We consider axisymmetric rotating convection in a cylindrical domain, focussing on the eye that can form at the centre of a cyclonic vortex. Upon increasing the thermal forcing we observe that the the system undergoes a Hopf bifurcation from a state with a steady eye to one in which the eye oscillates. For an aspect ratio, Ekman number, and Prandtl number of 0.1 we find that the critical Reynolds number at which this transition occurs is 398.

We examine the nature of the oscillations and propose that the behaviour results from an inertial wave trapped in the eye, with the frequencies falling within the expected range for inertial waves, and the oscillations displaying clear similarities to a standing inertial wave in a cylinder. We also examine the effect of Ekman number on the oscillation, finding that there is upper limit beyond which oscillations do not occur.