1. Introduction
It is crucial that climate models are able to accurately simulate the climate’s internal variability, in addition to the climate’s mean state and externally forced climate changes (IPCC 2013). For example, a correct representation of internal climate variability is needed in climate change detection and attribution studies. Such studies are based on signal-to-noise estimates for which the climate’s intrinsic low-frequency variability (LFV) must be estimated, at least in part, from long control integrations of climate models. Also for climate prediction studies a correct representation of the intrinsic climate variability is crucial such that internally generated sources of predictability can be exploited. Finally, the ability of models to make quantitative projections of changes in climate variability, including the statistics of extreme events under a warming climate, is dependent on an accurate representation of the climate’s internal variability.
The climate’s intrinsic LFV is typically described by large-scale modes of climate variability, which are often either statistical eigenmodes (e.g., EOFs) or dynamical eigenmodes (e.g., linear instability modes; see von Storch and Zwiers 1999; IPCC 2013; Dijkstra 2016). The modes of climate variability are characterized as large scale because they include large spatial structures such as basinwide coupled modes of ocean–atmosphere variability (e.g., El Niño–Southern Oscillation), Rossby wave trains, midlatitude jets and storm tracks, and so on.
In particular with respect to the ocean, a number of LFV modes (i.e., from multiannual to multidecadal time scales) have been described (Deser et al. 2010; Dijkstra 2016). It is clear from observations that multidecadal patterns of sea surface temperature variability exist, such as the Atlantic multidecadal oscillation (Schlesinger and Ramankutty 1994; Kerr 2000) and the Pacific decadal oscillation (Mantua et al. 1997; England et al. 2014). Most of these modes have a particular regional or even global manifestation whose amplitude can be larger than that of human-induced climate change. For example, intrinsic multidecadal variability of the ocean heat content has been held responsible for the relatively low recent trend in the global mean surface temperature anomaly, also referred to as the “global warming hiatus” (Meehl et al. 2011, 2013).
However, care is required when interpreting modes of climate variability since 1) their interpretation depends on how one separates modes of variability from forced changes in the time mean, 2) they may change drastically in space, structure, or probability distribution in response to climate change, and 3) in strongly nonlinear regimes they may be not strictly large scale but the large-scale structures can be entangled with smaller-scale structures such that some modes of climate variability may not be entirely representable in climate models with a too coarse spatial resolution.
Moreover, due to the fact that the real ocean dynamics resides in a highly turbulent regime (with a large Reynolds number leading to a high-dimensional unstable manifold on the attractor) it still remains hard to understand the exact physical mechanisms behind the ocean’s LFV (Berloff and McWilliams 1999a; Hogg et al. 2005; Dijkstra 2016). Plenty of model studies analyzing eddy-resolving ocean models show that LFV in such models is commonplace (Berloff and McWilliams 1999a; Hogg et al. 2005), and it is now known that the collective action of oceanic mesoscale eddies is one of the main drives of the midlatitude LFV (Kwon 2010). But at the same time the strong eddy field can obscure many features of the circulation, making it difficult to agree upon the mechanisms underpinning the variability (Hogg et al. 2005; Dijkstra 2016).
Central questions still need further clarification: Which part of the ocean’s LFV is completely intrinsic to the ocean and which part involves a dynamical coupling to the atmosphere? Which part of the ocean’s intrinsic LFV can be traced back to stationary modes at high viscosity (i.e., low-order bifurcations) and which part represents a genuinely eddy-driven turbulent phenomenon (i.e., physical mechanisms solely active at high Reynolds numbers) (Hogg and Blundell 2006; Berloff et al. 2007; Le Bars et al. 2016; Dijkstra 2016)?
Clarification of these questions is hampered by the fact that computational limitations force most studies on climate variability to employ climate models with ocean components that do not resolve the internal Rossby deformation radius (Hallberg 2013). In these coarse-resolution ocean models (typically operating at a horizontal resolution of 1°) usually deterministic eddy parameterizations are applied that are based on diffusive terms that aim to model the potential and kinetic energy transfer from the mean field to the eddy field. These diffusive eddy parameterizations achieve a reasonable representation of the time-mean effect of the mesoscale eddy field on the time-mean flow (Bryan et al. 2014; Griffies et al. 2015; Viebahn et al. 2016), but they are not able to excite the ocean’s internal LFV observed in eddy-resolving ocean model simulations (Le Bars et al. 2016). Consequently, the estimation of internal variability uncertainty (stemming from the chaotic nature of the system) in climate change detection or projections of climate change is still strongly hampered by model uncertainty (i.e., limitations of a model’s representation of the chaotic nature of the system) in many current climate change studies.
Hence, the search for suitable eddy parameterizations remains a challenging theoretical topic with clear practical dimension. Recently, efforts have been made toward eddy parameterizations that aim to step out of the diffusive parameterization framework and try to represent the eddy effects in terms of stochastic eddy forcing (Berloff 2005c; Grooms and Majda 2013; Porta Mana and Zanna 2014; Verheul et al. 2017). Stochastic climate modeling is based on the concept of scale separation in time (Franzke et al. 2015), namely, that the state vector of the system can be decomposed into fast modes and slow (low frequency) modes such that the time scales of these modes strongly differ. The impact of the fast modes on the slow modes appears as eddy forcing in the equations of motion for the slow modes. The development of stochastic climate models then proceeds by accounting for the effects of the unresolved fast modes in a stochastic fashion.
Moreover, for models formulated in physical space (like most ocean models) the essential difference between a high-resolution model and a low-resolution model is the extent of spatial information. The eddy forcing actually represents the impact of the spatially unresolved (or subgrid scale or small scale) processes on the spatially resolved (or larger scale) processes. Consequently, for models in physical space time-scale separation should imply scale separation in space. That is, the patterns associated with slow variability should exhibit strictly large-scale spatial structures whereas the patterns associated with fast variability should show strictly small-scale spatial structures. Otherwise, the slow modes and the fast modes cannot be disentangled on the low-resolution model grid.
