Editorial Type:
Article
Article Type:
Research Article

“Eddy” Saturation of the Antarctic Circumpolar Current by Standing Waves

Andrew L. Stewart aDepartment of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California

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Nicole K. Neumann bLamont-Doherty Earth Observatory, Columbia Climate School, Columbia University, Palisades, New York

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Aviv Solodoch aDepartment of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California

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Online Publication:
18 Apr 2023
Print Publication:
01 Apr 2023
DOI:
https://doi.org/10.1175/JPO-D-22-0154.1
Page(s):
1161–1181
Free access

Abstract

It is now well established that changes in the zonal wind stress over the Antarctic Circumpolar Current (ACC) do not lead to changes in its baroclinicity nor baroclinic transport, a phenomenon referred to as “eddy saturation.” Previous studies provide contrasting dynamical mechanisms for this phenomenon: on one extreme, changes in the winds lead to changes in the efficiency with which transient eddies transfer momentum to the sea floor; on the other extreme, structural adjustments of the ACC’s standing meanders increase the efficiency of momentum transfer. In this study the authors investigate the relative importance of these mechanisms using an idealized, isopycnal channel model of the ACC. Via separate diagnoses of the model’s time-mean flow and eddy diffusivity, the authors decompose the model’s response to changes in wind stress into contributions from transient eddies and the mean flow. A key result is that holding the transient eddy diffusivity constant while varying the mean flow very closely compensates for changes in the wind stress, whereas holding the mean flow constant and varying the eddy diffusivity does not. This implies that eddy saturation primarily occurs due to adjustments in the ACC’s standing waves/meanders, rather than due to adjustments of transient eddy behavior. The authors derive a quasigeostrophic theory for ACC transport saturation by standing waves, in which the transient eddy diffusivity is held fixed, and thus provides dynamical insights into standing wave adjustment to wind changes. These findings imply that representing eddy saturation in global models requires adequate resolution of the ACC’s standing meanders, with wind-responsive parameterizations of the transient eddies being of secondary importance.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Andrew L. Stewart, astewart@atmos.ucla.edu

Abstract

It is now well established that changes in the zonal wind stress over the Antarctic Circumpolar Current (ACC) do not lead to changes in its baroclinicity nor baroclinic transport, a phenomenon referred to as “eddy saturation.” Previous studies provide contrasting dynamical mechanisms for this phenomenon: on one extreme, changes in the winds lead to changes in the efficiency with which transient eddies transfer momentum to the sea floor; on the other extreme, structural adjustments of the ACC’s standing meanders increase the efficiency of momentum transfer. In this study the authors investigate the relative importance of these mechanisms using an idealized, isopycnal channel model of the ACC. Via separate diagnoses of the model’s time-mean flow and eddy diffusivity, the authors decompose the model’s response to changes in wind stress into contributions from transient eddies and the mean flow. A key result is that holding the transient eddy diffusivity constant while varying the mean flow very closely compensates for changes in the wind stress, whereas holding the mean flow constant and varying the eddy diffusivity does not. This implies that eddy saturation primarily occurs due to adjustments in the ACC’s standing waves/meanders, rather than due to adjustments of transient eddy behavior. The authors derive a quasigeostrophic theory for ACC transport saturation by standing waves, in which the transient eddy diffusivity is held fixed, and thus provides dynamical insights into standing wave adjustment to wind changes. These findings imply that representing eddy saturation in global models requires adequate resolution of the ACC’s standing meanders, with wind-responsive parameterizations of the transient eddies being of secondary importance.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Andrew L. Stewart, astewart@atmos.ucla.edu
Keywords: Southern Ocean; Channel flows; Momentum; Ocean dynamics; Eddies; Idealized models

1. Introduction

The response of the Southern Ocean to changes in the atmospheric state and circulation has been the subject of extensive scientific study in recent decades (see, e.g., Mayeweski et al. 2009; Rintoul 2018). This scientific scrutiny is motivated by the central role of the Southern Ocean, and particularly the Antarctic Circumpolar Current (ACC), in connecting the major ocean basins (Nowlin and Klinck 1986; Olbers et al. 2004) and closing the deep cells of the global overturning circulation (Marshall and Speer 2012; Talley 2013). Particular emphasis has been placed on the strengthening and southward shifting of the Southern Hemisphere westerlies that occurs with increased atmospheric CO2 concentrations, both as a consequence of anthropogenic influence (Hazel and Stewart 2019; Thompson and Solomon 2002) and naturally over paleoclimatic time scales (Toggweiler et al. 2006; Toggweiler 2009).

It is now a well-established result, both based on observations (Böning et al. 2008) and model simulations (Meredith and Hogg 2006; Hogg and Blundell 2006; Meredith et al. 2012), that the baroclinic transport of the ACC is approximately insensitive to changes in the westerly winds. This phenomenon, referred to as “eddy saturation” (Straub 1993; Hogg et al. 2008), apparently contradicts the historical conception of the ACC as a primarily wind-driven current (Munk and Palmén 1951; Nowlin and Klinck 1986). Indeed, idealized model experiments have shown that the ACC transport may remain insensitive to winds even at the limit of zero wind stress (Munday et al. 2013), provided that mesoscale eddies are sufficiently well resolved and that the meridional overturning circulation is sufficiently weak (Youngs et al. 2019).

Figure 1 illustrates schematically the mechanism via which eddy saturation occurs. This mechanism is most clearly isolated in a quasi-latitudinal coordinate system that follows mean streamlines of the ACC, which emphasizes the role of mesoscale eddies, rather than time-mean flows (Abernathey and Cessi 2014). In an equilibrium state, momentum is transferred from the winds into the upper ocean, and is then transferred down to the deep ocean (and ultimately the sea floor) via eddy interfacial form stress (IFS) (Marshall and Radko 2003; Abernathey and Cessi 2014). The eddy IFS results from baroclinic instabilities, which draw energy from the baroclinicity of the ACC (Treguier and McWilliams 1990; Youngs et al. 2017). Eddy saturation refers to the response of this system to a change in surface wind stress: increasing the surface wind stress leads to a commensurate increase in the eddy IFS, while the baroclinicity of the ACC remains unchanged. We note that this characterization of eddy saturation in terms of eddy IFS is equivalent (under quasigeostrophic scaling) to previous characterizations based on quasi-meridional Ekman versus eddy volume transports (e.g., Thompson and Naveira Garabato 2014; Youngs et al. 2019).

Fig. 1.
Fig. 1.

Schematic of the eddy saturation phenomenon, presented in a coordinate system that follows the time-mean geostrophic streamlines of the ACC, which for the upper ocean is the sea surface height (SSH). In each panel the tilted blue line schematically indicates an isopycnal surface, and the blue arrow (directed out of the page) indicates the near-surface flow of the ACC. (a) Momentum input by the wind (light blue arrow, out of the page) to the upper ocean is transferred downward by eddy interfacial form stress (IFS) to the deep ocean, where it is ultimately removed by topographic form stress and bottom friction. (b) Increases in the zonal surface wind stress lead to commensurate increases in the eddy IFS, with approximately no change in the baroclinicity of the ACC. Note that the geostrophic streamline-following coordinate system ensures there is no mean component of the IFS (see section 2b and Abernathey and Cessi 2014).

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0154.1

Previous studies of the eddy saturation phenomenon have proposed diverging mechanistic explanations. On one extreme, eddy saturation has been posited to result from changes in the efficiency of the transient eddy IFS (or, equivalently, the horizontal eddy buoyancy flux), characterized by the Gent–McWilliams diffusivity κ (Gent and McWilliams 1990; Gent et al. 1995). For example, the residual-mean theory of Marshall and Radko (2003, 2006) requires that κ scale linearly with the wind stress in order to preserve the baroclinicity of the ACC. Meredith et al. (2012) rationalized this linear scaling based on the mixing suppression theory of Ferrari and Nikurashin (2010). More recently, Marshall et al. (2017) and Mak et al. (2018) have developed a theory for changes in κ based on the eddy energy budget, constraining the eddy interfacial form stress following the geometric interpretation of Marshall et al. (2012) and Maddison and Marshall (2013). This theory explicitly predicts approximate independence of the ACC transport on wind stress, but relatively strong dependence of the ACC transport on friction at the sea floor (Marshall et al. 2017; Mak et al. 2018, 2022).

