  Anticyclonic vortices focus and trap near-inertial waves so that
near-inertial energy levels are elevated within the vortex core. Some aspects
of this process, including the nonlinear modification of the vortex by the
wave, are explained by the existence of trapped near-inertial eigenmodes. These
vortex eigenmodes are easily excited by an initial wave with horizontal scale
much larger than that of the vortex radius. We study this process using a
wave-averaged model of near-inertial dynamics and compare its theoretical
predictions with numerical solutions of the three-dimensional Boussinesq
equations. In the linear approximation, the model predicts the eigenmode
frequencies and spatial structures, and a near-inertial wave energy signature
that is characterized by an approximately time-periodic, azimuthally invariant
pattern. The wave-averaged model represents the nonlinear feedback of the waves
on the vortex via a wave-induced contribution to the potential vorticity that
is proportional to the Laplacian of the kinetic energy density of the waves.
When this is taken into account, the modal frequency is predicted to increase
linearly with the energy of the initial excitation. Both linear and nonlinear
predictions agree convincingly with the Boussinesq results.
