1. Introduction
Changes in wind stress over the Southern Ocean may be responsible for modulating the strength of the global meridional overturning circulation (MOC) (Toggweiler 2009). Such wind-induced changes in the MOC could help regulate glacial–interglacial cycles by venting CO2 from the deep ocean to the atmosphere (Toggweiler and Russell 2008; Anderson et al. 2009; Marshall and Speer 2011). The mechanism could also play an important role in future climate change; the westerlies appear to be shifting south because of greenhouse gas emissions and ozone depletion (Thompson and Wallace 2000; Marshall 2003; Polvani et al. 2011), and Toggweiler and Russell (2008) hypothesize that in response the MOC will strengthen, but by how much? To what extent is the Southern Ocean MOC controlled by the winds?
Since Johnson and Bryden (1989), we have recognized the existence of an eddy-driven overturning circulation in the Southern Ocean potentially large enough to completely cancel the wind-driven Ekman overturning. The actual MOC is the small residual between these two opposing circulations. Work by Toggweiler and Samuels (1998), Speer et al. (2000), and Marshall and Radko (2003, hereafter MR03) showed that, for realistically weak values of interior diapycnal mixing, the residual overturning transport in the subsurface Southern Ocean must proceed along mean isopycnal surfaces. The residual circulation can cross isopycnals in the surface diabatic layer, where cross-isopycnal advection can be balanced by direct diabatic forcing from the atmosphere (Marshall 1997). Therefore, from a diagnostic point of view, the strength and sense of the MOC can be inferred from surface buoyancy-flux data, as done by Speer et al. (2000) and Karsten and Marshall (2002b), independently of the wind stress. This thermodynamic perspective also implies that the MOC is sensitive to surface buoyancy fluxes, as hypothesized by Watson and Naveira Garabato (2006) or Badin and Williams (2010). Our goal here is to study the relationship between wind stress, overturning circulation, and surface buoyancy flux in a model that explicitly resolves mesoscale eddies, bypassing the need for any a priori assumptions about the eddy response.
On a related note, it is well established that coarse-resolution ocean models do not accurately simulate the response of the Southern Ocean overturning to changes in wind stress forcing when compared with eddy-resolving models. This is true of both realistic models (Hallberg and Gnanadesikan 2006; Farneti et al. 2010) and models with simplified geometry and forcing (Henning and Vallis 2005). In general, models that permit eddies seem less sensitive to changes in wind, whether the focus is the overturning circulation (as in the above works), the zonal transport (Hutchinson et al. 2010), or the transport of tracers such as anthropogenic carbon (D. Munday 2011, unpublished manuscript). Most of these results are ultimately due to compensation between the wind- and eddy-driven overturning circulations, which is more complete when mesoscale eddies are explicitly resolved rather than parameterized. The lack of a robust parameterization for mesoscale eddies is indicative of our incomplete understanding of the nature of eddy-driven circulations. Most recently, Viebahn and Eden (2010) studied the sensitivity of the residual MOC to the wind in an idealized model and found that changes in eddy kinetic energy (EKE) and eddy diffusivity play a central role in determining how the compensation occurs.
Our goal in this study is to further explore the physical mechanisms that determine the sensitivity of the residual MOC to changes in wind forcing. In particular, there are two questions not previously addressed that we wish to pursue here. First is the influence of the boundary condition for buoyancy. Second, we wish to develop a simple theory based on physical principles capable of explaining the MOC sensitivity. To study these issues, we reduce the system to its essential elements: an Ekman-driven and an eddy-driven circulation in a zonally symmetric channel with buoyancy forcing at the surface. This system was studied analytically by MR03, who invoked a closure for the eddies, but here we realize it as a high-resolution numerical model. The strength of the Ekman circulation obviously depends linearly on the winds; the strength of the eddy-driven circulation is determined by the geostrophic turbulent dynamics of the model. We vary the strength of the wind stress and diagnose the steady-state residual overturning circulation.