However, scale separation only holds for regimes in which scales are weakly coupled whereas in turbulent regimes different scales are strongly nonlinearly coupled. The lack of time-scale separation introduces non-Markovian memory effects and complicates the derivation of systematic parameterizations. The lack of scale separation in space implies that the dynamical modes are multiscale patterns both in the horizontal and vertical directions. For example, for the midlatitude ocean gyres it is found that due to the background flow most eigenmodes contain a large variety of scales (Shevchenko et al. 2016). In this case, the LFV is not a single-mode pattern, but rather a coherent pattern phenomenon consisting of a large number of short period phase-related eigenmodes interacting with each other. We note that this can apply to both (high resolution) statistical eigenmodes like EOFs (Gille and Kelly 1996) and linear eigenmodes on a background flow (Shevchenko et al. 2016). Obviously, the small-scale structures of the dynamical modes are not resolvable on a low-resolution model grid.
In this study, we approach the formulation of eddy parameterizations in the following way: First, we define the resolved dynamics and the corresponding instantaneous eddy forcing via spatial filtering (instead of, e.g., temporal averaging) such that we can account for the representation error of the equations of motion on the low-resolution model grid. Second, we represent the impact of the unresolved dynamics on the resolved dynamics (i.e., the eddy forcing) in terms of a series expansion in dynamical spatial modes that stem from the energy budget of the resolved dynamics. These so-called energy modes exhibit strictly large-scale spatial patterns and are equipped with a clear physical interpretation.
In section 2, the resolved dynamics and the related instantaneous eddy forcing are defined. We describe our eddy-resolving ocean model and its LFV in section 2a. Our spatial filtering approach is introduced in section 2b. The corresponding filtered equations of motion and the related eddy forcing terms are presented in section 2c, and in section 2d we analyze the resulting large-scale and small-scale energetics. Subsequently, section 3 deals with developing and testing closures with a focus on the baroclinic energy pathway. We show how we can test parameterizations in a high-resolution model (section 3a) and test the performance of the standard closure of the baroclinic energy pathway in ocean models [i.e., the Gent–McWilliams (GM) parameterization; Gent and McWilliams 1990] in section 3b. Finally, in section 3c we define and test the representation of the eddy forcing in energy modes with a focus on representing the baroclinic energy pathway. We end with a summary (section 4) and discussion (section 5).
2. Framework: Eddy forcing of the large-scale flow
Our general starting point is the following (see, e.g., Berloff 2005a): First, an eddy-resolving (ER) model is given (section 2a) in order to obtain a reference solution, say with state vector ψ, which contains both the large-scale and eddy components. Second, a non-eddy-resolving (non-ER) model is supposed to have the same general setup as the ER model (e.g., type of governing conservation equations, domain size, and boundary conditions; see section 2c), but the former has a significantly coarser horizontal grid resolution (by a factor of 10 in this study). Consequently, the non-ER model has far fewer degrees of freedom and it can only solve for the large-scale flow evolution. Moreover, the non-ER model may contain additional dynamical terms in the governing conservation equations (e.g., the current deterministic eddy parameterizations) that are supposed to parameterize part of the interactions between large-scale components and (sub)mesoscale eddy components.
Finally, the eddy forcing (EF) is a (not necessarily unique) dynamical term that still needs to be added to the governing conservation equations of the non-ER model at hand such that the non-ER solution, say with state vector 
a. Eddy-resolving ocean model exhibiting low-frequency variability
Figure 1 shows a snapshot (Fig. 1c) and a temporal average (Fig. 1d) of the upper-layer streamfunction [similar to Figs. 1a,c in Berloff (2005a), and Figs. 1a and 2a in Berloff (2005c)]. The upper-ocean time-mean circulation (Fig. 1d) consists of the southern (subtropical) and northern (subpolar) gyres that fill about 2/3 and 1/3 of the basin, respectively, which is consistent with the wind stress pattern. The time-mean flow is characterized by the Sverdrup balance in most parts of the basin. Only in regions related to the pair of the western boundary currents and their eastward jet (EJ) extensions do nonlinear and frictional terms become dominant (Pedlosky 1996). We note that for our specific model setup the boundary currents do not merge with each other but the subpolar gyre enters the subtropical region near the western boundary such that the point of separation from the coast of the subtropical western boundary current is pushed southward relative to the line of zero wind stress curl (similar to Berloff 2005a,c). This is a robust regime that appears at large Reynolds number in the stratified and baroclinically unstable double-gyre flow with no-slip boundary conditions (e.g., Haidvogel et al. 1992; Berloff and McWilliams 1999b; Siegel et al. 2001). In terms of the fluctuations, the basin can be partitioned into the more energetic “western” part, characterized by strong vortices, and the less energetic “eastern” part, dominated by the planetary waves [see Berloff et al. (2002) for details].
Large-scale component of an upper-layer streamfunction (a) snapshot and (b) time mean. (c),(d) The corresponding reference (i.e., unfiltered) upper-layer streamfunctions. Anomalies of the reference (i.e., unfiltered) upper-layer streamfunction [with respect to the time mean shown in (d)] corresponding to (e) a low and (f) a high in the low-frequency variability of PE (see Fig. 2a). The contour interval in all panels is 2.5 Sv (1 Sv ≡ 106 m3 s−1).
Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0117.1
Figure 2 shows the temporal evolution of the energetics of the reference simulation. The PE (Fig. 2a) exhibits clear cycles of decadal variability. The about 4 times smaller KE also shows cycles of decadal variability, which lags the variability in PE by 1–2 years. The wind energy input 
Time series of (a) energy reservoirs, (b) energy generation and dissipation, (c) conversion between large-scale and small-scale kinetic energy, (d) conversion between large-scale kinetic energy and large-scale available potential energy, (e) conversion between large-scale available potential energy and small-scale available potential energy, and (f) temporal tendency of the large-scale available potential energy reservoir. Note that the last three terms constitute the large-scale available potential energy budget [Eq. (27)].
Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0117.1
Figure 1 also shows upper-layer streamfunction anomalies corresponding to a low (Fig. 1e) and a high (Fig. 1f) in PE (see years 52 and 56 in Fig. 2a). These anomaly patterns are similar to those shown in Berloff et al. (2007) (see their Figs. 2 and 4), and demonstrate that the variability is concentrated around the subtropical EJ. More precisely, the decadal transitions are related to coherent meridional shifts and variations of the intensity of the subtropical EJ, likely governed by the nonlinear adjustment of the combined EJ–eddies system [see Berloff et al. (2007) for details].
b. Flow decomposition into large-scale and eddy components via spatial mode filtering
Scale-content (or image) consistency (SCC): The scale content of the spatial filter modes
Boundary conditions consistency (BCC): The large-scale flow is supposed to be a solution of the non-ER model equations and, hence, has to satisfy its boundary conditions. In the governing model equations, the differential operator with the highest-order derivative (typically related to dissipation) determines the number of boundary conditions that have to be specified. Consequently, the eigenmodes of this differential operator represent a set of spatial modes which are always able to satisfy the boundary conditions (i.e., span the correct function space), and, hence, represent the first choice if the Fourier modes (i.e., eigenfunctions of the Laplacian) cannot satisfy the boundary conditions.
Dynamical consistency (DC): The conservation equations governing the evolution of
Figure 3 shows selected leading eigenmodes (orthonormalized) of the bi-Laplacian with no-slip boundary conditions computed on the high-resolution (i.e., 10 km) grid. The overall structure (i.e., the scale content) of the 
Selected leading eigenmodes (orthonormalized) of the bi-Laplacian with no-slip boundary conditions computed on the high-resolution grid (i.e., 3492 grid points).
Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0117.1
First 300 eigenvalues of the bi-Laplacian (left axis) for the ER model (blue) and the non-ER model (red), and the relative difference (right axis), i.e., 
Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0117.1
Figure 1 also shows the corresponding snapshot (Fig. 1a) and temporal average (Fig. 1b) of the large-scale (i.e., filtered with 
c. Conservation equations of the large-scale flow
d. Lorenz energy cycle
Figure 5 shows the different terms of the LEC averaged in time (over 500 years of daily output) and summarizes the time-mean state and variance of the different energy reservoirs and energy pathways (for the reference simulation described in section 2a). The filtered terms 
Lorenz energy cycle (temporal average and standard deviation) of the two-layer QG model based on spatial filtering and 500 years of daily output. Shown are the reservoirs of large-scale available potential energy ([PE]), large-scale kinetic energy ([KE]), eddy available potential energy (PE′), and eddy kinetic energy (KE′), as well as the corresponding energy generation G, dissipation D, and conversion C terms.
Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0117.1
The overall picture is similar to the one found in realistic global ocean models (e.g., von Storch et al. 2012). Common in both the ocean and the atmosphere is that the dominant power pathway is the baroclinic pathway [PE] 
Figure 2 shows different terms of the LEC evolving in time. The variability in 
The large-scale wind energy input 
3. Closures for the baroclinic energy pathway
In stratified flows two distinctively different types of energy conversions between large-scale and eddy components exist: the energy conversion 
a. Testing closures in an eddy-resolving model
Then a parameterization of 
b. Standard GM parameterization
1) Constant GM diffusivity
Figure 6a shows time series of PE resulting from simulations in which the GM parameterization with a constant KGM is employed (blue and green lines). To assure numerical stability KGM ≥ 1500 m2 s−1 is necessary in our model.8 The GM parameterization does its job by extracting [PE] from the large-scale flow such that a statistical equilibrium results. However, the low-frequency variability exhibited by the reference simulation (black line) is absent. The dynamics exclusively reside below the PE-level of the low-PE regime of the reference simulation. That is, the low-frequency transitions in phase space to the high-PE regime are suppressed in case the GM parameterization with a constant KGM is used. Presumably, backscatter is necessary for the dynamics in order to be able to reach high-PE states.
(a) Potential energy corresponding to the reference simulation (black; same as in Fig. 2a), and for simulations in which the GM parameterization [Eq. (39)] is employed with KGM either a constant (blue, green) or given via Eq. (43) (red). (b) Estimated probability density function of KGM [computed via Eq. (43)] for a simulation in which the GM parameterization is employed (red) and for the reference simulation (black; the GM parameterization is not employed in the model but KGM is just diagnosed). The average and standard deviation are 1585 ± 1373 m2 s−1 (red) and 448 ± 697 m2 s−1 (black).
Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0117.1
2) Time-dependent GM diffusivity
Figure 6a shows the time series of PE resulting from a simulation in which the GM parameterization is employed with KGM obtained from projection [i.e., Eq. (43)] at every time step (red line). Now a form of low-frequency variability is indeed excited but the corresponding high-PE regime resides at and below the low-PE regime of the reference simulation (black line), and the PE variability has smaller variance (see also the second row of Table 1). The low-frequency variability actually oscillates around the PE level of the simulation in which the GM parameterization with KGM = 1500 m2 s−1 is employed (blue line). This is consistent with the fact that the time mean of KGM obtained from projection is given by 
Temporal average and standard deviation of PE, the GM diffusivity KGM, and the relative error of the eddy forcing for different model setups with 
Figure 6b shows the estimated pdf of KGM computed from Eq. (43) and either employed in the GM parameterization (red) or just diagnosed from the reference simulation (black). Both KGM distributions are unimodal and slightly positively skewed. Most striking, however, is that in both cases KGM captures a significant amount of negative values. Negative KGM values are not consistent with a diffusion model. Hence, the low-frequency variability (red line in Fig. 6a) presumably emerges from the wrong reason, namely, backscatter due to a negative diffusivity. We also note that the temporal average and standard deviation of KGM is significantly smaller when only diagnosed from the reference simulation (448 ± 697 m2 s−1) than when the GM parameterization is actually applied (1585 ± 1374 m2 s−1; see also the second row of Table 1). We discuss this difference in detail in the next section [sections 3c(3) and 3c(4)].
