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A Framework for Linear Stability Analysis of Finite-Area Vortices

In this investigation we revisit the question of the linear stability analysis of 2D steady Euler flows characterized by the presence of compact regions with constant vorticity embedded in a potential flow. We give a complete derivation of the linearized perturbation equation which, recognizing that the underlying equilibrium problem is of the free-boundary type, is done systematically using methods of the shape-differential calculus. Particular attention is given to the proper linearization of the contour integrals describing vortex induction. The thus obtained perturbation equation is validated by analytically deducing from it the stability analyses of the circular vortex, originally due to Kelvin (1880), and of the elliptic vortex, originally due to Love (1893), as special cases. We also propose and validate a spectrally-accurate numerical approach to the solution of the stability problem for vortices of general shape in which all singular integrals are evaluated analytically.