1. Introduction
Accurate representation of the mesoscale eddy field and its feedback onto the mean ocean state is one of the most pressing challenges for ocean modeling, especially in the ocean circulation models used for climate prediction, which generally lack explicit representation of the mesoscale eddy field. Over the past two decades, a widely adopted approach for parameterizing the missing eddy fluxes has been that due to Gent and McWilliams (1990, hereafter GM). The GM scheme parameterizes eddies through both a diffusion of tracers along neutral density surfaces (Redi 1982) and an eddy-induced circulation that acts to flatten neutral-density surfaces (Gent et al. 1995; McDougall and McIntosh 2001), thereby extracting available potential energy from the mean state. The adoption of GM resolved a number of known deficiencies in ocean circulation models by removing the spurious diapycnal water mass conversions that were prevalent in the existing eddy parameterization schemes (Danabasoglu et al. 1994).
A known deficiency of the existing GM-based eddy parameterizations is the very different response of the Southern Ocean circulation to changes in surface wind stress in models employing GM as compared with models with explicit eddies. Coarse-resolution models employing existing GM-based parameterizations are generally found to be more sensitive than eddy-permitting models to changing surface winds, though models employing eddy transfer coefficients that vary in three spatial dimensions are less sensitive than those that employ an eddy transfer coefficient that is varying in two spatial dimensions or is spatially constant (e.g., Farneti et al. 2015). One observed phenomenon is that the circumpolar transport increases with the strength of the surface wind forcing in coarse-resolution models employing existing GM-based parameterizations, whereas little sensitivity is observed in the equivalent models with explicit eddies (e.g., Munday et al. 2013; Farneti et al. 2015). This lack of sensitivity is known as eddy saturation (Hallberg and Gnanadesikan 2001; Tansley and Marshall 2001) and was first predicted on theoretical grounds by Straub (1993). Eddy saturation is generally found in models that at least partially resolve a mesoscale eddy field (e.g., Hallberg and Gnanadesikan 2006; Hogg and Blundell 2006; Hogg et al. 2008; Farneti and Delworth 2010; Farneti et al. 2010; Morrison and Hogg 2013; Munday et al. 2013; Hogg and Munday 2014) but not in models in which eddies are parameterized by GM-based schemes (e.g., Munday et al. 2013; Farneti et al. 2015).
A further discrepancy between eddy-permitting and coarse-resolution models employing existing GM-based parameterizations is the reduced sensitivity of the time-mean residual meridional overturning circulation to changing wind forcing obtained in eddy-permitting models (e.g., Meredith et al. 2012; Viebahn and Eden 2012; Morrison and Hogg 2013; Munday et al. 2013; Hogg and Munday 2014; Farneti et al. 2015). This reduced sensitivity in eddy-permitting models is known as eddy compensation (Viebahn and Eden 2012). Eddy compensation is less well understood than eddy saturation, depending in subtle ways on the vertical structure of the eddy response to changes in surface forcing (e.g., Morrison and Hogg 2013). The response is further complicated by the fact that the residual meridional overturning circulation is affected by bathymetric details (e.g., Hogg and Munday 2014; Ferrari et al. 2016; de Lavergne et al. 2017).
Generally, it is found that eddy-permitting calculations are strongly eddy saturated and partially eddy compensated. On the other hand, partial eddy saturation and eddy compensation can be obtained in a model that parameterizes eddies when the eddy transfer coefficient is allowed to vary in space and time (e.g., Gent and Danabasoglu 2011; Hofman and Morales Maqueda 2011; Farneti et al. 2015). The reader is referred to the work of Farneti et al. (2015) for a recent comprehensive comparison of global circulation ocean models at coarse resolutions and their ability to reproduce eddy saturation and eddy compensation.
Numerous papers have attempted to derive the functional dependence of the eddy transfer coefficient on the ocean state as a function of space and time from first principles (e.g., Treguier et al. 1997; Visbeck et al. 1997) and through diagnoses of numerical simulations (e.g., Ferreira et al. 2005; Ferrari et al. 2010; Bachman and Fox-Kemper 2013; Bachman et al. 2017). On the other hand, through a mixing length argument, Eden and Greatbatch (2008) proposed an eddy transfer coefficient that is related to the eddy kinetic energy (see also Cessi 2008; Marshall and Adcroft 2010; Jansen and Held 2014). This approach requires solving for the eddy kinetic energy through a prognostic eddy energy budget.