We find that increased eddy circulation does generally compensate for increased Ekman circulation under stronger winds. However, the degree of compensation depends on the surface boundary conditions. When the surface heat fluxes are held fixed, the residual MOC strength is relatively insensitive to the winds. With an interactive heat flux, we recover the results of Viebahn and Eden (2010): a residual MOC that increases weakly with the winds and whose sensitivity is set primarily by changes in eddy diffusivity. We develop a scaling theory for the eddy diffusivity dependence on the wind and apply this scaling to reconstruct the eddy response. This method yields a closed theory for the sensitivity of the residual MOC, which, despite many approximations, shows encouraging agreement with the results from the numerical model.
Section 2 describes the model setup, a reference solution, and the basic experimental results under differing values of wind stress. In section 3, we analyze the results in terms of the buoyancy budget and discuss the constraints imposed by the surface boundary condition for buoyancy. Section 4 describes a framework for understanding the MOC changes in terms of changes in Ekman circulation, isopycnal slope, and eddy diffusivity. Our scaling for the eddy diffusivity and the resulting MOC sensitivity estimates are then presented. We summarize the results and discuss their connection with the real ocean in section 5.
2. Experiments with numerical model
a. Modeling philosophy
The Southern Ocean is dominated by the Antarctic Circumpolar Current (ACC), a strong eastward flow in thermal-wind balance with the strong density front separating polar from tropical waters (Rintoul et al. 2001). This flow circumnavigates the globe and connects back on itself, inspiring a comparison with the large-scale atmospheric jets (Thompson 2008). Strong atmospheric westerly winds blow over the surface, driving an equatorward Ekman flow. The surface buoyancy flux—a combination of radiative, latent, and sensible heat fluxes as well as freshwater fluxes from evaporation, precipitation, and ice-related processes—is notoriously uncertain because of poor data sampling (Cerovečki et al. 2011). Nevertheless, the general pattern (shown in Fig. 1) indicates buoyancy loss in the extreme south polar regions, buoyancy gain on the poleward flank of the ACC, and buoyancy loss in some regions on the equatorward flank associated with mode water formation. Although the current meanders and splits as it makes its way around topographic features, authors such as de Szoeke and Levine (1981) and Ivchenko et al. (1996) have argued that, when the real ACC is described using a “streamwise average” view, the large-scale dynamics bear a close resemblance to zonally symmetric models.
Maps of the observed surface forcing in the Southern Ocean, averaged from the Common Ocean Reference Experiment (CORE2) dataset over the period 1949–2006 (Large and Yeager 2009): (left) The wind stress in N m−2, with the magnitude indicated by the colored shading and the direction by the arrows and (right) the buoyancy-equivalent heat flux in W m−2, which includes contributions from longwave and shortwave radiative fluxes; latent and sensible heat fluxes; and the buoyancy fluxes due to evaporation, precipitation, and runoff.
Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-023.1
Indeed, zonal channel models with highly idealized geometry form the foundation of contemporary theories of the Southern Ocean circulation, capturing the essential physics of the system and providing insight into important mechanisms (Munk and Palmén 1951; McWilliams et al. 1978; Marshall 1981; Johnson and Bryden 1989; Marshall 1997; Olbers et al. 2004; Marshall and Radko 2006, among many). The Southern Ocean MOC, however, exports and imports water from other ocean basins (e.g., Ganachaud and Wunsch 2000; Talley 2008). Antarctic Bottom Water (AABW) flows out of the Southern Ocean in the deepest layers. North Atlantic Deep Water (NADW) and Circumpolar Deep Water (CDW) flow in (poleward) at intermediate depths, and Antarctic Intermediate Water (AAIW) and Subantarctic Mode Water (SAMW) flow equatorward in the upper thermocline. As a result, channel-only models that attempt to investigate the Southern Ocean MOC without representing other basins find vanishingly weak deep residual circulations (Karsten et al. 2002; Kuo et al. 2005; Cessi et al. 2006; Cerovečki et al. 2009). Some authors have tackled this problem by attaching closed basins to their channels. This approach can certainly yield insights, but it also adds to the complexity of the problem by introducing gyre dynamics. When such basins are global scale, as in Wolfe and Cessi (2009), the computational cost of an eddy-resolving model becomes immense. When they are small (on the same order of the channel itself), as in Henning and Vallis (2005) and Viebahn and Eden (2010), the link with the real ocean is less clear.
