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Parameterizing mesoscale eddies in ocean circulation models remains an open problem due to the ambiguity with separating the eddies from large-scale flow, so that their interplay is consistent with the resolving skill of the employed non-eddy-resolving model. One way to address the issue is by using recently formulated dynamically filtered eddies. These eddies are obtained as the field errors of fitting some given reference ocean circulation into the employed coarse-grid ocean model. The main strengths are (i) no explicit spatio-temporal filter is needed for separating the large-scale and eddy flow components, (ii) the eddies are dynamically translated into the error-correcting forcing that perfectly augments the coarse-grid model towards reproducing the reference circulation. We uncovered physical properties of the eddies by interpreting involved nonlinear eddy/large-scale interactions via the classical flux-gradient relation. We described the eddies in terms of their full, space–time dependent transport tensor, which was made unique by constraining it to be the same for the potential vorticity, momentum and buoyancy fluxes. Both diffusive and advective parts of the transport tensor were found to be significant. The diffusive tensor component is characterised by polar eigenvalues and is further decomposed into isotropic and filamentation components. The latter component completely dominates, therefore, it should be taken into account by eddy parameterizations, which is not yet the case. We also showed that spatial inhomogeneities of the transport tensor components are important. Comparing these properties with those obtained for more common, locally filtered eddies revealed that they are distinctly different.

We present a comprehensive inter‐comparison of linear regression (LR), stochastic, and deep‐learning approaches for reduced‐order statistical emulation of ocean circulation. The reference data set is provided by an idealized, eddy‐resolving, double‐gyre ocean circulation model. Our goal is to conduct a systematic and comprehensive assessment and comparison of skill, cost, and complexity of statistical models from the three methodological classes. The model based on LR is considered as a baseline. Additionally, we investigate its additive white noise augmentation and a multi‐level stochastic approach, deep‐learning methods, hybrid frameworks (LR plus deep‐learning), and simple stochastic extensions of deep‐learning and hybrid methods. The assessment metrics considered are: root mean squared error, anomaly cross‐correlation, climatology, variance, frequency map, forecast horizon, and computational cost. We found that the multi‐level linear stochastic approach performs the best for both short‐ and long‐timescale forecasts. The deep‐learning hybrid models augmented by additive state‐dependent white noise came second, while their deterministic counterparts failed to reproduce the characteristic frequencies in climate‐range forecasts. Pure deep learning implementations performed worse than LR and its simple white noise augmentation. Skills of LR and its white noise extension were similar on short timescales, but the latter performed better on long timescales, while LR‐only outputs decay to zero for long simulations. Overall, our analysis promotes multi‐level LR stochastic models with memory effects, and hybrid models with linear dynamical core augmented by additive stochastic terms learned via deep learning, as a more practical, accurate, and cost‐effective option for ocean emulation than pure deep‐learning solutions. Plain Language Summary In weather and climate predictions, scientists use comprehensive ocean circulation models for representing the effects of the oceans on the atmosphere. These models simulate the three‐dimensional ocean dynamics using millions of variables and, thus, require significant computational resources and running time. Therefore, there is a need for low‐cost, data‐driven ocean models with fewer variables that can reproduce essential oceanic circulations with reasonable accuracy. There are several popular data‐driven approaches to build these models, but singling out the best one is difficult and significantly understudied. We have systematically assessed and compared the accuracy, stability, and computational cost of various data‐driven models against the linear regression—a fundamental and easy‐to‐implement deterministic model, that is, it provides a fixed output for a fixed input. We considered several stochastic and deep‐learning models for comparison; stochastic models combine a deterministic model with customized noise, whereas deep‐learning models train a complex network of neurons similar to the human brain. We found that the stochastic models that properly include the core dynamics, time‐delay effects, and model errors perform the best. The core dynamics provides the essential changes, time‐delay effects are the changes due to correlation between successive ocean states, and model errors provide other possible causes of changes. Key Points The multi‐level stochastic approach produces the most stable, accurate, and low‐cost emulator of a double‐gyre ocean model solution Artificial neural networks and long short term memory work better in a hybrid form with linear regression, providing the core dynamics, than in their standalone application Emulators incorporating memory effects and state‐dependent noise show enhanced performance and deep learning can learn these effects

