Strato-hyperbolic instability: a new mechanism of instability in stably stratified vortices

Abstract

The stability of stably stratified vortices is studied by local stability analysis. Three base flows that possess hyperbolic stagnation points are considered: the two-dimensional (2-D) Taylor–Green vortices, the Stuart vortices and the Lamb–Chaplygin dipole. It is shown that the elliptic instability is stabilized by stratification; it is completely stabilized for the 2-D Taylor–Green vortices, while it remains and merges into hyperbolic instability near the boundary or the heteroclinic streamlines connecting the hyperbolic stagnation points for the Stuart vortices and the Lamb–Chaplygin dipole. More importantly, a new instability caused by hyperbolic instability near the hyperbolic stagnation points and phase shift by the internal gravity waves is found; it is named the strato-hyperbolic instability; the underlying mechanism is parametric resonance as unstable band structures appear in contours of the growth rate. A simplified model explains the mechanism and the resonance curves. The growth rate of the strato-hyperbolic instability is comparable to that of the elliptic instability for the 2-D Taylor–Green vortices, while it is smaller for the Stuart vortices and the Lamb–Chaplygin dipole. For the Lamb–Chaplygin dipole, the tripolar instability is found to merge with the strato-hyperbolic instability as stratification becomes strong. The modal stability analysis is also performed for the 2-D Taylor–Green vortices. It is shown that global modes of the strato-hyperbolic instability exist; the structure of an unstable eigenmode is in good agreement with the results obtained by local stability analysis. The strato-hyperbolic mode becomes dominant depending on the parameter values.