Department of Mathematics

Vortex dynamics and vortex stability

The primary aim of the Vortex Dynamics and Vortex Stability Research Project is to develop the global stability theory and nonlinear stability theory for vorticity-dominated flows.


It was well said by Kuchemann in 1965 that “vortices are the sinews and muscles of fluid motions”. The vortex dynamics is the key for us to understand complicated unsteady fluid motions such as the vortex breakdown phenomenon and turbulence.  Some models describing the interaction of many vortices are also known to exhibit patterns that have been be observed in a wide variety of applications, from laboratory experiments involving electron columns to hurricanes, and thus provide a framework for rigorously analysing these behaviours.

The principal goals of this project are to develop stability theory for vorticity-dominated flows and to examine the types of patterns that can develop. In the case of the former, one main aim is to resolve major long-time standing fluid mechanics problems by using and developing modern mathematical methods.

More specifically, we are focusing on developing a unified global stability theory for vortex breakdown. To expand, the vortex breakdown, referred to as an abrupt burst of vortex flows, is a widespread phenomenon that critically affects a variety of flows of great importance in real world. We are working on developing a unified global stability theory, which may further extend the original global stability theory (Wang & Rusak 1995-1998) to more complicated and realistic flow situations. The goal is to establish a comprehensive vortex breakdown theory and to settle down this long-standing flow problem.

Secondly we are looking to develop the nonlinear stability theory for vortices.

Rayleigh established as a classical theory for the linear stability of infinite-long vortices in 1916. Szeri & Holmes 1988, among many others, have made efforts, using Arnold’ method, to develop a nonlinear stability theory for vortices, but only achieved a partial success. We have successfully developed the first clean nonlinear stability theory for vortices in recent years. We are working on further developing the theory and exploring the full physical implications of the nonlinear stability theory.


Researchers at The University of Auckland

Current Collarborators