The three-dimensional linear instability of Kirchhoff's elliptic vortex in an inviscid incompressible fluid is investigated numerically. Any elliptic vortex is shown to be unstable to an infinite number of short-wave bending modes, with azimuthal wave number m = 1. In the limit of small ellipticity, the axial wave number of each unstable mode approaches the value obtained by the asymptotic theory of Vladimirov and Il'in, indicating that the instability is caused by a resonance phenomena. As the ellipticity increases, the bandwidth broadens and neighboring bands overlap each other. The maximum growth rate of each mode, except for that of the longest one, agrees fairly with that of the elliptical instability modified by the influence of a Coriolis force. The growth rate of these three-dimensional modes are larger than those of the two-dimensional modes when ellipticity is smaller than a certain value.
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