## 1. Introduction

The uniqueness of ocean dynamics within the Antarctic Circumpolar Current (ACC) has long been recognized (Munk and Palmén 1951; Rhines and Holland 1979; Johnson and Bryden 1989). The ACC balances wind stress forcing in a different manner than subtropical gyre systems that develop dissipative western boundary currents along meridionally oriented obstructions. Overall, the general paucity of the land–ocean boundaries in the vicinity of the ACC prevents lateral dissipation from playing a leading role. Rather, both scaling (Munk and Palmén 1951) and observational (Johnson and Bryden 1989; Phillips and Rintoul 2000) analyses suggest that the eddy-induced, vertical transport of momentum is of central importance in balancing wind stress forcing within the ACC.

The importance of ocean eddies in the vertical transmission of momentum required to balance wind stress forcing has been considered from the outset. Even with minimal observational data, Munk and Palmén (1951) concluded that the viscous transfer of momentum based on plausible estimates of viscosity and vertical shear is insufficiently small to transmit the wind stress from the ocean surface to the ocean bottom. Rather, Munk and Palmén (1951, p. 54) suggest that “each layer induces, by turbulent interchange of momentum, motion in the layer beneath, and in this manner the wind stress is transmitted to the sea bottom.” This physical phenomenon is now typically referred to as interfacial form stress.

Since mesoscale eddies are very well approximated by geostrophic balance, the vertical transport of zonal momentum is equivalent to a meridional transport of buoyancy (Johnson and Bryden 1989). As a result, the importance of mesoscale eddies can be measured by either their vertical flux of eastward momentum or their meridional flux of buoyancy (McWilliams et al. 1978; Johnson and Bryden 1989; Phillips and Rintoul 2000; Marshall and Radko 2003; Howard et al. 2015). While both approaches offer insights into the role of eddies in the ACC, the two frameworks differ markedly in how we interpret and explain the role of mesoscale eddies in the ACC. The buoyancy flux framework leads to a perspective by which the Ekman layer–driven overturning is balanced by an eddy-induced overturning. This system is well described in Fig. 2 of Marshall and Radko (2003). But since the wind stress in and of itself is incapable of producing an overturning circulation, the buoyancy flux framework results in an explanation where two large overturning circulations exactly balance. So while geostrophic balance allows us to equate a wind stress forcing with an eddy-induced lateral flux of buoyancy, it is not clear that this leads to the most concise description of the underlying fluid dynamics.

Instead, we attempt below to further understand the influence of mesoscale eddies in circumpolar currents by diagnosing the balance of forces, that is, force equals mass times acceleration (*F* = *ma*), in the zonal direction of a circumpolar channel. Previously, such an analysis has been employed for idealized circumpolar currents (McWilliams et al. 1978) as well as the ACC (Mazloff et al. 2013), using both observational data (Johnson and Bryden 1989) and eddying ocean simulations (Gille 1997). Our goal is to contribute to this well-explored approach to understanding ocean dynamics in the ACC in two ways. Our first goal is to develop an analysis method that measures the role of eddies in the climate of circumpolar currents based on the exact mathematical framework developed in de Szoeke and Bennett (1993), Young (2012), and Maddison and Marshall (2013). Our second goal is to conduct the analysis with sufficient vertical resolution so that we can obtain an understanding of the force balance in different parts of the ocean water column.

In section 2, we review the three-dimensional, thickness-weighted averaged (TWA) equations that allow for the creation of exact ensemble mean-field equations. The idealized configuration used to study the force balance in an idealized circumpolar channel is described in section 3. The analysis of this system is presented in section 4, and conclusions are made in section 5.

## 2. Diagnosis of the TWA system

While the numerical model used to simulate the strongly eddying circumpolar current uses a standard Eulerian representation of the Boussinesq equations, the analysis of system dynamics is conducted entirely in buoyancy space. In particular, the goal is to diagnose the balance of forces in the buoyancy space representation of the TWA zonal momentum equation.

**e**

_{1}and

**e**

_{2}are orthogonal zonal and meridional vectors in buoyancy space, respectively. Buoyancy is defined by

*ρ*=

*ρ*

_{0}(1 −

*g*

^{−1}

*b*), where

*ρ*,

*ρ*

_{0}, and

*g*are the potential density, reference density, and gravity, respectively. Derivatives are indicated with a coordinate label subscript. The

*f*is the Coriolis parameter.

