## 1. Sallée–Rintoul model

*H*to the interior thermocline. The form of the subduction rate

*S*

_{b}reads as follows (Cushman-Roisin 1987; Marshall 1997):

*S*

_{b}(eddy), and Sallée et al. (2010, their section 4b) concluded that the eddy component “plays an order one role in the overall subduction in the Southern Ocean.” SR11 employed the following four-part model: 1) The adiabatic (A) regime was treated using the Gent–McWilliams (GM) streamfunction

**Ψ**

_{A}(Gent and McWilliams 1990):

*κ*

_{M}is the mesoscale diffusivity,

**s**(

**e**

_{z}= (0, 0, 1). 2) The diabatic (D) regime was parameterized as an extension of the A regime using

*T*(

*x*,

*y*,

*z*) was assumed to depend only on

*z*with the boundary conditions

*T*(0) = 0 and

*T*(A–D interface) = 1. 3) The A–D interface was taken at

*H*. 4) The mesoscale diffusivity was taken to be

^{−1}(Mazloff et al. 2010), which is hereinafter referred to as the Southern Ocean state estimate (SOSE). On the other hand, use of the diffusivities derived by Sallée et al. (2008) from surface drifter observations, yielded subductions that compared significantly better to the data. On those grounds, SR11 suggested a 10-fold increase of the mesoscale diffusivity in Eqs. (1.2)–(1.4).

Since the mesoscale diffusivity is but one component of a complete mesoscale parameterization, we suggest that the latter should first be assessed on its overall performance, the Antarctic Circumpolar Current (ACC) subduction being one of the tests. We employ the mesoscale models presented in Canuto et al. (2018a, hereinafter C18) and Canuto et al. (2019, hereinafter C19) that include recent theoretical and observational advances and that were assessed against a variety of data and the outputs of 17 other ocean general circulation models (OGCMs) (Griffies et al. 2009). The present model yields two main results: the ACC diffusivities compare well to those from drifter data (Sallée et al. 2008) and the ACC subduction rates are of the same magnitude as the SOSE data.

## 2. New mesoscale model

For the reader’s convenience, we have added appendix A with the relevant equations of C18 and C19.

### a. A regime

**u**

_{d}, which is a barotropic variable since it is the solution of an eigenvalue problem. The T/P-based conclusions by C11 confirmed the prediction and defined

**u**

_{d}as the most germane of all the nonlinear metrics. It must be noted that

**u**

_{d}cannot be identified with the Rossby phase velocity resulting from linear analysis and that does not reproduce altimetry data (Klocker and Marshall 2014). Figure 1 of C18 shows that the form of

**u**

_{d}given by Eq. (2.5) of C18 compares well to altimetry data. Second, the eddy-induced velocity is no longer given by the GM form alone since

**u**

_{d}introduces a second term:

*σ*

_{t}≡

*σ*

_{t}(1 +

*σ*

_{t})

^{−1}and

*σ*

_{t}=

*O*(1) is the turbulent Prandtl number. The implication of the new term in Eq. (2.1) was first studied in CD5 and more quantitatively in section 2f of C18, where it was shown that it

*lowers*the amount of energy that mesoscales draw from the mean potential energy, which in turn implies that the isopycnal slopes are steeper than in the GM model (see Fig. 4 of C18). This feature becomes relevant when studying, for example, the implications of the predicted increase of the wind stress that tends to steepen the isopycnal slopes (Gent 2016). The first GM term in Eq. (2.1) becomes the full eddy-induced velocity only at the steering level where

*f*< 0, one has [

**u**

^{+}⋅ ∇

*H*and thus subduction.