On the other extreme, eddy saturation has been posited to result from adjustment of the ACC’s time-mean standing meanders, i.e., standing Rossby waves (Marshall 2016; Bai et al. 2021). Thompson and Naveira Garabato (2014) noted that two-dimensional residual-mean theories of the ACC omit the zonally localized dynamics of the ACC’s standing meanders, where eddy activity is concentrated (Abernathey and Cessi 2014; Meredith 2016; Rintoul 2018). Based on analysis of a high-resolution global ocean model, Thompson and Naveira Garabato (2014) proposed that the ACC could adjust to changes in zonal wind stress via “flexing” of its standing meanders, i.e., changes in the meander shape and amplitude that serve to increase or decrease the efficiency of downward momentum transfer by mean IFS. Nadeau and Ferrari (2015) used channel model simulations to show that eddy saturation does not occur in the absence of bathymetric obstacles and standing meanders, and proposed that formation of closed gyres abutting the flow of the ACC plays a key role in balancing increases in zonal wind stress. Constantinou and Hogg (2019) showed that eddy saturation occurs even in barotropic channel model simulations, in which there can be no baroclinic instability, and thus concluded that barotropic flow–topographic interactions alone are sufficient to produce a saturation response.

Thus, there remains an outstanding question as to what extent eddy saturation occurs as a result of changes in the efficiency of transient eddy transfer, versus rearrangement of the ACC’s standing meanders. In this study we address this question by using an idealized isopycnal channel model to quantify the relative importance of transient eddy versus standing wave adjustment in eddy saturation. In section 2 we describe our numerical configuration, pose a decomposition of the eddy saturation response via the momentum balance along time-mean geostrophic streamlines, and present a method of diagnosing the transient eddy diffusivity. In section 3 we examine the saturation response simulated across a suite of model experiments, and apply our decomposition to distinguish the roles of transient eddy versus standing wave adjustment. Motivated by these findings, in section 4 we pose a quasigeostrophic standing wave theory of eddy saturation, and use it to draw insights into the dynamics of standing wave adjustment to wind changes. Finally, in section 5 we discuss our findings and conclude.

2. Numerical modeling approach and diagnostics

a. Model configuration

We designed our model configuration to minimize physical complexity while capturing the eddy saturation phenomenon, drawing inspiration from many previous studies of ACC dynamics (Ward and Hogg 2011; Nadeau and Ferrari 2015; Howard et al. 2015; Constantinou and Hogg 2019; Youngs et al. 2019). Specifically, we simulate the flow in a zonally reentrant walled channel of zonal length Lx = 3200 km, meridional width Ly = 1600 km, and depth H = 4000 m. The channel is posed on a midlatitude beta plane with Coriolis parameter f = f0 + β(y − Ly/2), where f0 = −1 × 10−4 s−1 is the reference Coriolis parameter and β = 1.5 × 10−11 m−1 s−1 is the Coriolis parameter gradient. To obstruct the zonal flow through the channel, we impose a single Gaussian bump in the sea floor [z = ηb(x)] at x = Xb = 1000 km of height Hb = 1000 m and width Wb = 150 km,
ηb=−H+Hbexp⁡[−(x−XbWb)2].
This geometry, shown in Fig. 2, crudely represents the presence of bathymetric obstacles that generate standing meanders in the ACC (e.g., Thompson and Naveira Garabato 2014; Nadeau and Ferrari 2015; Youngs et al. 2019).
Fig. 2.
Fig. 2.

Schematic of our model configuration. (a) Latitudinal profile of zonal wind stress applied to the upper isopycnal layer. (b) Snapshot of potential vorticity in the upper isopycnal layer of our reference simulation. Shading indicates the location of the sponge/restoring region at the northern boundary, while dark gray contours indicate the 3750-, 3500-, and 3250-m isobaths. (c) Zonal variation of the topographic elevation (brown) and the meridionally averaged isopycnal layer interface (blue) corresponding to the simulation snapshot shown in (b).

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0154.1

The density stratification is represented by two layers, each of constant density, as this is the minimum required to allow the development of baroclinic instability (Vallis 2006). The isopycnal velocities and layer thicknesses evolve according to conservation of momentum, which we write for convenience as
∂∂t(hkuk)+∇⋅(hkukuk)+(−1)kωu⋆+fz^×hu+hk∇Mk=δ1,kτwρ0x^−δ2,kτbρ0+∇⋅σk,
and conservation of volume,
∂hk∂t+∇⋅(hkuk)+(−1)kω=0.
We denote the isopycnal velocity in each layer as uk and the corresponding layer thickness as hk, where k = 1 corresponds to the upper layer and k = 2 to the lower layer. We make the rigid-lid approximation to exclude fast surface gravity waves, so the Montgomery potential Mk is defined as
Mk=π+δ2,kg′η.
Here π is the surface pressure (normalized by the reference density), η = −h1 is the elevation of the interface between the layers, g′ = gΔρ/ρ0 = 1 × 10−2 m s−2 is the reduced gravity, Δρ denotes the positive density difference between the layers, g is the gravitational constant, ρ0 = 1000 kg m−3 is a reference density, and δ denotes the Kronecker delta.
The flow in the upper layer is forced by a zonal wind stress τw(y)x^ whose latitudinal structure (see Fig. 2a) is given by
τw(y)=τwmaxsin⁡2(πyLy),
where x^ denotes a unit vector in the x direction. The flow in the lower layer is retarded by frictional stress τb at the sea floor. For all experiments described in the main text, we adopt a quadratic formulation of this frictional stress,
τb=ρ0Cd|u2|u2,
where Cd is the drag coefficient. We perform an analogous set of experiments, results of which are presented in appendix B, using a linear formulation of the frictional stress,
τb=ρ0rbu2,
where rb is the friction velocity. Over the northernmost 100 km of the model domain we impose a diapycnal velocity
ω=η−η0Trelax,
that serves to relax the layer interface toward a reference elevation η0 = −1750 m. The relaxation rate Trelax−1 varies linearly from 1 week−1 at the northern wall to zero at the edge of the restoring region. This relaxation is not strictly necessary in our model configuration because the flow can be configured to be perfectly adiabatic, but it serves to produce a consistent stratification at the northern edge of the domain across our experiments, without constraining the isopycnal slope across the channel. The diapycnal velocity also enters (2) in order to conserve momentum, multiplying an effective isopycnal velocity u⋆ that is assigned based on the layer that is “upwind” relative to the diapycnal velocity (LeVeque 2002),
u⋆={u1,ω<0,u2,ω>0.
Finally, we dissipate gridscale energy and enstrophy via a biharmonic viscous stress tensor, ∇ ⋅ σk, using a Smagorinsky prescription of the viscosity with dimensionless coefficient ASmag = 4 (Griffies and Hallberg 2000).

We solve (2)–(4) via forward numerical integration using the AWSIM model (Stewart and Dellar 2016). We adopt the same spatiotemporal numerical schemes as those discussed by Stewart et al. (2021). We discretize the model equations on a uniform horizontal grid of Nx × Ny = 512 × 256 points, corresponding to a grid spacing of approximately 6 km. The horizontal grid spacing is chosen to be much smaller than the baroclinic deformation radius, in order to ensure adequate resolution of mesoscale eddies (Hallberg 2013). For our chosen model parameters the baroclinic deformation radius is approximately 30 km, which is comparable to the first Rossby radius of deformation along the northern flank of the ACC (Chelton et al. 1998).

Motivated by the previous finding that bottom friction plays an important role in setting the ACC transport, we conduct a suite of experiments in which we covary the wind stress ( τwmax={0.01,0.013,0.017,0.022,0.03,0.039,0.05,0.07,0.1,0.13,0.17,0.22,0.3,0.39,0.5} N m−2) with either the quadratic drag coefficient (Cd = {0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4} × 10−3) or the linear drag velocity (rb = {2, 3, 4, 5, 6, 7, 8, 9, 10} × 10−4 m s−1). All other model parameters are held fixed. Each experiment is “spun up” at low resolution (Nx × Ny = 256 × 128 points) for either 200 years (for τwmax≥0.05 N m−2) or 400 years (for τwmax<0.05N m−2), because experiments with weak surface wind stress require longer to reach a statistically steady state (Constantinou and Hogg 2019). We then interpolate the model state to the high-resolution grid (Nx × Ny = 512 × 256 points) and continue the spinup for a further 100 years to reestablish statistical equilibrium; over the last 30 years integration, the linear trend in the annually computed zonal transport is not significant at the 5% level. Finally, we continue each high-resolution run for a further 30 years, which serves as the analysis period.

b. Zonal versus along-streamline momentum balance

In section 1 we framed the eddy saturation phenomenon in terms of the momentum balance of the ACC (see Fig. 1). In this subsection we show (consistent with previous studies, cf. Abernathey and Cessi 2014), that in streamline-following coordinates, eddy isopycnal form stress is almost entirely responsible for the downward transfer of momentum. Therefore, in this coordinate system the analysis of eddy saturation is simplified in that it requires consideration of only the eddy (and not the mean) isopycnal form stress term. We will take advantage of this result in section 2c to decompose the isopycnal form stress response (which results in eddy saturation) into contributions from changes in mean and transient motions.