Recently, Marshall et al. (2012) have developed a new energetically constrained eddy parameterization framework, here termed Geometry and Energetics of Ocean Mesoscale Eddies and Their Rectified Impact on Climate (GEOMETRIC). The eddy forcing in the momentum equation may be described in terms of an eddy flux tensor, whose entries may be written in terms of geometric parameters that depend on the eddy kinetic and eddy potential energy. A bound of the tensor in the quasigeostrophic limit in terms of the total (i.e., kinetic and potential) eddy energy results in an inferred GM eddy transfer coefficient that is entirely determined by the total eddy energy, the stratification, and an unknown nondimensional parameter that is bounded in magnitude by unity.
The efficacy of GEOMETRIC has been established through three proofs of concept:
In the linear Eady (1949) model of baroclinic instability, an analytical test case, GEOMETRIC produces the correct dimensional energy growth rate (Marshall et al. 2012).
In the fully turbulent nonlinear Eady spindown problem, as simulated by Bachman et al. (2017), the diagnosed eddy transfer coefficient from the numerical calculations is consistent with the eddy transfer coefficient predicted by GEOMETRIC across four orders of magnitude of the eddy transfer coefficient.
When applied to a two-dimensional model of the Antarctic Circumpolar Current with a domain-integrated eddy energy budget (Mak et al. 2017), GEOMETRIC produces eddy saturation, that is, a circumpolar volume transport that is insensitive to the magnitude of surface wind stress. This is due to an interplay between the zonal momentum budget and eddy energy budget (Marshall et al. 2017), the essential ingredients of which are preserved by GEOMETRIC.
However, GEOMETRIC has thus far not been implemented and tested in a three-dimensional ocean circulation model; this is the primary aim of the present study. First, we implement GEOMETRIC in an idealized three-dimensional channel, extending the calculations of Mak et al. (2017) in a two-dimensional channel model. Cases with both integrated and spatially varying parameterized eddy energy budgets are considered, comparing the results with those of eddy-permitting calculations. Second, we implement GEOMETRIC in a sector model with a basin and Southern Ocean reentrant channel, supporting an interhemispheric meridional overturning circulation in addition to a circumpolar current. The key advantage of the channel integrations is that we can afford to compare with eddy-permitting “model truths” than in the sector integrations, the latter taking far longer to equilibrate. Moreover, the channel model displays an interesting inverse sensitivity of thermal wind circumpolar transport to wind stress, which has not been previously documented but is reproduced by the GEOMETRIC parameterization. The key advantage of the sector integrations is that we are able, for the first time, to assess the extent to which GEOMETRIC is able to capture eddy compensation.
The article proceeds as follows. Section 2 outlines the GEOMETRIC approach and discusses the associated parameterized eddy energy budget. Implementation details relating to the parameterization schemes considered in this study are given in section 3. Results from GEOMETRIC are first presented for the channel model in section 4, followed by results in the sector model in section 5. The article concludes in section 6 with a summary of key findings and discussion of outstanding implementation challenges and future research questions.
2. GEOMETRIC
GEOMETRIC is a framework for parameterizing mesoscale eddies that preserves the conservation laws in the unaveraged equations of motion via closures that preserve the tensorial properties and symmetries possessed by the eddy flux tensor [see also related ideas in (see also related ideas in Marshall and Adcroft 2010). GEOMETRIC was originally derived under the quasigeostrophic approximation (Marshall et al. 2012), although elements of the framework generalize to the thickness-weighted averaged primitive equations (Maddison and Marshall 2013).
representation of the eddy-mean flow interaction through an eddy stress tensor that can be bounded in terms of the total eddy energy in the quasigeostrophic limit (Marshall et al. 2012) and
solution of a consistent eddy energy equation accounting for parameterized and resolved dynamical processes and their role in supplying or removing eddy energy from the relevant length scales (cf. Eden and Greatbatch 2008; Cessi 2008; Marshall and Adcroft 2010; Eden et al. 2014).
