Normal form of synchronization and resonance between vorticity waves in shear flow instability

Abstract

A minimal model of linearized two-dimensional shear instabilities can be formulated in terms of an 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 pseudoenergy 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 linearized 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 stable and an unstable star node yields the emergence of two neutral center fixed points of opposite circulation.