Two-dimensional vortex dynamics in a stratified barotropic fluid

Abstract

We show that two-dimensional ‘point’ vortex dynamics in both a polytropic fluid of γ = 3/2 and an isothermal fluid stratified by a constant gravitational field can be written in Hamiltonian form. We find that the formulation admits only one constant of the motion in addition to the Hamiltonian, so that two vortices are the most for which the motion is generally integrable. We study in detail the two-vortex problem and find a rich collection of behaviour: closed trajectories analogous to the circular orbits of the uniform-fluid two-vortex problem, open trajectories for which the self-propelled vortices scatter off each other, and both unstable and stable steadily translating pairs of vortices. Comparison is made to the case of two vortices in a uniform-density fluid bounded by a wall.