3.1. Eddy forcing and ECF

In this section we discuss traditional and novel approaches for defining the eddy forcing term in the large-scale dynamical equation provided that some reference flow can be decomposed into some large-scale truth (indicated by overbars) and eddies (indicated by primes)

(3.1a,b)\begin{equation} q(t,x,y,z)=\bar{q}+q' , \quad \psi(t,x,y,z)=\bar{\psi}+\psi' , \end{equation}

where the corresponding PV anomalies and velocity streamfunctions are related via the PV inversion (2.4)–(2.6). We assume here that everything so far happens on the fine grid, no coarse graining is involved, and later on the resulting eddy forcing will be related to the ECF. Note that, in general, both $\bar {q}$ and $q'$ have projections on the coarse grid; therefore, a projection of the former component does not represent projection of the full field $q$. This is apparent loss of information provided that the ultimate goal is to model $\bar {q}$ on the coarse grid and parameterize effects of $q'$. The starting point is the basic quasi-geostrophic dynamics, and here we focus on the upper layer and omit the layer index for simplicity

(3.2)\begin{equation} \frac{\partial q}{\partial t} +J(\psi,q) +\beta \frac{\partial \psi}{\partial x} = \nu \nabla^{4} \psi +W . \end{equation}

The next goal is to obtain a prognostic equation for $\bar {q}$, and for convenience we restrict the upcoming discussion to the PV anomaly, keeping in mind that the streamfunction is always inverted from it. Let us now consider the traditional filtering approach (e.g. Sagaut Reference Sagaut2006), in which the overbar indicates some localized spatial filtering $\bar {\,\cdot \,}$ used to obtain the large-scale component in (3.1a,b), and apply the filtering to the governing equation (3.2)

(3.3)\begin{equation} \frac{\partial \bar{q}}{\partial t} +J(\bar{\psi},\bar{q}) +\beta \overline{\frac{\partial \psi}{\partial x}} =\nu \overline{\nabla^{4} \psi} +\bar{W} +[ J(\bar{\psi},\bar{q}) -\overline{J(\psi,q)}] . \end{equation}

Note that, in the above equation, the filtering does not have to commute with the spatial derivatives (e.g. when the filter width varies in space). The term in the square brackets here is referred to as the eddy forcing and denoted as

(3.4)\begin{equation} EF^{(1)}=J(\bar{\psi},\bar{q}) -\overline{J(\psi,q)} , \end{equation}

where the superscript indicates that this is the first eddy forcing expression, which corresponds to the filtering approach. Equation (3.3) is diagnostic for $EF^{(1)}(t,x,y,z)$, and for this to happen it needs all 3 inputs: $\bar {q}$, $q'$ and $\bar {\,\cdot \,}$ (i.e. both the large-scale and eddy components, as well as the filter shape). On the other hand, if we treat this equation differently and aim to find $q'$, given the other 3 inputs – $EF^{(1)}$, $\bar {q}$ and $\bar {\,\cdot \,}$ – then, we are stuck, because unfiltering the Jacobian makes the whole effort numerically expensive and highly problematic, if not infeasible. To summarize, not only does traditional filtering have inherent loss of information, but also it does not easily allow us to recover eddies from the eddy forcing.

Let us now consider the alternative decomposition approach, which better fits our purposes. For this we substitute (3.1a,b) into (3.2) and rearrange the terms

(3.5)\begin{align} &\frac{\partial \bar{q}}{\partial t} +J(\bar{\psi},\bar{q}) +\beta \frac{\partial \bar{\psi}}{\partial x}=\nu \nabla^{4} \bar{\psi} +\bar{W}\nonumber\\ &\quad +\left[{-}J(\bar{\psi},q') -J(\psi',\bar{q}) -J(\psi',q') -\frac{\partial q'}{\partial t} -\beta \frac{\partial \psi'}{\partial x} +\nu \nabla^{4} \psi' +W'\right] . \end{align}

Compare this equation with (3.3) and note that it contains a different expression for the eddy forcing (in square brackets)

