The Okubo–Weiss criterion in hydrodynamic flows: geometric aspects and further extension

Abstract

The Okubo (1970 Deep Sea Res. 17 445)–Weiss (1991 Physica D 48 273) criterion, has been extensively used as a diagnostic tool to divide a two-dimensional (2D) hydrodynamical flow field into hyperbolic and elliptic regions and to serve as a useful qualitative guide to the complex quantitative criteria. The Okubo–Weiss criterion is frequently validated on empirical grounds by the results ensuing its application. So, we will explore topological implications into the Okubo–Weiss criterion and show the Okubo–Weiss parameter is, to within a positive multiplicative factor, the negative of the Gaussian curvature of the vorticity manifold. The Okubo–Weiss criterion is then reformulated in polar coordinates, and is validated for several examples including the Lamb–Oseen vortex, and the Burgers vortex. These developments are then extended to 2D quasi-geostrophic (QG) flows. The Okubo–Weiss parameter is shown to remain robust under the β-plane approximation to the Coriolis parameter. The Okubo–Weiss criterion is shown to be able to separate the 2D flow-field into coherent elliptic structures and hyperbolic flow configurations very well via numerical simulations of quasi-stationary vortices in QG flows. An Okubo–Weiss type criterion is formulate for 3D axisymmetric slows, and is validated via application to the round Landau–Squire Laminar jet flow.