## 1. Introduction

In the field of fluid dynamics and turbulence, formulating a closure for the governing equations has been a long-standing problem (Smagorinsky 1963; Launder et al. 1975). Resolving the flow down to the molecular scale where kinetic energy is dissipated to internal energy due to molecular viscosity is usually not feasible, whether in observations or a numerical model. Particularly in the field of geophysical fluid dynamics (GFD) where the scales of interest span up to *O*(1000) km, resolving the molecular scale is practically unachievable and will remain so for the foreseeable future. Due to the lack of resolution, a numerical model will only solve the governing equations for the “resolved” field, and some work has to be done to account for the “unresolved” field. A significant effort in GFD has been, therefore, to formulate a closure for the unresolved field, i.e., represent the unresolved field with the resolved momentum and/or tracer field (e.g., Mellor and Yamada 1982; Redi 1982; Gent and McWilliams 1990; Bachman et al. 2017).

The ocean component of climate models suffers from this issue of missing the unresolved dynamics because it barely resolves the mesoscale eddies [horizontal scale of *O*(20–100) km]. This is problematic because the unresolved (small-scale) field not only drains energy from the resolved (large-scale) field but also partially feeds back onto the resolved field via upscale momentum and buoyancy fluxes, and so modifies the dynamics of the large-scale flow (Vallis 2006; Lévy et al. 2012; Arbic et al. 2013; Aluie et al. 2018; Ajayi et al. 2021). Modeling studies with varying spatial resolution have shown that only partially resolving the mesoscale results in weaker mesoscale eddies, and consequently weaker feedback onto large-scale flows. It is also well known that mesoscale eddies exert a strong influence on oceanic jets such as the Gulf Stream (Chassignet and Xu 2017; Kjellsson and Zanna 2017; Chassignet and Xu 2021). Considering the impact of the jets on global tracer transport and air–sea interaction (Kelly et al. 2010; Tréguier et al. 2014; Jones and Cessi 2018; Bellucci et al. 2020), improving the representation of the eddy feedback onto the jet has climate implications. Hence, there has been a growing effort to represent the inverse cascade of kinetic energy otherwise lost to gridscale numerical viscosity at mesoscale-permitting resolution, a process often referred to as energy backscattering parameterizations (e.g., Zanna et al. 2017; Berloff 2018; Jansen et al. 2019; Bachman 2019; Juricke et al. 2019; Perezhogin 2019; Zanna and Bolton 2020, and references therein). Our study here is in the same realm of parameterization studies in which we aim to improve the large-scale state by parameterizing the net mesoscale feedback onto the large-scale flow.

Specifically, the goal of our study is to formulate a deterministic closure and hence a model for the eddy dynamics. Such an approach is not new; for example, Jansen et al. (2019), Juricke et al. (2019), and Perezhogin (2019) implemented a prognostic equation for the subgrid (unresolved) eddy energy and achieve the backscattering via a negative viscosity. One notable difference in our method is that while many previous studies have formulated their parameterizations based on a local closure (i.e., relating the eddy momentum/buoyancy flux locally at each grid point to the resolved momentum/buoyancy), we construct our closure by incorporating basin-scale information. This is motivated by the fact that Venaille et al. (2011) and Grooms et al. (2013) have shown that the eddy feedback on the large-scale flow is strongly nonlocal. We also focus on the subgrid potential vorticity (PV) equation rather than subgrid energy within the quasigeostrophic (QG) framework. The QG framework has been shown to be fruitful in examining the eddy–mean flow interaction and formulating eddy closures (e.g., Marshall et al. 2012; Porta Mana and Zanna 2014; Mak et al. 2016; Berloff 2018). In particular, Berloff et al. (2021) have shown some success in accounting for the nonlocal eddy feedback by solving for the subgrid QGPV equation [cf. Eq. (22)]. While our approach is similar, here, we propose an alternative strategy to achieve a deterministic closure for the subgrid PV. This approach of prognostically solving for the subgrid dynamics is sometimes referred to as super parameterization and has been commonly implemented for atmospheric or oceanic convection (e.g., Randall et al. 2003; Khairoutdinov et al. 2005; Campin et al. 2011; Beucler et al. 2020). In this paper, we will provide a proof of concept of this super parameterization approach with a QG model. The goal of this paper is indeed to see how a QG model can handle the small-scale eddy dynamics given a prescribed large-scale background flow.

The paper is organized as follows: we describe our QG model configuration in section 2 and in particular the (subgrid) eddy PV model in section 2b. We propose a closure for the subgrid PV model and detail on its performance in section 3. A proof of concept of a prognostic implementation of our super parameterization is given in section 4. We give our conclusions in section 5. The reader interested in reproducing our results will find all the technical details in the appendixes.

