Stability Analysis of the Q-Vortex: Critical Swirling Parameter Determination via Perturbation Theories and Resonant Triadic Perturbations in the Sub-Critical Region

We explore how the swirling parameter (q) of the Q-vortex influences its linear stability using degenerate and nearly-degenerate perturbation theories. Our study centers on the unstable inviscid modes of the Q-vortex, approximating variations in their mode shapes and eigenvalues as q changes through the non-degenerate perturbation theory. This approach is numerically implemented with our pseudospectral method for unbounded cylindrical domains, and our approximations closely match direct eigenvalue calculations. As q decreases, unstable modes approach neutral stability. At this stage, we employ nearly-degenerate perturbation theory to trace these modes to their fully neutral counterparts, allowing us to develop an efficient algorithm for determining the critical q value at which the Q-vortex transitions to stability. This method enables a more efficient sweeping of parameter space compared to traditional 'shooting' methods. Additionally, we identify resonant triads among these traced-back neutral modes and use them to perturb a Q-vortex with q below the critical value for linear stability. Our aim is to investigate the nonlinear evolution of these perturbations and their potential to induce unstable growth via triadic resonance in the sub-critical region.