Crucially, given knowledge of the total eddy energy and the stratification profile, the remaining unknown α, satisfying 
The remaining challenge in implementing GEOMETRIC is then to address ingredient 2, that is, to solve for the eddy energy field. Mak et al. (2017) implemented GEOMETRIC with a domain-integrated eddy energy budget while recognizing that, to delineate different dynamical regimes in more complex numerical ocean models, the eddy energy, and thus the associated parameterized eddy energy budget, should vary spatially. Solution of a prognostic equation for the eddy kinetic energy in three dimensions has been attempted by Eden and Greatbatch (2008). It is proposed here that the depth-integrated total eddy energy is solved for, as this offers both the conceptual and logistical simplicity of working in two rather than three dimensions and also avoids division by zero in (1) when the isopycnals are flat at some depth (but not if the isopycnals are flat throughout the water column).
Eddy energy budget
1) Advection
In observations, the eddy energy is seen to propagate at the velocity of the depth-mean flow, Doppler shifted by the intrinsic long Rossby phase speed (Klocker and Marshall 2014). In this study the contribution from the intrinsic long Rossby phase speed is not included (i.e., 
2) Source
In general the sources of eddy energy depend on multiple instability types associated with the ocean state. For the present study it is assumed that the primary source of eddy energy is associated with baroclinic instability (Charney 1948; Eady 1949). The term in (2) represents the loss of available potential energy due to the slumping of neutral density surfaces as represented by GM; see Marshall et al. (2017) and Mak et al. (2017) for further details. Sources of eddy energy from other instabilities may also be included in future parameterizations but are not considered in this present study.
3) Dissipation
The dissipation of eddy energy is complicated, involving a myriad of processes. These include bottom drag (e.g., Sen et al. 2008; Klymak 2018), lee wave radiation from the seafloor (e.g., Naveira Garabato et al. 2004; Nikurashin and Ferrari 2011; Melet et al. 2015), western boundary processes (Zhai et al. 2010), and loss of balance (e.g., Molemaker et al. 2005). Moreover, the eddy energy dissipation through these various processes will critically depend on the partition between eddy kinetic and eddy potential energy and the vertical structure of the eddy kinetic energy (Jansen et al. 2015; Kong and Jansen 2017). Each of these ingredients requires detailed investigation. Instead, a simple approach is followed here, representing eddy energy dissipation through a linear damping at a rate λ, recognizing that λ parameterizes all of the physics outlined above.
4) Diffusion
A Laplacian diffusion of eddy energy is incorporated following Eden and Greatbatch (2008). There are indications that the use of a Laplacian diffusion is an appropriate model of the divergence of the mean energy flux in an f-plane barotropic model of turbulence (Grooms 2015).
3. Parameterization implementation
To assess the performance of the proposed GEOMETRIC parameterization scheme outlined above, calculations employing models with eddies parameterized by different schemes are compared with calculations from equivalent models at eddy-permitting resolutions. The parameterization schemes employed in this study are as follows: GEOMloc, the GEOMETRIC parameterization scheme outlined in the previous section, with a depth-integrated but horizontally varying eddy energy equation; GEOMint, the GEOMETRIC parameterization scheme outlined in Mak et al. (2017), with a domain-integrated eddy energy equation; and the standard GM scheme with a prescribed 
a. GEOMloc
The first set of experiments employ GEOMETRIC locally in latitude and longitude, as detailed in section 2, with the eddy transfer coefficient computed as in (4), coupled to the parameterized eddy energy budget in (2). The GEOMloc scheme is implemented wholly within the GM/Redi package in MITgcm (Marshall et al. 1997a,b), building upon the existing implementation of Visbeck et al. (1997). First, 
The eddy energy budget in (2), discretized in space by centered second-order differencing, is time stepped with a third-order Adams–Bashforth scheme (started with forward Euler and second-order Adams–Bashforth steps) with the smoothed 
b. GEOMint
c. CONST
The coarse-resolution calculations GEOMint, GEOMloc, and CONST are compared to reference calculations at eddy-permitting resolutions (REF). To assess the performance of the parameterization variants, various diagnoses of the resulting time-averaged data are presented; unless otherwise stated, all subsequent figures and statements refer to the time-averaged data.