## 2. Model and methods

### a. The control run

*N*

^{2}and Coriolis parameter

*f*. Two ingredients are necessary to reproduce the double gyre pattern: the planetary vorticity must vary with latitude for the western boundary intensification and the wind forcing must be cyclonic in the northern part of the domain and anticyclonic in the southern part of the domain. To satisfy the first condition, we work with the

*β*-plane approximation such that the Coriolis parameter

*f*varies linearly with latitude. This sets the planetary scale

*L*=

_{β}*f*

_{0}/

*β*, which is large compared to the deformation scale

*R*=

_{d}*NH*/

*f*

_{0}, (with

*H*the depth of the ocean and

*f*

_{0}the average value of the Coriolis parameter in the domain). In this formalism, the main dynamical variable is the QG potential vorticity (PV) defined as

*ψ*the streamfunction, ∇

^{2}the horizontal Laplace operator, and

*A*

_{4}the biharmonic viscosity;

*r*the bottom friction coefficient which parameterizes a bottom Ekman layer (and is thus nonzero in the lower layer only); and

_{b}*F*the forcing resulting from an Ekman pumping in a thin Ekman layer at the surface and is thus nonzero in the upper layer only. We build the numerical version of this model in the Basilisk framework (Popinet 2015, basilisk.fr).

We solve Eqs. (5) and (1) in a horizontal square domain with side *L* = 5000 km and of vertical extension *H* = 5000 m. We discretize these equations with 512 × 512 horizontal points (which correspond to a horizontal resolution of slightly less than 10 km) and 4 vertical layers of thickness *h*_{1} = 238 m, *h*_{2} = 476 m, *h*_{3} = 953 m and *h*_{4} = 3333 m (from top to bottom). We adjust the background stratification *N*^{2} to mimic the stratification in middle of the subtropical gyre in the North Atlantic such that at each layer interface, we have *β* = 1.7 × 10^{−11} m^{−1} s^{−1}. For these parameters, the three deformation radii are *R _{d}*

_{1}= 25 km,

*R*

_{d}_{2}= 10 km, and

*R*

_{d}_{3}= 7 km. Note that these deformation radii correspond to the inverse squared eigenvalue of the vertical stretching operator. At this resolution we choose

*A*

_{4}= 6.25 × 10

^{9}m

^{4}s

^{−1}, and a spindown time scale

*r*= 1/166 days (which corresponds to a bottom Ekman layer of thickness

_{b}*δ*= 7.5 m). We solve the elliptic equation [Eq. (1)] with homogeneous Dirichlet boundary conditions

_{e}*ψ*= 0 on the sides (corresponding to no flux boundary condition) and homogeneous Neumann boundary conditions

*b*=

*ψ*= 0 at the top and bottom boundaries.

_{z}We use a cubic sine function in the definition of the wind in order to reproduce a narrow midlatitude atmospheric jet. For such a narrow jet, the boundary between the positive and negative area of the wind stress curl pattern is sharper than if we use the traditional cosine shape for the wind pattern. We choose *τ*_{0} = 0.25 N m^{−2}, which is an acceptable value for the difference between the maximum and minimum value of the wind in the North Atlantic (Josey et al. 2002). We have also kept the wind stress axisymmetric as our interest is on eddy time scales and not low-frequency variability (Berloff et al. 2007).

To integrate the model in time, we first perform a spinup phase of 80 years at low resolution (Δ*x* = 78.13 km) followed by another 80 years at the prescribed resolution (Δ*x* = 9.77 km). After this spinup of 160 years in total, the model is in a statistically steady state (i.e., *f*_{0}). Except in the region of the separated jet, the local Rossby number is much smaller than unity, consistent with the QG scaling. Henceforth, we refer to this run as the CTRL run.

### b. Mean flow and eddy models

The term

*κ*

_{GM}an eddy diffusivity coefficient and

Note that the presence of

### c. Mean flow and eddy dynamics in the full model

We first analyze the output of reference run (CTRL) which is a mesoscale-resolving simulation. We recall that this model solves the full PV equation [Eq. (5)]: we decompose the output of that simulation into a mean and an eddy flow. We perform this decomposition with a time mean. For the remainder of this study, the averaging operator *F*′ = 0, and so the time mean is similar to an ensemble mean here under the ergodic assumption (Galanti and Tsinober 2004). We will consider the mean and eddy flow diagnosed from the CTRL run as the “truth.” We will then use these diagnostics to validate the model of the eddy dynamics only (section 3).

We plot in Fig. 2a, a snapshot of the eddy kinetic energy in the upper layer. We find at least two distinct dynamical regimes: (i) the eddying jet with KE′ on the order of 0.5 m^{2} s^{−2} (corresponding to a velocity of |*u*′| ∼ 1 m s^{−1}). The intensity of the jet decreases downstream (eastward). (ii) A region with moderate eddies in the middle of each gyre; the magnitude of these eddies increases from east to west but their overall intensity is order KE′ ∼ 0.04 m^{2} s^{−2} (|*u*′| ∼ 0.2 m s^{−1}). There are other dynamical regions such as quiescent zone with no eddies at all at the same latitude as the jet but near the eastern boundary, and the regions near the northern and southern boundaries.