We now provide reference diagnostics of the mean circulation balance in our reference simulation, defined by a wind stress maximum of τwmax=0.1 N m−2, to match the order of magnitude of the maximum wind stress over the ACC (Large and Yeager 2009), and a quadratic bottom drag coefficient of Cd = 2 × 10−3 (Gill 1982; Arbic and Scott 2008). Figure 3 shows the time-mean momentum balance averaged zonally in both latitudinal and along-stream coordinates, and taken either over the full ocean depth or just the upper isopycnal layer. We separate “mean” versus “eddy” contributions to the momentum advection via thickness-weighted averaging (Young 2012; Aiki et al. 2016),
∇⋅(hkukuk¯)=∇⋅(hk¯u^ku^k)︸mean advection+∇⋅(hk¯uk†uk†^)︸eddy advection.
Here the overbar denotes a time average over the 30-yr simulation analysis period. The hat operator denotes a thickness-weighted average, with daggers denoting fluctuations,
uk^=hkuk¯hk¯,uk†=uk−uk^.
We separate mean and eddy components of the pressure gradient force (PGF) as
hk∇Mk¯=hk¯∇Mk¯︸mean PGF+hk′∇Mk′¯︸eddy PGF,
where primes denote deviations from the time average, i.e., •′=•−•¯. The resulting averaged momentum equations are given in full in appendix A.
Fig. 3.
Fig. 3.

Momentum balance in latitudinal vs streamline-following coordinates. (a) Time-mean equivalent sea surface height (SSH) in our reference simulation. The black contour indicates the core of the ACC, defined to be the median value of the SSH over the entire model domain. Dark gray contours indicate the 3750-, 3500-, and 3250-m isobaths. (b) Depth-integrated and (c) upper-isopycnal-layer momentum balances, integrated along lines of constant latitude. (d) Depth-integrated and (e) upper-isopycnal-layer momentum balances, integrated along contours of constant time-mean SSH. Explicit definitions of all of the terms in the momentum budget are given in appendix A. The black dashed line in (b)–(e) corresponds to the median SSH, as shown in (a). The red dotted line in (e) is the eddy interfacial form stress reconstructed from the time-mean flow and the diagnosed eddy diffusivity κ via (17).

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0154.1

Figures 3b and 3c show the zonally averaged, time-mean zonal momentum balance. Consistent with previous studies (e.g., Treguier and McWilliams 1990; Abernathey and Cessi 2014; Stewart and Hogg 2017), in this coordinate system the wind-input momentum is transferred downward almost entirely via the action of mean IFS, and then removed at the sea floor via topographic form stress (TFS),
τw(y)︸wind stress≈⟨pη¯∂xη¯⟩x︸mean IFS≈⟨pb¯∂xηb⟩x︸TFS.
Here ⟨•⟩x denotes a zonal average, pη = p|z=η is the pressure on the isopycnal interface, and pb = p|z=ηb is the pressure at the sea floor. Note that the forms of the IFS and TFS terms differ locally from those given in appendix A, but are equivalent when integrated over any closed contour
∮Cρ0h1∇M1⋅ds=∮Cpη∇η⋅ds,
∮Cρ0∑khk∇Mk⋅ds=∮Cpb∇ηb⋅ds,
where ds denotes an infinitesimal line element along the closed contour C.
As noted by Abernathey and Cessi (2014), integration along lines of constant latitude obscures the role of transient eddies. Figures 3d and 3e show the time-mean momentum balance integrated along contours of mean sea surface height, i.e., mean geostrophic streamlines, then divided by the zonal domain length Lx (see appendix A). As we employ the rigid-lid approximation (see section 2a), we define an equivalent sea surface height via the surface pressure as ζ = π/g. In this coordinate system the mean IFS vanishes identically, and the momentum balance approximately becomes (cf. Johnson and Bryden 1989)
∮ζ¯=ζ0τwx^⋅ds︸wind stress≈∮ζ¯=ζ0pη′∇η′¯⋅ds︸eddy IFS≈∮ζ¯=ζ0pb¯∇ηb⋅ds︸TFS.
Here ζ0 denotes the mean sea surface height contour along which the line integral is taken; the mean sea surface height in our reference experiment is shown in Fig. 3.

c. Separating transient eddy and mean flow contributions to eddy IFS

As discussed in section 1, it remains unclear to what extent eddy saturation occurs as a result of changes in the efficiency of transient eddies in transferring momentum downward via IFS versus adjustment of the ACC’s standing meanders. To distinguish between these contributions, we relate the eddy IFS locally to the vertical gradient of the mean flow (Greatbatch and Lamb 1990), which is equivalent (in the quasigeostrophic limit) to assuming a downgradient horizontal buoyancy flux (Gent et al. 1995),
τeddy≡pη′∇η′¯≡κρ0f2g′(u1^−u2^)+∇D.
The first term on the right-hand side of (16) is approximately nondivergent (under quasigeostrophic scaling, and for slow spatial variations in κ). We make the ansatz that this term accounts for the rotational component of the eddy IFS, and allow for an additional divergent component of the eddy IFS, ∇D (see Marshall and Shutts 1981). Substituting (16) into (15), we obtain
∮ζ¯=ζ0τeddy⋅ds=∮ζ¯=ζ0κ︸eddy efficiencyρ0f2 g′(u1^−u2^)⋅ds︸standing meander structure.
In section 3 we will use (17) to quantify the contributions of transient eddies versus standing meanders in adjusting the eddy IFS to changes in wind stress. To do this, we require spatially varying estimate of κ, which is defined implicitly by (16). We diagnose κ by taking the curl of (16) to eliminate the divergent component of the eddy IFS,
J(pη′,η′)¯≈κρ0f2g′∇×(u1^−u2^).
Here we have additionally neglected spatial variations of f and κ, under the assumption that they vary slowly relative to the mean flow. Although (18) could be used directly to calculate κ at each point in the model domain, x = x0, the result is excessively noisy. Instead, we invert (18) for κ via least squares linear regression of J(pη′,η′)¯ on (ρ0f2/g′)∇×(u1^−u2^) over all points within a 200-km radius of x0. The linear fits are statistically significant (p values typically many orders of magnitude smaller than 0.001) over almost all of the model domain.
Figure 4 shows the spatial distribution of κ in our reference simulation, as compared with the distribution of the depth-averaged eddy kinetic energy (EKE),
EKE=1∑khk∑k12hk¯u†2^.
Both the EKE and κ are elevated downstream of the bathymetric ridge, consistent with previous modeling studies and observations (e.g., Bischoff and Thompson 2014; Meredith 2016). This correspondence is visually evident in the “storm track” region (Bischoff and Thompson 2014), which we define as the uppermost quartile of the EKE over the entire model domain (see Fig. 4a). In this region there is generally little scatter in the linear regression that we use to estimate κ (Fig. 4c), supporting our assumption that κ is slowly varying relative to the mean flow. Note that the region within 200–300 km of the northern boundary also exhibits little scatter in the linear regression, but that this is an artifact resulting from the tendency of the sponge layer to create elongated zonal structures in both J(pη′,η′)¯ and (ρ0f2/g′)∇×(u1^−u2^).
Fig. 4.
Fig. 4.

Diagnostics of eddy activity in our reference simulation. (a) Depth-averaged eddy kinetic energy (EKE), (b) estimated transient eddy diffusivity, and (c) variance explained by our linear regression-based estimate of κ (see section 2c). In all panels the black contour bounds the fourth quartile of the EKE, which defines our storm track region. The gray contours indicate the 3750-, 3500-, and 3250-m isobaths.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0154.1

To further evaluate our approach to estimating κ, we compute the alongstream-averaged eddy IFS via (17). Figure 3e shows that this reconstruction only slightly underestimates the diagnosed alongstream-averaged eddy IFS, by around 10%–20% over the SSH contours that cross the bathymetric ridge (see Fig. 3a).

3. Saturation by transient eddies versus standing waves

In this section we utilize our suite of model experiments (see section 2a) to assess the extent to which adjustment of transient eddy behavior versus mean flow structure is responsible for eddy saturation. Briefly, we first show that the model’s baroclinic transport is indeed saturated in experiments with varying zonal wind stress, consistent with previous studies. We then show that the diagnosed variations in the transient eddy diffusivity (along with the eddy kinetic energy) suggests that transient eddies are not adjusting sufficiently rapidly to support the transport saturation. Finally, we use our decomposition of the eddy IFS (17) to verify that the saturation occurs primarily via adjustments of the standing waves, rather than of the transient eddies.

