Hydrodynamic Instability of Small Eddies in a Tornado Vortex

Abstract

A numerical method is shown for solving problems of nonaxisymmetric perturbations in incompressible, inviscid, swirling flow for arbitrary undisturbed azimuthal and axial flows. Exclusive attention is given to the hydrodynamic stability at large azimuthal and axial wavenumbers for a flow with a zone of constant and coinciding vertical and azimuthal vorticities. An infinite sequence of unstable modes is found having very large amplitude within the verticity zone. Analysis of a 3-level model confirms the instability in the 46-level numerical model and shows the instability criterion to be kr?z + n?? < 0, where k (>0) and n (≥0) are axial and azimuthal wavenumbers, respectively, ?z (>0) and ?? (≥0) are the corresponding vorticities, and r is radius. Maximum instability occurs approximately where k and n are related by 2kr?z + n?? = 0, provided ?? is small compared to ?z. Growth rates, for a tornado vortex of typical dimensions and vorticities, are significant fractions of a second.