Snapshots and time mean of potential energy and kinetic energy (m^{2} s^{−2}) diagnosed from the CTRL run. A snapshot of the (a) eddy kinetic and (b) potential are shown, along with (c),(d) their time means. The mean (e) kinetic and (f) potential energy are also shown.

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

Snapshots and time mean of potential energy and kinetic energy (m^{2} s^{−2}) diagnosed from the CTRL run. A snapshot of the (a) eddy kinetic and (b) potential are shown, along with (c),(d) their time means. The mean (e) kinetic and (f) potential energy are also shown.

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

Snapshots and time mean of potential energy and kinetic energy (m^{2} s^{−2}) diagnosed from the CTRL run. A snapshot of the (a) eddy kinetic and (b) potential are shown, along with (c),(d) their time means. The mean (e) kinetic and (f) potential energy are also shown.

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

We plot with the same color bar the eddy potential energy for the same snapshot (Fig. 2b). We observe that the magnitude of PE′ is similar to the magnitude of KE′ consistent with the QG scaling. We plot in Figs. 2c and 2d the mean eddy kinetic energy and mean eddy potential energy. The eddy potential energy and eddy kinetic energy exhibit similar patterns and are maximal in the jet. The maximum value of eddy energy in the jet area reflects the meandering jet. These meanders are strongest near the western boundary and decrease in amplitude moving east.

The energy stored in the mean flow exhibits a radically different pattern than the eddy energy (Figs. 2e,f). The QG model exhibits the standard result that most of the large-scale energy is stored in the form of potential energy. Note that the color bar in Fig. 2f is extended by a factor of 20 compared to the other plots because there is approximately 20 times more potential energy than kinetic energy in the large-scale flow. This result corresponds to the traditional view of the ocean circulation (

### d. Vorticity balance of the mean flow

For sufficiently long integration, the first term in the mean flow [Eq. (10)] will eventually vanish. There is thus a balance between the remaining terms of the mean PV equation. We only focus here on the rectification term that we plot in Fig. 3. We plot in Fig. 3a the raw estimate of

(a) The raw

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(a) The raw

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(a) The raw

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

It is also important to note that the pattern in Fig. 3a clearly has not converged because when we sum all the terms in Eq. (10), viz. *σ* is the standard deviation of the time series at a given point and *n* the number of samples. If we want the error bar to be 10% of the value of the mean *m*, the 95% confidence interval on the mean for that tolerance is given by *n* = 400*σ*^{2}/*m*^{2}. We get an estimate of *n* = 10^{5} samples to get this 10% precision for the mean. This corresponds to 10^{4} years of simulation which is clearly out of reach in the current setup. We have tested this using the 2740 years of output from Kondrashov and Berloff (2015) and found the convergence to be very slow (P. Berloff 2021, personal communication). The fact that such a long integration is required for accurate statistics is problematic from an eddy closure perspective, namely, the eddy statistics of today would depend on the dynamical state of the system thousands of years in the past. This conundrum also highlights the need for a closure for the eddy rectification.

## 3. The subgrid PV model

Our goal is now to see if we can approximate

*ψ*′), the

*diagnosed*eddy field from the CTRL run, and with a dagger (e.g.,

*ψ*

^{†}) the prognostic eddy dynamics that result from the

*explicit*time integration of the subgrid model [Eq. (12)] with the mean flow (

Definition of the notations.

### a. No eddy rectification forcing ($\mathcal{R}=0$ )

With the lack of a good predictor for the eddy rectification forcing, we can start by examining the subgrid model [Eq. (17)] *without* it on the right-hand side (viz.

For white noise initial conditions, we can decompose the run in several stages: we first observe a linear growth of the most unstable modes mainly in the jet and near the northern and southern boundary. The duration of this phase is on the same order of magnitude as the inverse linear growth rate (see appendix A, Fig. A1a). We then enter another transient phase during which a large-scale pattern emerges in the PV field, and after this transient phase, we reach a statistical steady state (i.e.,

(a),(b) Snapshots and (c),(d) time mean of kinetic and potential energy (m^{2} s^{−2}) diagnosed from the eddy model with no forcing (

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

(a),(b) Snapshots and (c),(d) time mean of kinetic and potential energy (m^{2} s^{−2}) diagnosed from the eddy model with no forcing (

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

(a),(b) Snapshots and (c),(d) time mean of kinetic and potential energy (m^{2} s^{−2}) diagnosed from the eddy model with no forcing (

