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Abstract Ocean general circulation models (OGCMs) are often used at horizontal resolutions that preclude the appearance of mesoscale eddies. The ocean mesoscale constitutes a significant component of ocean variability, and OGCMs whose resolutions are too coarse to represent the mesoscale are necessarily lacking this variability. In addition to being variable, the ocean mesoscale also induces variability on larger scales that could be resolved on a coarse grid, but coarse OGCMs often lack this variability too. This paper develops a stochastic parameterization that adds small increments to an OGCM's lateral velocity field, which excites natural modes of variability in the model. The rate at which these velocity increments add energy to the flow is tied to the rate at which the Gent‐McWilliams parameterization—a popular parameterization of the effect of mesoscale eddies on tracer transport—removes potential energy from the resolved scales. The stochastic parameterization is implemented in a non‐eddying OGCM, where it is shown to increase the variability significantly. 

Abstract Due to their limited resolution, numerical ocean models need to be interpreted as representing filtered or averaged equations. How to interpret models in terms of formally averaged equations, however, is not always clear, particularly in the case of hybrid or generalized vertical coordinate models, which limits our ability to interpret the model results and to develop parameterizations for the unresolved eddy contributions. We here derive the averaged hydrostatic Boussinesq equations in generalized vertical coordinates for an arbitrary thickness‐weighted average. We then consider various special cases and discuss the extent to which the averaged equations are consistent with existing ocean model formulations. As previously discussed, the momentum equations in existing depth‐coordinate models are best interpreted as representing Eulerian averages (i.e., averages taken at fixed depth), while the tracer equations can be interpreted as either Eulerian or thickness‐weighted isopycnal averages. Instead we find that no averaging is fully consistent with existing formulations of the parameterizations in semi‐Lagrangian discretizations of generalized vertical coordinate ocean models such as MOM6. A coordinate‐following average would require “coordinate‐aware” parameterizations that can account for the changing nature of the eddy terms as the coordinate changes. Alternatively, the model variables can be interpreted as representing either Eulerian or (thickness‐weighted) isopycnal averages, independent of the model coordinate that is being used for the numerical discretization. Existing parameterizations in generalized vertical coordinate models, however, are not always consistent with either of these interpretations, which, respectively, would require a three‐dimensional divergence‐free eddy tracer advection or a form‐stress parameterization in the momentum equations. 

Abstract There are two distinct parameterizations for the restratification effect of mesoscale eddies: the Greatbatch and Lamb (1990, GL90,https://journals.ametsoc.org/view/journals/phoc/20/10/1520-0485_1990_020_1634_opvmom_2_0_co_2.xml?tab_body=abstract-display) parameterization, which mixes horizontal momentum in the vertical, and the Gent and McWilliams (1990, GM90,https://journals.ametsoc.org/view/journals/phoc/20/1/1520-0485_1990_020_0150_imiocm_2_0_co_2.xml) parameterization, which flattens isopycnals adiabatically. Even though these two parameterizations are effectively equivalent under the assumption of quasi‐geostrophy, GL90 has been used much less than GM90, and exclusively inz‐coordinate models. In this paper, we compare the GL90 and GM90 parameterizations in an idealized isopycnal coordinate model, both from a theoretical and practical perspective. From a theoretical perspective, GL90 is more attractive than GM90 for isopycnal coordinate models because GL90 provides an interpretation that is fully consistent with thickness‐weighted isopycnal averaging, while GM90 cannot be entirely reconciled with any fully isopycnal averaging framework. From a practical perspective, the GL90 and GM90 parameterizations lead to extremely similar energy levels, flow and vertical structure, even though their energetic pathways are very different. The striking resemblance between the GL90 and GM90 simulations persists from non‐eddying through eddy‐permitting resolution. We conclude that GL90 is a promising alternative to GM90 for isopycnal coordinate models, where it is more consistent with theory, computationally more efficient, easier to implement, and numerically more stable. Assessing the applicability of GL90 in realistic global ocean simulations with hybrid coordinate schemes should be a priority for future work. 