Finally, we emphasize that the relative error 
c. Dynamical spatial mode representation of the eddy forcing based on energetics
Optimally, the spatial modes 
1) Specification of spatial energy modes
2) Selection of finite subset of spatial energy modes
In other words, the subspace spanned by 
The expansion coefficients in Eq. (56) are computed at each model time step by using ordinary least squares with respect to 
3) Approximation of the eddy forcing on the reference attractor
In the following we analyze the approximation of the eddy forcing 
(i) Relative error of eddy forcing
Figure 7a shows the time-mean relative error of the eddy forcing for the filter mode expansion either with GM term [black, Eq. (59)] or without GM term [red, Eq. (58)]. The two curves are almost identical, which demonstrates that the GM term is not able to significantly reduce the relative error of the eddy forcing. In other words, the GM field (as a direction in phase space) is largely orthogonal to the eddy forcing field. For both curves the decrease in relative error (i.e., the slope of the curve) is minimal at the beginning and monotonically increasing with increasing number of filter modes. The value of the relative error for the filter modes is on the order of 10−1 and only reaches a very small value [i.e., O(10−13)] when all filter modes are used. That is, the convergence of Eqs. (58) and (59) is slow.
Time-mean relative error of eddy forcing, 
Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0117.1
Figure 7b shows the time-mean relative error of the eddy forcing for the energy mode expansion Eq. (57) with Δτ = 3 h (blue), Δτ = 6 h (black), and Δτ = 12 h (magenta). Note the logarithmic scale on the ordinate. The effect of the GM term on the relative error is again very small such that the curves related to Eq. (56) are indistinguishable from the shown curves. In contrast to the filter modes, the decrease in relative error (i.e., the slope of the curve) is maximal at the beginning and monotonically decreasing with increasing number of energy modes (for comparison the curve of the filter modes is shown by the blue dashed line). With only four energy modes used in Eq. (56) the relative error drops to O(10−5) and for Δτ = 3 h a relative error of O(10−12) is reached with 30 energy modes. That is, the convergence of Eqs. (56) and (57) is very fast since adding energy modes reduces the order of magnitude of the relative error. Finally, it holds for the reference simulation that the smaller 
(ii) GM diffusivity
Figure 8a shows the time-mean GM diffusivity KGM for the filter mode expansion with GM term [blue, Eq. (59)]. The GM diffusivity KGM decreases nearly linearly due to the subsequent inclusion of more and more filter modes. However, the value of KGM remains on the order of 100 m2 s−1. Only when almost all filter modes are included the value of KGM becomes small and, hence, the impact of the GM term is insignificant.
Time mean of the GM diffusivity KGM for (a) the series expansion in Eq. (59) and (b) the series expansion in Eq. (57). Here “ref” refers to the reference simulation (
Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0117.1
Figure 8b shows the time-mean GM diffusivity KGM for the energy mode expansion with GM term [Eq. (57)] with Δτ = 3 h (blue), Δτ = 6 h (black), and Δτ = 12 h (magenta). The behavior of KGM resembles the behavior of the relative error (Fig. 7b). Note again the logarithmic scale on the ordinate. Including energy modes drastically reduces the value KGM, that is, by orders of magnitude. With only four energy modes used in Eq. (57) the value of KGM drops to 
4) Approximation of the eddy forcing in the presence of error perturbations
In this section we analyze the approximation of the eddy forcing 
(i) Relative error of eddy forcing
Figure 7a shows the time-mean relative error of the eddy forcing for the filter mode expansion either with the GM term [blue, Eq. (59); see also Table 1] or without the GM term [magenta, Eq. (58)]. The relative error for the simulations with error perturbations is slightly smaller than for the reference simulation (black and red curves). Nevertheless, the overall behavior is very similar to the reference simulation: The blue and magenta curves are nearly identical, which indicates that the GM term is not able to significantly reduce the relative error of the eddy forcing. For both curves the decrease in relative error (i.e., the slope of the curve) is minimal at the beginning and monotonically increasing with increasing number of filter modes. The convergence of Eqs. (58) and (59) is slow since the value of the relative error is on the order of 10−1 and only reaches a very small value [i.e., O(10−13)] when all filter modes are used.
A crucial point in Fig. 7a is that the filter mode expansion without the GM term [magenta, Eq. (58)] leads to a model blow-up if fewer than 10 filter modes are used (the magenta curve only starts when the number of modes = 10). On the other hand, the filter mode expansion with GM term [blue; Eq. (59)] leads to stable model simulations for any number of filter modes (see also Table 1). Hence, the effect of the GM term becomes clearer: the GM term cannot not significantly reduce the relative error of the eddy forcing but it can stabilize the model. In dynamical systems terms the GM term acts as a stabilizing direction in phase space. That is, the GM term cannot direct the system’s state along the attractor (it cannot excite the intrinsic low-frequency variability transitions in phase space as done by unstable directions) but it mainly keeps the system from diverging.
Figure 7b shows the time-mean relative error of the eddy forcing for the energy mode expansion Eq. (57) with Δτ = 3 h (red; see also Table 1), Δτ = 6 h (green), and Δτ = 12 h (cyan). Note the logarithmic scale on the ordinate. The effect of the GM term on the relative error is again very small such that the curves related to Eq. (56) are indistinguishable from the shown curves. On the other hand, the stabilizing effect of the GM term also appears for the energy modes: for the application of Eq. (56) (i.e., energy mode expansion without GM term) with only two energy modes the model blows up whereas for the application of Eq. (57) (i.e., energy mode expansion with GM term) the model is stable.
