Linear Stability of Hill's Vortex to Axisymmetric Perturbations

TL;DR

This paper analyzes the linear stability of Hill's vortex to axisymmetric perturbations using shape differentiation and spectral methods, revealing stable and unstable eigenvalues and singular eigenfunctions at stagnation points.

Contribution

It introduces a shape differentiation approach combined with spectral discretization to analyze the stability of Hill's vortex, refining classical results.

Findings

Identifies two stable and two unstable eigenvalues.

Eigenfunctions are singular, sharply peaked at stagnation points.

Provides a refined stability analysis of Hill's vortex.

Abstract

We consider the linear stability of Hill's vortex with respect to axisymmetric perturbations. Given that Hill's vortex is a solution of a free-boundary problem, this stability analysis is performed by applying methods of shape differentiation to the contour dynamics formulation of the problem in a 3D axisymmetric geometry. This approach allows us to systematically account for the effect of boundary deformations on the linearized evolution of the vortex under the constraint of constant circulation. The resulting singular integro-differential operator defined on the vortex boundary is discretized with a highly accurate spectral approach. This operator has two unstable and two stable eigenvalues complemented by a continuous spectrum of neutrally-stable eigenvalues. By considering a family of suitably regularized (smoothed) eigenvalue problems solved with a range of numerical resolutions we demonstrate that the corresponding eigenfunctions are in fact singular objects in the form of infinitely sharp peaks localized at the front and rear stagnation points. These findings thus refine the results of the classical analysis by Moffatt & Moore (1978).