In this paper, we investigate the three-dimensional stability of the Moore–Saffman elliptical vortex in a rotating stratified fluid. By means of an asymptotic analysis for long vertical wavelength perturbations and small Froude number, we study the effects of Rossby number, external strain, and ellipticity of the vortex on the stability of azimuthal modes $m\,{=}\,1$ (corresponding to a bending instability) and $m\,{=}\,2$ (corresponding to a twisting instability).
In the case of a quasi-geostrophic fluid (small Rossby number), the asymptotic results are in striking agreement with previous numerical stability analyses even for vertical wavelengths of order one. For arbitrary Rossby number, the key finding is that the Rossby number has no effect on the domains of long-wavelength instability of these two modes: the two-dimensional or three-dimensional nature of the instabilities is controlled only by the background strain rate $\gamma$ and by the rotation rate $\Omega$ of the principal axes of the elliptical vortex relative to the rotating frame of reference.
For the $m=1$ mode, it is shown that when $\Omega < -\gamma$, the vortex is stable to any long-wavelength disturbances, when $-\gamma < \Omega \lesssim 0$, two-dimensional perturbations are most unstable, when $0 \lesssim \Omega < \gamma$, long-wavelength three-dimensional disturbances are the most unstable, and finally when $\gamma < \Omega$, short-wavelength three-dimensional perturbations are the most unstable. Similarly, the $m=2$ instability is two-dimensional or three-dimensional depending only on $\gamma$ and $\Omega$, independent of the Rossby number. This means that if a long-wavelength three-dimensional instability exists for a given elliptical vortex in a quasi-geostrophic fluid, a similar instability should be observed for any other Rossby number, in particular for infinite Rossby number (strongly stratified fluids). This implies that the planetary rotation plays a minor role in the nature of the instabilities observed in rotating strongly stratified fluids.
The present results for the azimuthal mode $m=1$ suggest that the vortex-bending instabilities observed previously in quasi-geostrophic fluids (tall-column instability) and in strongly stratified fluids (zigzag instability) are fundamentally related.