1. Introduction
The Southern Ocean, and in particular the Antarctic Circumpolar Current (ACC), are key elements of the climate system. The ACC is driven by a combination of strong westerly winds and buoyancy forcing. Straub (1993) advanced the remarkable hypothesis that the equilibrated ACC zonal transport should be insensitive to the strength of the wind stress forcing. This insensitivity was later verified in eddy-resolving ocean models of the Southern Ocean and is now referred to as eddy saturation (e.g., Hallberg and Gnanadesikan 2001; Tansley and Marshall 2001; Hallberg and Gnanadesikan 2006; Hogg et al. 2008; Farneti et al. 2010; Meredith et al. 2012; Morrison and Hogg 2013; Munday et al. 2013; Abernathey and Cessi 2014; Farneti et al. 2015; Marshall et al. 2017). Some indications of eddy saturation are seen in observations (Böning et al. 2008; Firing et al. 2011).
There is evidence that the strength of the westerly winds over the Southern Ocean, which force the ACC, is increasing (Thompson and Solomon 2002; Marshall 2003; Yang et al. 2007; Swart and Fyfe 2012). Recently, Hogg et al. (2015) using satellite altimetry data identified that along with the strengthening of the westerlies comes also a linear trend of the Southern Ocean surface eddy kinetic energy (EKE), while the ACC zonal transport remains insensitive or even has decreased. This way, Hogg et al. (2015) concluded that the ACC is in an eddy-saturated state. Given the strengthening of the Southern Ocean westerly winds over the last decades, and the potential enhanced strengthening under global warming forcing (Bracegirdle et al. 2013), the question that naturally arises is how will the ACC transport respond? Thus, understanding the mechanisms behind eddy saturation is particularly relevant. This paper attempts to shed some more insight in the mechanism underlying eddy saturation.
Initially the explanation for eddy saturation given by Straub (1993) relied on baroclinic processes and on the existence channel walls. In the following detailed models of Nadeau and Straub (2009, 2012), Nadeau et al. (2013), and Nadeau and Ferrari (2015), the arguments for explaining eddy saturation were barotropic in heart. Specifically, Nadeau and Ferrari (2015) argued that the circulation can be decomposed to a circumpolar mode and a gyre mode (with the latter not contributing to the total transport). Nadeau and Ferrari (2015) showed that the wind stress curl spins barotropic gyres and, furthermore, that an increase of wind stress strength spins up the gyres while leaving the circumpolar mode intact; thus, the transport remains insensitive. Still, though, all those explanations relied in baroclinic instability for producing transient eddies to transfer the momentum from the surface of the ocean down to the bottom. On the other hand, Marshall et al. (2017) and Mak et al. (2017) recently showed that eddy saturation can emerge as a result of an eddy flux parameterization that was introduced by Marshall et al. (2012). In agreement with Straub (1993), Marshall et al. (2017) also relate the production of EKE to the vertical shear of the zonal mean flow. Again, baroclinic instability is identified as the main source of EKE. In this paper, we show that eddy saturation can be observed without baroclinic eddies, without channel walls, and without any wind stress curl, that is, without any gyres.
According to Johnson and Bryden (1989), different density layers are coupled via interfacial form stress that transfers momentum downward from the sea surface to the bottom. At the bottom it is topographic form stress that transfers momentum from the ocean to the solid earth (Munk and Palmén 1951). Note that only the standing eddies result in time-mean topographic form stress. Thus, it seems reasonable that topography should play a dominant role in understanding what sets up the total vertically integrated transport. There is a consensus that topography acts as the main sink in the zonal momentum balance [in agreement with Munk and Palmén (1951)]. But can the topography also have an active role in setting up the momentum balance, for example, by shaping the standing eddy field and its associated form stress? Abernathey and Cessi (2014) argued in favor of the active role the topography can have in setting up the momentum balance. They showed that isolated topographic features result in localized absolute baroclinic instability (i.e., baroclinic eddy growth in situ above the topographic features) and an associated almost-barotropic standing eddy field pattern. Transient eddies are suppressed away from the topographic features. Furthermore, Abernathey and Cessi (2014) demonstrated that the topographic feature results in the thermocline being shallower and the isopycnal slope being smaller compared to the flat-bottom case. Thus, the usual arguments assuming that the isopycnal slopes are so steep as to be marginal with respect to the flat-bottomed baroclinic instability cannot be invoked to explain the ACC equilibration (and thus neither eddy saturation) in a model configuration with localized topography. In addition, Thompson and Naveira Garabato (2014), and more recently Youngs et al. (2017), further emphasized the role of the standing eddies (or standing meanders) in setting up the momentum balance and the ACC transport. In this paper, we emphasize the role of the standing eddies and their form stress in determining transport in an eddy-saturated regime within a barotropic setting.
