Bifurcation of standing waves into a pair of oppositely traveling waves with oscillating amplitudes caused by a three-mode interaction

TL;DR

This paper discovers a new flow state in Taylor-Couette flow where two oppositely traveling waves oscillate in amplitude, resulting from a three-mode interaction causing a Hopf bifurcation.

Contribution

It introduces a novel oscillating wave state arising from a three-mode interaction, supported by numerical simulations and a coupled amplitude equation model.

Findings

Identified a flow state with oscillating traveling waves in Taylor-Couette system.

Demonstrated bifurcation scenario involving a nonlinearly excited mode.

Presented a three-mode amplitude equation model capturing the bifurcation.

Abstract

A novel flow state consisting of two oppositely travelling waves (TWs) with oscillating amplitudes has been found in the counterrotating Taylor-Couette system by full numerical simulations. This structure bifurcates out of axially standing waves that are nonlinear superpositions of left and right handed spiral vortex waves with equal time-independent amplitudes. Beyond a critical driving the two spiral TW modes start to oscillate in counterphase due to a Hopf bifurcation. The trigger for this bifurcation is provided by a nonlinearly excited mode of different symmetry than the spiral TWs. A three-mode coupled amplitude equation model is presented that captures this bifurcation scenario. The mode-coupling between two symmetry degenerate critical modes and a nonlinearly excited one that is contained in the model can be expected to occur in other structure forming systems as well.