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

Everywhere in the domain, the mean kinetic energy in this subgrid run (Fig. 4c) is weaker than the mean eddy kinetic energy diagnosed from the CTRL run (Fig. 2c), namely,

The isotropic wavenumber spectra taken over the whole domain for (a) kinetic and (b) potential energy in the first layer. The energies diagnosed from the CTRL run are shown in solid black, from the subgrid model with no forcing in dotted red (

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

The isotropic wavenumber spectra taken over the whole domain for (a) kinetic and (b) potential energy in the first layer. The energies diagnosed from the CTRL run are shown in solid black, from the subgrid model with no forcing in dotted red (

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

The isotropic wavenumber spectra taken over the whole domain for (a) kinetic and (b) potential energy in the first layer. The energies diagnosed from the CTRL run are shown in solid black, from the subgrid model with no forcing in dotted red (

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

In the eddy run, we still see a local kinetic energy (KE^{†}) maximum in the middle of the domain where the mean jet is and we also observe deformation radius size eddies in the rest of the gyre (Fig. 4a). Such difference between

We also plot in Fig. 6a the eddy streamfunction for the same snapshot as the one plotted in Fig. 2, and in Fig. 6b the subgrid streamfunction of the subgrid model for the same snapshot as in Fig. 4. This plot confirms the differences already highlighted of a weaker baroclinicity in the eddy run and also shows that large-scale Rossby waves present in the eddy field diagnosed from the CTRL run (*ψ*′; Fig. 6a) are not present in the eddy model (*ψ*^{†}; Fig. 6b). This is probably because Rossby waves in the full model are triggered by intense eddies in the meandering jet. Since this model only produces mild eddies, there are no Rossby wave that will emerge in the eddy model. Another possibility is that Rossby waves are excited by the winds [*F* in Eq. (9)], which project themselves onto the temporally varying fields of *ψ*′, whereas the subgrid model (*ψ*^{†}) has no input to excite such waves.

(a) The eddy streamfunction *ψ*′ diagnosed from the CTRL run and (b) subgrid streamfunction *ψ*^{†} simulated from the subgrid model with no forcing (

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

(a) The eddy streamfunction *ψ*′ diagnosed from the CTRL run and (b) subgrid streamfunction *ψ*^{†} simulated from the subgrid model with no forcing (

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

(a) The eddy streamfunction *ψ*′ diagnosed from the CTRL run and (b) subgrid streamfunction *ψ*^{†} simulated from the subgrid model with no forcing (

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

The interesting point is that without the eddy rectification forcing, the large-scale pattern in *ψ*^{†} that emerges corresponds to the cyclonic gyre (in blue) is in the southern part of the domain and the anticyclonic gyre (red) is in the northern part of the domain (Fig. 6b), which is precisely the opposite from the streamfunction in the CTRL run. We interpret this large-scale pattern in *ψ*^{†} as the result of the rectification of the large-scale flow by small-scale eddies: the eddies tend to create a flow that opposes the large-scale forcing from the CTRL output (*ψ*^{†}) is not a fair reproduction of the eddy dynamics in the CTRL run (*ψ*′; Fig. 6). We show in section 3b, however, that we have some success in recovering the eddy dynamics from the dagger fields by parameterizing the eddy rectification forcing.

We now focus on the rectification term

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

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This experiment suggests that eddy dynamics feedback onto the large-scale dynamics via the inverse cascade. In the eddy model, this feedback on the large-scale potential energy concurs to flatten isopycnal surfaces and effectively shuts off the generation of eddies via baroclinic instability. We conclude that although the term

### b. Parameterizing the eddy rectification forcing ($\mathcal{R}$ )

*ψ*as

*ψ*

^{*}are respectively the large-scale and small-scale components of the field

*ψ*. Based on Pedlosky’s scale decomposition, the large-scale flow evolves on a slow time scale and the small-scale flow evolves on a fast time scale (Pedlosky 1987). We accomplish such scale decomposition by enforcing

*q*

^{†}to remain a small-scale field

*q*

^{†}=

*q*

^{†*}is satisfied for all time. Because of the equivalence between the slow time scale and large-scale spatial scale, our hope is that enforcing Eq. (19) will be equivalent to enforcing

*τ*; see appendix B, section b, for details on the numerical implementation). With this parameterization of the rectification term, we can already anticipate that the spatial filtering strategy will not work well in the region of the separated jet where there is no clear scale separation between the eddy flow and the mean flow (cf. Jamet et al. 2021). However, as we shall see, this strategy works well in the rest of the domain.

_{f}We illustrate the effect of the spatial filter operator [Eq. (18)] in Fig. 8 where we plot the same subgrid streamfunction as the one used in Fig. 4 along with its large-scale and small-scale component. We do this scale separation by applying a low-pass filter with a discrete wavelet transform (numerical details of the implementation are provided in appendix B). In Fig. 8, we use a cutoff length scale of *λ _{c}* = 500 km. In the large-scale pattern, we recognize a cyclonic and anticyclonic gyre, and a weak jet in the middle that we described earlier.