Lately, there has been increasing evidence arguing for the importance of the barotropic processes in determining the ACC transport. Ward and Hogg (2011) studied the ACC equilibration using a multilayer primitive equation wind-driven model with an ACC-type configuration starting from rest. Following turn-on of the wind a strong barotropic current forms within several days that is able to transfer most of the imparted momentum to the bottom; only after several years does the momentum start being transferred vertically via interfacial form stress. The fast response is that a bottom pressure signal arises a few days after turn-on, and the associated topographic form stress couples the ocean to the solid earth. Subsequently, for about 10 years, and contrary to the statistical equilibrium scenario described by Johnson and Bryden (1989), interfacial from stress transfers momentum vertically from the bottom upward. At equilibrium both eastward momentum is transferred from the surface downward and also westward momentum from the bottom upward. On the other hand, studies using in situ velocity measurements, satellite altimetry, and output from the Southern Ocean State Estimate (SOSE) also argue in favor of the importance of the bottom velocity component of the ACC transport, which comes about from the barotropic component of the flow1 (Rintoul et al. 2014; Peña Molino et al. 2014; Masich et al. 2015; Donohue et al. 2016). For example, using measurement from the cDrake experiment, Donohue et al. (2016) estimated that the bottom velocity component of the ACC transport accounts for about 25% of the total transport.
Constantinou and Young (2017) discussed the role of standing eddies using a simple barotropic model forced by an imposed steady wind stress and retarded by a combination of bottom drag and topographic form stress (Hart 1979; Davey 1980; Holloway 1987; Carnevale and Frederiksen 1987). Constantinou and Young (2017) used a random, monoscale topography and argued that a critical requirement for eddy saturation is that the ratio of planetary potential vorticity (PV) gradient to topographic PV gradient is large enough so that the geostrophic contours (i.e., the contours of the planetary PV plus the topographic contribution to PV) are open in the zonal direction. Here, we demonstrate that what matters for eddy saturation is not the actual value of the ratio of planetary PV over topographic PV but rather the structure of the geostrophic contours themselves.
The main goal of the present paper is to provide insight on how eddy saturation is established in this barotropic setting. We do that by comparing numerical results using two simple sinusoidal topographies. We show that this barotropic model without baroclinic instability can exhibit impressive eddy saturation, provided that the geostrophic contours are open (see section 3). We do not claim here that baroclinic processes are not important for setting up the momentum balance in the Southern Ocean. Instead, we emphasize the role of barotropic dynamics and the fact that we can still observe eddy saturation without baroclinicity. In this barotropic model, transient eddies arise as an instability caused by the interaction of the large-scale zonal flow with the topography. The simple topography used here allows us to study in detail this barotropic–topographic instability that gives rise to transient eddies and thus provide insight on how eddy-saturated states appear (see section 5). We show, in this way, that topography does not only have a passive role in setting the zonal momentum balance by acting as the main sink of zonal momentum, but it can also have an active role by producing transient eddies that shape the standing eddy field and its associated form stress.
2. Setup
The model formulated in (2) is the simplest model that can be used to investigate beta-plane turbulence above topography driven by a large-scale zonal wind stress applied at the surface of the fluid. It has been used in the past for studying the interaction of zonal flows with topography (Hart 1979; Davey 1980; Holloway 1987; Carnevale and Frederiksen 1987) and recently by Constantinou and Young (2017) for studying the geostrophic flow regimes above random monoscale topography.
