Transverse normal modes of finite vortex arrays

The normal modes of stationary vortex patterns in two dimensions are studied for arbitrary dissipation in both the unbounded and circle-bounded cases. All eigenvectors are given for the ground-state patterns of $N=2\mathrm{}$ through 12 vortices and $N=19\mathrm{}$; all frequencies are given for $N=2\mathrm{}$ through 20. The complete eigenfrequency spectrum of single-ring patterns is given for all $N$. The integrals of unbounded motion are generalized to include dissipation. It is shown that angular momentum is strictly confined to the breathing mode of all unbounded patterns and of all bounded, single-ring patterns. Comparisons are made with infinite-vortex lattice modes, with a continuum model, and with earlier predictions of pattern stability and mode frequencies.