The overall behavior of the relative error for the simulations with error perturbations (red, green, cyan) is similar to the results of the reference simulation (blue, black, magenta). That is, the relative error decreases much faster (adding energy modes reduces the order of magnitude of the relative error) than for the filter modes (shown for comparison by the blue dashed line). However, because of the induced error perturbations the decrease in relative error is weaker than for the reference simulation. For example, for 30 energy modes and Δτ = 3 h the relative error is O(10−5) instead of O(10−12) for the reference simulation. Moreover, the impact of 
(ii) GM diffusivity
Figure 8a shows the time-mean GM diffusivity KGM for the filter mode expansion with GM term [red, Eq. (59); see also Table 1]. The behavior is largely similar to the results of the reference simulation (blue), namely, the GM diffusivity KGM decreases nearly linearly due to the subsequent inclusion of more and more filter modes. However, the value of KGM is significantly larger (about one order of magnitude) than when diagnosed from the reference simulation. This in accordance with the interpretation of the GM term as a stabilizing direction in phase space because in the presence of error perturbations (driving the system away from the attractor) the eddy forcing will project more on stable directions (driving the system back to the attractor). In other words, in the presence of error perturbations the GM term has work to do.
Figure 8b shows the time-mean GM diffusivity KGM for the energy mode expansion with GM term [Eq. (57)] with Δτ = 3 h (red; see also Table 1), Δτ = 6 h (green), and Δτ = 12 h (cyan). The behavior is largely similar to the results of the reference simulation (blue, black, magenta): including energy modes drastically reduces the value KGM, that is, by orders of magnitude. It also largely holds that the smaller 
(iii) Time series of potential energy
Figure 9 shows time series of PE related to simulations employing the filter mode expansion with GM term [red, Eq. (59); see also Table 1] and the energy mode expansion with GM term and Δτ = 3 h [blue, Eq. (57); see also Table 1]. For comparison the time series of the PE of the reference simulation is shown in black.
Time series of PE for the reference simulation (black) and for simulations with 
Citation: Journal of Physical Oceanography 49, 4; 10.1175/JPO-D-18-0117.1
The energy mode expansion exhibits monotonic and fast convergence behavior in terms of PE (i.e., low-frequency variability). If only two energy modes are used (Fig. 9d) the PE variability is still significantly different from the reference PE. Intense low-frequency variability is present but it is situated between the low-PE regime of the reference simulation and another very-low-PE regime. Already with four energy modes in the expansion the high-PE regime of the reference simulation is regularly reached (not shown). But the low-PE regime is still bit lower than for the reference case. For six or more energy modes (Figs. 9f,h,j) the PE variability of the reference simulation appears to be essentially recovered.
As expected, the situation is different for the filter mode expansions. The convergence behavior is nonmonotonic. Even for 20 filter modes (Fig. 9i) the PE variability is significantly different from the reference simulation. When using filter modes it appears to be difficult to reach the high-PE regime of the reference simulation. Either the PE variance is significantly smaller than for the reference simulation (Figs. 9c,g) or the low-PE regime is lower than for the reference simulation (Figs. 9e,i). This is also visible in Table 1.
4. Summary
The three key points of this study can be summarized as follows: First, we propose a new approach to parameterizing subgrid-scale processes. In this approach the impact of the unresolved dynamics on the resolved dynamics (i.e., the eddy forcing) is represented by a series expansion in dynamical spatial modes stemming from the energy budget of the resolved dynamics. More precisely, the so-called energy modes are directly obtained from the equations of motion of the resolved flow by identifying the integral kernels that lead to the different reservoir, generation, dissipation, and conversion terms in the large-scale energy budget. Hence, the energy modes exhibit strictly large-scale patterns and they are equipped with a clear physical interpretation in terms of energetics. Convergence toward the eddy forcing is accomplished via delay embedding by including additional realizations of these fields further in the past. We also note the relation to the Mori–Zwanzig formalism, which indicates that the representation of unresolved physics needs to include a memory term that involves the past history of the resolved physics. For the two-layer QG ocean model considered in this study, we demonstrate that the convergence of a series expansion in the energy modes is orders of magnitude faster than the convergence of a series expansion in Fourier-type modes. That is, the eddy forcing can be accurately approximated with a very limited number of energy modes, which enables a feasible parameterization.
Second, we explore a novel way to test parameterizations in models. The resolved dynamics and the corresponding instantaneous eddy forcing are defined via spatial filtering, which accounts for the representation error of the equations of motion on the low-resolution model grid. In this way closures can be tested within the high-resolution model. Whereas in low-resolution models all energy pathways between large-scale and eddy components must be parameterized simultaneously, testing parameterizations in the high-resolution model offers the possibility to isolate the effects of a single parameterization (related to a single energy pathway) while the other large-scale eddy energy conversions are correctly computed. For the two-layer QG ocean model considered in this study, we focus on parameterizations of the baroclinic energy pathway while the barotropic energy pathway is correctly computed by the high-resolution model.
Third, we test the standard closure of the baroclinic energy pathway in the ocean components of state-of-the-art climate models [i.e., the Gent–McWilliams (GM) parameterization with a scalar diffusivity] in the high-resolution QG ocean model considered in this study. It turns out that the GM field steers trajectories along a stabilizing direction in phase space. That is, the GM field does not project well on the eddy forcing (it exhibits a very high relative error) and fails to excite the model’s intrinsic low-frequency variability (i.e., it is not able to propagate the model’s state along the correct attractor e.g., along an unstable direction). The GM field mainly stabilizes the model. That is, if the representation of the eddy forcing is very inaccurate (e.g., small number of modes used in expansion) the GM term performs the necessary dissipation of available potential energy such that the model does not diverge.
