Foundations of Eddy Parameterizations

To advance the development and assessment of parameterizations of eddy transport in numerical models, the creation and application of a new theoretical framework is proposed. The framework is not a new parameterization, but rather a set of principles linking physical meaning to parameterization form, including extant as well as yet-unformulated variations. The result will be a gain in perspective that will support and spur parameterization progress and evolution.

Drawing on ocean physics together with methods from differential geometry and stochastic differential equations that are not yet widely applied in this field, we will incorporate three essential elements: (i) explicit and rigorous definitions for filtering and averaging operations, clearly delineating what is meant by “mean”; (ii) a flexible approach to stochastic modeling that relaxes the unrealistic constraints implicit in existing parameterizations on eddy path smoothness and dependence on local conditions; and (iii) the use of coordinate-independent representations to express the physical essence of parameterizations free from any particular choice of model geometry, together with formal rules for realizing each formulation in any given coordinate system. This combination is termed the Explicit, Stochastic, Coordinate-independent or ESC framework.

This work will have immediate practical benefits: clarifying the link between parameterization forms and conserved properties (energy, enstrophy, etc.); speeding the identification and assessment of discretization errors in multiple model systems; uniquely identifying the advective and diffusive components of parameterized fluxes across implementations; and accounting for the impact of the observed range of high-frequency trajectory behavior on parameterization structure. This work will provide an avenue for the cross-evaluation of existing or proposed parameterizations from first principles, reducing the reliance on simulations for that purpose. Two significant software packages that will result from this work are (i) symbolic algebra software for implementing eddy parameterizations in an arbitary model coordinate system and choice of discretization, and (ii) numerical code for error diagnostics in model implementation. The theoretical orientation of this project is timely, as it is synergistic with the applied focus of ongoing parameterization testing and improvement efforts, as well as with an ongoing mathematical effort; close collaboration with these groups will ensure the ready accessibility and impact of our results.

The aspiration is to bridge the gap between fundamental theory and concrete applications. In so doing, the proposed new conceptual tools will enable the study of eddy fluxes—one of the most consequential areas of research in ocean physics—to evolve at a more rapid and efficient pace. Building on experience in developing parameterizations both for coarse and eddy-permitting resolutions, and facilitating the blossoming efforts in machine learning, this work will yield tools that will apply both to the present and future generations of models.