Particle description of the interaction between wave packets and point vortices

TL;DR

This paper develops a comprehensive model of wave packet and point vortex interactions on the ocean surface, revealing how wave-induced drifts can stabilize vortex configurations and providing insights into ocean surface dynamics.

Contribution

It introduces a generalized Lagrangian framework for wave-vortex interactions, extending classical models to include wave action, phase, and vorticity fields, and analyzes their stability and drift effects.

Findings

Wave packets induce a net drift on vortices similar to Darwin drift.

Wave packet interactions can stabilize otherwise unstable vortex streets.

Numerical simulations show long-term stable configurations relevant to the upper ocean.

Abstract

This paper explores an idealized model of the ocean surface in which widely separated surface-wave packets and point vortices interact in two horizontal dimensions. We start with a Lagrangian which, in its general form, depends on the \emph{fields} of wave action, wave phase, stream function, and two additional fields that label and track the vertical component of vorticity. By assuming that the wave action and vorticity are confined to infinitesimally small, widely separated regions of the flow, we obtain model equations that are analogous to, but significantly more general than, the familiar system consisting solely of point vortices. We analyze stable and unstable harmonic solutions, solutions in which wave packets eventually coincide with point vortices (violating our assumptions), and solutions in which the wave vector eventually blows up. Additionally, we show that a wave packet induces a net drift on a passive vortex in the direction of wave propagation which is equivalent to Darwin drift. Generalizing our analysis to many wave packets and vortices, we examine the influence of wave packets on an otherwise unstable vortex street and show analytically, according to linear stability analysis, that the wave packet induced drift can stabilize the vortex street. The system is then numerically integrated for long times and an example is shown in which the configuration remains stable, which may be particularly relevant for the upper ocean.