# On the normal form of synchronization and resonance between vorticity waves in shear flow instability

## TL;DR

This paper models shear flow instability as a nonlinear dynamical system of vorticity wave interactions, revealing a phase-locking resonance mechanism and a new bifurcation type through a Hamiltonian action-angle framework.

## Contribution

It introduces a generalized Hamiltonian formulation of vorticity wave interactions and uncovers a novel inhomogeneous bifurcation in shear flow instability analysis.

## Key findings

- Normal modes correspond to fixed points of the dynamical system.
- The phase-locking depends on the cosine of the relative phase.
- A new bifurcation involving the annihilation of fixed points was identified.

## Abstract

A central mechanism of linearised two dimensional shear instability can be described in terms of a nonlinear, action-at-a-distance, phase-locking resonance between two vorticity waves which propagate counter to their local mean flow as well as counter to each other. Here we analyze the prototype of this interaction as an autonomous, nonlinear dynamical system. The wave interaction equations can be written in a generalized Hamiltonian action-angle form. The pseudo-energy serves as the Hamiltonian of the system, the action coordinates are the contribution of the vorticity waves to the wave-action, and the angles are the phases of the vorticity waves. The term "generalized action-angle" emphasizes that the action of each wave is generally time dependent, which allows instability. The synchronization mechanism between the wave phases depends on the cosine of their relative phase, rather than the sine as in the Kuramoto model. The unstable normal modes of the linearised dynamics correspond to the stable fixed points of the dynamical system and vice versa. Furthermore, the normal form of the wave interaction dynamics reveals a new type of inhomogeneous bifurcation -- annihilation of a pair of stable and unstable fixed points yields the emergence of two neutral center fixed points of opposite circulation.