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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.
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Comparing Two Parameterizations for the Restratification Effect of Mesoscale Eddies in an Isopycnal Ocean Model
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.
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- Award ID(s):
- 1912332
- PAR ID:
- 10479292
- Publisher / Repository:
- DOI PREFIX: 10.1029
- Date Published:
- Journal Name:
- Journal of Advances in Modeling Earth Systems
- Volume:
- 15
- Issue:
- 12
- ISSN:
- 1942-2466
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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