5. Discussion
Finally, we elaborate on open issues of this study and related future research directions.
a. Self-consistent closure of the baroclinic energy pathway
A closure of the baroclinic energy pathway is self-consistent if it does not involve the actual (“true”) baroclinic eddy forcing. However, in this study we still use 
For example, a problem related to these issues and well-known in statistics and machine learning is the issue of overfitting versus underfitting or the bias/variance trade-off. Low-bias approaches can usually give accurate representation of the data but produce large variances. In contrast, models with higher bias produce lower variances but less accurate representations. Regularization methods introduce bias into the regression solution that can reduce variance considerably. In this way the behavior of the expansion coefficients becomes simpler and easier to model but the approximation becomes less accurate.
b. Self-consistent closure of the barotropic energy pathway
To make the equations for the large-scale flow [Eqs. (13) and (14)] completely self-consistent one also has to specify a self-consistent closure of the barotropic energy pathway (i.e., 
c. Dynamical systems analysis of the large-scale flow in the turbulent regime
As soon as adequate closures for both the baroclinic and the barotropic energy pathways are available it is in principle possible (i.e., feasible due to low model resolution) to analyze the dynamics of the large-scale flow in the turbulent regime in a systematic way. In the case of deterministic closures this is related to the existence of multiple equilibria, stability properties, bifurcations, and chaotic attractors (Dijkstra 2005). In the case of stochastic closures the investigation will be from the perspective of random dynamical systems, which is related to stochastic bifurcations (i.e., changes in the probability density function), pullback attractors, and invariant measures (Dijkstra 2013). We note that in case of low model resolutions a whole set of numerical techniques to investigate transitions in stochastic dynamical systems becomes feasible (Dijkstra et al. 2016). For example, it becomes possible to numerically solve the stochastic partial differential equations (SPDEs) via dynamical mode expansions (Sapsis and Lermusiaux 2009) and to investigate the interaction of external noise forcing with internal nonlinear variability in the turbulent regime (Sapsis and Dijkstra 2013).
d. Comparison with other approaches to eddy parameterization
In this study, we compared turbulence closures based on energy modes with the GM eddy parameterization approach. We focused on a positive and spatially constant KGM because it is straightforward to diagnose (e.g., not entering issues around rotational eddy fluxes), and, more importantly, because a spatially homogenous KGM is still regularly applied in state-of-the-art realistic ocean models. On the other hand, the estimation and performance of a spatially inhomogeneous (and possibly tensor-valued) KGM remains a crucial topic (Eden et al. 2007, 2009; Viebahn and Eden 2010). The relation between energy modes and the GM parameterization, as well as other approaches to eddy parameterization (Porta Mana and Zanna 2014; Jansen and Held 2014; Bachman et al. 2018), will hopefully be further elucidated in future studies.
e. More realistic ocean model configurations
The ocean model considered in this study is situated at the more idealized end in the hierarchy of ocean models. Several features and processes must be included in order to make the details more realistic. These include higher vertical resolution, diabatic terms like buoyancy forcing and buoyancy sinks, and realistic topography and coastlines. We are currently extending our results to a three-layer model including realistic topographic interactions. Eventually, one also has to consider the primitive equations in order to be able to investigate global realistic ocean models. The corresponding energy budgets are more complicated but detailed analyses are becoming available nowadays (von Storch et al. 2012; Wu et al. 2017; Jüling et al. 2018).
f. Climate model simulations subject to intrinsic (eddy driven) low-frequency variability
Finally, when adequate closures for the energy pathways in realistic ocean models are available then long-period low-resolution climate model simulations exhibiting eddy-driven low-frequency variability become possible. This is crucial since then issues related to anthropogenic climate change (forced variability) versus intrinsic low-frequency variability (internal variability) can be addressed in a statistically significant manner.
Acknowledgments
The authors thank Inti Pelupessy, Alexis Tantet, and Fred Wubs for helpful discussions. This research is funded by the Netherlands Organization for Scientific Research (NWO) through the Vidi project “Stochastic models for unresolved scales in geophysical flows.” HD acknowledges support by the Netherlands Earth System Science Centre (NESSC), financially supported by the Ministry of Education, Culture and Science (OCW), Grant 024.002.001.
REFERENCES
Bachman, S. C., J. A. Anstey, and L. Zanna, 2018: The relationship between a deformation-based eddy parameterization and the LANS-α turbulence model. Ocean Modell., 126, 56–62, https://doi.org/10.1016/j.ocemod.2018.04.007.
Berloff, P., 2005a: On dynamically consistent eddy fluxes. Dyn. Atmos. Oceans, 38, 123–146, https://doi.org/10.1016/j.dynatmoce.2004.11.003.
Berloff, P., 2005b: On rectification of randomly forced flows. J. Mar. Res., 63, 497–527, https://doi.org/10.1357/0022240054307894.
Berloff, P., 2005c: Random-forcing model of the mesoscale oceanic eddies. J. Fluid Mech., 529, 71–95, https://doi.org/10.1017/S0022112005003393.
Berloff, P., and J. C. McWilliams, 1999a: Large-scale, low-frequency variability in wind-driven ocean gyres. J. Phys. Oceanogr., 29, 1925–1949, https://doi.org/10.1175/1520-0485(1999)029<1925:LSLFVI>2.0.CO;2.
Berloff, P., and J. C. McWilliams, 1999b: Quasigeostrophic dynamics of the western boundary current. J. Phys. Oceanogr., 29, 2607–2634, https://doi.org/10.1175/1520-0485(1999)029<2607:QDOTWB>2.0.CO;2.
Berloff, P., J. C. McWilliams, and A. Bracco, 2002: Material transport in oceanic gyres. Part I: Phenomenology. J. Phys. Oceanogr., 32, 764–796, https://doi.org/10.1175/1520-0485(2002)032<0764:MTIOGP>2.0.CO;2.
Berloff, P., A. Hogg, and W. Dewar, 2007: The turbulent oscillator: A mechanism of low-frequency variability of the wind-driven ocean gyres. J. Phys. Oceanogr., 37, 2363–2386, https://doi.org/10.1175/JPO3118.1.