Inspired by the Southern Ocean, we take L = 775 km, H = 4 km, ρ0 = 1035 kg m−3, f0 = −1.26 × 10−4 s−1, and β = 1.14 × 10−11 m−1 s−1. Also, we take 
An important factor controlling the behavior of the flow is the structure of the geostrophic contours, that is, the level sets of βy + η(x, y). Figure 1 shows the structure of geostrophic contours for the two topographies 
The structure of the geostrophic contours 
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
The structure of the geostrophic contours
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
The structure of the geostrophic contours
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
All solutions presented in this paper employ 5122 grid points; this resolution allows about six grid points within 
3. Results
a. Variation of the time-mean large-scale flow with wind stress forcing
Figure 2 shows how the time-mean, large-scale flow 
The equilibrated, time-mean, large-scale flow 
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
The equilibrated, time-mean, large-scale flow
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
The equilibrated, time-mean, large-scale flow
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
In the remainder of this subsection we describe in detail the qualitative features of the flow on the lower and upper branches as well as the transition from one branch to the other for the two topographies.
For weak wind stress forcing values (i.e., on the lower branch) the equilibrated solutions of (2) are time independent without any transient eddies. These steady states have a large-scale flow U and an associated stationary eddy flow field ψ that both vary linearly with wind stress forcing F. As a result, EKE varies quadratically with wind stress, that is, as F2. Figure 3a shows how EKE varies with wind stress forcing for topography 
(a) Variation of the equilibrated EKE and the standing eddy EKE (sEKE) with wind stress forcing for the topography 
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
(a) Variation of the equilibrated EKE and the standing eddy EKE (sEKE) with wind stress forcing for the topography
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
(a) Variation of the equilibrated EKE and the standing eddy EKE (sEKE) with wind stress forcing for the topography
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
For wind stress forcing values beyond the threshold of the lower-branch instability, the flow above the two topographies is qualitatively very different: topography 
For topography 
The eddy saturation regime in Fig. 2a terminates at 
On the other hand, the case with topography with closed geostrophic contours 
b. The flow regimes
As described in the previous subsection, we distinguish three qualitatively different flow regimes for each topography. There exist, for both topographies, a lower-branch flow regime for weak wind stress forcing and an upper-branch flow regime for strong wind stress forcing. These flow regimes consist of steady flows without any transient eddies. In between the lower- and upper-branch flow regimes there exists a regime in which the flow has a transient component: the eddy saturation regime for topography 
(a) The energy spectra for three typical cases using topography 
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
(a) The energy spectra for three typical cases using topography
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
(a) The energy spectra for three typical cases using topography
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
As in Fig. 4, but for three typical cases using topography 
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
As in Fig. 4, but for three typical cases using topography
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
As in Fig. 4, but for three typical cases using topography
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
The eddy field ψ in the eddy-saturated regime is characterized as two-dimensional turbulence. The wind stress F directly drives the large-scale flow U in (2b), and, in turn, U drives the eddy field through the term U∂xη in (2a). This leads to a forward transfer of energy from the largest possible scale to the eddies on the length scale of the topography 
c. Further characteristics of the eddy saturation regime for topography
It has been noted that in the eddy saturation regime even though the total ACC transport varies very little with wind stress, the domain-averaged EKE is approximately linearly related to the wind stress (see, e.g., Hallberg and Gnanadesikan 2001, 2006). Figure 3a shows how this is reflected in this barotropic model; indeed, in the eddy saturation regime the EKE varies with wind stress forcing at a rate much slower than quadratic. It is also apparent from Fig. 3a that strong transient eddies characterize the eddy saturation regime; this can be seen by the diminishing of the standing eddy EKE (sEKE), that is, the EKE that results from the standing eddies alone.
