An Analytical Criterion for Centrifugal Instability in Non-Axisymmetric Vortices

etc15 Tracking Number 341

Non-axisymmetric vortices are ubiquitous in nature; examples include polar vortices in planets, the giant red spot in Jupiter, tornadoes and cyclones on Earth, mesoscale eddies in the ocean. Turbulent flows are furthermore known to be dominated by small- and large-scale vortex structures. Owing to the wide range of applications, knowledge of conditions under which a given vortex becomes unstable is beneficial. Here, the centrifugal instability of two-dimensional, non-axisymmetric vortices in the presence of an axial flow $(w)$ and a background rotation $(\Omega_z)$ is studied using the local stability approach. The local stability approach, based on geometric optics and similar in formulation to the rapid distortion theory \cite{bib:godeferd2001}, considers the evolution of shortwavelength perturbations along streamlines in the base flow. This approach, developed by Lifschitz $\&$ Hameiri \cite{bib:lifschitz1991}, is particularly useful for base flows for which a global stability analysis is computationally expensive. A sufficient criterion for centrifugal instability in an axisymmetric vortex with $(w)$ and $(\Omega_z)$ is first derived by analytically solving the local stability equations for wave vectors that are periodic upon evolution around a closed streamline. This criterion is then heuristically extended to non-axisymmetric vortices and written in terms of integral quantities on a streamline. The criterion is then shown to be accurate in describing centrifugal instability over a reasonably large range of parameters that specify Stuart vortices and Taylor-Green vortices.