(3.6)\begin{equation} EF^{(2)} ={-}J(\bar{\psi},q') -J(\psi',\bar{q}) -J(\psi',q') -\frac{\partial q'}{\partial t} -\beta \frac{\partial \psi'}{\partial x} +\nu \nabla^{4} \psi' +W' . \end{equation}

Note, that the eddy forcing definitions are equivalent if the filtering commutes with the spatial derivatives. Let us assume this for our purposes. There is also a mismatch between $\bar {W}$ and $W$ in the governing equations but it can be easily absorbed into one of the eddy forcing definitions (plus, in our case, it is very small because the wind forcing is essentially a smooth large-scale function). Thus, up to a close approximation (that can be easily dealt with and can change none of our conclusions and methods), we have: $EF^{(1)}=EF^{(2)}=EF$. Note that (3.6) is also diagnostic for $EF$, given $\bar {q}$ and $q'$, but it does not need any additional information on the filter shape. This is an advantage because (3.6) can be now solved for $q'$, given $EF^{(2)}$ and $\bar {q}$ – this is what we need for our purposes. To summarize, the decomposition approach still has the inherent problem of the loss of information but now it allows us to find the eddies given the eddy forcing, and there is no need to know the filter.

Suppose that we have a situation (3.5) for the grid that we consider as a coarse one, indicated by subscript ‘$c$’. Also, let us now assume that the filtered field $\bar{q}$ is known as the large-scale flow component but the exact filter shape is not known, and the full flow is also not known. In other words, let us take $\bar {q}=q$, where the latter is the reference ‘true’ flow field projected on the coarse grid. Then, we can write down (on the coarse grid, as indicated by subscript ‘$c$’) the following equations:

(3.7)\begin{equation} \frac{\partial \bar{q}}{\partial t} +J_c(\bar{\psi},\bar{q}+\beta y) =\nu \nabla_c^{4} \bar{\psi} +\bar{W} +EF_c(t,x,y), \end{equation}

and

(3.8)\begin{equation} \frac{\partial q'}{\partial t} +J_c(\bar{\psi},q') +J_c(\psi',\bar{q}) +J_c(\psi',q'+\beta y) -\nu \nabla_c^{4} \psi' -W' ={-}EF_c(t,x,y) . \end{equation}

Note that, despite similarities of these equations with (3.5) and (3.6), there are fundamental differences now: both equations are formulated on the coarse grid; all overbarred fields are the ‘true’ reference fields projected on the coarse grid; there is no explicit information on the filtering applied; and the primed fields are the unknown quantities to be found. Next, (3.7) can be used to find the eddy forcing (from the overbarred input), and now we refer to it as the ECF. This change of notation is justified by the fact that EF deals with real eddies, which are a component of the reference flow, whereas ECF deals with field error, which is reconstructed on top of the reference flow. The corresponding ECF can be used as the input for (3.8) to find the $q'$ eddies, which we refer to as the field errors (and the corresponding streamfunction $\psi '$ is found by the inversion). Note that no information on the filter shape was ever needed, and this is the fundamental advantage of the whole approach. Instead of starting from the full flow and filtering it out into the large-scale and eddy components, we started from the full flow declared as the large-scale field and used the coarse-grid model to reconstruct the field error, which is not a physical object despite being a viable alternative to previously used decomposition approaches. Let us stress that, in the classical flow decomposition scenario, the starting point is some given $q;$ then, $\bar {q}$ is found by a filtering and $q'$ is found as the residual. In the approach proposed in this paper, the starting point is some given $\bar {q};$ then, $q'$ is found dynamically and $q$ is found as the sum of both. Note, however, that in the classical approach $q$, $\bar {q}$ and $q'$ exist as functions on the high-resolution grid, while in the new approach they only exist on the coarse-model grid.