To evaluate the extent to which our idealization of the ACC is eddy saturated, we examine the sensitivity of both the total transport and its barotropic/baroclinic components, defined respectively as
Ttot=⟨∫0Ly∑khkuk¯⟩x,
Tbt=⟨∫0Lyu2∑khk¯⟩x,
Tbc=Ttot−Tbt.
We define the lower-layer flow as the barotropic component of the zonal flow, and then average the associated barotropic transport zonally to obtain Tbt, with Tbc computed as a residual.

In Fig. 5 we plot the dependence of these transports on the wind stress maximum and the quadratic drag coefficient. The total volume transport is generally substantially lower than the observed transport of the ACC (Whitworth and Peterson 1985; Donohue et al. 2016), but is of the correct order of magnitude, and is comparable to previously reported transports in channel model simulations (e.g., Stewart and Hogg 2017; Youngs et al. 2019). Figure 5 shows that the baroclinic transport is approximately independent of wind stress, whereas the barotropic transport increases with wind stress, and thus so does the total transport. This is consistent with some previous modeling studies (Nadeau and Ferrari 2015; Youngs et al. 2019), whereas others have found that the total transport is approximately independent of the wind stress (Munday et al. 2013; Marshall et al. 2017). Similar to Marshall et al. (2017), the total, barotropic, and baroclinic transports all increase with quadratic drag coefficient, although for drag coefficients ≳ 2 × 10−3 this sensitivity is relatively weak. For the strongest wind stresses and weakest bottom friction coefficients examined here, the barotropic transport actually begins to decrease with wind stress, and may even become negative. This flow reversal appears to be associated with extreme strengthening of deep gyres in the lee of the ridge (Nadeau and Ferrari 2015), but the specific mechanism via which it occurs and its relevance to the dynamics of the ACC are left as a topic for future investigation.

Fig. 5.
Fig. 5.

Dependence of the (a) total transport, (b) barotropic transport, and (c) baroclinic transport on the wind stress maximum and on the quadratic drag coefficient in our channel model simulations. Note the nonlinear abscissa.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0154.1

To obtain preliminary insights into how transient eddies and standing meanders/waves change while preserving the baroclinic flow, we now examine the response of the transient eddy and standing wave activity to variations in the wind stress. Figure 6a shows the sensitivity of κ, spatially averaged over the storm track region in each simulation, to changes in the wind stress maximum and quadratic drag coefficient. Least squares logarithmic fits over all of our simulations yield approximate scalings of κ∼τw0.6 and EKE∼τw0.7. Including random multiplicative errors with average magnitudes of 10% in the estimate of κ, motivated by our comparison between the diagnosed and reconstructed eddy IFS in Fig. 2, does not change the scaling of κ with τw to within one decimal place. In contrast, residual-mean theory (Marshall and Radko 2003) predicts that κ must scale linearly with τw in order to preserve the baroclinic transport. Thus our diagnostics suggest that adjustments in the efficiency of downward transient eddy momentum transfer (i.e., of κ) are not sufficient to produce the eddy saturation response in these simulations. Figure 6c shows the sensitivity of the domain-averaged standing wave kinetic energy (SKE), which we simply estimate as the depth-averaged mean kinetic energy,
SKE=1∑khk∑k12hk¯uk^2.
In contrast to the EKE, the SKE scales approximately linearly with wind stress, with SKE ∼ τ1.1. This is qualitatively consistent with flexing of the standing meander in response to increased zonal wind stress (Thompson and Naveira Garabato 2014), though the change in SKE could also arise due to spinup of the mean gyres that abut the circumpolar flow in Fig. 3 (Nadeau and Ferrari 2015).
Fig. 6.
Fig. 6.

Sensitivity of transient eddies and standing waves to the wind stress maximum and the quadratic drag coefficient. (a) Transient eddy diffusivity κ averaged over the storm track region (see Fig. 4), (b) domain-averaged eddy kinetic energy, and (c) domain-averaged standing wave energy.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0154.1

To quantify the relative roles of transient eddy versus standing wave adjustment in eddy saturation, we now separately examine their contributions to the changes in eddy IFS that occur in response to wind stress perturbations. The rationale for this approach is that the wind-input momentum along mean streamlines is primarily transferred downward toward the sea floor via eddy IFS [cf. Eq. (15)]. To demonstrate that this holds across the parameter space examined in this study, in Fig. 7a we quantify the contributions of wind stress, eddy IFS, and eddy advection (cf. Fig. 3) to the alongstream momentum balance. We plot these contributions as functions of the wind stress maximum, holding the quadratic drag coefficient fixed at Cd = 2 × 10−3. For each simulation we select a mean streamline that tracks the core of the zonally reentrant flow by taking the median of the time-mean SSH over the entire model domain, e.g., as shown in Fig. 3a. For all of the wind stresses examined here, the upper-layer momentum balance along this contour is primarily between the wind stress and the eddy IFS, although horizontal redistribution of momentum by eddies balances up to 25% of the wind stress. We therefore conclude that (15) holds approximately across our suite of simulations.

Fig. 7.
Fig. 7.

(a) Upper-layer momentum budget along a streamline through the core of the circumpolar flow (see Figs. 3a,e) in experiments with varying wind stress maxima τwmax and fixed quadratic drag coefficient Cd = 2 × 10−3. The “total eddy force” is the sum of the eddy IFS and eddy momentum flux convergence. (b) Reconstructions of the eddy IFS obtained via (17) in which we independently vary the mean flow while holding the spatial distribution of κ fixed and equal to that of the reference simulation, κref (blue points); and varying κ while holding the mean flow u¯ fixed and equal to that of the reference simulation, u¯ref (red points). The quadratic drag coefficient is held fixed and equal to Cd = 2 × 10−3. (c) Reconstruction of the eddy IFS via (17) across our entire suite of simulations with fixed κ and varying mean flow. (d) As in (c), but with fixed mean flow and varying κ.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0154.1

To quantify the contributions of transient eddies versus standing waves to changes in eddy IFS, we use (15) to produce estimates of the eddy IFS resulting from independent perturbations of κ and the mean flow, respectively. For notational clarity we first rewrite (17) as
EIFS=F[κ,u¯],
where EIFS denotes the alongstream-averaged eddy IFS, the functional F is defined by the right-hand side of (17), and u¯ is shorthand for the time-mean flow in both isopycnal layers. We then define reference spatial distributions of κ and u¯ from our reference experiment (see section 2b), which we denote as κref and u¯ref, respectively. Finally, for each of our experiments we define two reconstructions of the eddy IFS,
EIFS|κ=κref(n)=F[κref,u¯(n)],
EIFS|u¯=u¯ref(n)=F[κ(n),u¯ref].
Here the superscript (n) is an arbitrary index to distinguish different experiments, and κ(n) and u¯(n) denote the eddy diffusivity and mean flow diagnosed from the nth experiment. Thus, the reconstructions EIFS|κ=κref(n) provide an estimate of the eddy IFS response that results from varying only the mean flow, while the reconstructions EIFS|u¯=u¯ref(n) provide an estimate of the eddy IFS response that result from varying only the eddy diffusivity.

Figure 7b demonstrates this approach, focusing on the same subset of our simulations as shown in Fig. 7a. Holding the mean flow fixed and varying κ yields a reconstructed eddy IFS that increases with the wind stress, but with a substantially smaller slope than the diagnosed eddy IFS. In particular, for wind stresses larger than the reference value, the reconstructed eddy IFS is almost invariant under increases of the wind stress. This implies that changes in transient eddy diffusivity alone fail to capture the eddy saturation response. In contrast, holding κ fixed and varying the mean flow yields a reconstructed eddy IFS that closely tracks the diagnosed IFS.

Figures 7c and 7d expand the scope of this analysis to include our entire suite of simulations with varying wind stress maximum and quadratic drag coefficient. These reconstructions exhibit the same qualitative pattern as Fig. 7b, despite significant scatter associated with the varying quadratic drag coefficient. To provide a quantitative assessment of these reconstructions, we note that the eddy IFS spans orders of magnitude, and thus we quantify the root-mean-square difference between the logarithms of the diagnosed and reconstructed eddy IFS (the LRMSE). Consistent with our inference based on visual inspection, the LRMSE of EIFS|u¯=u¯ref(n) is substantially higher than that of EIFS|κ=κref(n) (∼0.30 versus ∼0.14). These diagnostics indicate that the eddy saturation response occurs primarily as a result of adjustment of the mean flow, i.e., of the standing meander, rather than of the efficiency of eddy momentum transfer.