Equation (5) is a statement for conservation of buoyancy

**i**,

**j**,

**k**denoting standard Cartesian unit vectors [see Eq. (53) from Young (2012)].

One subtlety in the analysis of the TWA system is the outcropping problem. As described by Young (2012), imagine an observer positioned at a fixed *z* position *z* position *ζ* of this observer will naturally change as the observer’s buoyancy coordinate moves up and down in the water column. So long as the buoyancy value of the observer is contained in the water column for all time, the ensemble-mean *z* positon of the coordinate is defined without ambiguity. However, the observer’s buoyancy value may not exist in the fluid at their *z* position of the ocean surface when their buoyancy value is lighter than the surface buoyancy value. Therefore, we assume that all buoyancy coordinates lighter than those at the ocean surface are stacked on the surface. Data samples that record *ζ* as residing at the ocean surface are just as valid in the estimate of the ensemble mean as data samples of *ζ* from the ocean interior [for more detail, see appendix A, Eqs. (A1)–(A3)]. Our diagnosis of the TWA system follows the Lorenz convention for outcropped buoyancy coordinates and differs from that used by Mazloff et al. (2013), where outcropped buoyancy coordinates are excluded from the averaging procedure.

*x*direction; the system is zonally symmetric. We denote the zonal-averaging operator with

The

## 3. Modeling system and simulation configuration

### a. Modeling system

The ocean component of the Model for Prediction Across Scales (MPAS-Ocean; Ringler et al. 2013) is used as the modeling system to produce the simulation described below. MPAS-Ocean is built on a mimetic, finite-volume discretization in the horizontal (Thuburn et al. 2009; Ringler et al. 2010) with an arbitrary Lagrangian–Eulerian (ALE) vertical coordinate (Petersen et al. 2015). The horizontal discretization possesses discrete analogs of Kelvin’s circulation theorem and conservation of mechanical energy, making the model well suited for the simulation of highly rotating flows. MPAS-Ocean solves the Boussinesq equations where scalars are expressed in flux form and are advected with a monotone transport algorithm (Skamarock and Gassmann 2011). Thus, the tracer advection algorithm locally conserves volume-weighted tracer concentration and is bounds preserving.

A full listing of configuration parameters is provided in Table 1. The model is configured with 5-km resolution in the horizontal and 100 vertical layers. Since the horizontal domain is planar, we tessellate the 1000 km × 2000 km region with 92 000 regular hexagons arranged on a 200 × 460 grid. The ALE vertical coordinate is specified to mimic the traditional *z*-star coordinate (Adcroft and Campin 2004) with the layer thickness ranging from 0.63 m at the surface to 92.1 m at the bottom where the maximum ocean depth is 2500 m. Just over half of the 100 layers are contained in the top 250 m of the fluid.

Parameters used in simulation.

The cascade of enstrophy to the grid scale is removed from the system with a variant of the biharmonic Laplacian operator where the hyperviscosity on the rotational and divergent parts of the flow are 7.8 × 10^{8} m^{4} s^{−1} and 7.8 × 10^{9} m^{4} s^{−1}, respectively. This results in the simulation producing enstrophy and energy power spectra with slopes of approximately −1 and −3, respectively, extending down to wavelengths of 20 km or about four grid lengths.

The surface boundary layer is modeled using the K-profile parameterization (KPP) from Large et al. (1994). The background diffusivity and viscosity are set to 5.0 × 10^{−6} m^{2} s^{−1} and 1.0 × 10^{−4} m^{2} s^{−1}, respectively. The bottom boundary layer stress is parameterized with quadratic bottom drag using a coefficient of 3.0 × 10^{−3}.

### b. Simulation configuration

The configuration for the zonal idealized Southern Ocean (ZISO) follows closely that used by Abernathey et al. (2011), Stewart and Thompson (2013), and Saenz et al. (2015). The test case is zonally uniform with respect to all boundary conditions and forcing. The 1000 km spanned in the zonal direction *L*_{x} is periodic, and the 2000 km spanned in the meridional direction *L*_{y} is bounded by walls with a no-slip boundary condition.

*h*(

*y*) is specified aswhere

*H*= 2500 m is the maximum depth,

*H*

_{s}= 500 m is the shelf depth,

*Y*

_{s}= 250 km is the center

*y*position of the shelf, and

*W*

_{s}= 500 km is the center

*y*position of the shelf break.

