Perturbation analysis of triadic resonance in columnar vortices: selection rules and the roles of external forcing and critical layers

Abstract

The remarkable robustness of isolated columnar vortices suggests the existence of fundamental constraints that prevent spontaneous disintegration. In this work, we investigate the weakly nonlinear stability of such flows, demonstrating that the triadic resonance of wave modes is governed by a set of hydrodynamic ``selection rules''. By employing a multi-scale perturbation analysis, we prove that resonant interactions between smooth neutral modes, specifically regular Kelvin waves and discrete critical layer modes with passive singularities, are strictly conservative and confined to the Manley--Rowe relations. Using wave pseudoenergy within a large-$k$ WKBJ framework, we show that these rules topologically prohibit intrinsic instability, analogous to the forbidden transitions of quantum mechanics. Consequently, the breakdown of a columnar vortex requires a specific symmetry-breaking mechanism to overcome this barrier. We identify and analyse two distinct pathways for this violation: (1) \textit{Parametric instability}, where external forcing acts as an active energy pump; using a robust tuning method based on non-degenerate perturbation theory, we generalize classical elliptical instability to arbitrary driving frequencies and identify new instability configurations involving dicrete critical layer modes. (2) \textit{Active critical layers}, where an embedded singularity breaks the Hermitian symmetry of the operator, enabling the extraction of mean-flow energy via a wave-mean resonance. These findings provide a theoretical guidance for flow control, suggesting that the mitigation of aircraft wake vortices requires either tuned external forcing or the ``engineering'' of critical layers (e.g., via thermal stratification) to trigger the forbidden transitions.