  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.
