Nonlinear stability of axisymmetric swirling flows

Abstract

We establish sufficient conditions for the nonlinear stability of incompressible, inviscid, swirling flows using the Arnol’d energy-Casimir method. We derive an axisymmetric Lie-Poisson bracket and work with equations of motion in swirl-function-vortex-density form. The flows and perturbations we consider may have axial variations. The formulation is closely analogous to that of two-dimensional, stratified, Boussinesq flows considered by Abarbanel et al . [ Phil. Trans. R. Soc. Lond. A 318, 349-409 (1986)) and a high-wavenumber cut off is necessary to overcome indefiniteness, as in that case. We give several examples of columnar swirling flows and discuss the relation of our results to linear stability studies of swirling flows.