4. A theory of standing wave saturation

The diagnostics presented in section 3 indicate that eddy saturation occurs primarily as a result of adjustment of standing meanders in response to changes in zonal wind stress. Motivated by this finding, we now pose a quasigeostrophic standing wave theory of eddy saturation. This theory simultaneously supports the conclusions drawn from our simulations and yields insight into the dynamics of eddy saturation.

a. Theoretical model formulation

Our theoretical model closely follows those derived in several recent studies (Abernathey and Cessi 2014; Constantinou and Young 2017; Bai et al. 2021), which in turn build on earlier work by Davey (1980). Under the assumptions that the flow is quasigeostrophic and slowly varying in the meridional direction, this formulation simplifies conservation of potential vorticity to a one-dimensional wave equation, complemented by a zonal momentum equation that constrains the zonal mean flow. The resulting system can, in principle, be solved analytically to produce explicit predictions of the circumpolar transport and the structures of the standing meanders/waves. Below we discuss the key steps required to derive the model equations, and then discuss our method of solution.

We start with the two-layer quasigeostrophic analog of (2)–(4) (e.g., Pedlosky 1987), neglecting diapycnal fluxes and the hyperviscous stress tensor. Taking a time average, these equations become
∂qk¯∂t+J(ψk¯,qk¯)+∇⋅uk′qk′¯=−δ1,kρ0H1∂τw∂y−δ2,kρ0H2∇×τb¯
where the potential vorticity (PV) in each layer is given by
qk=∇2ψk+βy+(−1)kLk−2(ψ1−ψ2)+δ2,kf0H2ηb.
Here ψk is the geostrophic streamfunction, uk=−∇×ψkz^ is the geostrophic velocity, J(•,•) denotes the Jacobian operator, the deformation radius in each layer is defined as Lk=g′Hk/|f0|, and the reference layer thicknesses are H1 = 1500 m and H2 = 2500 m . Unless otherwise stated, the model parameters and geometry match those given in section 2a. To make the theory analytically tractable, we use the linear formulation (7) of the bottom frictional stress.
The transient eddy PV flux in (24) can be decomposed, following some manipulations, as
∇⋅uk′qk′¯=∇⋅uk′ξk′¯︸vorticity flux divergence+(−1)(k−1)Lk−2J(ψ1′,ψ2′)¯︸isopycnal pressure torque,
where ξk=∇×uk=∇2ψk is the relative vorticity. Following (18), we assume that the isopycnal pressure torque acts down the vertical gradient of the relative vorticity, i.e.,
f02g′J(ψ1′,ψ2′)¯=f02g′κ(ξ1¯−ξ2¯),
where κ is taken to be a constant. We further assume that the horizontal fluxes of relative vorticity are also directed down the mean relative vorticity gradient,
uk′ξk′¯=−ν∇ξ¯k,
where ν is a constant horizontal eddy viscosity. Bai et al. (2021) showed that adding a horizontal eddy viscosity to a barotropic standing wave model removes the indeterminacy of the solution that occurs when dissipation can only occur via bottom friction (see also Constantinou and Young 2017). Substituting (26)–(28) and (7) into (24) yields the following evolution equation for the mean PV,
∂qk¯∂t+J(ψk¯,qk¯)=ν∇2ξk¯+(−1)kLk−2κ(ξ1¯−ξ2¯)−δ1,kρ0H1∂τw∂y−δ2,krbρ0H2ξ2¯.
We now partition the mean streamfunction into a zonal mean component ⟨ψk¯⟩x(y,t) and perturbation component ψ˜k(x,y,t),
ψk¯=⟨ψk¯⟩x+ψ˜k,
and we define the zonal-mean zonal velocity in each layer as Uk=−∂y⟨ψk¯⟩x. We then assume that the time-averaged system is in steady state, i.e., ∂t ≡ 0, and that the flow is slowly varying in y, i.e., ∂y ≪ ∂x, such that all derivatives of ψ˜k and Uk with respect to y can be neglected. After further manipulations, (29) may be rewritten in the form of a linear wave equation,
Uk∂x3ψ˜k+Lk−2(−1)k(U2∂xψ˜1−U1∂xψ˜2)+β∂xψ˜k+δ2,kU2f0H2∂xηb=−δ2,krbH2∂x2ψ˜2+ν∂x4ψ˜k+(−1)kκLk−2∂x2(ψ˜1−ψ˜2).
We derive the corresponding zonal momentum equations by multiplying (29) by Hk, integrating from y′ = 0 to y′ = y and zonally averaging, where y′ is a variable of integration. Assuming that the meridional velocity and wind/frictional stresses are zero at y = 0, this yields
τw=EIFS+SIFS,
EIFS+SIFS=TFS+ρ0rbU2,
where
SIFS=ρ0f02g′⟨ψ˜1∂xψ˜2⟩x,
EIFS=ρ0κ(y)f02g′(U1−U2),
TFS=ρ0f⟨ψ˜2∂xηb⟩x,
are the standing wave interfacial form stress (SIFS), eddy interfacial form stress (EIFS), and topographic form stress (TFS), respectively. Taken together, (31), (32a), and (32b) comprise four equations for the four unknowns: the zonal mean velocities Uk and the perturbation streamfunctions ψ˜k.

Note that we use a distinct eddy diffusivity κ(y) in (33b), from that appearing in (31). The rationale for this is that (32a) and (32b) describe the zonal momentum balance zonally averaged along latitude lines, rather than the momentum balance averaged along mean streamlines. Based on our simulations (see Fig. 3), we therefore expect the SIFS, rather than the EIFS, to balance the wind stress in (32a). In our simulations, the eddy diffusivity obtained by zonally averaging eddy IFS and zonal velocity [cf. Eq. (16)] across a latitude band is much smaller than the diffusivities diagnosed following the method discussed in section 2c. Based on the diagnostics presented in appendix C, we select κ(y) = 80 m2 s−1, whereas based on Fig. 4a, we select κ = 400 m2 s−1. If we were to choose κ(y) to be as large as κ then we would find that EIFS made an O(1) contribution to the theoretical momentum balance (not shown), conflicting with our simulation results (Fig. 3). The eddy viscosity of ν = 2000 m2 s−1 is chosen to be as small as possible while preserving the stability of our numerical solutions, discussed below.

Although in principle analytical progress toward a solution of (31)–(33c) can be achieved via a zonal Fourier transform (Bai et al. 2021), in practice the resulting equations yield little additional physical insight. We therefore solve the equations numerically via MATLAB’s least squares trust-region reflective algorithm. In all cases presented here, the optimized solution yielded a difference between the left- and right-hand sides of (32a) and (32b) of no more than 10−7 N m−2, which is five orders of magnitude smaller than our smallest wind stress.

In Fig. 8a we plot the equivalent sea surface height predicted by our theory, for comparison with the time-mean simulated sea surface height shown in Fig. 3a. In Fig. 8b we directly compare the predicted and diagnosed sea surface height and isopycnal elevation along the midline of the channel, y = Ly/2. Here we have solved (31)–(33c) using the corresponding wind stress at each latitude [cf. Eq. (5)]. We use the same reference wind stress maximum as in section 2a ( τwmax=0.1Nm−2) and set the linear friction velocity to rb = 4 × 10−4 m s−1. All other model parameters are assumed to be independent of latitude. The resulting mean flow resembles a standing meander in the lee of the bathymetric ridge, but exhibits qualitative differences from the mean flow in our simulations. Notably, although the amplitude of the standing wave is similar over the ridge, the theoretical standing wave visibly persists over a longer zonal distance. The theoretical zonal-mean upper-layer flow is also larger, as indicated by the larger meridional sea surface height gradient in Fig. 8 than in Fig. 3. These differences likely arise because, under the assumption of a flow that is slowly varying in y, the theoretical solution is not constrained by the meridional walls. Furthermore, the theoretical solution cannot form “gyres” in the lee of the ridge (Nadeau and Ferrari 2015), and thus the topographic form stress must be established entirely by the standing Rossby wave. The theoretical baroclinic and barotropic transports are approximately 242 and 15 Sv (1 Sv ≡ 106 m3 s−1), respectively. These transports are of the same orders of magnitude as those measured in the ACC (Donohue et al. 2016) but several times higher than the corresponding transports from our simulations (see Fig. 5). We conclude that our theoretical solution qualitatively resembles the circulation of the ACC and produces a zonal transport of the same order of magnitude as observations, but that caution is required in comparing its predictions with the channel model simulations presented in sections 2 and 3.