(a) Low-pass and (b) high-pass filtered subgrid streamfunction diagnosed from the subgrid model with

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

(a) Low-pass and (b) high-pass filtered subgrid streamfunction diagnosed from the subgrid model with

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

(a) Low-pass and (b) high-pass filtered subgrid streamfunction diagnosed from the subgrid model with

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

To use this spatial filter as a parameterization of *λ _{c}* or spatially varying

*λ*. Henceforth, we present our best results obtained with nonuniform

_{c}*λ*, and now briefly describe the reasons for opting for a nonuniform

_{c}*λ*. We see in Fig. 2a that the patch of high-eddy kinetic energy has horizontal dimensions on the order of 1000 km. In the region of the separated jet, there is thus no clear scale separation between the eddy flow and the mean flow. To a certain extent, this corroborates what we observe in the instability analysis (appendix A). In Fig. A1, we see that in the region of the separated jet, the most unstable mode has a characteristic length scale

_{c}*λ*= 300 km compared to the most unstable length scale in the return flow which is

*λ*= 230 km. We use this information to build a filter with nonuniform length scale in the form of

*λ*=

_{c}*αλ*, and we set

*α*= 4.5 to get

*λ*∼

_{c}*O*(1000) km in the area of the return flow. We plot in Fig. 9 the final map of

*λ*which corresponds to a smoothed version of the most unstable length scale (see appendix A). As desired,

_{c}*λ*has values on the order of 1000 km with a maximum of 1350 km in the region of the separated jet and a minimum of 850 km near the northeast and southeast corners.

_{c}The cutoff length scale (*λ _{c}*) in meters based on the instability length scale.

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

The cutoff length scale (*λ _{c}*) in meters based on the instability length scale.

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The cutoff length scale (*λ _{c}*) in meters based on the instability length scale.

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From here on, when we refer to the subgrid model [Eq. (17)],

We plot the energy diagnostics in Fig. 10. Comparing Figs. 10c and 10d with Figs. 4c and 4d, we see that using this

(a),(b) Snapshots and (c),(d) time means of potential energy and kinetic energy (m^{2} s^{−2}) diagnosed from the eddy model where

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

(a),(b) Snapshots and (c),(d) time means of potential energy and kinetic energy (m^{2} s^{−2}) diagnosed from the eddy model where

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

(a),(b) Snapshots and (c),(d) time means of potential energy and kinetic energy (m^{2} s^{−2}) diagnosed from the eddy model where

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

(a)

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(a)

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(a)

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Joint histogram of the spatially smoothed

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Joint histogram of the spatially smoothed

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Joint histogram of the spatially smoothed

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

## 4. Modification of the mean flow due to the eddy rectification term

The procedure described in the previous section demonstrates that the subgrid model can fairly reproduce the “true” eddy dynamics given a prescribed background flow. There is one caveat, however, which is precisely the specification of this background flow. Indeed, from an eddy parameterization perspective, taking

### a. Noneddying full model and mesoscale-resolving subgrid model

To see how the eddy model performs in the more realistic situation where *x* ≈ 78.13 km, increase the biharmonic viscosity to *A*_{4} = 6.25 × 10^{11} m^{4} s^{−1}, and also use a harmonic viscosity with *A*_{2} = 1000 m^{2} s^{−1}. Henceforth, we call this configuration the REF run. In this coarse-resolution model, the flow converges to a stationary state with almost no variability. This mean flow has less potential energy than the CTRL run and the midlatitude eastward jet is very weak (see Fig. 13). Note that we spun up the coarse full model with white noise initial conditions but without any rectification term.

(a) Snapshot of the EKE of the eddy model driven by the low-resolution background flow, namely, the outputs from REF. The contours show the time-mean reference streamfunction from the low-resolution REF run. (b)

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

(a) Snapshot of the EKE of the eddy model driven by the low-resolution background flow, namely, the outputs from REF. The contours show the time-mean reference streamfunction from the low-resolution REF run. (b)

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

(a) Snapshot of the EKE of the eddy model driven by the low-resolution background flow, namely, the outputs from REF. The contours show the time-mean reference streamfunction from the low-resolution REF run. (b)

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

The subgrid model itself is still Eq. (17) which now takes the time mean of the coarse model as barred variables, and for *λ _{c}* = 1000 km (simply because the unstable modes of the mean flow of REF exhibit an almost uniform pattern for both the instability time scale and the instability length scale). A snapshot of the eddies and diagnosed eddy rectification from the eddy model are shown in Fig. 13. The eddy activity resemble the CTRL run near the western boundary but lacks the signature in the separated jet region (Figs. 2a and 13a). As a consequence, the eddy rectification of the separated jet in the domain interior is negligible (Fig. 13b).

