despite the changes in the background state, the baroclinic transport in this experiment remains very similar to the experiment without downstream topography (shown in Fig. 17).

We have performed a large number of experiments with different downstream topographies, by extending the topographic seamount southward across Drake Passage and varying the height of the downstream ridge. The downstream topography has a clear impact on the time-mean circulation in all cases. In many of these experiments we find a small change in the baroclinic transport, typically of order 5%. In some cases we find a decrease in transport and in others we find an increase, dependent on the influence of the topographic obstacles on the structure, and hence stability, of the the time-mean circulation. However, the precise nature of the change is difficult to predict.

## 5. Conclusions

The relation between the transport of the ACC and surface wind forcing has been frequently discussed in the oceanographic literature. In this paper we have sought to examine whether there is a straightforward relation in the context of an idealized, two-layer representation of the Southern Ocean. Our main results are as follows.

For weak wind stress, as predicted by Straub (1993), the baroclinic (i.e., thermal wind) component of the eastward transport can be approximately predicted from the critical vertical shear required for baroclinic instability.

For stronger wind stresses and in the special case of a flat-bottomed channel, the eastward baroclinic transport is roughly proportional to the square root of the wind stress, as predicted by Johnson and Bryden (1989).

The circumpolar transport is not proportional to the wind stress curl, as predicted by Sverdrup balance theories. This has been demonstrated by Samelson (1999), Gnanadesikan and Hallberg (2000), and Gent et al. (2001) in models with thermodynamic forcing. Here we have shown this is the case even for a purely wind-driven current.

In general, in the presence of topographic and land barriers, the circumpolar transport is dependent on a complicated interaction between wind forcing, eddy fluxes, topographic effects, and friction. There appears to be no simple relation between the circumpolar transport and the wind forcing, although the sensitivity is greatly reduced compared with that predicted by Johnson and Bryden (1989).

The theory and results discussed in this paper suggest that it may be helpful to think of the ACC as a “free mode,” which is then modified by Sverdrup balance (as proposed by Hughes 1997). In a free mode, the transport is determined by a delicate integral balance between weak forcing and dissipation. Consequently, the ACC transport is likely to be extremely sensitive to a wide range of parameters in numerical models. A similar sensitivity is seen in the inertial recirculation of the Gulf Stream (e.g., Pedlosky 1996 and references therein). It is interesting to note that there is relatively little attention focused on predicting the transport of the Gulf Stream recirculation, while we are still seeking a definitive answer to the question of what controls the ACC transport. If free-mode ideas are pertinent to the ACC, then this issue may not be well posed until the geostrophic eddy field is far better resolved by general circulation models.

The ACC transport will also be influenced by diapycnal processes, as argued by Gnanadesikan and Hallberg (2000). This is because changes in the meridional density gradients associated with surface thermodynamic forcing directly affect the zonal velocity of the ACC through thermal wind balance. The theoretical framework presented here can be easily extended to include thermodynamic forcing. Indeed, diapycnal fluxes simply appear as an additional term on the right-hand side of (7) and, in principle, can simply be added to the surface wind forcing. However, due to the nonlinear relation between surface wind forcing and circumpolar transport, we suggest that the addition of buoyancy forcing is unlikely to significantly modify the overall circumpolar transport.

In summary, it seems unlikely that there is a simple relationship between ACC transport and wind stress. However our numerical experiments suggest that the circumpolar transport is significantly less sensitive to surface wind forcing than predicted by the baroclinic adjustment theory of Johnson and Bryden (1989). The reduced sensitivity arises due to changes in the structure of the mean circulation with stronger wind forcing, which acts to increase the effective dissipation of the current.

## Acknowledgments

We thank Chris Hughes, David Webb, Roger Samelson, and two anonymous reviewers for helpful comments on preliminary drafts. We are also grateful to Alistair Adcroft for providing us with the three-dimensional multigrid solver. This study was supported by the U.K. Natural Environment Research Council, Grant GR3/10157.

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### APPENDIX

#### Derivation of the Upper Layer Mass Budget

The momentum equation can be written

where the symbols are defined in sections 2 and 3. Taking the time average of (A1) gives

and rearranging for **u**, we obtain

If we multiply (A3) by *h* and evaluate the net transport across a time-mean Bernoulli contour, the first term on the right-hand side vanishes, and so

The mass budget in the upper layer can therefore be written

## Footnotes

*Corresponding author address:* Dr. David P. Marshall, Dept. of Meteorology, University of Reading, Earley Gate, P.O. Box 243, Reading RG6 6BB, United Kingdom. Email: davidm@met.rdg.ac.uk

^{1}

There are several different definitions of “barotropic” and “baroclinic” components in use in oceanography. Here we define the barotropic component as the bottom velocity component multiplied by the ocean depth, and the baroclinic component as the total depth-integrated transport less the barotropic component.