We choose to address this problem in a novel way: by including a narrow “sponge layer” along the channel’s northern boundary, in which the temperature is relaxed to a prescribed exponential stratification profile. This diabatic forcing provides a return pathway for deep residual overturning, which otherwise would not be able to cross isopycnals. Physically, the sponge layer encapsulates all the diabatic processes occurring outside of the Southern Ocean, such as deep-water formation by air–sea heat fluxes in the North Atlantic or diapycnal mixing in the abyss. The disadvantage of this method is that the stratification at the northern boundary cannot change significantly. The advantage is that it provides a clean, simple framework in which to investigate nonzero residual circulations, focusing on the dynamics of the channel alone rather than the complex teleconnections of the global problem (Wolfe and Cessi 2011). In combination with appropriate surface wind and buoyancy forcing, we will see that this configuration can produce realistic overturning cells.
Given the many idealizations made in constructing our model, we must interpret our results with care. We emphasize that our goal is not to make quantitative predications for the real global ocean–atmosphere system; rather, we hope to gain insight into the underlying physical mechanisms that govern this system in order to inform the interpretation of more realistic models and observations.
b. Model physics and numerics
The basic physical system simulated by our model is a Boussinesq fluid on a beta plane with a linear equation of state and no salinity. The model is forced mechanically by a surface stress and thermodynamically by a surface heat flux as well as by the aforementioned sponge-layer restoring. Mechanical damping is provided by linear bottom drag; there is no topography. Key physical and numerical parameters are given in Table 1.
Parameters used in the numerical model reference experiment.
The model code is the Massachusetts Institute of Technology general circulation model (MITgcm), a general-purpose primitive equation solver (Marshall et al. 1997a,b). The domain is a Cartesian grid 1000 km long (i.e., zonal direction), 2000 km wide (i.e., meridional direction), and 2985 m deep. Although this domain is relatively narrow, the zonal symmetry means that a larger domain would not alter the results and would only add computational cost. The domain size does not appear to constrain the eddy size, because a typical eddy size is ~200 km. We resolve the first baroclinic deformation radius (approximately 15 km in the center of the domain), employing 5-km horizontal resolution and with 30 vertical levels, with spacing increasing from 10 m at the surface to 280 m at the bottom. A realistically effective diapycnal diffusivity (κυ = 0.5 × 10−5 m s−2) is maintained thanks to the second-order-moment advection scheme of Prather (1986) (see also Hill et al. 2011). To maintain a surface mixed layer, we employed the K-profile parameterization (KPP) mixing scheme (Large et al. 1994). In our case, this scheme simply acts to mix tracers and momentum over a layer of roughly 50-m depth.
The model was spun up from rest for approximately 200 yr until it reached a statistically steady state, as indicated by the mean kinetic energy. A typical eddy temperature field from the equilibrated state is shown in Fig. 2. Averages were performed over 20-yr intervals. In cases where parameters were changed, the model was allowed to reach a new equilibrium before taking an average.
A 3D snapshot of the model’s temperature field, revealing the mesoscale eddy field. The temperatures range from 0° to 8°C. Overlaid on top are depictions of the wind stress and heat-flux surface forcing. To the right is the zonal- and time-mean zonal velocity 
Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-023.1
c. The zonal momentum balance
Likewise, as discussed in detail in section 4, the barotropic component of the flow does not participate in the eddy energy cycle, and thus we expect the eddy-driven circulation to be similar with or without topography. Experiments performed with a topographic ridge (but not described further here) support the conclusion that the presence of topography strongly damps the barotropic zonal flow but does not affect the MOC. We therefore expect that conclusions drawn from our model about the MOC can still apply to the real Southern Ocean, especially to the portion of the flow that occurs above major topographic features.
d. Residual overturning circulation
The MOC is characterized by three distinct cells, as shown in Fig. 3. In the interior of the domain, away from the surface and the sponge layer, the MOC is directed along mean isopycnals: that is, 
The residual MOC streamfunction Ψiso (left) as originally diagnosed in isopycnal coordinates and (right) mapped back to depth coordinates. The units are Sv, and the contour interval is 0.1 Sv. The solid black line in (left) indicates the mean SST, and the gray lines are the 5% [in (left) only] and 95% levels of the SST CDF. The dotted black is the southern boundary of the sponge layer. The mean T contours are also shown in (right) in black, and the contour interval is 0.5°C.
Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-023.1
The surface heat flux is specified as a fixed function of latitude; consequently, the heat flux is felt by all isopycnals that graze the surface at that particular latitude. The cumulative distribution function (CDF) of surface temperature tells how likely a particular temperature is to be found at the surface and thus be exposed to diabatic transformation. Superimposed on Fig. 3 are the 5% and 95% values of T from the surface temperature CDF. (The mean SST is very close to the median value.) Nearly all of the diabatic MOC transport (i.e., advection across mean isopycnals) takes places in between these values. When plotted in z coordinates, the 95% CDF value is an effective measurement of the depth of the surface diabatic layer; below it, the contours of Ψiso and 
(left) The Eulerian-mean streamfunction 
Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-023.1
e. Model response to wind changes
We now examine the MOC sensitivity to altered wind stress. We consider two principal cases. First, the surface buoyancy flux is held fixed as the winds are varied. In the second set of experiments, we employ an interactive, relaxation-type boundary condition. The fixed-flux boundary condition is a justifiable one for freshwater and incoming shortwave radiation, but an interactive boundary condition is more appropriate for sensible and latent heat (Haney 1971). The results are summarized in Fig. 5, where the strengths of the upper and lower cells in each experiment are plotted on a single graph. Because Ψiso is roughly constant along isopycnals below the diabatic layer, we diagnosed the transports by simply finding the maximum and minimum values of Ψiso below 500 m at y = 1800 km, 100 km south of the edge of the sponge layer. We will henceforth refer to these maximum and minimum values of Ψiso as MOCupper and MOClower. Besides the individual upper and lower cells, there is a third relevant quantity: the total volume flux of upwelled deep water, MOCupwell = MOCupper − MOClower. This value is also shown in Fig. 5, along with the strength of 
A summary of the MOC cell strength in all of the different experiments. The Ekman circulation 
Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-023.1
In general, the various MOC values appear to have linear dependence on the wind. This is not a universal rule for all possible models and ranges of parameters (e.g., Viebahn and Eden 2010), but it is an accurate and useful approximation for our particular experiments. This simplification allows us to characterize the MOC sensitivities in a single number by a simple least squares linear fit applied to Fig. 5. The slope ∂MOC/∂τ0 gives a sense of how strongly each cell depends on the wind. These values are given in the first column of Table 2, along with the value of R2 for the regression. The R2 values reveal that the linear fit is very good in most cases.
Linear MOC dependence on wind ∂MOC/∂τ0, as determined by least squares fit. The value of R2 for the linear regression is given in parenthesis, a measure of the goodness of fit. The values are computed at fixed points in space near where maxima and minima of Ψiso occur: z = −477 m, y = 1150 km (upper cell), and y = 300 km (lower cell). The first column shows Ψiso, and the second column shows 
1) Fixed flux boundary condition
The MOC transports are rather insensitive to the wind in the fixed flux experiments. Here, MOClower shows no correlation with τ0, varying in a narrow range about 0.4 Sv; MOCupper is quite weak for the weakest winds (τ0 = 0.05 and 0.1 N m−2), but for the rest of the experiments (0.125 N m−2 ≥ τ0 ≥ 0.3 N m−2), the changes in MOCupper are slight: it increases only from 0.5 to 0.6 Sv over this range. The linear fit for ∂MOCupper/∂τ0 (Table 2) shows a sensitivity ¼ of that of the Ekman circulation. Examination of the structure of Ψiso show that, for weak winds, the upper cell becomes confined more and more to the surface diabatic layer and does not reach the interior. Because MOClower does not change, MOCupwell follows the changes in MOCupper. The small changes in residual MOC reflect the fact that, as the magnitude of 
2) Relaxation boundary condition
The results of these experiments are also shown in Fig. 5. The changes are significantly larger than the fixed-flux case. Both MOCupper and MOClower increase with stronger winds; this means a strengthening of the upper cell (because it is positive: i.e., clockwise) and a weakening of the lower (negative, counterclockwise) cell. The linear fit (Table 2) shows that MOCupper is nearly twice as sensitive as the fixed-flux case. Because the changes in 
Viebahn and Eden (2010) performed a very similar experiment, simulating only an upper cell and using a relaxation boundary condition for buoyancy. Their results are broadly consistent with ours: a sensitivity of the residual circulation much weaker than the sensitivity of the Ekman circulation. However, they observed decreasing sensitivity with increasing winds, whereas the trend in our MOCupper appears quite linear. This qualitative difference is most likely attributable to the different northern boundary; they had a small, unforced basin attached to the northern edge of their channel, rather than a sponge layer.
