Three-Wave Resonance in Neutrally Stable Wake Vortices

We examine the weakly nonlinear dynamics of linearly-neutrally-stable columnar vortices, whose nonlinear instabilities could help mitigate aircraft wake hazards. Generally, neutral eigenmodes must be in resonance to interact nonlinearly, where the lowest-order resonance in the wake vortex flows involves triplets of eigenmodes, with quadratic nonlinearities. Adapting a multi-scale perturbation approach, the evolution of the resonant triads is found to follow the classical "three-wave equations", and secondary instabilities such as the elliptical instability are recovered by setting one of the modes to be dominant. When dissipation is absent, triadic resonance of non-singular modes obeys the conservation laws known as the Manely-Rowe relations, which bound the triad's amplitudes for all time unless its resonance is of an explosive kind. Using a spectral method for unbounded domains and non-degenerate perturbation theory, the resonant triads of non-singular inviscid modes of the Lamb-Oseen vortex are quickly located, but none are found to be explosive. Therefore, we examine triads in which the dominant mode is a singular critical-layer mode that is regularized by viscosity, and we exploit a pseudospectral code that has tunable parameters to accurately and efficiently compute the regularized critical-layer modes.