Applied Mathematics Seminar | Stephanie Waterman, A geometric decomposition of eddy-mean flow interactions

MC 6486

Speaker

Stephanie Waterman
Department of Earth, Ocean & Atmospheric Sciences,
University of British Columbia, Vancouver

Title

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. 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. Exploiting the geometric interpretation of this object yields the so-called “geometric decomposition” of eddy-mean flow interactions, a framework in which scale interactions are expressed in terms of the eddy energy together with geometric parameters describing average eddy shape and orientation. 

In this talk I will present the geometric decomposition of eddy-mean flow interactions, and illustrate it with an application to an unstable jet. Specifically I will show that in the barotropic (2D) case, 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 average eddy orientation; and 2. variations in the anisotropic part of the eddy kinetic energy, a function of the average eddy 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: a significant simplification to the representation of the eddy forcing. In an extension to the 3D case, I will show how the average eddy ellipsoid encodes useful information on both the dominant orientation of the eddy momentum and buoyancy fluxes, and the partitioning of eddy energy between kinetic and potential forms. As in the 2D case, spatial patterns in eddy ellipsoid geometry can be directly linked to the eddy forcing.

This framework has the potential to offer new insights into eddy-mean flow interactions in a number of ways. For example first, it identifies the ingredients of the eddy motion that have a mean flow forcing effect. Second, it links eddy effects to spatial patterns of variance ellipse geometry that can suggest the mechanisms underpinning these effects. 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.