Abstract We describe a new way to apply a spatial filter to gridded data from models or observations, focusing on low‐pass filters. The new method is analogous to smoothing via diffusion, and its implementation requires only a discrete Laplacian operator appropriate to the data. The new method can approximate arbitrary filter shapes, including Gaussian filters, and can be extended to spatially varying and anisotropic filters. The new diffusion‐based smoother's properties are illustrated with examples from ocean model data and ocean observational products. An open‐source Python package implementing this algorithm, called gcm‐filters, is currently under development. 

Abstract The Gent–McWilliams parameterization is commonly used in global ocean models to model the advective component of tracer transport effected by unresolved mesoscale eddies. The vertical structure of the transfer coefficient in this parameterization is studied using data from a 0.1° resolution global ocean‐ice simulation. The vertical structure is found to be well approximated by a baroclinic mode structure with no flow at the bottom, though horizontal anisotropy is crucial for obtaining a good fit. This vertical structure is motivated by reference to the vertical structure of mesoscale eddy velocity and density anomalies, which are also diagnosed from the data. 

Abstract Energy exchanges between large-scale ocean currents and mesoscale eddies play an important role in setting the large-scale ocean circulation but are not fully captured in models. To better understand and quantify the ocean energy cycle, we apply along-isopycnal spatial filtering to output from an isopycnal 1/32° primitive equation model with idealized Atlantic and Southern Ocean geometry and topography. We diagnose the energy cycle in two frameworks: 1) a non-thickness-weighted framework, resulting in a Lorenz-like energy cycle, and 2) a thickness-weighted framework, resulting in the Bleck energy cycle. This paper shows that framework 2 is more useful for studying energy pathways when an isopycnal average is used. Next, we investigate the Bleck cycle as a function of filter scale. Baroclinic conversion generates mesoscale eddy kinetic energy over a wide range of scales and peaks near the deformation scale at high latitudes but below the deformation scale at low latitudes. Away from topography, an inverse cascade transfers kinetic energy from the mesoscales to larger scales. The upscale energy transfer peaks near the energy-containing scale at high latitudes but below the deformation scale at low latitudes. Regions downstream of topography are characterized by a downscale kinetic energy transfer, in which mesoscale eddies are generated through barotropic instability. The scale- and flow-dependent energy pathways diagnosed in this paper provide a basis for evaluating and developing scale- and flow-aware mesoscale eddy parameterizations. Significance Statement Blowing winds provide a major energy source for the large-scale ocean circulation. A substantial fraction of this energy is converted to smaller-scale eddies, which swirl through the ocean as sea cyclones. Ocean turbulence causes these eddies to transfer part of their energy back to the large-scale ocean currents. This ocean energy cycle is not fully simulated in numerical models, but it plays an important role in transporting heat, carbon, and nutrients throughout the world’s oceans. The purpose of this study is to quantify the ocean energy cycle by using fine-scale idealized numerical simulations of the Atlantic and Southern Oceans. Our results provide a basis for how to include unrepresented energy exchanges in coarse global climate models. 

Flow configurations that maximize the instantaneous rate of conversion from potential to kinetic energy are sought using a combination of analytical and numerical methods. A hydrostatic model is briefly investigated, but the presence of unrealistic ageostrophic flow configurations renders the results unrealistic. In the quasigeostrophic (QG) model, flow configurations that locally optimize the conversion rate are found, but it remains unclear if these flow configurations produce the global maximum conversion rate. The difficulty is associated with the fact that in the QG model, the vertical velocity is a quadratic function of the QG streamfunction, which renders the conversion rate a cubic function of the QG streamfunction. For these locally maximal conversion rates, the rate of conversion depends on the horizontal length scale of the flow: for scales larger than the deformation radius, the maximal rates are small and decrease as the horizontal scale increases; for scales smaller than the deformation radius, the maximal conversion rate rises until it becomes comparable to the maximal rate at which potential energy can be extracted from the mean flow.