Interpreting Reynolds stress from perspective of eddy geometry

Abstract

Geometric features in oceanic mesoscale eddies such as tilt and anisotropy can influence the properties of the Reynolds stress that provides feedback between the eddies and the background flow. By regarding an eddy as a wave, previous studies have parameterized the Reynolds stress based on the equivalence in the tilt angle between the phase of the eddy stream functions and the variance ellipse for the Reynolds stress (RS-ellipse). However, the wave assumption cannot predict the anisotropy of the RS-ellipse, and also largely simplifies the eddy geometry, which would naturally be an ellipsoid rather than a wave. The present study explores the shape relation between elliptical eddies and the RS-ellipse, by mathematically reformulating the Reynolds stress based on the eddy shape. The new formula reveals that the shape relation is regulated by the horizontal extent of the occurrence probability distribution (PDF) of the eddy, and that the shape of the eddy and RS-ellipse are identical at the place of maximum PDF when the horizontal scale of the PDF is sufficiently larger than the size of the eddy. A similar tendency is found in eddies detected by satellite altimetry in the Kuroshio Extension jet region. A detailed analysis of the PDF in this region shows that the tilts of the eddies are likely to be consistent with the destabilization effect on the jet, suggesting a strong relation between the eddy geometry and the jet's stability in this region. These findings may open a path toward a new method to parameterize the Reynolds stress with the background state, exploiting the shape equivalence between the eddies and the RS-ellipse.