In conclusion, we have found a rich structure in the spinwave modes of an individual vortex, and have shown how these modes are related to the instability of in-plane vortices to become out-of-plane vortices at . This information can be valuable for improving the description of the dynamics of interacting vortices in terms of their positions, and, the internal vibrational motions we have found. Because this is a zero-temperature single-vortex calculation, we can only speculate that it might be possible for the vortex instability to affect dynamical correlation functions. If there is an effect, it would be most prominent in the correlation function of the out-of-plane spin components, especially for a material whose anisotropy constant is near . It will be a future challenge to consider whether the frequency spectra we have found can be related to these dynamic correlations in thermal equilibrium in a quantitative way.