On the Linear Stability of the Lamb-Chaplygin Dipole

We consider the linear stability of the Lamb-Chaplygin dipole with respect to circulation-preserving perturbations in 2D Euler flows. It is demonstrated that this equilibrium solution 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. 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 found 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.