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Ocean mesoscale eddies are often poorly represented in climate models, and therefore, their effects on the large scale circulation must be parameterized. Traditional parameterizations, which represent the bulk effect of the unresolved eddies, can be improved with new subgrid models learned directly from data. Zanna and Bolton (ZB20) applied an equation‐discovery algorithm to reveal an interpretable expression parameterizing the subgrid momentum fluxes by mesoscale eddies through the components of the velocity‐gradient tensor. In this work, we implement the ZB20 parameterization into the primitive‐equation GFDL MOM6 ocean model and test it in two idealized configurations with significantly different dynamical regimes and topography. The original parameterization was found to generate excessive numerical noise near the grid scale. We propose two filtering approaches to avoid the numerical issues and additionally enhance the strength of large‐scale energy backscatter. The filtered ZB20 parameterizations led to improved climatological mean state and energy distributions, compared to the current state‐of‐the‐art energy backscatter parameterizations. The filtered ZB20 parameterizations are scale‐aware and, consequently, can be used with a single value of the non‐dimensional scaling coefficient for a range of resolutions. The successful application of the filtered ZB20 parameterizations to parameterize mesoscale eddies in two idealized configurations offers a promising opportunity to reduce long‐standing biases in global ocean simulations in future studies. 

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 The use of coarse resolution and strong grid‐scale dissipation has prevented global ocean models from simulating the correct kinetic energy level. Recently parameterizing energy backscatter has been proposed to energize the model simulations. Parameterizing backscatter reduces long‐standing North Atlantic sea surface temperature (SST) and associated surface current biases, but the underlying mechanism remains unclear. Here, we apply backscatter in different geographic regions to distinguish the different physical processes at play. We show that an improved Gulf Stream path is due to backscatter acting north of the Grand Banks to maintain a strong deep western boundary current. An improved North Atlantic Current path is due to backscatter acting around the Flemish Cap, with likely an improved nearby topography‐flow interactions. These results suggest that the SST improvement with backscatter is partly due to the resulted strengthening of resolved currents, whereas the role of improved eddy physics requires further research. 

Abstract. We describe an idealized primitive-equation model for studying mesoscale turbulence and leverage a hierarchy of grid resolutions to make eddy-resolving calculations on the finest grids more affordable.The model has intermediate complexity, incorporating basin-scale geometry with idealized Atlantic and Southern oceans and with non-uniform ocean depth to allow for mesoscale eddy interactions with topography.The model is perfectly adiabatic and spans the Equator and thus fills a gap between quasi-geostrophic models, which cannot span two hemispheres, and idealized general circulation models, which generally include diabatic processes and buoyancy forcing.We show that the model solution is approaching convergence in mean kinetic energy for the ocean mesoscale processes of interest and has a rich range of dynamics with circulation features that emerge only due to resolving mesoscale turbulence. 

Abstract We address the question of how to use a machine learned (ML) parameterization in a general circulation model (GCM), and assess its performance both computationally and physically. We take one particular ML parameterization (Guillaumin & Zanna, 2021,https://doi.org/10.1002/essoar.10506419.1) and evaluate the online performance in a different model from which it was previously tested. This parameterization is a deep convolutional network that predicts parameters for a stochastic model of subgrid momentum forcing by mesoscale eddies. We treat the parameterization as we would a conventional parameterization once implemented in the numerical model. This includes trying the parameterization in a different flow regime from that in which it was trained, at different spatial resolutions, and with other differences, all to test generalization. We assess whether tuning is possible, which is a common practice in GCM development. We find the parameterization, without modification or special treatment, to be stable and that the action of the parameterization to be diminishing as spatial resolution is refined. We also find some limitations of the machine learning model in implementation: (a) tuning of the outputs from the parameterization at various depths is necessary; (b) the forcing near boundaries is not predicted as well as in the open ocean; (c) the cost of the parameterization is prohibitively high on central processing units. We discuss these limitations, present some solutions to problems, and conclude that this particular ML parameterization does inject energy, and improve backscatter, as intended but it might need further refinement before we can use it in production mode in contemporary climate models. 

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.