We solved (3.7) for each layer over the full record (100 years) of the reference ‘truth’, and the resulting ECF solution was saved daily (or hourly), as needed for further analyses. In practice, we estimated the time derivative in (3.7) with the centred finite difference of the true PV anomaly. The other option to obtain the same (up to numerical errors) result is to integrate the coarse-grid model over a short time interval to find the corresponding residual from the truth and to divide this residual by the time interval, so that the resulting mid-interval ECF is evaluated. We tested this option and report here that it can be more practical for significantly more complicated, comprehensive OGCMs. For both algorithms, we found that the resulting ECF is insensitive to the time interval, as long as it is less than approximately 2 h – this value can be interpreted as the temporal resolution requirement for the time derivative in the ECF dynamics discussed further below. Finally, the ECF is cubically interpolated in time for further purposes, such as the time integration of the augmented coarse-grid model or solving for the field errors.

The resulting ECF field (figure 2) is characterized mostly by the nearly grid-scale anomalies in the immediate vicinity of the meandering eastward jet and most prominent vortices, and also located along the basin boundaries. This is expected because the coarse-grid derivatives have the largest errors on steep gradients, which characterize all these flow features and locations. We also note that some of the same-sign ECF anomalies are organized in groups spanning several grid intervals – this indicates the multi-scale nature of the field to be taken into account by eddy parameterizations. The main difference between the upper- and deep-ocean ECFs is mostly in terms of the magnitude, which tends to decrease with depth by one order of magnitude. We note that the ECF can be decomposed into rigorously defined components with specific dynamical meanings in accord with (3.8). In order to do this, first, we have to find the field errors, therefore, this analysis is deferred to § 4.2.

Figure 2.

The ECF augmenting the coarse-grid model towards the reference true flow. Top row shows fields describing the upper-ocean ECF: (a) instantaneous snapshot, (b) time mean and (c) standard deviation. Bottom row shows the same but for the deep-ocean ECF. The zonal and meridional distances along the basin boundaries are as in figure 1, and are not shown from here on.

The governing coarse-grid field error model (3.8) can be viewed as the original quasigeostrophic dynamics that was modified as follows: (i) it is augmented with the additional Jacobians $J(\bar {\psi }_i, q'_i)$ and $J(\psi '_i, \bar {q}_i)$, which involve advection of the field error PV anomaly by the large-scale flow, and advection of the reference PV anomaly by the field error; and (ii) it is driven by the ECF history instead of the wind curl $W(x,y)$. Note that the field error dynamics is fully nonlinear, and its linearized properties are fundamentally different from the classical linearization around the state of rest, due to the large-scale flow effects imposed via the additional Jacobians. It is worth noting the difference between (3.8) and the approach taken in Berloff (Reference Berloff2005a,Reference Berloffb), where only the Jacobian terms were taken into account in defining the eddy forcing of the coarse-grid dynamics, and the eddies were defined as the residual between the reference flow and the coarse-grid model solution. Note that, for (3.8), the implemented boundary conditions, mass constraint and elliptic PV inversion problem are the same, as in the original double-gyre model (§ 2; Karabasov et al. Reference Karabasov, Berloff and Goloviznin2009), and each involved equation is discretized and solved on the coarse grid, and with the same numerical methods.

There is a subtlety with the initial conditions for (3.8), and it is worth discussing here. For the reference field error solution, we assume the initial condition to be the state of rest (in each fluid layer), but, in principle, it can be anything, thus resulting in a different field error solution. We consider this issue later (§ 4.1) and show that, regardless of the initialization, the field errors always reach the same statistical equilibrium.