*τ*

_{0}= 0.2 N m

^{−2},

*L*

_{s}= 800 km,

*W*

_{sf}= 600 km, and

*τ*

_{f}= −0.05 N m

^{−2}. The flux of temperature

*T*

_{f}across the ocean surface is computed as

*T*

_{f}= −

*p*(

*T*−

*T*

_{r}), where

*p*is the piston velocity with a value of 1.0 × 10

^{−5}m s

^{−1}andwith constants

*T*

_{m}= 3.0°C,

*T*

_{a}= 1.0°C, and

*T*

_{b}= 2.0°C.

*T*

_{b}of the form

*T*

_{b}= −(

*T*−

*T*

_{i})/

*τ*

_{i}, where

*T*

_{i}is the interior restoring temperature and

*τ*

_{i}is the interior restoring time scale. Along the southern boundary on the shelf for

*y*= 0 km interior restoring is used to mimic the production of deep bottom water, and interior restoring of temperature along the northern boundary is used to transform northward-flowing deep bottom water into southward-flowing middepth water for

*y*= 2000 km, namely,with an interior restoring time scale ofwhere

*z*

_{e}= 1000 m,

*L*

_{e}= 80 km, and

*L*

_{y}= 2000 km.

*ρ*

_{ref}= 1025.0 kg m

^{−3},

*α*= 0.255 kg m

^{−3}°C

^{−1}, and

*T*

_{ref}= 19.0°C.

The simulation is started from rest with a uniform temperature of 3.0°C. The model is integrated for 100 yr at 20-km resolution and then interpolated to the 5-km resolution mesh and integrated for another 35 yr. The analysis is conducted on the last 20 yr of the simulation.

## 4. Results

Figure 1 shows a snapshot of relative vorticity at a level of *z* = −100 m with a color range spanning −6.0 × 10^{−5} to +6.0 × 10^{−5} s^{−1}. Relatively large eddies fill the portion of the domain forced by westerly winds for 1000 < *y* < 1800 km. Within this region the Rossby radius of deformation (RRD) of the first baroclinic mode is approximately 20 km, so eddies in this portion of the domain are well resolved by 5-km grid spacing. Eddy activity on the shelf is also apparent but at significantly smaller spatial scales. On the shelf the RRD is approximately 5 km, so eddy activity is only marginally resolved in this region. These two regions of eddy activity are separated by a quiescent region between 400 < *y* < 600 km that is associated with eastward- and westward-alternating surface front currents (SFCs) that are qualitatively similar to the Antarctic Slope Front.

The climatological state is produced by averaging over the last 20 yr of the simulation. In addition, all data are averaged zonally. Figure 2 shows the climatological surface heat flux *Q*, which results from the linear surface restoring. The amplitude of the surface heat flux ranges around approximately ±10 W m^{−2} in a sinusoidal structure with net cooling/warming southward/northward of *y* = 1200 km. For reference, Fig. 2 also shows the zonal wind stress along the second *y* axis.

### a. Eulerian-averaged ZISO climate

Figure 3 shows the zonally averaged zonal velocity (colors) and zonally averaged temperature (contours) produced through Eulerian averaging (i.e., data averaged at fixed *x*, *y*, and *z* coordinates). The climate is similar to that produced in Stewart and Thompson (2013). The simulation is dominated by what we will refer to as the idealized circumpolar current (ICC) that spans approximately one-half of the meridional extent of the domain. An eastward jet with a zonal velocity of approximately 0.35 m s^{−1} is present at the surface and is located directly beneath the maximum in wind stress forcing near *y* = 1400 km. At the core of the ICC, the zonal velocity has vertical shear of approximately 8.0 × 10^{−5} s^{−1}, resulting in a change in zonal velocity of 0.2 m s^{−1} from top to bottom that is in thermal wind balance. The deep-water formation at the southern boundary of the shelf produces density currents that flow north along the ocean bottom. The northward density currents, along with geostrophy, produce westward zonal flow along the shelf break with a velocity of −0.15 m s^{−1}. Multiple SFCs across the shelf break are indicated by the eastward- and westward-alternating surface velocity. These fronts are qualitatively similar to the Antarctic Slope Front.

