The Ellipsoidal Vortex: A Novel Approach to Geophysical Turbulence

William J Mckiver

doi:10.1155/2015/613683

Abstract

We review the development of the ellipsoidal vortex model within the field of geophysical fluid dynamics. This vortex model is built on the classical potential theory of ellipsoids and applies to large-scale fluid flows, such as those found in the atmosphere and oceans, where the dynamics are strongly affected by the Earth's rotation. In this large-scale limit the governing equations reduce to the quasi-geostrophic system, where all the dynamics depends on a single scalar field, the potential vorticity, which is a dynamical marker for vortices. The solution of this system is achieved by the inversion of a Poisson equation, that in the case of an ellipsoidal vortex can be solved exactly. From this ellipsoidal solution equilibria have been determined and their stability properties have been studied. Many studies have shown that this ellipsoidal vortex model, while being conceptually simple, is an extremely powerful tool in eliciting some of the fundamental characteristics of turbulent geophysical flows.

Highlights

In turbulent flow there exist structures whose collective interactions are at the very heart of the dynamics of turbulence— vortices
We review the development of the ellipsoidal vortex model within the field of geophysical fluid dynamics
We reviewed the recent history of the theory of ellipsoidal vortices in the field of geophysical fluid dynamics

Summary

Introduction

In turbulent flow there exist structures whose collective interactions are at the very heart of the dynamics of turbulence— vortices. One of the earliest incarnations of such an approach was that of Kirchhoff [6], who studied an isolated two-dimensional patch of uniform vorticity bounded within an ellipse He showed that this is an exact analytical solution to the Euler equations which rotates with a constant angular velocity. Further work by Love [7] showed that there existed stable equilibria when this vortex is subjected to disturbances The extension of these early works to the threedimensional case of an ellipsoidal vortex had to wait almost a century for the works of Zhmur and Pankratov [8, 9] and Meacham [10] who extended this approach to the quasigeostrophic system [11], a set of equations relevant to largescale geophysical fluid dynamics.

The Quasi-Geostrophic System

Potential Theory of an Isolated Ellipsoidal Vortex

Evolution of a Vortex in a Background Shear Flow

Equilibria and Stability

Multiple Ellipsoidal Vortices

Conclusions