Given the already obtained history of the ECF, we solved (3.8) over the available 100 years of data with the same numerical algorithms as the ones employed for the double-gyre model. We used the ECF saved on daily intervals, and cubically interpolated it in time, in order to have it available on each (much shorter) time step of the field error model. These are all technical details that can be selected and tuned to fit the purpose, whereas our main objective is to demonstrate the new concepts and outline the framework. On the conceptual level, the general idea of solving some dynamical model (e.g. Grooms & Majda Reference Grooms and Majda2014; Grooms, Majda & Smith Reference Grooms, Majda and Smith2015) or statistical model (e.g. Ryzhov et al. Reference Ryzhov, Kondrashov, Agarwal, Berloff and McWilliams2020) to predict the field error (i.e. eddies) for follow-up parameterization purposes is not new. However, our approach has several novel and important aspects: (i) the dynamical field error equation is exact (i.e. it operates without any loss of information, additional assumptions or approximations) up to the numerical accuracy, (ii) the field error is forced by the known ECF history (as opposed to a random forcing), (iii) the ECF is objectively the optimal one, in the sense that it perfectly augments the coarse-grid model and allows us to reproduce the whole reference truth completely and (iv) no local filtering of the eddies is ever needed at any stage of the proposed procedure.

To summarize, in this section we outlined the methodology. The main steps were as follows: (i) preparing the reference true flow on the coarse grid; (ii) obtaining the ECF used to augment the coarse-grid model towards the reference truth; (iii) using the dynamical field error model to translate the ECF into the field error. In the next section we systematically analyse the obtained field errors and use them to interpret the ECF.

4.1. Statistical properties of field errors

The obtained field errors can be characterized as follows. First, once we start integrating (3.8) from the state of rest, the field errors undergo an adjustment period of approximately 200 days, after which they reach statistical equilibrium, which is clearly seen in the energy time series (figure 3). Second, the time series of the equilibrated field error energy is highly correlated (with a correlation of approximately 0.4) with the energy of the reference true flow; and on the interdecadal time scale it exhibits robust low-frequency variability characteristic of the large-scale flow evolution (figure 3b). This indicates that the eddies are playing the key role in maintaining the intrinsic large-scale low-frequency variability (see Berloff & McWilliams (Reference Berloff and McWilliams1999a); and Berloff et al. (Reference Berloff, Hogg and Dewar2007) for its properties and underlying mechanism). We also note that the field error potential energy is approximately twice as large as the kinetic energy, which indicates that the field error scales are noticeably larger than the first baroclinic Rossby deformation radius, and the field error itself is significantly baroclinic.

Figure 3.

Field error energy time series and power spectra. This is an illustration of the field error spin-up process, statistical equilibration and modulation in terms of the low-frequency variability. (a) Time series of field error kinetic, potential and total energies in the upper layer. (b) Power spectra of the time series; the broad peaks correspond to the large-scale interdecadal variability of the gyres, which imprints itself on modulation of the eddy population (Berloff & McWilliams Reference Berloff and McWilliams1999a; Berloff et al. Reference Berloff, Hogg and Dewar2007). Acronyms: $EPE_1$, $EKE_1$ and $ETE_1$ indicate (field) error potential, kinetic and total energies in the upper layer (indicated by the subscript). Black line indicates the total energy of the reference true flow (in (a) it is multiplied by 0.1); the corresponding energy of the non-augmented coarse-grid model is approximately half of it (not shown). Note that the characteristic spin-up time of the eddies is approximately 0.5 years (barely seen near time zero in (a)), after which the eddies reach statistical equilibrium.

Analysis of the spatial structure of the field errors tells us a few things about them (figures 4 and 5). First, we note that, although the field errors occupy the whole basin, their intensity is concentrated around the eastward jet, and this is more so in the upper ocean. This behaviour is similar to what is always seen with locally filtered eddies (§ 4.3), and it can be taken as a simple sanity check. We have carried out formal evaluation of the involved scales from the two-dimensional Fourier power spectra. The field error isotropic scales corresponding to the PV anomaly have maximum power for the wavelengths of approximately 250 km (approximately 8 grid intervals). The spectrum is broad, with significant power values (more than 10 % of the maximum power) occupying wavelengths from 60 to 350 km (from 2 to 12 grid intervals). These values are significantly longer than the nominal coarse-grid resolution interval (30 km) and indicate the ‘grey’ zone of scales. From the corresponding power spectra of the velocity streamfunction, we concluded that its length scales are longer than those of the PV anomaly by 10 %–20 %. Second, the PV cores of the field errors are typically resolved by less than 10 coarse-grid intervals (figure 4a), implying that they belong to the ‘grey zone’ of partial resolution, which arises from the fact that well-resolved flow features typically require approximately 10 grid intervals to represent them (Karabasov et al. Reference Karabasov, Berloff and Goloviznin2009). The relative vorticity variance (not shown) is concentrated near the eastward jet separation point, suggesting the dominance of the smallest and most intense eddies there; and the dynamic pressure variance dominates downstream of the separation point, suggesting active baroclinic processes there. The field error potential energy is approximately twice as large as the kinetic energy (both are not shown), which is consistent with the energy time series (figure 3). The former field has standard deviation similar to that in figure 4(c), and the latter one to that in figure 4( f).