Bryan, F. O., P. R. Gent, and R. Tomas, 2014: Can Southern Ocean eddy effects be parameterized in climate models. J. Climate, 27, 411–425, https://doi.org/10.1175/JCLI-D-12-00759.1.
Deser, C., M. A. Alexander, S.-P. Xie, and A. S. Phillips, 2010: Sea surface temperature variability: Patterns and mechanisms. Annu. Rev. Mar. Sci., 2, 115–143, https://doi.org/10.1146/annurev-marine-120408-151453.
Dijkstra, H. A., 2005: Nonlinear Physical Oceanography. Springer, 532 pp.
Dijkstra, H. A., 2013: Nonlinear Climate Dynamics. Cambridge University Press, 367 pp.
Dijkstra, H. A., 2016: A normal mode perspective of intrinsic ocean-climate variability. Annu. Rev. Fluid Mech., 48, 341–363, https://doi.org/10.1146/annurev-fluid-122414-034506.
Dijkstra, H. A., and J. P. Viebahn, 2015: Sensitivity and resilience of the climate system: A conditional nonlinear optimization approach. Commun. Nonlinear Sci. Numer. Simul., 22, 13–22, https://doi.org/10.1016/j.cnsns.2014.09.015.
Dijkstra, H. A., and Coauthors, 2016: A numerical framework to understand transitions in high-dimensional stochastic dynamical systems. Dyn. Stat. Climate Syst., 1, dzw003, https://doi.org/10.1093/climsys/dzw003.
Eden, C., R. J. Greatbatch, and J. Willebrand, 2007: A diagnosis of thickness fluxes in an eddy-resolving model. J. Phys. Oceanogr., 37, 727–742, https://doi.org/10.1175/JPO2987.1.
Eden, C., M. Jochum, and G. Danabasoglu, 2009: Effects of different closures for thickness diffusivity. Ocean Modell., 26, 47–59, https://doi.org/10.1016/j.ocemod.2008.08.004.
England, M. H., and Coauthors, 2014: Recent intensification of wind-driven circulation in the Pacific and the ongoing warming hiatus. Nat. Climate Change, 4, 222–227, https://doi.org/10.1038/nclimate2106.
Franzke, C., T. O’Kane, J. Berner, P. Williams, and V. Lucarini, 2015: Stochastic climate theory and modeling. Wiley Interdiscip. Rev.: Climate Change, 6, 63–78, https://doi.org/10.1002/wcc.318.
Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr., 20, 150–155, https://doi.org/10.1175/1520-0485(1990)020<0150:IMIOCM>2.0.CO;2.
Gille, S. T., and K. A. Kelly, 1996: Scales of spatial and temporal variability in the Southern Ocean. J. Geophys. Res., 101, 8759–8773, https://doi.org/10.1029/96JC00203.
Gottwald, G., D. Crommelin, and C. Franzke, 2017: Stochastic climate theory. Nonlinear and Stochastic Climate Dynamics, C. Franzke and T. J. O’Kane, Eds., 209–240.
Griffies, S. M., and Coauthors, 2015: Impacts on ocean heat from transient mesoscale eddies in a hierarchy of climate models. J. Climate, 28, 952–977, https://doi.org/10.1175/JCLI-D-14-00353.1.
Grooms, I., and A. Majda, 2013: Efficient stochastic superparameterization for geophysical turbulence. Proc. Natl. Acad. Sci. USA, 110, 4464–4469, https://doi.org/10.1073/pnas.1302548110.
Haidvogel, D., J. C. McWilliams, and P. R. Gent, 1992: Boundary current separation in a quasigeostrophic, eddy-resolving ocean circulation model. J. Phys. Oceanogr., 22, 882–902, https://doi.org/10.1175/1520-0485(1992)022<0882:BCSIAQ>2.0.CO;2.
Hallberg, R., 2013: Using a resolution function to regulate parameterizations of oceanic mesoscale eddy effects. Ocean Modell., 72, 92–103, https://doi.org/10.1016/j.ocemod.2013.08.007.
Hogg, A., and J. Blundell, 2006: Interdecadal variability of the Southern Ocean. J. Phys. Oceanogr., 36, 1626–1645, https://doi.org/10.1175/JPO2934.1.
Hogg, A., P. D. Killworth, J. Blundell, and W. K. Dewar, 2005: Mechanisms of decadal variability of the wind-driven ocean circulation. J. Phys. Oceanogr., 35, 512–531, https://doi.org/10.1175/JPO2687.1.
IPCC, 2013: Climate Change 2013: The Physical Science Basis. T. F. Stocker et al., Eds., Cambridge University Press, 1535 pp.
Jansen, M. F., and I. M. Held, 2014: Parameterizing subgrid-scale eddy effects using energetically consistent backscatter. Ocean Modell., 80, 36–48, https://doi.org/10.1016/j.ocemod.2014.06.002.
Jüling, A., J. Viebahn, S. Drijfhout, and H. Dijkstra, 2018: Energetics of the Southern Ocean Mode. J. Geophys. Res. Oceans, 123, 9283–9304, https://doi.org/10.1029/2018JC014191.
Kerr, R. A., 2000: A North Atlantic climate pacemaker for the centuries. Science, 288, 1984–1985, https://doi.org/10.1126/science.288.5473.1984.
Kwon, Y.-O., 2010: Role of the Gulf Stream and Kuroshio–Oyashio systems in large-scale atmosphere–ocean interaction: A review. J. Climate, 23, 3249–3281, https://doi.org/10.1175/2010JCLI3343.1.
Le Bars, D., J. P. Viebahn, and H. A. Dijkstra, 2016: A Southern Ocean Mode of multidecadal variability. Geophys. Res. Lett., 43, 2102–2110, https://doi.org/10.1002/2016GL068177.