### b. Thickness-weighted averaged ZISO climate

The remainder of the results are presented and discussed in buoyancy coordinates using the thickness-weighted averaging approach discussed in section 2. Since the simulations use a linear equation of state with constant salinity, surfaces of constant temperature are also surfaces of constant buoyancy. These surfaces are depicted as contours in Fig. 3. As described in appendix A, at intervals of 3 days during the course of the last 20 yr of the simulation, the model state is interpolated onto 100 predefined buoyancy surfaces that entirely span the buoyancy space of the simulation.

The TWA time- and zonal-mean zonal velocity is shown in Fig. 4 in colors. The range of buoyancy surfaces that exist in any given fluid column in the simulation varies in time. The black contours in Fig. 4 depict the fraction of time a given buoyancy surface exists, that is, the probability of existence *z* position of a buoyancy coordinate *y* = 1400 km, nearly a third of the buoyancy values ever occupied by the fluid have values of

The TWA meridional flow is shown in Fig. 5 as colors and, in addition, the zero contour of the TWA meridional velocity is shown as a thick gray line. Also shown in Fig. 5 is the probability at which a given buoyancy coordinate is at the surface. The probability at surface value ^{1} Across the entire meridional extent of the domain, the TWA meridional velocity is toward the south in the lightest buoyancy classes visited by the fluid at each ^{−2}, which corresponds to depths between

### c. Zonal force balance of the ICC

With this summary of the zonal- and time-mean climate of the TWA system, we turn our attention to the TWA zonal momentum equation and the balance of forces within the ICC. Figure 6 shows the terms of the zonal- and time-mean TWA zonal momentum equation after averaging meridionally over the box shown in Fig. 5. This box is 200 km wide and is centered on the maximum wind stress forcing at *y* = 1400 km. The five terms in the zonally averaged TWA zonal momentum equation (17) are shown in Fig. 6, along with the residual error. The mean depths of

The surface layer occupies the range ^{−2}, the acceleration due to wind stress continues to increase because *τ*, this leads to a progressively larger wind stress acceleration as the buoyancy classes get lighter.

Within the upper portion of the surface layer, at mean depths between 0 and 60 m, meridional advection of thickness-weighted potential vorticity (MAPV; Fig. 6, blue line) acts to accelerate

In the upper portion of the surface layer, the wind stress and MAPV are both acting to accelerate ^{−1} day^{−1}. Only at the very bottom of the surface layer does the eddy-induced force act in concert with the wind stress to balance the MAPV. The eddy force also has a zero-value node at the bottom of the surface layer.

The upwelling zone is the portion of the buoyancy space with

The approximations we have made to diagnose the nonlinear bottom drag force based on mean quantities are not sufficient to close the force balance within the bottom boundary layer with the same level of precision as within the surface layer [see Eqs. (B5) and (B6)]. The zonal component of bottom stress is a nonlinear term based on correlations between kinetic energy and zonal velocity. We attempt to approximate this nonlinear term based on mean quantities. Furthermore, we assume that all of the bottom stress is deposited into the single, ensemble-mean buoyancy layer that rests on the bottom. As a result our approximation of the bottom drag force is only qualitatively correct. But we do have sufficient fidelity to conclude that the bottom drag force and northward-flowing geostrophic velocity are acting to decelerate the zonal flow while the eddies act in opposition to accelerate the zonal flow.

### d. Eddy fluxes of zonal momentum

The eddy-induced force in the TWA zonal velocity equation is clearly playing a leading role in the force balance within the ICC. To better understand how eddies are contributing to the equilibrated ^{2} s^{−2} on the southward side of the ICC. The meridional flux of zonal momentum is the opposite sign of

The vertical flux of zonal momentum is the primary mechanism by which eddies participate in the force balance of the ICC. The vertical flux of zonal momentum shown in Fig. 8 has been scaled by the reference density in order to convert to units of newtons per square meter. In this way we can get a sense of the magnitude of the vertical flux of zonal momentum by comparing to the zonal wind stress, which has a characteristic value of 0.2 N m^{−2}. Throughout the broad region of the ICC, the vertical flux of zonal momentum is always negative. We interpret the negative values as the flux of positive (eastward) zonal momentum in the downward direction toward more negative buoyancy values. Except for a small pause at the bottom of the surface layer, the downward flux of eastward momentum grows continuously with increasing depth in the vicinity of the ICC. Starting at a value of zero in the light buoyancy classes that are never occupied, the downward flux of eastward momentum grows to a value of 0.20 N m^{−2} at the bottom of the surface layer. Below the surface layer, the downward flux continues to increase while traversing the upwelling zone. Finally, the downward flux of eastward momentum converges within the bottom boundary layer and its value returns to zero in the heavy buoyancy classes that are never occupied by the fluid.

