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Geometric Decomposition of Eddy Feedbacks in Barotropic Systems

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  • 1 Department of Earth, Ocean and Atmospheric Sciences, University of British Columbia, Vancouver, British Columbia, Canada, and Climate Change Research Centre and ARC Centre of Excellence for Climate System Science, University of New South Wales, Sydney, New South Wales, Australia
  • | 2 NorthWest Research Associates, Redmond, Washington
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Abstract

In oceanic and atmospheric flows, the eddy vorticity flux divergence—denoted “F” herein—emerges as a key dynamical quantity, capturing the average effect of fluctuations on the time-mean circulation. For a barotropic system, F is derived from the horizontal velocity covariance matrix, which itself can be represented geometrically in terms of the so-called variance ellipse. This study proves that F may be decomposed into two different components, with distinct geometric interpretations. The first arises from variations in variance ellipse orientation, and the second arises from variations in the kinetic energy of the anisotropic part of the velocity fluctuations, which can be seen as a function of variance ellipse size and shape. Application of the divergence theorem shows that F integrated over a closed region is explained entirely by separate variations in these two quantities around the region periphery. A further decomposition into four terms shows that only four specific spatial patterns of ellipse variability can give rise to a nonzero eddy vorticity flux divergence. The geometric decomposition offers a new tool for the study of eddy–mean flow interactions, as is illustrated with application to an unstable eastward jet on a beta plane.

Corresponding author address: Stephanie Waterman, Department of Earth, Ocean and Atmospheric Sciences, University of British Columbia, Room 2020, Earth Sciences Building, 2207 Main Mall, Vancouver, BC V6T 1Z4, Canada. E-mail: snw@alum.mit.edu

Abstract

In oceanic and atmospheric flows, the eddy vorticity flux divergence—denoted “F” herein—emerges as a key dynamical quantity, capturing the average effect of fluctuations on the time-mean circulation. For a barotropic system, F is derived from the horizontal velocity covariance matrix, which itself can be represented geometrically in terms of the so-called variance ellipse. This study proves that F may be decomposed into two different components, with distinct geometric interpretations. The first arises from variations in variance ellipse orientation, and the second arises from variations in the kinetic energy of the anisotropic part of the velocity fluctuations, which can be seen as a function of variance ellipse size and shape. Application of the divergence theorem shows that F integrated over a closed region is explained entirely by separate variations in these two quantities around the region periphery. A further decomposition into four terms shows that only four specific spatial patterns of ellipse variability can give rise to a nonzero eddy vorticity flux divergence. The geometric decomposition offers a new tool for the study of eddy–mean flow interactions, as is illustrated with application to an unstable eastward jet on a beta plane.

Corresponding author address: Stephanie Waterman, Department of Earth, Ocean and Atmospheric Sciences, University of British Columbia, Room 2020, Earth Sciences Building, 2207 Main Mall, Vancouver, BC V6T 1Z4, Canada. E-mail: snw@alum.mit.edu
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