### b. Impact of the rectification on the large-scale flow

To see how we can use this eddy parameterization for coarse-resolution models, we turn our attention to Eq. (10). The only difference between this equation and the full model is the presence of the rectification term

When we integrate in time the coarse-resolution model with the rectification term, the circulation changes in a few places. We plot in Figs. 14a and 14b the change in the streamfunction when we force the coarse model with

Colors indicate the difference in streamfunction between the coarse reference run with ^{2} s^{−1}). Contours indicate the streamfunction of the low-resolution REF run. (a) The run with

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

Colors indicate the difference in streamfunction between the coarse reference run with ^{2} s^{−1}). Contours indicate the streamfunction of the low-resolution REF run. (a) The run with

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

Colors indicate the difference in streamfunction between the coarse reference run with ^{2} s^{−1}). Contours indicate the streamfunction of the low-resolution REF run. (a) The run with

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

Since the resolution of the full model is noneddying, a common eddy parameterization to implement would be the GM parameterization [cf. Eq. (11)]. We implement it in the QG model and we use a diffusivity coefficient (*κ*_{GM} = 1000 m^{2} s^{−1} applied only to buoyancy, equivalently the layer thickness in quasi geostrophy; cf. Uchida et al. 2021a). As GM is intended to mimic the baroclinic process of reducing PE, it would tend to further weaken the separated jet, which is what we see over the entire domain (blue in the subtropical and red in the subpolar gyre; Fig. 14c). The two runs with the eddy rectification forcing, on the other hand, tends to sharpen and strengthen the jet upon separation near the western boundary as we see between the meridional extent of 150–350 km (Figs. 14b,c). In other words, our closure captures the energy backscattering from the “subgrid” eddies onto the coarse full flow as they would if the eddy model were run until it reaches statistical convergence (see the similarity between Figs. 14b,c). The benefit of using *O*(10^{2}). We have shown that for a noneddying resolution, our closure provides a potential path forward to go beyond GM.

## 5. Conclusions and discussion

In this study, we have examined the eddy rectification term, which encapsulates the net eddy feedback onto the mean flow, from a quasigeostrophic (QG) double gyre simulation. In doing so, we decompose the QG potential vorticity (PV) into its mean flow, defined by a time mean, and eddies as the fluctuations about the mean. This paper is an attempt to estimate the rectification term *ψ*′), perhaps warrants some attention (Figs. 6 and 8a). We have shown that approximating the eddy rectification forcing with the spatially filtered eddy PV (

Once the eddy rectification forcing is estimated from the (subgrid) eddy model [

As a first step toward a PV-based coupled closure, we have emphasized the importance of solving the subgrid model explicitly and provided a proof of concept by solving the “partially” coupled system within the QG framework. We emphasize “partially” as the eddy rectification forcing we gave the full model at noneddying resolution was the time mean of the rectification predicted from the subgrid model (*q*^{total} = *q*^{coarse} + *q*^{†} at each time step where *q*^{coarse} here is the full PV resolved at coarse resolution. In such case, the total eddy kinetic energy would become

We also tested a case where the full model was mesoscale-permitting (Δ*x* = 19.5 km; *A*_{4} = 6.25 × 10^{11} m^{4} s^{−1}). The idea was to examine how an eddy model would perform if the full model also partially resolved the eddies. We followed the same procedure as described in section 4: (i) run the full model without the rectification (

While we have attempted to design a deterministic super parameterization where one explicitly solves the subgrid processes, it is possible that we are facing the limit of deterministic closures for the mesoscale-permitting regime. Stochastic and/or machine learning approaches may need to be considered (Bauer et al. 2020; Guillaumin and Zanna 2021; Frezat et al. 2021). Nonetheless, we have shown that our closure improves the eddy model in representing the eddies in comparison to them diagnosed from a mesoscale-resolving full model. Lastly, one may ask how our results can be extended to primitive equation models. In primitive equations, the eddy Ertel PV flux encapsulates the eddy feedback onto the mean flow (Young 2012). In other words, a closure based on Ertel PV may allow one to capture the eddy variability in a primitive eddy model.

As an alternative to our spatial filtering approach, we hypothesize that it is possible to obtain the rectification term through iteratively solving for Eq. (12) as the Fixed-Point Theorem would predict. As we discussed in section 3a, the subgrid model without any forcing term (*J*(*ψ*^{†}, *q*^{†}) on the left-hand side of Eq. (12) (Fig. 7). The idea is then to rerun the subgrid model with this first guess as the forcing term [

## Acknowledgments.

We would like to acknowledge the editor Paola Cessi along with Elizabeth Yankovsky and two other anonymous reviewers for their comments, which led to significant improvements in the manuscript. We wish to thank Antoine Venaille, Laure Saint Raymond, and William K. Dewar for their insightful comments and suggestions. This work has been supported by the French national program LEFE/INSU. Uchida acknowledges support from the French ‘Make Our Planet Great Again’ (MOPGA) initiative managed by the Agence Nationale de la Recherche under the Programme d’Investissement d’Avenir, with the reference ANR-18-MPGA-0002.

