The linear stability of unbounded strained vortices in a stably stratified fluid is investigated theoretically. The problem is reduced to a Floquet problem which is solved numerically. The three-dimensional elliptical instability of Pierrehumbert type [Phys. Rev. Lett. 57, 2157 (1986)] is shown to be suppressed by the stable stratification and it disappears when the Brunt- Väisälä frequency exceeds unity. On the other hand, two classes of new instability mode are found to occur. One appears only when the Brunt-Väisälä frequency is less than 2, whereas the other persists for all values of the Brunt-Väisälä frequency. The former mode is related to a parametric resonance of internal gravity waves, and the latter modes are related to superharmonic parametric instability.
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