Investigating stability and finding new steady vortex flows through numerical bifurcation approaches
In 1875, Lord Kelvin stated an energy-based argument for determining equilibrium and stability in vortex flows. The possibility of implementing Kelvin’s argument, in the form of a simple bifurcation approach, had been the subject of debate. In this talk, we build on results from dynamical systems theory to show that, by constructing solution families through isovortical rearrangements, one obtains a bifurcation diagram that contains stability information. To detect bifurcations to new equilibrium families, we propose calculating vortices that have been made “imperfect” through the introduction of asymmetries in the vorticity field. The resulting approach can in principle be used to determine the number of positive-energy (likely unstable) modes for each solution belonging to a family of steady vortices. However, to compute full bifurcation diagrams, we must numerically solve the Euler equations in an unbounded domain, without prescribing any symmetry. These requirements pose challenges in achieving convergence (due to degeneracies in the Euler equations) and accuracy (as small-scale features develop). We introduce a numerical method that overcomes these limitations. We apply the overall numerical bifurcation methodology to several families of flow equilibria, including elliptical vortices, opposite-signed vortex pairs (of both rotating and translating type), single and double vortex rows, as well as gravity waves. For all flows, we discover new families of equilibria, which exhibit lower symmetry, and obtain stability properties in agreement with detailed linear stability analyses. We conclude by highlighting general solution features that still require explanation.