## Data availability statement.

The open-source software for the QG model can be found at github.com/bderembl/msom. It was developed as a module of Basilisk (available at basilisk.fr). Simulation outputs are available upon request.

## APPENDIX A

### Linear Stability Analysis

*ψ*′ by one Fourier component

*k*,

*l*, and

*ω*are the vertical structure of the Fourier mode, the zonal, meridional, and temporal wavenumber, respectively. We span the (

*k*,

*l*) space in order to find

*ω*, which are the eigenvector and the eigenvalue of the equation. If the imaginary part of

*ω*is negative, the corresponding mode is exponentially decaying and the solution is stable but if the imaginary part of

*ω*is positive, the solution is unstable. In the (

*k*,

*l*) space, the most unstable mode corresponds to the solution for which Im(

*ω*) is maximum. We call

*k*and

_{m}*l*, the zonal and meridional wavelength of that most unstable mode, and

_{m}*λ*in Fig. A1. One first important information from these plots is that the large-scale solution is unstable almost everywhere in the domain (except in the small white area at

*y*= 2500 km near the eastern boundary). This was not obvious a priori because we computed the most unstable mode with the same viscosity as the CTRL run and viscosity is known to damp instabilities. We divide the time scale pattern into three distinct dynamical regimes: the western boundary and the intergyre jet which have the fastest growing mode (order 20 days), the return flow near the northern and southern boundary for which the instability time scale is order 60 days, and the rest of the domain for which the instability time scale is greater than 115 days (the color bar saturates beyond this value). We do not consider the instability with long time scale because such long time scales are much bigger than the eddy time scale and become irrelevant for the eddy dynamics (local instability analysis is probably not relevant in areas with such long time scales). The instability length scale is noisier but overall in the area where

(a) Time scale and (b) length scale of the most unstable mode (computed at every fourth grid point).

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

(a) Time scale and (b) length scale of the most unstable mode (computed at every fourth grid point).

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

(a) Time scale and (b) length scale of the most unstable mode (computed at every fourth grid point).

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

When we compare these plot with Fig. 2c, there does not seem to be an obvious link between the local instability parameter and the observed eddy kinetic energy. The path of the jet has a wider signature in the

We use these two fields to build the length scale cutoff of the spatial filter. We start by simply setting *λ _{c}* =

*λ*. However, we argue against using the raw value of

*λ*as shown in Fig. A1b as this field is noisy and also because some instabilities are not relevant to the dynamics. The instabilities irrelevant to mesoscale dynamics occur in places where the instability time scale is greater than the advection time scale (which is on the order of 20 days in most of the gyre, not shown). To get rid of the nonrelevant unstable modes, we adjust the value of

*λ*to 225 km everywhere where

_{c}*αλ*over which we propagate the value of

_{c}*λ*. We take

_{c}*α*= 4.5. This is done to let enough space for all instabilities to develop around the formation site. Several halos overlap at one point and so for each point we retain the maximum value of all halos that are present at that point. We smooth the final map to damp the halo pattern that may have persisted. We plot the final map of

*λ*in Fig. 9.

_{c}## APPENDIX B

### Numerical Implementation

#### a. Spatial filter

The discrete wavelet transform bears some resemblance with the multigrid solver. We define a set of grids from the finest model resolution 2* ^{n}* × 2

*to the coarsest resolution 2° × 2° (one grid point). In our high-resolution model (512 × 512), there are*

^{n}*n*+ 1 = 10 sets of grids. The two key operations in the filtering procedure are

the restriction

$\mathcal{C}$ for which we coarsen a field by averaging 4 neighboring points andthe prolongation

$\mathcal{P}$ for which we refine a field by linear interpolation of neighboring points.

*ψ*is defined on a grid of level

^{l}*l*(2

*× 2*

^{l}*). Then we have*

^{l}*l*as

Hence from the wavelet coefficients, one can reconstruct the field at the finest grid with an iterative procedure. The wavelet coefficients at level *l* hold the information about the structure of the field at length scale of the grid size Δ*l*. To high-pass filter a field with a cutoff length scale *λ _{c}* = Δ

*k*, we simply need to set to zero the wavelet coefficients

*ψ*for

^{l}*l*<

*k*. In the case where

*λ*varies smoothly in space, we can zero the wavelet coefficients locally only.