### b. D regime

*T*(

*x*,

*y*,

*z*) whereby the streamfunction is considered an extension of the first of Eq. (1.2) in the form Eq. (1.3):

*T*(

*x*,

*y*,

*z*) depends on

*x*,

*y*,

*z*, thus far it has always been taken to be a function of

*z*only, an assumption that has the following implication. Consider the second relation in Eq. (2.4):

*T*(

*x*,

*y*,

*z*) depends only on

*z*makes the first term on the right-hand side of Eq. (2.5) vanish which affects the subduction rates. As for

*T*(

*z*), SR11 adopted a straight line with the conditions

*T*(0) = 0 to ensure that

*w*

^{+}(0) = 0 [see Eq. (1.4) of C18]. If the A–D interface is denoted by

*h*, matching Eq. (2.3) with Eq. (1.2) requires that

*T*(

*h*) = 1, but the choice of

*h*is not trivial. For example, Gnanadesikan et al. (2007) concluded that a tapering approach yielded OGCMs results that were “disconcerting” because of the strong dependence of

*h*on the isopycnal slope at that depth. Last, while tapering functions may work as a numerical device, the physical content of the D regime can hardly be represented that way since, as discussed in section 1b of C18, the A and D regimes satisfy very different conservation laws, that is, potential vorticity in the A regime with an inverse energy cascade and relative vorticity in the D regime with enstrophy cascade.

*T*(

*x*,

*y*,

*z*) is a function of

*x*,

*y*,

*z*that is no longer arbitrary but is given by the mesoscale model itself. Figure 7 of C18 shows that Eq. (2.8) yields results lower than the commonly used straight line.

### c. Extent of the D regime

*h*is not determined by a mesoscale model, SR11 [their Eq. (4)] and Sallée et al. [2010, their Eq. (11)] assumed

*O*(1). Such a depth is called the boundary layer depth (HBL) and is location dependent. While the choice of HBL as the lower limit of

*h*is well motivated, it is still not sufficient since one also needs to know how deep h can be. In that respect, we suggest that

*h*should be

*less than the depth of the thermocline*since at that depth the stratification would be too strong for the D regime to exist. We thus suggest the following heuristic expression:

*h*that they called

*H*

_{1,2}; the upper bound was the depth of peak stratification as in Eq. (2.10) but the lower bound was still taken to be the mixed layer depth rather than the HBL. In Fig. 1 we plot the ratio

*h*/

*H*in the ACC, where

*h*is computed using Eq. (2.10) and the mixed layer depth

*H*is computed from the potential density criterion Δ

*σ*= 0.03 kg m

^{−3}. The results in Fig. 1 show that in the majority of locations

*h*>

*H*or

*h*≈

*H*, in accordance with previous authors (e.g., Mensa et al. 2013; Veneziani et al. 2014) who found

*h*>

*H*. At the same time, the results also show that it is possible that

*h*<

*H*in some locations, as suggested by an anonymous referee.

### d. Mesoscale diffusivity

*κ*

_{M}is a key ingredient in any mesoscale parameterization, and the difficulties in determining it are demonstrated by the variety of suggestions that were made, for example, in section 3c of Sallée et al. (2010). Thus far, all of the suggested expressions were heuristic, and one can surmise the following time sequence of models of increasing physical content:

*N*

^{2}was considered to be a proxy for the eddy kinetic energy. While an improvement, it does not provide the full

*x*,

*y*dependence shown by the T/P data (Scharffenberg and Stammer 2010), which can only be obtained by constructing the last entry in Eq. (2.11), a model of the 3D

*κ*

_{M}(

*x*,

*y*,

*z*).

*κ*

_{M}(

*x*,

*y*,

*z*) model depend on how accurately the key ingredient, the eddy kinetic energy

*K*(

*x*,

*y*,

*z*), reproduces the WOCE (2002) data for the vertical profile and the T/P data (Scharffenberg and Stammer 2010) for the

*x*,

*y*surface values. Canuto and Dubovikov [1996, their Eq. (24)] derived the expression for the turbulent viscosity felt by an eddy of size

*ℓ*caused by all the eddies smaller than

*ℓ*. Section 2 of C19 discusses how that expression is applied to the present oceanic context and further shows how it contains the well-known mixing length theory as a particular case. The structure of the mesoscale diffusivity given by Eq. (2.5) of C19 is

*α*≅ ½ represents the departure from the mixing length theory, as explained in Eqs. (2.3) and (2.4) of C19;