This study aims to understand the ocean’s circulation, which is characterized by the presence of multiple alternating zonal jets and transient mesoscale eddies, by systematic analysis of the underlying linear dynamics of this system. For this purpose, properties of the linear normal modes such as growth rates, dispersion, spatial structure, and nonlinear self-interactions are explored for a hierarchy of idealized, vertically, and horizontally sheared flows with increasing complexity. The authors find that large-scale background vertical shear, alternating multiple zonal jets, bottom friction, and the Reynolds number have important effects on these modes. This study hypothesizes that when these effects are taken into account, the linear results can be used to predict many properties of nonlinear mesoscale eddies. This hypothesis is confirmed in Part II of this paper.

We present a comprehensive study of the mesoscale eddy forcing in the ocean by proposing spatially local filtering of the high-resolution double-gyre ocean circulation solution into its large- and small-scale (eddy) components. The large-scale component is dominated by the mid-latitude gyres, the western boundary currents and their highly transient eastward jet extension; the eddy component is concentrated around the eastward jet and strongly interacts with it. The proposed decomposition method achieves flow filtering based on the local spatial correlations. This is different from the existing decomposition methods, e.g. classical Reynolds decomposition and moving-average (spatial) filtering with a constant filter size based on the first baroclinic Rossby deformation radius. Next, we characterize the dynamical impacts of the resulting eddy forcing on the large-scale flow in terms of their mutual time-lagged spatial correlations, formulated as product integral characteristics. Its temporal statistics uncover robust causality between the eddy forcing and the induced large-scale potential vorticity anomalies – referred to as the eddy backscatter. The results also prove the significance of the transient eddy forcing and the time lag dependence of the eddy backscatter. We argue that these properties are to be considered by eddy parametrization schemes. We further used the decomposed eddy fields to augment a coarse-resolution ocean model. The augmented solution statistically reproduces the missing eastward jet extension, enhances the eddy activities around it and recovers the essential large-scale low-frequency variability. This justifies a reduced-order statistical emulation of the eddies – an emerging methodology for including eddy effects in non-eddy-resolving ocean models.

Several turbulent flow regimes of an idealized ocean circulation model are systematically analyzed in physical and spectral domains. Zonal dispersion properties of transient mesoscale eddies are described by the zonal wavenumber/temporal frequency spectra. The eddy patterns and the corresponding nonlinear eddy forcing exerted on the jets are examined by filtering different parts of the spectra. Comparison of the outcome of this nonlinear analysis with the properties of the linearized solutions (obtained in Part I of this paper) demonstrates that a very substantial part of the flow dynamics is controlled by the underlying linear dynamics. This result supports the hypothesis that it is possible to describe most of the oceanic mesoscale eddies as a wave turbulence phenomenon.

Multiple alternating zonal jets observed in the ocean are studied with an idealized quasigeostrophic model of flow in a zonal channel. The jets are maintained by the eddies generated by the imposed, supercritical background flow. The formation, nonlinear dynamics and equilibration of the jets are explained in terms of linear stability arguments and nonlinear self-interactions of the linear eigenmodes. In the proposed mechanism, energy of the background flow is released to the primary instability mode with long meridional and short zonal length scales. This mode undergoes secondary, transverse instability that sets the meridional scale of the emerging multiple zonal jets. This instability channels energy into several weakly damped zonal eigenmodes that amplify the jets. The emerging jets feed back on the instabilities through the partial meridional localization of the most unstable eigenmodes.

