Abstract
The linear stability of a doubly periodic array of vortices to three-dimensional perturbations is studied. The instabilities are separated into symmetric and antisymmetric modes. For two-dimensional disturbances only the symmetric mode is found to be unstable. The antisymmetric mode shows a peak in growth rate at long wavelengths. This is attributed to the Crow instability. For short wavelengths both symmetric and antisymmetric modes are found to have similar growth rates. This is attributed to the elliptical instability and it is found to occur even when the vortex cells are square, a physical explanation for which is provided, but for elongated vortex cells the growth rates are higher. Viscosity is found to have a strong stabilizing influence on short wavelength perturbations.
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