A diagnostic framework linking eddy flux ellipse with eddy-mean energy exchange

Abstract The performance of non-eddy-resolving numerical models requires a good understanding, with appropriate representation, of eddy-mean flow feedbacks. To help improve our understanding of these feedbacks, we propose here a diagnostic framework \textcolor{black}{that links eddy geometry to the eddy-mean energy exchange terms from the Lorenz energy diagram}. In particular, this framework provides explicit mathematical formulas that link eddy-mean energy exchange rates with both mean state structure and the properties of eddy momentum/buoyancy ellipses. Complimenting previous works that interpret the mean flow shear cases (e.g., ${\partial \overline{u}}/{\partial y}$ and ${\partial \overline{u}}/{\partial z}$) using eddy geometry, this framework also gives a geometric interpretation of the horizontal normal strain component (e.g., ${\partial \overline{u}}/{\partial x}$) of eddy-mean kinetic energy exchange rates ($M_{K_E}$). To test this framework, we use the ECCO2 (Estimating the Circulation and Climate of the Ocean, phase 2) state estimate. In the ECCO2 state estimate, the horizontal normal strain component of the globally integrated $M_{K_E}$ has a magnitude comparable to that of the shear strain component. Regional application of this framework shows that the quasi-meridional orientation of eddy momentum ellipses in the downstream Kuroshio Extension is consistent with the dominance of the horizontal normal strain component of $M_{K_E}$ there. Our key findings \textcolor{black}{also hold} in a submesoscale-permitting model. This framework offers a tool for interpreting eddy-mean energy exchange and eddy ellipse patterns. It may also offer guidance for the parameterizations of eddy stress and eddy-mean flow feedbacks.