Results both from quasigeostrophic models (Hogg and Blundell 2006; Nadeau and Straub 2012; Nadeau and Ferrari 2015) and also primitive equation models (Marshall et al. 2017) demonstrated the somehow counterintuitive fact that in the eddy saturation regime the total transport increases with increasing bottom drag. Hogg and Blundell (2006) suggested that this effect comes about because increasing the bottom drag decreases the strength of the transient eddies that are responsible for transferring momentum to the bottom where the topographic form stress acts; thus, the strength of the zonal current increases. Nadeau and Straub (2012) on the other hand, argued that increase of the bottom drag damps the gyre circulation and its associated form stress; thus, the zonal current increases to provide the drag needed to balance the momentum imparted by the wind stress.
In the barotropic model studied here, we also show that transport increases with increasing bottom drag in the eddy saturation regime. Figure 3b shows how the time-mean, large-scale flow 
4. The effect of the transient eddies on the standing eddy field
(a),(b) A comparison of the eddy flux divergence ∇⋅
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
(a),(b) A comparison of the eddy flux divergence ∇⋅
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
(a),(b) A comparison of the eddy flux divergence ∇⋅
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
More importantly, however, notice how for the case with cos(14x/L) topography shown in Fig. 6a, the eddy PV fluxes are offset from the topography contours by one-eighth wavelength (see Fig. 6c). This offset in the eddy forcing induces a standing flow that projects onto the sin(14x/L) rather than the cos(14x/L), that is, projects onto 
5. The stability of the lower- and upper-branch solutions for topography
For topography of the form η = η0cos(mx) [as is (3a)], we can understand both the transitions among the three flow regimes as well as the role that the transient eddies play on eddy saturation by studying a three-dimensional invariant manifold of model (2).
Starting with the work by Hart (1979), the stability of solutions (9) to perturbations that lie within this invariant manifold has been extensively studied (Hart 1979; Pedlosky 1981; Källén 1982; Rambaldi and Flierl 1983; Yoden 1985). Here, we study the stability of (9) to any general perturbation that may or may not lie within the invariant manifold. Charney and Flierl (1980) studied the stability of the inviscid and unforced version of (12), that is, with μ = F = 0 and assuming the large-scale mean flow Ue to be a given external parameter of the problem. Here, we perform the stability of the forced–dissipative problem. Thus, we use Ue that results from a steady solution of (2) [or, for 
a. Stability calculation
b. Stability results
The region of stability of steady states (9), both within and outside the low-dimensional manifold [i.e., with respect to both (11) and (12)], is marked with the thick semitransparent curve in Fig. 2a. The onset of this barotropic–topographic instability explains (i) the appearance of transient eddies as wind stress is increased and (ii) the termination of the upper-branch solution as wind stress is decreased.
Figure 7a shows the large-scale flow Ue of the steady states (9) as a function of wind stress forcing F, together with their stability as predicted by (11) and (12). Also shown in Fig. 7a for comparison are the numerical results of (2). It is clear that the transition from the lower branch to the turbulent regime, in which eddy saturation is observed, is triggered by the instability caused by (12) rather than by (11). This instability of the lower branch first occurs at 
(a) The steady solutions (9) for topography 
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
(a) The steady solutions (9) for topography
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
(a) The steady solutions (9) for topography
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
Next, we would like to investigate how this instability contributes to the occurrence of eddy saturation. What is interesting here is how the instability maximum growth rate varies with the large-scale flow Ue. Figure 7b shows that in the regime in which we find eddy saturation the instability maximal growth rate caused by (12) increases dramatically with U; growth rates increase by a factor of 1000 when U increases only by a factor of 15. The transient eddy source thus increases significantly for only small changes in U. In this way, and by analogy with arguments for baroclinic eddy saturation, larger wind stress forcing implies the need for stronger eddies, but much stronger eddies can be produced with just a slight increase in Ue, thus leading to eddy saturation. Furthermore, the above argument suggests that the flow adjusts to a state close to marginal stability for this barotropic–topographic instability, similarly to the baroclinic marginal stability argument first invoked by Stone (1978).
