  In a uniformly rotating fluid, inertial waves propagate along rays that are
inclined to the rotation axis by an angle that depends on the wave frequency.
In closed domains, multiple reflections from the boundaries may cause inertial
waves to focus on to particular structures known as wave attractors. Such
structures have previously been studied from a theoretical point of view, in
laboratory experiments, in linear numerical calculations and in some recent
numerical simulations. In the present paper, two-dimensional direct numerical
simulations of an inertial wave attractor are presented. In the linear regime,
we first recover the results of the linear calculations and asymptotic theory
of Ogilvie (2005) who considered a prototypical problem involving the focusing
of linear internal waves into a narrow beam centred on a wave attractor in a
steady state. The velocity profile of the beam and its scalings with the Ekman
number, as well as the asymptotic value of the dissipation rate, are found to
be in agreement with the linear theory. We also find that, as the beam builds
up around the wave attractor, the power input by the applied force reaches its
limiting value more rapidly than the dissipation rate, which saturates only
when the beam has reached its final thickness. In the nonlinear regime, the
beam is strongly affected and becomes unstable to a subharmonic instability.
This instability transfers energy to secondary waves possessing shorter
wavelengths and lower frequencies. The onset of the instability of a narrow
inertial wave beam is investigated by means of a separate linear analysis and
the results, such as the onset of the instability, are found to be consistent
with the global simulations of the wave attractor. The excitation of such
secondary waves described theoretically in this work has also been seen in
recent laboratory experiments on internal gravity waves.