The theory behind GEOMETRIC applies to the GM eddy transfer coefficient and not to the enhanced eddy diffusion of tracers along isopycnals (e.g., Redi 1982). While GM and Redi diffusion are often implemented in the GM/Redi tensor together (e.g., Griffies 1998; Griffies et al. 1998), the corresponding coefficients are not the same (e.g., Abernathey et al. 2013). In all calculations presented here, the Redi diffusion coefficient is prescribed to be 
In the coarse-resolution calculations, the parameters α and λ in GEOMint, GEOMloc, and 
4. Channel configuration
a. Setup and diagnostics
As an extension of the f-plane, zonally averaged channel model of the Antarctic Circumpolar Current presented in Mak et al. (2017), a three-dimensional idealized channel configuration on a β plane is considered. The configuration is essentially a shorter version of the channel configuration reported in Munday et al. (2015) and Marshall et al. (2017), with no continental barriers. The domain is 4000 km long and 2000 km wide, with a maximum depth of 3000 m. The model employs a linear equation of state with temperature only and an implicit free surface. A ridge with a height of 1500 m and width of 800 km blocks 
For the eddy-permitting reference calculations termed REF, the horizontal grid spacing is uniform at 10 km. A control simulation with control peak wind stress 
For the models with parameterized eddies, the horizontal grid spacing is 100 km, except at the northern boundary where the grid spacing is 50 km so as to have at least three grid points over the region with enhanced vertical temperature diffusivity. A control GEOMint calculation with 
Sensitivity experiments are then carried out in which either the magnitude of wind stress or the eddy energy dissipation (linear bottom drag for REF and λ for GEOMint and GEOMloc) are varied. The relevant parameter values are documented in Table 1.
Parameter values that are employed for the channel experiments. The control simulations employ the boldface values of 
b. Summary of key results
The key results, examining the effects of varying wind stress and eddy energy dissipation, are presented in Fig. 1. As a summary, in this channel setup, the total transport of REF decreases with increasing wind stress and increases with increased linear bottom drag. The total transport is composed principally of transport due to thermal wind, as seen in Figs. 1c and 1d. The changes in the thermal wind transport are reflected in the resulting 
Diagnosed transport (Sv; 1 Sv ≡ 106 m3 s−1) and thermocline depth (m) in the channel model, for varying wind stress and varying eddy energy dissipation. Showing (a),(b) total transport, (c),(d) thermal wind transport, and (e),(f) thermocline depth.
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Diagnosed transport (Sv; 1 Sv ≡ 106 m3 s−1) and thermocline depth (m) in the channel model, for varying wind stress and varying eddy energy dissipation. Showing (a),(b) total transport, (c),(d) thermal wind transport, and (e),(f) thermocline depth.
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Diagnosed transport (Sv; 1 Sv ≡ 106 m3 s−1) and thermocline depth (m) in the channel model, for varying wind stress and varying eddy energy dissipation. Showing (a),(b) total transport, (c),(d) thermal wind transport, and (e),(f) thermocline depth.
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
c. Detailed results from perturbation experiments
1) Varying wind stress experiments
First, it is interesting to note that, even in REF, the total transport decreases with increasing wind, and the transport is significant even at zero wind stress. The latter is due to the enhanced vertical temperature diffusivity near the northern boundary, which acts to maintain a stratification at depth and, together with surface restoring of temperature, results in tilting isopycnals and thus a thermal wind transport (e.g., Hogg 2010; Munday et al. 2011). In this model, the thermocline becomes shallower with increasing wind. As a result, the geostrophic flow occupies a smaller region even though the peak geostrophic flow speed may be larger, resulting in a smaller integrated thermal wind transport. The decreased thermocline depth with increasing wind is partially due to the choice of imposing high vertical diffusivity near the northern boundary; such behavior is not observed when a fully dynamical basin sets the northern channel stratification (as in the sector configuration in the next section) or when the northern boundary temperature is relaxed to a prescribed profile [as in, e.g., Abernathey and Cessi (2014), though they employ a flux boundary condition at the ocean surface].
