The Geometric Decomposition of Eddy Feedbacks: Insights into the Energetics of Scale Interactions

Stephanie Waterman, University of New South Wales, Climate Change Research Centre & ARC Centre of Excellence for Climate System Science, Sydney, Australia; University of British Columbia, Earth, Ocean and Atmospheric Sciences, Vancouver, BC, Canada, Jonathan M Lilly, NorthWest Research Associates, Redmond, WA, United States and Kial Douglas Stewart, Australian National University, Research School of Earth Science, Canberra, ACT, Australia
Abstract:
Understanding eddy-mean flow interactions is a long-standing problem in geophysical fluid dynamics, with modern relevance to the task of representing eddy effects in coarse resolution ocean models while preserving their dependence on the underlying dynamics of the flow field. A promising approach is to express the eddy forcing of the mean flow in the form of gradient operators applied to an eddy stress tensor. Such a formulation yields what we term the “geometric decomposition” of eddy feedbacks, a framework in which eddy-mean flow interactions are expressed in terms of the eddy energy together with geometric parameters describing the average variance ellipse (or in 3D, ellipsoid) shape and orientation. This framework has the potential to offer new insights into eddy-mean flow interactions in a number of ways: 1. it identifies the components of the eddy motions that can force mean flows; 2. it links eddy effects to the spatial patterns of eddy geometry, providing potential insight into the mechanisms underpinning these effects; and 3. it illustrates the importance of resolving the characteristic shape and orientation of eddy fluctuations, and not just the eddy energy, to accurately represent eddy feedback effects.

Here we consider the utility of the geometric decomposition in elucidating the energetics of scale interactions, with a particular focus on the insights that this decomposition can provide on mean-eddy energy transfers, as well as the bounds that the eddy energy places on the eddy forcing of the mean flow. For example, the tilts of the variance ellipses provide an intuitive description of the exchange of energy between the eddy- and mean- flows, while the ellipse anisotropies are interpretable as geometric expressions of the bounds that the eddy kinetic and potential energies place on eddy fluxes, and ultimately on the eddy forcing of the mean flow. These important concepts will be both discussed and illustrated.

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