Instability of an elliptical flow: Weakly nonlinear analysis

Elliptical flows are of interest since they appear in various vortical flows in which circular vortices are deformed by straining flow. We study the nonlinear evolution of an elliptical flow by weakly nonlinear analysis. Two sets of amplitude equations are derived for different situations. First, the evolution of bending waves is considered. Nonlinear interaction of the two base Kelvin waves results in cubic nonlinear terms, which leads to saturation of the elliptical instability. Next, the secondary instability is considered. Three Kelvin waves, one of which is a bending wave, form a resonant triad thanks to freedom of wavenumber shift. As a result three-wave equations augmented with linear terms are obtained as amplitude equations. They explain the previous numerical results on the secondary instability obtained by Kerswell J. Fluid Mech., 382, pp. 283-386, 1999.