*r*

_{d}is the Rossby deformation radius,

*K*(

*x*,

*y*,

*z*) is the 3D eddy kinetic energy, and

*ϖ*(

**u**

_{d},

*K*) represents the interaction of mesoscales with the mean velocity

*K*(

*x*,

*y*,

*z*) and the barotropic mesoscale drift velocity:

*K*was derived to be

*B*

_{1}(

*z*)is the first baroclinic mode (Wunsch 1997),

*K*

_{s}(

*x*,

*y*) was discussed in detail in section 4 of C19, with the result given by their Eq. (4.10) and the assessment against T/P data (Scharffenberg and Stammer 2010) shown in Figs. 9 and 10 of C19. It is relevant to point out that both vertical and horizontal components of

*K*(

*x*,

*y*,

*z*) were expressed analytically. Last, the function

*ϖ*(

**u**

_{d},

*K*) was derived to be

**u**

_{D}|

^{2}K

^{−1}, (2.15) recovers the heuristic expression used by Bates et al. (2014). Figure 5 of C19 shows the comparison of (2.12) with North Atlantic Tracer Release Experiment (NATRE) data (Ferrari and Polzin 2005).

## 3. OGCM results from the C18 and C19 parameterizations

In addition to the tests discussed above, we used the new mesoscale parameterization in the GISS-ER stand-alone OGCM (see appendix B) under Core Ocean-Reference Experiment phase 1 (CORE-I) forcing (Griffies et al. 2009). The 500-yr run yielded the results in Figs. 12–17 of C19 showing the global ocean temperature, the Atlantic overturning circulation, the meridional heat transport, and the Drake Passage transport, all of which were compared with the results of 17 previous OGCMs. Figure 18 of C19 shows how the model reproduces the winter ACC mixed layer depths.

## 4. Mesoscale diffusivity and subduction rates

The mesoscale diffusivities derived by Sallée et al. (2008) using surface drifter data were larger than those used in the SR11 model and reproduced more closely the SOSE data. This motivated SR11 to suggest to boost the diffusivity in Eqs. (1.2)–(1.4) 10-fold. Since the subduction rates we obtain shown in Fig. 2 reproduce satisfactorily the SOSE data, it remains to be shown that the mesoscale diffusivities predicted by the present model reproduce the surface drifter data. Before we do so, we need to remark that the reason to study the case in Fig. 2c with *w*^{+} = 0 was to highlight the contribution of *w*^{+} since Hiraike et al. (2016), using an eddy-resolving simulation, reported that the *w*^{+} contribution is large; indeed, Fig. 2c shows that with *w*^{+} = 0, the resulting subduction rates do not reproduce the SOSE data. Next, consider Fig. 3. The 3D diffusivities of this model shown in the left panel compare well to the results in Fig. 3 of Sallée et al. (2008); for completeness, the right panel shows the 2D diffusivities used by SR11.

## 5. Conclusions

The two models for the mesoscale diffusivity Eq. (1.4) employed by SR11 yielded subduction rates smaller than SOSE data by an order of magnitude. On the other hand, the mesoscale diffusivities derived by Sallée et al. (2008) from surface drifter data were larger than those in Eq. (1.4) and reproduced more closely the data. Thus, SR11 suggested boosting the diffusivity in Eq. (1.4) 10-fold. In this work, we used the mesoscale parameterizations presented in C18 and C19, whose implications were assessed against a variety of data before being used in the subduction problem that represents an additional test of the C18 and C19 parameterizations. Use of the latter reproduced satisfactorily topology (subduction equatorward and obduction poleward) and magnitudes of the SOSE data. Last, because it was previously shown (Canuto et al. 2018b) that submesoscales also produce sizeable subduction but with a topology different than that of mesoscales, a complete picture will require that mesoscales and submesoscales be considered together.

Author Canuto thanks Profs. Y. Tanaka and D. Marshall for informative correspondence. The authors thank two referees for questions that helped to sharpen our presentation. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Center for Climate Simulation (NCCS) at the Goddard Space Flight Center.