Despite the perhaps unexpected sensitivity to changing wind forcing in REF, it is encouraging to see that both GEOMint and GEOMloc are able to reproduce the analogous sensitivities, particularly in the thermal wind transport and thermocline depth diagnostics. The agreement between the results with GEOMint and GEOMloc and those with REF is less satisfactory at lower winds where the thermocline is deeper and, correspondingly, the transport is noticeably larger; the causes of these discrepancies remain to be investigated further. In contrast, the standard CONST calculations display opposite sensitivity in the transport and thermocline depth. Figure 2 shows the zonally averaged temperature profile and zonal flow of the eddy-permitting calculation and coarse-resolution calculations. The GEOMint and GEOMloc calculations are able to capture the changes in the stratification displayed by REF, especially in the upper ocean, in terms of the morphology and location of the temperature contours. An examination of the absolute difference in zonally averaged zonal velocity (not shown) shows the largest discrepancies lie within the high vertical temperature diffusivity region, where the coarse-resolution calculations generally have weaker zonal mean flows.
Zonally averaged zonal velocity (shaded; 
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Zonally averaged zonal velocity (shaded;
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Zonally averaged zonal velocity (shaded;
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
2) Varying eddy energy dissipation experiments
With increased bottom drag, the total transport of REF increases, consistent with the results of Marshall et al. (2017). The rationale is that increased eddy energy dissipation requires steeper isopycnals for the eddy energy to be replenished through baroclinic instability. This leads to increased thermal wind transport, and is consistent with the diagnostics displayed in Figs. 1b, 1d, and 1f. This feature of increased thermal wind transport is reproduced by the GEOMint and GEOMloc calculations, and is consistent with the findings of Mak et al. (2017).
d. Impact on the diagnosed eddy energy and 
Diagnosed outputs relating to the parameterization variants for the channel model, for (left) varying wind stress and (right) varying eddy energy dissipation, showing (a),(b) domain-averaged eddy energy (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Diagnosed outputs relating to the parameterization variants for the channel model, for (left) varying wind stress and (right) varying eddy energy dissipation, showing (a),(b) domain-averaged eddy energy (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Diagnosed outputs relating to the parameterization variants for the channel model, for (left) varying wind stress and (right) varying eddy energy dissipation, showing (a),(b) domain-averaged eddy energy (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
For GEOMint and GEOMloc, the resulting 
Depth-averaged (left),(center) total eddy energy (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Depth-averaged (left),(center) total eddy energy (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Depth-averaged (left),(center) total eddy energy (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
For changing eddy energy dissipation, while the sensitivity of the diagnosed 
Figure 4 shows the spatially varying depth-averaged eddy energy and 
In ocean observations, mesoscale eddies within the core of the Antarctic Circumpolar Current are observed to propagate eastward at a speed consistent with advection by the background depth-mean flow, Doppler shifted by the westward propagation at the intrinsic long Rossby phase speed (Klocker and Marshall 2014). An interesting observation here in this channel model is that the eddy energy is extended too far to the east in GEOMloc. The inclusion of westward propagation by the intrinsic long Rossby wave speed may remedy this deficiency by offsetting the contribution from the background mean flow advection.
In the control and large-eddy energy dissipation case in GEOMloc, the resulting 
For completeness, the eddy energy fields at the large eddy energy dissipation values are also included. At larger r, the dominant contribution of the eddy energy in REF comes from the EPE. On the other hand, increasing λ in GEOMloc appears to instead concentrate the eddy energy around the ridge, with an increase in the magnitude over the ridge. The eddy energy pattern is not entirely different from the control case and in fact resembles well the general EKE pattern of REF (not shown), possibly indicating that the scheme as implemented is able to better capture changes in EKE pattern.
