  Motivated by a recent demonstration of cyclic addition of quantized vorticity
into a Bose-Einstein condensate, the vortex pump, we study dynamical
instabilities and core sizes of giant vortices. The core size is found to
increase roughly as a square-root function of the quantum number of the vortex,
whereas the strength of the dynamical instability either saturates to a fairly
low value or increases extremely slowly for large quantum numbers. Our studies
suggest that giant vortices of very high angular momenta may be achieved by
gradually increasing the operation frequency of the vortex pump.
