The evolution of a Lamb-Oseen vortex is studied in a stratified rotating fluid under the complete Coriolis force. In a companion paper, it was shown that the non-traditional Coriolis force generates a vertical velocity field and a vertical vorticity anomaly at a critical radius when the Froude number is larger than unity. Below a critical non-traditional Rossby number (based on the horizontal component of background rotation), a two-dimensional shear instability was next triggered by the vorticity anomaly. Here, we test the robustness of this two-dimensional evolution against small three-dimensional perturbations. Direct numerical simulations (DNS) show that the two-dimensional shear instability then develops only in an intermediate range of non-traditional Rossby numbers for a fixed Reynolds number . For lower , the instability is three-dimensional. Stability analyses of the flows in the DNS prior to the instability onset fully confirm the existence of these two competing instabilities.

In addition, stability analyses of the local theoretical flows at leading order in the critical layer demonstrate that the three-dimensional instability is due to the shear of the vertical velocity. For a given Froude number, its growth rate scales as 2/3 / , whereas the growth rate of the two-dimensional instability depends on / 2 , provided that the critical layer is smoothed by viscous effects. However, the growth rate of the three-dimensional instability obtained from such local stability analyses agrees quantitatively with those of the DNS flows only if second order effects due to the traditional Coriolis force and the buoyancy force are taken into account. These effects tend to damp the three-dimensional instability.