On the Linear Stability of the Lamb-Chaplygin Dipole

TL;DR

This paper investigates the linear stability of the Lamb-Chaplygin dipole, revealing its inherent instability through spectral analysis and numerical simulations, and highlighting the subtle nature of this instability involving infinite-dimensional effects.

Contribution

The study provides a detailed spectral analysis of the Lamb-Chaplygin dipole's linear stability, including derivation of a simplified linearized equation and identification of unstable eigenmodes.

Findings

The flow is linearly unstable.

Unstable eigenmodes are localized near the vortex boundary.

Eigenvalues are embedded in the essential spectrum, indicating subtle instability.

Abstract

The Lamb-Chaplygin dipole (Lamb1895,Lamb1906,Chaplygin1903) is one of the few closed-form relative equilibrium solutions of the 2D Euler equation characterized by a continuous vorticity distribution. We consider the problem of its linear stability with respect to 2D circulation-preserving perturbations. It is demonstrated that this flow is linearly unstable, although the nature of this instability is subtle and cannot be fully understood without accounting for infinite-dimensional aspects of the problem. To elucidate this, we first derive a convenient form of the linearized Euler equation defined within the vortex core which accounts for the potential flow outside the core while making it possible to track deformations of the vortical region. The linear stability of the flow is then determined by the spectrum of the corresponding operator. Asymptotic analysis of the associated eigenvalue problem shows the existence of approximate eigenfunctions in the form of short-wavelength oscillations localized near the boundary of the vortex and these findings are confirmed by the numerical solution of the eigenvalue problem. However, the time-integration of the 2D Euler system reveals the existence of only one linearly unstable eigenmode and since the corresponding eigenvalue is embedded in the essential spectrum of the operator, this unstable eigenmode is also shown to be a distribution characterized by short-wavelength oscillations rather than a smooth function. These findings are consistent with the general results known about the stability of equilibria in 2D Euler flows and have been verified by performing computations with different numerical resolutions and arithmetic precisions.