When shown in buoyancy coordinates, it is not possible to accurately sense the zonal force produced by the vertical divergence of the eastward stresses shown in Fig. 8. This is because, as shown in Eq. (17), the divergence of a vertical flux in buoyancy space is weighted by thickness, differentiated with respect to buoyancy, then deweighted by thickness. An alternative approach is to notice that *z* coordinates, the vertical flux of zonal momentum appears far more uniform in the vertical, especially within the core of the ICC. The vertical flux of zonal momentum increases rapidly from the surface down to

### e. Heat balance in the surface layer

*S*

_{f}expressed asThe meridional surface volume flux integrates the TWA meridional velocity across the surface layer weighted by

*S*

_{f}has units of meters squared per second and represents the surface layer meridional volume flux per unit length in the zonal direction.

*y*axis in Fig. 10 shows the net surface heat flux

*Q*in watts per square meter. The

*y*position of

*Q*= 0 corresponds to the

*y*position of

*S*

_{f}= 0. Given that the meridional gradient of surface temperature is positive definite and highly uniform across nearly the entire meridional extent of the domain (see Fig. 3),

*S*

_{f}= 0 must align with

*Q*= 0 unless there is nonnegligible heat flux across the bottom of the surface layer. But Fig. 5 indicates that the TWA meridional velocity, and thus the heat transport, at the bottom of the surface layer is essentially zero across most of the ICC. So, on average, for

*y*positions greater than 1250 km, the surface layer carries colder waters northward to balance the heat flux into the ocean. And, in turn, for

*y*positions less than 1250 km, the surface layer carries warmer waters southward to balance the heat flux out of the ocean. We hypothesize that the dominant balance within the surface layer can be expressed asimplying that the surface heat flux is primarily balanced by the surface layer meridional advection of the meridional temperature gradient. Order of magnitude estimates indicate that such a balance is realized in the ICC. For example, between

*y*= 1250 and 1600 km the simulation has a characteristic surface layer meridional flux of 0.25 m

^{2}s

^{−1}and a characteristic meridional temperature gradient of 1°C per 350 km. This results in an advective heat flux of 3 W m

^{−2}, which is very close to the surface heat flux averaged between

*y*= 1250 and 1600 km (see Fig. 10).

## 5. Conclusions

Interpreting eddy–mean flow interactions through the lens of the thickness-weighted averaging approach results in a simple, yet complete, explanation of the equilibrium in zonally symmetric circumpolar currents. The powerful perspective provided by the TWA analysis stems from the concise and exact decomposition of the full three-dimensional Boussinesq equations into mean and eddy components and by conducting the analysis in buoyancy space.

Supporting previous thinking from Tréguier et al. (1997) and Marshall and Radko (2003), a key finding from the TWA analysis is the identification of the surface layer based on ventilation as opposed to traditional boundary layer dynamics. The surface layer is defined as the set of buoyancy coordinates that ever reside at the ocean’s surface. This is the set of buoyancy values that directly feel the nonconservative forcing due to wind stress and surface heat flux. Tréguier et al. (1997) suggest a similar definition where the boundary between the adiabatic interior and diabatic surface zones should be placed between sporadic and uninterrupted isopycnal layers. Marshall and Radko (2003, p. 2344) suggest that the mixed layer depth should likely be defined as “the *z*-position of the deepest isopycnal that occasionally grazes the surface.” By construction the surface layer isolates the diabatic ocean layer from the adiabatic region of the ocean. While such a definition is naturally accommodated in buoyancy coordinates by finding the lightest, uninterrupted buoyancy value at each

Figure 10 lends additional support to the notion of a dynamically isolated surface layer. The linear restoring to a prescribed surface temperature results in net heat flux in/out of the ocean over the northern/southern half of the domain. By integrating the TWA meridional velocity across the surface layer we were able to elucidate the primary heat (buoyancy) balance. While the TWA meridional velocity is both southward and northward within the surface layer, the integrated surface layer meridional flux *S*_{f}, shown in Fig. 10, is strongly related to the surface heat flux *Q*, shown in the same figure. The zero value of *Q* corresponds to the zero value of *S*_{f}. North/south of this nodal point, positive/negative values of *S*_{f} carry cold/warm water to balance the positive/negative surface heat flux. Our conjecture is that the primary balance equation has the form of ** τ** but is instead determined by the surface flux of buoyancy. Surface stress can certainly indirectly modify the surface-integrated buoyancy balance through modification of the buoyancy coordinate outcropping positions. This relationship is similar to that derived by Marshall and Radko (2003) in their Eq. (11). While Marshall and Radko (2003) conduct their analysis in

*z*coordinates, which precludes defining the surface layer based on ventilation, the analysis conducted above in buoyancy coordinates supports their hypothesis developed using

*z*coordinates.

