UC San Diego

The motion of buoyant vortices is studied using reduced-order models. For a vortex filament, i.e. a curve in three-dimensional space along which vorticity is concentrated as a delta function, the evolution of the filament is calculated using a desingularized Biot–Savart integral. Buoyancy is added using a momentum balance argument while surface tension is also included. A set of equations that couples the transverse motion of the buoyant vortex filament and the axial flow within its core is derived. The new model is verified in an asymptotic limit by comparing it to the previous analytical solution for a thin vortex ring. In another approximation, axisymmetric contour dynamics is implemented to model a buoyant vortex ring of deforming core. A vortex patch is enclosed by a vortex sheet which emerges from baroclinic generation of vorticity. The evolution of the vortex sheet strength is derived from the Euler equation, and represents the effects of density, buoyancy and surface tension on the axisymmetric vortex ring. Numerical calculations for the integro-differential equations are carried out until the curvature singularity of vortex sheet evolution leads to a blowup. Finally, a linear stability analysis is performed for vortices in the presences of density and surface tension. The basic state solution is given as a perturbation series in a small parameter representing curvature of the vortex or a weak strain acting on it. With the small parameter in the basic state, instabilities are triggered by the resonances between two neutrally stable Kelvin waves for the unperturbed vortex. Those parametric instabilities are the curvature instability and the Moore–Saffman–Tsai–Widnall instability, corresponding to the parameter being curvature or a strain. The effects of density and surface tension on these instabilities are studied by calculating the growth rate and the instability bandwidth in various wavenumber regimes.