Figure 4.

Field errors and their statistics in the upper ocean. PV anomaly $q'_1\!:$ (a) instantaneous snapshot, (b) time mean, (c) standard deviation. Velocity streamfunction $\psi '_1$ (i.e. dynamic pressure): (d) instantaneous snapshot, (e) time mean, ( f) standard deviation. The flow anomalies are present all over the basin, but their intensity is concentrated around the eastward jet; the relative vorticity and stretching components of the field error PV are of similar size.

Figure 5.

Field error and its statistics in the deep ocean. Velocity streamfunction $\psi '_2$ (i.e. dynamic pressure): (a) instantaneous snapshot, (b) time mean and (c) standard deviation.

Figure 6.

The leading 9 EOF patterns of the obtained field error characterized by the coherent eddy-like anomalies with alternating sign and straddling the eastern jet extension. Panels show the upper-ocean streamfunction EOFs ranked from (a) the first to (i) the ninth, respectively. The percentages of the total variance are indicated in each panel. Units are non-dimensional and the same across all panels.

A particular important property of the field error is its time-mean component. It is weak relative to the transient component but still significant, as can be judged by its values (figure 4b,e). The time-mean anomalies are characterized by the presence of the westward jet and recirculations, which are opposite to the large-scale eastward jet extension of the western boundary currents and its adjacent recirculations. These features are totally different from those obtained by local eddy filtering (e.g. Berloff Reference Berloff2018; see also § 4.3). In the deep ocean (figure 5) the field errors are more spread out around the basin, and their time-mean field is more complicated. In the interior of the basin, it exhibits zonally elongated anomalies corresponding to very latent (i.e. weak relative to the ambient large-scale circulation and transient eddies) alternating jets embedded in the gyres (studied by Chen, Kamenkovich & Berloff Reference Chen, Kamenkovich and Berloff2016). Near zonal basin boundaries, these elongated anomalies become comparable in strength to the transient field errors. Since these results are coarse model dependent, inAppendix A we consider a non-eddy-resolving configuration of the ocean model and discuss the resulting characteristics and robustness of the field errors.

We analysed the field error patterns by means of standard statistical analysis in terms of the empirical orthogonal functions (EOFs) and their principal components (Preisendorfer Reference Preisendorfer1988). The layer-wise EOFs were calculated for the streamfunction fields and ranked in terms of their contributions to the total variance. Typical numbers of the EOFs needed to cover 60 % and 90 % of the total variance are approximately 70 and 300, respectively. This information tells us that the field error is statistically complicated and multi-patterned. The leading EOFs (figure 6) come mostly in pairs with quadratures and correspond to the propagating meanders of the eastward jet. The higher-rank EOFs are more complicated and involve north–south shifts and intensity changes of the eastward jet, all coupled to the propagating anomalies. All these patterns are qualitatively (but not quantitatively) similar to the corresponding eddy patterns defined from the Reynolds averaging and extracted by other statistical means (Kondrashov, Chekroun & Berloff Reference Kondrashov, Chekroun and Berloff2018).

In Appendix B we discuss various sensitivities of the field error and ECF and show that these fields are statistically robust. In § 4.3 we discuss the main differences between the field error and locally filtered eddies in more detail.