3. The surface buoyancy boundary condition
Our experiments make it clear that a residual overturning driven by a fixed buoyancy flux is less sensitive to the winds than one with an interactive buoyancy flux. In this section, we seek to understand this behavior diagnostically through the residual buoyancy budget, using the framework of MR03.
a. Transformed Eulerian-mean buoyancy budget
We begin by reviewing some essential elements of TEM theory (Andrews and McIntyre 1976; Andrews et al. 1987; Treguier et al. 1997; Plumb and Ferrari 2005). The reader is referred to MR03 for a complete discussion of the theory in the context of ACC dynamics.
The terms in the approximate form of the Marshall–Radko balance (17). The plot is shown as a function of T on the bottom of the x axis, but it can also be considered a function of y, whose corresponding values are shown at the top of the x axis.
Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-023.1
b. Buoyancy-flux sensitivity to winds
The residual buoyancy budget as expressed by (15) or (17) already reveals the strong constraint imposed on the MOC by a fixed surface buoyancy flux: because the term B cannot change, changes in the MOC must be accompanied by changes in 
As described above for the reference case, we can diagnose the forcing terms B and D from each of our experiments to understand how these terms change with the wind; this is shown in Fig. 7, which contains contour plots of B and D as functions of y and τ0. Also plotted are contours of the zonal-mean SST, from which it is easy to see the changes in 
The forcing terms of the surface residual buoyancy budget (22) for changing values of wind τ0, expressed in units of W m−2 equivalent by multiplying by ρ0cp(gα)−1. Shown are (a) the air–sea buoyancy flux B for the fixed-flux case and (c) the diabatic eddy flux D. (b),(d) As in (a),(c), but for the relaxation surface boundary condition. The thin black lines are contours of the zonal-average SST, and the contour interval is 0.5°C, from which changes in the surface buoyancy gradient ∂bs/∂y can be inferred. The thick dashed black lines indicate the boundaries of the regions of applied surface heating and cooling from the reference experiment τ0 = 0.2 N m−2.
Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-023.1
The changing air–sea buoyancy flux B in the relaxation case is evident in Fig. 7b. The flux is everywhere increasing as the winds increase, in accord with the fact that SSTs are decreasing [see (9); SSTs also change in the fixed-flux case; however, because the flux is not interactive, this has no effect on B]. This is completely consistent with the increased upper-cell transport and decreased lower-cell transport. In comparison with the fixed-flux case, the changes in 
Dependence of the air–sea buoyancy flux B on wind stress was observed by Badin and Williams (2010) in a similar yet coarse-resolution model. Their study also noted the sensitivity of B to the choice of Gent–McWilliams eddy-transfer coefficient. In our interactive buoyancy-flux experiments, both the eddy transfer and the buoyancy flux are free to respond to changing winds, resulting in a tangled equilibration problem. The diagnostics presented in this section merely show how the buoyancy budget is consistent with the residual circulation; they do not explain the magnitude of the sensitivity. For that, we need to look closer at the eddy circulation itself.
