The stability of a horizontal stratified vortex

A trailing vortex behind a wing moving through a stratified fluid will act to twist the density field into a pattern where the density profile continuously overturns along the axis of the vortex. This configuration is approximated here with a density field that is overturning periodically along the axis of the vortex. The matching velocity field is found approximately assuming weak stratification and constant axial vorticity at leading order, and confining the flow to a fixed radius. This base flow is shown to be unstable to a wave triad consisting of two disturbance waves and a component of the base flow as the third `wave'. Three components of this base flow lead to instability: 1) the twirling component, 2) the streaming component, and 3) the rolling component. All three instabilities depend strongly on the axial length for the density field to overturn (the pitch). The twirling and rolling instabilities are important when the pitch is small, and they also depend on the Froude number. The streaming instability is dominant when the pitch is large, and is independent of the Froude number.