4.2. Eddy forcing components

One of the important issues within the scope of our study is understanding which dynamical terms of the field error model (3.8) are most important. We diagnosed all of them, once the field error was found, and extracted both the time-mean and standard deviation patterns of each term (not shown). First, the two linear advection terms tend to balance each other, that is, they are nearly equal but of opposite sign. Their sum is comparable in size to and negatively correlated with the nonlinear advection term, although there is no clear functional relation. Overall, at the leading order, the total combined advection tends to balance the ECF. An immediate conclusion from this is that the simplistic approach taken in Berloff (Reference Berloff2005a,Reference Berloffb), where eddy forcing has been reduced to the combined advection, was largely correct. (In that study the eddies were defined differently, and the coarse-grid model solution was not forced to be perfectly augmented.) However, it was inaccurate in the sense that, unlike the current study, there was no prognostic eddy model.

Next, we analysed a relative impact of the linear non-advective term in (3.8) on the overall balance, and found that it is by one order of magnitude smaller than the impact of the combined advection, and the $\beta$-term is the leading one in it. From this assessment, we cannot conclude that the linear part can be neglected as it makes the whole field error model prognostic. On the other hand, we report that the field error model is a balanced one, with the main balance between the advection and ECF. We concluded that the background flow advection acts in (3.8) at the leading order, and, therefore, it cannot be neglected. (We also verified this directly by removing these terms and recomputing the field error. The outcome is that the field error anomalies become significantly delocalized from the eastward jet, cover the whole basin and start resembling mesoscale basin modes.) Finally, we comment (without detailed analysis leading us far beyond the scope of this paper) that the field error model (3.8) can be stripped of the tendency term and even all the linear non-advective terms – this would make it purely balanced and diagnostic; perhaps, with a modest and acceptable loss of accuracy. This development would be along the ideas proposed by Chekroun, Liu & McWilliams (Reference Chekroun, Liu and McWilliams2017), but finding efficient methods for solving the corresponding equation may be a problem.

The next issue on our agenda is assessment of the robustness of the augmentation of the coarse-grid model by the ECF. Since it is estimated approximately (although in a controlled way), we had to verify that the corresponding augmented coarse-grid model solution indeed stays close to the reference truth in the presence of numerical errors. There are two things we learned about this. First, we found that if the ECF is supplied as a discrete history of the external forcing, then, after approximately 2 years, the augmented solution diverges from the reference truth due to the chaotic sensitivity and growth of the solution errors. This is because the ECF is weak and gentle, and if its pattern loses proper relation with the large-scale flow, then its augmenting efficiency decreases and eventually disappears. This is also consistent with the above-discussed significant sensitivities to the initial conditions and details of the large-scale flow. Second, we estimated the ECF interactively, by running the coarse-grid model and augmenting its prediction to the reference truth. In this case, subtle differences due to the numerical errors are absorbed in the new ECF history, which deviates from the reference one, but remains statistically identical to it. To estimate robustness of the coarse-grid model solution to errors in ECF, we interactively multiplied the latter by devaluating factors of 0.1, 0.01 and 0.001, on each hourly time step, and assessed how close the augmented solution follows the reference truth for the full duration of the data (100 years). In the case of 0.1, the augmented solution was nearly identical (0.99 correlation) to the reference truth, and only in the last case it significantly deviated (0.96 correlation), with the errors reaching the level of the reference field error. To summarize, the interactive ECF robustly augments the coarse-grid model, even in the presence of significant, intentionally introduced errors.