4. Constraints on the eddy circulation
In this section, we seek to understand what sets the strength of the eddy circulation. This discussion is most relevant to the interactive buoyancy-flux experiments, whose residual circulation cannot be assumed a priori based on knowledge of the buoyancy flux. The essential question is, how well can we estimate the sensitivities of Ψiso reported in Table 2 based on first principles?
a. Decomposing the eddy circulation: Slope and diffusivity
Viebahn and Eden (2010) found that changes in s were very small compared to changes in K and that the changes in Ψ* could therefore be attributed primarily to changes in K. To test this idea in our model, we calculate 
Fractional changes in (a) eddy circulation 
Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-023.1
The relative insensitivity of s seems somewhat inevitable given the boundary conditions. Because the buoyancy is relaxed to prescribed values at both the surface and the northern boundary, the large-scale isopycnal slope is effectively prescribed as well (of course, small changes in surface buoyancy are necessary to bring about changes in heat flux, as seen in Fig. 7). Only isopycnals that do not outcrop are unconstrained on the southern edge, resulting in higher values of Δs/sref in the far southern part of the domain. Viebahn and Eden (2010) found Δs to be small in an experiment with no sponge layer.
Focusing on the same surface (z = −477 m), we can use (22) to directly estimate MOCupper and MOClower by picking points in y that correspond with the maximum and minimum values of Ψiso (these points do not move significantly in space with changes in τ0). By calculating Δs and ΔK at these points, we can evaluate (22). The linear MOC sensitivities produced in this way are given in the third column of Table 2. These sensitivities agree very well with the values given by Ψiso, indicating that (22) is a good approximation.
Given the observed smallness of Δs, we can ask, to what extent is the sensitivity of the MOC due to ΔK? To answer this question, we evaluate (22) with Δs = 0 and compute the linear sensitivity. For comparison, we also do the opposite, setting ΔK = 0 and using only Δs. The results, given in the fourth column of Table 2, indicate that ΔK is the dominant factor in the upper-cell sensitivity in both the fixed-flux and relaxation experiments. Especially in the relaxation experiment, the sensitivity due to Δs alone is close to the 
b. Eddy diffusivity dependence on wind stress
Given the prominent role of ΔK in determining the MOC sensitivity, we focus now on understanding its scaling behavior with the winds. As a starting point, we plot the full K(y, z) for three different values of τ0 in Fig. 9. In general, K is positive nearly everywhere and appears intensified very near the surface and toward the bottom, with a minimum at middepth. The details of the vertical structure of K are interesting but are not our focus here (a paper on this topic is in preparation). For now, we simply note that the spatial structure does not change qualitatively with τ0, allowing us to imagine a fixed spatial structure that simply scales with τ0 (Viebahn and Eden 2010 found a strikingly similar spatial pattern). Many studies, including MR03 and Visbeck et al. (1997), have assumed that K itself is proportional to s. Instead, we employ a mixing-length theory, which relates K to the eddy kinetic energy and thus to the mechanical energy balance.
Flux-gradient buoyancy diffusivity K(y, z) for (left to right) three different wind strengths and fixed-flux boundary condition, shown with contour interval 500 m2 s−1. The black contours are the mean isotherms, and the contour interval is 0.5°C.
Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-023.1
Mixing-length theory (Taylor 1921; Prandtl 1925) claims that the eddy diffusivity can be expressed as a characteristic eddy velocity Ve times an eddy length scale Le, such that 
Globally averaged EKEs diagnosed from the model. The quantities plotted are the barotropic EKE (square), the bottom EKE (circle), and the EKE implied by the diffusivity K (triangle), assuming a constant mixing length of 30 km. The black line is the EKE predicted by the scaling relation (27). The fixed-heat-flux experiments are white, whereas the relaxation experiments are gray.