5. Sector configuration
A sector with a reentrant channel connected to an ocean basin is considered to allow for the possibility of an interhemispheric residual meridional overturning circulation (RMOC). A growing number of analyses and results from eddy-permitting numerical models suggests that while the circumpolar transport is largely insensitive to changes in wind forcing, the RMOC shows some sensitivity to changes in wind forcing (e.g., Hogg et al. 2008; Farneti and Delworth 2010; Farneti et al. 2010; Farneti and Gent 2011; Gent and Danabasoglu 2011; Meredith et al. 2012; Morrison and Hogg 2013; Munday et al. 2013; Farneti et al. 2015). A sector configuration allows for study of whether the GEOMint and GEOMloc have the potential to reproduce both eddy saturation and eddy compensation in a more complex and realistic setting.
a. Setup and diagnostics
In this instance, the eddy-permitting reference calculation REF has a 
For the coarse-resolution calculations, the horizontal spacing is 
Varying wind stress and varying eddy energy dissipation experiments are carried out but the latter only for the GEOMint and GEOMloc calculations owing to computational constraints. The relevant parameter values are documented in Table 2.
Parameter values that are employed for the sector experiments. The control simulations employ the boldface values of 
b. Response of the circumpolar transport
Figure 5 shows the diagnosed circumpolar transport and pycnocline depth for varying wind stress and eddy energy dissipation values. To summarize, for varying wind stress, the eddying calculation REF possesses a circumpolar transport that displays weak dependence on the peak wind stress and may be described as eddy saturated. The pycnocline depth is also only weakly dependent on varying peak wind stress. Assuming again that the circumpolar transport is dominated by thermal wind transport and noting that isopycnals are essentially pinned at the outcropping regions, increases in pycnocline depth are linked directly to increased circumpolar transport via increases in the tilt of isopycnals and thermal wind balance.
Diagnosed transport (Sv) and pycnocline depth (m), for varying wind stress and eddy energy dissipation, showing (a),(b) total circumpolar transport and (c),(d) pycnocline depth of the basin.
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Diagnosed transport (Sv) and pycnocline depth (m), for varying wind stress and eddy energy dissipation, showing (a),(b) total circumpolar transport and (c),(d) pycnocline depth of the basin.
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Diagnosed transport (Sv) and pycnocline depth (m), for varying wind stress and eddy energy dissipation, showing (a),(b) total circumpolar transport and (c),(d) pycnocline depth of the basin.
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
With this in mind, the CONST calculations are categorically not eddy saturated, displaying large sensitivity of the circumpolar transport and pycnocline depth to changing wind forcing. On the other hand, both the circumpolar transport and pycnocline depth in GEOMint and GEOMloc display weak sensitivity to changing wind stress, far more consistent with the REF case. It is interesting to note that at slightly lower winds, the pycnocline depth of GEOMint and GEOMloc increases, with a corresponding signal in the diagnosed circumpolar transport, much like the channel configuration (see Fig. 1e). Increasing λ in the GEOMint and GEOMloc calculations increases the circumpolar transport and pycnocline depth. Again, the rationale is that increased eddy energy dissipation requires steeper isopycnals for the eddy energy to be replenished through baroclinic instability, leading to larger circumpolar transport through thermal wind balance.
c. Response of the meridional overturning circulation
For the varying wind stress experiments, while the GEOMint and GEOMloc calculations are eddy saturated, the associated sensitivity in the RMOC remains to be investigated. The diagnosed RMOCs for varying wind stress are shown in Fig. 6. Focusing first on the control case for REF (Fig. 6b; cf. Fig. 8c of Munday et al. 2013), it may be seen that the RMOC consists of two main cells: (i) an upper positive cell that represents the model analog of North Atlantic Deep Water (NADW) downwelling in the Northern Hemisphere, upwelling in the Southern Ocean and returning northward in surface layers; (ii) a lower negative cell that represents the model analog of Antarctic Bottom Water (AABW), established by the convective activity occurring in the southern edges of the domain, spreading northward at depth, upwelling, and returning southward. Additionally, there is an Antarctic Intermediate Water (AAIW) negative cell, located slightly north of the NADW upwelling region, characterized by shallow convection.