Fig. 8.
Fig. 8.

(a) Equivalent sea surface height predicted by our quasigeostrophic standing wave theory (see section 4a). Dark gray contours indicate the 3750-, 3500-, and 3250-m isobaths. Note the different color axis range from Fig. 3a. (b) Comparison of the theoretically predicted sea surface height (amplified by a factor of 1000 for visibility) and isopycnal interface elevation along y = Ly/2 with those diagnosed from our reference simulation.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0154.1

b. Regimes of standing wave saturation

We now show that our theory captures the eddy saturation phenomenon and exploit its analytical simplicity to draw some insights into the dynamics of the saturation response. We obtain solutions of (31)–(33c) over the same range of wind stresses and linear friction velocities as discussed in section 2a. To aid our dynamical interpretation, rather than solve the theoretical model equations at all latitudes in our model domain, we instead solve the equations at a single latitude using the meridionally averaged wind stress. For the wind stress profile (5), this corresponds to setting
τw=12τwmax
in Eq. (32a). Thus the resulting solutions should be thought of as an approximation to the meridionally averaged standing wave behavior, although there are substantial qualitative departures of the theoretical solutions from our simulations, as discussed in section 4a. To obtain equivalent zonal transports, we multiply the depth-integrated mean zonal velocity (i.e., H1U1 + H2U2) by Ly = 1600 km.

In Fig. 9 we plot the sensitivity of the total transport, baroclinic transport, barotropic transport, and standing wave kinetic energy to changes in wind stress and friction velocity, analogous to Figs. 5 and 6c. For wind stresses ≳ 0.3 N m−2, the total zonal transport is approximately independent of the wind stress; this contrasts with our simulations, in which the baroclinic transport is independent of the wind stress. In this regime there is a relatively small increase in the barotropic transport with wind stress, accompanied by a compensating decrease in baroclinic transport. We note that the theory overpredicts that barotropic transport only by a factor of ∼2 (cf. Figs. 9b and 5b), whereas it overpredicts the baroclinic transport by a factor of ∼7 (cf. Figs. 9c and 5c). This discrepancy is likely related to the assumption of small meridional gradients in the theory, which eliminates the influence of meridional walls and precludes the formation of gyres in the lee of the ridge, as discussed in section 4a.

Fig. 9.
Fig. 9.

Dependence of the (a) total transport, (b) barotropic transport, (c) baroclinic transport, and (d) standing wave kinetic energy on the wind stress maximum and on the linear friction velocity in our quasigeostrophic theory. Each point corresponds to a solution of Eqs. (31)–(33c). The “equivalent wind stress maximum” defines the wind stress profile via (5), which is then meridionally averaged and used to force the theoretical momentum equation (32a). Note the nonlinear abscissa on all axes, and the nonlinear ordinate in (d).

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0154.1

The theoretically predicted total and baroclinic transports increase with the linear drag coefficient, with the total transport increasing from ∼285 Sv for rb = 2 × 10−4 m s−1 to ∼350 Sv for rb = 1 × 10−3 m s−1. Thus the theory appears to capture the “frictional control” of the zonal transport exhibited in our simulations and previous work (Marshall et al. 2017), despite mesoscale eddies being entirely parameterized. In the eddy-saturated regime the standing wave kinetic energy increases approximately linearly with the wind stress, consistent with our simulations (see Fig. 6d). For wind stresses ≲ 0.03 N m−2, the total and barotropic transports both increase linearly with the wind stress, while the standing wave kinetic energy rapidly decays to zero as τw → 0, approximately scaling as SKE∼τw7.

To aid in the interpretation of these results, in Fig. 10 we plot the SIFS and TFS terms from (32a) to (32b), normalized by the wind stress. Figure 10a shows that in all of these solutions, TFS is almost entirely responsible for removing the wind-input momentum at the sea floor; with bottom friction [rightmost term in (32b)] playing a negligible role (cf. Munk and Palmén 1951). Note that in Fig. 10, the contribution of bottom friction to the momentum balance is given by one minus the “total” curve, i.e., [(1 − TFS)/τw]. This should not be confused with the contribution of bottom friction to establishing the TFS, which is plotted in yellow and discussed in the following subsection. In the saturated regime ( τwmax≳0.03Nm−2) the wind stress is primarily balanced by SIFS; this is to be expected based on our model simulations, because our theoretical momentum balance is averaged along latitude lines (see Fig. 3). Thus, provided that the standing waves are primarily responsible for the downward transfer of wind-input momentum, i.e.,
τw≈SIFS≈TFS,
the theoretical model solutions exhibit saturation of the zonal transport.
Fig. 10.
Fig. 10.

Black curves show the magnitudes of (a) topographic form stress (TFS) and (b) standing wave interfacial form stress (SIFS) to the zonal momentum balance in our quasigeostrophic theory, normalized by the zonal wind stress [Eqs. (32a) and (32b)]. Colored curves show the decomposition of the TFS into contributions from viscous, diffusive, and frictional energy dissipation [Eqs. (37) and (38)]. We hold the linear friction velocity constant and equal to rb = 4 × 10−4 m s−1 for clarity. The “equivalent wind stress maximum” defines the wind stress profile via (5), which is then meridionally averaged and used to force the theoretical momentum equation (32a).

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0154.1

Figures 10a and 10b show that both saturation and the approximate momentum balance (35) fail for small wind stresses ( τwmax≲0.04Nm−2) because the wind stress becomes balanced by EIFS instead of SIFS. The approximate wind stress required for this to occur can be estimated by computing the EIFS associated with the baroclinic shear in the saturated regime, i.e., U1−U2∼0.1ms−1, which yields EIFS=EIFSsat≈0.008Nm2. Figure 9b shows that this EIFS is small compared with the SIFS in the saturated regime, but for a sufficiently small wind stress we will reach a situation where τw ∼ EIFSsat. When this occurs, SIFS can no longer primarily balance the wind stress and/or the baroclinic shear must decrease. To obtain an estimate for the wind stress at which this transition occurs, we use (32a) to predict that the EIFS will balance 50% of the wind stress when (1/2)τw≈EIFSsat≈0.016Nm−2, i.e., for τwmax≈0.032Nm−2. This prediction agrees approximately with Fig. 9b. In this regime the transport is not saturated; instead, there is a linear dependence of the transport on wind stress (Fig. 9a). This occurs because we have held the meridional eddy diffusivity κ(y) constant. Assuming that EIFS ≈ τw in the limit of small wind stress, we expect the baroclinic shear to scale as
U1−U2→g′f02ρ0κ(y)τwasτw→0.
Figure 9 shows that this scaling accurately captures the total and baroclinic transports predicted by our theory in the limit of weak wind stress.

