Diagnostic Framework Linking Eddy Flux Ellipse with Eddy-Mean Energy Exchange

Abstract

The design of non-eddy-resolving numerical models requires a good understanding and an appropriate representation of the eddy-mean flow feedback. To understand this feedback, we propose a diagnostic framework that links eddy geometry with the eddy-mean energy exchange terms in the Lorenz energy diagram. This framework provides explicit mathematical formulas that link eddy-mean energy exchange rates with both the mean state structure and the properties of eddy momentum ellipses and eddy buoyancy ellipses. Considering that the mean flow contains both along- and cross-stream variations, we decompose the eddy-mean kinetic energy exchange term into 3 components: one associated with the cross-stream variation in mean flow (MC), one associated with the along-stream variation in mean flow (MA), and one associated with the variation in mean flow (MR). We also state the corresponding geometric formulas. The geometric interpretation of MC is consistent with barotropic instability theories and the literature on eddy geometry. As for MA, the weakening (strengthening) of mean flow in the along-stream direction corresponds to eddy kinetic energy generation (decay) through MA. MA and a portion of MR are related under the quasi-geostrophic assumption. From a global integral perspective, both the along-stream and cross-stream variations in the mean flow contribute considerably to eddy-mean kinetic energy exchange. At the Kuroshio Extension, both the mean state energy level and eddy energy level are key to shaping the spatial pattern of eddy-mean energy exchange. This framework offers a tool for geometrically interpreting eddy-mean energy exchange, which may offer guidance for eddy parameterizations.