# The motion of buoyant point vortices

## TL;DR

This paper extends the Hamiltonian framework to include buoyancy effects in point vortices, systematically analyzing their dynamics, stability, and chaotic behavior in various vortex configurations.

## Contribution

It introduces a novel formulation for buoyant vortices, explores their interactions, and identifies conditions for bounded or unbounded motion and chaos.

## Key findings

- Two-vortex systems can be bounded or unbounded with a clear criterion.
- Maximum height achievable by buoyant vortex pairs is theoretically established.
- Chaotic motion is demonstrated in the buoyant three-vortex system.

## Abstract

A general formulation is presented for studying the motion of buoyant vortices in a homogeneous ambient fluid. It extends the well-known Hamiltonian framework for interacting homogeneous point vortices to include buoyancy effects acting on the vortices. This is then used to systematically examine the buoyant 1-, 2-, and 3-vortex problems. In doing so we find that 2 buoyant vortices may either evolve as a pair in bounded circular orbits, or as two independent unbounded vortices that drift apart, and a criteria is found to distinguish these cases. Special attention is given to the buoyant vortex couple, consisting of two vortices of equal and opposite circulation, and equal buoyancy anomaly. We show that a theoretical maximum height is generally possible for the rise (or fall) of such couples against buoyancy forces. Finally, the possibility and onset of chaotic motions in the buoyant 3-vortex problem is addressed. In contrast to the homogeneous 3-vortex problem, the buoyant vortex system shows evidence that chaos is present. We also demonstrate the chaotic advection of tracer parcels arising from the flow field induced by just 2 buoyant vortices.