Resonant instability in two-dimensional vortex arrays by Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proceedings A Volume 467(2128):1164-1185 April.

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Resonant instability in two-dimensional vortex arrays by Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proceedings A Volume 467(2128): April 8, 2011 ©2011 by The Royal Society

Possible eigenvalue behaviour for (a) an exchange of stability (giving rise to a non-propagating unstable eigenmode), and (b) a Hamiltonian Hopf bifurcation (leading to an oscillatory instability). Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467: ©2011 by The Royal Society

Schematic of perturbations involving (a) a displacement or (b) deformation of the vortex cores. m denotes the azimuthal wavenumber of the perturbation. Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467: ©2011 by The Royal Society

Perturbation involving an overall displacement of the configuration, schematically illustrated through the case of a co-rotating vortex pair. Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467: ©2011 by The Royal Society

Schematic of possible phase relations between disturbances on three vortices. Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467: ©2011 by The Royal Society

(a) Angular velocity for the elliptical model (dashed line) and the full solution (solid line) for three identical co-rotating vortices. Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467: ©2011 by The Royal Society

Selected eigenvalues for three co-rotating vortices, showing the prediction for the development of the first resonance from the elliptical model (a), together with results from an accurate linear stability analysis (b). Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467: ©2011 by The Royal Society

Eigenvalues for three co-rotating vortices. Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467: ©2011 by The Royal Society

Close-up view of the second resonance occurring for three vortices. Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467: ©2011 by The Royal Society

Imaginary part of the overall-displacement eigenvalue σI and of the angular velocity Ω of the configuration. Paolo Luzzatto-Fegiz, and Charles H. K. Williamson Proc. R. Soc. A 2011;467: ©2011 by The Royal Society