# APPENDIX A

## Mesoscale Parameterization for Coarse-Resolution OGCMs

The mesoscale parameterizations in C18 and C19 is summarized as follows. The OGCMs solve the equations for the mean momentum and mean arbitrary tracers, the latter being both active tracers (such as *T*, *S*) and passive tracers (such as CO_{2}) that have different parameterizations. In C18 and C19 we treated the effect of mesoscales on an arbitrary tracer since the parameterization of the mesoscale-induced momentum fluxes (Reynolds stresses) are not yet available (work is in progress). Diabatic (D) and adiabatic (A) regimes are governed by different conservation laws and have different parameterizations.

### a. General relations

#### 1) D regime

**e**

_{z}=(0, 0, 1), and

*Q*represents external forcing. The

*horizontal*flux is given by

*κ*

_{M}is the mesoscale diffusivity discussed below. The

*vertical tracer flux*is given by

*N*is the Brunt–Väisälä frequency,

**s**is the slope of the isopycnals, and

*h*

_{*}denotes the depth of the D regime. The function Φ(

*z*) allows one to match the flux at

*h*

_{*}with that of the A regime. We have

*r*

_{d}is the first Rossby deformation radius. In the case of buoyancy

#### 2) A regime

**E**in the mean tracer equation that now reads as follows:

**U**

^{+}= (

**u**

^{+},

*w*

^{+}) is the nondivergent, 3D eddy-induced velocity, and the Redi flux is

*K*is the eddy kinetic energy. Because of the smallness of this term, it will be neglected hereinafter.

#### 3) Eddy-induced velocity

*σ*≡

*σ*

_{t}(1 +

*σ*

_{t})

^{−1},

*σ*

_{t}is the turbulent Prandtl number of

*O*(1),

**= ∇**

*β**f*is the Rossby phase velocity, and

*H*

_{b}is the ocean depth.

#### 4) Drift velocity

**s**

_{*}is the isopycnal slope at

*h*

_{*}.

### b. Mesoscale diffusivity κ_{M}

*α*≃ ½.

### c. Eddy kinetic energy

*x*,

*y*,

*z*) is the vertical profile and

*K*

_{s}(

*x*,

*y*) is the surface value. We have

*B*

_{1}(

*z*)is the first baroclinic mode solution of the eigenvalue problem

_{z}

*B*

_{1}= 0 at

*z*= −

*H*

_{b}, 0, and

*B*

_{1}(0) = 1. The surface kinetic energy

*K*

_{s}was derived to be

### d. Depth of the D regime

### e. Implementation in an OGCM

# APPENDIX B

## The OGCM

We employed the 3D diffusivity tensor for an arbitrary tracer given in section 7 of C18, the mesoscale diffusivity Eq. (A.14), and the KPP vertical mixing scheme (Large et al. 1994) in the GISS ER-model which is the ocean component of the coupled NASA GISS model E (Russell et al. 1995, 2000; Liu et al. 2003). An early version of the revised E2-R code was run in a stand-alone mode (Danabasoglu et al. 2014). It employs a mass coordinate approximately proportional to pressure with 32 vertical layers with thickness from ≈12 m near the surface to ≈200 m at the bottom. The horizontal resolution is 1.25° (longitude) by 1° (latitude). It is a fully dynamic, non-Boussinesq, mass-conserving free-surface ocean model using a quadratic upstream scheme for the horizontal advection of tracers and a centered difference scheme in the vertical. An 1800-s time step is used for tracer evolution. Sea ice dynamics, thermodynamics, and ocean–sea ice coupling are represented as in the CMIP5 model-E configuration (Schmidt et al. 2014), save that here ice is on the ocean model grid. To force the model, we used the CORE-I Protocol (Griffies et al. 2009) with fluxes obtained from bulk formulae, the inputs to which are the ocean model surface state and atmospheric conditions derived from a synthesis of observations that repeat the seasonal cycle of a “normal year.” The results we present correspond to the output of the final 20 winters (July–September) of a 500-yr run.

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