c. Dynamics of standing wave saturation

The above analysis demonstrates that saturation of the zonal transport can indeed occur entirely via adjustments of standing waves, as suggested by our diagnostics in section 3. However, even from this heavily simplified theory one cannot trivially discern what sets the zonal transport, nor mechanistically how the standing waves adjust such that the SIFS compensates changes in the wind stress. To yield further insight into this process, we use the standing wave energy budget to relate the SIFS and TFS to energy dissipation, building on earlier work by Abernathey and Cessi (2014). Multiplying the upper-layer wave equation [Eq. (31) with k = 1] by ∂xψ˜1 and zonally averaging, we obtain, after some manipulations,
SIFS=ρ0νH1U1⟨(∂x2ψ˜1)2⟩x︸viscous+ρ0κf2g′U1[⟨(∂xψ˜1)2⟩x−⟨∂xψ˜2∂xψ˜1⟩x]︸diffusive.
This equation relates the SIFS directly to dissipative processes associated with the eddy viscosity and the eddy diffusivity. Thus, the SIFS can only occur in the presence of (parameterized) transient eddy PV fluxes (cf. Treguier and McWilliams 1990; Straub 1993). Similarly, we multiply the lower-layer wave equation [Eq. (31) with k = 2] by ∂xψ˜2 and zonally average to obtain, after further manipulations,
TFS=rbU2⟨(∂xψ˜2)2⟩x︸frictional+ν[H2U2⟨(∂x2ψ˜2)2⟩x+H1U1⟨(∂x2ψ˜1)2⟩x]︸viscous+κf2g′[1U2⟨(∂xψ˜2)2⟩x+1U1⟨(∂xψ˜1)2⟩x−(1U2+1U1)⟨∂xψ˜2∂xψ˜1⟩x]︸diffusive.
This equation relates the TFS directly to dissipative processes associated with the linear friction, the eddy viscosity, and the eddy diffusivity. Physically, the relationships (37) and (38) arise because the SIFS and TFS extract energy from the mean flow and convert it to standing wave energy. This standing wave energy source must be balanced by corresponding sinks, which are provided either by frictional energy dissipation or via conversion to transient eddy energy.
In Fig. 10 we quantify the contributions of bottom friction, eddy viscosity, and eddy diffusivity to the TFS and SIFS, computed via (37) and (38). In the eddy-saturated regime ( τwmax≳0.04Nm−2), the TFS is primarily (∼70%) supported by viscous dissipation, with diffusive and frictional dissipation making relatively minor contributions. Consistent with this, Bai et al. (2021) found that including a horizontal eddy viscosity substantially enhanced the TFS in a similar, barotropic standing wave theory, and was necessary to produce qualitative agreement between the theory and numerical simulations. Physically, this implies that the production of standing wave energy associated with topographic form stress is primarily removed via (parameterized) eddy viscous dissipation. The SIFS is consistently supported in almost equal parts by viscous and diffusive dissipation. Given that (35) holds approximately, and given that the upper-layer zonal velocity is approximately independent of the wind stress (because U1 ≫ U2 and U1 − U2 is insensitive to the wind stress), these diagnostics imply that the rate of standing wave energy dissipation must increase linearly with the wind stress. Noting that the second “diffusive” term in (37) is empirically negligible compared with the first (not shown), this implies that the SKE scales as
SKE∼τwmax.
This linear relationship between the standing wave kinetic energy and the wind stress is consistent both with our theoretical solutions in the eddy-saturated regime (Fig. 9d) and with our channel model simulations (Fig. 6c).

5. Discussion and conclusions

This work was motivated by the divergent previous explanations of the eddy saturation phenomenon, i.e., the approximate independence of the ACC transport to changes in the mean zonal winds. Specifically, various previous studies have either posited that eddy saturation occurs as a result of changes in the efficiency of eddy transfer (Marshall and Radko 2003; Meredith et al. 2012; Marshall et al. 2017; Mak et al. 2018), or argued that it occurs as a result of flexing of the ACC’s standing meanders (Thompson and Naveira Garabato 2014; Nadeau and Ferrari 2015; Constantinou and Hogg 2019). As is evident from the alongstream-averaged momentum balance utilized in this study, eddies play a central role in the downward transfer of wind-input momentum (see Figs. 1 and 3), or equivalently the southward transport of heat (Vallis 2006). This is consistent with previous studies showing that eddy heat fluxes across mean streamlines are approximately equal to the combined heat flux due to standing plus transient eddies across lines of constant latitude (Marshall et al. 1993; Abernathey and Cessi 2014). The focus of this study is on the mechanisms via which this downward transfer of momentum along mean streamlines by transient eddies responds to changes in the surface wind stress. As shown in section 2b, changes in this transient eddy momentum transfer result from a combination of 1) changes in the efficiency with which the eddy field transfers momentum down the vertical gradient in the mean flow and 2) a restructuring of the mean flow that allows the eddies to transfer more momentum downward in a circumpolar integral. The former corresponds to an increase in the eddy diffusivity κ as discussed in section 2c. The latter may correspond to a combination of lengthening of the standing meanders (Thompson and Naveira Garabato 2014) and spinup of gyres abutting the circumpolar flow (Nadeau and Ferrari 2015), which we collectively refer to as an adjustment of the “standing waves.”

In this study we sought to distinguish between these previously proposed mechanisms of eddy saturation via analysis of hundreds of simulations using an eddy-resolving, two-layer channel model (section 2a). These simulations exhibit saturation of the zonal transport (Fig. 5), with changes in wind-input momentum being accommodated by changes in the eddy IFS along mean streamlines (Fig. 7). To isolate the mechanism of eddy saturation, we therefore decomposed the alongstream-averaged eddy IFS into multiplicative contributions from the eddy diffusivity and the structure of the mean flow [Eq. (17)]. We then separately diagnosed κ and the mean flow from each of our model simulations, allowing us to create approximate reconstructions of the alongstream-averaged eddy IFS (Fig. 3). Via reconstructions based on independent variations of κ and the mean flow, we showed that varying the mean flow alone approximately reconstructs the diagnosed eddy IFS across our suite of experiments (Fig. 7). In contrast, varying κ alone yields a much less accurate reconstruction of the diagnosed eddy IFS; indeed, for a realistic range of wind stresses ( ≳0.1Nm−2) the eddy IFS is approximately insensitive to variations in κ. We therefore concluded that, at least in this suite of model simulations, eddy saturation is primarily attributable to adjustment of standing waves, rather than transient eddies.

Motivated by this finding, we posed a quasigeostrophic theory of our idealized ACC in which κ is held constant, and thus saturation can only occur via adjustment of standing waves (section 4a). This theory is distinguished from previous studies (e.g., Abernathey and Cessi 2014) primarily via the separate treatment of eddy vertical momentum transfer, proportional to κ, and lateral eddy momentum fluxes, proportional to the eddy viscosity ν. However, the assumptions underpinning the theory produce qualitative differences from our channel model simulations (cf. Figs. 3a and 8). Thus, while the theory serves to demonstrate that eddy saturation can occur purely via adjustment of standing waves (Fig. 9), it is difficult to directly compare its predictions with the diagnostics from our simulations. Furthermore, one might hope that the relative simplicity of our theory might facilitate the derivation of further simplified scalings, for example to elucidate the parameter dependence of the zonal transport. Though our efforts have failed to acquire such insights thus far, further study of the model equations may prove fruitful. The theory nonetheless offers transferable insights into the dynamics of standing wave-induced saturation; for example, it predicts that the standing wave kinetic energy must increase linearly with the wind stress in order to produce a saturated zonal transport (Figs. 6c and 9d and section 4c).

A caveat of our overall approach is that the posing of our channel model simulations and quasigeostrophic theory is highly idealized, which raises questions regarding the transferability of our findings to more realistic model configurations and to nature. Future studies could test our conclusions by extending the eddy IFS decomposition (section 2b) and mean flow eddy diffusivity perturbation analysis (section 3) to more realistic model configurations. A specific caveat of this perturbation analysis is that it simplistically perturbs the mean flow or κ at each horizontal point in space (see Fig. 7), which could skew the resulting eddy IFS calculations in some situations. For example, if we applied a wind perturbation of sufficient magnitude to substantially shift the path of the mean flow over the topographic ridge, then the mean flow streamlines in that simulation may no longer traverse the region of maximum EKE and κ in the reference simulation (Fig. 4), and this may be expected to bias the calculation toward a smaller eddy IFS. This may explain why the reconstructed eddy IFS with fixed κ and varying mean flow underpredicts the diagnosed eddy IFS in the experiments with the largest wind stresses (Figs. 7b,c). It may be possible to circumvent such issues via more dynamically based perturbations; for example, one might be able to empirically derive a relationship between κ and the mean flow ( u¯) in each experiment, i.e., κ≈κ(u¯), and then attempt to reconstruct the eddy IFS by independently perturbing u¯ and the functional relationship κ(u¯) across all of the experiments.

It also remains to be understood why this and previous model studies have exhibited apparently contradictory mechanisms of eddy saturation. For example, previous modeling studies that have argued for saturation by transient eddy adjustment (Meredith et al. 2012; Munday et al. 2013; Marshall et al. 2017; Mak et al. 2018) have reported that the EKE scales linearly with the wind stress, whereas in our simulations it scales sublinearly (Fig. 6). A common feature of these previous studies is that they use a linear formulation of the bottom friction; this, in part, motivated us to conduct a parallel suite of experiments with linear bottom friction (see appendix B). For weak winds stresses ( ≲0.1Nm−2), independently varying κ more accurately reproduces the changes in the diagnosed alongstream eddy IFS than does independently varying the mean flow, indicating that the saturation may be a result of transient eddy adjustments. However, for a realistic range of wind stresses ( ≳0.1Nm−2) the transport saturation appears to be better explained by variations in the mean flow. These experiments suggest that the formulation of bottom friction may be an essential element of eddy saturation. Further work is required to assess whether this does indeed explain the differences in saturation behavior proposed in previous modeling studies, and perhaps to assess which formulation of bottom friction is most accurate at the scales of motion represented by the models (Arbic and Scott 2008).