_{c}#### b. Computation of $\mathcal{R}$

*q*

^{†}as shown in Eq. (20). However, the filtering operation can be numerically expensive. Also, because the large-scale component of

*q*

^{†}grows on a slow time scale, we chose to periodically (every three days) remove the large-scale component of

*q*

^{†}in Eq. (17). We chose this 3-day period because it is comparable to the eddy time scale and was short enough compared to the time needed for large-scale mode to build up observed in Fig. 8a, which is on the order of years. Last, we found that removing the large-scale component of

*q*

^{†}is less efficient than removing the large-scale component of

*ψ*

^{†}and then applying the linear operator

*q*

^{†}and then invert the elliptic equation [Eq. (1)] to compute

*ψ*

^{†}}, we observed a spurious large-scale component in

*ψ*

^{†}. Hence, every three days, we add the term

*t*) and then set

*n*=

_{f}*τ*/Δ

_{f}*t*. Therefore, the cumulative effect of

This time scale separation is similar to ocean models where the barotropic and baroclinic modes are solved with different time stepping (cf. Marshall et al. 1997). The relaxation by our parameterization damps the large-scale component of *q*^{†}, i.e.,

## APPENDIX C

### The Subgrid Model at Coarser Resolution with a Prescribed Background Flow

Given that the prognostic subgrid model [Eq. (17)] solved at mesoscale-resolving resolution is the best our method can achieve (section 3b), we examine the sensitivity of how our closure scales at coarser resolutions. We ran two additional cases of the subgrid model with the resolution of ∼19.5 and ∼39 km (256 and 128 grid points, respectively) keeping the parameters identical to the mesoscale-resolving run except for numerical viscosity. As noted earlier, the first deformation radius is around 25 km, so the two resolutions can be considered mesoscale permitting (Hallberg 2013). The biharmonic viscosities were *A*_{4} = (6.25, 31.25) × 10^{10} m^{4} s^{−1}, respectively. The mean flow and length scale of the spatial filter (*λ _{c}*) were provided by coarse graining them with a 2 × 2 and 4 × 4 boxcar filter, respectively. While we acknowledge there may be more sophisticated approaches to filter the background flow (Aluie et al. 2018; Grooms et al. 2021), the boxcar filter is the simplest operator that commutes with spatial derivatives, and additional terms owing to noncommutative properties between the filter and derivatives do not arise upon coarse graining the background flow.

We show in Fig. C1 the time mean of the eddy kinetic and potential energies from the two runs at coarser resolutions. Notably, the run with 256 grids and eddy rectification forcing performs better than the highest-resolution eddy model without the forcing (Figs. 4 and C1a,b) with the energy levels similar to the eddy energies diagnosed from the CTRL run in the separated jet region (Fig. 2). We also see this from the wavenumber spectra where in the spatial range of ∼300 km, the level of EKE is similar between KE^{†} and KE′ (Fig. 5). Moving to the coarsest resolution, we see that the jet penetration into the gyre deteriorates due to insufficient resolution and high viscosity prohibiting the instabilities to grow (Figs. C1c,d). The lack of energy is apparent in the wavenumber spectra where they fall off too quickly with wavenumber (Fig. 5).

The time mean of kinetic and potential energy (m^{2} s^{−2}) diagnosed from the eddy model at coarser resolutions with the varying spatial filter. The energies from the run with 256 grids in (a) and (b) and 128 grids in (c) and (d) are shown.

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

The time mean of kinetic and potential energy (m^{2} s^{−2}) diagnosed from the eddy model at coarser resolutions with the varying spatial filter. The energies from the run with 256 grids in (a) and (b) and 128 grids in (c) and (d) are shown.

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

The time mean of kinetic and potential energy (m^{2} s^{−2}) diagnosed from the eddy model at coarser resolutions with the varying spatial filter. The energies from the run with 256 grids in (a) and (b) and 128 grids in (c) and (d) are shown.

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

With the numerical viscosity as a tuning parameter, we end this appendix by showing the dependency of the system on it. Figure C2 shows the ratio between domain-integrated EKE diagnosed from the CTRL run and respective mesoscale-permitting subgrid models plotted against the numerical viscosity. The runs we show in Fig. C1 were taken from the runs with the highest viscosity, respectively. As we decrease the viscosity, the level of EKE increases as expected, with the run with 128 grids showing a strong dependency. While the subgrid model with a prescribed double-gyre background flow could be run stably with small numerical viscosity in respect to its resolution, the poorly resolved instabilities tended to excite Rossby waves in the gyre interior (not shown), which accumulated at the western boundary (the western boundary current is too zonally broad in Fig. C1c). This caused the domain integrated EKE to be larger than that diagnosed from the CTRL run, namely, values larger than unity in Fig. C2. The transition of the dynamical regime from Rossby waves to mesoscale eddies depending on model resolution has also been documented in realistic ocean simulations (Constantinou and Hogg 2021).

A scatterplot showing the ratio between area integrated

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

A scatterplot showing the ratio between area integrated

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

A scatterplot showing the ratio between area integrated

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0217.1

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