While the configuration is idealized, the large-scale structure of the TWA meridional velocity is broadly similar to that observed in the Southern Ocean. Namely, the production of bottom water on the shelf flows northward along the ocean floor with southward, upwelling flow residing above and extending across the majority of the meridional domain. In addition, across the ICC portion of the domain we find that the TWA meridional velocity is northward in the bottom half of the ventilation-defined surface layer. So if we move in the positive meridional direction along, for example, the *b* = −0.002 m s^{−2} surface in Fig. 5, we know that there has to be a zero crossing of the TWA meridional velocity. But the following question remains: why does the zero crossing of the TWA meridional velocity coincide almost exactly with the very bottom of the ventilation-defined surface layer? The zero crossing cannot occur inside the ventilation-defined surface layer because this would imply that the coldest waters to ever reach the surface would, on average, come from the north. Given the zonal symmetry of the system and the linear restoring of surface buoyancy, such a scenario is not dynamically tenable. Alternatively, the zero crossing cannot occur outside of the ventilation-defined surface layer due to continuity. Just outside of the ventilation-defined surface layer the system is adiabatic, and conservation of mass equation (15) implies *b* surface to be implausible within the adiabatic region of the fluid. So it might be that the idealized nature of this system constrains the zero crossing of the TWA meridional velocity to coincide with the bottom of the ventilation-defined surface layer. But the dynamical simplicity of the crossing to be coincident with the interface of the diabatic and adiabatic regions of the fluid suggests that the finding might be applicable to more realistic ocean conditions.

The analysis clearly highlights the leading role that mesoscale eddies play in the force balance of the circumpolar current. The eddies participate in the force balance through the differential vertical flux of eastward momentum. While the importance of downward transport of vertical momentum by eddies in the ACC is well accepted (Munk and Palmén 1951; Johnson and Bryden 1989; Gille 1997; Phillips and Rintoul 2000), by diagnosing the Eliassen–Palm flux tensor ∇⋅

In this idealized configuration, the bottom of the ventilation-defined surface layer coincides with the location where all of the forces in the zonal momentum balance equation [Eq. (17)] are very nearly equal to zero. By definition, the wind stress force is zero at the bottom of the surface layer. As explained above, the bottom of the surface layer also coincides with a node in the meridional velocity and thus a node in the force produced by MAPV. This then constrains the eddy force to also be zero in order for equilibrium in the balance of forces to be obtained.

While we have not used the theoretical constructs associated with Ertel potential vorticity (EPV) to interpret the eddy–mean flow interaction within the ICC, the eddy force ∇⋅

We have demonstrated that the TWA framework developed by de Szoeke and Bennett (1993), Young (2012), and Maddison and Marshall (2013) is exceptionally powerful in its ability to exactly and concisely express the role of eddies within the mean, climatological ocean state. Development of variations and extensions of the broad class of thickness-weighted approaches to the eddy–mean flow decomposition suggests that even further advances are possible (e.g., Aiki and Greatbatch 2014; Aoki 2014). In reference to eddy–mean flow decomposition, Rhines and Holland (1979) suggest that our goal in developing equation sets that capture eddy–mean flow interaction is to have a mean-field equation in which eddy effects are as transparent as possible. We find that the TWA framework is possibly optimal in this regard.

Still, it is important to note that our in situ implementation of the TWA analysis framework is incomplete as compared to the possibilities offered by the underlying theory. First, because of the amount of data required to conduct the analysis, the only viable path to analysis of global eddying simulations is through an in situ implementation where the diagnostics are computed during the simulation (see appendix A). In terms of analysis, we are only diagnosing the zonal-mean TWA system [see Eq. (17)], whereas the TWA framework can be applied to understand eddy–mean flow interaction in the full 3D system. We have made assumptions about how and where the nonconservative surface forcing and bottom drag are modeled in the force balance [see Eqs. (B4) and (B6)]. We have also not addressed more complicated buoyancy forcings, such as penetrative solar heating, because of the idealized nature of this study. Finally, the TWA analysis framework can be used to measure the spurious contributions to the diabatic vertical velocity in the ocean interior due to imperfect numerics. We have not accounted for spurious numerical buoyancy fluxes in our analysis. While incomplete, the analysis framework is sufficient to explain eddy–mean flow interaction in an idealized circumpolar channel. We see no impediment to a complete in situ TWA analysis framework that could be applied to realistic global eddying ocean configurations.