RMOC streamfunction (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
RMOC streamfunction (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
RMOC streamfunction (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
For the control wind forcing, the global morphology of the RMOC appears to be well captured in all the coarse-resolution calculations, as seen in Figs. 6e, 6h, and 6k for GEOMint, GEOMloc, and CONST, respectively. The main differences arise in the lack of an excursion of the RMOC above the time- and zonal-mean surface density in the north and in the details of the AABW negative cell. The former is because there are no explicit mesoscale eddies in the coarse-resolution calculations. The latter, on the other hand, likely depend on both the eddy induced circulation and convective processes; a discussion of the latter difference is deferred to the discussion section.
When varying wind stress, the changes in the RMOC displayed by REF are largely matched by GEOMint and GEOMloc. With no wind forcing, the NADW positive cell is approximately of the same magnitude and with similar extents into the Southern Hemisphere. With large wind stress forcing, increases in magnitude and extent of both the NADW positive cell and AABW negative cell are seen. Both GEOMint and GEOMloc struggle to reproduce the latitudinal extent and the strength of the AABW negative cell. However, both GEOMint and GEOMloc certainly appear to provide improvements over CONST; where the latitudinal extent of the NADW with zero wind forcing differs significantly from REF, there is increased noise in the AABW cell, and the NADW cell spans over a smaller set of water mass classes with large wind stress forcing. The enhanced level of noise in and just north of the channel region in CONST coincides, and is consistent, with increased convective activity in the same regions, where the prescribed 
d. Impact on the diagnosed eddy energy and
Figure 7 shows the domain-averaged eddy energy 
Diagnosed outputs relating to the parameterization variants for the sector model, for (left) varying wind stress and (right) varying eddy energy dissipation, showing (a),(b) domain-averaged eddy energy (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Diagnosed outputs relating to the parameterization variants for the sector model, for (left) varying wind stress and (right) varying eddy energy dissipation, showing (a),(b) domain-averaged eddy energy (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Diagnosed outputs relating to the parameterization variants for the sector model, for (left) varying wind stress and (right) varying eddy energy dissipation, showing (a),(b) domain-averaged eddy energy (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Depth-averaged (left),(center) total eddy energy (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Depth-averaged (left),(center) total eddy energy (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
Depth-averaged (left),(center) total eddy energy (
Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-18-0017.1
With increasing eddy energy dissipation, increasing λ results in decreased 
Finally, Fig. 8 shows the depth-averaged total eddy energy field and the 
At large wind stress forcing, a recirculation region is seen north of the circumpolar current in both REF and GEOMloc. The eddy energy is large in the circumpolar current, with correspondingly large 
6. Discussion and concluding remarks
This article has described the implementation of Geometry and Energetics of Ocean Mesoscale Eddies and Their Rectified Impact on Climate (GEOMETRIC) in a three-dimensional primitive equation ocean model. The GEOMETRIC framework utilizes the Gent–McWilliams eddy parameterization but with the eddy transfer coefficient prescribed as 
On the other hand, this study has highlighted several subtleties that need to be addressed. The following discussions will focus on details of the parameterization, but it is recognized that other model details such as bathymetry play a central role in shaping the RMOC (e.g., Hogg and Munday 2014; Ferrari et al. 2016; de Lavergne et al. 2017; Holmes et al. 2018) and will also affect the overall model response.
