Minimal nonlinear dynamical system for the interaction between vorticity waves and shear flows

This study is a direct follow-up of the paper by Heifetz and Guha [Phys. Rev. E 100, 043105 (2019)] on a minimal nonlinear dynamical system, describing a prototype of linearized two-dimensional shear instability. In that paper, the authors describe the instability in terms of an action at a distance between two vorticity waves, each of which propagates counter to its local mean flow as well as counter to the other. Here we add to the model the effect of mutual interaction between the waves and the mean flow, where growth of the waves reduces the mean shear and vice versa. This addition yields oscillatory Hamiltonian dynamics, including states of phase slipping and libration with finite-size wave amplitude oscillations. We find that wave???mean-flow dynamics emerging from unstable normal modes in the linearized stage are doomed to librate around the antiphased neutral configuration in which the waves hinder each other???s counterpropagation rate. We discuss as well how the given dynamics relates to familiar models of phase oscillators.

Automated summary

This study is a direct follow-up to the previous paper by Heifetz and Guha, exploring a minimal nonlinear dynamical system that describes a linearized two-dimensional shear instability. By adding the effect of mutual interaction between the waves and the mean flow, the study reveals oscillatory Hamiltonian dynamics with phase slipping and finite-size wave amplitude oscillations. It further demonstrates that wave-mean flow dynamics resulting from unstable normal modes in the linearized stage exhibit oscillations around an antiphased neutral configuration where the waves hinder each other's counterpropagation rate. The study also discusses the connection between the dynamics observed and familiar models of phase oscillators.