The stability calculations presented in this section are based on the fact that the topography does not depend on y. However, the same flow transitions occur for flows with open geostrophic contours above complex topographies. In that case, finding the lower- and upper-branch equilibrium steady states is much more painful since some of the Jacobian terms, for example, 
6. Discussion
The results reported here demonstrate that eddy saturation can occur even without baroclinicity. A bare barotropic setting is capable of producing an eddy saturation regime in which the transport is insensitive to wind stress forcing. The model shows in addition some other features of eddy saturation that were previously observed in more elaborate ocean models; EKE varies close to linearly with wind stress forcing (instead of quadratically) and also transport increases with increasing bottom drag.
In this barotropic model with no baroclinic eddies, the flow relies on the barotropic–topographic instability to produce transient eddies. For the y-independent topography with open geostrophic contours 
The factor that controls the appearance or not of eddy-saturated states in a barotropic setting is whether or not the geostrophic contours are open. This was demonstrated here using two simple sinusoidal topographies that bare this distinction. However, the bare existence of the eddy saturation in this simple setting depends on the presence of a transient eddy source (i.e., of an instability). The range of wind stress forcing values where eddy saturation could potentially occur is limited by the range of wind stress forcing values for which the barotropic–topographic instability occurs. The latter crucially depends on the height of the topography. For the flat-bottom case with h = 0, all geostrophic contours are open, but there is no eddy saturation since there is no instability. Figure 8 shows the steady states (9) together with their stability (similarly to Fig. 7a) for various topography heights. The range of wind stress forcing values that produces instability scales roughly as 
Steady solutions (9) for topography 
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
Steady solutions (9) for topography
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
Steady solutions (9) for topography
Citation: Journal of Physical Oceanography 48, 2; 10.1175/JPO-D-17-0182.1
We note that the flow characteristics that were described in section 3 are not quirks of the simple sinusoidal topographies in (3). The eddy saturation regime, the drag crisis, and multiple equilibria that are present here when geostrophic contours are open have been all also recently found to exist in this model for a flow above a random monoscale topography with open geostrophic contours (Constantinou and Young 2017) and also above a multiscale topography that has a k−2 power spectrum (not reported).
In conclusion, the results presented here emphasized that barotropic processes might play a role in shaping the zonal momentum balance in the ACC and in producing eddy saturation. In particular, barotropic processes are responsible at least for the component of the transport that is related to the bottom flow velocity of the ACC. [This bottom velocity component of the ACC transport accounts for about 25% of the total transport according to Donohue et al. (2016).] Even though the ACC is strongly affected by baroclinic processes, our results here reinforce the increasing evidence arguing for the importance of the barotropic mechanisms in determining the ACC transport (Ward and Hogg 2011; Thompson and Naveira Garabato 2014; Youngs et al. 2017). Additionally, the results of this paper argue that topography does not only have a passive role in the momentum budget acting merely as a sink for zonal momentum, but it can also have an active role by producing transient eddies that shape the standing eddy field and its form stress.
Acknowledgments
I am mostly grateful to William Young for his support and insightful comments. Discussions with Anand Gnanadesikan, Petros Ioannou, Spencer Jones, Sean Haney, Cesar Rocha, Andrew Thompson, Gregory Wagner, and Till Wagner are greatly acknowledged. Also, I thank Louis-Philippe Nadeau and David Straub for their constructive review comments. This project was supported by the NOAA Climate and Global Change Postdoctoral Fellowship Program, administered by UCAR’s Cooperative Programs for the Advancement of Earth System Sciences. The author also acknowledges partial support from the National Science Foundation under Award OCE-1357047.
APPENDIX
Stability of Steady States (9) to Perturbations outside the Low-Dimensional Manifold
(12) is homogeneous in y, that is, if ϕ(x, y, t) is a solution so is ϕ(x, y + a, t) for any a; and
the coefficients of (12) remain unchanged under
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Traditionally, the baroclinic component of the flow is obtained through the thermal–wind relationship after assuming zero flow at the bottom. Thus, any transport caused by the bottom flow is not thought as part of the baroclinic component of the ACC transport.
A MATLAB code used for solving (2) is available at the github repository: https://github.com/navidcy/QG_ACC_1layer.
For a more detailed discussion on Bloch eigenfunctions the reader is referred, for example, to the textbook by Ziman (1972) or to appendix D in the thesis of Constantinou (2015).