Mantua, N. J., S. R. Hare, Y. Zhang, J. M. Wallace, and R. Francis, 1997: A Pacific interdecadal climate oscillation with impacts on salmon production. Bull. Amer. Meteor. Soc., 78, 1069–1079, https://doi.org/10.1175/1520-0477(1997)078<1069:APICOW>2.0.CO;2.
Meehl, G. A., J. M. Arblaster, J. T. Fasullo, A. Hu, and K. E. Trenberth, 2011: Model-based evidence of deep-ocean heat uptake during surface-temperature hiatus periods. Nat. Climate Change, 1, 360–364, https://doi.org/10.1038/nclimate1229.
Meehl, G. A., A. Hu, J. Arblaster, J. Fasullo, and K. E. Trenberth, 2013: Externally forced and internally generated decadal climate variability associated with the interdecadal Pacific oscillation. J. Climate, 26, 7298–7310, https://doi.org/10.1175/JCLI-D-12-00548.1.
Olbers, D., J. Willebrand, and C. Eden, 2012: Ocean Dynamics. Springer, 704 pp.
Pedlosky, J., 1996: Ocean Circulation Theory. Springer, 455 pp.
Pope, S., 2000: Turbulent Flows. Cambridge University Press, 771 pp.
Porta Mana, P., and L. Zanna, 2014: Toward a stochastic parameterization of ocean mesoscale eddies. Ocean Modell., 79, 1–20, https://doi.org/10.1016/j.ocemod.2014.04.002.
Sapsis, T., and P. Lermusiaux, 2009: Dynamically orthogonal field equations for continuous stochastic dynamical systems. Physica D, 238, 2347–2360, https://doi.org/10.1016/j.physd.2009.09.017.
Sapsis, T., and H. Dijkstra, 2013: Interaction of additive noise and nonlinear dynamics in the double-gyre wind-driven ocean circulation. J. Phys. Oceanogr., 43, 366–381, https://doi.org/10.1175/JPO-D-12-047.1.
Schlesinger, M. E., and N. Ramankutty, 1994: An oscillation in the global climate system of period 65–70 years. Nature, 367, 723–726, https://doi.org/10.1038/367723a0.
Shevchenko, I. V., P. S. Berloff, D. Guerrero-López, and J. E. Roman, 2016: On low-frequency variability of the midlatitude ocean gyres. J. Fluid Mech., 795, 423–442, https://doi.org/10.1017/jfm.2016.208.
Siegel, A., J. B. Weiss, J. Toomre, J. C. McWilliams, P. S. Berloff, and I. Yavneh, 2001: Eddies and vortices in ocean basin dynamics. Geophys. Res. Lett., 28, 3183–3187, https://doi.org/10.1029/1999GL011246.
Starr, V., 1968: Physics of Negative Viscosity Phenomena. McGraw-Hill, 256 pp.
Takens, F., 1981: Detecting strange attractors in turbulence. Dynamical Systems and Turbulence, D. Rand and L.-S. Young, Eds., Lecture Notes in Mathematics Series, Vol. 898, 366–381.
Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, 745 pp.
Verheul, N., J. Viebahn, and D. Crommelin, 2017: Covariate-based stochastic parameterization of baroclinic ocean eddies. Math. Climate Wea. Forecasting, 3, 90–117, https://doi.org/10.1515/mcwf-2017-0005.
Viebahn, J., and C. Eden, 2010: Towards the impact of eddies on the response of the Southern Ocean to climate change. Ocean Modell., 34, 150–165, https://doi.org/10.1016/j.ocemod.2010.05.005.
Viebahn, J., A. von der Heydt, D. Le Bars, and H. Dijkstra, 2016: Effects of Drake Passage on a strongly eddying global ocean. Paleoceanography, 31, 564–581, https://doi.org/10.1002/2015PA002888.
von Storch, H., and F. W. Zwiers, 1999: Statistical Analysis in Climate Research. Cambridge University Press, 499 pp.
von Storch, J.-S., C. Eden, I. Fast, H. Haak, D. Hernández-Deckers, E. Maier-Reimer, J. Marotzke, and D. Stammer, 2012: An estimate of the Lorenz energy cycle for the World Ocean based on the STORM/NCEP simulation. J. Phys. Oceanogr., 42, 2185–2205, https://doi.org/10.1175/JPO-D-12-079.1.
Wouters, J., and V. Lucarini, 2013: Multi-level dynamical systems: Connecting the Ruelle response theory and the Mori–Zwanzig approach. J. Stat. Phys., 151, 850–860, https://doi.org/10.1007/s10955-013-0726-8.
Wu, Y., Z. Wang, and C. Liu, 2017: On the response of the Lorenz energy cycle for the Southern Ocean to intensified westerlies. J. Geophys. Res. Oceans, 122, 2465–2493, https://doi.org/10.1002/2016JC012539.
Identity with respect to time evolution is meant in a statistical/dynamical sense.
Note that this criterion is expressed here with respect to the equations in physical space. With respect to the equations in modal/wavenumber space it says that the constant interaction coefficients (obtained by computing the amplitude equations of the individual modes) related to the resolved large-scale modes should not significantly change whether computed from the high-resolution or low-resolution representation of the modes.
Orthonormalized in the streamfunction norm (equivalent to PE norm), 
Note that there is a subtlety here: We assume that the terms 
In our model setup the two conversion terms are identical in the time mean (see Fig. 5) due to the absence of buoyancy sources/sinks.
Note that 
Note that this estimation is not affected by rotational eddy fluxes since it is not computed on the level of fluxes [like Eq. (37)] but on the level of dynamical terms appearing in the PV budget.
Note that the model blows up if 
Dynamical modes are time-dependent and budget-based in the sense that their computation explicitly involves the governing conservation equations (see, e.g., Dijkstra 2016). In contrast, for example, statistical modes (e.g., EOFs) are data based and not budget based.
Loosely speaking, these are Fourier-type modes. More precisely, the spatial filter modes 