4.3. Comparison with statistically filtered eddies

It is important to understand all systematic differences between the field error and locally filtered eddies, because, if these differences were not significant, then our results would be incremental. For this purpose, at each time instance we obtained locally filtered eddies by applying square running-average filtering with the width of 3.75 first baroclinic Rossby radii (i.e. 150 km) to the fine-grid reference eddy-resolving solution and then subsampled the resulting filtered eddies, as well as the large-scale component, on the coarse grid. Both wider and narrower filter widths were also explored, but the outcomes were qualitatively similar. Near the basin boundaries the filtering width was reduced, so that it remained always twice the distance to the nearest boundary point. The outcome of the filtering was decomposition of the flow solution into the large-scale and eddy components. The former component is just a smoother version of the reference truth, therefore, we do not discuss it, and further below refer to it as the ‘locally filtered true large-scale flow’. We used this flow for computing the new field error, and found that it is structurally and statistically similar to the reference one but has larger amplitude. Thus a smoother large-scale flow corresponds to more energetic eddies (this is also consistent with the non-eddy-resolving model analyses in Appendix A). Statistically filtered eddies also have this property, simply because a wider filter size moves a larger fraction of the total flow into the eddies.

The central result is that the locally filtered eddies are qualitatively and profoundly different from the field error (figure 7), and the main differences are as follows. First, typical amplitudes of the locally filtered eddies are approximately 5 times smaller than those of the field error. This is rather surprising, given that both types of eddies are meant to augment the coarse-grid model (note that only the field error actually achieves this) and be eventually parameterized. From this, we again conclude that spatial structure is more important than the amplitude of the eddies. Second, the locally filtered eddies have a robustly present ribbon-like feature that straddles the eastward jet and contains PV anomalies of the same polarity as the nonlinear recirculation zones adjacent to it. This feature is robustly present even in the time-mean locally filtered eddies. These ribbons are due to the smoothing effect of the filtering on the sharp PV gradient corresponding to the eastward jet core. In principle, this effect can be reduced by modifying the filter size, but it remains to be researched how to do this modification objectively (e.g. by basing the filter size on the local correlation length scale). Around the eastward jet, the field errors are characterized by the time-mean flow anomaly of the opposite sign to the adjacent recirculation zones (figure 4). Third, the locally filtered eddies have more of a multi-scale character, as expressed by the simultaneous presence of both small-scale vortices and zonally large-scale ribbon-like anomalies. From these two aspects, we reinforce our earlier conclusion about the ultimate importance of the spatial eddy patterns, rather than the eddy intensity. Next, combined net advection in (3.8) is also approximately 5 times weaker and significantly different from those of the locally filtered eddies. Recalling that the ECF perfectly augments the coarse-grid model, we conclude that the eddy forcing structure is a lot more important than its amplitude, which is similar to our earlier statement about the eddies. Further analyses, going beyond the scope of this paper, are needed to understand structural differences between the ECF and locally filtered eddy forcings.

Figure 7.

Comparison of locally filtered eddies and field errors for the filtered and unfiltered ‘true’ flow. Snapshots of the upper-ocean streamfunctions at the same but randomly chosen time (set by the reference true flow). (a) Field error and (b) locally filtered eddies for the reference solution; (c) field error recomputed for the ‘locally filtered true large-scale flow’. Note that the locally filtered eddies have a totally different pattern and approximately 5 times smaller amplitude than their dynamical counterparts.

How do the statistical eddies and field errors depend on the filter size? We compared outcomes from the filter sizes of 150 and 300 km, which are equivalent to 3.75 and 7.5 $Rd_1$, respectively. These results illuminated the following two dependencies, which are illustrated by figure 8. First, the intensity of statistical eddies expectedly increases with filter width, but the field error behaves in the opposite way. Since any parameterization is desired to be more skilful than forceful, we interpret this result in favour of the field error. Second, we find that with heavier filtering the statistical eddies contain an increasingly larger fraction of the eastward jet and its adjacent recirculations, whereas the field errors do not show a hint of this behaviour. This can be also interpreted in favour of the field errors, because the eastward jet and recirculations can be resolved rather than parameterized, in principle.

Figure 8.

Comparison of locally filtered eddies and field errors for different filters. Upper row: filter size is 150 km; lower row: filter size is 300 km. Upper-ocean snapshots of velocity streamfunction are shown. Left panels: large-scale flow; middle panels: statistically filtered eddies; right panels: field errors.