TY - JOUR

T1 - On the normal form of synchronization and resonance between vorticity waves in shear flow instability

AU - Heifetz, Eyal

AU - Guha, Anirban

N1 - Funding: Alexander von Humboldt foundation.

PY - 2019/10

Y1 - 2019/10

N2 - 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 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 actionangle” 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.

AB - 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 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 actionangle” 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.

U2 - 10.1103/PhysRevE.100.043105

DO - 10.1103/PhysRevE.100.043105

M3 - Article

C2 - 31770928

VL - 100

JO - Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E: Statistical, Nonlinear, and Soft Matter Physics

SN - 1539-3755

IS - 4

M1 - 043105

ER -