Most of the nearly zonal, multiple, alternating jets observed in the oceans are latent, that is, their amplitudes are weak relative to the ambient mesoscale eddies. Yet, relatively strong jets are often observed in dynamical simulations. To explore mechanisms controlling the degree of latency, we analyse solutions of an idealized, eddy-resolving and flat-bottom quasigeostrophic model, in which dynamically generated mesoscale eddies maintain and interact with a set of multiple zonal jets. We find that the degree of the latency is controlled primarily by the bottom friction: the larger the friction parameter, the more latent are the jets; and the degree of the latency is substantial for a realistic range of the oceanic bottom friction coefficient. This result not only provides a plausible explanation for the latency of the oceanic jets, but it may also be relevant to the prominent atmospheric multiple jets observed on giant gas planets, such as Jupiter. We hypothesize that these jets can be so strong because of the relative absence of the bottom friction. The mechanism controlling the latency in our solutions is understood in terms of the changes induced in the linear eigenmodes of the time–mean flow by varying the bottom friction coefficient; these changes, in turn, affect and modify the jets. Effects of large Reynolds numbers on the eddies, jets, and the latency are also discussed.

Multiple alternating zonal jets observed in the ocean are studied with an idealized quasigeostrophic zonal-channel model, with the supercritical, zonal background flow imposed. Both eastward and westward background flows with vertical shear are considered. The underlying nonlinear dynamics is illuminated with analysis of the vertical-mode interactions and time-mean eddy fluxes. Interactions between the vertical modes are systematically studied. The barotropic component of the jets is maintained by both barotropic–barotropic and baroclinic–baroclinic time-mean interactions; thus, the barotropic component of the jets cannot be accurately simulated with a randomly forced barotropic model. The roles of the vertical-mode interactions in driving the baroclinic component of the jets are also characterized. Not only the first but also the second baroclinic mode is found to be important for maintaining the baroclinic component of the jets, whereas the barotropic component of the jets is maintained mostly by the barotropic and first baroclinic modes. The properties of the eddy forcing were systematically studied. It is shown that the baroclinic component of the jets is maintained by Reynolds stress forcing and resisted by form stress forcing only in the eastward background flow. In the westward background flow, the jets are maintained by form stress forcing and resisted by Reynolds stress forcing. The meridional scaling and kinematical properties of the jets are studied as well as the roles of meridional boundaries. The Rhines scaling for meridional spacing of the jets is not generally confirmed, and it is also shown that there are multiple stable equilibria with different numbers of the time-mean jets. It is also found that the jets are associated with alternating weak barriers to the meridional material transport, but the locations of these barriers are not unique and depend on the direction of the background flow and depth. Finally, if the channel is closed with meridional walls, then the jets become more latent but the eddy forcing properties do not change qualitatively.

This paper studies the large-scale low-frequency variability of the wind-driven midlatitude ocean gyres and their western boundary currents, such as the Gulf Stream or Kuroshio, simulated with the eddy-resolving quasi-geostrophic model. We applied empirical orthogonal functions analysis to turbulent flow solutions and statistically extracted robust and significant large-scale decadal variability modes concentrated around the eastward jet extension of the western boundary currents. In order to interpret these statistical modes dynamically, we linearized the governing quasi-geostrophic equations around the time-mean circulation and solved for the corresponding full set of linear eigenmodes with their eigenfrequencies. We then projected the extracted decadal variability on the eigenmodes and found that this variability is a multimodal coherent pattern phenomenon rather than a single mode or a combination of several modes as in the flow regimes preceding developed turbulence.

Intrinsic low-frequency variability is studied in the idealized, quasigeostrophic, midlatitude, wind-driven ocean gyres operating at large Reynolds number. A robust decadal variability mode driven by the transient mesoscale eddies is found and analyzed. The variability is a turbulent phenomenon, which is driven by the competition between the eddy rectification process and the potential vorticity anomalies induced by changes of the intergyre transport.