While the calculations with GEOMETRIC appear to capture the bulk morphological changes of the RMOC over changing wind stress forcing, there are features that are at odds with the reference calculation, notably in the strength and extent of the modeled AABW. A candidate in improving the RMOC is to incorporate a vertically varying eddy response. While this study presents results for a vertically uniform eddy transfer coefficient 
A set of calculations with the structure function 
An interhemispheric RMOC for the control simulation is recovered in sample calculations with a larger imposed 
While slumping of isopycnals in baroclinic instability and eddy-induced stirring along isopycnals [as parameterized by Gent and McWilliams (1990) and Redi (1982), respectively] are often implemented together (e.g., Griffies 1998; Griffies et al. 1998), in this study 
As discussed in the text, while eddy saturation is not expected to depend to leading order on the lateral redistribution of eddy energy (Mak et al. 2017), other details such as the model RMOC and western boundary currents may do so. In the present implementation of GEOMETRIC, eddy energy is advected by the depth-mean flow only, and the resulting eddy energy signature is generally found to have a more eastward extension in GEOMint than the corresponding eddy-permitting calculation. While the magnitude of eddy energy diffusion will have a role in redistributing the parameterized eddy energy, an obvious question is whether inclusion of a westward advective contribution at the long Rossby phase speed (consistent with Chelton et al. 2007, 2011; Zhai et al. 2010; Klocker and Marshall 2014) can remedy the overly eastward extension of the eddy energy signature. Taking the linear eddy energy damping rate employed here at 
Perhaps the most poorly constrained aspect of the present implementation of GEOMETRIC is the treatment of eddy energy dissipation. Dissipation of mesoscale eddy energy can be through a myriad of processes such as bottom drag (e.g., Sen et al. 2008; Klymak 2018), lee wave radiation (e.g., Naveira Garabato et al. 2004; Nikurashin and Ferrari 2011; Melet et al. 2015), western boundary processes (Zhai et al. 2010), and loss of balance (e.g., Molemaker et al. 2005), all of which vary in time, space, and magnitude. Given the overwhelming complexity and the uncertainty in representing such energy pathways, the choice of linear damping of eddy energy at a constant rate over space is chosen to represent the collective effect of the aforementioned processes. With this choice, it is found that coarse-resolution models with GEOMETRIC are able to reproduce the broad sensitivities of the circumpolar transport and pycnocline depth obtained in the eddy-permitting reference for varying wind stress and eddy energy dissipation.
On the other hand, the sensitivity of the domain-averaged eddy energy magnitude, while reasonable in the varying wind stress experiments, is at odds in the varying dissipation experiments in the channel configurations. Further investigation is required to reproduce the eddy energetic sensitivities displayed in eddy-permitting reference calculations. Additionally, while the Reynolds stresses have been neglected such that it is only buoyancy fluxes that have been closed, the inclusion of Reynolds stresses are known to be important for shaping the mean flow of inertial jets (e.g., Hughes and Ash 2001; Li et al. 2016; Tamarin et al. 2016) and for flows over variable bottom topography (e.g., Wang and Stewart 2018). The inclusion of a closure of Reynolds stresses within the GEOMETRIC framework is not pursued here but clearly is an important topic for future investigation.
In closing, with the important caveat that there are many details that can be improved upon, the results of this study lend further support to the GEOMETRIC framework as a viable parameterization scheme to better represent mesoscale eddies in coarse-resolution models, such as reproducing more accurately the response of the large-scale ocean state with explicit eddies to changes in forcing. For implementation into a global circulation ocean model, the primary change required is to couple a depth-integrated eddy energy budget to the existing Gent–McWilliams module. Diagnoses of eddy energetics via observations (e.g., Zhai and Marshall 2018, manuscript submitted to Geophys. Res. Lett.), idealized turbulence models (Grooms 2015, 2017), and ocean-relevant simulations (e.g., Stewart et al. 2015; Youngs et al. 2017) will provide a first constraint on how to improve the representation of the advection and dissipation of eddy energy, aiding in a more accurate and useful representation of the ocean climatological response. In terms of approach, the GEOMETRIC framework underlines the need to shift the focus from how to close for eddy buoyancy fluxes, to also developing improved representations of the eddy energetics and associated eddy energy pathways.
Acknowledgments
This work was funded by the UK Natural Environment Research Council Grant NE/L005166/1 and NE/R000999/1 and utilized the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk). DRM is supported by the U.K. Natural Environment Research Council (ORCHESTRA; Grant NE/N018095/1). The authors thank the two referees for comments that greatly improved the presentation of the article. The lead author thanks Gurvan Madec for discussions relating to energetic pathways. All data used herein are archived in the Edinburgh DataShare service (at https://doi.org/10.7488/ds/2297).
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