Citation: Journal of Physical Oceanography 41, 12; 10.1175/JPO-D-11-023.1
c. Predicting the MOC sensitivity
5. Discussion and conclusions
One important conclusion of this study is that the sensitivity of the Southern Ocean MOC to the winds depends on the surface boundary condition for buoyancy. This is not an immediately intuitive result, because the winds are a purely mechanical forcing. However, it becomes clear once one considers the TEM (or, equivalently, isopycnal average) point of view expressed by (15): in a quasi-adiabatic ocean interior, the residual MOC is primarily set by diabatic water-mass transformation at the surface, and, if the winds are unable to alter the transformation rates (as in the fixed-buoyancy-flux case), the sensitivity of the MOC is weak. In fact, evidence of this point emerges from the existing literature when comparing different models. For instance, Hallberg and Gnanadesikan (2006) used a predominantly fixed-flux surface boundary condition and found a relatively weak sensitivity of the residual MOC to increased winds. In contrast, Wolfe and Cessi (2010) used a relaxation boundary condition and found much greater sensitivity; in certain locations, they found an increase in residual MOC transport almost equal to the increase in Ekman transport, the upper limit of the sensitivity. The increased transport was accompanied by increased transformation in both southern and northern high latitudes. Although our model contains only an ACC channel, it manages to qualitatively reproduce the behavior of both these two different models just by changing the surface boundary condition. Similar conclusions were reached by Bugnion et al. (2006), using an adjoint method in a coarse-resolution model, and by Badin and Williams (2010).
The surface boundary condition of the real ocean is mixed. Certain contributions to the air–sea buoyancy flux, such as net shortwave radiation and precipitation, are largely independent of the SST and surface winds. Latent and sensible heat fluxes, on the other hand, are interactive (Haney 1971). For the winds to play a strong role in modulating the residual MOC, as envisioned by Toggweiler and Russell (2008), our study suggests that the interactive fluxes must dominate. It should therefore be a top priority to continue to improve our understanding of the processes that determine the air–sea buoyancy flux in the Southern Ocean—including sea ice processes, which we have completely neglected—and whether these components are sensitive to changes in wind or other climate changes.
Of the various simplifications we have made, perhaps the most restrictive and unrealistic is the fixed stratification imposed by the northern boundary sponge layer. In fact, many of the related studies we have cited have focused explicitly on the question of what sets the stratification (MR03; Henning and Vallis 2005; Wolfe and Cessi 2010). In the analytical model of MR03, the thermocline depth was found to be proportional to 
Finally, we developed a scaling theory for the eddy diffusivity and used it estimate the MOC sensitivity. Traditionally, scaling theories for eddies have been based on ideas from linear baroclinic instability, and the eddy diffusivity is assumed to be somehow proportional to the isopycnal slope (Green 1970; Stone 1972; Killworth 1997; Visbeck et al. 1997). Although baroclinic instability plays a crucial role in the energy cycle of our model, linear theory cannot predict the fully equilibrated eddy energy. Instead we have followed some of the ideas developed by Cessi (2008), invoking the mechanical energy balance to gain insight into the eddy energy and diffusivity. Consequently, our scaling theory for the eddy diffusivity (27) includes a dependence on both the wind stress parameter τ0 and the bottom drag rb but not on the isopycnal slope s. The scaling shows good agreement with the GCM results. Furthermore, we think it represents a promising way forward in understanding the role of eddies in the equilibration of the Southern Ocean.
We have examined only steady states, but the time-dependent response to wind changes is important and interesting. Meredith and Hogg (2006) have suggested Southern Ocean eddies can respond very fast (~1 yr) to changes in wind, whereas Treguier et al. (2010) found that the interannual MOC variability in a realistic model was dominated by Ekman transport, with little eddy compensation. This issue deserves further study as well.
Acknowledgments
Raf Ferrari and Malte Jansen made several helpful suggestions. Comments from Paola Cessi, two anonymous reviewers, and the editor greatly improved the manuscript. The observational wind and buoyancy flux data are from the Research Data Archive (RDA), which is maintained by the Computational and Information Systems Laboratory (CISL) at the National Center for Atmospheric Research (NCAR). NCAR is sponsored by the National Science Foundation (NSF). The original data are available from the RDA (http://dss.ucar.edu) in dataset ds260.2.
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This choice of parameters corresponds to a sensitivity of ∂Qeff/∂Ts ~ 15 W m−2 K−1.
Ferrari and Nikurashin (2010) recently refined the idea to include the modulation of Le by the presence of mean flows, and there is indeed mounting evidence that the spatial variations in K in the Southern Ocean are modulated by the strong jets found there (Marshall et al. 2006; Smith and Marshall 2009; Abernathey et al. 2010; Naveira Garabato et al. 2011).
This is equivalent to assuming that the baroclinic transport in the model is “saturated” (Straub 1993), which is indeed the case for our experiments.