# A generalized action-angle representation of wave interaction in stratified shear flows

## TL;DR

This paper introduces a generalized action-angle Hamiltonian framework for analyzing wave interactions in stratified shear flows, highlighting the role of pseudo-energy and wave phases in stability and resonance phenomena.

## Contribution

It develops a novel formalism expressing wave interactions in stratified shear flows using action-angle Hamilton equations, linking to resonance instability mechanisms.

## Key findings

- Provides a compact Hamiltonian formulation of wave dynamics.
- Relates the formalism to resonance instability mechanisms.
- Highlights the time-dependent nature of wave actions.

## Abstract

In this paper we express the linearized dynamics of interacting interfacial waves in stratified shear flows in the compact form of action-angle Hamilton equations. The pseudo-energy serves as the Hamiltonian of the system, the action coordinates are the contribution of the interfacial waves to the wave-action, and the angles are their phases. The term "generalized action-angle" aims to emphasize that the action of each wave is generally time dependent and this allows instability. An attempt is made to relate this formalism to the action at a distance resonance instability mechanism between counter-propagating vorticity waves via the global conservations of pseudo-energy and pseudo-momentum.