In addition to partly reconciling previous explanations of the eddy saturation phenomenon, these findings also have implications for coarsely resolved ocean/climate model simulations that must parameterize the effects of mesoscale eddies (Gent and McWilliams 1990; Gent et al. 1995). If eddy saturation is primarily the result of transient eddy adjustments, then this motivates the use of parameterization schemes that allow κ to adapt to changes in surface wind stress (Marshall and Radko 2003; Mak et al. 2018, 2022). In contrast, if saturation is primarily the result of standing wave adjustments then much simpler parameterization schemes may suffice, provided that the standing waves are resolved. Consistent with the latter, Farneti et al. (2015) found that an ensemble of interannually forced, coarse-resolution global ocean simulations consistently exhibited independence of the baroclinic transport from the wind stress. Kong and Jansen (2021) compared simulations with resolved and parameterized eddies in an idealized Southern Ocean sector model. They found that even a coarse simulation with a constant κ had a similar wind forcing response to that of an eddy-resolving simulation, consistent with our idealized quasigeostrophic theory. These findings suggest that the key to representing eddy saturation in coarse ocean/climate models is to resolve or parameterize the standing wave response, rather than the transient eddy response, to wind changes.

Acknowledgments.

This material is based in part upon work supported by the National Science Foundation under Grants OCE-1751386 and OPP-2023244, and by the National Aeronautics and Space Administration ROSES Physical Oceanography program under Grant 80NSSC19K1192. This work used the Extreme Science and Engineering Discovery Environment (XSEDE; Towns et al. 2014), which is supported by National Science Foundation Grant ACI-1548562. The authors thank two anonymous reviewers for comments that improved the submitted manuscript and editor Paola Cessi for handling the peer-review process.

Data availability statement.

The AWSIM model code used in this study can be obtained from https://github.com/andystew7583/AWSIM. The codes used to configure and analyze our model simulations are available via https://doi.org/10.5281/zenodo.6850435.

APPENDIX A

Momentum Balance in Latitudinal and Streamline Coordinates

In this appendix we provide complete expressions for the isopycnal momentum balances discussed in section 2b and presented in Fig. 3. In all of these expressions the hyperviscous stress tensor and the diapycnal (restoring) velocity, all of which contribute negligibly to the zonally or alongstream-averaged momentum balance. We neglect the Coriolis term because there is no overturning circulation in our model simulations, and thus this term is also negligible.

a. Zonally averaged momentum balance

The depth-integrated, zonally averaged zonal momentum balance is given by
∂∂t∑k⟨hk¯uk^⟩x︸tendency≈−∑k∂∂y⟨hk¯υk^uk^⟩x︸mean advection−∑k∂∂y⟨hk¯υk†uk†^⟩x︸eddy advection−∑k⟨hk¯∂xMk¯⟩x︸topographic form stress+τwρ0︸wind stress−1ρ0⟨τb(x)¯t⟩x︸bottom friction,
and the upper-layer zonally averaged zonal momentum balance is given by
∂∂t⟨h1¯u1^⟩x︸tendency≈−∂∂y⟨h1¯υ^1u^1⟩x︸mean advection−∂∂y⟨h1¯υ1†u1†^⟩x︸eddy advection−⟨h1¯∂xM1¯⟩x︸mean IFS−⟨h1′∂xM1′¯⟩x︸eddy IFS+τwρ0︸wind stress

b. Alongstream-averaged momentum balance

To present the alongstream-averaged momentum balance, we first define the alongstream-averaging operator for an arbitrary vector a,
⟨a⟩M=1Lx∮ζ¯=ζ0a⋅ds.
The depth-integrated, alongstream-averaged momentum balance is then
∂∂t∑k⟨hk¯uk^⟩M︸tendency≈−∑k⟨∇⋅(hk¯u^ku^k)⟩M︸mean advection−∑k⟨∇⋅(hk¯uk†uk†^)⟩M︸eddy advection−⟨hk¯∇Mk¯⟩M︸topographic form stress+⟨τwρ0x^⟩M︸wind stress−⟨τbρ0⟩M︸bottom friction,
and the upper-layer alongstream-averaged momentum balance is
∂∂t⟨h1¯u1^⟩M︸tendency≈−⟨∇⋅(h1¯u1^u1^)⟩M︸mean advection−⟨∇⋅(h1¯u1†u1†^)⟩M︸eddy advection−⟨h1′∇M1′¯⟩M︸eddy IFS+⟨τwρ0x^⟩M︸wind stress.

APPENDIX B

Simulations with Linear Bottom Friction

In this appendix we reproduce the results presented in section 3 using diagnostics from our suite of experiments with a linear, rather than quadratic, formulation of the bottom friction.

Figure B1 shows the sensitivity of the total, barotropic and baroclinic transports to variations in the wind stress and linear friction velocity. These sensitivities closely resemble those shown in Fig. 5, albeit with a somewhat larger range of transports in response to the range of friction velocities explored here.

Fig. B1.
Fig. B1.

As in Fig. 5, but with diagnostics drawn from our suite of channel model simulations with a linear formulation of the bottom friction.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0154.1

Figure B2 shows the sensitivity of the transient eddy diffusivity (averaged over the storm track region), the domain-averaged eddy kinetic energy, and the domain-averaged standing wave kinetic energy to variations in the wind stress and linear friction velocity. These sensitivities also qualitatively resemble their counterparts in Fig. 6. However, in these simulations κ and EKE increase with wind stress slightly faster than in the simulations with quadratic friction: least squares fits over all of our simulations yield κ ∼ τ0.7 and EKE ∼ τ0.8.

Fig. B2.
Fig. B2.

As in Fig. 6, but with diagnostics drawn from our suite of channel model simulations with a linear formulation of the bottom friction.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0154.1

Figure B3 shows our reconstruction of the eddy IFS resulting from independent variations of the eddy diffusivity and the mean flow across our suite of simulations. Similar to our simulations with quadratic bottom friction, the eddy IFS consistently dominates the alongstream momentum balance. However, the results of our eddy IFS reconstructions differ substantially from our experiments with quadratic bottom friction, with distinct behaviors for wind stresses larger than the reference case versus smaller than the reference case. For larger wind stresses, holding κ fixed and varying the mean flow provides a more accurate reconstruction of the eddy IFS, although the accuracy is lower than found in the quadratic friction experiments (see Fig. 7). For smaller wind stresses, the result is reversed: holding the mean flow fixed and varying κ provides a more accurate reconstruction of the eddy IFS. In contrast, holding κ fixed and varying the mean flow leads to the eddy IFS decreasing too rapidly with the wind stress.

Fig. B3.
Fig. B3.

As in Fig. 7, but with diagnostics drawn from our suite of channel model simulations with a linear formulation of the bottom friction. In this figure we have added dotted lines to indicate our reference experiment.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0154.1

APPENDIX C

Eddy Interfacial Form Stress Along Latitude Lines

Our quasigeostrophic theory of eddy saturation (see section 4) employs a different eddy diffusivity κ(y) to parameterize transient eddy transfer of momentum down the vertical gradient of the zonal-mean flow. Here we justify this choice via analysis of the zonally integrated zonal momentum balance. Figure C1a shows the domain-averaged zonal wind stress, zonal component of the mean IFS and zonal component of the eddy IFS for experiments with varying wind stress maxima and constant quadratic drag coefficient Cd = 2 × 10−3. Consistent with Fig. 3, the downward transfer of momentum is consistently dominated by the mean IFS, with the eddy IFS reaching at most 20% of the wind stress for very weak wind stresses. We estimate κ(y) using the domain-averaged zonal component of the eddy IFS and the domain-averaged zonal mean flow via
κ(y)=g′ρ0f2∫0Lydy⟨pη′∂xη′¯⟩x∫0Lydy⟨u1^−u2^⟩x.
Figure C1 shows that the diagnosed κ(y) across our suite of experiments with varying wind stresses and quadratic drag coefficients. For wind stress maxima smaller than ∼0.17 N m−2, κ(y) typically lies between 0 and 100 m2 s−1, depending on the drag coefficient. For larger wind stress maxima the diagnosed κ(y) varies widely with the drag coefficient, with various experiments seemingly simulating upgradient momentum transfer by transient eddies in a domain average.
Fig. C1.
Fig. C1.

Diagnostics of the zonally integrated eddy interfacial form stress (IFS) in our simulations with quadratic bottom friction. (a) Domain-integrated zonal wind stress, zonal component of the mean IFS, and zonal component of the eddy IFS in simulations with varying wind stress and fixed quadratic drag coefficient Cd = 2 × 10−3. (b) Meridional eddy diffusivity (see appendix C) as a function of the wind stress and the quadratic drag coefficient.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0154.1

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