All of the results obtained above will have to be reconsidered in more complex ocean configurations. In particular, it is unclear if the ventilation-defined surface layer will remain dynamically isolated in more complex geometries with more complex forcing. Also, evidence clearly shows that the bottom form stress force plays a leading role in the force balance at the bottom of the Southern Ocean (Johnson and Bryden 1989; Gille 1997; Phillips and Rintoul 2000). Since this configuration employs bottom bathymetry without variations in the zonal direction, bottom form stress cannot contribute to the zonal force balance. Our next step is to add ridges and plateaus to this idealized configuration in order to quantify how the force balance is impacted by bottom form stress. While demonstrating the robustness of our results within increasingly complex physical settings is required, we suspect that our findings with respect to the diabatic and force balance in the ventilation-defined surface layer will prove useful in describing the role of mesoscale eddies in the climate of the Southern Ocean.

This work is part of the “Multiscale Methods for Accurate, Efficient, and Scale-Aware Models of the Earth System” project, supported by the U.S. Department of Energy’s Office of Science program for Scientific Discovery through Advanced Computing (SciDAC). Code developments and simulations relied heavily on the work of the MPAS dynamical core development team at LANL and NCAR, and in particular the contributions from the MPAS-Ocean development team at LANL. The analysis presented above greatly benefited by comments from Andy Hogg and from two anonymous reviewers.

# APPENDIX A

## Calculation of the Eliassen–Palm Flux Tensor

This appendix describes how the terms in the thickness-weighted averaged Boussinesq equations are constructed based on an ensemble of ocean simulation states measured with Eulerian coordinates.

The TWA discrete coordinate system is *h*, *i*, and *j* subscripts, respectively, as would be expected. But increasing *k* moves downward in the flow and thus toward more negative buoyancy values. Assume that the ensemble is composed of *N* members. The ensemble members sample across simulations at fixed

Now assume that the states are provided in a discrete Eulerian coordinate system composed of *A* are interpolated to buoyancy coordinates, that is, *I* is the vertical interpolation operator. Buoyancy coordinate values

*z*position of the

At a given *A* is recorded, might be outcropped some of the time, none of the time, or all of the time. If *k* and *k* + 1, are outcropped, the *z* positions of both buoyancy surfaces are the same as the sea surface height. The Lorenz convention requires that these outcropped *z* positions be included as valid data used to produce the ensemble average defined in Eq. (A1) that is used, subsequently, in Eq. (A3).

*N*= 2000 ensemble members are included in the computation of

^{2}This in situ computation of

*N*members and we wish to add an additional member, we simply use

# APPENDIX B

## Calculation of Momentum and Diabatic Forcing in Buoyancy Coordinates

*b*

_{min}(

*t*

_{h},

*x*

_{i},

*y*

_{j}) and

*b*

_{max}(

*t*

_{h},

*x*

_{i},

*y*

_{j}) to be the minimum and maximum buoyancy values in each water column. Now define a mask

*ϕ*is valued 1 within the buoyancy space occupied by the fluid and 0 otherwise. Using Eq. (A1),

*ϕ*. In our simulation,

Given the linear equation of state with constant salinity, a buoyancy layer is equivalent to a constant temperature layer. So at any point we can exchange *T*_{f} at the ocean surface is *T*_{f} = −*p*_{T}(*T*_{s} − *T*_{r}), where *p*_{T} is the piston velocity, *T*_{r} is the restoring temperature, and *T*_{s} is the ocean surface temperature. The surface buoyancy flux is linearly related to the surface temperature flux as *b*_{f} = *T*_{f}*αg*/*ρ*_{0}. When

*ϖ*appearing in the TWA equation is thickness weighted, that is,

*τ*, but it is even more straightforward since the wind stress does not vary with

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^{1}

As discussed in appendix B,

^{2}

Note that the in situ approach to computing