Ingredients of the Eddy Soup: A Geometric Decomposition of Eddy-Mean Flow Interactions

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 models while preserving their dependence on the underlying dynamics of the flow field. Exploiting the recognition that the velocity covariance matrix/eddy stress tensor that describes eddy fluxes, also encodes information about eddy size, shape and orientation through its geometric representation in the form of the so-called variance ellipse, suggests a potentially fruitful way forward.

Here we present a new framework that describes eddy-mean flow interactions in terms of a geometric description of the eddy motion, and illustrate it with an application to an unstable jet. Specifically we show that the eddy vorticity flux divergence F, a key dynamical quantity describing the average effect of fluctuations on the time-mean flow, may be decomposed into two components with distinct geometric interpretations: 1. variations in variance ellipse orientation; and 2. variations in the anisotropic part of the eddy kinetic energy, a function of the variance ellipse size and shape. Application of the divergence theorem shows that F integrated over a region is explained entirely by variations in these two quantities around the region’s periphery.

This framework has the potential to offer new insights into eddy-mean flow interactions in a number of ways. It identifies the ingredients of the eddy motion that have a mean flow forcing effect, it links eddy effects to spatial patterns of variance ellipse geometry that can suggest the mechanisms underpinning these effects, and finally it illustrates the importance of resolving eddy shape and orientation, and not just eddy size/energy, to accurately represent eddy feedback effects. These concepts will be both discussed and illustrated.